metal forming and the finite element method

METAL FORMING AND THE FINITE-ELEMENT METHOD

OXFORD SERIES ON ADVANCED MANUFACTURING

SERIES EDITORS

J. R. CROOKALLMILTON C. SHAW

1. William T. Harris. Chemical Milling: the Technology of CuttingMaterials by Etching (1976)

2. Bernard Crossland. Explosive Welding of Metals and its Applications(1982)

3. Milton C. Shaw. Metal Cutting Principles (1984)

4. Shiro Kobayashi, Soo-Ik Oh, Taylan Altan. Metal Forming and theFinite-Element Method (1989)

METAL FORMING AND THEFINITE-ELEMENT METHOD

SHIRO KOBAYASHISOO-IK OH

TAYLAN ALTAN

New York OxfordOXFORD UNIVERSITY PRESS

1989

Oxford University PressOxford New York Toronto

Delhi Bombay Calcutta Madras KarachiPetaling Jaya Singapore Hong Kong Tokyo

Nairobi Dar es Salaam Cape TownMelbourne Auckland

and associated companies inBerlin Ibadan

Copyright © 1989 by Oxford University Press, Inc.

Published by Oxford University Press, inc.,200 Madison Avenue, New York, New York 10016

Oxford in a registered trademark of Oxford University Press

All rights reserved. No part of this publication may be reproduced,stored in a retrieval system, or transmitted, in any form or by any means,

electronic, mechanical, photocopying, recording, or otherwise,without the prior permission of Oxford University Press.

Library of Congress Cataloging-in-Publication Data

Kobayashi, Shiro.Metal forming and the finite-element method /

Shiro Kobayashi, Soo-Ik Oh, Taylan Allan.p. cm. — (Oxford series on advanced manufacturing;

4) Bibliography: p. Includes index.ISBN 0-19-504402-9

1. Metal-work—Mathematical models. 2. Finite-element method.I. Oh, Soo-Ik. II. Allan, Taylan. III. Title. IV. Series.

TS213.K56 1989671'.072'4—dc!9 88-11995

CIP

1 3 5 7 9 8 6 4 2

Printed in the United States of Americaon acid-free paper

PREFACE

The application of computer-aided engineering, design, and manufactur-ing, CAE/CAD/CAM, is essential in modern metal-forming technology.Thus, process modeling for the investigation and understanding ofdeformation mechanics has become a major concern in research, and thefinite-element method (FEM) has assumed increased importance, particu-larly in the modeling of forming processes.

There are many excellent textbooks on the principles and fundamentalsof metal forming, but only a few describe the application of FEM to theanalysis and simulation of metal-forming processes.

The main purpose of this book is to present the fundamentals andapplications of FEM in metal-forming analysis and technology. The bookis primarily written for graduate students and researchers. However, itshould also be useful to practicing engineers who have a good backgroundin FEM and who are interested in applying this technique to the analysis ofmetal-deformation processes.

In the application of FEM to metal forming, there are two formulations,namely, flow formulation and solid formulation. Flow formulation assumesthat the deforming material has a negligible elastic response, while solidformulation includes elasticity. Despite recent advances, the application ofsolid formulation to the analysis of metal-forming problems remainslimited. On the other hand, flow formulation has found applications in awide variety of important forming problems. This book, therefore, ismainly devoted to the applications that are based on flow formulation(purely plastic and viscoplastic). However, recent advances achieved insolid formulation have made it applicable to the analysis of some formingproblems. In order not to neglect these investigations, comparisons ofsolutions based on both formulations, solid and flow, are presented inChapter 16.

The book begins with a general background on the subject in Chapter 1.The description of metal-forming processes is given in Chapter 2, andChapter 3 details important technological aspects of these processes.Chapters 4 and 5 present the theory of plasticity and methods of analysis asapplied to metal forming. The FEM formulations are described inChapters 6 and 7, and the applications of the method to the analyses ofvarious forming processes are presented in Chapters 8 through 11. Chapter

vi Preface

12 presents a thermo-viscoplastic analysis and Chapters 13, 14, and 15include developments in the areas of deformation of porous materials,three-dimensional problems, and preform design. The book concludes withChapter 16, in which further developments are discussed, along with theoutline of solid formulation and comparison of the results by both solidand flow formulations.

Although this book primarily deals with metals, some of the principlesand solution techniques should be applicable to deformation analyses ofother materials, such as polymers and composites.

Sincere thanks are due to a number of individuals. First of all, we wishto express our appreciation to Professor E. G. Thomsen, ProfessorEmeritus, University of California at Berkeley, who helped us to devoteour careers to research in metal forming. We also thank Professor M. C.Shaw, Arizona State University, for his encouragement and support inwriting this book, and Professor W. Johnson, Emeritus Professor, Univer-sity of Cambridge, for his critical comments during the preparation of themanuscript.

The senior author wishes to thank his former graduate students in theDepartment of Mechanical Engineering, University of California atBerkeley, who have contributed to the advances in the application of FEMto metal forming.

The contents of this book are largely the results of research supportedby the Air Force Wright Aeronautical Laboratory, the National ScienceFoundation, and the Army Research Office, and their support isacknowledged.

We also thank Mr. and Mrs. Joe Bavonese for typing the manuscript.

Berkeley S. K.Columbus S. O.May 1988 T. A.

CONTENTS

Symbols, xiii

1. Introduction, 1

1.1 Process Modeling, 11.2 The Finite-Element Method, 31.3 Solid Formulation and Flow Formulation, 41.4 Metal Forming and the Finite-Element Method, 5

References, 6

2. Metal-Forming Processes, 8

2.1 Introduction, 82.2 A Metal-Forming Operation as a System, 82.3 Classification and Description of Metal-Forming Processes, 11

References, 24

3. Analysis and Technology in Metal Forming, 26

3.1 Introduction, 263.2 Flow Stress of Metals, 283.3 Friction in Metal Forming, 303.4 Temperatures in Metal Forming, 333.5 Impression and Closed-Die Forging, 353.6 Hot Extrusion of Rods and Shapes, 363.7 Cold Forging and Extrusion, 393.8 Rolling of Strip, Plate, and Shapes, 413.9 Drawing of Rod, Wire, Shapes, and Tubes, 45

3.10 Sheet-Metal Forming, 47References, 52

4. Plasticity and Viscoplasticity, 54

4.1 Introduction, 544.2 Stress, Strain, and Strain-Rate, 544.3 The Yield Criteria, 584.4 Equilibrium and Virtual Work-Rate Principle, 61

viii Contents

4.5 Plastic Potential and Flow Rule, 634.6 Strain-Hardening, Effective Stress, and Effective Strain, 664.7 Extremum Principles, 684.8 Viscoplasticity, 70

References, 72

5. Methods of Analysis, 73

5.1 Introduction, 735.2 Upper-Bound Method, 745.3 Hill's General Method, 785.4 The Finite-Element Method, 835.5 Concluding Remarks, 88

References, 88

6. The Finite-Element Method—Part I, 90

6.1 Introduction, 906.2 Finite-Element Procedures, 906.3 Elements and Shape Function, 946.4 Element Strain-Rate Matrix, 1016.5 Elemental Stiffness Equation, 108

References, 110

7. The Finite-Element Method—Part II, 111

7.1 Numerical Integrations, 1117.2 Assemblage and Linear Matrix Solver, 1157.3 Boundary Conditions, 1177.4 Direct Iteration Method, 1217.5 Time-Increment and Geometry Updating, 1237.6 Rezoning, 1267.7 Concluding Remarks, 129

References, 129

8. Plane-Strain Problems, 131

8.1 Introduction, 1318.2 Finite-Element Formulation, 1318.3 Closed-Die Forging with Flash, 1338.4 Sheet Rolling, 1378.5 Plate Bending, 1418.6 Side Pressing, 148

References, 149

9. Axisynunetric Isothermal Forging, 151

9.1 Introduction, 1519.2 Finite-Element Formulation, 151

Contents ix

9.3 Compression of Solid Cylinders and Heading of Cylindrical Bars,153

9.4 Ring Compression, 1599.5 Evaluation of Friction at Tool-Workpiece Interface, 1639.6 Forging and Cabbaging, 165

References, 172

10. Steady-State Processes of Extrusion and Drawing, 174

10.1 Introduction, 17410.2 Method of Analysis, 17410.3 Bar Extrusion, 17610.4 Bar Drawing, 17810.5 Multipass Bar Drawing and Extrusion, 18310.6 Applications to Process Design, 186

References, 187

11. Sheet-Metal Forming, 189

11.1 Introduction, 18911.2 Plastic Anisotropy, 19011.3 In-plane Deformation Processes, 19211.4 Axisymmetric Out-of-plane Deformation, 19511.5 Axisymmetric Punch-Stretching and Deep-Drawing Processes,

20111.6 Sheet-Metal Forming of General Shapes, 20611.7 Square-Cup Drawing Process, 21011.8 Nonquadratic Yield Criterion, 217

References, 220

12. Thermo-Viscoplastic Analysis, 222

12.1 Introduction, 22212.2 Viscoplastic Analysis of Compression of a Solid Cylinder, 22312.3 Heat Transfer Analysis, 22512.4 Computational Procedures for Thermo-Viscoplastic Analysis,

22712.5 Applications, 22912.6 Concluding Remarks, 240

References, 242

13. Compaction and Forging of Porous Metals, 244

13.1 Introduction, 24413.2 Yield Criterion and Flow Rules, 24513.3 Finite-Element Modeling and Numerical Procedures, 24613.4 Simple Compression, 24913.5 Axisymmetric Forging of Flange-Hub Shapes, 253

Contents

13.6 Axisymmetric Forging of Pulley Blank, 25613.7 Heat Transfer in Porous Materials, 25913.8 Hot Pressing Under the Plane-Strain Condition, 26213.9 Compaction, 266

References, 272

14. Three-Dimensional Problems, 275

14.1 Introduction, 27514.2 Finite-Element Formulation, 27614.3 Block Compressions, 27814.4 Square-Ring Compression, 28414.5 Simplified Three-Dimensional Elements, 28714.6 Analysis of Spread in Rolling and Flat-Tool Forging, 28914.7 Concluding Remarks, 295

References, 296

15. Preform Design in Metal Forming, 298

15.1 Introduction, 29815.2 Method for Design, 29815.3 Shell Nosing at Room Temperature, 30115.4 Plane-Strain Rolling, 30515.5 Axially Symmetric Forging, 30915.6 Hot Forming, 31515.7 Concluding Remarks, 318

References, 320

16. Solid Formulation, Comparison of Two Formulations, andConcluding Remarks, 321

16.1 Introduction, 32116.2 Small-Strain Solid Formulation, 32116.3 Large Deformation: Rate Form, 32316.4 Large Deformation: Incremental Form, 32616.5 Comparison with Rigid-Plastic (Flow) Solutions, 32716.6 Concluding Remarks, 334

References, 335

Appendix. The FEM Code, SPID (Simple Plastic IncrementalDeformation), 338

A.1 Introduction, 338A.2 Program Structure, 339

x

Contents xi

A.3 Input and Output Files, 340A.4 Input Preparations, 340A.5 Description of the Major Variables, 342A.6 Program Listing, 343A.7 Example Solution, 364

Index, 371

This page intentionally left blank

SYMBOLS

A Cross-sectional areaA Function of relative

density for porousmaterials

A0 Initial cross-sectionalarea

AjN area contribution of thejth element to node N

B Function of relativedensity for porousmaterials

B Breadth

B0 Initial breadthB Strain-rate matrix

C ConstantCr Class of functions with

continuous derivatives ofall orders up to andincluding r

C Volumetric strain-ratevector

C Heat capacity matrix

D Diameter

D0 Initial diameterD Effective strain-rate

coefficient matrixE Young's modulusE(eij) Work functionE Energy-rateEtj Lagrangian strainF Coefficient of anisotropyF(aij) Function of stressesF, TractionG Shear modulusG Coefficient of anisotropyH Coefficient of anisotropy

H

H0

H1H(e)

HH

/,

/2

/3

/I

h

/3

J

K

K

K,LL1, L2, L]

Lijki-Lijkl

M

M

N

N

P

P

Height

Initial heightFinal height

Work-hardening functionTime derivative of height

Increment of heightLinear invariant of stresstensor

Quadratic invariant ofstress tensor

Cubic invariant of stresstensor

Linear invariant ofdeviatoric stress tensorQuadratic invariant ofdeviatoric stress tensorCubic invariant ofdeviatoric stress tensor

Jacobian of coordinatetransformation

Penalty constant

Stiffness matrix

Heat conduction matrixCoefficient of anistropyArea coordinateSmall-strain moduliConstitutive moduli

Coefficient of anisotropy

Gradient matrix of shapefunction vector N


Shape function matrix

Load

Effective strain-ratematrix

XiV

p.

QR

R

Ra

R0

R0

Ri

Re

Rn

RD

RP

SSSc

SD

SF

S1

Su

S,

TT

Ta

Tb

Td

Te

TR

Ts

Symbols

Element of strain-ratematrix B

Heat flux vector

Roll radius

Relative density ofporous materials

Average relative densityof porous materialsInitial relative density ofporous materialsInitial radius

Internal radius of ringsand tubes

Radius of extruded ordrawn bars

Radius of neutral point inring compressionDie corner radius

Punch radiusMicrostructureSurfaceSurface of tool-workpiece contact

Surface of discontinuity

Surface where traction isprescribed

Internal surfaceSurface where velocity isprescribed

Surface where heat flux isprescribed

ThicknessTemperature

Nodal-point temperature

Temperature of basemetal in porous materials

Die temperature

Environmentaltemperature

Apparent temperature ofporous materials

Surface temperature

Tw

T

T

UD

U0

U1UP

V

V0

Vb

K

AF

WW0

waW

wp

WP

W0

xa

Y

Y0

Yb

YR

Ya

Z«

a

c

Workpiece temperature

Time derivative oftemperature

Coordinatetransformation matrixDie or roll velocityEntrance velocity inrolling

Exit velocity in rolling

Punch velocityVolumeInitial volumeVolume of base metal inporous materials

Volume of void in porousmaterials

Volume change

WidthInitial widthAverage widthTime derivative of width

Total plastic work perunit volumePlastic work-rate per unitvolumeWork-rate per unitvolume in reference state

Element of strain-ratematrix BYield stress in uniaxialtensionInitial yield stressYield stress of base metalin porous materials

Apparent yield stress ofporous materials



Height-to-diameter ratioSpecific heat

Symbols xv

Cd

Cb

cu

CR

d

e

ffsf

f(ogg(o

h

hc

Wlub

h

h(o

k

ky

k1kR

kb

I

I

10

Specific heat of diematerial

Specific heat of basemetal in porous materials

Specific heat of void inporous materials

Apparent specific heat ofporous materials

Punch depth in sheet-metal forming

Engineering strain

Engineering strain-rate


Frictional stress

Nodal-point force vector

Yield function


Scalar function of stressinvariants

Heat transfer coefficient

Heat transfer coefficientat tool-workpiececontact surface

Heat transfer coefficientof lubricant


Scalar function of stressinvariants

Shear yield stress

Apparent shear yieldstress of porous materials

Thermal conductivity

Apparent thermalconductivity of porousmeterials

Thermal conductivity ofbase metal in porousmaterials

Gage length in tensiletest


Initial gage length intensile test

Unit tangent vector

Friction factor

Strain-rate exponent


Strain-hardeningexponent


Unit normal to thesurface

Pressure

Average pressure

Die pressure in drawing

First Piola-Kirchhoffstress

Heat generated throughfriction

Heat flux across surface

SqShape functions

r-Value in sheet forming

r-Values in the rolling,45°, and transversedirections, respectively

Heat generation-rate

Second Piola-Kirchhoffstress

Time

Time-increment

Velocity component

Velocity component atthe orth node

Initial velocity

Relative sliding velocity

Velocity componentnormal to a surface

Velocity componenttangent to a surface

Velocity discontinuity

Relative sliding velocityat nodal point

Velocity vector at nodalpoint

emmmn

n

n

P

Pa

PD

Pa

qf

qn

qr

rx, r45, ry

r

sij

tAf

uiui( )

MO

«Jun

u,

AM

v,

V

e

XVI

V0

Av

wiw,

Xa> Jai ^ex

aa

ft

Y, Y', Y"

Sh

<50

<5a|8

e

e

Eb

EN

EijEij

Symbols

Initial velocity vector atnodal pointVelocity corrections ofnodal valuesVirtual velocityWeight factorsx, y, z-Coordinates ofcrth nodeDie semi-angleDeceleration coefficientCoupling coefficient intemperature calculationViscosity coefficientsRadial displacement inbore expandingRadial displacement inflange drawingKronecker deltaEmissivityEffective strainEffective strain of basemetal in porous materialsEffective strain value atnode NStrain-rateInfinitesimal strainVolumetric strain-ratePlastic strain-rateElastic strain-rateEffective strain-rateEffective strain-rate ofbase metal in porousmaterials

Apparent effective strain-rate of porous materials

Limiting strain-rate

Natural coordinate

-Coordinate of orthnode

Function of relativedensity in porousmaterials

•nria

eK

AA

d)i

Vv

££»

n

n6jtdn{i)

6nD

6jTP

djTsF

diti

5nSc

P

Po

Pb

PdPR

Pv

a

Natural coordinate^-Coordinate of aihnodeAngleHeat generationefficiency factorLagrangian multiplierProportionality factor(rate) in flow rulesProportionality factor(infinitesimal) in flowrulesCoefficient of frictionPoisson's ratioNatural coordinate^-Coordinate of athnodePlane of zero mean stressin stress spaceFunctionalVariation of functional ir<5jr-value at ;'th elementTerm due to deformationenergy-rate in dnPenalty term in dnTerm due to traction indjiTerm that includesLagrangian multiplier indnTerm due to friction indjtDensityInitial densityDensity of base metal inporous materialsDensity of die materialApparent density ofporous materialsDensity of void in porousmaterialsStephan-Boltzmanconstant

ev

EH"V£

Eb

-ER

EO

C

£*

n

xvii

Bulge function in simplecompression

Strain-rate sensitivityfunctionRate of rotation

Symbols

*(*)

0(F)

W,j

Cauchy stressDeviatoric stress

Effective stress, flowstress

Mean stressKirchhoff stress

Shear traction in Hill'smethod

Oij

°n'o

om

T,i

T/

METAL FORMING AND THEFINITE-ELEMENT METHOD

1INTRODUCTION

1.1 Process Modeling

In the late 1970s and early 1980s the use of computer-aided techniques(computer-aided engineering, design, and manufacturing) in the metal-forming industry increased considerably. The trend seems to be towardever wider application of this technology for process simulation andprocess design.

A goal in manufacturing research and development is to determine theoptimum means of producing sound products. The optimization criteriamay vary, depending on product requirements, but establishing anappropriate criterion requires thorough understanding of manufacturingprocesses. In metal-forming technology, proper design and control re-quires, among other things, the determination of deformation mechanicsinvolved in the processes. Without the knowledge of the influences ofvariables such as friction conditions, material properties, and workpiecegeometry on the process mechanics, it would not be possible to design thedies and the equipment adequately, or to predict and prevent theoccurrence of defects. Thus, process modeling for computer simulation hasbeen a major concern in modern metal-forming technology. Figure 1.1.indicates the role of process modeling in some detail.

In the past a number of approximate methods of analysis have beendeveloped and applied to various forming processes. The methods mostwell known are the slab method, the slip-line field method, the visioplas-ticity method, upper- (and lower-) bound techniques, Hill's generalmethod, and, more recently, the finite-element method (FEM).

In the slab method, the workpiece being deformed is decomposed inseveral slabs. For each slab, simplifying assumptions are made mainly withrespect to stress distributions. The resulting approximate equilibriumequations are solved with imposition of stress compatibility between slabsand boundary tractions. The final result is a reasonable load predictionwith an approximate stress distribution.

The slip-line field method is used in plane strain for perfectly plasticmaterials (constant yield stress) and uses the hyperbolic properties that thestress equations have in such cases. The construction of slip-line fields,although producing an "exact" stress distribution, is still quite limited in

1

Geometrical parameters

Tool geometry

Workpiece geometry

Process Parameters

Die and tool motion

Temperature

Lubrication

Material

Material Parameters

Workhardening

Strain rate sensitivity

Anisotropy

Temperature

PROCESS MODELING OUTPUT

Process Analysis

and Optimization

Loads, energies,stresses, strains,

temperature, metal

flow (geometrical

change)

Determination

of process

geometry andprocess

performancecondition

Primary product

requiremements

Forming limits

Secondary product

requirements

Tolerance, surfaceproperty etc.

CONSTRAINTS!

FIG. 1.1 Block diagram for process design and control in metal forming.

Equipment

Capacity

limitations

INPUT

Introduction 3

predicting results that give good correlations with experimental work.From the stress distributions, velocity fields can be calculated throughplasticity equations.

The visioplasticity method originated by Thomsen et al. [1] combinesexperiment and analysis. After the velocity vectors have been determinedfrom an actual test, strain-rates are calculated and the stress distributionsare obtained from plasticity equations. The method has helped to obtainreliable solutions in detail for processes in which the experimentaldetermination of the velocity vectors was possible.

The upper-bound method requires the guessing of admissible velocityfields, among which the best one is chosen by minimizing total potentialenergy. Information leading to a good selection of velocity fields comesfrom experimental evidence and experience. This method, with ex-perience, can deliver a fast and relatively accurate prediction of loads andvelocity distributions.

Hill [2] has given a general method of analysis for metal-workingprocesses when the plastic flow is unconstrained. The method is based on acriterion of approximation derived from the interpretation of the virtualwork-rate principle. The method was applied to the analysis of compres-sion with barreling, spread in bar drawing, and thickness change in tubesinking.

The upper-bound method and Hill's general method are outlined inmore detail, with illustrative examples in relation to the finite-elementmethod, in Chap. 5. For further reference, the books that provide a wealthof solutions to many metal-forming problems using the above methods arelisted in the References [3-21].

These methods have been useful in predicting forming loads, overallgeometry changes of deforming workpieces, and qualitative modes ofmetal flow, and in determining approximate optimum process conditions.However, accurate determination of the effects of various process para-meters on the detailed metal flow became possible only recently, when thefinite-element method was developed for the analyses. Since then, thefinite-element method has assumed steadily increased importance insimulation of metal-forming processes. Among the books cited, however,only a few discuss the application of the finite-element method tometal-forming processes.

1.2. The Finite-Element Method

The finite-element technique, whose engineering birth and boom in the1960s was due to the application of digital computers to structural analysis,has spread to a variety of engineering and physical science disciplines inthe last decade.

The basic concept of the finite-element method is one of discretization.The finite-element model is constructed in the following manner [22]. Anumber of finite points are identified in the domain of the function, and the

4 Metal Forming and the Finite-Element Method

values of the function and its derivatives, when appropriate, are specifiedat these points. The points are called nodal points. The domain of thefunction is represented approximately by a finite collection of subdomainscalled finite elements. The domain is then an assemblage of elementsconnected together appropriately on their boundaries. The function isapproximated locally within each element by continuous functions that areuniquely described in terms of the nodal-point values associated with theparticular element. The path to the solution of a finite-element problemconsists of five specific steps: (a) identification of the problem; (b)definition of the element; (c) establishment of the element equation; (d)the assemblage of element equations; and (e) the numerical solution of theglobal equations. The formation of element equations is accomplishedfrom one of four directions: (1) direct approach; (2) variational method;(3) method of weighted residuals; and (4) energy balance approach.

The basis of finite-element metal-flow modeling, for example, using thevariational approach is to formulate proper functionals, depending uponspecific constitutive relations. The solution of the original boundary valueproblem is obtained by the solution of the dual variational problem inwhich the first-order variation of the functional vanishes. Choosing anapproximate interpolation function (or shape function) for the fieldvariable in the elements, the functional is expressed locally within eachelement in terms of the nodal-point values. The local equations are thenassembled into the overall problem. Thus, the functional is approximatedby a function of global nodal-point values. The condition for this functionto be stationary results in the stiffness equations. These stiffness equationsare then solved under appropriate boundary conditions. The basic mathe-matical description of the methods, as well as the solution techniques, aregiven in several books (for example, References [23, 24, 25]).

The main advantages of the finite-element method are: (1) the capabilityof obtaining detailed solutions of the mechanics in a deforming body,namely, velocities, shapes, strains, stresses, temperatures, or contactpressure distributions; and (2) the fact that a computer code, once written,can be used for a large variety of problems by simply changing the inputdata.

1.3 Solid Formulation and Flow Formulation

In the analysis of metal forming, plastic strains usually outweigh elasticstrains and the idealization of rigid-plastic or rigid-viscoplastic materialbehavior is acceptable. The resulting analysis based on this assumption isknown as the flow formulation [26]. In other applications, phenomenaassociated with elasticity cannot be neglected. In the so-called solidformulation [27], the material is considered to behave as an elastic-plasticor elastic-viscoplastic solid.

The original problem associated with the deformation process ofmaterials is a boundary-value problem. For the deformation process of

Introduction 5

rigid-viscoplastic materials the boundary-value problem is stated asfollows: at a certain stage in the process of quasistatic distortion, the shapeof the body, the internal distribution of temperature, the state ofinhomogeneity, and the current values of material parameters are sup-posed to be given or to have been determined already. The velocity vectoru is prescribed on a part of surface Su together with traction F on theremainder of the surface, SF. Solutions to this problem are the stress andvelocity distributions that satisfy the governing equations and the boundaryconditions.

In the solid approach, the boundary value problem is stated such that, inaddition to the current states of the body, the internal distribution of thestress also is supposed to be known and the boundary conditions areprescribed in terms of velocity and traction-rate. Distributions of velocityand stress-rate (or displacement and stress-increment) are the solutions tothe problem.

The solid formulations of the finite-element method for metal-formingproblems have been based on the use of the Prandtl-Reuss equations forelastic-plastic materials. The formulation is given in the rate form andassumes the infinitesimal theory of deformation. In analyzing metal-forming processes, however, the elastic-plastic finite-element method withinfinitesimal formulation has severe drawbacks. The large amount ofrotation involved in metal forming rules out infinitesimal analysis, andlarge-deformation analyses also have some difficulties in reproducingobserved phenomena, such as folding in compression of solid cylinders(see Chap. 9). Furthermore, the nature of elastic-plastic constitutiveequations requires short time steps in nonsteady-state analysis, a require-ment that is severe when the body goes from elastic to plasticdeformations.

A simplified solution to this problem is to neglect the elastic portion ofdeformation and treat all plastic deformation as a flow problem. Ingeneral, this makes an infinitesimal analysis feasible and large stepspossible.

1.4 Metal Forming and the Finite-Element Method

The application of the finite-element method to metal-forming problemsbegan as an extension of structural analysis technique to the plasticdeformation regime. Thus, early applications of the finite-element methodto metal-forming problems were based on the plastic stress-strain matrixdeveloped from the Prandtl-Reuss equations. Hydrostatic extrusion,compression, and indentations were analyzed using this matrix and theinfinitesimal variational formulations.

An analysis method in the area of metal-forming application, in manycases, can be justified only by its solution reliability and computationalefficiency. This realization has led to the development of numericalprocedures based on the flow formulation. Initial applications of the


rigid-plastic finite-element method to metal-forming processes were mainlyin the analysis of compression and other simple processes. Since thoseearly days, many developments of the numerical techniques have occurred,as well as the continuous growth in the field of applications. Althoughadvances have been made in recent years, the application of solidformulation to metal-forming problems is limited. On the other hand, flowformulation has found applications for a wide variety of important formingproblems.

A most important improvement was the inclusion of the effects ofstrain-rate and temperature in material properties and of thermal couplingin the solution. This development has extended the finite-element analysisinto the warm and hot working range. A further important step in thedevelopment of analysis procedures was the development of a user-oriented general-purpose program. The natural course of development ofthe technique has been in the analysis of two-dimensional and axiallysymmetric problems, and most recent developments emphasize the ap-plication of the finite-element method to three-dimensional problems. Afurther development is the unique application of the finite-element methodto preform design in metal forming. Extensive references on the use of thefinite-element method to metal-forming applications can be found in theReferences [28, 29].

References

1. Thomsen, E. G., Yang, C. T., and Bierbower, J. B., (1954), "An Experimen-tal Investigation of the Mechanics of Plastic Deformation of Metals," Univ.California Pub. Engg., Vol. 5.

2. Hill, R., (1963), "A General Method of Analysis of Metal-Working Proc-esses," /. Mech. Phys. Solids, Vol. 11, p. 305.

3. Hill, R., (1950), "The Mathematical Theory of Plasticity," Oxford UniversityPress, London.

4. Prager, W., and Hodge, P. G., Jr., (1951), "Theory of Perfectly PlasticSolids," Chapman and Hall, London.

5. Hoffman, O., and Sachs, G., (1953), "Introduction to the Theory of Plasticityfor Engineers," McGraw-Hill, New York.

6. Unksov, E. P., (1961), "An Engineering Theory of Plasticity," Butterworths,London.

7. Johnson, W., and Mellor, P. B., (1973), "Engineering Plasticity," VanNostrand and Reinhold, London.

8. Ford, H., and Alexander, J. M., (1963), "Advanced Mechanics of Materials,"Longmans Green, London.

9. Alexander, J. M., and Brewer, R. C., (1963), "Manufacturing Properties ofMaterials," Van Nostrand, London.

10. Thomsen, E. G., Yang, C. T., and Kobayashi, S., (1963), "Mechanics ofPlastic Deformation in Metal Processing," Macmillan, New York: Macmillan-Collier, London.

11. Kalpakjian, S., (1967), "Mechanical Processing of Materials," Van Nostrand,Princeton, NJ.

12. Avitzur, B., (1968), "Metal Forming and Processes," McGraw-Hill, NewYork.

Introduction 7

13. Johnson, W., and Kudo, H., (1962), "The Mechanics of Metal Extrusion,"Manchester University Press, Manchester, UK.

14. Johnson, W., Sowerby, R., and Haddow, J. B., (1970), "Plane-StrainSlip-Line Fields," American Elsevier, New York.

15. Johnson, W., Sowerby, R., and Venter, R. D., (1982), "Plane Strain Slip LineFields for Metal Deformation Processes," Pergamon Press, Oxford.

16. Blazynski, T. Z., (1976), "Metal Forming," Wiley, New York.17. Rowe, G. W., (1977), "Principles of Industrial Metalworking Processes,"

Edward Arnold, London.18. Slater, R. A. C., (1977), "Engineering Plasticity," Wiley, New York.19. Hosford, W. F., and Caddell, R. M., (1983), "Metal Forming; Mechanics and

Metallurgy," Prentice-Hall, Englewood Cliffs, NJ.20. Altan, T., Oh, S. I., and Gegel, H., (1983), "Metal Forming; Fundamentals

and Applications," American Society for Metals, Metals Park, Ohio.21. Boer, C. R., Rebelo, N., Rystad, H., and Schroder, G., (1986), "Process

Modelling of Metal Forming and Thermomechanical Treatment," Springer-Verlag, Berlin.

22. Mote, C. D., Jr., (1980), "Introduction to the Finite Element Method,"Lecture Note, University of California at Berkeley.

23. Zienkiewicz, O. C., (1977), "The Finite Element Method," 3rd Edition,McGraw-Hill, New York.

24. Strang, G., and Fix, G. J., (1973), "An Analysis of the Finite ElementMethod," Prentice-Hall, Englewood Cliffs, NJ.

25. Huebner, K. H., (1975), "The Finite Element Method for Engineers," Wiley,New York.

26. Zienkiewicz, O. C., (1984), Flow Formulations for Numerical Solutions ofForming Processes, "Numerical Analysis of Forming Processes," edited by J.F. T. Pittman et al., Wiley, New York, p. 1.

27. Nagtegaal, J. C., and Veldpaus, F. E., (1984), On the Implementation ofFinite Strain Plasticity Equations in a Numerical Model, "Numerical Analysisof Forming Processes," edited by J. F. T. Pittman et al., Wiley, New York, p.351.

28. Kobayashi, S., (1982), "A Review on the Finite Element Method and MetalForming Process Modeling," /. Appl. Metal Working, Vol. 2, p. 163.

29. Kobayashi, S., (1985), "Metal Forming and the Finite Element Method—Pastand Future," Proceedings of the 25th Int. Conf. Mach Tool Des. Res., April,Birmingham, p. 17.

2METAL-FORMING PROCESSES

2.1 IntroductionIn metal forming, an initially simple part—a billet or sheet blank, forexample—is plastically deformed between tools (or dies) to obtain thedesired final configuration. Thus, a simple part geometry is transformedinto a complex one, in a process whereby the tools "store" the desiredgeometry and impart pressure on the deforming material through thetool-material interface.

The physical phenomena constituting a forming operation are difficult toexpress with quantitative relationships. The metal flow, the friction at thetool-material interface, the heat generation and transfer during plasticflow, and the relationships between microstructure/properties and processconditions are difficult to predict and analyze. Often, in producing discreteparts, several forming operations (preforming) are required to transformthe initial "simple" geometry into a "complex" geometry, without causingmaterial failure or degrading material properties. Consequently, the mostsignificant objective of any method of analysis is to assist the formingengineer in the design of forming and/or preforming sequences. For agiven operation (preforming or finish-forming), such design essentiallyconsists of (1) establishing the kinematic relationships (shape, velocities,strain-rates, strains) between the deformed and undeformed part, i.e.,predicting metal flow; (2) establishing the limits of formability or produci-bility, i.e., determining whether it is possible to form the part withoutsurface or internal defects; and (3) predicting the forces and stressesnecessary to execute the forming operation so that tooling and equipmentcan be designed or selected.

For the understanding and quantitative design and optimization ofmetal-forming operations it is useful (a) to consider a metal formingprocess as a system and (b) to classify these processes in a systematic way.

2.2 A Metal-Forming Operation as a SystemA metal-forming system comprises all the input variables relating the billetor blank (geometry and material), the tooling (geometry and material), theconditions at the tool-material interface, the mechanics of plastic deforma-tion, the equipment used, the characteristics of the final product, and

8

Metal-Forming Processes 9

Billet(l) Process (2,3,4,5) Product (6)

FIG. 2.1 Illustration of metal forming system using closed-die forging as an example: 1,billet; 2, dies; 3, interface; 4, deformation mechanics; 5, forming machine; 6, product; 7,environment [2].

finally the plant environment in which the process is being conducted [1].Such a system is illustrated in Fig. 2.1, using impression die forging as anexample [2].

The "systems approach" in metal forming allows study of the effects ofprocess variables on product quality and process economics. The key to asuccessful metal-forming operation, i.e., to obtaining the desired shapeand properties, is the understanding and control of metal flow. Thedirection of metal flow, the magnitude of deformation, and the tempera-tures involved greatly influence the properties of the formed components.Metal flow determines both the mechanical properties related to localdeformation and the formation of defects such as cracks or folds at orbelow the surface. The local metal flow is in turn influenced by the processvariables, which are discussed below.

Material Variables

For a given material composition and deformation/heat-treatment history(microstructure), the flow stress (or effective stress), and the workability(or formability) in various directions (anisotropy), are the most importantmaterial variables in the analysis of a metal-forming process.

For a given microstructure, the flow stress is expressed as a function ofstrain, strain-rate, and temperature. To determine the actual functionalrelationship, it is necessary to conduct torsion, plane-strain compression,and uniform axisymmetric compression tests. Workability or formability isthe capability of a material to deform without failure; it depends on (1)conditions existing during deformation processing (such as temperature,rate of deformation, stresses, and strain history), and (2) material variables


(such as composition, voids, inclusions, and initial microstructure). In hotforming processes, temperature gradients in the deforming material (forexample, due to local die chilling) also influence metal flow and failurephenomena.

Tooling and Equipment

The selection of a machine for a given process is influenced by the time,accuracy, and load-energy characteristics of that machine. Optimumequipment selection requires consideration of the entire forming system,including lot size, conditions at the plant, environmental effects, andmaintenance requirements, as well as the requirements of the specific partand process under consideration.

The tooling variables include (1) design and geometry, (2) surface finish,(3) stiffness, and (4) mechanical and thermal properties under conditionsof use.

Friction

The mechanisms of interface friction are very complex. One way ofexpressing friction quantitatively is through a friction coefficient n, or afriction shear factor m. There are various methods of evaluating friction,i.e., estimating the value of \n or m. Tests most commonly used are thering and spike tests for massive forming and the plane-strain-draw andstretch-draw tests for sheet forming [2].

Deformation MechanicsIn forming, material is deformed plastically to generate the shape of thedesired product. Metal flow is influenced mainly by (1) tool geometry, (2)friction conditions, (3) characteristics of the stock material, and (4)thermal conditions existing in the deformation zone. The details of metalflow influence the quality and properties of the formed product and theforce and energy requirements of the process. The mechanics of deforma-tion, i.e., the metal flow, strains, strain-rates, and stresses, can beinvestigated by process modeling. Some analysis methods for processmodeling are outlined in Chap. 1 (see Section 1.1), and process modelingby the finite-element method is the main subject of this book.

Product PropertiesThe macro- and microgeometry of the product, i.e., its dimensions andsurface finish, are influenced by process variables. The processing condi-tions (temperature, strain, and strain-rate) determine the microstructuralvariations taking place during deformation and often influence finalproduct properties. Consequently, a realistic systems approach mustinclude consideration of (1) the relationships between properties andmicrostructure of the formed material and (2) the quantitative influencesof process conditions on metal flow and resulting microstructures.


TABLE 2.1 Classification of Massive Forming Processes [3]

Forging Rolling Extrusion Drawing

Closed-die forgingwith flash

Closed-die forgingwithout flash

CoiningElectro-upsettingForward extrusion

forgingBackward extrusion

forgingHobbingIsothermal forgingNosingOpen-die forgingOrbital forgingP/M forgingRadial forgingUpsetting

Sheet rolling Nonlubricated hotShape rolling extrusionTube rolling Lubricated direct hotRing rolling extrusionRotary tube Hydrostatic extrusion

piercingGear rollingRoll forgingCross rollingSurface rollingShear forming

(flow turning)Tube reducing

DrawingDrawing with

rollsIroningTube sinking

2.3 Classification and Description of Metal-Forming Processes [3]

The metal-forming processes may be classified into two broad categories:

1. Massive forming processes (Table 2.1)2. Sheet-metal forming processes (Table 2.2)

In both cases, the surfaces of the deforming material and of the tools are

TABLE 2.2 Classification of Sheet-metal Forming Processes [3]

Bending and Straight FlangingBrake bendingRoll bending

Surface Contouring of SheetContour stretch forming

(stretch forming)AndroformingAge formingCreep formingDie-quench formingBulgingVacuum forming

Linear ContouringLinear stretch forming

(stretch forming)Linear roll forming

(roll forming)

Deep Recessing and FlangingSpinning (and roller flanging)Deep drawingRubber pad formingMarform processRubber diaphragm hydroforming

Shallow RecessingDimplingDrop hammer formingElectromagnetic formingExplosive formingJoggling


in contact, and friction between them has a major influence on the process.In massive forming, the input material is in billet, rod, or slab form, and aconsiderable increase in the surface-to-volume ratio occurs in the formedpart. In sheet forming, a sheet blank is plastically deformed into athree-dimensional object without any significant changes in sheet thicknessand surface characteristics.

Processes that fall under the category of massive forming processes(Table 2.1), have the following distinguishing features:

• The workpiece undergoes large plastic deformation, resulting in anappreciable change in shape or cross section.

• The portion of the workpiece undergoing permanent (plastic) defor-mation is generally much larger than the portion undergoing elasticdeformation; therefore, elastic recovery after deformation isnegligible.

The characteristics of sheet-metal forming processes (Table 2.2) are:

• The workpiece is a sheet or a part fabricated from a sheet.• The deformation usually causes significant changes in shape, but not

in cross section, of the sheet.• In some cases, the magnitudes of permanent plastic and recoverable

elastic deformations are comparable; therefore, elastic recovery orspringback may be significant.

Most significant metal-forming processes are listed in Tables 2.1 and 2.2[3]. Selected massive and sheet forming processes are described in Figs 2.2through 2.29 [2-10].

FIG. 2.2 Closed-die forging without flash. A billet with carefully controlled volume isdeformed (hot or cold) by a punch in order to fill a die cavity without any loss of material.The punch and the die may be made of one or several pieces.

FIG. 2.4 Coining opeation. Coining is a closed-die forming operation, usually performedcold, in which all surfaces of the work are confined or restrained, resulting in a well-definedimprint of the die on the workpiece. It is also a restriking operation used to sharpen or changean existing radius or profile.

FIG. 2.5 Forward extrusion forging. A punch compresses a billet (hot or cold) confined in acontainer so that the billet material flows through a die in the same direction as the punch.

13

FIG. 2.3 Closed-die forging with flash. A billet is formed (hot) in dies (usually with twohalves) such that the flow of metal from the die cavity is restricted. The excess material isextruded through a restrictive narrow gap and appears as flash around the forging at the dieparting line.

FIG. 2.6 Backward extrusion forging. A moving punch applies a steady pressure to a slug(hot or cold) confined in a closed die, and forces the metal to flow around the punch in adirection opposite the direction of punch travel.

14

(A) (B)

FIG. 2.7 Robbing (A) in a container and (B) without restriction. Robbing is the process ofindenting or coining an impression into a cold or hot die block by pressing with a punch.

FIG. 2.8 Nosing. Nosing is a hot or cold forming process in which the open end of a shell ortubular component is closed by axial pressing with a shaped die.

FIG. 2.9 Open-die forging. Open-die forging is a hot forming process in which metal isshaped by hammering or pressing between flat or simple contoured dies.

15

FIG. 2.10 Various stages in orbital forging processes. Orbital forging is the process offorming shaped parts by incrementally forging (hot or cold) a slug between an orbiting upperdie and a nonrotating lower die. The lower die is raised axially toward the upper die, which isfixed axially but whose axis makes orbital, spiral, planetary, or straight-line motions.

FIG. 2.11 Radial forging of a shaft. This hot or cold forming process utilizes two or moreradially moving anvils, or dies, for producing solid or tubular components with constant orvarying cross sections along their lengths.

16

FIG. 2.12 Upsetting with flat-heading tool. Upsetting is the process of forming metal (hot orcold) so that the cross-sectional area of a portion, or all, of the stock is increased.

FIG. 2.13 Schematic of the rolling process for sheet and plates. Sheet and plate rolling is ahot or cold forming process for reducing the cross-sectional area of the stock with the use ofrotating rolls. In general, the rolled material elongates and spreads simultaneously while thecross-sectional area is reduced.

FIG. 2.14 Roll passes for rolling an angle (L) shape. Shape rolling is a cold or hot formingprocess for reducing as well as shaping the cross section of the metal stock by passing itthrough a series of rotating sets of rolls with appropriately shaped grooves.

17

FIG. 2.15 Principles of ring rolling. Ring rolling is a process whereby a hollow circular blank(cold or hot) is formed into a ring. A main roll presses on the outside diameter of the blank,which is supported by a mandrel on the inside diameter. Shaped cross sections are obtainedby appropriate contouring of the mandrel and the roll. The height of the ring is controlled byauxiliary rolls.

FIG. 2.16 Rotary tube piercing, a, Double-conical working rolls; b, guide roll; c, billet; d,conical piercer point. A hollow is formed by peripherally rolling a cylindrical hot billet over aconical piercer point. The billet is driven by a pair of cone-shape rolls, set askew to thelongitudinal axis of the billet. The frictional load between the rolls and the billet causes thebillet to rotate and forces it to advance longitudinally over the piercer point.

18

(a) (b)

FIG. 2.17 Shear forming from a plate, (a) Start, (b) partially or completely formed part.Shear forming is a process for hot or cold seamless shaping of dished parts by the combinedforces of rotation and pressure. This process differs from spinning principally in that itreduces the thickness of the formed part.

FIG. 2.18 Tooling and metal flow for direct and indirect extrusion process. The product ofdesired cross section is obtained by forcing a heated billet through a die without lubricatingthe billet, the container, or the die. In the direct extrusion process, the product is extruded inthe direction of ram movement. When the product is extruded in a direction opposite that ofram travel, the process is called indirect extrusion.

DIRECT EXTRUSION INDIRECT EXTRUSION

19

FIG. 2.19 Hot extrusion setup using glass lubrication. The heated billet is forced through adie, using some form of lubrication, to obtain a product of desired cross section. Glass is themost widely used lubricant for extruding long lengths from steels and high-temperaure alloyson a production basis.

(a) (b)

FIG. 2.20. Drawing of (a) rod or wire and (b) tube. Drawing is the process of reducing thecross-sectional area and/or the shape of a rod, bar, tube or wire (cold or hot) by pullingthrough a die.

20

FIG. 2.21 Schematic of ironing. Ironing is the process of smoothing or thinning the wall of ashell or cup (cold or hot) by forcing the shell through a die with a punch.

FIG. 2.22 Tube sinking process. Tube sinking is the process of sizing the outside diameter ofa tube by drawing the tube (cold or hot) through a die without supporting the tube internallywith a mandrel.

A B C D

FIG. 2.23 Typical brake-bending operations. (A) Air bending; (B) air rounding, (C) diebending; (D) die rounding. Brake bending is a forming operation widely used for forming flatsheets into linear sections, such as angles, channels and hats. There are two typicalbrake-forming setups: air bending and die bending. In air bending, the workpiece issupported only at the outer edges so that the length of the ram stroke determines the bendangle of the part. In die bending, the sheet is forced into a female die cavity of the requiredpart angle.

21

Workpiece

Bending roll

Driven rolls

Support rolls

FIG. 2.24 Roll bending (three-roll forming). Roll bending gives a curvature to a sheet, bar,or shaped section by bending it between two or three cylindrical rolls that can be adjusted.

FIG. 2.25 Roll forming. This process is used to produce long components of various crosssections. The sheet metal is formed by passing it through a succession of progressively shapedpower-driven contoured rolls.

22

(a) Roller prof i les forone pass

(b) Various passes for rollforming "HAT" section

1C) Id)FIG. 2.26 Various spinning operations, (a) Hollow shape forming; (b) bulging, (c) reducing,(d) threading. Spinning is the process of shaping seamless dished parts by the combinedforces of rotation and pressure. Spinning does not result in any change in thickness.

FIG. 2.27 Deep drawing. (A) First draw; (B) redraw; (C) reverse draw. In deep drawing, asheet blank (hot or cold), usually subjected to a peripheral hold-down pressure, is forced by apunch into and through a die to form a deep recessed part having a wall thicknesssubstantially the same as that of the blank. This process is used to produce cylindrical orprismatic cups with or without a flange on the open end. Cups or tubes can be sunk orredrawn to increase their length and to reduce their lateral dimensions.

23


FIG. 2.28 Rubber-pad forming, (a) With rubber punch; (b) with rubber pad. This is aforming operation for producing shallow parts. A rubber pad is attached to the press slideand becomes the mating die for a punch, or group of punches, which has been placed on thepress bed or plate. The rubber pad is confined in a container (pad holder), and the entireslide with attached pad holder is forced against the tools, usually by hydraulic pressure. Asthe slide descends, the pliable but virtually incompressible rubber fills the space between theslide and the dies and forces the metal to take the exact contours of the dies.

BEGINNING OF FORMING COMPLETE FORMING

FIG. 2.29. Rubber-diaphragm hydroforming. In this process, the blank is held between adiaphragm, which closes the ram pressure chamber, and a blank holder. A male punch worksagainst the diaphragm, and the metal is shaped by balancing the pressure of the ram chamberagainst the pressure of the press base chamber on which the punch is mounted.

References1. Altan, T., Lahoti, G. D., and Nagpal, V., (1981), "Systems Approach in

Massive Forming and Application to Modeling of Forging Processes," /. Appl.Metal Working, ASM, Vol. 1, No. 2, p. 29.

2. American Society for Metals, (1961), "Metals Handbook", Eighth Edition,Vol. 1 (Properties and Selection of Metals) and Vol. 4 (Forming), Americansociety for Metals, Metals Park, OH.

3. Altan, T., Oh, S. I., and Gegel, H., (1983), "Metal Forming: Fundamentalsand Applications," ASM International, Metals Parks, OH.

4. International Institution for Production Engineering Research, (1962), Dic-tionary of Production Engineering, Vol. 1 (Forming and Drop Forging), Vol. 3


(Sheet Metal Forming), and Vol. 5 (Cold Extrusion and Upsetting), Verlag,W. Girardet, Essen.

5. Lahoti, G. D., and Altan, T., (1976), "Design of Dies for Radial Forging ofRods and Tube," Technical Paper MF76-390, Society of ManufacturingEngineers, Dearborn, MI.

6. Aluminium, American Society for Metals, (1967), Vol. 3 (Fabrication andFinishing), edited by K. R. Van Horn, American Society for Metals, MetalsPark, OH, p. 81.

7. Geleji, A., (1967), "Forge Equipment Rolling Mills and Accessories,"Akademiai Kiado, Budapest.

8. Beyon, R. E., (1956), "Roll Design and Mill Layout," Association of Iron andSteel Engineers, Pittsburgh.

9. Sachs, G., (1951), "Principles and Methods of Sheet-Metal Fabricating,"Reinhold, New York.

10. Lange, K., (1972), "Lehrbuch der Umformtechnik/Textbook of FormingTechnology" (in German), Springer-Verlag, Berlin.

ANALYSIS AND TECHNOLOGYIN METAL FORMING

3.1 IntroductionThe design, control, and optimization of forming processes require (1)analytical knowledge regarding metal flow, stresses, and heat transfer, aswell as (2) technological information related to lubrication, heating andcooling techniques, material handling, die design and manufacture, andforming equipment.

The purpose of using analysis in metal forming is to investigate themechanics of plastic deformation processes, with the following majorobjectives.

• Establishing the kinematic relationships (shape, velocities, strain-rates, and strains) between the undeformed part (billet, blank, orpreform) and the deformed part (product); i.e., predicting metal flowduring the forming operation. This objective includes the prediction oftemperatures and heat transfer, since these variables greatly influencelocal metal-flow conditions.

• Establishing the limits of formability or producibility; i.e., determin-ing whether it is possible to perform the forming operation withoutcausing any surface or internal defects (cracks or folds) in thedeforming material.

• Predicting the stresses, the forces, and the energy necessary to carryout the forming operation. This information is necessary for tooldesign and for selecting the appropriate equipment, with adequateforce and energy capabilities, to perform the forming operation.

Thus, the mechanics of deformation provides the means for determininghow the metal flows, how the desired geometry can be obtained by plasticdeformation, and what the expected mechanical properties of the pro-duced part are.

For understanding the variables of a metal-forming process, it is best toconsider the process as a system, as illustrated in Fig. 2.1 in Chap. 2. Theinteraction of most significant variables in metal forming are shown, in asimplified manner, in Fig. 3.1. It is seen that for a given billet or blankmaterial and part geometry, the speed of deformation influences strain-rate and flow stress. Deformation speed, part geometry, and die tempera-ture influence the temperature distribution in the formed part. Finally,

26

3

Analysis and Technology in Metal Forming 27

FIG. 3.1 Simplified illustration of the interactions between major process variables in metalforming

flow stress, friction, and part geometry determine metal flow, formingload, and forming energy.

In steady-state flow (kinematically), the velocity field remains un-changed, as is the case in the extrusion process (Fig. 3.2B); in nonsteady-state flow, the velocity field changes continuously with time, as is the casein upset forging (Fig. 3.2A) [1].

(A) (B)

FIG. 3.2 Metal flow in certain forming processes. (A) Non-steady state upset forging; (B)steady-state extrusion [1].


The state of deformation in a plastically deforming metal is fullydescribed by the displacements, velocities, strains, and strain-rates. Thereare several approximate methods for analyzing metal-forming problems.They are briefly outlined in Section 1.1 of Chap. 1, and the upper-boundmethod and Hill's general method are further illustrated in Chap. 5.Details of some of these methods are given in the metal-forming bookslisted in Chap. 1, and the forming processes, with emphasis on technologi-cal aspects, are described in Reference [1]. It is to be noted that everymethod of analysis requires as input (1) a description of the materialbehavior under the process conditions, i.e., flow stress data, and (2) aquantitative value to describe the friction, i.e., the friction factor m, or thefriction coefficient /j,. These two quantities themselves—flow stress andfriction—must be determined by experiment and are difficult to obtainaccurately. Thus, in addition to simplifications and approximations as-sumed in the methods, any errors in flow stress measurements oruncertainties in the value of the friction factor are expected to influencethe reliability of the results of analysis. Johnson and Sowerby [2] reviewedrecent analytical researches into drawing, extrusion, rolling, forging, andsheet-metal forming in the context of the limitations imposed by tech-nological considerations.

3.2 Flow Stress of MetalsThe yield stress of a metal under uniaxial conditions, as a function ofstrain, strain-rate, and temperature, can also be considered as the flowstress (or the effective stress). The definition of the effective stress as arepresentative stress under combined loading is given in Chap. 4.

The flow stress o is important because in metal-forming processes theforming loads and stresses depend on (1) part geometry, (2) friction, and(3) the flow stress of the deforming material. The flow stress of a metal isinfluenced by:

• Factors unrelated to the deformation process, such as a chemicalcompositon, metallurgical structure, phases, grain size, segregation,and prior strain history

• Factors explicitly related to the deformation process, such as tempera-ture, degree of deformation, and rate of deformation. The degree ofdeformation and rate of deformation under general loading aremeasured by the effective strain £ and the effective strain-rate e,respectively, and their definitions are also given in Chap. 4

Thus, the flow stress a can be expressed as a function of temperature T,strain e, strain-rate e, and microstructure S:

In hot forming of metals at temperatures above the recrystallizationtemperature, the influence of strain on flow stress is insignificant, and the


influence of strain-rate (i.e., rate of deformation) becomes increasinglyimportant. Conversely, at room temperature (i.e., in cold forming), theeffect of strain-rate on flow stress is negligible, and the effect of strain onflow stress (i.e., strain hardening) is most important. The degree ofdependence of flow stress on temperature varies considerably amongdifferent materials. Therefore, temperature variations in a forming opera-tion can have quite different effects on load requirements and on metalflow for different materials. For instance, a drop of approximately 100°F inthe hot forming temperature (from 1700 to 1600°F) would result in a 40%increase in flow stress for titanium alloy Ti-8Al-lMo-lV. The increase inflow stress that would result from the same temperature drop, 100°F withinthe hot working range (from 2200 to 2100°F), would be only about 15% forAISI type 4340 steel [3].

To be useful in metal-forming analyses, the flow stresses of metals mustbe determined experimentally for the strain, strain-rate, and temperatureconditions that exist in metal-forming processes. The methods mostcommonly used for obtaining flow stress data are tensile, uniformcompression, and torsion tests. The compression test is particularly simple,and therefore it is very widely used. In this test, the flat platens and thecylindrical sample are maintained at the same temperature, so that diechilling, with its influence on metal flow, is prevented. To be applicablewithout errors or corrections, the cylindrical workpiece must be upsetwithout any barreling; i.e., the state of uniform deformation in theworkpiece must be maintained. Barreling is prevented by using adequatelubrication, e.g., Teflon or machine oil at room temperature and, at hotworking temperatures, graphite in oil for aluminium alloys, and glass forsteel, titanium, and high-temperature alloys [4]. The load and displace-ment or specimen height are measured during the test. From thisinformation the flow stress is calculated at each stage of deformation, or,for increasing strain, at a strain-rate given by the ratio of the instantaneousram speed to specimen height.

At room temperature the flow stresses of most metals (except that oflead, for example) are only slightly strain-rate dependent. Therefore, anytesting machine or press can be used for the compression test, regardless ofits ram speed.

At hot working temperatures, i.e., above the recrystallization tempera-ture, the flow stresses of nearly all metals are very much strain-ratedependent. Therefore, whenever possible, these temperature range com-pression tests are conducted on a machine that provides a velocity-displacement profile such that the constant-strain-rate condition can bemaintained throughout the test. Mechanical cam-activated presses calledplastometers or hydraulic programmable testing machines (MTS, forexample) [5] are used for this purpose. In order to maintain nearlyisothermal and uniform compression conditions, the test is conducted in afurnace or a fixture. The specimens are lubricated with appropriatelubricants—for example, oil-graphite for temperatures up to SOOT and

FIG. 3.3 Flow stress vs. strain, and strain-rate vs. strain, for type 403 stainless steel at 1800,1950 and 2050°F (tests were conducted in a mechanical press where e was not constant) [4].

glass for temperatures up to 2300T. The fixture and the specimens areheated to test temperature and then the test is initiated. Examples ofhigh-temperature uniaxial flow stress data are given in Figs. 3.3 and 3.4.

3.3 Friction in Metal FormingFriction conditions at the die-material interface greatly influence metalflow, formation of surface and internal defects, stresses acting on the dies,and load and energy requirements. There are three basic types oflubrication that govern the frictional conditions in metal forming [5, 6].

1. Under dry conditions, no lubricant is present at the interface andonly the oxide layers present on the die and workpiece materials mayact as a "separating" layer. In this case friction is high, and such asituation is desirable in only a few selected forming operations, suchas hot rolling of plates and slabs and nonlubricated extrusion ofaluminium alloys.



FIG. 3.4 Flow stress vs. strain, and strain-rate vs. strain, for Waspaloy at 1950, 2050 and2100°F (tests were conducted in a mechanical press where e was not constant) [4].

2. "Hydrodynamic" conditions exist when a thick layer of lubricant ispresent between the dies and the workpiece. In this case the frictionconditions are governed by the viscosity of the lubricant and by therelative velocity between the die and the workpiece. The viscositiesof most lubricants decrease rapidly with increasing temperature.Consequently, in most practical high-speed forming operations, suchas strip rolling and wire drawing, the hydrodynamic conditions existonly within a certain regime of velocities, where the interfacetemperatures are relatively low [6].

3. "Boundary" lubrication is the most widely encountered situation inmetal forming. Increases in temperature at the interface and therelatively high forming pressures do not usually allow the presence ofa hydrodynamic lubrication regime. Boundary lubrication, on theother hand, does not lend itself to reliable analysis.

Consequently, most of the knowledge on metal-forming lubricationis empirical, with very little analysis-based information.


In most forming applications, the lubricity of a lubricant is the singlemost significant factor, since it directly determines the interface friction. Inorder to evaluate the performances of various lubricants and to be able topredict forming pressures, it is necessary to express the interface frictionquantitatively, in terms of a factor or a coefficient. The friction shearstress, fs, is most commonly expressed as

p being a compressive normal stress to the interface, or as

k being the shear strength of the deforming material, where 0 < ra < 1.Studies in forming mechanics indicate that eq. (3.3) adequately repre-

sents the friction condition in bulk forming processes while eq. (3.2) iscommonly used for representation of friction in sheet-metal forming. Areason for this is that the compressive normal stress at the interface insheet-metal forming is much smaller in magnitude, in comparison with thatin bulk deformation processes. For various forming conditions, the valuesof m vary as follows:

• m = 0.05-0.15 in cold forming of steels, aluminium alloys, andcopper, using conventional phosphate-soap lubricants or oils

• m = 0.2-0.4 in hot forming of steels, copper, and aluminum alloyswith graphite-based (graphite-water or graphite-oil) lubricants

• m= 0.1-0.3 in hot forming of titanium and high-temperature alloyswith glass lubricants

• m = 0.7-1.0 when no lubricant is used, e.g., in hot rolling of plates orslabs and in nonlubricated extrusion of aluminium alloys

In determining the friction factor m for hot forming, in addition tolubrication effects, the effects of die chilling or heat transfer from the hotmaterial to colder dies must be considered. Therefore, the lubrication testsused for determining friction factors must include both lubrication anddie-chilling effects. Consequently, in hot forming, a good test must satisfyas well as possible the following requirements.

• The specimen and die temperatures must be approximately the sameas those encountered in the actual hot forming operation.

• The contact time between specimen and tools under pressure must beapproximately the same as in the forming operation of interest.

• The ratio of the newly generated deformed surface area to originalsurface area of the undeformed specimen must be approximately thesame as in the process investigated.

• The relative velocity between deforming metal and dies should haveapproximately the same magnitude and direction as in the formingprocess.

Lubricity, as defined by the friction factor m, is most commonly


measured by using the ring test. In the ring test, a flat ring-shapedspecimen is compressed to a known reduction. The change in internal andexternal diameters of the forged ring is very much dependent on thefriction at the die-workpiece interface. If friction were zero, the ringwould deform in the same way as a solid disk, with each element flowingradially outward at a rate proportional to its distance from the center.With increasing deformation, the internal diameter of the ring is reduced iffriction is high, and is increased if friction is low. Thus, the change in theinternal diameter represents a simple method for evaluating interfacefriction [5,6].

3.4 Temperatures in Metal FormingIn metal-forming processes, both plastic deformation and friction contrib-ute to heat generation. Approximately 90-95% of the mechanical energyinvolved in the process is transformed into heat. In some continuousforming operations such as drawing and extrusion, performed at highspeeds, temperature increases of several hundred degrees may be in-volved. A part of the generated heat remains in the deformed material,another part flows into tooling, while still a further part may flow into theundeformed portion of the material. The temperatures developed in theprocess influence lubrication conditions, tool life, and the properties of thefinal product, and, most significantly, determine the maximum deforma-tion speed that can be used for producing sound products withoutexcessive tool damage. Thus, temperatures generated during plasticdeformation greatly influence the productivity of metal-forming processes[5].

The magnitudes and distribution of temperatures depend mainly on:

• The initial material and die temperatures• Heat generation due to plastic deformation and friction at the

die-material interface• Heat transfer between the deforming material and the dies and

between the material and the environment (air or coolant)

In actual forming operations there is temperature gradient in deformingmaterial and in the dies. The temperature distributions encountered informing operations for producing discrete parts, such as die forging,upsetting, and deep drawing, are quite different from the temperatureincreases found in quasicontinuous deformation processes such as wiredrawing, rolling, and extrusion. In forming operations of the former type,e.g., in cold forging, the metal flow is kinematically nonsteady state.Deformation takes place in a relatively short period of time, i.e., fromseveral milliseconds to a fraction of a second. The deforming material is incontact with the dies during this short period. After the part has beenformed and removed from the die, the dies can cool off during aconsiderable period of time, until the next part is loaded into them.


DISPLACEMENT. HQ-H, MM

FIG. 3.5 Load vs. displacement curves obtained in closed-die forging of an axisymmetricsteel part (dimensions in inches) at 2012°F in three different machines with different initialvelocities, Vpi [5].

In continuous forming operations, e.g., wire drawing, the metal flow isnearly steady state. The deforming material is continuously in contact withthe die and there is a cumulative temperature increase that significantlyinfluences die life, production rate, and the quality of drawn material.

The influence of temperatures in metal-forming operations is mostdramatic in hot forming operations, where the contact time under pressurebetween the deforming material and the dies is the most significant factorinfluencing temperature conditions. This is illustrated in Fig. 3.5, wherethe load-displacement curves are given for hot forging of a steel part usingdifferent types of forging equipment [5]. These curves illustrate that owingto strain-rate and temperature effects, for the same forging process,different forging loads and energies are required by different machines.For the hammer, the forging load is initially higher owing to strain-rateeffects, but the maximum load is lower than for either hydraulic or screwpresses. The reason for this is that in the presses the extruded flash coolsrapidly, whereas in the hammer the flash temperature remains nearly thesame as the initial stock temperature.

Thus, in hot forming, not only the material and the formed shape butalso the type of equipment used (rate of deformation and die-chillingeffects) determine the metal-flow behavior and the forming load andenergy required for the process. Surface tearing and cracking or develop-ment of shear bands in the formed material often can be explained by

HQ-H. INCH


excessive chilling of the surface layers of the formed part near thedie-material interface.

3.5 Impression and Closed-Die ForgingIn impression or closed-die forging, two or more dies are moved towardeach other to form a metal billet, at a suitable temperature, in a shapedetermined by the die impressions. In most practical hot forging opera-tions, the temperature of the workpiece materials is higher than that of thedies. Metal flow and die filling are largely determined by (1) the flow stressand formability, (2) the friction and heat transfer at the die-materialinterface, and (3) the complexity of the forging shape. For a given metal,both flow stress and forgeability (formability in forging) are influenced bythe metallurgical structure, temperatures, strains, strain-rates, and stressesthat occur during deformation.

The main objective of forging process design is to ensure adequate flowof the metal in the dies so that the desired finished part geometry can beobtained without defects and with prescribed properties. Metal flow isgreatly influenced by part or die geometry. Often several operations(preforming or blocking) are needed to achieve gradual flow of metal froman initially simple shape (cylinder or round-cornered square billet) into themore complex shape of the final forging.

The steps involved in designing a forging process are seen in Fig. 3.6. Inthe finish-forging die, the flash geometry is selected such that the flash isencouraged to restrict metal flow into the "flash gutter," outside of the diecavity, as seen in Fig. 3.7 [3]. This results in an increase in the forgingstresses. Therefore, it is necessary on one hand to decrease the flashthickness or increase the flash width, while on the other maintaining thedie stresses at a level such that the dies are not damaged. Computerizedupper-bound and slab techniques can be used for this purpose.

Design of blocker and preform geometries is the most critical part offorging die design. The blocker operation has the purpose of distributingthe metal adequately within the blocker (or preform) to achieve thefollowing objectives:

• Filling the finisher cavity without any forging defects• Reducing the amount of material lost as forging flash• Reducing die wear by minimizing metal movement in the finisher die• Providing the required amounts of deformation and grain flow so that

desired forging properties are obtained

Traditionally, blocker dies and preforms are designed by experienceddie designers and are modified and refined by die try outs. The initialblocker design is based on several empirical guidelines. These guidelinesdepend on the material used and on the forging machine utilized.

At present, computer-aided design (CAD) of blocker cross sections canbe carried out using interactive graphics. However, this method still


CAD/CAM PROCEDURE OFFORGING DIE DESIGN

FIG. 3.6 Outline of a CAD/CAM procedure for forging die design and manufacture.

employs the same empirical relationships listed above, but stored in aquantitative manner in the computer memory.

As illustrated in Fig. 3.6, the ultimate process and die design in forgingrequires nonisothermal flow simulation via 2D and 3D FEM techniques.Thus, dies can be designed better and the need for experimentation isreduced.

3.6 Hot Extrusion of Rods and ShapesExtrusion is used to produce long and straight sections of constant crosssection. There are basically three variations of extrusion, depending on thelubrication technique used. In the nonlubricated extrusion process (Fig.3.8A), a flat-face die is used, and the material flows by internal shear andcauses a "dead-metal zone" to form in front of the extrusion die. In


FIG. 3.7 Metal flow and load-stroke curve in impression-die forging: (A) upsetting, (B)filling, (C) end, (D) load-stroke curve.

lubricated extrusion (Fig. 3.8B), a suitable lubricant is present between theextruded billet and the extrusion tooling, i.e, the container and the die.The third and most recently developed technique is hydrostatic extrusion(Fig. 3.8C), in which a fluid film between the billet and the tooling exertspressure on the deforming billet. Hydrostatic extrusion is used only inunusual applications for extruding special alloys, composites, or cladmaterials, where adequate lubrication cannot be easily provided byconventional lubrication techniques [5].

Metal flow during extrusion varies considerably, depending on thematerial, friction, and the shape of the extruded section. In extrusion ofaluminium alloys, temperatures developed during the process, greatlyinfluence the maximum ram speed. This is especially true in extrusion ofhard aluminium alloys. Temperatures are determined by simultaneous (a)

(A) (B) (C)

FIG. 3.8 Schematic illustrations of the nonlubricated (A), lubricated (B), and hydrostatic (C)extrusion processes.


heat generation due to plastic deformation and friction, (b) heat transfer atbillet-tooling interface and heat conduction within the billet and thetooling, and (c) heat transported with the extruded product.

In hot extrusion of aluminum and copper alloys, container lubrication isnot used and the dies are of the "flat-face" type, with the die openingimparting the desired section geometry to the extrusion. In extrusion ofsteels, titanium alloys, and other high-temperature materials, glass- orgraphite-based lubricants are used. The dies have some sort of "smoothentry" design to provide for easy metal flow and to avoid severe internalshear, or formation of a dead-metal zone, during extrusion. "Smoothentry" dies are also used successfully for extruding composite materials. Inthese applications, internal shear, which occurs in extrusion with flat dies,must be avoided in order to maintain the integrity and the uniformity ofthe composite structure.

In today's industrial practice, the design of extrusion dies, whether ofthe "flat-face" or the "smooth entry" type, is still an art rather than ascience. Die design for a new extrusion is developed from previousexperience and through costly experimentation and in-plant trials. Thus,process and die development may require relatively long periods of timeand may tie up extrusion presses that should preferably be used for actualproduction. A scientific design of the extrusion process could use FEM toestablish the following variables [5]:

• Optimum number of shaped orifices in the die; as seen in Fig. 3.9, inaluminum extrusion several sections are often extruded simul-taneously [7]

• Location of the orifices relative to the billet axis for uniform flowthrough each orifice

• Orientation of the orifices• Modification of the shape of the orifices to correct for thermal

shrinkage and die deflection under load

FIG. 3.9 Schematic illustration of a flat-face die for extrusion of "T" sections [7].


• Determination of die bearing lengths for balancing metal flow to avoidthe twisting and bending of the extrusion emerging from each orifice

• Determination of the extrusion load: quite difficult since the extrudedsections are usually nonsymmetrical, resulting in a complex 3D metalflow in the deformation zone

Some semiempirical analytical techniques have been developed forcomputer-aided design of extrusion dies. However, these techniques areapproximate and could be vastly improved by developing FEM-basedcomputer programs for metal-flow simulation.

3.7 Cold Forging and ExtrusionCold extrusion is a special type of forging process in which cold metal isforced to flow plastically under compressive force into a variety of shapes.These shapes are usually axisymmetric with relatively small nonsymmetri-cal features, and, unlike impression die forging, the process does notgenerate flash. The terms cold forging and cold extrusion are often usedinterchangeably and refer to well-known forming operations such asextrusion, upsetting or heading, coining, ironing, and swaging. Severalforming steps are used to produce a final part of relatively complexgeometry, starting with a slug or billet of simple shape, as shown in Fig.3.10 [8].

In warm forging, the billet is heated to temperatures below therecrystallization temperature, for example, up to 700-800°C (1292-1479°F)for steels, in order to lower the flow stress and the forging pressures. Incold forging, the billet or the slug is at room temperature whendeformation starts.

In cold forging, the tool stresses are quite high, in the order of250-350 ksi (1724-2413 MN/m2). Consequently, the prediction of theforming load and stresses is quite important for die design and machineselection. In addition, the distribution of the strains in each deformationstages is significant, since it determines the hardness distribution as well asthe formability of the part. Both these variables, strains and stresses, areinfluenced by the following process parameters:

• Area reduction. The extrusion load increases with increasing reductionin cross-sectional area because the strain increases with reduction.

• Die geometry (angle, radii). The die geometry directly influencesmaterial flow, and therefore it affects the distribution of the effectivestrain and flow stress in the deformation zone. In forward extrusion,for a given reduction, a larger die angle increases the volume of metalundergoing shear deformation and results in an increase in formingload. On the other hand, the length of the die decreases, which resultsin a decrease in die friction load. Consequently, for a given reductionand given friction conditions there is an optimum die angle thatminimizes the extrusion load.

£

FIG. 3.10 Schematic illustration of forming sequences in cold forging of a gear blank [8]. Left to right: sheared blank, simultaneous forward rod andbackward cup extrusion, forward extrusion, backward cup extrusion, simultaneous upsetting of flange, and coining of shoulder.


• Extrusion velocity. With increasing velocity, both the strain-rate andthe temperature generated in the deforming material increase. Theseeffects counteract each other, and consequently the load in coldextrusion is not affected significantly by the extrusion velocity.

• Lubrication. Improved lubrication lowers the container friction force,and the die friction force, resulting in lower extrusion loads.

• Workpiece material. The flow stress of the billet material directlyinfluences the loads necessary for homogeneous and shear deforma-tions. The prior heat treatment and/or any prior work hardening alsoaffect the flow stress of a material. Therefore, flow stress valuesdepend not only on the chemical composition of the material but alsoon its prior processing history.

• Billet dimensions. In forward extrusion, an increase in billet lengthresults in an increase in container friction load. In backward extru-sion, the billet length has little effect on the extrusion load.

The quantitative influence of the process variables discussed above uponformability, product properties, and die stresses can be best predicted byusing FEM-based codes. While there have been analyses of forgingoperations by FEM, practical applications are still needed for improvedprocess and die design.

3.8 Rolling of Strip, Plate, and Shapes

Most engineering metals, such as aluminum alloys, copper alloys, andsteels, are first cast into ingots and are then further processed by hotrolling into "blooms," "slabs" and "billets." These are known as semi-finished products because they are subsequently rolled into other productssuch as plate, sheet, tube, rod, bar, and structural shapes.

The primary objectives of the rolling process are to reduce the crosssection of the incoming material while improving its properties and toobtain the desired section at the exit from the rolls. The process can becarried out hot, warm, or cold, depending on the application and thematerial involved. The literature on rolling technology, equipment, andtheory is very extensive because of the significance of the process[9,10,11]. Many investigators prefer to divide rolling into cold and hotrolling processes. However, from a fundamental point of view, it is moreappropriate to classify rolling processes on the bases of the complexity ofmetal flow during the process and the geometry of the rolled product.Thus, rolling of solid sections can be divided into the following categories,as illustrated in Fig. 3.11.

1. Uniform reduction in thickness with no change in width. This is thecase in strip, sheet, or foil rolling where the deformation is only inthe directions of rolling and sheet thickness and there is nodeformation in the width direction (plane-strain deformation). Thistype of metal flow exists when the width of the deformation zone is atleast about 10 times the length of that zone (Fig. 3.HA).


2. Uniform reduction in thickness with an increase in width. This type ofdeformation occurs in rolling of blooms, slabs, and thick plates. Thematerial is elongated in the rolling (longitudinal) direction, is spreadin the width (transverse) direction, and is compressed uniformly inthe thickness direction (Fig. 3.11B).

3. Moderately nonuniform reduction in cross section. In this case thereduction in the thickness direction is not uniform. The metal iselongated in the rolling direction, is spread in the width direction,and is reduced nonuniformly in the thickness direction. Along thewidth, metal flow occurs only toward the edges of the section.Rolling of an oval section in rod rolling, or rolling of an airfoilsection (Fig. 3.11C), would be considered to be in this category.

4. Highly nonuniform reduction in cross section. In this type ofdeformation, the reduction is highly nonuniform in the thicknessdirection. A portion of the rolled section is reduced in thicknesswhile other portions may be extruded or increased in thickness (Fig.3.11D). As a result, in the width (lateral) direction metal flow may betoward the center. Of course, in addition, metal flows in thethickness direction as well as in the rolling (longidutinal) direction.

The strip rolling process is schematically illustrated in Fig. 3.12. A verylarge number of books and papers have been published on the subject ofstrip rolling. The most rigorous analysis was performed by Orowan [12]and has been applied and computerized by various investigators. Morerecent studies consider elastic flattening of the rolls and temperatureconditions that exist in rolling [13,14]. The roll separating force and theroll torque can be estimated with various levels of approximation by the

FIG. 3.11 Four types of metal flow in rolling: (A) strip, (B) plate, (C) simple shape, (D)complex shape (broken and solid lines illustrate the sections before and after deformation,respectively).


FIG. 3.12 Schematic representation of strip rolling (strip has constant width).

slab method, the upper-bound method, or the slip-line method of analysis.Most recently, computerized numerical techniques have been used toestimate metal flow, stresses, roll separating force, temperatures, andelastic deflection of the rolls.

As seen in Fig. 3.12, because of volume constancy, the velocity of thedeforming material in the x, or rolling, direction must steadily increasefrom entrance to exit. At only one point along the roll-strip interface isthe surface velocity of the roll equal to the velocity of the strip. This pointis called the neutral point or neutral plane, indicated by N in Fig. 3.12.

The interface frictional stresses are directed from the entrance and exitplanes toward the neutral plane because the relative velocity between theroll surface and the strip changes its direction at the neutral plane. Thisinfluences the distribution of the rolling stresses.

In the rolling of thick plates, metal flow occurs in three dimensions, ascan be seen in Fig. 3.11B. The rolled material is elongated in the rollingdirection as well as spread in the lateral, or width, direction. Spread inrolling is usually defined as the increase in width of a plate or slab as apercentage of its original width. The spread increases with increasingreduction and interface friction, decreasing plate width-to-thickness ratio,and increasing roll-diameter-to-thickness ratio. In addition, the free edgestend to bulge with increasing reduction and interface friction. Thethree-dimensional metal flow that occurs in plate rolling is difficult toanalyze. Therefore, most studies of this process have been experimental in


FIG. 3.13 Schematic illustration of five different roll-pass designs for a steel angle section[18].

nature, and several empirical formulae have been established for estimat-ing spread [15]. Attempts were also made to predict elongation and spreadtheoretically [16, 17].

Rolling of shapes, also called caliber rolling, is one of the most complexdeformation processes (Figs. 3.11C and D). A round or round-corneredsquare slab is rolled in several passes into (a) relatively simple sectionssuch as rounds, squares, or rectangles, or (b) complex sections such asL,U,T,H, or other irregular shapes. For this purpose, certain intermediateshapes or passes are used, as shown in Fig. 3.13 for rolling of anglesections [18]. The design of these intermediate shapes, i.e. roll pass design,is based on experience and differs from one company to another, even forthe same final rolled section geometry. Relatively few quantitative data onroll pass design are available in the literature.

At the present stage of technology, the process variables are consideredin roll pass design by using a combination of empirical knowledge, somecalculations, and some educated guesses. A methodical way of designingroll passes requires not only an estimate of the average elongation, butalso the variation of this elongation, within the deformation zone. Thedeformation zone is limited by the entrance, where a prerolled shapeenters the rolls, and by the exit where the rolled shape leaves the rolls.The deformation zone is cross-sectioned with several planes. The rollposition and the deformation of the incoming billet are investigated at eachof these planes. Thus, a more detailed analysis of metal flow and animproved method for designing the configuration of the rolls are possible.It is evident that this technique can be drastically improved and madeextremely efficient by use of computer-aided techniques and, ultimately,by FEM.

In recent years, most companies that produce shapes have computerizedtheir roll pass design procedures for rolling rounds [19,20] or structuralshapes [21, 22]. In most of these applications, the elongation per pass andthe distribution of the elongation within the deformation zone for eachpass are predicted by using an empirical formula. If the elongation per pass


is known, it is then possible, by use of computer graphics, to calculate thecross-sectional area of a section for a given pass, i.e., reduction and rollgeometry. The roll geometry can be expressed parametrically, i.e., interms of angles, radii, etc. These geometric parameters can then be variedto "optimize" the area reduction per pass and to obtain an acceptabledegree of "fill" of the roll caliber used for that pass.

The above discussion illustrates that, except in strip rolling, the metalflow in rolling is in three dimensions, i.e., in the thickness, width, androlling directions. Determinations of metal flow and rolling stresses in 3Drolling, i.e., shape rolling, are very important in designing rolling mills andin setting up efficient production operations. However, the theoreticalprediction of metal flow in such complex cases is nearly impossible at thistime. Numerical techniques are being developed in an attempt to simulatemetal flow in such complex rolling operations. The FEM-based computercodes offer excellent potential for predicting metal flow and stresses in 3Drolling operations.

3.9 Drawing of Rod, Wire, Shapes, and Tubes

Drawing is one of the oldest metal-forming operations and has majorindustrial significance. This process allows excellent surface finishes andclosely controlled dimensions to be obtained in long products that haveconstant cross sections. In drawing, a previously rolled, extruded, orfabricated product with a solid or hollow cross section is pulled through adie at a relatively high speed. In drawing of steel or aluminum wire, forexample, exit speeds of several thousand feet per minute are verycommon. The die geometry determines the final dimensions, the cross-sectional area of the drawn product, and the reduction in area. Drawing isusually conducted at room temperature using a number of passes orreductions through consecutively located dies. At times, annealing may benecessary after a number of drawing passes before the drawing operation iscontinued. The deformation is accomplished by a combination of tensileand compressive stresses that are created by the pulling force at the exitfrom the die, by the back-pull tensile force that is present betweenconsecutive passes, and by the die configuration.

In wire or rod drawing, the section is usually round but could also beshaped. In cold drawing of shapes, the basic contour of the incoming shapeis established by cold rolling passes that are usually preceded by annealing.After rolling, the section shape is refined and reduced to close tolerancesby cold drawing, as shown in Fig. 3.14 [23]. Here again, a number ofannealing steps may be necessary to eliminate the effects of strainhardening, i.e., to reduce the flow stress and increase the ductility.

In tube drawing without a mandrel, also called tube sinking, the tube isinitially pointed to facilitate feeding through the die; it is then reduced inoutside diameter while the wall thickness and the tube length areincreased. The magnitudes of thickness increase and tube elongation


FIG. 3.14 Cold rolled (round to triangle) and cold drawn shape, requiring numerousannealing steps [23].

depend on the flow stress of the drawn part, die geometry, and interfacefriction.

Drawing with a fixed plug is widely known and used for drawing oflarge-to-medium-diameter straight tubes. The plug, when pushed into thedeformation zone, is pulled forward by the frictional force created by thesliding movement of the deforming tube. Thus, it is neccesary to hold theplug in the correct position with a plug bar. In drawing of long andsmall-diameter tubes, the plug bar may stretch and even break. In suchcases it is advantageous to use a floating plug. This process can be used todraw any length of tubing by coiling the drawn tube at high speeds of up to2000 ft/min. In drawing with a moving mandrel, the mandrel travels at thespeed at which the section exits from the die. This process, also calledironing, is widely used for thinning of the walls of drawn cups or shells, forexample, in the production of beverage cans [24] or artillery shells [5].

The principle of a can ironing press is illustrated in Fig. 3.15. In thefigure, the press is horizontal, and the ram has a relatively long stroke andis guided by the hydrostatic bushing (A). The front seal (B) preventsmixing of the ironing lubricant with the hydrostatic bushing oil. With theram in the retracted position, the drawn cup is automatically fed into thepress, between the redraw die (D) and the redraw sleeve (C). The redrawsleeve centers the cup for drawing and applies controlled pressure while

FIG. 3.15 Schematic illustration of multiple-die ironing operation for manufacturing bever-age cans [24].


the cup is drawn through the first die (D). As the ram proceeds, the redrawncup is ironed by passing through the carbide dies (E), which graduallyreduce the wall thickness. The ironed can is pressed against the domingpunch (T), which forms the bottom shape of the can. When the ram startsits return motion, the mechanical stripper (G), assisted by the air stripper(F), removes the can from the ironing punch (H). The punch is made ofcarbide or cold-forging tool steel. The stripped can is automaticallytransported to the next machine for trimming of the top edge of the canwall to a uniform height.

3.10 Sheet-Metal FormingThe products made by sheet-metal forming processes include a largevariety of shapes and sizes, ranging from simple bends to doublecurvatures with shallow or deep recesses. Thus the scope of sheet-metalforming is very broad. Sachs [25] described systematically the principles ofthe numerous sheet-metal forming methods, emphasizing their similaritiesand differences. Some important processes are described and significantvariables are discussed in the References [26-28].

Yoshida [29] proposed a classification of general press-forming processesbased on the governing deformation mechanisms. The basic mechanismsare stretching, drawing, and bending. Depending upon the shape and therelative dimensions of the blank and the tool, one or more basicmechanisms is predominantly involved in sheet metal deformation.

The limits of sheet-metal forming are determined by the occurrence ofdefects, such as wrinkles and ruptures in the blank. Some defects observedin cup drawing are shown by the sketches in Fig. 3.16 [28]. The limitingdrawing ratio, which is the ratio of the maximum possible blank dimensionto the dimension in a drawn part, is a measure of the limit in the drawingrange. The occurrence of wrinkles further restricts the drawing range.Wrinkling may form either in the flange over the die surface or in theblank around the die shoulder. Wrinkling over the flat face of the die canbe avoided by applying the blank holding force, and the possibility ofwrinkling of the blank around the die shoulder can be reduced by takingthe radius of the die corner large enough compared with the blankthickness.

In stretching of sheet metals over the punch, the rupture of the blankoccurs over the punch profile. Necking precedes eventual rupture.Therefore, the forming limit is governed by the condition of instability,and the site of necking initiation depends on the friction condition at thepunch-workpiece interface. In flange stretching (or hole expansion),fracture occurs at the edge of the flange. Necking, however, does notalways initiate from the periphery of the flange but, for some materials,starts at some distance from the edge. The stress and strain fields in theabove stretching processes are nonuniform, and the analysis for theinstability condition must take this nonuniformity into account.


EARING

FIG. 3.16 Some defects observed in cup drawing [28].

Bending operations are involved in all complex stamping. In bending, incontrast to most stretching operations, there is a severe gradient of stressthroughout the thickness of the material. On the outside of the bend thestress is tension and on the inside it is either compression or a reducedlevel of tension. The severity of the tensile strains depends on the radius,angle, and length of the bend. Fracture occurs on the tensile side bythinning and fracture.

Formability of sheet metals, as the measure of the ability of the metal toundergo the desired shape change without failure, is often evaluated bysimple tests such as a tensile test. The parameters obtained by simpletension, namely, degree of anisotropy, the workhardening rate of stress-strain relationship, the maximum uniform elongation, and the maximumnominal stress, are related to formability. However it is not possible toevaluate accurately the formability of the materials in terms of theseparameters. Thus, for complete assessment of the formability, the directmethods, such as the Ericksen test, Swift cup test, and Fukui conical cuptest [30] have been used for determination of formability.

An important development in representation of the formability of sheetmetals is the forming limit diagram [31]. In this diagram the major andminor surface strains at a critical site are plotted at the onset of visible,localized necking in a deformed sheet, and the locus of strain combinations

FRACTURE

WRINKLES

FIG. 3.17 Forming limit diagram and deformed rectangular strip and circular blank withcircular cutoff [34].

49

Minor Strain (%)


that will produce failures in an actual forming operation can be drawn.Figure 3.17 shows the forming limit diagram for different materials.

Experimental methods are used to construct the diagram. The Nakajimamethod [32] uses a hemispherical punch and rectangular blanks withvarious widths and lubrication conditions. Analogously to the Nakajimamethod, Hasek [33] used hemispherical punch stretching of circular blankswith various circular cutoffs. The method of Hasek has an advantage ofeliminating or reducing the wrinkling phenomena that occur at the edgesof rectangular strips in the Nakajima method. In Fig. 3.17, deformedspecimens (FEM simulation [34]) in the above two methods are alsoshown.

Friction at the punch-workpiece interface is an important factor forformability, and it changes the strain-path of a critical site of the sheet inthe forming limit diagram of Fig. 3.17. Material parameters, such asstrain-hardening and strain-rate sensitivity, are important for formability.Another important factor in sheet-metal forming is the anisotropy(direction-dependent properties) of the sheet metal. Earing in deepdrawing of cups is due to anisotropy (see Fig. 3.16). Anisotropy alsoinfluences formability. For example, the limiting drawing ratio in cupdrawing increases with increasing r-value, which is the ratio of the widthstrain to the thickness strain in uniaxial tension and is a measure ofanisotropy in the thickness direction [26].

In sheet-metal forming, considerations of residual stresses and spring-back are particularly important. Consequently elastic properties of sheetmetals cannot be neglected in the analysis of problems in which theseconsiderations are of major concern.

Although simple analytical technique considers only the stress com-ponents in the plane of the sheet (plane-stress situation), complexity ofproduct shapes and deformation in forming make it difficult for research inanalytical methods to advance technology that would be useful inmetal-stamping plants [35].

Among many areas of research need, die design and process design areof particular technological importance and two specific problems, namely,design of drawbeads in stamping operations and design of multistageforming, are given as examples.

In a die-formed part the presence of sloping walls that are not confinedbetween the die and the punch introduces the danger of buckling orwrinkling. Generally, this tendency to wrinkle increases with increasingunsupported area and with decreasing metal thickness. To avoid wrinkling,large parts made from thin metal must be formed with a considerablyhigher hold-down force. In addition, beads are frequently added to themating die and hold-down surfaces, either all around or only in the areaswhere wrinkling would otherwise develop. A drawbead is shown in Fig.3.18 in the drawing of a box-shaped part [25]. Beads retard and control theflow of sheet metal into the die cavity. Design of drawbeads includes thedetermination of their orientation, configuration, and location.


FIG. 3.18 Draw beads in drawing a box-shaped part [25].

In most sheet-metal parts, the shape is formed almost entirely in the firstoperation. Subsequent processes are generally trimming, piercing, flang-ing, and some minor restriking of detail. There are instances, however, inwhich the formability of the metal is such that it is impossible to reach thefinal form in one operation, and a number of intermediate shapes arerequired. An example is shown in Fig. 3.19 for forming an automobile

FIG. 3.19 Multistage forming processes for an automobile wheel center [36],


wheel center [36]. The problem is to determine the number of preformingoperations and the configuration for each preform.

The two examples of design in sheet-metal forming described present achallenge to the power of numerical analysis to make a significantcontribution to the advancement of technology.

References

1. Lange, K., (editor), (1972), "Study Book of Forming Technology," (inGerman), Vol. I, II and III, Springer-Verlag, Berlin.

2. Johnson, W., and Sowerby, R., (1978), "Metal Forming Processes: Analysisand Technology," ASME, AMD, Vol. 28, p. 1.

3. Allan, T., and Boulger, F. W., (1973), "Flow Stress of Metals and ItsApplication in Metal Forming Analyses," Trans. ASME, J. Engr. Ind., Vol.95, p. 1009.

4. Douglas, J. R., and Allan, T., (1975), "Flow Stress Determination for Melalsal Forging Rates and Temperalures," Trans. ASME, J. Engr. Ind., Vol. 97, p.66.

5. Allan, T., Oh., S. I, and Gegel, H., (1983), "Melal Forming: Fundamenlalsand Applications," ASM International, Metals Park, OH.

6. Schey, J. A. (editor), (1983), "Metal Deformation Processes: Friction andLubrication," Marcel Dekker, New York, 1970; superseded by Schey, J. A.,"Tribology in Metalworking: Lubrication, Friction and Wear," AmericanSociely for Melals, Melals Park, OH.

7. Billhardt, C. F., Nagpal, V., and Allan, T., (1978), "A Computer GraphicsSystem for CAD/CAM of Aluminum Exlrusion Dies," SME Paper MS78-957.

8. Sagemuller, F., (1968), "Cold Impact Exlrusion of Large Formed Parts,"Wire, No. 95, p. 2.

9. Dieter, G. E., (1961), "Mechanical Metallurgy," McGraw-Hill, New York,Chapter 29, p. 488.

10. Geleji, A., (1967), "Forge Equipmenl, Rolling Mills and Accessories,"Akademiai Kiado, Budapesl.

11. Larke, E. C., (1957), "The Rolling of Strip, Sheet and Plate," Chapman andHall, London.

12. Orowan, E., (1943), "The Calculation of Roll Pressure in Hoi and Cold FlalRolling," Proc. Inst. Mech. Eng., Vol. 150, p. 140.

13. Alexander, J. M., (1972), "On Ihe Theory of Rolling," Proc. R. Soc. London,Series A, Vol. 326, p. 535.

14. Laholi, G. D., Shah, S. N., and Allan, T., (1978), "Computer Aided Analysisof the Deformations and Temperalures in Strip Rolling," Trans. ASME, J.Engr. Ind., Vol. 100, p. 159.

15. Ekelund, S., in H. Neumann, (1963), "Roll Pass Design," (in German), VEBDeulscher Verlag, Leibzig, p. 48.

16. Oh, S. I., and Kobayashi, S., (1975), "An Approximate Melhod forThree-Dimensional Analysis of Rolling," Int. J. Mech. Sci., Vol. 17, p. 293.

17. Neumann, H., (1969), "Design of Rolls in Shape Rolling," (in German), VEBDeulscher Verlag, Leibzig.

18. Schulza, A., (1970), "Comparison of Roll Pass Designs Used for RollingAngle Sections," (in German), Stahl and Eisen, Vol. 90, p. 796.

19. Neumann, H., and Schulze, R., (1974), "Programmed Roll Pass Design forBlocks," (in German), Neue Hutte, Vol. 19, p. 460.

20. Suppo, U., Izzo, A., and Diana, P., (1973), "Eleclronic Computer Used inRoll Design Work for Rounds," Der Kalibreur, Vol. 19, p. 3.


21. Metzdorf, J., (1981), "Computer Aided Roll Pass Design—Possibilities ofApplication," (in German and French), Der Kalibreur, No. 34, p. 29.

22. Schmeling, F., (1982), "Computer Aided Roll Pass Design and Roll Manufac-turing," (in German), Stahl und Eisen, Vol. 102, p. 771.

23. "Rathbone Cold Drawn Profile Shapes and Pinion Rods," Rathbone Corpora-tion, Palmer, MA.

24. Brochure of the Standum Company, Compton, CA.25. Sachs, G., (1951), "Principles and Methods of Sheet Metal Fabricating,"

Reinhold, New York.26. Schey, J., (1987), "Introduction to Manufacturing Processes," 2nd edition,

McGraw-Hill, New York.27. Kalpakjian, S., (1984), "Manufacturing Processes for Engineering Materials,"

Addison-Wesley, Reading, MA.28. Johnson, W., and Mamalis, A. G., (1978), "Aspects of the Plasticity

Mechanics of Some Sheet Metal Forming Processes," Hellenic Steel publica-tions, Thessaloniki, Greece.

29. Yoshida, K., (1959), "Classification and Systemization of Sheet Metal Press-forming Process," Scientific Papers of the Institute of Physical and ChemicalResearch, Vol. 53, No. 1514, p. 126.

30. Fukui, S., Kudo, H., Yoshida, K., and Okawa, H., (1952), "A Method forTesting of Deep-Drawability of Sheet Metals," Report of the Institute ofScience and Technology, University of Tokyo, Vol. 6, p. 351.

31. Keeler, S. P., and Backofen, W. A., (1963), "Plastic Instability and Fracturein Sheets Stretched Over Rigid Punches," Trans. Am. Soc. Metals, Vol. 56, p.25.

32. Nakajima, K., Kikuma, T., and Hasuka, K., (1968), "Study on the For-mability of Steel Sheets," Yawata Technical Report, No. 264, p. 141.

33. Hasek, V., (1978), "Untersuchung und Theoretische Beschreibung wichtigerEinflussgrossen auf das Grenzformanderungsschaubild," Institute of MetalForming Report, University of Stuttgart, West Germany, p. 213.

34. Toh, C. H., (1983), "Process Modeling of Sheet Metal Forming of GeneralShapes by the Finite Element Method based on Large Strain Formulations,"Ph.D. dissertation, Department of Mechanical Engineering, University ofCalifornia at Berkeley.

35. Koistinen, D. P., and Wang, N-M, (editors), (1978), "Mechanics of SheetMetal Forming," Plenum Press, New York and London.

36. Duncan, J. L., and Sowerby, R., (1987), "Review of Practical ModelingMethods for Sheet Metal Forming," Proc. 2d Int. Conf. Tech. Plasticity,Stuttgart, p. 615.

4PLASTICITY AND VISCOPLASTICITY

4.1 Introduction

The theory of plasticity describes the mechanics of deformation inplastically deforming solids, and, as applied to metals and alloys, it isbased on experimental studies of the relations between stresses and strainsunder simple loading conditions. The theory described here assumes theideal plastic body for which the Bauschinger effect and size effects areneglected. The theory also is valid only at temperatures for whichrecovery, creep, and thermal phenomena can be neglected. The basictheory of classical plasticity is described by Hill [1], and also in References[2-5], in addition to the books listed in Chap. 1. A concise description ofthe general plasticity theory necessary for metal forming is given in thebook by Johnson et al. [6]. In this chapter, certain important aspects of thetheory are presented in order to elucidate the developments of thefinite-element solutions of metal-forming problems discussed in this book.

First, various measures of stress and strain are introduced. Then, thegoverning equations for plastic deformation and principles that are thefoundations for the analysis are described. The extension of the theory ofplasticity to time-dependent theory of viscoplasticity is outlined in Section4.8. Particular references are made, in Sections 4.3 through 4.7, to thebooks by Hill [1] and by Johnson and Mellor [7], and to the section ongeneral plasticity theory in the book by Johnson et al. [6].

4.2 Stress, Strain, and Strain-Rate

The basic quantities that may be used to describe the mechanics ofdeformation when a body deforms from one configuration to anotherunder an external load are the stress, strain, and strain-rate. Variousmeasures of these quantities are defined, depending upon how closelyformulations represent actual situations. Although it is not possible toprovide the complete mathematical formulations in one-dimensional de-formation, these measures are introduced for the case of simple uniaxialtension.

Consider the uniaxial tension test of a round specimen whose initiallength is /0 and cross-sectional area is A0. The specimen is stretched in the

54

Plasticity and Viscoplasticity 55

FIG. 4.1 Uniaxial tension, (a) Tension specimen; (b) stress-strain curves.

axial direction by the force P to the length / and the cross-sectional area Aat time t, as shown in Fig. 4.1. The response of the material is recorded asthe load-displacement curve, and converted to the stress-strain curve asshown in the figure. The deformation is assumed to be homogeneous untilnecking begins.

There are two modes of description of the deformation of a continuousmedium, the Lagrangian and the Eulerian. The Lagrangian descriptionemploys the coordinates Xt of a typical particle in the reference (orundeformed) state as the independent variables, while in the case ofEulerian description the independent variables are the coordinates xt of amaterial point in the deformed state. When the deformation is in-finitesimal, where products of the derivatives of the displacements can beneglected, we need make no distinction between the two.

In the infinitesimal deformation theory, the stresses and strain-rates (orinfinitesimal strains) are expressed with respect to a fixed coordinatesystem in the material configuration at a time under consideration. In


uniaxial tension, they are defined by

stress

infinitesimal strain ds

where the dot denotes the time derivative. The stress defined in eq. (4.1) iscalled true stress or Cauchy stress. The total amount of deformation ismeasured by integrating infinitesimal strain as

and is called logarithmic or natural strain.In the Lagrangian description of finite deformation, the measures of

stress, strain, and strain-rate are expressed as follows.Let the position of a particle in the deformed configuration at time t be

designated by

where A' is a reference position of a particle and t is the time. In uniaxialtension let X be directed along the longitudinal axis of the specimen (seeFig. 4.1); then

Extension is defined as the displacement gradient relative to the referenceposition and is expressed by

This is the engineering strain.The Lagrangian strain component Eu is defined by

(see Reference [8] for geometrical interpretation of eq. (4.6)). Thestrain-rate components are the time derivatives of strain components givenby eqs. (4.5) and (4.6). They are

and

s t r a a i n - r a t e


since

The Piola-Kirchhoff stress tensor is defined as force intensity acting inthe deformed configuration but measured per unit area of the referenceconfiguration, while the Cauchy stress is denned as force per unit area inthe deformed state. In the uniaxial tension, the engineering stress

corresponds to a component of the nonsymmetric (or first) Piola-Kirchhoffstress tensor. Correspondence between stress and strain-rate measures isestablished such that the rate of work per unit volume (W0) in thereference configuration is the product of stress and strain rate. From eqs.(4.7) and (4.8),

Then the stress measure corresponding to the Lagrangian strain rate En is

The stress given by eq. (4.10) corresponds to a component of thesymmetric (or second) Piola-Kirchhoff stress tensor. (See References[9,10] for general background on finite deformation.)

For the analysis of metal forming processes, flow formulation is based oninfinitesimal deformation theory, while solid formulation considers finitedeformation. Consideration of finite deformation with flow formulation isgiven in Chap. 11 in connection with sheet-metal forming, and solidformulation is outlined in Chap. 16, where comparison of solutionsaccording to the two formulations are discussed. Since flow formulation isused mostly throughout this book, further detail on stress and strain-ratemeasures in the infinitesimal deformation theory is given.

The strain-rate tensor [etj], where i,j=x,y,z, is symmetric and thetensor components are defined by


where ut are velocity components and y,7 are engineering shear strain-ratecomponents. Using suffix notation, eq. (4.11) can be written as

where a comma denotes differentiation with respect to the coordinates thatfollow.

The Cauchy stress tensor [ofi], where i, j = 1,2,3 or x, y, z, is alsosymmetric and is represented by the nine components as

The stress may also be specified by the three principal components, or bythe three tensor invariants. The principal stresses (oly o2, cr3) are the rootsof the cubic equation

where Ilt I2, and 73 are quantities independent of the direction of the axeschosen and called the three invariants of the stress tensor oti. They aredefined by the relations

The first (linear) and second (quadratic) invariants have particular physicalsignificance for the theory of plasticity.

4.3 The Yield Criteria

A yield criterion is a law defining the limit of elasticity under any possiblecombination of stresses. It is expressed by

A function of stresses /(<?,y) is called yield function. The suitability of anyproposed yield criterion must be verified by experiment.

For isotropic materials, plastic yielding can depend only on themagnitude of three principal stresses and not on their directions. Then anyyield criterion is expressible in the form

From the experimental fact that the yielding of a material is, to a firstapproximation, unaffected by a moderate hydrostatic pressure, or tension,it follows that yielding depends only on the principal components


(oi, o2', cr3') of the deviatoric stress tensor

where am = \(a\ + o2 + o3) is the hydrostatic component of the stress anddij ( = 1 for i=j and = 0 for i¥=j) is the Kronecker delta. The principalcomponents of the deviatoric stress tensor are not independent, sinceai + a2 + <73' is identically zero.

The yield criterion then reduces to the form

where

Two Criteria

Two simple criteria have been in extensive use for the analysis of metaldeformation. Tresca's criterion (shear stress criterion) given in 1864 is

with o1 > CT2 > o3. It may alternatively be written in the form of eq. (4.19)in terms of J2 and J3, but the results are complicated and not useful.

Another criterion was proposed by Heuber (1904), by von Mises (1913),and by J. C. Maxwell in a letter to Kelvin in 1856 [7]. It has beentraditionally called the von Mises criterion or the Huber-Mises criterion,but may be appropriately called the Maxwell-Heuber-Mises criterion.(We refer to the distortion energy criterion according to a physicalinterpretation of the criterion suggested by Hencky in 1924.) The criterionstates that yielding occurs when J2 reaches a critical value, or, in otherwords, that the yield function/of eq. (4.19) does not involve J3. It can bewritten in the alternative forms

or

or

where A; is a parameter regulating the stress scale and depending on thematerial property. In the suffix notation used in eq. (4.21a), a recurringletter suffix indicates that the sum must be formed of all terms obtainableby assigning to the suffix the values x, y, z (or 1, 2, 3).

The constant in eqs. (4.20) and (4.21) may be determined from simplestates, such as in uniaxial tension. At yielding in simple tension, cr, = Yand 02 = o3 = 0. Therefore, eqs. (4.20) and (4.21) may be written as


and

respectively. Parameter k in eq. (4.21) can be identified as the shear yieldstress by considering yielding in simple shear and k = Y/\/3 according tothe criterion (4.21).

It must be noted that the yield criteria defined by eqs. (4.22) mustdepend on the previous process of plastic deformation (strain hardening).If we assume that hardening occurs if, and only if, plastic work is done,then the assumption that the yield criterion is independent of thehydrostatic component implies that there is no volume change duringplastic deformation.

Geometrical RepresentationA state of stress is completely specified by the values of the three principalcomponents. Then any stress state may be represented by a vector in athree-dimensional stress space, where_the principal stresses are taken asCartesian coordinates. In Fig. 4.2, OS is the vector (olt o2, o3) and itscomponent, OP, is the vector representing the deviatoric stress(oi, o2', <V). OP always lies in the plane n whose equation is ol + o2 +a3 = 0. The_hydrostatic component (om, om, am) of the stress is repre-sented by PS, which is perpendicular to the plane n.

A yield criterion, which is independent of the hydrostatic component ofstress, is represented by a curve C in the plane Jt. The yield locuscorresponding to the shear stress criterion is a regular hexagon, while it isobvious from the relation in eq. (4.21a) that the locus of the distortionenergy criterion is a circle of radius \/2 k or V2/3 Y. By selecting thevalues of the constant in eq. (4.20) and k in eq. (4.21), according to eq.

FIG. 4.2 Geometrical representation of a plastic state of stress in (tr1; a2, a3) space.


FIG. 4.3 Yield locii on the Jt-plane for distortion energy criterion and maximum shear stresscriterion.

(4.22), the two criteria can be made to agree with each other and withexperiment for uniaxial tension. The two loci are shown in Fig. 4.3.

For most metals the distortion energy criterion fits the data more closelythan the shear stress criterion. Furthermore, the distortion energy criterionis continuous and convenient to use in numerical analysis. Therefore, thedistortion energy criterion is exclusively used in this book. Other yieldcriteria, such as those for anisotropic sheet materials and for porousmetals, are introduced in Chaps. 11 and 13, respectively.

4.4 Equilibrium and Virtual Work-Rate Principle

Equilibrium Equations

In the rectangular Cartesian coordinate system, the equilibrium equations,if the body force is neglected, are given by

In the notation used in eq. (4.24), a recurring letter suffix indicates thesum, as explained for eq. (4.21a), and a comma denotes partialdifferentiation as explained for eq. (4.12).

With suffix notation, eqs. (4.23) are written as


Equilibrium with Tractions

The stress along the boundary surface S is in equilibrium with an appliedtraction Ft (force per unit surface area). Equilibrium of the stresses iswritten as

where «, is the unit outward normal to the surface. Writing eq. (4.25a), inunabridged notation in the two-dimensional case (see Fig. 4.4), one has

where the components of the unit outward normal ny are given by(dyldl, dxldl}.

Virtual Work-rate Principle

The virtual work-rate principle states that for the stress field that is inequilibrium within the body and with applied surface tractions, thework-rate inside the deforming body equals the work-rate done by thesurface tractions for all velocity fields that are continuous and continuouslydifferentiable (virtual velocity fields).

Let Ofj be any stress field that is in equilibrium and vv, be any virtualvelocity field. Then the principle is expressed by

where V is the volume of the body and S is the surface. Since otj is

FIG. 4.4 Equilibrium of surface tractions.


symmetric, eq. (4.26a) is also written as

where e,y is the strain-rate derivable from vv, according to eiy = 2(wy + wjti).For proof, note that

and

For the two-dimensional case, eq. (4.26b) can be written in unabridgedform as

4.5 Plastic Potential and Flow Rule

Hooke's law is well known for describing the relationships betweenstresses and corresponding deformation in the elastic deformation regime.When deformation extends to the plastic range, the stress and plastic strainrelationships are derived using the concept of plastic potential.

Plastic Potential and Flow Rule

The ratios of the components of the plastic strain-rate e/ (or infinitesimalplastic strain de^p) are defined by

or

where g and h are scalar functions of the invariants of deviatoric stressesand / is the yield function (if / = 0, neutral loading, and / < 0, unloading).The function g(a,y) is called the plastic potential. Although eq. (4.27a) iswritten in the rate form, the relations between stress and strain areindependent of time.


Assuming a simple relation g =/, eq. (4.27) becomes

where A or dX is a positive proportionality constant, being equal to hf orh df. Equation (4.28) is a flow rule associated with the yield function /(o,-,-).In differentiation of /(a,y) with respect to shear stress components, inunabridged form, note that, for example,

The plastic strain-rate can be represented in the same (al, o2, o3) space by afree vector G(e1

p, e2p, £3

P), where the factor G is introduced to obtain thedimension of stress. This vector lies in the jr-plane since k-f + s2

p + £3P =

0 (no volume change). Because df/do^ are the direction cosines of theoutward normal to the yield surface /(%) = C, the plastic strain-ratevector is normal to the yield locus at point (cr/, o2'', o3) in the n plane.The plastic strain-rate vector associated with the yield locus of thedistortion energy criterion is shown by PQ in Fig. 4.5. The concept ofplastic potential and its associated flow rule also apply to the shear stresscriterion. It will not be discussed here, however, since the shear stresscriterion is not used in this book, as stated in Section 4.3. It should benoted that the yield locus must be convex (concave to the origin) at allpoints if the stress corresponding to a given strain-rate is to be unique.

Maximum Plastic Work Principle

The maximum plastic work principle follows from the flow rule and theconvexity of the yield locus, and is

where otj is a yield state of stress, ei/> is the associated strain-rate, and a,,*

FIG. 4.5 Normality of the strain-rate vector to the yield locus.


FIG. 4.6 Two states of stress and a strain-rate vector.

is any other stress state represented by a point either on or inside the yieldsurrace.

Since £,yp is parallel to the outward normal to the yield locus at the point

Oi/, (0i/ ~ Oij*)£ijp is proportional to the scalar product of the outwardnormal to the yield locus at the point a// with the chord joining a^*' to o^'(see Fig. 4.6). Therefore, eq. (4.29) holds and the equality sign applieswhen <T,/ = ai*' or cr,-, and ot* differ only by a uniform hydrostatic stress.Drucker [11] arrived at eq. (4.29) from a definition of a stable plasticmaterial.

The Prandtl-Reuss Equation and the Levy-Mises EquationWith the yield function given by f{otj) =J2 = 2<V<V> (eq. 4.21a), eq.(4.28) becomes, in the rate form,

since

noting that the repeated subscripts k and / indicate summation with respectto these quantities.

Equation (4.30) can be written as

Combining the elastic strain-rate components e^ and the plastic strain-ratecomponents eff, according to e,y = e,/ + e^, we obtain the Prandtl-Reussequations for elastic-plastic solids, as


where G, E, and v are shear modulus, Young's modulus, and Poisson'sratio, respectively.

For rigid-plastic materials, the assumption is that e;; = E/, and Levy-Mises equations are obtained by removing the superscripts/) in eq. (4.31).They are expressed in terms of stress components oif by three equations ofthe type

and three of the type

The Levy-Mises equations (4.33) will be used extensively throughout thebook. In the following sections and in the chapters to follow, the plasticstrain-rate and the infinitesimal plastic strain are indicated withoutsuperscript p, unless otherwise specified.

4.6 Strain-Hardening, Effective Stress, and Effective Strain

When a real metal is deformed at room temperature, its resistance tofurther deformation increases (strain hardening). A hypothesis that thedegree of hardening is a function of the plastic work has been mentionedin Section 4.3. For a mathematical formulation of srain hardening, it isassumed that the final yield locus is the same, no matter by whatstrain-path a given stress state is reached (isotropic hardening).

The total plastic work per unit volume during a certain finite deforma-tion is

where the integral is taken over the actual strain-path.The hypothesis that the radius of the yield locus of the distortion energy

criterion is a function only of Wn may be written as

where a is written for Y or \/3 k, and known as the flow stress, theeffective stress, generalized stress, or equivalent stress.

Another hypothesis for strain-hardening relates a to a certain measureof the total plastic deformation. A quantity de, known as the effective,generalized, or equivalent infinitesimal plastic strain is defined according to

The effective strain e = J de, integrated over the strain-path, provides ameasure of the plastic distortion. It is assumed that strain-hardeningcharacteristics can be formulated by


where H is a certain function depending on the metal concerned.Equation (4.37) is most frequently used for process analysis in metal

forming. Methods of experimental determination of stress-strain pro-perties are discussed in Chap. 3 and are also described in the book byJohnson and Mellor [7].

It should be noted that an explicit expression for the effective strain canbe obtained when the principal axes of successive strain-increments do notrotate relative to the element, and, further, when the components of anystrain-increments bear constant ratios to one another. It should also benoted that, although quanties J d£t can always be formed (de, =infinitesimal principal strain), even where the principal axes rotate, theycannot generally be evaluated explicitly, nor do they possess any geometri-cal significance. With the measures of o and de (and therefore e, effectivestrain-rate), the proportionality factor A in the Levy-Mises equations(4.33) can be expressed by

(4.38)

The derivation of eq. (4.38) is as follows. The plastic work-rate per unitvolume, using the flow rule (4.28), is written as

Because the yield function f{aij) is a homogeneous function of degree 2,namely, /(to,y) = ^/(cr/,), and using Euler's theorem for homogeneousfunctions,i.e.,if is a homogeneous function of degree n, then

the plastic work rate becomes

from which eq. (4.38) results for the yield function /(a,y) of eq. (4.21a),and a denned by eq. (4.35). The expression for e in terms of strain-ratecomponents, corresponding to the effective stress o of eq. (4.35), can beobtained by considering the inversion of the flow rule. Then

from which

In unabridged form, eq. (4.39a) is


The numerical factors in the definition of a and e have been chosen so thatthe functional relationship between a and J df. is identical with therelationship between true stress and logarithmic strain in a tension orcompression test.

4.7 Extremum Principles

By analogy with extremum principles in the theory of elasticity, tworelated extremum principles pertaining to a consistent state are proven fora rigid-plastic material that undergoes plastic deformation under pre-scribed surface traction over the surface SF and prescribed velocities overthe surface Su. A complete solution to this problem consists of anequilibrium stress field and an associated velocity field satisfying theboundary conditions, and the stress field satisfies the yield criterion wheredeformation occurs and does not violate the yield criterion in the rigidregions. The uniqueness theorem proves that the stress field of thecomplete solution is uniquely determined in the deforming region. Wherethe deformation mode is not uniquely defined, it must be compatible withthe stress field.

The First Extremum Principle

Let Ojj, Uj denote a complete solution to the plastic deformation problemof a rigid-plastic solid under the prescribed boundary conditions. Let a,y*be an equilibrium stress field satisfying the stress boundary conditions onSF and nowhere violating the yield criterion (statically admissible). Theprinciple of virtual work-rate gives

Then

according to the maximum plastic work principle. Since Fj = F* on SF,where S = SF + Su,

Equation (4.42) is the first extremum principle and it states that a lowerbound to the rate of work of the actual surface traction can be obtainedfrom a statically admissible stress field.

In the quasi-static flow of a rigid plastic solid, velocity discontinuity ispermissible. When the velocity field u{ contains surfaces of discontinuity,the principle of virtual work-rate gives, instead of eq. (4.40),


FIG. 4.7 Conditions along a surface of velocity discontinuity.

where SD is a surface of velocity discontinuity, T* is the shear stresscomponent of oif*, and k is the shear yield stress along the surface of SD,where |A«| is the amount of tangential velocity discontinuity across SD.

Equation (4.43) is obtained by applying the virtual work-rate principleto parts (1) and (2) in Fig. 4.7 separately, and then adding with thecondition that the traction to part (1), Fj'\ and to part (2), Fp', along thesurface SD are in equilibrium. Because the second term on the left-handside of eq. (4.43) is always positive, eq. (4.41) and therefore eq. (4.42) alsohold for this case.

It should be mentioned that a surface of stress discontinuity in anequilibrium state of stress is also permissible, but that its presence does notaffect the relationships described above. Further, note that a surface ofstress discontinuity cannot be the surface of velocity discontinuity in theassociated stress and velocity fields, because physical interpretations arethe limits of the elastic region and large plastic deformation regions forstress and velocity discontinuities, respectively.

The Second Extremum Principle

In the second extremum principle, let «,•* be a velocity field that satisfiesthe incompressibility and the velocity boundary conditions on Su. Such avelocity field is said to be kinematically admissible. The principle of virtualwork-rate gives

where i is the shear stress component of the stress a{j of the completesolution along the velocity discontinuity surface 5D* with the amount oftangential discontinuity |Aw*| in the velocity field «,*. If u* is the actual


solution, then T = k. Thus

and also

from the maximum plastic work principle, where o^* satisfies the yieldcriterion and is associated with Et*. Therefore, we have

The surface S = SF + Su and on Su, ut* = uh consequently

Equation (4.46) is the second extremum principle and it shows that anupper bound to the rate of work of the actual surface traction on Su can beobtained from a kinematically admissible velocity field. A differentapproach to the extremum principles is given in the References [12,13].

4.8 Viscoplasticity

The mathematical theory of plasticitiy adequately describes the time-independent aspect of the behavior of materials but is inadequate foranalysis of time-dependent behavior. An approach to achieving a satisfac-tory formulation for time-dependent behavior has been to generalizeplasticity to cases within the strain-rate-sensitive range. One such generali-zation has been provided by the theory of viscoplasticity. The history ofthis theory goes back to as early as 1922 [14]. Since then, generalizations ofearly versions have been achieved and various forms of the theory ofviscoplasticity have been provided (summarized, for example, by Perzyna[15] and by Cristescu [16]). In this section, we describe an approach to theconstruction of equations for rigid viscoplastic materials. Then, theextremum principle is presented as the basis for the finite-elementformulation of viscoplastic flow analysis.

Constitutive Equations

What follows is based on work by Perzyna. We consider a rigid viscoplasticmaterial and apply the infinitesimal theory for each incremental deforma-tion. Perzyna introduced a function F(<7y) such that

where k is the static yield stress in shear. Considering F(cTy) as the functionsimilar to the plastic potential in the theory of plasticity, the constitutive


equation is expressed as

where y" is a viscosity constant of the material, and ($(F)} is a function ofF such that

Then

where a = V(3/2) {o^'o^'}1'2 so that 0 is identical to the yield stress inuniaxial tension, and Y = V3 k is the static yield stress in tension.

Squaring both sides of eq. (4.49),

and using t = V(2/3) {%£,•,• }1/2, we obtain

From eqs. (4.49) and (4.50), the constitutive equation becomes

with a = V(3/2) H'<V>1/2 and £ = V(2/3) {e^}1/2

Equation (4.51) is formally identical to the Levy-Mises equations. Theeffective stress o in eq. (4.51), however, depends on the strain-rate-dependent function $, which is to be determined by the properties of thematerial under consideration. If we choose the function $ = ((a/ Y) — l)1/m,for example, then

from eq. (4.50). Equation (4.52) is a familiar rate-dependence law and theexponent m is the strain-rate sensitivity index.

Extremum Principle

Hill [17] derived the (second) extremum principle associated with thedeformation process of the rigid-viscoplastic materials. Among all thepossible constitutive equations, we will restrict our attention to the casewhere there exists the work function £(£</), sucn that


and where E is convex. The existence of work function £(f,y) is ensured ifOtj' is a single-valued function of e,y, satisfying doti'/dekl = dok,'13^.Also, £ is a convex, if

where et* is a strain-rate field derived from an admissible velocity fieldu*. With this restriction, we have the following relation,

where e^, «,- are actual quantities and the starred quantites are kinemati-cally admissible ones. It can be shown that the solution is uniquelydetermined at points of the body where E is strictly convex, but notnecessarily in all respects elsewhere.

References

1. Hill, R., (1950), "The Mathematical Theory of Plasticity," Oxford UniversityPress, London.

2. Mendelson, A., (1968), "Plasticity: Theory and Application," MacMillan, NewYork.

3. Szczepinski, W., (1979), "Introduction to the Mechanic of Plastic Forming ofMetals," Sijthoff & Noordhoff International Publishers, The Netherlands.

4. Yamada, Y., (1965), "Mechanics of Plastic Deformation," Nikkan KogioShinbun-Sha, Tokyo.

5. Chakrabarty, J., (1987), "Theory of Plasticity," McGraw-Hill, New York.6. Johnson, W., Sowerby, R., and Venter, R. D., (1982), "Plane Strain Slip Line

Fields for Metal Deformation Processes," Pergamon Press, Oxford, UK.7. Johnson, W., and Mellor, P. B., (1973), "Engineering Plasticity," Van

Nostrand Reinhold, London.8. Sokolnikoff, I. S., (1956), "Mathematical Theory of Elasticity," McGraw-Hill,

New York.9. Naghdi, P. M., and Trapp, J. A., (1975), "The Significance of Formulating

Plasticity Theory with Reference to Loading Surfaces in Strain Space," Int. J.Engg. Sci., Vol. 13, p. 785.

10. Green, A. E., and Naghdi, P. M., (1965), "A General Theory of anElastic-Plastic Continuum," Arch. Rational Mech. Anal., Vol. 18, p. 251.

11. Drucker, D. C., (1951), "A More Fundamental Approach to Plastic StressStrain Relations," Proc. 1st U.S. Natl. Congress, Appl. Mech. ASME, p. 487.

12. Drucker, D. C., Greenburg, H. J., and Prager, W., (1951), "The SafetyFactor of an Elastic-Plastic Body in Plane Stress," Trans. ASME., J. Appl.Mech., Vol. 18, p. 371.

13. Drucker, D. C., Prager, W., and Greenburg, H. J., (1952), "Extended LimitDesign Theorems for Continuous Media," Q. Appl. Math., Vol. 9, p. 381.

14. Bingham, E. C., (1922), "Fluidity and Plasticity," 1st edition, McGraw Hill,New York, p. 125.

15. Perzyna, P., (1966), "Fundamental Problems in Viscoplasticity," Adv. App.Mech., Vol. 9, p. 243.

16. Cristescu, N., (1967), "Dynamic Plasticity," North-Holland, Amsterdam.17. Hill, R., (1956), "New Horizons in the Mechanics of Solids," J. Mech. Phys.

Solids, Vol. 5, p. 66.

5METHODS OF ANALYSIS

5.1 Introduction

In Chap. 4 we described fundamentals for the analysis of metal-formingprocesses and derived some useful principles. The governing equations forthe solution of the mechanics of plastic deformation of rigid-plastic andrigid-viscoplastic materials are summarized as follows:

Yield critertion

Constitutive equations:

with

Compatiblity conditionms:

The unknowns for the solution of a quasi-static plastic deformation processare six stress components and three velocity components. The governingequations are three equilibrium equations, the yield condition, and fivestrain-rate ratios derived from the flow rule. The boundary conditions areprescribed in terms of velocity and traction. Along the tool-workpieceinterface, the velocity component is prescribed in the direction normal tothe interface and the traction is specified by the frictional stress in thetangential direction.

Since it is difficult to obtain a complete solution that satisfies all of thegoverning equations, various approximate methods have been devised,depending upon the assumptions and approximations. Some of them havebeen mentioned in Chap. 1.

The basic principles and concepts involved in the finite-element methodare the variational principle and discretization. In the course of developingthe analysis methods, the variational principle has played a significant partin expanding analysis capabilities to prediction of phenomena of industrialimportance. These methods are specifically the upper-bound method andHill's general method, leading to the finite-element method. In this

73

Equilbrium equations:


chapter, the two methods are briefly outlined with examples. The basicformulations for the finite-element method are given in Section 5.4.Discretization of these basic formulations and numerical solution tech-niques are discussed in Chaps. 6 and 7.

5.2 Upper-Bound Method

The two extremum principles discussed in Chap. 4 provide an upper boundand a lower bound to the rate of work of the unknown surface tractions onSu. Often the velocity on Su is uniform; then the extremum principles canbe used to obtain lower and upper bounds for the forming load. For manymetal-forming operations, no exact solutions, even for the load to causeunconstrained plastic deformation, are available. If the two bounds can bemade sufficiently close to each other, an almost exact solution can beestimated. However, in the case of metal-forming problems, it is wellknown that a good lower bound is difficult to obtain [lj. Consequently, theupper-bound method was used in most applications. Even though theadvantage of bracketing the exact solution is usually abandoned, upperbounds are valuable to mechanical or production engineers, since theyhave to estimate a load that is required to perform a forming operation.

The second extremum principle states (see Chapter 4) that

where u* is a kinematically admissible velocity field, e,y* is the strain-ratefield derivable from «*; |Aw*| is the amount of velocity discontinuity alongthe surface of discontinuity S^, and otl* satisfies the yield criterion and isassociated with £,-,*.

In plane-strain problems an admissible velocity field can consist ofrigid-block sliding along the lines of velocity discontinuity. Then, since£,y* = 0 in V and usually Ft = 0, except along the tool-workpiece interfacewhere the magnitude of the frictional stress fs, acting in the oppositedirection to the relative sliding, is prescribed, the upper-bound calculationbecomes very simple according to

where \us*\ is the amount of relative sliding along the tool-workpieceinterface Sc.

Solutions with velocity fields, using rigid-block sliding, are presented formany metal-forming problems in the books by Johnson and Mellor [2] andby Johnson and Kudo [3]. Also, extension of the concept of velocitydiscontinuities to formation of "plastic hinges" found its applications tomany plate-bending problems, where determination of plate-collapse loadsis of primary importance [2].

Methods of Analysis 75

Perhaps the first successful use of kinematically admissible velocity fieldsin predicting the mean pressure to extrude round bars is that of Johnsonand Kudo [4]. Kudo [5, 6] has contributed immensely to this field and hiswork has helped to set the direction for many theoretical studies inextrusion and forging. The books by Avitzur [7, 8] show the quantity ofeffort required in arriving at some solutions and a variety of problems inwhich the upper-bound solutions yield useful information. The principlesapplied in all cases are basically identical. Differences in approach are onlyseen in constructing velocity fields, and the calculation of the upper-boundload follows the same procedure. In the following, construction of anadmissible velocity field and the upper-bound calculation are illustrated inan example [9].

Compression of a Solid Cylinder Between Flat Rough Dies

In the compression of a cylinder shown in Fig. 5.la, the deformation issymmetric about the r-axis. Hence the forming pressure is the same as thatobtained for the compression of a cylinder between the top rough die(frictional stress equals to the shear yield stress k) and the bottom smoothdie (frictional stress is zero) for the height H and the diameter D. Anassumed velocity field is shown in Fig. 5.1b. The deforming workpiece isdivided into two regions by a velocity discontinuity curve (1-4). The

FIG. 5.1 (a) Compression of a solid cylinder between flat rough dies, (b) An admissiblevelocity field.


velocity components in each region are assumed to be

and

In eq. (5.3), the exponent n is a parameter with which a minimum upperbound is sought, and it is entirely different from n used as a strain-hardening exponent in denning the stress-strain property of a material. Ineach region, the radial velocity component ur is assumed and the axialcomponent uz is determined so that it satisfies the incompressibilitycondition given by

and the boundary conditi

From the condition that the normal velocity component must becontinuous across the curve of discontinuity, we have dz/dr = Aw r/Aw^,along the curve (1-4), where A denotes the difference of velocitycomponents between the two regions. Integration results in the equationfor the discontinuity curve as

In eqs. (5.3) and (5.4), a = 2H/D with dimensions as shown in Fig. 5.la.It is to be noted that the line of discontinuity becomes straight for n = 0and the velocity field becomes continuous throughout the body for n = 1.

An upper bound to the rate of total forming energy E is obtained bycalculating the energy rate of deformation of a continuously deformingbody and the energy-rates due to shearing along the surface of velocitydiscontinuity, based on an assumed velocity field. The upper bound to theforming pressure, then, can be found from the equation

where A is the projected area of the die surface on a plane perpendicularto the motion of this surface upon which the forming pressure acts, and UD

is the die velocity.The region 1 in Fig. 5.1b moves as a rigid body and no energy-rate

dissipation occurs along the die-workpiece interface. The energy rate forthe continuously deforming region (124) is written as

and

in Region 1

in Region 2

where

77

according to the velocity field in region 2 given by eq. (5.3).In order to simplify the calculation, an approximation is made for the

effective strain-rate as

The energy-rate dissipated as a result of the velocity discontinuity |A«|along the curve (1-4) is given by

Thus the upper bound solution of the forming pressure becomes

Equation (5.5) is plotted as a function of 2H/D in Fig. 5.2, with the

FIG. 5.2 Comparison of upper bounds to the average forming pressure for the compressionof a cylinder.

Methods of Analysis


value of n that makes pja minimum, and is compared with the pressuresfor n = 0 and n = 1.

Remarks

The technique outlined above together with the increased use of compu-ters led to the development of generalized upper-bound approaches to awider class of problems. Examples of such approaches are the analysis ofaxisymmetric forging of complex shapes [10], estimation of lateral spreadin plate rolling [11] and in forging [12], and upper bounds to extrusion anddrawing through dies of various shapes [13-16].

In these examples, construction of admissible velocity fields becomesmore complicated and upper-bound calculations must often resort tonumerical means. A question arises, then, whether pursuing this approachis worth the effort, because detailed flow is much more important thanforces. Kudo [17] surveyed the development and use of the upper-boundmethod and concluded that the approach should continue to be useful inteaching, in workshop trials, and in research on bulk forming processes.Also, it will be noted in Section 5.5 that the finite-element methodemerges from the principle involved in the upper-bound method and fromthe concept of discretization.

5.3 Hill's General Method

Hill [18], combining flexibility with rigorous principles, proposed a newmethod of analysis applicable to any technological forming process. Hesystematized the details of procedure and illustrated the method withpreliminary analyses of inhomogeneous compression, bar drawing, andforging. Ideally, a good method of approximation must be capable ofreproducing and predicting the main phenomena, and of delivering reliableinformation about loads and principal dimensional changes. It should alsobe governed by well-defined general principles, but be flexible enough tocover all normal situations, arbitrary material properties, different kinds offriction, or any tool shape. Hill believed that the proposed method comesclose to the ideal and indicated that the prospects were encouraging, butthe final assessment awaits a longer and more diverse investigation.

Since 1963, however, with few exceptions, no further investigation onthe method has appeared in the literature. Nagamatsu et al. [19] andMurota et al. [20] analyzed plane-strain and axisymmetric compression byHill's method. Lahoti and Kobayashi [21,22] carried out the analysis ofring compression with barreling, spread in Stechel rolling, and thicknesschange in tube sinking. In the following, the basic ideas of the method areoutlined and the procedure of the method is illustrated in an example.

The Method of AnalysisSuppose we have some piecewise-continuously differentiable distributionof stress a^ in a region V, together with traction F, over its surface 5. Then


the converse of the virtual work-rate principle is stated as follows: Theconsidered distribution o(i is in equilibrium within a body (neglecting thebody force) and with applied surface tractions given by Fh if

for all virtual velocity fields w,- that are continuous and continuouslydifferentiable. Introducing an approximate stress field that may notnecessarily satisfy equilibrium in eq. (5.6a) and applying the divergencetheorem, we obtain

where CT,-,M/ is the traction on S computed from the considered stress field.If eq. (5.6b) holds for all virtual velocity wjt then dOfj/dXi = 0 in V anda,y«, = FJ on S. Equations (5.6a) and (5.6b) thus are equivalent. Equation(5.6b) asserts that a sufficient condition for the considered stress distribu-tion to satisfy the required statical conditions is that the divergences shouldhave zero net work-rate in the stated class of virtual motions, or thedivergences should be "orthogonal" to the virtual motions.

The interpretation of this theorem leads to a criterion of approximation,namely, the static conditions for the approximating field o^ can beregarded as closely satisfied overall when eq. (5.6a) is satisfied for asufficient subclass of virtual orthogonalizing motion w,, with the traction Ff

prescribed on SF and computed from the approximating field otj on theremainder of the surface.

For metal-forming processes, the surface 5 of the deforming zoneconsists of three distinct parts, S = Sc + SF + Sj, where Sc adjoins a tool orcontainer, SF is unconstrained, and S/ is the junction with the rigid zone;on surface SF, ordinarily the traction is zero; the frictional constraint oversurface Sc is represented by a constant frictional stress mk (where k is theshear yield stress of the deforming body) or a Coulomb coefficient offriction (i.

The selection criterion for the approximating field a,-,- is then expressedbv

for constantfrictional stress

for Coulombfriction

for a sufficient subclass of virtual orthogonalizing motion w,, where T(-denotes the surface traction computed from the considered field o^; n; isthe local unit outward normal; and /, is a unit tangent vector opposite insense to the relative velocity of sliding in the approximating field.


The initial procedure of the method is to choose a class of velocity fieldsfrom which the approximate stress fields can be determined. The chosenvelocity fields must satisfy all kinematic conditions. The associated stress(given by the material constitutive law) is either determined uniquely, or atleast determined to within a hydrostatic pressure if the material isincompressible. The associated distribution of stress in the deformationzone of a chosen velocity field will generally not satisfy all the staticalrequirements. In applying the selection criterion (5.7), the orthogonalizingfamily Wj must be sufficiently wide and extensive to identify a singleapproximating velocity field in the particular class constructed for satisfy-ing the kinematic conditions. If the class is defined by equations involvingan unknown function of just one position variable, then the orthogonaliz-ing family must also involve an arbitrary function of one variable. Clearly,the easiest choice of orthogonalizing motions is the class of approximatingvelocity fields itself; or one can use differences of pairs of these fields.

Once a family is chosen, the calculus-of-variations technique is appliedto eqs. (5.7), treating w, as a variation. We obtain a system of equilibriumequations and boundary conditions, suited to the particular approximatingclass and uniquely determining its best member.

Illustrative Example—Compression of Solid Circular CylinderA solid circular cylinder with diameter D and height 2H is compressedbetween parallel rigid platens having a constant friction mk (see Fig. 5.la).It is required to calculate the incipient barreling of the free surface. Takeaxes (r, z) with the origin at the half height and z along the direction ofcompression.

Since longitudinal nonuniformity is the feature under investigation, theapproximating class of velocity field must contain an unspecified functionof z. The simplest construct for the velocity components is

with 0(—z) = -0(z) and (j)(H) = l, where the prime impliesdifferentiation.

For symmetry about the mid-plane, </> is an odd function of z withcontinuous first and second derivatives at least.

To match the class (5.8), we must choose a comparable family of virtualorthogonalizing motions, also involving an arbitrary function of z. Take

with V(~2) = ~'<P(Z)-Noting that Sj = 0, nf is a unit vector in the direction of z, {«,} =

{nr, ng, nz} = {0, 0, 1}, and /, is a unit vector in the direction of r,


{lj} = (lr, le, 4} = {-1, 0, 0}, the criterion (5.7) can be written as

Substitution of eq. (5.9) into eq. (5.10) results in

where o'z — oz — j(az + or + og) and P is the compression load given by

Integrating by parts and noting that 0, eq. (5.11) becomes

Since i/>' is completely arbitrary, it follows that

and

at any z, which integrates to

In deriving eq. (5.14), the yield condition in the form of a'2= — §Y,where Y denotes the yield stress in uniaxial compression, is used as a firstorder of approximation.

Evaluating eq. (5.14) at z=H and using eq. (5.13), we obtain theexpression for the compression load P as

By replacing mk with fiY in eq. (5.15), the familiar Siebel formula isobtained.

and thus

Substitution of eq. (5.16) into eq. (5.14) with the help of eq. (5.15), resultsin

where

Integration of eq. (5.17), with 0(0) = 0 and $(//) = !, determines thebulge function <f>(z) as

Substituting eq. (5.18) into the approximate velocity field (5.8), theincipient of bulge formation can be described by the radial velocitycomponent at r = D/2. Figure 5.3 shows the profile of the radial velocitycomponent along the free surface for H/D = 0.5 and several values offriction factor m. In order to trace the development of bulge, calculationmust be repeated step by step, updating the current geometricalconfiguration.

To illustrate the method, the formulation given in this example uses asimpler approximating velocity field than that given by Hill.

RemarksIt appears that the class of problems to which the Hill method can beapplied effectively is one for which the determination of the geometricalchanges and load are the main objectives. This class of problems can begrouped into three areas: (1) simple upsetting of rings and blocks, (2)flat-tool forging and its allied problems, (3) steady-state processes withunknown steady-state configurations. The method is well-based on mathe-matically sound principles and is amply flexible in introducing varioussimplifying assumptions into the analysis. Actual procedure using themethod usually involves extensive calculations and sometimes encountersoverwhelming difficulties in obtaining final solutions. However, as pointed


To obtain the differential equation for (j>, Trz is calculated from theapproximating field (5.8), using the flow rule as


FIG. 5.3 Computed profile of the radial velocity component along the free surface incompression.

out in Section 5.5, the basis of the finite-element method is the discreterepresentation of the principles and ideas given in Hill's method.

5.4 The Finite-Element Method

The four approaches used for the derivation of the basic equations for thefinite-element analysis have been stated in Chap. 1. They are (1) the directapproach, (2) the variational method, (3) the method of weightedresiduals, and (4) the energy balance approach. In the following, thevariational method and a special case of the weighted residual method aredescribed.

Basis for the Finite-Element Formulation

The variational approach is based on one of two variational principles. Itrequires that among admissible velocities ut that satisfy the conditions ofcompatibility and incompressibility, as well as the velocity boundaryconditions, the actual solution gives the following functional (function offunctions) a stationary value:

and

for rigid-plastic materials

for rigid-viscoplastic materials

84

where o is the effective stress, e is the effective strain-rate, Ff representssurface tractions, and E(eLj) is the work function (see eq. (4.53) in Chap.4). The solution of the original boundary-value problem is then obtainedfrom the solution of the dual variational problem, where the first-ordervariation of the functional vanishes, namely,

where o = O(E) and a = o(e, e) for rigid-plastic and rigid-viscoplasticmaterials, respectively. The incompressibility constraint on admissiblevelocity fields in eq. (5.20) may be removed by introducing a Lagrangemultiplier A [23, 24] and modifying the functional (5.19) by adding theterm J Ac,, dV, where ev = ea, is the volumetric strain-rate. Then,

Another way of removing the constraint is to use the penalized form ofthe incompressibility [25] as

where K, a penalty constant, is a very large positive constant.In eqs. (5.21) and (5.22), dut and <5A are arbitrary variations and de and

dev are the variations in strain-rate derived from dut. Equation (5.21) or(5.22) is the basic equation for the finite-element formulation.

It should be noted that under the plane-stress condition that applies tothe analysis of sheet-metal forming (Chap. 11), the change of sheetthickness is determined from the incompressibility condition and theconstraint is not imposed to admissible velocities. Also, the incompres-sibility condition does not apply to the deformation of porous materials,discussed in Chap. 13.

An alternative approach to eq. (5.20) is to begin with a weak form of theequilibrium equations, namely,

where duf is an arbitrary variation in «,-. Equation (5.23) becomes

Note that eq. (5.24) is the expression of the virtual work-rate principle, ifdut is considered to be the virtual motion (see eq. (4.26) in Chap. 4).

Using the symmetry of the stress tensor and the divergence theorem, eq.(5.24) results in

Metal Forming and the Finite-Element Method


For the surface integral term in eq. (5.25), the boundary conditions thatdiij = 0 on Su (essential boundary condition) and o^n, = Ft on SF

(suppressible boundary condition), are imposed.Substitution of a,y = a,/ + <5,yOm, where am = \okk, the hydrostatic stress,

into eq. (5.25) gives

From the constitutive equation and the definition of e, it is seen that,a// (5e,y = ode, and the final form of eq. (5.26) suitable for the finite-element formulation becomes

with the incompressibility constraint given by £„ = 0 in V. The incompres-sibility constraint is removed by modifying eq. (5.27a) as

By comparison of eqs. (5.21) and (5.27b), the Lagrangian multiplier A ineq. (5.21) is identified as the mean stress om. Further, eq. (5.22) isregained by the use of om = (K/2)£v in eq. (5.27b), thus providing theinterpretation of K as a constant similar to the bulk modulus.

Equation (5.21) or (5.22) is the basic equation for the finite-elementdiscretization. Once the solution for the velocity field that satisfies thebasic equation is obtained, then corresponding stresses can be calculatedusing the flow rule and the known mean stress distribution.

Comments on Basic Equations

The duality of the boundary-value problem and the variational problemcan be seen clearly by considering the construction of the functional (5.19).If the functional is n(u), then dn(u) = 0 breaks down into two parts, thatof the domain and that of the boundary. Suppose that the functional Jt(u)is given by

where x is the coordinate, u is the field variable and u' is the derivative ofu with respect to x. Then

Integrating the second term by parts, we have


Thus,

dn(u) = Euler Equation} du dV + {the boundary term)

The Euler equation is a differential equation taken over the domain Vthat is argued to be zero because of the arbitrariness of du. Given thedifferential equations, the functional n and the boundary terms can oftenbe constructed by manipulation. For our boundary value problem,

The essential boundary condition that <5w, = 0 on Su is imposed and theboundary requirement on SF are suppressed in eq. (5.28), since duf isarbitrary on SF. The manipulation of the first term of eq. (5.28) followseqs. (5.23), (5.24), and (5.25). Then, eq. (5.28) becomes

from which the functional (5.19) is obtained.The construction of the functional discussed above also shows that the

variational formulation involves redundancy in manipulation, compared tothe alternative approach. Interpretation of eq. (5.23) leads to the conceptof error minimization. Galerkin's method is the obvious discretization ofeq. (5.23), and is a specific method of using weighted residuals where theweighting functions are the same as the approximating functions.

Boundary ConditionsThe traction boundary condition on SF is either zero-traction or ordinarilyat most a uniform hydrostatic pressure. However, the boundary conditionsalong the die-workpiece interface are mixed. Furthermore, in general,neither velocity nor force (including magnitude and direction) can beprescribed completely along this interface, because the direction of thefrictional stress is opposite to the direction of the relative velocity betweenthe deforming workpiece and the die, and this relative velocity is notknown a priori. Situations exist, e.g., in extrusion and drawing, in whichthe direction of metal flow relative to the die is known. This class ofproblems can be solved if the magnitude of the frictional stress fs is givenaccording to the well-known Coulomb law, fs = ftp, or the friction law ofconstant factor m, expressed by fs = mk (where k = Y/VJ); here, p is thedie pressure and k is the shear yield stress.

For problems such as ring compression, rolling, and forging, theunknown direction of the relative velocity between the die-workpieceinterface makes it difficult to handle the boundary condition in astraightforward manner. A unique feature of this type of problem is thatthere exists a point (or a region) along the die-workpiece interface where


the velocity of the deforming material relative to the die becomes zero (seeSection 3.8 of Chap. 3), and the location of this point (or region) dependson the magnitude of the frictional stress itself. In order to deal with thesesituations, a velocity-dependent frictional stress is used as an approxima-tion to the condition of constant frictional stress. At the interface Sc thevelocity boundary condition is given in the direction normal to theinterface by the die velocity, and the traction boundary condition isexpressed by

where fs is the frictional stress, € is the unit vector in the opposite directionof relative sliding, us is the sliding velocity of a material relative to the dievelocity (relative sliding velocity), and u0 is a small positive numbercompared to us. The approximate expression (5.30) for a constantfrictional stress has been used for the smooth transition of the frictionalstress in the range near the neutral point [26].

Treatment of a Rigid RegionThe principles described in the preceding sections apply to a domain inwhich the entire body is deforming plastically. In metal-forming processes,however, situations do arise in which rigid zones exist, and unloadingoccurs. The rigid zones are characterized by a very small value of effectivestrain-rate in comparison with that in the deforming body. If these portionsare included within the control volume V (the volume included in thestatement of boundary value problems), the value of the first term of thebasic equation (5.21) or (5.22) cannot be uniquely determined because ofthe undefined value of the effective stress when the effective strain-rateapproaches zero.

This difficulty is removed by assuming that the stress-strain-raterelationship in eq. (5.1c) is approximated by

where £0 takes an assigned limiting value, say 10~3 [27]. This presumedstress-strain-rate relationship is equivalent to the assumption of a New-tonian fluid-like material behavior for nearly-rigid regions. For theseregions, the first term of the basic equation, Svode dV, is then replacedby

Thus, the finite-element discretization process is based on eq. (5.21) or(5.22), with eq. (5.30) in mind for the die-workpiece interface boundarycondition, and eq. (5.31) for the regions that are considered to be nearlyrigid.


5.5. Concluding Remarks

The upper-bound method is based on one of the extremum principles. Theconventional minimization technique in the method is associated with aweaker statement in the form of a variational principle, namely, eq. (5.20).Equation (5.20) also serves as a basis for the finite-element discretization.Thus both methods are devised essentially from the same principle.However, admissible velocity fields in the upper-bound approach aretraditionally formulated in closed forms, and thus a considered class ofvelocity fields is limited. In the finite-element approach, because ofdiscrete representation of approximating velocity fields, the class ofconsidered velocity fields is much wider.

In Hill's general method, Hill stated that we have considerable freedomof choice for orthogonalizing virtual motions and suggested that the easiestchoice is the class of approximating fields itself; alternatively, one can usedifferences of pairs of these fields, or infinitesimal variations within theclass. When the variations within the approximating field are chosen asorthogonalizing virtual motions in Hill's general method, the variationalprinciple, eq. (5.20), is supplied. Thus, it is clear that only the concepts ofdiscretization and discrete representation of approximate velocity fieldsdistinguish the finite-element method from the others.

References

1. Kobayashi, S., and Thomson, E. G., (1965), "Upper and Lower BoundSolutions to Axisymmetric Compression and Extrusion Problems," Int. J.Mech. ScL, Vol. 7, p. 127.

2. Johnson, W., and Mellor, P. B., (1973), "Engineering Plasticity," VanNostrand Reinhold, London.

3. Johnson, W., and Kudo, H., (1962), "Mechanics of Metal Extrusion,"Manchester University Press, Manchester, UK.

4. Johnson, W., and Kudo, H., (1960), "Use of Upper Bound Solutions. . .ForAxisymmetric Extrusion Processes," Int. J. Mech. Sci., Vol. 1, p. 175.

5. Kudo, H., (1960), "Some Analytical and Experimental Studies of Axisym-metric Cold Forging and Extrusion—I," Int. J. Mech. Sci., Vol. 2, p. 102.

6. Kudo, H., (1961), "Some Analytical and Experimental Studies of Axisym-metric Cold Forging—II," Int. J. Mech. Sci., Vol. 3, p. 91.

7. Avitzur, B., (1968), "Metal Forming: Processes and Analysis," McGraw-Hill,New York.

8. Avitzur, B., (1983), "Handbook of Metal Forming Processes," Wiley, NewYork.

9. Kobayashi, S., (1964), "Upper-Bound Solution to Axisymmetric FormingProblems," Trans. ASME, J. Engr. Ind., Vol. 83, p. 326.

10. McDermott, R. P., and Bramley, A. N., (1974), "An Elemental Upper-BoundTechnique for General Use in Forging Analysis," Proc. 15th M.T.D.R. Conf.,p. 437.

11. Oh, S. I., and Kobayashi, S., (1975), "An Approximate Method for aThree-Dimensional Analysis of Rolling," Int. J. Mech. ScL, Vol. 17, p. 293.

12. Juneja, B. L., (1973), "Forging of Polygonal Discs with Barrelling," Int. J.Machine Tool Des. Res., Vol. 13, p. 87.


13. Juneja, B. L., and Prakash, R., (1975), "An Analysis for Drawing andExtrusion of Polygonal Sections," Int. J. Machine Tool Des. Res., Vol. 15, p.1.

14. Nagpal, N., and Allan, T., (1975), "Analysis of Three-Dimensional MetalFlow in Extrusion of Shapes with the Use of Dual Stream Functions," Proc.3rd NAMRC, Carnegie-Mellon University, p. 26.

15. Yang, D. Y., and Lee, C. H., (1978), "Analysis of Three-DimensionalExtrusion of Sections Through Curved Dies by Conformal Transformation,"Int. J. Mech. ScL, Vol. 20, p. 541.

16. Kiuchi, M., (1984), "Overall Analysis of Non-Axisymmetric Extrusion andDrawing," Proc. 12th NAMRC, Michigan Technological Univ. Houghton,Michigan, p. 111.

17. Kudo, H., (1985), Upper Bound Approach to Metal Forming Processes—ToDate and in the Future, "Metal Forming and Impact Mechanics," edited byS. R. Reid, Pergamon Press, Oxford, p. 19.

18. Hill, R., (1963), "A General Method of Metal-Working Processes," /. Mech.Phys. Solids, Vol. 11, p. 305.

19. Nagamatsu, A., Murota, T., and Jimma, T., (1970), "On the Non-UniformDeformation of Block in Plane Strain Compression Caused by Friction," Bull.JSME, Vol. 13, p. 1389.

20. Murota, T., Jimma, T., and Nagamatsu, T., (1966), "A Theoretical Analysisof Circular Cylinder in Axial Compression with Friction," Proc. 16th JapanNatl. Congr. Appl. Mech., p. 141.

21. Lahoti, G. D. and Kobayashi, S., (1974), "On Hill's General Method ofAnalysis for Metal-Working Processes," Int. J. Mech. ScL, Vol. 16, p. 521.

22. Lahoti, G. D., and Kobayashi, S., (1974), "Flat-Tool Forging," Proc. 2dNorth American Metal Working Res. Conf., May, University of Wisconsin-Madison, p. 73.

23. Washizu, K., (1968), "Variational Methods in Elasticity and Plasticity,"Pergamon Press, Oxford.

24. Lee, C. H., and Kobayashi, S., (1973), "New Solutions to Rigid-PlasticDeformation Problems Using a Matrix Method," Trans. ASME, J. Engr. Ind.,Vol. 95, p. 865.

25. Zienkiewicz, O. C., (1977), "The Finite Element Method," 3d Edition,McGraw-Hill, New York.

26. Chen, C. C., and Kobayashi, S., (1978), "Rigid-Plastic Finite-ElementAnalysis of Ring Compression," Application of Numerical Methods toForming Processes, ASME, AMD, Vol. 28, p. 163.

27. Chen, C. C., Oh, S. I., and Kobayashi, S., (1979), "Ductile Fracture inAxisymmetric Extrusion and Drawing, Part 1: Deformation Mechanics ofExtrusion and Drawing," Trans. ASME, J. Engr. Ind., Vol. 101, p. 23.

6THE FINITE-ELEMENT METHOD—PART I

6.1 Introduction

The concept of the finite-element procedure may be dated back to 1943when Courant [1] approximated the warping function linearly in each of anassemblage of triangular elements to the St. Venant torsion problem andproceeded to formulate the problem using the principle of minimumpotential energy. Similar ideas were used later by several investigators toobtain the approximate solutions to certain boundary-value problems. Itwas Clough [2] who first introduced the term "finite elements" in the studyof plane elasticity problems. The equivalence of this method with thewell-known Ritz method was established at a later date, which made itpossible to extend the applications to a broad spectrum of problems forwhich a variational formulation is possible. Since then numerous studieshave been reported on the theory and applications of the finite-elementmethod.

In this and next chapters the finite-element formulations necessary forthe deformation analysis of metal-forming processes are presented. For hotforming processes, heat transfer analysis should also be carried out as wellas deformation analysis. Discretization for temperature calculations andcoupling of heat transfer and deformation are discussed in Chap. 12. Moredetailed descriptions of the method in general and the solution techniquescan be found in References [3-5], in addition to the books on thefinite-element method listed in Chap. 1.

6.2 Finite-Element Procedures

The path to the solution of a problem formulated in finite-element form isdescribed in Chap. 1 (Section 1.2). Discretization of a problem consists ofthe following steps: (1) describing the element, (2) setting up the elementequation, and (3) assembling the element equations. Numerical analysistechniques are then applied for obtaining the solution of the globalequations. The basis of the element equations and the assembling intoglobal equations is derived in Chap. 5 (eq. (5.20) and eqs. (5.21) or(5.22)).

The solution satisfying eq. (5.20) is obtained from the admissible

90

The Finite-Element Method—Part I 91

velocity fields that are constructed by introducing the shape function insuch a way that a continuous velocity field over each element can bedenned uniquely in terms of velocities of associated nodal points. In thedeformation process shown in Fig. 6.1, the workpiece is divided intoelements, without gaps or overlaps between elements. In order to ensurecontinuity of the velocities over the whole workpiece, the shape function isdefined such that the velocities along any shared element-side areexpressed in terms of velocity values at the same shared set of nodes(compatibility requirement). Then a continuous velocity field over thewhole workpiece can be uniquely defined in terms of velocity values atnodal points specified globally.

We define a set of nodal point velocities in a vector form as

where the superscript T denotes transposition and N = (total number ofnodes) x (degrees of freedom per node).

An admissibility requirement for the velocity field is that the velocityboundary condition prescribed on surface Su (essential boundary condi-tion) must be satisfied. This condition can be imposed at nodes on Su by

FIG. 6.1 Finite-element mesh and nodal point specifications in a forming process.


assigning known values to the corresponding variables. It is to be notedthat the incompressibility condition is not required for defining a velocityfield in the formulation of eq. (5.21) or (5.22).

Equations (5.20) and (5.21) or (5.22) are now expressed in terms ofnodal point velocities v and their variations 6v. From arbitrariness of 6v{,a set of algebraic equations (stiffness equations) are obtained as

where (/) indicates the quantity at the/th element. The capital-letter suffixsignifies that it refers to the nodal point number.

Equation (6.2) is obtained by evaluating the (dn/dvj) at the elementallevel and assembling them into the global equation under appropriateconstraints.

In metal-forming, the stiffness equation (6.2) is nonlinear and thesolution is obtained iteratively by using the Newton-Raphson method.The method consists of linearization and application of convergencecriteria to obtain the final solution. Linearization is achieved by Taylorexpansion near an assumed solution point v = v0 (initial guess), namely,

where Au/ is the first-order correction of the velocity v0. Equation (6.3)can be written in the form

where K is called the stiffness matrix and f is the residual of the nodal pointforce vector.

Once the solution of eq. (6.4) for the velocity correction term Av isobtained, the assumed velocity v0 is updated according to vn + <x Av, wherea is a constant between 0 and 1 called the deceleration coefficient. Iterationis continued until the velocity correction terms become negligibly small.The Newton-Raphson iteration process is shown schematically in Fig. 6.2.It is seen from the figure that convergence of Newton-Raphson iterations,the initial guess velocity should be close to the actual solution. When adeformation process is relatively simple, the initial guess velocity can beprovided, for instance, by the upper-bound method. However, if theprocess is complex and obtaining a good initial guess solution is difficult,then the use of the direct iteration method discussed in Section 7.4 ofChap. 7 may be appropriate.

Two convergence critera may be used. One measures the error norm ofthe velocities, ||Av||/||v||, where the Euclidean vector is defined as||v|| = (vTv)1/2, and requires such an error norm to decrease from iterationto iteration. The other criterion requires the norm of the residualequations, ||3jr/3v||, to decrease.

In general terms, the first criterion is most useful in the early stages of


FIG. 6.2 Schematic representation of the Newton-Raphson method: (a) convergence; (b)divergence.

iteration, when the velocity field is still far from the solution. The secondtest is most useful when slightly ill-conditioned systems reach the finalstage of iterations. The final solution is considered to be achieved when theerror norm reaches a specified small value, say 5 x 1(T5.

The finite-element method procedures outlined above are implementedin a computer program in the following way.

1. Generate an assumed solution velocity.2. Evaluate the elemental stiffness matrix for the velocity correction

term Av in eq. (6.4).3. Impose velocity conditions to the elemental stiffness matrix, and

repeat step 2 over all elements defined in the workpiece.4. Assemble elemental stiffness matrix to form a global stiffness

equation.


5. Obtain the velocity correction terms by solving the global stiffnessequation.

6. Update the assumed solution velocity by adding the correctional termto the assumed velocity. Repeat steps 2 through 6 until the velocitysolution converges.

7. When the converged velocity solution is obtained, update thegeometry of the workpiece using the velocity of nodes during a timeincrement. Steps 2 through 7 are repeated until the desired degree ofdeformation is achieved.

The order of the certain steps mentioned above may change dependingupon the computer programming practice.

The above procedure applies to the analysis of nonsteady-state-processes. For steady-state processes, updating the geometry of theworkpiece is not necessary. The procedure for steady-state processes isdescribed in Chap. 10 for the analysis of axisymmetric extrusion anddrawing.

6.3 Elements and Shape Function

The order of a boundary-value problem is denned as that of the highestderivative of the field variable. It is 2 in our problem, and the integrand ineq. (5.20) is of order 1. Thus, the admissibility requirements are that thevelocity must have a continuous first-order derivative within the element(completeness requirement) and the velocity should be continuous at thesubdomain interfaces (compatibility requirement). Symbolically, a functionof class Cr has continuous derivatives of all orders up to and including r.Then shape functions for the velocity field within the element must be ofclass Cl for completeness, and the compatibility of elements requires themto be of class C° globally.

The geometry of an element, in general, is uniquely defined by a finitenumber of nodal points (nodes). The nodes are located on the boundary ofthe element or within the element, and the shape function defines anadmissible velocity field locally in terms of velocities of associated nodes.Thus, elements are characterized by the shape and the order of shapefunctions.

In the finite-element method, interpolation of a scalar function f(x, y)defined over an element is introduced in a form

where fa is a function value associated with ath node, and qa(x, y) is theshape function. It is, in general, a polynomial function of x and y definedover the element in such a way that


FIG. 6.3 Area coordinate system of the triangular element.

where (xe, y@) is the coordinates of /3th node and 6a/i is the Kroneckerdelta. Owing to the property of the shape functions given by eq. (6.6), /a

in eq. (6.5) has the value of the function / at (xa, ya) and the fa areindependent of each other.

There are various types of elements, depending upon the shape of theelement and the polynomial order of shape funtions. In the following, theelements used in this book are discussed.

Triangular Element FamilyIn the triangular element family, it is convenient to define shape functionsin the area coordinate system, L1; L2, L3. The area coordinates for atriangle, as shown in Fig. 6.3, are defined by the following linear relations:

where (xa, ya) are the coordinates of a corner of the triangle. It can bereadily shown that an alternative definition of the coordinate of a point Pcan be given by the ratio of the area of the shaded triangle to that of thetotal triangle as

Solving eq. (6.7) for Lt, L2 and L3 gives

96

where

and


A = area of (123)

With a linear triangular element with nodes at its corners (see Fig. 6.4a),the shape functions qa are linear and are given by

A quadratic triangular element has primary nodes at the corners andsecondary nodes at the mid-sides (see Fig. 6.4b) and the shape functionsare

The admissible velocity field within the element can be represented, forboth linear and quadratic elements, by

where u^a\ u^ are the velocity components of the <*th node.Elements that use the same shape functions as the coordinate transfor-

mation are called isoparametric elements. Linear triangular elements,owing to eqs. (6.7) and (6.10), are isoparametric. However, with quadraticshape functions, isoparametric elements are curved triangular elements.

FIG. 6.4 (a) Linear triangular element; (b) quadratic triangular element.


FIG. 6.5 (a) Natural coordinate and rectangular parent element; (b) isoparametric element(mapped on Cartesian coordinate; quadrilateral element); (c) shape function.

In this book, the linear triangular element is used for the analysis ofplate-bending under plane-strain conditions in Chap. 8. It is also used forthe analysis of sheet-metal forming in Chap. 11.

Rectangular Element Family

The shape functions of rectangular elements are, in general, denned in aparametric form over a domain —!<£<!, — l < r j < l in a naturalcoordinate system (|f, r/). The element defined in the natural coordinatesystem is sometimes called the parent element. The simplest of therectangular elements is the 4-node linear element shown in Fig. 6.5a. Theshape functions, qa, which are bilinear in £ and r/, are defined as

where (£„., r]a) are the natural coordinates of a node at one of its corners.


The value of the shape function, given by eq. (6.13), is shown schemati-cally in Fig. 6.5c. Admissible velocity fields can be defined uniquely overthe rectangular element by the nodal velocity components as

where (u^a\ u(ya)) is the velocity at the <*th node and summation is over all

four nodes.Coordinate transformation from the natural coordinate (£, 77) to the

global coordinate (x, y) is denned by

where (xa, ya) are the global coordinates of the o-th node. Since thecoordinate transformation (6.15) uses the same shape functions (6.14), thelinear element is isoparametric and takes quadrilateral shape in theCartesian map, as shown in Fig. 6.5b.

Elements with shape functions of higher-order polynomials can bedefined in a similar manner. Figure 6.6 shows a quadratic rectangularelement with 8 nodes in the natural coordinate and global coordinatesystems. The shape functions are defined by

corner nodes:

mid-side nodez;

Rectangular quadratic elements can also be defined by 9 nodes byadding an internal node to an 8-node quadratic rectangular element. Theshape functions of the 9-node quadratic element can be derived from thoseof 8-node element. It may be of interest to note that the internal node doesnot affect the velocity distributions at the element boundary and theelement shape. The rectangular element with 9 nodes is said to be of theLagrangian family, since the shape functions are derived from Lagrangepolynomials [6].

The isoparametric element can be distorted, for instance as shown inFig. 6.6b, for a quadratic rectangular element by mapping according to eq.(6.15) with the shape functions given by eq. (6.16). The distorted shapes of


(b)

FIG. 6.6 Eight-node quadratic rectangular element: (a) natural coordinate and parentelement; (b) isoparametric element after Cartesian mapping.

the isoparametric elements are sometimes useful, since they allow moreflexibility and convenience in mesh generation. On the other hand, thereare some limitations on the element geometry. To be a valid element, thecoordinate transformation given by eq. (6.15) should be one-to-one, or theJacobian of the coordinate transformation (see eq. (6.29)) should bepositive at all points within the element. The validity of the elementgeometry is important, since elements deform considerably in metal-forming simulation.

Elements used for axisymmetric deformation are the same as two-dimensional elements in terms of shape functions and coordinate transfor-mations of the element. However, the axisymmetric element representsthe cross section of a torus (ring element), while the other two dimensionalelements represent the cross section of a straight bar. The strain-ratedefinition and the volume integration procedure are therefore different.


The 4-node linear isoparametric element (quadrilateral element) is usedextensively in this book for the analyses of two-dimensional and axisym-metric deformation processes.

Three-Dimensional Brick ElementThe three-dimensional brick element is a natural extension of thetwo-dimensional linear element. The parent element is defined over adomain -!<£<!, -1 < 77 < 1, — 1 < £ :£ 1, in natural coordinate systemwith a node at each corner. The shape functions are defined as

In eq. (6.17), (£a, rja, £a) are the coordinates of the orth node in thenatural coordinate system. Figure 6.7 shows the brick element and its

(b)

FIG. 6.7 Three-dimensional brick element in natural and Cartesian coordinate systems.


node-ordering convention. The velocity field can be expressed by

where (u^a\ u^, u^') is the velocity of the oth node. The coordinatetransformation is given by

where (xa, ya, za) are the coordinates of the o-th node.The use of the brick element is shown in Chap. 14 for the analysis of

three-dimensional deformation.

Note on Notation

In expressing elemental functions, we treat each velocity component as ascalar. Sometimes it is more convenient to express a velocity field in avector form as

where superscript T denotes transposition. The use of the transpose is onlyfor the convenience of expressing formulations in matrix forms.

The vectors u and v are defined by their components according to

OT ={«,(£,»»), «,(£»,)}

VT= {Mi1), <>, U?\ Uf>, ...} = {Vl,V2,V3,...}

respectively, for the case of two-dimensional deformation.The shape functions are also arranged in the matrix form as

for the two-dimensional case.

6.4 Element Strain-Rate Matrix

In Chap. 4, the strain-rate components in Cartesian coordinate system aredefined by


It was also shown, in Section 6.3, that the admissible velocity for all typeof elements can be expressed by

It is seen from eq. (6.22) that strain rate components can be evaluated ifdqa/dXi is known.

For the Cartesian coordinate system, we denote the coordinate jc, by(x, y, z) for three-dimensional deformation, by (r, z, 0) for axisymmetricdeformation, and by (x, y) for two-dimensional deformation.

Let Xa, Ya, and ZK be denned as

c

It is convenient to arrange the strain-rate components in a vector form.For two-dimensional elements and axially symmetric deformation, thestrain-rate components can be written as

for plane-stresss

Substituting eq. (6.21) into eq. (6.20), we have

Then the strain-rate components given by eq. (6.22) are expressed by


for plane- strain

and

for axisymmetricdeformation

In eq. (6.25), u^ and u2 correspond to ux and uy, respectively, fortwo-dimensional deformation, and Pa is zero for plane-strain and the rowof £3 is deleted for plane-stress deformation. For the axially symmetriccase, MI and u2 represent ur and uz, respectively, and Pa becomes qalr.

Equation (6.25) can be written in the matrix form as

where B is called the strain-rate matrix and written as

The number of columns of the B matrix is determined by the number ofdegrees of freedom allowed to the element.

The evaluation of the strain-rate matrix, or of Xa, Ya, and Za, requiresthe differentiation of shape functions with respect to the global coordinate.

Substituting eqs. (6.23) into eqs. (6.24a, 6.24b and 6.24c), the strain-ratevectors are represented in a unified form, as


Since the shape functions are expressed in the natural coordinate system, itis necessary to express the global derivatives in terms of the derivativeswith respect to the natural coordinate. Consider a coordinate transforma-tion given by eq. (6.19), where shape functions are denned in the naturalcoordinate system. Then the derivatives of the shape functions with respectto the natural coordinate system can be expressed, using the chain rule, as

where J is the Jacobian matrix of the coordinate transformation, given by

Then the derivatives in eq. (6.23a) can be obtained as

where J"1 is the inverse of J.It may be mentioned that in plane-strain deformation, the strain rate e3

is not necessary, since it is always zero. However, it is convenient toinclude £3 in eq. (6.25) so that the strain rate matrix B of the plane straindeformation has the same form as that of the axisymmetric deformation asshown in eq. (6.27).

Triangular Element Family

The strain-rate matrix of the triangular family of elements can be derivedby applying eq. (6.30) to the shape functions given in eqs. (6.10) and(6.11). Since the area coordinates are not independent of each other, wecan eliminate L3 from the expressions of qa by using L3 = 1 — Ll — L2.Equation (6.30) can be written for triangular element as

where |J| is the determinant of the Jacobian matrix and expressed by

The strain-rate matrix of a linear triangular element, shown in Fig. 6.4a,can be obtained in a closed form by substituting q1 = L1, q2= L2,


q3 = 1 — L! - L2, and is written as follows:

where

Note that |J| is twice of the area of the triangle. It may also be noted thatall the strain-rate components of a linear triangular element are constantover one element, since Xa and Ya of eq. (6.33) are not functions of thearea coordinates.

The strain-rate matrix of the quadratic triangular element can be derivedin a similar manner to that shown above. However, the expressions for Xa

and Ya are much more complex, and it is easier to evaluate thesenumerically following procedures similar to those used for the linearelement. It may be also mentioned that the strain-rate is not constant andvaries within the quadratic element.

Rectangular Element Family

For the rectangular family of elements, Xa and Ya in eq. (6.25) can bewritten as

where |J| is the determinant of the Jacobian matrix of eq. (6.15) and isexpressed by

For a quadrilateral element with the numbering of nodes shown in Fig.6.5c, Xa, Ya, and |J| can be expressed in the closed form as

By using procedures similar to those of rectangular element family, we canderive the differential operator as

where IJI , the determinant of the Jacobian matrix, is given by


and |J| is expressed by

where xu =x,-Xj and yu = y,- y,.For higher-order elements, it is easier to evaluate Xa and Ya numerically

according to eq. (6.30).

Three-Dimensional Brick Element

The strain-rate matrix, or Xa, Ya, and Za for a three-dimensional brickelement, can be derived by extending the derivation for the rectangularfamily elements. Rewriting eq. (6.22) in the matrix form, we have

with

The evaluation of Xa, Ya, and Za in eq. (6.36) can be made by using eqs.


(6.37) and (6.38). The strain rate matrix B becomes

Matrices of Effective Strain-Rate and Volume Strain-RateIn the finite-element formulation for the analysis of metal forming, theeffective strain-rate £ and the volumetric strain-rate EV are frequently used.Therefore, it is necessary to express the effective strain-rate and volumetricstrain-rate in terms of the strain-rate matrix.

The effective strain-rate is defined in terms of strain-rate components,according to eq. (4.39) in Chap. 4, as

or, in the matrix form

The diagonal matrix D has f and 3 as its components; corresponding tonormal strain-rate and engineering shear strain-rate, respectively. Substi-tution of eq. (6.26) into eq. (6.40) gives

where P = BTDB.The matrix D in eq. (6.40) takes different forms depending upon the

expression of the effective strain-rate, in terms of strain-rate components.For example, the effective strain-rate in plane-stress problems is expressedin a different form from that of plane-strain problems, although thedefinition of the effective strain-rate is identical in both cases. The matrixD written for plane-stress problems is not diagonal (Chap. 11). Theexpression of the effective strain-rate also depends on the yield criterion.Thus, the matrix D is different for anisotropic materials (Chap. 11) and forporous materials (Chap. 13), and is described in corresponding chapters.

The volumetric strain-rate £„ is given by

and expressed by

with C, = Bu + B2i + B3I, where Bu is an element of the strain-rate matrixB.


6.5 Elemental Stiffness Equation

It can easily be seen from the way in which the element was introducedthat the global integrals over the whole workpiece stem from the assemblyof integrals over the local domain of disjoint finite elements. Therefore, itis convenient to evaluate the stiffness matrix given by eq. (6.3) at theelemental level, and to assemble into a global stiffness matrix.

Of the two variational formulations (5.21) and (5.22) in Chap. 5, we firstdiscuss the penalty function method, eq. (5.22). Denote the first, second,and third terms (including signs) of eq. (5.22) dnD, dnp, 6j[Sp,respectively. In metal forming, the boundary conditions along the die-workpiece interface are mixed (see Section 5.4 of Chap. 5). Therefore,along the interface Sc the treatment of traction depends on the frictionrepresentation. This aspect of the surface integral terms is discussed inChap. 7.

Using the discrete representation of the quantities involved in dn thatare developed in Sections 6.3 and 6.4, we can express the integrals of dn interms of nodal-point velocities. Equation (6.3) then becomes

Evaluating stiffness matrices at the elemental level from eqs. (6.43) and(6.44), assembling them for the whole workpiece, we obtain a set ofsimultaneous linear equations (6.4).

When effective strain-rate e approaches 0, or becomes less than apreassigned value e0, we have, by eq. (5.31) in Chap. 5,

It should be noted that the term (—djtSp/dvj) is the applied nodal pointforce and that dnDl' dv, + djtp/dv, is the reaction nodal point force.

The second derivatives of n are expressed as

where


where OO/EO = constant. The derivatives of nD can be expressed by

The penalty constant K and the limiting strain rate £0 are introducedrather arbitrarily for computational convenience. However, proper choicesof these two constants are important in successful simulation of metal-forming processes. A large value of K is, in general, preferred to keep thevolume strain-rate close to zero. However, too large a value of K maycause difficulties in convergence, while too small a K results in unaccept-ably large volumetric strain. Numerical tests show that an appropriate Kvalue can be estimated by restricting volumetric strain rate to 0.0001—0.001times the average effective strain-rate.

The limiting strain-rate, e0, under which the material is considered to berigid, has been introduced to improve the numerical behavior of therigid-plastic formulation [7]. Too large a value of the limiting strain-rateresults in a solution in which the strain-rate of the rigid zone becomesunacceptably large. On the other hand, if we choose too small a value oflimiting strain-rate, then the convergence of the Newton-Raphson methoddeteriorates considerably. Numerical tests show that an optimum resultcan be obtained by choosing the limiting strain rate as -^ of the averageeffective strain-rate.

Equation (5.21) is the basic formulation using a Lagrange multiplier inorder to remove the incompressibility constraint. It replaces dnp in eq.(5.22) by <5jrA = J <5(Aew) dV. The Lagrange multiplier A is treated as anindependent variable and a function of material points denned over theworkpiece. The Lagrange multipliers are defined at points where thevolume constancy is enforced and are introduced at the reduced integra-tion points (see Section 7.1 of Chap. 7). For example, one variable isassigned for each linear rectangular [8] and three-dimensional brickelement, and four A are needed for a quadratic rectangular element.

Since 6A is also arbitrary in eq. (5.21), we obtain a set of simultaneousequations given by

The first equation of (6.47) is nonlinear and eq. (6.3) applies forlinearization. The additional terms necessary for evaluation of matrices inthe stiffness equation are

Then eqs. (6.3) and (6.47) can be arranged in the system of equations

Each coefficient appearing in eqs. (6.48) can be evaluated at the elementallevel and assembled into the global stiffness equation.

It is difficult to assess which method, the penalty function method or theLagrange multiplier method, is better in the finite-element implementa-tion. However, it can be mentioned that the Lagrange multiplier methodintroduces a larger number of independent variables. Also, the diagonalterms of the stiffness matrix corresponding to the A always become zeroand special attention is required during the assembly of the stiffness matrixso that an equation corresponding to any A does not become the first onein the stiffness equations after applying the boundary conditions.

References

1. Courant, R., (1943), "Variational Methods for the Solution of Problems ofEquilibrium and Vibrations," Bull. Amer. Math. Soc., Vol. 49, p. 1.

2. Clough, R. W., (1960), "The Finite Element Method in Plane Stress Analysis,"/. Struct. Div., ASCE, Proc. 2nd Conf. Electronic Computation, p. 345.

3. Bathe, K. J., and Wilson, E. L., (1976), "Numerical Methods in Finite-ElementAnalysis," Prentice-Hall, Englewood Cliffs, NJ.

4. Oden, J. T., (1972), "Finite Element of Nonlinear Continua," McGraw-Hill,New York.

5. Desai, C. S., and Abel, J. F., (1972), "Introduction to the Finite ElementMethod," Van Nostrand Reinhold, New York.

6. Hildebrand, F. B., (1974), "Introduction to Numerical Analysis," 2d Edition,McGraw-Hill, New York.

7. Chen, C. C., Oh, S. I., and Kobayashi, S., (1979), "Ductile Fracture inAxisymmetric Extrusion and Drawing, Part I: Deformation Mechanics ofExtrusion and Drawing," Trans. ASME, J. Engr. Ind., Vol. 101, p. 23.

8. Lee, C. H., and Kobayashi, S., (1973), "New Solutions to Rigid-PlasticDeformation Problems Using a Matrix Method," Trans. ASME, J. Engr. Ind.,Vol. 95, p. 865.

Metal Forming and the Finite-Element Method110

7THE FINITE-ELEMENT METHOD—PART II

7.1 Numerical IntegrationsNumerical integration is an important part of the finite-element technique.As seen in Section 6.5 of Chap. 6, volume integrations as well as surfaceintegrations should be carried out in order to represent the elementalstiffness equations in a simple matrix form. In deriving the variationalprinciple, it is implicitly assumed that these integrations are exact.However, exact integrations of the terms included in the element matricesare not always possible. In the finite-element method, further approxima-tions are made in the procedure for integration, which is summarized inthis section.

Integration Formula

Numerical integration requires, in general, that the integrand be evaluatedat a finite number of points, called Integration points, within the integra-tion limits. The number of integration points can be reduced, whileachieving the same degree of accuracy, by determining the locations ofintegration points selectively. In evaluating integration in the stiffnessmatrices, it is necessary to use an integration formula that requires theleast number of integrand evaluations. Since the Gaussian quadrature isknown to require the minimum number of integration points, we use theGaussian quadrature formula almost exclusively to carry out the numericalintegrations.

Consider a one-dimensional scalar function/(:c) defined in — l<x<l .In the Gaussian quadrature, the integration of f ( x ) can be evaluated [1] by

where xt is the coordinate of an integration point, w, is a weight factor, andthe summation is carried out over n (order of integration) integrationpoints. Assuming f(x) as a polynomial function, the formula given by eq.(7.1) provides the exact integration of polynomial of degree (2n — 1).

The optimum integration points and corresponding weight coefficientsare given in Table 7.1 [1]. The integration of higher-degree polynomialfunctions can be found elsewhere [1,2].

Ill


TABLE 7.1 Integration Points and Weight Factors of theGaussian Quadrature Formula [1]

n

1

2

3

4

X,

0

±0.577350269189626

±0.774 596 669 241 4830.000000000000000

±0.861 136311594053±0.339981043584856

w,

2.000000000000000

1.000000000000000

0.555 555 555 555 5560.888888888888889

0.3478548451374540.652 145 154 862 546

n = number of integration points.

x, = coordinate of integration points.

w, = integration weight factors.

The integration of a scalar function /(£, 77) defined by the naturalcoordinates (—1 < £ < 1, — 1 < 77 < 1) in the two-dimensional space can beobtained by applying the integration formula successively, namely,

and, in the three-dimensional space (-1 < g < 1, - l< j j< l , -!<£<!);

For a function f ( x , y) defined over an isoparametric element, integrationoff(x, y) can be evaluated as

where |/(£, T])\ is the determinant of the Jacobian of the coordinatetransformation matrix and given by eq. (6.34b).

For a function, f ( r , z), defined over an isoparametric axisymmetricelement, the volume integration over 1 radian is evaluated by

where /"(£/, f]j) is the radial position of the integration point.

The Finite-Element Method—Part II 113

The integration over the triangular family elements can be evaluated as

where (Lu, L2I, L3f) is the area coordinate of an integration point [3].The integration formula given by eq. (7.6) gives the exact integration of

a polynomial function of degree n. The integration points and weightfactors of the Gaussian quadrature formula for triangular elements arelisted in Table 7.2 [1].

Application of Integration FormulasThe use of Gaussian quadrature and isoparametric elements in FEMformulation will introduce errors. Usually, it is true that a higher-orderintegration is suggested in order to obtain accurate evaluations ofintegrands. But a large portion of the computer execution time is spent inperforming this numerical integration. Therefore, a proper choice shouldbe the lowest possible order of integration that does not introduce mucherror into the results.

The minimum order of integration that ensures convergence has beenestablished for different types of elements. For a linear rectangularelement, 2 x 2 integration points are sufficient to guarantee convergence.For a three-dimensional brick element, it is necessary to use 2 x 2 x 2integration points. The required integration order for quadratic elements is3x3. For the triangular element family, one-point integration is sufficientfor a linear element, while three-point integration is required for aquadratic element. It is also known that increasing the order of integrationabove the required minimum does not necessarily improve the solutionaccuracy. In determining the order of integration, possible overconstraintsimposed by the FEM formulation must be also considered. For example,the incompressibility requirement cannot be satisfied at all points in a

TABLE 7.2 Numerical Integration Formulas for Triangular Elements [1]

Number ofIntegrationPoints

1

3

4

IntegrationOrder

Linear

Quadratic

Cubic

Coordinates(^,, L2, L3)

i i i3> 3> 3

U,oo,UI n 12 > « > 2

1 1 13> 3' 3

0.6,0.2,0.20.2,0.6,0.20.2,0.2,0.6

WeightFactors

1ii3132748

254825482548


rectangular linear element except for uniform deformation [4]. In order torelieve this overconstraint, one-point integration is used for the volumetricstrain-rate term in a linear rectangular element. This is known as thereduced integration scheme. The reduced integration in a linear rectangu-lar element, in effect, imposes the volume constancy averaged over thatelement.

For higher-order elements, it is found that the integration order of thedilatational contribution may be reduced by 1 to obtain the best results.That is, the volumetric strain-rate term is integrated at 2x2 integrationpoints for a quadratic rectangular element and at one point for a quadratictriangular element. For the linear triangular element, which gives constantstrain-rate over the element, the reduced integration technique cannot beapplied. This is why the linear triangular element is not used for the case inwhich volume constancy is required. When the linear triangular elementwith the regular integration order is used for forming simulation, themodel tries to achieve the volume constancy averaged over severalelements. It has been shown that if four linear triangular elements arearranged to form a rectangle, then the volume constancy is satisfied overthe four elements within the rectangle [4]. It may be also mentioned thatthe number of Lagrange multipliers necessary to enforce volume constancyshould be the same as the number of the reduced integration points of theelement used.

The Gaussian integration points are the natural locations to evaluate thestrain-rate and stress within the element, since all the necessary informa-tion to calculate the strain-rate and stress are already evaluated at thesepoints. It is shown [5], however, that the calculated stress values at theregular integration points deviate considerably from the actual ones whenvolume constancy is enforced. On the other hand, stress and strain-ratesevaluated at the reduced integration points are known to be moreaccurate. Consequently, throughout this book, reduced integration pointsare used for stress and strain-rate evaluations.

The integrations of the boundary terms can also be evaluated by theGaussian quadrature. It is shown that a large number of integration pointsare necessary when the integrand contains higher-order polynomials. Thefriction term given by eq. (5.30) cannot be approximated by a low-orderpolynomial over the usual range of the sliding velocity [6] and it isnecessary to use a higher-order integration formula. It may sometimes bemore convenient to use Simpson's formula to calculate the frictionalenergy rate along the tool-workpiece interface [7], since this formula canbe applied to represent the local behavior of the integrand more accuratelyby subdividing the integration range. Simpson's formula is given by

where a and b are the limits of the variable of each subdivision.


7.2 Assemblage and Linear Matrix SolverElement AssemblageThe concept of assemblage can be simply explained. When the elementequations are assembled, the global constraints are that (1) the globalnodal-point velocities are identical to the nodal-point values of theelements surrounding that nodal point, and (2) the global nodal-pointforces are the sums of the elemental nodal-point forces. However, theassembly procedure requires extensive bookkeeping in computer im-plementation. In order to handle this extensive bookkeeping, it iscustomary to assign sequential numbers to nodes and elements, in order toidentify them in the finite-element method model. Figure 7.1 shows anexample of node and element numbering. Here, nodes are numbered 1through 30 and elements are numbered 1 through 19. To further identifythe nodes associated with each element, we define an array called theelement connectivity for each element. For example, the connectivity ofelement 8 is defined as (16,10,11,17), where the nodes identified bynumbers define the element. It is also customary to use a predefinedordering convention for a type of element in defining the connectivity.Ordering conventions for different types of elements are indicated inSection 6.3.

The velocity field that was constructed in Section 6.3 is defined in such away that the change of a velocity component at a node influences thevelocity field only over the elements that share the node. For example, if avelocity component of node 10 in Fig. 7.1 changes, then the velocity field isaffected only over the elements 4, 5, 7, and 8, while it is unaffected over allother elements. According to eq. (6.3) in Chap. 6, we have the globalstiffness equation corresponding to node 10 as

FIG. 7.1 Example of node and element numbering.


Since the stiffness matrix has been evaluated at the elemental level,evaluated elemental stiffness matrices should be assembled as shown in eq.(7.8).

Often the assembled stiffness equations are numbered in the order ofnodal-point numbering and the elemental stiffness equations are numberedin the order of element connectivity, so that the connectivity of eachelement can be used to correlate the corresponding equations andvariables in global and elemental stiffness equations.

It is also seen from eq. (7.8) that the stiffness equation corresponding tonode 10 contains the variables of nodes associated with the elementssurrounding node 10. That is, the equation contains the velocity com-ponents of nodes 5, 6, 7, 9, 10, 11, 15, 16, and 17. Therefore, the globalstiffness matrix is a sparse matrix because of the limited range of influenceof a node velocity on the admissible velocity field. The sparseness of theglobal stiffness matrix can be utilized to reduce the computational effort insolving the linear equations. Thus, the stiffness matrix is arranged in abanded matrix form, as shown in Fig. 7.2, by proper numbering of nodes.

Gaussian Elimination

Completing assemblage of elements, the global stiffness equation can bewritten in the form of eq. (6.4) in Chap. 6, namely,

The most common procedure of solving eq. (7.9) is Gaussian elimina-tion. The reader may refer to any book on linear algebra or thefinite-element method for the Gaussian elimination technique [for ex-ample, 8, 9].

In the solution process, solving linear matrix equations takes a substan-tial amount of computer time. It is therefore necessary to program a linearequation solver in the most efficient way. Depending upon the computerprogramming technique, the linear matrix solver may take several different

FIG. 7.2 Banded stiffness matrix.


forms. The simplest way is to store the stiffness matrix in a banded matrixform, and to apply Gaussian elimination over the maximum band width. Ina banded matrix solver, the same number of matrix elements is stored foreach equation. In the skyline solver, K is stored column-wise, from thediagonal matrix element to the last nonzero element in the column [8].

The number of operations necessary to solve a linear matrix can beestimated by ^nm\, where n is the total number of equations and mk is thehalf band width of the matrix. Here, one operation consists of onemultiplication (or division), which is almost always followed by anaddition. The estimated number of operations shows that the computertime necessary to solve the matrix equation is proportional to the square ofthe half band width. It is therefore a must to number the nodes in such away that the assembled stiffness equation has minimum band width.

The variables of the stiffness matrix can be "eliminated" when thestiffness matrix is assembled partially. That is, when the stiffness equationfor node / is completely assembled, then the corresponding variable can beexpressed by other variables, although the assembly is not complete. Thefrontal solution technique is a linear equation solver based on theelimination of variables during assembly. The frontal solution technique[10] is known to be advantageous when the stiffness matrix with minimumband width is still sparse. Such cases arise for the three-dimensionalfinite-element method. In the frontal method, element numbering, but notnode numbering, influences the matrix solution efficiency; therefore, newnodes can be added without renumbering the nodes.

7.3 Boundary Conditions

Since the boundary condition along the tool-workpiece interface Sc ismixed, it is convenient to write the boundary surface S in three distinctparts

Imposition of the traction boundary conditions on SF is straightforward.Recalling the boundary integral 6nSp or the first derivative of nSp withrespect to a node velocity component, the traction boundary condition isimposed in the form of nodal-point forces. It should be mentioned that thesame nodal-point force can be obtained with different tractiondistributions.

The velocity boundary conditions on Su are essential boundary condi-tions. In the finite-element discretization, the velocity boundary conditionis enforced only at nodes on Su, and the velocity along the element-side isdetermined automatically in terms of velocities of nodes and elementshape functions. For the node at which the velocity is defined, the velocitycorrection AuM is zero. Consequently, the corresponding stiffness equationshould be removed. The simplest way to implement this procedure is toreplace the corresponding rows and columns by zeros and to set the


diagonal term to 1, as shown below:

On the surface Sc, the traction is prescribed in the tangential directionand the velocity is prescribed in the normal direction to the interface.When the interface direction is inclined with respect to the globalcoordinate axis, the coordinate transformation of the stiffness matrix uponthe inclined direction is necessary in order to impose mixed boundaryconditions.

Consider a velocity vector v in the global coordinate system and thecorresponding vector v' in the inclined boundary coordinates. Then thevector is transformed from the global to the local coordinate system by

where d is measured from the x axis in the global coordinate system to thex' axis of the local coordinate system in counterclockwise direction. In thethree-dimensional coordinate system, the transformation matrix is ex-pressed by

where T^ is a direction cosine between the coordinate axis of xt and x'j. Thetransformation matrix for all nodes on the surface Sc can be constructed as

and the stiffness equation (7.9) is transformed to

where T is the coordinate transform matrix.Similarly, the nodal-point force vector f is transformed to f according to

In the two-dimensional coordinate system, the transformation matrix ofnode / is written as


The velocity boundary condition at the tool-workpiece interface is givenby

where UD is the tool velocity and n is the unit normal to the interfacesurface.

In the direction of the relative sliding between the die and theworkpiece, the frictional stress /, is prescribed as the traction boundarycondition. The frictional stress is usually represented according to theCoulomb law or as a constant frictional stress, as discussed in Chap. 3. Thefriction representation by a constant friction factor m (eq. (3.2) in Chap. 3)is approximated by eq. (5.30) in Chap. 5, in order to deal withneutral-point problems in metal forming.

Equation (5.30) expresses that the magnitude of the frictional stress isdependent on the magnitude of the relative sliding and that their directionsare opposite to each other. Then, the relationship can be written as

The approximation of the frictional stress by the arctangent function ofthe relative sliding velocity eliminates the sudden change of direction ofthe frictional stress mk at the neutral point. Figure 7.3 shows thearctangent function of the relative sliding as an approximate representationof the constant frictional stress. The figure shows that the frictional stressapproaches mk asymptotically as the relative sliding velocity us increases.However, the frictional stress, fs approximated by eq. (7.12) deviatesconsiderably from the value of mk as us approaches zero. It may be notedthat the value of u0 was introduced arbitrarily for performing numerical

RELATIVE S L I D I N G VELOCITY U s / D I E VELOCITY

FIG. 7.3 The arctangent function for the friction representation.


calculations and that the choice of u0 could have a significant influence onthe reliability of the solution. It is seen from the figure that the ratio us/u0

should be equal to or larger than 10 in order to attain the friction valuewithin 9% of the one originally intended. On the other hand, if we choosethe ratio too large, then the sudden change of the frictional stress near theneutral point can cause difficulties in numerical calculation [11]. Since theorder of magnitude of us is 0.1 (with the unit die velocity), a recommendedvalue for «0 is 10~3 ~ 10~4-

For the discretization, consider a die and an element that is in contactwith the die, as shown in Fig. 7.4. The boundary condition normal to thecontact surface is enforced at the contact nodes. Also, the relative slidingvelocity at the nodes vs can be evaluated. It should be noted, however,that the element-side cannot be made to conform to the die surface.However, it may be assumed that the relative sliding velocity us can beapproximated in terms of nodal-point values vsa by using a shape functionof the elements as

where the subscript a denotes the value at aih node.In deriving the stiffness equation, dn (see eq. (6.43) in Chap. 6) should

include the term dnSc, and the final form of the stiffness equation shouldcontain the terms

FIG. 7.4 An element in contact with the die.


and

in addition to those given in eqs. (6.43) and (6.44) in Chap. 6.The finite-element method approximation of the boundary conditions

introduces errors to the solution of the boundary-value problem. Note thatthe surface integration in eqs. (7.14) and (7.15) is carried out over theelement surface rather than the actual die surface. When linear elementsare used with a curved die, the interface area represented by elements isalways smaller than the actual interface area, and the effect of friction inthe analysis is always smaller than the actual. For deformation processesthat are sensitive to friction, this type of error could be quite serious.

Also, the velocity boundary condition imposed by the finite-elementmodel can be considerably different from that of the actual problem. Inexpressing the sliding velocity by eq. (7.13), it was assumed that themismatch angle between the element-side and the tangent direction of thedie at contact node (angle 6 in Fig. 7.4) is small. When this angle ofmismatch is large, the deformation mode is not modeled correctly. Theerrors resulting from the boundary conditions imposed by the finite-element method can be minimized by increasing the number of elements atthe boundary.

7.4 Direct Iteration MethodThe convergence of the Newton-Raphson method, which is described inSection 6.2 of Chap. 6, is usually very good, provided that the "initialguess" is in the vicinity of the solution. Also, the Newton-Raphsonmethod, in general, takes a small number of iterations if it is convergent.However, it may not be easy to obtain a good initial guess velocity whenthe deformation process is complex.

Another technique for solving a nonlinear equation is the direct iterationmethod [6,12]. In the direct iteration method, it is assumed that theconstitutive equation ((5.1c) in Chap. 5) is linear during each iteration.Then a/e in dnD/dv, in eq. (6.43) is assumed to be constant during eachiteration. The nonlinear friction term expressed by eq. (7.14) is alsoapproximated by a linear relationship (viscous friction coefficient) betweenthe frictional stress and the relative sliding velocity. Then the stiffnessequation resulting from dn = 0 becomes linear.

The computational procedures of the direct iteration method are asfollows:

1. Assign an assumed strain-rate e for each element. If a previoussolution or iteration is not available, assign a constant averagestrain-rate to each element. If a previous solution or iteration isavailable, then use the strain-rate obtained previously for eachelement.


2. Assign an assumed sliding velocity to each element-side that is incontact with a die. If a previous solution or iteration is not available,assign a constant average sliding velocity to all relevant element-sides. If it is available, use a sliding velocity that is obtained fromprevious solution or iteration.

3. Calculate alt at each integration point of the element, where a isevaluated for e assigned in step 1.

4. Calculate the viscous friction coefficient for each die contact sidefrom the linear relationship between frictional stress and the relativesliding velocity.

5. Evaluate the stiffness matrix and obtain a velocity solution.6. Calculate the strain-rate for each element by using the velocity

solution of step 5.7. Calculate the sliding velocity for each die contact element-side.8. Check whether solution converges, using convergence criteria.9. If the solution does not converge, go to step 3.

Figure 7.5 shows schematically the direct iteration solution procedure. Itmay be noted, from steps 1 and 2, that the direct iteration method doesnot require any "initial guess velocity." For metal-forming applications,the direct iteration method converges fast towards the solution during theearlier stages of iteration. However, as the solution point is approached,the convergence becomes very slow. It seems that the best computationalefficiency can be obtained by using (1) the direct iteration method forgenerating the initial guess and for cases where the Newton-Raphsonmethod does not converge, and (2) the Newton-Raphson method for allother cases.

NODE VELOCITY

FIG. 7.5 Schematic representation of the direct iteration method.


7.5 Time-Increment and Geometry UpdatingWhen the velocity solution is obtained, then the deformed geometry of theworkpiece, in the case of two dimensions, for example, can be obtained byupdating the coordinates of the nodes (Lagrangian mesh system) by

where (x,, y/) are the coordinates of node /, t0 is the time at currentconfiguration, and Af is the time-increment. The strain is updated in asimilar manner from the strain-rate solution. In general, the timeincrement At can be determined by considering several factors, such as thetime (A£die) necessary for a next free node to contact the die surface, adesired maximum strain-increment (Afstrain), and a maximum allowabletime-increment (A?a). The actual time-increment is determined by takingthe minimum of A£die, Afstrain, and Afa. The time necessary for a next freenode to contact the die can be determined by calculating these timeincrements for all free nodes and choosing the minimum time-increment.The time-increment required to limit the maximum strain-increment canbe readily obtained from strain-rate solutions. The maximum allowabletime-increment is given rather arbitrarily. However, consideration of theerror in the volume constancy is a factor for determining its magnitude.

During the time increment Af, elements lose volume after the geometryis updated. Consider a two-dimensional element, as shown in Fig. 7.6,where the element (1234) with a width of W and a height H is deformed tothe shape (12'3'4') after a time increment At. The volume constancyrequires that

FIG. 7.6 Two-dimensional uniform deformation of a rectangular element.


where the dot represents the time derivative. After the time increment Af,the volume change can be calculated by

where V and AF are the volume of the element and the amount of volumechange, respectively. Equation (7.18) suggests that updating by eq. (7.16)always results in a volume loss during a time-increment and that thevolume loss rate is proportional to the square of (A///H). Assuming that(A/////) is constant at all increments, we have, after n increments,

The total volume loss given by eq. (7.19) is plotted in Fig. 7.7 as a functionof total reduction in height and of &.H/H. It is seen from the figure that0.7% volume is lost at 50% reduction in height if (A/f///) is 0.01, and thevolume loss reaches 2.3% at 90% reduction in height with (AH/H) of0.01. Therefore, it is necessary to use (A/////) less than 0.01 in order tomaintain less than 2% volume loss at 90% reduction in height. It is alsoseen that volume loss increases considerably with larger step size.

In case of axisymmetric deformation, the volume of an element (solidcylinder) is represented by V — nR2H. The volume loss rate can be

FIG. 7.7 Volume loss during plane-strain FEM simulation as function of deformation forvarious step sizes of A/////.

or


FIG. 7.8 Volume loss during plane-strain FEM simulation as function of deformation forvarious step sizes of A///W0.

calculated by

Equation (7.20) shows that the volume loss in axisymmetric deformation isabout 25% less than that in plane-strain deformation.

Taking the step size proportional to the current height H, a total of 225steps are required to reach 90% reduction in height with (A///H) = 0.01.When the step size is controlled by a constant A////f,,, where H0 is theinitial height, the volume loss is accelerated as deformation proceeds. Asshown in Fig. 7.8, the effective (A///H) becomes larger as H is reduced.

The volume loss rate, due to geometry update with finite time-incrementestimated above, is based on uniform deformation. In the simulation ofpractical forming processes, the amount of volume loss will vary within theworkpiece. However, the estimated volume loss rate based on the uniformdeformation is a good reference for determining the maximum allowabletime-increment for simulation of practical forming processes.

Maintaining the volume loss within certain small percentage of the totaldeforming volume is a serious consideration in the prediction of proper diefilling and defect formation, which are important in process design.


7.6 RezoningIn practical forming processes, deformation is usually very large, and it isnot uncommon to encounter effective strain with magnitudes of 2 or more.Moreover, the relative motion between the die surface and the deformingmaterial is also large. Such large deformations and displacements cause thefollowing computational problems during FEM simulation with a Lagran-gian mesh:

• Difficulties in incorporating the die boundary shape into the FEMmesh, with increasing relative displacement between die andworkpiece

• Difficulties in accommodating the considerable change of deformationmode with one mesh system

• Formation of unacceptable element shapes with negative Jacobian dueto large local deformation

In order to overcome these difficulties, it is necessary periodically toredefine the mesh system [13,14]. The rezoning consists of two proce-dures. One is the assignment of a new mesh system to the workpiece andthe other is the transfer of information from the old to the new meshthrough interpolation. Generation of the new mesh is essentially the sameas the initial mesh generation performed by using a commercially availablemesh generator. The field variables, which depend on deformation history,are effective strains and temperatures, and they must be interpolated onthe new mesh.

Temperatures are given at nodal points in the finite-element method, asdiscussed in Chapter 12 for thermo-viscoplastic analysis. Thus, thetemperature distribution is expressed by using element shape functionsover the whole workpiece. Interpolation from the old mesh to the newmesh is done simply by evaluating the temperature at the new nodelocations.

Interpolation of effective strain requires an additional step, sinceeffective strains are given at the reduced integration point of each element.Therefore, before interpolation it is necessary to obtain the effective strainvalues at nodal points from the values given at the regular integrationpoints. Thus, effective strain distribution can be expressed by nodal-pointvalues and element shape functions.

For a linear rectangular element, one value of effective strain is given atthe center of each element. Among the various methods tested [13], itappears that the area-weighted average method is the most convenient andprovides sufficient accuracy for remeshing in metal-forming simulations. Inthe area-weighted averaging scheme, the nodal value is determined on thebasis of the average of the adjacent element values weighted by theassociated element size. Consider a node N surrounded by adjacentelements, as shown in Fig. 7.9. The nodal value of the effective strain at


FIG. 7.9 Node N surrounded by adjacent elements for area-weighted average.

node N can be obtained by

where EN is the effective strain value at node N and £7 is the effective strainvalue at the center of element / that surrounds the node N. AjN is the areacontribution of ;th element to node N and is denned by

where qN is the element shape function of element ;' at node N. Thesummation of eq. (7.21) is done over all the elements that surrounds thenode N. Once the effective strains are determined at all nodes, the straindistribution over each element can be denned by

where qa is the element shape function. The interpolation from the oldmesh to the new mesh is done by evaluating the effective strain value atthe integration point locations of the new mesh system.

As can be expected, the area-weighted average method, in general,provides very accurate interpolation results for internal nodes when thetrend of the field variable distribution is relatively well defined by thevalues at surrounding elements. The method often fails at the workpieceboundary when the field variable has large gradients, since the boundarynodes do not have a sufficient number of surrounding elements. However,for all practical purposes, the method provides sufficient accuracy formetal-forming simulations. Other methods that improve the accuracy ofinterpolation at the workpiece boundary have been suggested and arediscussed in Reference [13].

The rezoning procedure has been applied to simulations of various


RnoiusmmFIG. 7.10 FEM mesh assignment for rezoning.

metal-forming processes, and an example of a disk forging process withflash [15] is shown in Fig. 7.10. The figure shows the distorted mesh of thedisk forging process (upper half) at 65% reduction in height. It shows thatthe elements are highly distorted near the die corner radius between thebore and flange joint. Such a highly distorted grid requires remeshing.However, in this example, the remeshing procedure was necessary notonly because of the large element distortion, but also because of the lack

FIG. 7.11 Example of strain contours before and after rezoning.


of degrees of freedom near the flash. The new mesh is also shown in Fig.7.10 (bottom half). Note that the new mesh system allows far moredegrees of freedom near the flash. Interpolation of effective strain wascarried out by the area-weighted average method and the results are shownin Fig. 7.11. It is seen from the figure that the strain distributions arealmost identical before and after rezoning.

7.7 Concluding Remarks

In Chaps. 6 and 7', discretization of the basic equations for the finite-element method developed in Chap. 5 was formulated. The solutiontechniques for the finite-element formulations were explained, and impor-tant aspects of numerical procedures for metal-forming process analysiswere presented in some detail.

We did not discuss the programming of the method, because it is outsideof the scope of this book. However, since it is an important part of thefinite-element method, a simple FEM code with the program description isgiven in the Appendix.

In the following chapters, in which applications of FEM to variousmetal-forming processes are discussed, some finite-element formulationsare recapitulated and explained in further detail.

References

1. Zienkiewicz, O. C., (1977), "The Finite Element Method," 3d Edition,McGraw-Hill, New York.

2. Strang, G., and Fix, G. J., (1973), "An Analysis of the Finite ElementMethod," Prentice-Hall, Englewood Cliffs, NJ.

3. Cowper, G. R., (1973), "Gaussian Quadrature Formulas for Triangles," Int. J.Num. Meth. Engr., Vol. 7, p. 405.

4. Nagtegaal, J. C., Parks, D. M., and Rice, J. C., (1974), "On NumericallyAccurate Finite Element Solutions in the Fully Plastic Range," Comput. Meth.Appl. Mech. Eng., Vol. 4, p. 153.

5. Zienkiewicz, O. C., and Hinton, E., (1976), "Reduced Integration, FunctionSmoothing and Non-Conformity in Finite Element Analysis," /. Franklin Inst.,Vol. 302, p. 443.

6. Oh, S. I., (1982), "Finite Element Analysis of Metal Forming Problems withArbitrarily Shaped Dies," Int. J. Mech. Sci., Vol. 24, p. 479.

7. Lee, G. J., and Kobayashi, S., (1982), "Spread Analysis in Rolling by theRigid-Plastic Finite Element Method," Numerical Method in Industrial Form-ing Processes, p. 777.

8. Bathe, K. J., (1982), "Finite Element Procedures in Engineering Analysis,"Prentice-Hall, Englewood Cliffs, NJ.

9. Noble, B., (1969), "Applied Linear Algebra," Prentice-Hall, EnglewoodCliffs, NJ.

10. Irons, B. M., (1970), "A Frontal Solution Program for Finite ElementAnalysis," Int. J. Num. Meth. Engr., Vol. 2, p. 5.

11. Chen, C. C., and Kobayashi, S., (1978), "Rigid-plastic Finite Element MethodAnalysis of Ring Compression," Application of Numerical Method to FormingProcesses, ASME, AMD, Vol. 28, p. 163.


12. Lyness, J. F., Owen, D. R. J., and Zienkiewicz, O. C., (1974), "FiniteElement Analysis of Steady Flow of Non-Newtonian Fluid Through ParallelSided Conduits," Int. Symp. on Finite Element Method in Flow Problems,Swansea, p. 489.

13. Oh, S. I., Tang, J. P., and Badawy, A., (1984), "Finite Element MeshRezoning and its Applications to Metal Forming Analysis," Proc. of 1st ICTPConference, Tokyo, p. 1051.

14. Badawy, A., Oh, S. I., and Allan, T., (1983), "A Remeshing Technique forthe FEM Simulation of Metal Forming Processes," Proc. Int. Computer Engr.Conf., ASME, Chicago, IL., p. 143.

15. Ficke, J. A., Oh, S. I., and Malas, J., (1984), "FEM Simulation of Closed DieForging of Titanium Disk using ALPID," Proc. of NAMRC XII, Houghton,ML, p. 166.

8PLANE-STRAIN PROBLEMS

8.1 IntroductionThis chapter is concerned with the formulations and solutions for planeplastic flow. In plane plastic flow, velocities of all points occur in planesparallel to a certain plane, say the (jc, y) plane, and are independent of thedistance from that plane. The Cartesian components of the velocity vectoru are ux(x, y), uy(x, y), and uz = 0.

For analyzing the deformation of rigid-perfectly plastic and rate-insensitive materials, a mathematically sound slip-line field theory wasestablished (see the books on metal forming listed in Chap. 1). Thesolution techniques have been well developed, and the collection ofslip-line solutions now available is large [1]. Although these slip-linesolutions provide valuable insight into deformation modes and formingloads, slip-line field analysis becomes unwieldy for nonsteady-state prob-lems where the field has to be updated as deformation proceeds to accountfor changes in material boundaries. Furthermore, the neglect of work-hardening, strain-rate, and temperature effects is inappropriate for certaintypes of problems. Many investigators, notably Oxley and his co-workers,have attempted to account for some of these effects in the construction ofslip-line fields. However, by so doing, the problem becomes analyticallydifficult, and recourse is made to experimental determination of velocityfields, similarly to the visioplasticity method. Some of this work issummarized in Reference [2]. The applications of the finite-elementmethod are particularly effective to the problems for which the slip-linesolutions are difficult to obtain.

8.2 Finite-Element FormulationThe finite-element formulation specific to plane flow is recapitulated here.The variational form as a basis for finite-element discretization is given by

for the Lagrange multiplier method, where A is the Lagrange multiplier, or

131


FIG. 8.1 Quadrilateral element and natural coordinate system.

for the penalty function constraints, where K is a large positive constant.In equations (8.la) and (8.1b), EV is the volumetric strain-rate, and theeffective strain-rate t includes only nonzero strain-rate components,namely, ex, £y, and yxy.

A quadrilateral element (linear isoparametric element) is used fordiscretization and is shown in Fig. 8.1. The velocity field u is approximatedby shape functions in terms of nodal point velocity values as

where

with

according to eq. (6.13) in Chap. 6.For convenience of combined programming of plane-strain and axisym-

metric problems (see eq. (6.24b) in Chap. 6), the strain-rate vector isformed according to

then

Plane-Strain Problems 133

where Xa and Ya are expressed by eq. (6.35) for the node numberingshown in Fig. 8.1.

The volumetric strain-rate ev is expressed by

where

The effective strain-rate e, in a discrete form, is denned by

by putting P = BTDB.The matrix D in eq. (8.5) is given by

Then the basic equation, for example, eq. (8.1b) is discretized and a set ofnonlinear simultaneous equations (stiffness equations) is obtained from thearbitrariness of <5v as

with P, C, and N defined above. For the solution procedure, eq. (8.6) islinearized according to eq. (6.3) in Chap. 6.

8.3 Closed-Die Forging with Flash [11]

Forging in closed dies is an important operation in shaping metals intouseful objects. The process involves the compression of a billet, usuallybetween two dies. The metal flow is restricted to fill the closed die cavity,and excess material flows through the gap between the closing dies andforms a flash. In order to completely fill the die cavity, more material thanthe actual volume of the finish forged part is needed. The flash due to thisexcess material is subsequently trimmed from the forging.

At the finishing stage of a closed-die forging operation, the actual plasticdeformations are mainly at the flash portion, with a minor amount of flowof the forging material to fill the die cavity. Solutions at this stage,assuming that the cavity is filled and that material flows into the flash, werebased on the slab method [3,4,5]. The upper-bound method was alsoapplied to forging problems by several researchers [6, 7, 8]. In addition tothe mathematical analysis, experimental studies can be found in severalreferences. Kasuga et al. [9] observed the material flow in plane-strain and


FIG. 8.2 Schematic diagram of plane-strain closed-die forging.

axisymmetric rib-web type forgings. Lyapunov and Kobayashi [10] con-ducted experiments to examine the metal flow in plane-strain closed-dieforgings.

A schematic diagram of the apparatus used in the experiment byLyapunov and Kobayashi is shown in Fig. 8.2. In this study, a leadspecimen 50.8mm (2.0 in.) in height and width, with a square cross-section, is placed inside the cavity between the upper and lower dies. Thecross-section of the cavity, when two dies are closed, consists of a flangewith a tapered rectangular cross-section and a tapered shaft extended inthe axial direction.

The results obtained from this experiment, including grid patterns,velocity fields, load-displacement curves, flash dimensions, and the heightvariations of the specimen, were compared with the results of rigid-plasticfinite-element analysis [11].

The specimen used in the analyses is made of pure lead, which ischaracterized by a rigid-perfectly plastic (i.e., nonwork-hardening) mate-rial behavior with the constant flow stress Y0 assumed to be 2500 psi(17.236 MN/m2). Two extreme cases of friction conditions were con-sidered: the first assumed perfect lubrication at the interfaces of the dieand workpiece so that no frictional force exists; the second assumed asticking condition that implies that once the material touches the diesurface the material is completely adhered to the die.

The initial mesh system consists of 208 elements interconnected at 238nodal points. A very fine mesh was used for elements located below thefillet of the upper die and also for elements along the free surface. Thenonsteady-state forging process was analyzed in a step-by-step mannerwith a die displacement at each step that was 1% of the initial height of thespecimen. It is to be noted that additional elements were provided at thestage of flash formation.


The computed results were obtained in terms of grid distortion, velocitydistribution, load-displacement curve, flash geometry, and axial heightvariations.

Predicted and experimental velocity distributions, for the case of stickingfriction, are given in Fig. 8.3. It is seen that the computed velocitydistributions are in good agreement with the experimental observations.Material particles near the upper fillet move sideways in the earlier stagesand change direction downward in subsequent stages. The particles at thecore portion remain stationary until the flash is formed and then begin tomove upwards when the lower die cavity is almost filled.

When the side cavity of the upper die is essentially filled, the materialstarts to flow through the gap between the upper and lower dies, and asmall portion of the flash is formed. This metal flow into the flash enhancesthe filling of the lower die cavity. A comparison of grid distortions betweentheory and experiment at this stage is shown in Fig. 8.4. The figuredemonstrates that the agreement between theory (sticking) and experi-ment is good and reveals the deforming region at the stage of flashformation.

The study of closed-die forging suggested the need for (1) handling ofcomplex die geometry more efficiently, and (2) establishment of aremeshing scheme for simulations of severe deformation (such as flashformation).

FIG. 8.3 Velocity distributions in closed-die forging: (a) FEM (sticking friction); (b)experiment [10].


FIG. 8.4 Grid patterns in closed-die forging. Upper: experiment (33rd step) [10]. Lower:FEM (32nd step, dark portion indicates rigid elements).

A significant advance in analysis technique along these lines wasaccomplished by Oh [12] who improved the rigid-plastic finite-elementmethod to handle arbitrarily shaped boundary conditions and to generateautomatically an initial guess for a solution. These studies resulted in thedevelopment of a general-purpose program named ALPID.

A discussion of remeshing is presented in Section 7.6 of Chap. 7. Thescheme described in Chap. 7 was incorporated into the ALPID program byWu, Oh, and Altan [13]. Mori et al. [14] simulated the axisymmetricbackward extrusion stage of a forging sequence by using spatially fixedelements as a remeshing scheme.


8.4 Sheet Rolling

Rolling is one of the oldest processes used in the metal-working industry.In view of the tremendous amount and wide variety of rolled productsmanufactured each year, rolling can be considered to be one of the mostimportant forming processes. For more than half a century, numerousanalytical and experimental investigations have been carried out on rolling.The slab, the slip-line, and the upper-bound methods have been widelyused in theoretical analyses.

In recent years, several attempts have been made to solve the rollingproblem by using the finite-element technique. Zienkiewicz et al. [15] andDawson [16] have dealt with the rolling problem in terms of viscoplasticity;Dawson, moreover, incorporated temperature effects into the computa-tion. Shima et al. [17] analyzed several cases of rolling using therigid-plastic model; they did not include work-hardening behavior. All thestudies cited above used the plane-strain assumption, and no comparisonswere made between these computed results and experimentalobservations.

The plane-strain rolling problem was solved by the rigid-plastic finite-element method by Li and Kobayashi [18], on the basis of the infinitesimaltheory of plastic deformation. As a comprehensive investigation on rolling,a series of cases with different dimensions and material properties werenumerically analyzed and the results were compared with the experimentsfound in the literature. The process of rolling is shown schematically inFig. 8.5.

The process variables investigated were material properties, roll dia-meter, the initial and final workpiece thicknesses, and the frictionconditions. Most of the process conditions were taken from the literaturein order to compare computed and experimental results.

The analysis was performed using both nonsteady-state and steady-stateprocedures. For the nonsteady-state procedure, deformation of the work-piece was simulated in a step-by-step manner. The solution scheme for thesteady-state procedure is described in Chap. 10.

The mesh system and the boundary conditions used for the analysis areshown in Fig. 8.5, the frictional stress was assumed to be velocitydependent and represented by eq. (5.30) in Chap. 5.

The computations performed for each case resulted in the evaluations ofthe velocity field, the grid distortion pattern, the distributions of stresses,strain-rates and total effective strain, the normal pressure variation alongthe roll-workpiece interface, and the roll-separating force and the rolltorque.

Comparisons were made between the computed results and the ex-perimental data on contact pressure, roll separating force, and torque.Al-Slehi et al. [19] not only measured the roll separating force and the rolltorque, but also measured the contact pressure distribution and thecoefficient of friction with the pressure-pin technique.


For steady-state approach

FIG. 8.5 (a) Geometry and external forces in rolling, (b) Mesh systems used fornonsteady-state and steady-state computations.

The classical slab method gives the shape of the stress distribution,known as the "friction hill," with maximum pressure located at the neutralpoint, on both sides of which the pressure decreases monotonically.However, in some circumstances the pressure distribution curves wereobserved to have double peaks. A slip-line solution [20] also predicted thepressure drop in the middle of the arc of contact. In the computed results,double-peak pressure distribution curves, as well as friction-hill type


4 dec,

FIG. 8.6 Comparison of measured [19] and computed roll-workpiece contact pressuredistributions: (a) P = 1.79 and (b) P = 13.18 for aluminum; (c) P = 1.85 and (d) P = 11.57 forcopper.

distributions, were obtained. Figure 8.6 shows the pressure variations forsome typical cases, comparing the experimental and computed results.

An important parameter that determines the deformation mode is theratio of the roll-workpiece contact length to the initial workpiecethickness. This ratio is equivalent to the parameter P defined byP = R(H0-Hl)/Hl = (Reduction)(R/H0). In Fig. 8.6a, the pressure dis-tribution curve shows maxima near entrance and exit with a pressure dropin the middle of contact length. This is characteristic for small values of theparameter P. For large values of the parameter P, Fig. 8.6b shows afriction-hill type pressure distribution. There is considerable discrepancy inmagnitude between the computed values and the experimental measure-ments. The same observations apply to the pressure distributions for adifferent material given in Figs. 8.6c and d. As typified by the resultsshown in Fig. 8.6, two modes of deformation are observed. In one thecontact pressure distribution shows double peaks and deformation is moreinhomogeneous, and in the other the pressure distribution is a friction-hilltype and homogeneous deformation is dominant.

For the roll separating force and the torque in rolling aluminum andcopper, the comparisons between the computed results and the ex-perimental values [19] are shown in Fig. 8.7. The computed results for theroll separating force are smaller than the measured values, while the


FIG. 8.7 Comparison of measured [19] and computed roll separating force and roll torque;(a) and (b) for aluminum; (c) and (d) for copper.

computed torque values generally overestimate the actual values. Similarcomparisons of the roll separating force and the torque for steel [21] areshown in Fig. 8.8. In these cases, the predictions are lower than theexperimental values.

Discrepancies between the computed and measured values of contactpressure, roll separating force, and torque shown in Figs. 8.6-8.8 may beattributed to the rigid-roll assumption in the finite-element analysis and tothe uncertainty in friction modeling. It is a known fact that roll flattening is

Reduction

Plane-Strain Problems

R / H 0 = 12.5 R/H0 - 39

141

FIG. 8.7 (continued)

an important aspect in the analysis of rolling and contributes to an increasein the contact pressure; hence the actual roll separating force should belarger than that obtained with the assumption of rigid-rolls.

8.5 Plate Bending

The air bending operation is a simple but widely used process in thesheet-metal industry. Air bending implies that the shape taken up by themetal depends on the punch position and not on the die shape. Althoughthe process is simple, the bending operation presents several technical


FIG. 8.8 Comparison of measured [19] and computed roll separating force and roll torquefor steel; (a) and (b) with R/Ha = 65; (c) and (d) with R/Hn = 130.

problems, such as the limitations in flexibility and the deviations in bendangle, due to elastic spring-back. There are also shape inaccuracies thatarise from edge distortion and punch shape.

Several analytical solutions for sheet bending and spring-back have beenpublished, but they are usually based on simple beam theory and includean assumption about displacement distribution in the thickness direction orin the plate direction, or in both. A different approach to this problem is toapply plate bending theory, or shell theory, by using the finite-elementmethod [22]. Another approach [23] is to use the finite-element method byconsidering the sheet as a bulk material. Hibbit et al. [24], using thefinite-element formulation, solved a three-point bending problem withcounter-pressure at the bottom of a sheet.

Reduction



The analysis of sheet bending as a bulk deformation process wasperformed by the rigid-plastic finite-element method using the incrementaltheory, and by the elastic-plastic finite-element method with large-strainformulation [25]. Some of the rigid-plastic loading solutions are shownhere and compared with those of the elastic-plastic solution. In Chap. 16,trie large-strain formulation for the elastic-plastic finite-element method isdescribed and more detailed comparisons of the two solutions are given.

Sheet bending, as shown in Fig. 8.9, is considered to take place underthe plane-strain condition. The punch and the die are assumed to be rigid.The workpiece material is aluminum alloy 2024-0 and isotropic. Thenormalized dimensions (taking the workpiece thickness T = 1) are: dieopening, W = 7.5, punch radius, RP = 1.8, and die corner radius, RD =1.2. It is assumed that there is no frictional stress acting along the punchand die surfaces.


FIG. 8.9 (a) Schematic diagram of sheet bending operation used in the computation withT = l, W = 7.5, RP = 1.8, RD = 1.2. (b) Boundary conditions in sheet bending.

Since the process is symmetric with respect to AB in Fig. 8.9b, theanalysis is performed for one half of the workpiece. The boundaryconditions along AB are u x=0 and Fy=Q, where ut,Fi denote velocity andtraction respectively. Along GH, where the circular punch and workpieceare in contact, un = Up cos 9 and F/ = 0. The subscript n denotes thedirection normal to the circular arc GH, / is the tangential direction, andUP is the punch velocity.

On the free surfaces, BC, DE, EF, FG, HA, the boundary conditionsare simply Fx = 0 and Fy=0. The boundary conditions on the surface, CD,are written as «„ = 0 and Ft — 0.

Solutions during loading were obtained for the punch loads, the stress


and strain distributions, and the deformed geometries at each stage of theprocess.

Figure 8.10 compares the grid distortion patterns obtained from rigid-plastic and elastic-plastic formulations. The black area of the workpieceunder the punch represents the deforming zone in the rigid-plastic analysisand the plastic zone in the elastic-plastic analysis. From the elastic-plasticcalculations, it can be seen that yielding begins at the outer fiber near thebending axis and spreads gradually toward the rest of the sheet. The trend

Elasto-Plastic Rigid-Plastic

FIG. 8.10 Comparison of grid distortions by rigid-plastic and elastic-plastic (elasto-plastic)formulations for punch displacements: (a) 0.6, (b) 1.2, (c) 1.8, (d) 2.25, (e) 2.8, (f) 3.2. Thedarkened area denotes the plastically deforming (plastic) zone.


is the same as that predicted by the elementary beam theory. As theprocess continues, a well-defined wedge-shaped elastic-plastic boundarymoves away from the bending axis. But, after a certain punch displace-ment, the boundary remains nearly stationary. The rigid-plastic calculationdoes not show the well-defined continuous spread of the plastic zone. Itcan be seen, however, that the size of the deforming zone determined bythe rigid-plastic analysis compares well with that of the plastic zoneobtained in the elastic-plastic analysis. It can also be seen from the figurethat unloading takes place near the bending axis during the later stages ofdeformation in the elastic-plastic case, and that the rigid zone appearsduring the same stage in the rigid-plastic case.

Another interesting feature in the deformation mode of sheet bending isthat the bend radius of the workpiece does not follow that of the punch;the workpiece separates from the punch and only a small portion of theworkpiece is in contact with it, resulting in shape inaccuracies in the bentsheet. This is shown quantitatively in Fig. 8.11, where the bend angle andthe clearance between the punch pole and the workpiece are plottedagainst the punch displacement for both analyses. The figure shows thatthe curves for bend angle vs. punch displacement obtained from the twoformulations are in almost perfect agreement. From the figure it can beseen that workpiece separation from the punch pole occurs when thepunch displacement reaches 1.48 according to the elastic-plastic calculationand 1.68 according to the rigid-plastic analysis. The clearances that arecalculated from the two formulations show the same trend, but therigid-plastic formulation results in slightly larger clearance than that ofelastic-plastic analysis.

PUNCH DISPLACEMENT

FIG. 8.11 Relationship of bend angle to punch displacement, and of clearance between thepunch poles and the workpiece to punch displacement.


Elasto-Plastic Rigid-PlasticFIG. 8.12 Comparison of distributions of normalized bending stress (as/initial yield stress)and effective strain by rigid-plastic and elastic-plastic (elasto-plastic ) formulations at the endof loading (punch displacement = 3.2).

Figure 8.12 shows the distributions of bending stress and the effectivestrain at the end of loading, for rigid-plastic and elastic-plastic materials.From the figure it can be seen that the bending stress increases toward theouter surface and toward the punch axis, as expected. It can be also seenthat the neutral line (os = 0) shifts toward the punch near the punch axisand is at about half of the thickness in other locations. The effective straindistributions show a similar pattern. The figure also reveals that thedistributions of bending stress and effective strain for both analyses agreewith each other very well, with minor differences.

Unloading and resulting spring-back and residual stresses are discussedin Chap. 16.


8.6 Side PressingIn side pressing, a long circular cylinder is compressed sideways (perpen-dicular to the cylinder axis) between two flat dies. This process has beenshown to be very useful in studying flow behavior at hot workingtemperatures. Experiments show that hot isothermal side pressing of(a + /J)Ti-6242 alloy, for example, leads to stable deformation, while thatof ()8)Ti-6242 produces severe shear bands and unstable flow. This is dueto two completely different deformation modes and to the differencesbetwen the flow stress behaviors of the two structures; that is, the (a + /3)microstructure exhibits a stable strain-rate hardening, while the (/3)microstructure shows substantial amounts of strain softening behavior.

In order to estimate the strain concentration, an isothermal rigid-viscoplastic finite-element method (see Chap. 12 for viscoplastic analysis)was used to simulate the side pressing of cylinders with two different flowstress characteristics corresponding to the two microstructures [26]. The

(a)

(b)

FIG. 8.13 ^umparison of predicted strain-rate (per second) distributions and experimentallydetermined flow localization in side pressing of Ti-6242-0.1Si at 913°C (1675T): (a) (a + ft)microstructure; (b) (/J) microstructure [26],


diameter of the specimen was 10.2 mm (0.40 inch) and the deformationwas assumed to be plane-strain (zero strain in the axial direction). Thetemperature during deformation was 913°C (1675°F). Figure 8.13 showsthe experimentally observed transverse sections of side-pressed cylindersof (a + /?) and (/3) microstructures and the predicted effective strain-ratedistributions for both cases. It can be seen that the method effectivelypredicts, for the same alloy and forming conditions, detailed variations inmetal flow that are due to differences in microstructure and flow stressbehavior.

References

1. Johnson, W., Sowerby, R., and Venter, R. D., (1982), 'Plane-Strain Slip-LineFields for Metal Deformation Processes," Pergamon Press, Oxford.

2. Oxley, P. L. B., and Hastings, W. F., (1976), "Minimum Work as a PossibleCriterion for Determining the Frictional Conditions at the Tool/Chip Interfacein Machining," Phil. Trans. R. Soc., London, Series A. Vol. 282, p. 565.

3. Kobayashi, S., Herzog, R., Lapsley, J. T., Jr., and Thomsen, E. G., (1959),"Theory and Experiment of Press Forging Axisymmetric Parts of Aluminumand Lead," Trans. ASME, J. Engr. Ind., Vol. 81, p. 228.

4. Kobayashi, S., and Thomsen, E. G., (1959), "Approximate Solutions to aProblem of Press Forging," Trans. ASME, J. Engr. Ind., Vol. 81, p. 217.

5. Akgerman, N., and Altan, T., (1972), "Modular Analysis of Geometry andStresses in Closed-Die Forging: Application to a Structural Part," Trans.ASME, J. Engr. Ind., Vol. 94, p. 1025.

6. Johnson, W., (1958), "Over-Estimates of Load for Some Two-DimensionalForging Operations," Proc. 3d U.S. Congr. Appl. Mech., ASME (New York),p. 571.

7. Kudo, H., (1958), "Studies on forging and Extrusion Process, I," Koken-shuho, Tokyo University, Vol. 1, p. 37.

8. McDermott, R. P., and Bramley, A. N., (1974), "An Elemental Upper-BoundTechnique for General Use in Forging Analysis," Proc. 15th Int. MTDRConference, Birmingham, England, p. 437.

9. Kasuga, Y., Tsutumi, S., and Saiki, H., (1974), "Material Flow in SunkenForging Dies," /. Japan Soc. Tech. Plast., Vol. 15, p. 147.

10. Lyapunov, N. L, and Kobayashi, S., (1974), "Metal Flow in Plane-StrainClosed-Die Forging," Proc. of 5th North Amer. Metalworking Res. Conf.NAMRC, p. 114.

11. Chen, C. C., and Kobayashi, S., (1980), Rigid-Plastic Finite Element Analysisof Plane-Strain Closed-Die Forging, "Process Modeling," ASM, Metals Park,Ohio, p. 167.

12. Oh, S. I., (1982), "Finite Element Analysis of Metal Forming Processes withArbitrary Shaped Dies," Int. J. Mech. ScL, Vol. 24, p. 479.

13. Wu, W. T., Oh, S. L, and Altan, T., (1984), "Investigation of DefectFormation in Rib-Web Type Forging by ALPID," Proc. 12th NAMRC, p.159.

14. Mori, K., Osakada, K., and Fukada, M., (1983), "Simulation of Severe PlasticDeformation by Finite Element Method with Spatially Fixed Elements," Int. J.Mech. ScL, Vol. 25, p. 775.

15. Zienkiewicz, O. C., Jain, P. C., and Onate, E., (1978), "Flow of SolidsDuring Forming and Extrusion: Some Aspects of Numerical Solutions," Int. J.Solids Structures, Vol. 14, p. 15.


16. Dawson, P. R., (1978), Viscoplastic Finite Element Analysis of Steady-StateForming Processes Including Strain History and Stress Flux Dependence,"Applications of Numerical Methods to Forming Processes," ASME, AMD,Vol. 28, p. 55.

17. Shima, S., (1980), "Rigid-Plastic Finite Element Analysis of Strip Rolling,"Proc. 4th Int. Conf. on Prod. Eng., p. 82.

18. Li, G.-J., and Kobayashi, S., (1982), "Rigid-Plastic Finite Element Analysis ofPlane Strain Rolling," Trans. ASME, J. Engr. Ind., Vol. 104, p. 55.

19. Al-Salehi, F. A., Firbank, T. C., and Lancaster, P. R., (1973), "AnExperimental Determination for the Roll Pressure Distributions in ColdRolling," Int. J. Mech. ScL, Vol. 15, p. 693.

20. Firbank, T. C., and Lancaster, P. R., (1965), "A Suggested Slip-Line Field forCold Rolling with Slip Friction," Int. J. Mech. ScL, Vol. 7, p. 84.

21. Shida, S., and Awazuhara, H., (1973), "Rolling Load and Torque in ColdRolling," /. Japan Soc. Tech. Plast., 14, p. 267.

22. Popov, E. P., Khojesteh-Bakht, M., and Yaghmai, (1967), "Analysis ofElastic-Plastic Circular Plates,"/. Eng. Mech. Div. Proc., ASCE, p. 49.

23. Cupka, V., Miyamoto, H., Miyoshi, T., and Suzuki, K., (1974), "Analysis ofCounterpressure Bending by FEM," Proc. Int. Conf. Prod. Engr., p. 257.

24. Hibbitt, H. D., Marcal, P. V., and Rice, J. R., (1970), "A Finite ElementFormulation for Problems of Large Strain and Large displacement," Int. /.Solids Structures, Vol. 6, p. 1069.

25. Oh, S. I., and Kobayashi, S., (1980), "Finite Element Analysis of Plane-StrainSheet Bending," Int. J. Mech. ScL, Vol. 22, p. 583.

26. Oh, S. I., Lahoti, G. D., and Allan, T., (1981), "ALPID—A General PurposeFEM Program for Metal Forming," Proc. NAMRC-IX, State College,Pennsylvania, p. 83.

9AXISYMMETRIC ISOTHERMAL FORGING

9.1 Introduction

According to Spies [1], the majority of forgings can be classified into threemain groups. The first group consists of compact shapes that haveapproximately the same dimensions in all three directions. The secondgroup consists of disk shapes that have two of the three dimensions (lengthand width) approximately equal and larger than the height. The thirdgroup consists of the long shapes that have one main dimension sig-nificantly larger than the two others. All axially symmetric forgings belongto the second group, which includes approximately 30% of all commonlyused forgings [2]. A basic axisymmetric forging process is compression ofcylinders. It is a relatively simple operation and thus it is often used as aproperty test and as a preforming operation in hot and cold forging. Theapparent simplicity, however, turns into a complex deformation whenfriction is present at the die-workpiece interface. With the finite-elementmethod, this complex deformation mode can be examined in detail. In thischapter, compression of cylinders and related forming operations arediscussed. Since friction at the tool-workpiece interface is an importantfactor in the analysis of metal-forming processes, this aspect is also givenparticular consideration. Further, applications of the FEM method forcomplex-shaped dies are shown in the examples of forging and cabbaging.

9.2 Finite-Element Formulation

Finite-element discretization with a quadrilateral element is similar to thatgiven in Chap. 8. The cylindrical coordinate system (r, 9, z) is used insteadof the rectangular coordinate system. The element is a ring element with aquadrilateral cross-section, as shown in Fig. 9.1.

The £ and rj of the natural coordinate system vary from -1 to 1 withineach element. An arbitrary point (r, z) inside the element can beexpressed in terms of the natural coordinate (£, rj), and the coordinatetransformation is expressed by

151


FIG. 9.1 Quadrilateral element and natural coordinate system.

where ra, za (a = 1, . . . ,4) are the positions of the four surrounding nodalpoints of an element in the global coordinate system. For an isoparametricelement of Fig. 9.1, discussed in Chap. 6, transformation functions qa ineq. (9.1) are the same as shape functions defined by eq. (8.2b) in Chap. 8.

Then

and the matrix IMT is identical to eq. (8.2a) in Chap. 8.The strain-rate vector is defined (see eq. (6.24c) in Chap. 6) by

where

The strain-rate matrix B in eq. (9.3) is given by

Axisymmetric Isothermal Forging 153

and

and

Equations (9.4b) and (9.4c) are obtained by replacing (x, y) in eqs. (6.35a)in Chap. 6 by (r, z).

The determinant of the Jacobian of transformation |J| is expressed byeq. (6.35b) in Chap. 6 with (r, z) replacement. The matrix D and thevector C in the expressions of the effective strain-rate and the volumetricstrain-rate are the same as those given by eqs. (8.4) and (8.5) in Chap. 8.

The stiffness equations based on, for example, eq. (8.la) in Chap. 8,become

and

where the subscript j indicates the element number and M is the totalnumber of elements. Linearization of eq. (9.5) is made according to eq.(6.48) in Chap. 6.

9.3 Compression of Solid Cylinders and Heading of CylindricalBars

In compression of circular, solid cylinders between parallel, flat dies, thedeformation is homogeneous when there is no friction, but with frictionthe distribution of the compressive stress is nonuniform, and the averagecompression stress differs from the flow stress. Furthermore, the freesurface barrels, and fracture may occur at the barreled surface.

The complexity of nonuniform deformation is not only represented bybarreling of the free surface, but is further indicated by the fact that a partof the initially free surface comes into contact with the die duringcompression. This flow phenomenon is known as "folding." The mode ofdeformation is also influenced by the workpiece geometry. With specimensof small height-to-diameter ratio (say, less than 1.6), barreling results in asingle bulge, the maximum diameter being at the equatorial surface. For


specimens of larger height-to-diameter ratio, however, the mode ofdeformation is different and a double bulge is sometimes observed alongthe free surface [3, 4].

Applications of the elastic-plastic finite element method to this problem,based on the infinitesimal deformation theory, revealed that the analysisdid not produce the phenomenon of folding. This shortcoming was themain motivation for developing the finite-element formulation based onrigid-plastic material behavior [5]. The method was applied to obtaincomplete rigid-plastic solutions in upsetting solid cylinders under variousinterface friction conditions by using the work-hardening property of SAE1040 steel at room temperature [6]. The detailed mechanics were investig-ated by this method and the theoretical results predicted the observedfolding phenomenon remarkably well, as seen in Fig. 9.2.

An important objective of the analysis of simple compression is topredict the possible occurrence of fracture at the free surface. Since thefracture condition can be observed for cracks occurring at free surfaces,upsetting of cylindrical specimens has been used extensively by severalinvestigators for the study of ductile fracture and workability of materials.

Workability of materials implies the extent to which materials are ableto deform without cracks during a mechanical working process. Kudo andAoi [3], in their investigation of carbon steel, directly measured theequatorial free surface strains in upsetting of solid cylinders. The frictioncondition at the interface was controlled qualitatively by using grooveddies and conical dies, with and without lubrication. Stresses at the freesurface were derived from measurements of the strains, and the fracturewas observed for various stress and strain histories at the critical site.

With the finite-element method, strain paths of a critical site can becomputed under various friction conditions, and they are compared withexperimental results in Fig. 9.3. Combining the observed fracture condi-tions and the computed results on deformation characteristics, it waspossible to predict workability in terms of reduction in height at fracturefor various friction conditions and initial workpiece dimensions, as shownin Fig. 9.4.

The analysis of compression of tall cylinders required the developmentof a method that could handle the treatment of rigid zones. With thisdevelopment the analysis was performed for compression of cylinders withheight-to-diameter ratio larger than unity [7]. The grid distortions at 36and 66% reductions in height for H0/D0 = 2.5 are shown in Fig. 9.5. Thedouble-bulge formation on the free surface, typical of compression of tallcylinders, is quite noticeable at a reduction of 36%.

Rigid zones are also present in the heading process, which is describedin Chap. 2. Although cold heading is a very common metal-workingoperation, only a few theoretical studies have been undertaken toinvestigate the details of this process [8-10]. The finite-element solutionswere obtained and some of the computed results were compared withexperiments [11].


FIG. 9.2 Grid distortions in simple compression at 50% reduction in height for the twofriction conditions.

The deformation pattern during the heading operation can be observedin detail by examining the effective strain distributions, which are plottedon the right-hand side in Fig. 9.6 at 60% reduction in height. An importantfeature of the strain distributions predicted at various stages is that thepattern does not seem to change much as the reduction in height increases.As seen in the figure, there is a strain concentration at the edge of thecontact surface on the top die, and on the contact surface at the bottom

FIG. 9.3 Strain paths of a point at the equatorial free surface in simple compression forinitial height-to-diameter ratio of 1.0 [6].

156


FIG. 9.4 Workability of SAE 1040 (annealed) steel at room temperature in slow-speedupsetting.

die. Otherwise, the maximum strain is near the axis and almost atmid-height of the upset head. Since hydrostatic pressure seems to play animportant role in ductile fracture, it is instructive to examine thedistribution of am. This distribution is plotted on the left-hand side of Fig.9.6. Although the pattern of the effective strain distribution remainsalmost unchanged with increasing reduction in height, the hydrostaticstress distribution changes a great deal. A very important observation,from the fracture point of view, is that with increasing reduction in heightthe hydrostatic pressure increases near the axis of the specimen, whereas itdecreases near the free surface. In fact, the hydrostatic stress becomestensile at higher reductions. This explains why, in heading operations,

66 Percent Reduction

FIG. 9.5 Grid distortions in simple compression at 36 and 66% reductions in height for initialheight-to-diameter ratio of 2.5.

FIG. 9.6 Effective strain (e) contours and hydrostatic pressure (—a m /Y 0 ) contours in headingat 60% reduction in height.

158


FIG. 9.7 Experimental and theoretical flow lines in heading at 31% reduction, 47%reduction, and 63% reduction (60%, theory) [11].

surface cracking is the predominant failure mode rather than the formationof internal cracks.

A significant comparison between theory and experiment can be made interms of the grid distortion pattern. The flow patterns obtained ex-perimentally by etching the formed specimens are shown in Fig. 9.7 atapproximately 31%, 47%, and 63% reductions in height. The figure alsoshows the grid patterns obtained theoretically at 31%, 47%, and 60%reductions. It can be seen that the distortion of axial lines predicted by thetheory is almost identical to the flow lines observed experimentally.

9.4 Ring compressionWhen a short ring specimen is compressed between two flat, parallelplatens (Fig. 9.8a) the diameter of inner surface either increases ordecreases as the height of the specimen is reduced, depending on thefriction condition at the interface. The inner diameter of the ring is


(C )

FIG. 9.8 Two deformation modes in ring compression, (a) Compression geometry; (b)situation with low interface friction; (c) situation with high interface friction.

increased if the interface friction is low (Fig. 9.8b) and it is decreased ifthis friction is high (Fig. 9.8c). Because the change in internal diameter ofthe compressed ring is sensitive to friction at the die-workpiece interface,ring compression has been widely used as a test to evaluate the frictioncondition in metal-forming processes.

To determine the friction condition quantitatively, however, the re-lationship between the geometrical change of the workpiece and thefriction condition at the tool-workpiece interface must be established.Since Kunogi [12] analyzed the process mathematically, this aspect of ringcompression has been the subject of many investigations in the past (seeAvitzur [13]). Most of the mathematical methods are based on anapproximate stress analysis or on the upper-bound analysis using relativelysimple velocity fields. While these theories provided useful information,more accurate and realistic solutions were needed for wider and meaning-ful application of the ring test in practice.

In implementing the finite-element method for analyzing metal-formingproblems, particular attention must be paid to boundary conditions. There


is a class of problems such as ring compression, rolling, and forging, inwhich there exists a "neutral" point (or a region) along the die-workpieceinterface. At this point the velocity of the deforming material relative tothe die velocity becomes zero, and the frictional stress usually changes itsdirection. As discussed in Chap. 5, Chen and Kobayashi [14] used thevelocity-dependent frictional stress expressed by eq. (5.30) for analyzingthese problems. This approach was successful in analyzing plane-strainsheet rolling (Chap. 8) and also ring compression. The finite-elementsolution of ring compression was obtained in terms of geometrical changesof the specimen, velocity distribution, grid distortions, and stress andstrain distributions under various friction conditions for several materials.Some of the computed results are shown in Fig. 9.9 in terms of changes ofminimum internal diameters as functions of reduction in height. The figureshows that the geometrical changes differ for the same m value, depending

REDUCTION IN HEIGHT,(H0-H)/HO x IOO%

FIG. 9.9 Comparison of the finite-element solution with the upper bound solutions for ringcompression. Experimental data: O, copper; •, copper; A, pure aluminum [17].


upon the selection of the initial or current shear yield stress k. It is alsoseen that the geometrical changes are affected by the material property, asindicated by the curves calculated with ra = 1.0 for copper and forannealed aluminum.

The problem of ring compression was also analyzed by the elastic-plasticfinite-element method. Hartley et al. [15,16] developed a method forintroducing friction into the finite-element analysis by the inclusion of alayer of elements. The stiffness of these elements was modified by afunction of the interfacial shear factor and the method was applied to theanalysis of ring compression.

FIG. 9.10 Grid distortions in ring compression at 50% reduction in height for variousm-values: (a) m = 0.12 (pure aluminum); (b) m=0.25 (annealed Al 1100); (c) m = 0.6(copper), (d) m - 1.0 (copper).


The finite-element method not only provides overall quantities, such aschanges in specimen dimensions and load-displacement curves, but it alsoreveals the details of the deformation characteristics. Figure 9.10 showsgrid distortions at 50% reduction in height under various interface frictionconditions. At low friction, the outward flow dominates, but withincreasing friction, the neutral flow point appears along the die-workpieceinterface, resulting in both inward and outward barreling. Furthermore,for high friction, folding occurs at the inner surface as well as the outersurface. The deformation characteristics shown in Fig. 9.10 are inqualitative agreement with experimental observations reported in theliterature [17].

9.5 Evaluation of Friction at the Tool-Workpiece Interface

In analyzing the mechanics of metal-forming processes, a realistic frictioncondition must be specified in order for a theory to yield a reliablesolution. Also, in practice, understanding and controlling friction oftenlead to successful metal-forming operations. Consequently, considerableeffort has been devoted to determining satisfactorily the friction conditionin various metal-forming processes [18]. Yet, the mechanism of friction isstill not well known and, as Thomsen [19] has pointed out, friction remainsone of the most elusive variables in metal forming technology.

It has been demonstrated that in simple upsetting, for an assumedfrictional stress distribution, the finite-element method is capable ofaccurately computing, among other details of the deformation mechanics,the relative displacement distributions at the die-workpiece interface. Thissuggests that the method also permits, in turn, the calculation of thefrictional stress distribution when the displacement distributions at theinterface are measured and used as the boundary condition for computa-tion [20]. Such a study has been conducted. The experiments consisted ofmeasuring the displacement distributions at the die-workpiece interface atvarious reductions in height when upsetting solid circular cylindersbetween two rigid, flat, and parallel dies. The specimens of 20.32mm(0.8 in.) diameter and 20.32mm (0.8 in.) height were prepared from analuminium alloy 7075-T6. A series of indentation marks with 1.016mm(0.04 in.) spacing were made diametrically at one end of the specimenusing the Vickers Micro-hardness Tester, as shown schematically in Fig.9.1 la. Also, short lines parallel to the end-plane were scribed on thecylindrical surface near the end-plane in order to determine the displace-ment distribution when this region comes into contact with the die. Theresults are examined in Figs. 9.lib, and c for two types of representationof the friction conditions. One is for the coefficient of friction /* and theother for the friction factor m. It is seen in the figure that the variations for(j, and m are similar, but that the curve for jit shows more uniformdistribution.

In ring compression, friction values are evaluated by comparing the

(c)

FIG. 9.11 (a) Solid circular cylinder specimen of aluminum alloy 7075-T6 with indentedmarks and scribed lines; distributions of coefficient of friction n and friction factor m; (b) withwax as a lubricant; (c) without any lubricant [20].

164


experimental data with the so-called calibration curves, as shown in Fig.9.9. This procedure, however, raises some questions regarding theaccuracy and efficiency of this evaluation scheme, because the curves arenot unique and depend on strain, strain-rate, and the thermal characteris-tics of the material, and also on the specimen geometry. Furthermore,experimental data usually do not follow the trend of predicted curve for aconstant friction value. Thus, Hwang and Kobayashi [21] proposed amethod of direct evaluation of the friction value from the experimentalmeasurements in ring tests. The method uses the finite-element technique,and includes fitting curves to the experimental data with an iterationscheme for evaluating a current friction value based on measured changesof the ring dimensions. This scheme for evaluating friction was applied tothe experimental data of DePierre and Gurney [22] for annealed aluminum1100, for a ring geometry of 6:2:1 (outer diameter:inner diameter:thick-ness). One set of experiments was conducted with Johnson's wax as alubricant and another without a lubricant.

The results for the case with a lubricant are shown in Fig. 9.12a. Theupper part shows experimental points and the fitted curve, and the lowerpart shows the computed variation of the friction factor m, which onlyranges from m = 0.1 to 0.2. The results for the case without a lubricant areshown in Fig. 9.12b. It can be seen that the variation in friction value islarge and the estimation of friction from the calibration curves generatedfor constant friction values results in an erroneous interpretation of theexperimental data.

9.6 Forging and Cabbaging

Axisymmetric Spike Forging

In spike forging, a cylindrical billet is forged in an impression diecontaining a central cavity. The material flows radially as well as axiallyinto the upper cavity, forming a spike. The height of the spike depends on(1) the geometrical dimensions of the dies, (2) the initial workpiece, and(3) the interface friction. Experiments show that spike height increaseswith increasing friction if the other process conditions remain unchanged[23].

Figure 9.13a shows the undeformed workpiece geometry with 16nine-node elements (see Section 6.3 of Chap. 6), with the top and bottomdies at their initial positions. The material used in these calculations was(or + j8) Ti-6242-0.1Si at 954°C (1750°F). Figure 9.13b shows the effect offriction on metal flow. The die velocity used for these simulations was25.4 mm/s (1.0 in./s). The predictions shown in the figure were in excellentagreement with experiments [24].

Cabbaging [25]In the initial stage of backward extrusion, called cabbaging, a round billetis placed in a container. A punch is used to upset and partially pierce one

(b)FIG. 9.12 Variation of friction value determined from experimental data [22]: (a) lubricatedwith wax; (b) without any lubricant.

166

FIG. 9.13 Axisymmetric spike forging: (a) undeformed grid; (b) deformation at a die strokeof 0.58H0 for two m values (H0 is initial billet height) [24].

167


end of the billet. Thus, the outside diameter of the billet is sized, and thepierced recess on top of the billet provides punch guidance for thesubsequent backward extrusion operation. The cabbaging operation hasbeen simulated using the actual production conditions with minor assump-tions. The undeformed billet had a diameter of 73.7mm (2.94 in.) and aheight of 354mm (13.85 in.) The billet material was AISI 1046 steel, andthe operation was done at 1100°C (2012°F). The frictional shear factorused in the analysis was m = 0.3. The punch speed used for the simulationwas 59.3 mm/s (2.3 in/s).

Figure 9.14 shows the undeformed grid lines, the calculated griddistortions at various punch displacements, and the predicted punch loadversus displacement curve. Because of the unusually high strain concentra-tion, the element near the punch tip underwent too much distortion, as canbe seen in Fig. 9.14. A partial "remeshing" near the punch tip was done ata punch displacement of 91.4 mm (3.6 in.).

Compressor Disk Forging [26]

A simulation of a compressor disk forging from Ti 6242 alloy is illustratedin Fig. 9.15. Because of symmetry, only a quarter of the disk is shown. Thecylindrical preform shape used in this analysis was 158.8mm (6.5 in.) in

PUNCH DISPLACEMENT (INCH)

FIG. 9.14 Predicted grids and punch load-displacement curve for the cabbaging process [25].


(c) Predicted Effective Strain Distribution at 70 Percent Reduction

FIG. 9.15 Results of compressor disk forging simulation [26].

diameter and 63.5 mm (2.5 in.) in height. The velocity of the upper dieused in the simulation was 5.1 mm/min (0.2in./min). The bottom die wasstationary.

Forging was done isothermally at 900°C (1650°F), with an averagenominal strain-rate of about 0.175mm"1. The results show that thefinite-element method can be used effectively for simulating disk-typeforgings and for predicting strains, strain-rates, and stresses for a givenpreform shape.


Flashless Forging of a Gear Blank

In practical metal-forming operations, deformations are usually very large.As a result, during simulation of a forming operation these large strainsdistort the initial mesh to such a degree that a new mesh (remesh) isnecessary in order to continue the simulation. For this purpose a"remeshing" method has been developed [27] (see Section 7.6 of Chap. 7).

To evaluate the remeshing algorithm, a simulation of a gear blankforging was conducted, using the following process variables:

• The undeformed cylindrical billet had a height of 169.16mm (6.66 in.)and a diameter of 69.98mm (2.755 in.) (Figure 9.16).

• The billet material was AISI 8620 steel and the forging temperaturewas 1100°C (2012°F). The flow stress data for 8620 steel were obtainedby performing isothermal uniform compression tests at strain rates of25-30s"1 and 2.5-3s~1. The flow stress a was calculated using theexpression a = Ce"em (ksi), where C, n and m were given as 10.92,

FIG. 9.16 Metal flow patterns in flashless forging of a gear blank at 0, 40, 60, and 78%reductions in initial billet height.

Axisymmetric Isothemal Forging 171

0.0, and 0.134, respectively.• The velocity of the upper die used in the analysis was 127.0mm/s

(5.0in./s), and the bottom die was stationary.• The friction factor used in the analysis was chosen as m = 0.4.

Figure 9.16 shows the metal-flow patterns after 0, 40, 60, and 78%reductions in initial billet height. It should be noted that at the 60%reduction the element at the lower right-hand corner was distorted to sucha degree that it was impossible to continue the simulation withoutgenerating a new mesh. A new mesh was generated at the 61% reductionin billet height, as shown in Fig. 9.17. The strain values of the last stepbefore remeshing (at 61% reduction) were then interpolated on the nodesof the new mesh. The simulation was continued using the new mesh. Twomore remeshings were necessary to complete the simulation up to thecomplete die fill and to obtain the results shown in Fig. 9.18.

FIG. 9.17 Distorted grid (right half) and "remeshed" grid (left half) at 61% reduction inbillet height for flashless forging.

FIG. 9.18 Grid distortions near completion of die filling for flashless forging.


References1. Spies, K., (1957), "The Preforms in Closed-Die Forging and Their Preparation

by Reducer Rolling," (in German) Doctoral Dissertation, Technical UniversityHannover.

2. Lange, K., (1958), "Closed-Die Forging of Steel," (in German), Springer-Verlag, Berlin.

3. Kudo, H., and Aoi, K., (1967), "Effect of Compression Test Condition uponFracturing of a Medium Carbon Steel—Study on Cold Forgeability Test: PartII," (in Japanese), J. Japan Soc. Tech. Plast., Vol. 8, p. 17.

4. Nagamatsu, A., Murota, T., and Jimma, T., (1971), "On the NonuniformDeformation of Material in Axially Symmetric Compression Caused byFriction," Bull. JSME, Vol. 14, p. 331.

5. Lee, C. H., and Kobayashi, S., (1971), "Analysis of Axisymmetric Upsettingand Plane-strain Side-pressing of Solid Cylinders by the Finite-elementMethod," Trans. ASME, J. Engr. Ind., Vol. 93, p. 445.

6. Kobayashi, S., and Lee, C. H., (1973), "Deformation Mechanics andWorkability in Upsetting Solid Circular Cylinders," Proc. North AmericanMetalworking Research Conference, Vol. 1, p. 185.

7. Shah, S. N., Lee, C. H., and Kobayashi, S., (1974), "Compression of TallCircular, Solid Cylinders Between Parallel Flat Dies," Proc. InternationalConference on Production Engineering, Tokyo, p. 295.

8. Lecocq, A. G., (1971), "Stresses in the Shank of a Bolt During ColdHeading," Wire (English version of Draft) Coburg, Vol. 115, p. 197.

9. Thomason, P. F., (1969-70), "The Effect of Heat Treatment on the Ductilityin a Cold Heading Process," Proc. Inst. Mech. Engr., Vol. 184, p. 875.

10. Gill, F. L., and Baldwin, W. M., (1964), "Proper Wire Drawing ImprovesCold Heading Process," Metal Progress, Vol. 85, p. 83.

11. Shah, S. N., and Kobayashi, S., (1974), "Rigid-Plastic Analysis in ColdHeading by the Matrix Method," Proc. 15th International Machine ToolDesign Research Conf., p. 603.

12. Kunogi, M., (1954), "On Plastic Deformation of Hollow Cylinders underAxial Compressive Loading,"/. Sci. Res. Inst., Tokyo, p. 2.

13. Avitzur, B., (1983), "Handbook of Metal Forming Processes," Wiley, NewYork.

14. Chen, C. C., and Kobayashi, S., (1978), "Rigid-plastic Finite-element Analysisof Ring Compression," Application of Numerical Methods to FormingProcesses, ASME, AMD, Vol. 28, p. 163.

15. Hartley, P., Sturgess, C. E. N., and Rowe, G. W., (1979), "Friction inFinite-element Analysis of Metal-forming Processes," Int. J. Mech. Sci., Vol.21, p. 301.

16. Hartley, P., Sturgess, C. E. N., and Rowe, G. W., (1979), "An Examinationof Frictional Boundary Conditions and Their Effect in an Elastic-plasticFinite-element Solution," Proc. MTDR, p. 157.

17. Male, A. T., and DePierre, V., (1970), "The Validity of MathematicalSolutions for Determining Friction from the Ring Compression Test,"Lubrication Technology, p. 389.

18. Schey, J. A., Ed., (1970), "Metal Deformation Processes—Friction andLubrication," Marcel Dekker, New York.

19. Thomsen, E. G., (1969), "Friction in Forming Processes," Annals CIRP, Vol.17, p. 149.

20. Oh, S. I., Thomsen, E. G., and Kobayashi, S., (1975), "Calculation ofFrictional Stress Distributions at the Die-Workpiece Interface in SimpleUpsetting," Proc. 3d North American Metalworking Research Conference, p.159.


21. Hwang, S. M., and Kobayashi, S., (1983), "A Note on Evaluation of InterfaceFriction in Ring Tests," Proc. NAMRC XI, University of Wisconsin, Madison,WI, p. 193.

22. DePierre, V., and Gurney, F., (1972), "A Method for Determination ofConstant and Varying Friction Factors During Ring Compression Tests," AirForce Materials Laboratory, Report AFML-TR-72-37.

23. Oh, S. I., (1982), "Finite Element Analysis of Metal Forming Problems withArbitrarily Shaped Dies," Int. J. Mech. Sci., Vol. 17, p. 293.

24. Oh, S. I., Lahoti, G. D., and Altan, T., (1981), "ALPID—A General PurposeFEM Program for Metal Forming," Proc. NAMRC-IX, State College, PA, p.83.

25. Oh, S. I., Lahoti, G. D., and Altan, T., (1982), "Analysis of BackwardExtrusion Process by the Finite Element Method," Proc. NAMRC—X,Hamilton, Canada, p. 143.

26. Oh, S. I., Park, J. J., Kobayashi, S., and Altan, T., (1983), "Application ofFEM Modeling to Simulate Metal Flow in Forging a Titanium Alloy EngineDisk," Trans. ASME, J. Engr. Ind., Vol. 105, p. 251.

27. Badawy, A., Oh, S. I., and Altan, T., (1983), "A Remeshing Technique forthe FEM Simulation of Metal Forming Processes," Proc. Int. Comp. Eng.Conf., ASME, Chicago, p. 143.

10STEADY-STATE PROCESSES

OF EXTRUSION AND DRAWING

10.1 Introduction

Except at the start and the end of the deformation, processes such asextrusion, drawing, and rolling are kinematically steady state. Steady-statesolutions in these processes are needed for equipment design and diedesign and for controlling product properties.

A variety of solutions for different conditions in extrusion and drawinghave been obtained by applying the slip-line theory and the upper-boundtheorems [1-3]. Early applications of the finite-element method to theanalysis of extrusion [4-6] have been for the loading of a workpiece thatfits the die and container, and for the extrusion of a small amount of itrather than extruding the workpiece until a steady state is reached. Anexception is the work by Lee et al. [7] for plane-strain extrusion withfrictionless curved dies using the elastic-plastic finite-element method. Inview of the computational efficiency, various numerical procedures par-ticularly suited for the analysis of steady-state processes have beendeveloped by several investigators [8-14]. Shah and Kobayashi [8]analyzed axisymmetric extrusion through frictionless conical dies by therigid-plastic finite-element method. The technique involves construction ofthe flow lines from velocities and integration of strain-rates numericallyalong flow lines to determine the strain distributions. An improvement ofthe method was made by including friction at the die-workpiece interface.The steady-state deformation characteristics in extrusion and drawing wereobtained as functions of material property, die-workpiece interfacefriction, die angle, and reduction [15].

10.2 Method of AnalysisIn kinematically transient or nonsteady-state forming problems, a meshthat requires continuous updating (Lagrangian) is used. In steady-stateproblems, a mesh fixed in space (Eulerian) is appropriate, since theprocess configuration does not change with time. For steady-state prob-lems whose solutions depend on the loading history or strain history of thematerial, combined Eulerian-Lagrangian approaches are necessary[16,17]. In deformation of rigid-plastic materials under the isothermalconditions, the solution obtained by the finite-element method is in terms

174

Steady-State Processes of Extrusion and Drawing 175

of velocities and, hence, strain-rates. In the nonsteady-state processes, theeffective strain-rates are added incrementally for each element to deter-mine the effective strains after a certain amount of deformation. For theanalysis of a steady-state process, a technique is needed for determiningthe distributions of effective strain if the deforming material is a strain-hardening material. The following method is developed to obtain the flowlines, distorted grid pattern, and the effective strain distribution.

In the finite-element technique, the strain-rates for the elements areassumed to be the values at the centers of the elements; the velocities areknown at the nodal points of the elements. The coordinates of theseelement centers and nodal points are also known. First, a point is selectedalong the entrance boundary of the deformation zone where the effectivestrain is zero. From the known coordinates of this point, components ofthe velocity and the effective strain-rate are determined by interpolationfrom surrounding nodal-point values and from surrounding element-centervalues, respectively. These velocities are then incrementally added to thecoordinates of the point to determine its new position. The effectivestrain-rate multiplied by the time-increment is added to the effective strainto determine the effective strain at the new location of the point, and theprocedure is repeated. The procedure of incrementally adding the velocityand the strain-rate is continued until the point reaches the exit boundary ofthe deformation zone. Beyond the exit, the point moves with a velocityequal to the exit velocity, and the effective strain remains unchanged.Starting with a different point at the entrance boundary, another flow lineand the values of effective strain along this flow line can be determined in asimilar manner. This procedure yields the entire network of grid distor-tions and the effective strain distribution.

In nonsteady-state processes, the effect of material work-hardening canbe readily incorporated into the analysis by computing the incrementalstrains and modifying the flow stress at each deformation step according tothe increase in the total effective strain. In the analysis of steady-stateprocesses, however, the flow stress distribution must be consistent with thefinal effective strain distribution according to the work-hardening charac-teristics of the material. This requirement can be achieved by using thefollowing computational procedure. During iterations to obtain a converg-ing solution, the flow lines corresponding to the latest velocity field areconstructed after each iteration. The network of grid distortions and theeffective strain distributions are then determined from these flow lines.The effective strains for all elements are then interpolated from thesevalues and, using a given stress-strain relationship, corresponding flowstress distributions are determined for the elements. Using this new flowstress distribution, the next iteration is carried out to determine thevelocity field, and the same procedure is repeated until a convergedsolution is obtained. The solution depends not only on the velocitydistribution but also on the flow stress distribution. Therefore, when thevelocity solution converges, the flow stress and effective strain distributions


also match each other according to the stress-strain behavior of thematerial.

10.3 Bar ExtrusionBoundary Conditions and Mesh SystemThe boundary conditions and the mesh system used to analyze extrusionthrough conical dies are shown in Fig. 10.1 [15]. The material in thecontainer moves axially with a uniform velocity of unit magnitude. Thecontainer is assumed to be frictionless, and along the conical die surfacesthe tangential traction is equal to the frictional stress at the die-workpieceinterface. The extruded material moves axially with a uniform velocitywhose magnitude is determined by the area reduction and the incompres-sibility relationship. Also, there are no tractions on the surface of theextruded part. Along the axis of symmetry of extrusion, the conditions aresuch that the shear traction and the radial velocity must vanish.

It must be noted that in performing the computations the die cornerswere slightly modified by connecting with a straight line the two materialnodal points located closest to the die corners. This modification was madein order to avoid singularities in the velocity components near the diecorners.

FIG. 10.1 Boundary conditions and mesh system for steady state axisymmetric extrusionanalysis.


ResultsThe computations were carried out under various extrusion processconditions for SAE 1112 steel and aluminum alloy 2024-T351. Since thesolution with nonwork-hardening material was used as the initial guess inanalyzing the extrusion of work-hardening materials, the results fornonwork-hardening materials are also discussed. The computation wasperformed for each solution until the error norm of ||Av||/||v|| = 0.000 08was reached. The number of iterations to reach the above convergencedepends on the initial guess, but the average number of iterations requiredfor the final solution was about 25-30.

The predicted results were obtained for average extrusion pressure,normal pressure distribution on the die, grid distortions, and for velocity,stress, and strain-rate distributions. Some of these are discussed below.

Detailed differences in deformation and f}ow behavior, due to materialproperties and friction at the die-workpiece interface, are clearly indicatedin calculated grid distortions in Fig. 10.2. The steady-state grid distortionpatterns are compared for nonwork-hardening and work-hardening (SAE1112 steel) cases for two friction conditions. Figure 10.2a shows thedistortion of the grid lines that are originally perpendicular to the axis ofthe workpiece. It is seen that there is a cusp or double peak on the axis inthe extruded part for both the work-hardening and nonwork-hardeningmaterials. However, the magnitude of the cusp is smaller with the formermaterial. With increasing friction at the die-workpiece interface, the

FIG. 10.2 Grid distortion patterns for the extrusion of nonwork-hardening (right halves) andwork-hardening (left halves) materials with (a) frictional stress fs = 0, and (b) fs = 0.4Y0: diesemi cone angle, 30°, R0/R,, = 2.37.


double-peak or cusp type distortion disappears almost completely for bothwork-hardening and nonwork-hardening materials as seen in Fig. 10.2b.

A typical effective strain distribution is shown for a nonwork-hardeningmaterial in Fig. 10.3. The strain has the highest value near the die and thelowest value near the extrusion axis throughout the deformation zone. Theeffective strain distribution at the exit section is also shown in the figure,which indicates that the strain has its lowest value near the extrusion axisand increases towards the extrusion surface.

In general, the velocity distributions calculated for aluminum alloy2024-T351 do not differ significantly from those obtained for SAE 1112steel. Direct comparisons of the computed velocity fields with thosemeasured in a visioplasticity study conducted by Shabaik and Thomsen[18] was possible, and such a comparison is shown in Fig. 10.4. Thecomputed results are given for the case of a frictionless die. Thecomparison of the computed and measured velocity distributions in Fig.10.4 reveals that the theoretical results, in both velocity components, arein agreement with the experimental results in magnitude as well as intrends.

The distribution of the mean stresses for the aluminum alloy is alsosimilar to the pattern obtained for SAE 1112 steel. As the reductiondecreases, the mean stress increases and becomes tensile in the zone nearthe center, as shown in Fig. 10.5. The computations also reveal that withincreasing friction at the die-workpiece interface the mean stress becomesmore compressive throughout the deformation zone. The tensile stresses,in the central zone of the extrusion, are responsible for the occurrence ofcenter bursting cracks.

10.4 Bar DrawingThe drawing process is characterized by smaller die angles and smallerreductions than those used in extrusion. Initially, it appears that thedifferences between extrusion and drawing are merely in geometricalquantities and hydrostatic stress components (extrusion being essentially aprocess of compression, and drawing a process of tension). However, thisis not the case and the finite-element results obtained in extrusion cannotbe extrapolated to obtain results in drawing by simply taking into accountthe geometrical conditions and the concepts of pushing in extrusion andpulling in drawing.

Computational Conditions and Procedures

The material selected for the computation was SAE 1144 cold-drawn steel[15]. The computation was performed for semi-die angles a of 6° and 8°,two friction conditions, and several reductions.

For computational purposes, the solution for extrusion with a = 45° andRn/Re = 1.25 (/?o = billet radius, Re = radius of extruded bar) was used asan initial guess for drawing with a = 6° and R(,/Re = 1.2 by modifying the

FIG. 10.3 A typical effective strain distribution in extrusion of a nonwork-hardening material.

FIG. 10.4 Comparison of (a) computed and (b) experimental [18] (K = 0 and 20 correspondto Z/R0 = Q and 1.0, respectively) velocity distributions in extrusion; R0/Re = 2, semi dieangle a = 45° (frictionless in computation).

180


FIG. 10.5 The mean stress distribution in extrusion through a frictionless die at R0/RC =1.25.

solution according to geometrical proportions. Thirty-four iterations wererequired to achieve the converged solution. This solution was then used asan initial guess for the computation of all the other drawing conditions.The convergence was excellent and only 6—10 iterations were necessary toobtain the solutions in most cases.

Results

The results obtained for drawing are examined in detail for a = 6°. Thevariation of effective strain distribution in drawn bars is similar to thatfound in extrusion, but the magnitudes are smaller owing to smaller dieangles and reductions in drawing. By comparing the results of drawingwith those of extrusion, it is found that the die angle is the most importantvariable in controlling the nonuniformity of the strains and properties inextruded or drawn bars.

The die pressure distribution is plotted for various reductions and fortwo die friction conditions in Fig. 10.6. A higher die pressure is obtainedfor a smaller reduction, as was the case for extrusion. However, contraryto the case in extrusion, the die pressure is higher in frictionless drawingthan in drawing with friction. These findings on die pressure are inagreement with experimental results [19] and with the results predicted bythe slip-line theory. Further observations of the results for drawing andcomparison of the results with those of extrusion revealed the following.

Although materials properties somewhat influence metal flow, as seen ingrid distortion patterns, their effects on the overall deformation and flowbehavior appear to be insignificant in extrusion and drawing processes.


FIG. 10.6 The die pressure distributions in drawing for various reductions and for the two diefriction conditions (die semi-cone angle = 6°).

Among other variables, friction at the die-workpiece interface plays animportant role in determining the detailed mechanics in these processes.With increasing friction, the degree of grid distortion becomes larger andthe size of the deformation zone expands in both processes, but the effectof friction on nonuniformity of product properties is less significant. Themain difference between the extrusion process and the drawing process canbe found in the die pressure distribution and the distribution of hydrostaticstress components. In both processes, the die pressure decreases withincreasing reduction in area. However, the die pressure is greater forlarger interface friction in extrusion, while the reverse is true in drawing.

182


Similarly, in extrusion the hydrostatic stress component decreases withincreasing interface friction, but in drawing this value increases withincreasing friction.

10.5 Multipass Bar Drawing and ExtrusionIn practice, many bar drawing and extrusion processes are multipassoperations. Internal fracturing (center bursting) in these processes usuallyoccurs after significant area reduction through several passes. In order topredict and prevent the occurrence of such defects, it is necessary todetermine the deformation characteristics involved in multipass opera-tions. For this purpose, commercially pure aluminum was considered forsimulation. The effect of work-hardening was examined in single-passoperations, and the effect of flow stress distributions on the flow patternwas revealed by analyzing two-pass drawing and extrusion processes [20].

ExtrusionIn Fig. 10.7, the steady-state grid distortion patterns are compared for thetwo passes. In both extrusion and drawing, deformation behavior during

FIG. 10.7 Comparison of grid distortions in extrusion of pure aluminum for the first andsecond passes (/, = 0).


FIG. 10.8 Stress and srain histories of a material point along the axis of (frictionless)extrusion.

the second pass is generally closer to that of nonwork-hardening materials,in comparison with the first-pass deformation. The critical site for theoccurrence of center bursting in extrusion is along the axis of extrusion.The stress and strain histories are important in applying ductile fracturecriteria. Figure 10.8 shows these stress and strain histories of a pointmoving along the axis of extrusion for the frictionless case. The axial stressbecomes tensile over a certain range of deformation and a larger tensionzone is obtained for nonwork-hardening materials than for work-hardeningmaterials. The tensile stress increases during the second pass. It was foundthat the interface friction had the effect of decreasing the axial stress, thussuppressing the possibility of producing center bursting. These results arequalitatively in agreement with experimental observations.

Drawing

The hydrostatic component distributions in two-pass drawing are given inFig. 10.9. The top figure is for nonwork-hardening material and is valid forboth first and second passes. The rigid zone boundaries are also shown inthe figure. The size of the deformation zone in the first pass is larger withwork-hardening materials than that for nonwork-hardening materials.However, the deformation zone becomes smaller in the second pass and isalmost identical to that for nonwork-hardening materials. The straindistributions of the work-hardening materials during the second pass arecloser to those for a nonwork-hardening material, although the differencesbetween the strains for the first pass of work-hardening materials and ofnonwork-hardening materials are not great. The hydrostatic stress com-ponent distributions revealed that the region under the die is in a state ofcompressive hydrostatic stress, while the core and exit regions are in astate of tensile hydrostatic stress. The mean stress becomes more tensile

FIG. 10.9 Hydrostatic pressure distributions in two-pass drawing (die semi-cone angle, 6°R0/«e = l.l,/s = 0.4F0).

185


FIG. 10.10 Stress and strain histories of a material point along the axis of drawing.

with larger die friction. This is contrary to the results obtained inextrusion. The stress and strain histories along the axis of drawing areplotted in Fig. 10.10. The peak axial stress appears near the entrance; forextrusion it is observed near the exit. The axial stress is greater fornonwork-hardening materials than for work-hardening ones during the firstpass, but increases during the second pass. This is also true for extrusion,as seen in Fig. 10.8, suggesting an increased possibility of center burstformation in multipass processes.

10.6 Applications to Process DesignAn important application of metal-flow simulation is determining theprocess conditions necessary to prevent the formation of defects. Thepredicted stress-strain histories along the axis of extrusion and drawing,discussed above, when combined with the ductile fracture criterion, can beutilized in selecting the preferred process conditions, such as reduction inarea, semi-die angles and die-workpiece interface friction, so that centerbursting can be avoided [21].

Another significant application concerns computer-aided design ofextrusion dies. Streamlined die design is now being •considered in extrusionof certain structural shapes and workpiece combinations for a number ofapplications in the aerospace industry. These components generally havecomplex part geometries, and the workpiece material in general has poorworkability. This combination of conditions can be found in almost everystructural alloy family, including aluminum, steel, titanium, and nickel-based alloys. Failure in these alloy systems is usually attributed tocenter-burst phenomena or porosity due to the decohesion of second-phaseparticles from the matrix material. The use of the finite-element method in


metal-flow simulation for the optimum die design in extrusion has beeninvestigated by Gegel et al. [22]. They performed extrusion simulationswith a variety of die geometries, including shear, conical, parabolic, andstreamlined dies. They have developed various optimization criteria, andwith these criteria and metal-flow simulations they successfully designedstreamlined dies for extrusion. Research in this area will result in furtherapplications of finite-element metal-flow analysis for process design andoptimization.

References1. Johnson, W., Sowerby, R., and Venter, R. D., (1982), "Plane-Strain Slip Line

Fields for Metal Deformation Processes," Pergamon Press, Oxford.2. Johnson, W., and Kudo, H., (1962), "The Mechanics of Metal Extrusion,"

Manchester University Press, Manchester, UK,3. Avitzur, B., (1983), "Handbook of Metal Forming Processes," Wiley, New

York.4. Murota, T., Jimma, T., and Kato, K., (1970), "Analysis of Axisymmetric

Extrusion," Bull. JSME, Vol. 13, p. 1366.5. Iwata, K., Osakada, K., and Fujino, S., (1972), "Analysis of Hydrostatic

Extrusion by the Finite Element Method," Trans. ASME, J. Engr. Ind., Vol.94, p. 697.

6. Lee, C. H., Iwasaki, H., and Kobayashi, S., (1973), "Calculation of ResidualStresses in Plastic Deformation Processes," Trans. ASME, J. Engr. Ind., Vol.95, p. 283.

7. Lee, E. H., Mallet, R. L., and Yang, W. H., (1976), "Stress and DeformationAnalysis of the Metal Extrusion Process," SUDAM No. 76-2, StanfordUniversity, June.

8. Shah, S. N., and Kobayashi, S., (1977), "A Theory on Metal Flow inAxisymmetric Piercing and Extrusion," J. Prod. Engr., Vol. 1, p. 73.

9. Zienkiewicz, O. C., and Godbole, P. N., (1974), "Flow of Plastic andViscoplastic Solids with Special Reference to Extrusion and Forming Proc-esses," Int. J. Num. Meth. Engr., Vol. 8, p. 3.

10. Zienkiewicz, O. C., Jain, P. C., and Onate, E., (1978), "Flow of SolidsDuring Forming and Extrusion; Some Aspects of Numerical Solutions," Int. J.Solids Structures, Vol. 14, p. 15.

11. Zienkiewicz, O. C., Onate, E., and Heinrich, J. C., (1978), "Plastic Flow inMetal Forming, I. Coupled Thermal Behavior in Extrusion. II Thin SheetForming," Applications of Numerical Methods to Forming Processes, ASME,AMD, Vol. 28, p. 107.

12. Dawson, P. R., (1978), "Viscoplastic Finite Element Analysis of Steady StateForming Processes including Strain History and Stress Flux Dependence,"Applications of Numerical Methods to Forming Processes, ASME, AMD,Vol. 28, p. 55.

13. Dawson, P. R., and Thompson, E. G., (1978), "Finite Element Analysis ofSteady-State Elasto-Visco-Plastic Flow by the Initial Stress-Rate Method," Int.J. Num. Meth. Engr., Vol. 12, p. 47.

14. Derbalian, K. A., Lee, E. H., Mallet, R. L., and McMeeking, R. M., (1978),"Finite Element Metal Forming Analysis with Spacially Fixed Mesh," Appli-cations of Numerical Methods to Forming Processes, ASME, AMD, Vol. 28,p. 39.

15. Chen, C. C., Oh, S. I., and Kobayashi, S., (1979), "Ductile Fracture in


Axisymmetric Extrusion and Drawing—Part I, Deformation Mechanics ofExtrusion and Drawing," Trans. ASME, J. Engr. Ind., Vol. 101, p. 23.

16. Schreura, P. J. G., Veldpaus, F. E., and Brekelmans, W. A. M., (1982), AnArbitrary-Eulerian-Lagrangian Finite Element Model for the Simulation ofGeometrical Non-Linear Hyper-Elastic and Elasto-Plastic Deformation Proc-esses, in "Numerical Method in Industrial Forming Processes," (Edited by J.F. T. Pittman et al.), Pineridge Press, Swansea, UK, p. 491.

17. Huetink, J., (1982), Analysis of Metal Forming Processes Based on aCombined Eulerian-Lagrangian Finite Element Formulation, in "NumericalMethod in Industrial Forming Processes," (Edited by J. F. T. Pittman et al.),Pineridge Press, Swansea, UK, p. 501.

18. Shabaik, A. H., and Thomsen, E. G., (1968), "Investigation of the Applica-tion of the Visioplasticity Methods of Analysis to Metal DeformationProcesses," Final Report—Part II, Department of the Navy.

19. Wistreich, J. G., (1955), "Investigation of the Mechanics of Wire Drawing,"Proc. Inst. Mech. Engr., Vol. 169, p. 654.

20. Chen, C. C., and Kobayashi, S., (1978), "Deformation Analysis of Multi-PassBar Drawing and Extrusion," Annals CIRP, Vol. 27, No. 1, p. 151.

21. Oh, S. I., Chen, C. C., and Kobayashi, S., (1979), "Ductile Fracture inAxisymmetric Extrusion and Drawing, Part 2: Workability in Extrusion andDrawing," Trans. AMSE, J. Engr. Ind., Vol. 101, p. 36.

22. Gegel, H. L., Malas, J. C., Gunasekera, J. S., Morgan, J. T., and Doraivelu,S. M., (1984), "Computer-Aided Design of Extrusion Dies by Metal-FlowSimulation," AGARD Lecture Series, No. 137, Process Modeling Applied toMetal Forming and Thermomechanical Processing, North Atlantic TreatyOrganization, p. 8-1.

11SHEET-METAL FORMING

11.1 IntroductionThe stress-state is said to be plane when the direction normal to the planeis a principal stress direction and the magnitude of the stress in thisdirection is zero. This situation occurs when a sheet is loaded along itsedges in the plane of the sheet. In-plane deformation of sheet metal, suchas bore expanding and flange-drawing, is an example of plane-stressproblems. For out-of-plane deformation of sheet metals, such as punchstretching, sheet bending, and cup drawing, a simple analytical method isthe use of membrane theory. This theory neglects stress variations in thethickness direction of a sheet and considers the distribution of stresscomponents only in the plane of the sheet. Thus, the basic formulations forthe analysis of both in-plane and out-of-plane deformations contain onlythe stress components acting in the plane of the sheet. However, theanalysis of out-of-plane deformation requires consideration of largedeformation, while the infinitesimal theory is applicable for in-planedeformation analysis.

Many materials employed in engineering applications possess mechani-cal properties that are direction-dependent. This property, termedanisotropy, stems from the metallurgical structure of the material, whichdepends on the nature of alloying elements and the conditions ofmechanical and thermal treatments. Metal sheets are usually cold-rolledand possess different properties in the rolled and transverse directions.Therefore, in sheet-metal forming in particular, the effect of anisotropy onthe deformation characteristics may be quite appreciable and important.

In the past the calculation of the detailed mechanics of large plasticdeformation of metal sheets has been achieved with some success bynumerical methods. However, without exception, these studies have dealtwith deformations that possess a high degree of symmetry, and wereconcerned with the anisotropy existing only in the direction of sheetthickness (normal anisotropy). Methods that are capable of solvingnonaxisymmetric problems in forming of anisotropic sheet metal are stillbeing sought. The finite-element method is one of those methods. It wasapplied to the elastic-plastic analysis of nonaxisymmetric configurations ofsheet stretching with normal anisotropy by Mehta and Kobayashi [1].Yamada [2] presented a stress-strain matrix for a material that is

189


elastically isotropic, and that obeys Hill's anisotropic yield criterion in theplastic range. He treated the incipient deformation of a circular blank ofanisotropic material in the flange drawing process. For the deformationanalysis of a metal sheet having three mutually perpendicular axes ofanisotropy (orthotropic material), the rigid-plastic finite-element methodhas been applied to plane-stress bore expanding and flange drawing [3,4].To analyze out-of-plane forming processes, Wang [5] proposed a methodof solution in which two spatially independent variables are required todefine the geometry. The method is based on a variational procedure andassumes that the material of the sheet is rigid-plastic. On the sameprinciple, Kim et al. [6, 7] analyzed three processes, namely, the bulging ofa sheet subjected to hydrostatic pressure, the stretching of a sheet with ahemispherical punch, and the deep drawing of a sheet with a hemisphericalpunch.

Using the elastic-plastic approach, complete solutions of stretch-formingand deep-drawing problems, taking into account the contact problem atthe blank holder, die, die profile, and punch head, were obtained by Win[8]. On the basis of the nonlinear theory of membrane shells, Wang andBudiansky [9] developed a procedure for calculating the deformations inthe stamping of sheet metal by arbitrarily shaped punches and dies.

Onate and Zienkiewicz [10] presented a finite-element formulationbased on an extension of the general viscoplastic flow theory forcontinuum problems to deal with thin shells.

Toh and Kobayashi [11,12] analyzed sheet-metal forming processes,axially symmetric and nonsymmetric, by the finite-element method basedon the membrane theory. The finite-element model takes into account therigid-plastic material characteristics and includes the normal anisotropy ofthe sheet metal as well as the finite deformation that occurs during thesheet-forming process.

In this chapter, first the yield function for anisotropic materials isdefined and in-plane deformation processes are analyzed using theinfinitesimal theory. Then the large-strain formulation using membranetheory is described for the analysis of out-of-plane deformation processes.The formulation is extended to sheet-metal forming of general shapes andapplied to square-cup drawing. Finally, in Section 11.8 we discuss thenonquadratic yield function for anisotropic materials.

11.2 Plastic Anisotropy

Considering states of anisotropy that possess three mutually orthogonalplanes of symmetry at every point, the simplest yield criterion foranisotropic materials is one that reduces to the distortion energy criterionwhen the anisotropy is vanishingly small. Hill [13] assumed the yieldcriterion to be quadratic in the stress components as follows:

Sheet-Metal Forming 191

where the orthotropic axes are taken as the coordinate axes (x, y, z) andF, G, H, L, M, N are anisotropy parameters. The parameters in eq. (11.1)are not definite but their ratios are for defining the behavior of a givenmaterial. Thus, eq. (11.1) is rewritten as

where /(cry) is the plastic potential, o is the effective stress, and /, g, h, I,m, n are pure numbers. The coefficients/, g, h, I, m, n in eq. (11.2) aresubjected to conditions depending upon the selection of a as a referencematerial property.

If Yx, Yy, Yz are the tensile yield stresses in the principal directions ofanisotropy, it is easily shown from eq. (11.2) that

then

Under plane-stress conditions (oz = TZX = ryz = 0), eq. (11.2) reduces to

It should be noted again that the ratios of anisotropic parameters, nottheir absolute values, in eq. (11.4a) define the state of anisotropy. This isobvious, if we use, for example, the condition of eq. (11.3b) correspondingto the definition of a expressed by eq. (11.3a), and write eq. (11.4a) in theform

Supposing that/(cr,y) in eq. (11.4a) is the plastic potential, the strain-raterelations from the flow rule, referred to the principal axes of anisotropy,are

If a in eq. (11.2) is defined as

and then substituting eq. (11.6) into eq. (11.4a). We have

Frequently, the anisotropy of a material is represented by the strain ratiosknown as /"-values. For a tensile specimen cut at an angle a to the rollingdirection (x direction)

where o is the tensile yield stress. The ratio of the transverse to thethrough-thickness strain (or strain-rate) is

Substitution of eq. (11.5), using the stresses given by eq. (11.8), into (11.9)results in

where rx, r45 and ry are the /--values for a = 0° (x direction), 45°, and 90° (ydirection), respectively. For the anisotropy to be rotationally symmetricabout the z axis, f=g and n=f + 2h (and / = m) in addition to thecondition given by eq. (11.3b). If there is complete spherical symmetry, orisotropy,

11.3 In-plane Deformation ProcessesFinite-Element FormulationUnder the plane-stress condition, the mean stress om is expressed in termsof strain-rate components. Therefore, the Lagrange multiplier is not


The effective strain-rate £ in eq. (11.5), corresponding to the effectivestress a denned in eq. (11.4a) can be obtained first by inverting eq. (11.5)as

From eq. (11.10)


necessary, and the variational formulation becomes

f

The volume constancy is not required in the assumed velocity fieldbecause ez is now treated as a dependent variable. For an isoparametricquadrilateral element, shown in Fig. 8.1 (Chap. 8), the matrices necessaryfor discretization of eq. (11.11) in the two-dimensional case have beenderived in Chap. 6, and some of them are recapitulated in Chap. 8.However, the matrix D in the definition of e according to

differs for anisotropic materials from that for isotropic materials. Withreference to eq. (11.7), the matrix D for orthotropic materials, becomes

Since/ + g + h = 3, eq. (11.13a), in terms of r-values, becomes

If we select a = Yx = Yy as a reference material property and note that0£ = Wp (plastic work-rate), we have

and

with the strain-rate vector defined by eT= {ex, ey, yxy}.In the case of normal anisotropy (isotropy in the plane), f=g and

n =f + 2h or, in terms of r-values, rx = ry = r45 = r and h/f = h/g = r andnig = 1 + 2r. Thus, the effective stress and the effective strain-rate fromeqs. (11.4a) and (11.7), respectively, become


The expressions for the matrix D corresponding to eq. (11.13a), (11.13b),and (11.13c), from eq. (11.12), become, respectively,

and

For isotropic materials the r-value in eqs. (11.14) is equal to unity. It is tobe noted that the strain-rate component ez is eliminated in expressing ethrough the condition that ez = —(ex + ey) for the case of plane stress.

Bore Expanding and Flange Drawing

The problems considered for the analysis are bore expanding and flangedrawing; these are shown schematically in Fig. 11.1.

In the bore-expanding process, a circular blank with a concentric hole isstretched radially. The stretching is accomplished by drawing the blankthrough a die opening, using a flat-bottomed punch (Fig. 11.la). The outerperiphery of the blank is fixed by a holder. To analyze the expansion of thehole of radius /?0 in the blank of initial thickness T0, a uniform radialdisplacement 6b is applied along the boundary of radius Rb. The specimendimension selected for the analysis is Rb/Rn

= 4.4.For flange drawing, the process is simulated by considering a circular

sheet of initial radius Rb that is subjected to a uniform radial displacementtowards its center, along the circular boundary of constant radius R0 (Fig.ll.lb). Similarly to bore expanding, this boundary condition is imposed byapplying the displacement 60 along the boundary of radius R0. Thedrawing ratio selected for the analysis was Rb/R0 = 2.5.

The stress-strain property of the material used for the computation isexpressed by

with F0 = 23,600psi (162.7 MN/m2).Two materials were considered for the calculation. One is a material

with planar anisotropy with r-values of rx = 1.69, r45 = 1.43, and ry = 2.24,which correspond to those of Al-killed steel. The other material is ofnormal anisotropy with rave = \(rx + 2r45 + ry) = 1.70 and has the identicaleffective stress-strain curve.


( a ) BORE E X P A N D I N G ( b ) F L A N G E D R A W I N G

FIG. 11.1 The processes of bore expanding and flange drawing.

In Fig. 11.2, the strain distributions for normal anisotropy and those inthe direction a = 45° for planar anisotropy are compared (ah the anglefrom the principal axis of anisotropy). The distributions in the directions ofa = 5° and 85° are almost identical to those obtained for normal aniso-tropy. The formation of "ears" and "hollows" in flange drawing is shownin Fig. 11.3a and the corresponding strain contours are given in Fig. 11.3b.The ears appear along the axes of anisotropy and a hollow is formed atapproximately a = 47° at an early stage and changes its location slightlynear the end of the process. According to Hill [13], the principal axes ofstress and strain-increment coincide for the orientations a = 0°, a ( = 49°),and 90° for the present planar anisotropy; the ears are formed at a = 0°and 90°, and a hollow at a = (90 - a)° = 41°. The location of the hollow,shown in Fig. 11.3, is not in agreement with Hill's prediction, but is closerto that for minimum r-value (a = 51°).

11.4 Axisymmetric Out-of-plane DeformationVariational Formulation

The variational formulation used in the in-plane deformation process givenby eq. (11.11) is inadequate for analyzing out-of-plane deformation


R A D I U S / R0

FIG. 11.2 Strain distributions at various deformation stages of bore expanding.

processes such as punch-stretching and cup-drawing operations. Consider adeformation of a circular sheet of radius R, as shown in Fig. 11.4. Thevelocityt of the sheet is given by

where «0 is a constant.It is easily shown that all the components of strain rate, denned by

£ij = 2(MI,/ + "/,;)> become zero, even though the length of the sheet wouldchange after a finite time interval A?. This suggests that there is at leastone deformation mode that cannot be determined by the variationalprinciple formulated by eq. (11.11). The zero-energy deformation mode

tFrom Section 11.4 through Section 11.7, where finite strain formulation is utilized, u, isused as a component of displacement, ut for a velocity component, and v, for a nodal pointdisplacement.

FIG. 11.3 (a) Deformed grid pattern in flange drawing, (b) Effective strain contours in flangedrawing.

197


Z

o RFIG. 11.4 Out-of-plane deformation of a ring element.

where dn-m denotes the variational functional for infinitesimal deformationgiven by eq. (11.11).

In axisymmetric deformation of a membrane, the principal stressdirections remain the same with respect to the material fiber, i.e., themeridian and circumferential directions, during deformation. We definethe logarithmic strain-increment during the time-increment Af as

In eq. (11.16), A*0 and As are the lengths of an incremental line elementin the meridian direction before and after the time increment, respectively,and R0 and R are the radial positions of a material point before and afterthe time increment, respectively. The components of the strain-rate can beexpressed by

For axisymmetric deformation, the sheet is assumed to have normalanisotropy, represented by the r-value, and the constitutive equation,corresponding to eq. (11.6), can be expressed by

given in Fig. 11.4 is the result of incomplete description of deformation bythe membrane theory.

To obtain an adequate description utilizing the membrane theory, it isassumed that the mode of deformation can be determined by thevariational functional integrated over the finite time-increment Af, namely,


where

and

Note that the components of strain-rate and stress are functions of time,and that the coordinate axis with which these quantities are denned rotateswith the material fiber. Owing to the work-hardening, a in eq. (11.18b) isalso a function of time and is expressed as

where H' = do/de and is the slope of the true stress vs. true strain curve ofthe material.

Substituting eqs. (11.17) and (11.19) into eq. (11.15), we can readilyobtain the variational functional integrated over a time-increment At as

where T is the thickness of the sheet, A is the surface area of the sheet,and the effective strain increment AE is expressed as

assuming that strain-rate ratios remain constant during a time-incrementAnneq. (11.18c).

Finite-Element Formulation

In the finite-element formulation, the sheet geometry is approximated by aseries of conical frustums, as shown in Fig. 11.5, treating each conicalfrustum as a line element. The displacement within the element isexpressed by

where u^a) and u^ are the radial and axial displacements at the ath node,respectively, and the summation is performed for the two nodes. Theshape function qa is written in the natural coordinates as

where -l<f <1.Since the element used is linear, the meridian strain-increment is

constant in each element. The lengths of the undeformed and deformed


z

FIG. 11.5 Approximation of the sheet geometry by a series of conical frustrums.

elements, sn and s, are expressed by

The radial positions of a material point before and after the deformation,,/?0 and R, can be expressed by

The logarithmic strain-increment can be calculated by simply substitutingeqs. (11.22) and (11.23) into eq. (11.16). The effective strain-incrementcan be written in vector form as

where AET = {A.E,, A£e} and D is a matrix given by

similar to eq. (11.14c).The solution procedure for the functional given by eq. (11.20) is similar

to that described in Chap. 6. The first derivative of the functional withrespect to nodal displacement v, can be expressed by

where

Note that the derivatives of strain components are given by

Assembling the eqs. (11.24) and (11.25) for all elements in the finite-element scheme, we obtain the linear simultaneous equations

The solution of eq. (11.26) can be obtained by the technique described inChap. 7. The finite-strain formulation described in this section was derivedby Kim et al. [6,7]. This formulation is valid for rate-insensitive materials.Similar formulations that are based on rigid-viscoplastic material havebeen derived by Rebelo and Kobayashi [14], and also by Park et al. [15].In both formulations, strain-rate is defined by A£/Af, which is an averagestrain rate over a time-interval A?.

11.5 Axisymmetric Punch-Stretching and Deep-Drawing Processes

Punch-stretching and deep-drawing processes with hemispherical punchesare analyzed by using the formulation described in Section 11.4 [6,7].

Boundary Conditions with Hemispherical Punch

The boundary conditions in these processes are prescribed not only bytractions and incremental displacements but also by their ratios. Further,the radial and axial positions of the material elements in the contact regionare not independent; they are related to each other through a mathemati-cal expression representing the geometrical requirement that they must beactually on the surface of the punch head. The hemispherical surface of thepunch head (Fig. 11.6) can be expressed by

where r0 and z0 are radial and axial positions of a nodal point at thepresent configuration, ur and uz are the increments of radial and axial


The second derivative of the functional can be written as

where


z

FIG. 11.6 Geometrical requirement for a node on the contact surface.

displacements, and c is a parameter related to the punch depth d as shownin Fig. 11.6.

Recall that the finite-element formulation has already been linearizedand what it really solves for are the perturbation terms. Therefore, we alsolinearize the boundary condition (11.27) in terms of perturbation (orcorrection) of displacements ur and uz, as

where starred (*) quantities are initial guesses and Awr, A«2 are perturba-tions. By rearranging eq. (11.28) we have

where

and

In order to implement Coulomb friction between sheet and punch, atangential frictional stress is assumed and a converged solution is obtained.Then, from the computed nodal-point forces, the friction coefficient iscalculated. If the computed friction coefficient is not what was intended,then tangential force is modified and the solution process is repeated. It


should be noted here that the correction of frictional force requires themodification only in the F vector in eq. (11.20). The deformation step iscontrolled by the punch-head increment. The optimum step size wasshown to be the one that results in the effective strain-increment of 0.04[16].

Punch StretchingThe present rigid-plastic finite-element method has been compared else-where [6] with the finite-difference methods by Wang and by Woo. Theagreement of the solutions from the two numerical methods was excellent.Here, the rigid-plastic finite-element solutions are first compared withexperiments. The effect of interface friction on the detailed mechanics isthen examined by evaluating the results of the computations.

1. Comparison with experiments. For soft copper (experiment byKaftanoglu and Alexander [17]), the parameters are:

• Stress-strain characteristics: o = 68394(0.0122 + g)° 3789 psi

= 471.56(0.0122 + ef3789 MN/m2

• Thickness: 1.219mm (0.048 in.)• Friction condition: PFTE film lubricant• Radius of the sheet: 18.21 mm (0.717 in.)• Punch radius: 16.51mm (0.65 in.)

Kaftanoglu reports that the friction condition changes with deformationand measures three different friction coefficients: ^ = 0.2 at stage 1,/* = 0.135 at stage 2, and // = 0.07 at stage 3. To include the variation ofthe friction coefficient in the analysis, more information on the frictionhistory is needed. Therefore, as a representative value, we use the mean ofthree values of the friction coefficient, ^ = 0.135, for our computation.Figures 11.7 and 11.8 show the distribution of the circumferential strain

FIG. 11.7 Comparison of the circumferential strain distributions between computed andexperimental [17] results, at three punch positions.


FIG. 11.8 Comparison of the computed results with experiment [17] for thickness straindistribution, at three punch positions.

and thickness strain. The agreement between the experimental data andthe numerical solution is reasonable, considering the fact that the exactfriction condition is not known.

2. Effect of friction. The parameters used for computation are:

• Stress-strain characteristics: a = Ce° 23 (C a constant)• Normal anisotropy: r = 1.27• Punch radius: Rp = 1.0• Radius of the blank: Rb = 1.0

The material that corresponds to the characteristics above is aluminum-killed steel. With other parameters held fixed, three different frictioncoefficients, n = 0.1, 0.2, and 0.3, are assumed. As shown in Fig. 11.9, ingeneral friction does not affect the punch load, even though punch loadslightly increases with the increase in friction at later stages of deforma-tion. The arrows in Fig. 11.9 indicate the limiting stage when the polarregion of the sheet bounded by the ring element becomes rigid. This rigidzone is observed numerically when the computed value of the effectivestrain-increment is vanishingly small.

The effect of friction on the strain distributions is shown in Fig. 11.10.At a given punch depth, lower friction gives more thinning over the zonethat is in contact with the punch, while less thinning is observed in theunsupported region. Over the unsupported region, the frictional effect onthe strain distribution is not as significant as in the contact region, and atthe initial stages of deformation it is almost unnoticeable.

Deep Drawing with Hemispherical Punch [6]

In deep drawing, a circular sheet of metal is placed between the blankholder and the die and then fully drawn into the shape of a cup by a punch(see Fig. 2.27 in Chap. 2 for the process description). The formability ismeasured by the maximum blank size that can be drawn without a failure,

205

FIG. 11.9 Punch load vs. punch depth in deep drawing with hemispherical punch fordifferent coefficients of friction.

or more often, by the ratio of the blank diameter to the punch diameter.This ratio is called the limiting drawing ratio and the test used to determinethe limiting drawing ratio is called the Swift test.

Deep drawing is not only a useful method of material testing, but alsoone of the basic operations in sheet-metal stamping. In practice, variousshapes may be used for the bottom of the punch; however, most past

CURRENT RADIUS /BLANK RADIUS

FIG. 11.10 Effect of friction on thickness strain distribution at punch depth (d/R ) = 0.25and 0.45.

Sheet-Metal Forming


investigations have been on deep drawing with a flat-bottomed punch.A difference between deep drawing with a hemispherical punch and

punch stretching with a round die corner is the presence of the flange,which is free to slide over the die. The blank holding force is implementedin the formulation as a tangential friction force acting on the last nodelocated at the rim of the sheet. The increment of deformation is controlledby the punch movement.

Woo [18] analyzed drawing with hemispherical punch by dividing deepdrawing into two component processes of (1) pure radial drawing over theflange and (2) punch stretching over the hemispherical punch in hisanalysis. He used the finite-difference method, and also conductedexperiments. The parameters used by Woo are:

• Material: Soft copper• Stress-strain characteristics: a = 5.4 + 27.8(e)0504 ton/in2

= (0.082 08 + 0.422 568e0 504)xlO9 N/m2(£<0.36)

= 5.4 + 24.4(e)°-375 ton/in2

= (0.082 08+ 0.3708e0'375)xlO9 N/m2(g>0.36)

• Blank radius: 55.88mm (2.2 in.), thickness: 0.889mm (0.035 in.)• Radius of die throat: 53.92mm (2.123 in.)• Radius of die profile: 12.7mm (0.5 in.)• Radius of punch head: 25.4mm (1.0 in.)• Blank holding force: 500kg (0.5 ton.)

As shown in Figs. 11.11 and 11.12, the solution by the rigid-plastic FEMis in excellent agreement with the experiment for the flange part: however,over the punch head it predicts more straining than does the experimentwhen the friction coefficient of 0.04, as Woo suggested, is assigned for thecontact regions over the punch head and over the die. The deviation of thenumerical solution from the experimental data becomes larger as deforma-tion progresses, which is reflected in the punch load vs. punch depthrelationship shown in Fig. 11.13.

The lubricant used in the experiment was graphite in tallow. In theanalysis the practical difficulty always lies in the determination of areasonable value of friction coefficient, because in real sheet-metal formingoperations, the friction coefficient is difficult to measure and may evenchange during deformation.

11.6 Sheet-Metal Forming of General Shapes

Variational FormulationAs shown in Section 11.4 for axisymmetric deformation, the solution of theout-of-plane deformation of general shapes can be obtained through the


FIG. 11.11 Distribution of thickness strains for [ip=Q.Q4, ;irf = 0.04, and comparison withexperiment [18].

functional integrated over the finite time duration At. It has been shown inSection 11.4 that the variational functional dn is integrated over the finitetime-increment A/ can be expressed in terms of the logarithmic strain-increment, provided that the directions of principal strains are known inadvance. For a general out-of-plane deformation, the principal stretchdirection is not known in advance owing to the shear strain component inthe plane of the sheet.

FIG. 11.12 Distribution of circumferential strains for ftp=0.04, jUrf = 0.04, and comparisonwith experiment [18].


Punch Depth

FIG. 11.13 Punch load vs. punch depth, comparison of predictions with experiment [18].

In order to simplify the variational functional, it is assumed that therigid-plastic constitutive equation given by eq. (11.6) holds between thePiola-Kirchhoff stress (second) stj and the Lagrangian strain-rate Etj withthe reference state at t0, which is the beginning of the small but finitetime-increment Af. The Piola-Kirchhoff stress and Lagrangian strain-ratewere introduced in Chap. 4. The components of Lagrangian strain areexpressed by

where «, is a component of displacement. Note that the displacement of aparticle at t = t0 is zero. The component of the Lagrangian strain-rate isexpressed by

where the dot indicates the time derivative. The Piola-Kirchhoff stress sapis related to the Cauchy stress a,-,- according to

where / is the determinant of the deformation gradient matrix and thedeformation gradient xit<x is expressed by

Noting that o,-, = s/,- at t = t0, the variational functional integrated over atime-increment At is obtained, similarly to eq. (11.20), as


where h' = ds/dE, and s is defined in terms of s,y in the same form for agiven by eq. (11.13c).

The slope h' is related to H'( = dolde) according to h' =H' -2o(t0).The derivation of this relationship can be found in References [11] and[19].

The effective strain-increment can be written in the matrix form as

where

The matrix D in eq. (11.35) is identical to eq. (11.14c). Once the solutionof the displacement increment is obtained, the geometry is updated. Theupdated geometry becomes the reference state for the next time incrementAf.

Finite-Element Formulation

Let the sheet-metal domain be decomposed into an assemblage of lineartriangular elements. A local Cartesian coordinate system is assigned toeach element in such a way that the (x-y) coordinate plane coincides withthat of the element plane and the z axis is normal to the element plane.The displacement field within the element can be described by

where qa is the shape function of the linear triangular element given in eq.(6.10) in Chap. 6. Equation (11.36) can be rewritten in a matrix form as

where vT={41), u«\ «<'>, u™, uf\ u(?, u?\ uf\ u<3>}.The component of displacement gradient can be written as

where Xa and Ya are defined in eq. (6.33a) in Chap. 6.The components of Lagrangian strain-increment defined by eq. (11.30)


are expressed in terms of displacement gradient by

The first derivative of the functional, eq. (11.34), with respect to thenodal displacement can be written

where

etc.Note that 3AE/3u7 is also a function of the nodal displacements,

because of the nonlinear terms involved in the expression of AE. Thesolution of eq. (11.39) is obtained iteratively by the Newton-Raphsonmethod described in Chap. 7. The details of the stiffness matrix evaluationcan be found in Reference [19].

Since the element stiffness equations are evaluated based on the localcoordinate system, the elemental stiffness matrices should be transformedinto the global coordinate system before they are assembled.

11.7 Square-Cup Drawing Process [11,12]The finite-element formulation presented in Section 11.6 has been appliedto the square-cup drawing process. Figure 11.14 shows a schematic view ofthe square-cup drawing process. In the simulation of this process, theCoulomb friction law is assumed at the sheet-tool interface. The blank-

Here, we have


FIG. 11.14 Schematic view of the square-cup drawing process.

holding force is treated as a concentrated vertical load acting on the nodalpoints that are located at the outer rim of the blank.

Two sets of computational conditions were used in the finite-elementsimulations. One is cup drawing from a square blank and the other is cupdrawing from various blank shapes.

1. Cup drawing with square blanks• Material: aluminum-killed steel• Stress-strain characteristic: a = 107 183£° 228 psi

= 739£a228MN/m2

• r- Value: 1.6• Blank size: 110 x 110 mm (4.33 x 4.33 in.)• Material thickness: 0.86mm (0.034 in.)• Punch size, a: 40 x 40 mm (1.575 X 1.575 in.)• Punch radius, Rp: 5mm (0.197 in.)• Punch corner radius, Rc: 3.2mm (0.126 in.)• Die opening, b: 42.5 x 42.5 mm (1.67 x 1.67 in.)• Die corner radius, RD: 5mm (0.197 in.)• Coefficients of friction: up=0.2, uD=0.04 for punch and die

respectively.• Blank holding force: 500 kg.

The FEM mesh that was used in the square-cup drawing simulation isshown in Fig. 11.15. Figures 11.16 and 11.17 show the thickness straindistributions across the diagonal and the transverse directions of a square

Forming and the Finite-Element Method

FIG. 11.15 Finite-element mesh used in the square-cup drawing simulation.

cup, respectively. The data obtained experimentally by Thomson [20] arealso included. Maximum thinning occurs around the regions of punch anddie radii, particularly in the corner radii of the forming tools. The trends ofthe strain distributions predicted by FEM are consistent with the ex-perimental data. However, different strain magnitudes are observed in thenumerical prediction. The discrepancy between the predicted and ex-perimental results may be attributed to the larger cup height achieved inthe experiment and the coefficients of friction used in the numericalcomputation. The variations of the strain distributions across the diagonalof the drawn cup are quite drastic as compared to those in the transversedirection. Hence, it is obvious that the potential failure site in deepdrawing of a square cup is located along the diagonal of the drawn cup andin the vicinity of the die corner where thinning is maximum. This fact hasbeen shown experimentally by several investigators [21,22]. Figure 11.18shows a square cup formed by the finite-element simulation at a punchdepth of 20.2 mm (0.795 in.).

2. Cup drawing with various blank shapesThe material flow, final flange configurations, punch load, and the punch

depth are noted and compared by using various blank shapes of identicalinitial surface area. The material and process variables used in the

212

initial Distance from Center of Blank,mm

FIG. 11.16 Comparison of the numerical solutions with the experimental data [20] forthickness strain distributions across the diagonal of a drawn cup.

FIG. 11.17 Comparison of the numerical solutions with the experimental data [20] forthickness strain distributions in the transverse direction of a drawn cup.

213


1/4 OF SQUARE CUP

FIG. 11.18 The square cup formed by the FEM simulation [19].

simulations are:

• Sheet material: AISI 304 stainless steel• Stress-strain characteristic, CT = 219.6£°'43ksi

= 1514e° 43 MN/m2

• r-value: 1.025• Sheet thickness: 0.76 mm (0.03 in.)• Coefficient of friction: \ip = \ID = 0.04• Other dimensions for the punch and die are the same as those used in

drawing with square blanks.

Figure 11.19 shows the blank shapes and corresponding finite-elementmeshes used in the process simulations. The same coefficient of frictionwas used between the punch head, the die surface, and the sheet metal.The deformation is controlled by incremental punch advancement and theblank holding force is set equal to zero. The results of the computation arecompared with experiments [12].

The punch load versus displacement curves are shown in Fig. 11.20 forvarious blank shapes. The finite-element solutions and experimental data

FIG. 11.19 Blank shapes and finite-element domains used in the square-cup drawingsimulations: (a) square; (b) octagonal; (c) circular.

215


Punch Depth, mm

FIG. 11.20 Load-displacement curves in square-cup drawing processes with various initialblank shapes.

are in good agreement in the early stages of deformation. However, forpunch depth larger than 6mm (0.236 in.), the discrepancies between thetheory and experiment become increasingly significant. This may beattributable to the fact that the lubricant is squeezed out at the corners offorming tools as the deformation proceeds, and the friction becomesappreciably higher than the value of fi = 0.04 assumed in the simulations.Another source for discrepancy may be the fact that constant-gap blankholding was used in experiments, while the blank holding force is assumedto be zero in computations. It is noted, however, that the load-displacement curves predicted by the finite-element method are consistentwith the experimental data, in that the circular blank requires the largestload to deform as compared to other shapes, particularly at larger punchpenetration.

Strain distributions of the drawn cups across the transverse and diagonaldirections obtained by the finite-element method are shown in Fig. 11.21for various blank shapes. The strain distributions across the cup walls arealmost identical, but different amounts of thickening result in flangeportions due to the blank shapes. Comparisons of the deformed flange


FIG. 11.21 Thickness strain distributions in square cup. (a) Transverse direction; (b)diagonal direction [12].

contours predicted by the finite-element method with the experimentalresults are shown in Fig. 11.22 for various blank shapes. Owing to thedifferences in metal flow rates, depending on the blank shape, the resultingflange configurations differ. The earing behavior is most significant in asquare blank, whereas, in a circular blank, earing is reduced to a minimumat the same stage of deformation. The predicted flange configurations arein excellent agreement with the experimental results. Photographs ofcompletely drawn square cups with various blank shapes are shown in Fig.11.22. It is to be noted that blanks with orientations of 0°, 45°, and 90° withrespect to the rolling direction were tested, but the results are of nosignificant difference since AISI 304 sheet exhibits little planar anisotropicbehavior.

11.8 Nonquadratic Yield CriterionFor in-plane isotropy (/ = g), eq. (11.4a) reduces, in terms of principalstresses, to

FIG. 11.22 Comparisons of the deformed flange shapes predicted by FEM (dotted line) andexperiment (solid line) [12].

218


Equation (11.5) becomes

Under uniaxial tension (a1, o2) = (ou, 0), then eqs. (11.41) and (11.42)give, respectively,

Under equibiaxial tension, (a

where r is the ratio of transverse to through-thickness strain-rates under ou.Woodthorpe and Pearce [23] found r < 1 but ab/ou>l for some

materials. This pairing of inequalities is in conflict with eq. (11.44). Thus,Hill [24] proposed the following nonquadratic yield criterion that allowsflexibility than the quadratic:

where loading is coaxial with the orthotropy. The coefficients /, g, etc.characterize the anisotropy, a is a scaling factor with units of stress, andm > 1 to ensure convexity of the yield locus. Hill suggested four simpleversions of eq. (11.45) for planar isotropy and particular values ofcoefficients.

Parmar and Mellor [25, 26] used one of the special cases for predictingthe limit strains occurring in frictionless in-plane stretching and fordetermining the plastic stress and strain distributions in bore expanding.The special case was obtained by taking a = b = 0 and / = g = 0 in eq.(11.45), as

We have, instead of eq. (11.43),

Then eq. (11.46) becomes

and instead of eq. (11.44), we now have

The predicted behavior is therefore anomalous with m such that ob/ou< 1,when r > 1 or ob/ou > 1 when r < 1.

andwe have


Recently, Kobayashi et al. [27] examined four special cases suggested byHill, using experimental data for various anisotropic sheet metals. Theyconcluded that all four cases give reasonable results from a practical pointof view and that there is a preference for the case given by eq. (11.46).

The strain-rate ratio associated with eq. (11.48) is readily obtained fromthe flow rule as

with the ordering of o-i > o2. In eq. (11.50) the effective strain-rate e isintroduced from

oj = 0^1 + CT2e2 = 1(0! + o2)(i-l + e2) + 1(0! - CT^E! - £2).

The explicit formula for e can be obtained by eliminating ol and o2 fromeqs. (11.48) and (11.50) as

Wang [28] implemented the yield criterion, eq. (11.46), into the finite-element procedure and applied the method for hemispherical punchstretching to assess the effect of m-value on the resulting straindistributions.

References1. Mehta, H. S., and Kobayashi, S., (1971), "Finite-element Analysis and

Experimental Investigation of Sheet-metal Stretching," Rep. No. MD 71-2,University of California.

2. Yamada, Y., (1969), "Recent Japanese Developments in Matrix DisplacementMethod for Elasto-Plastic Problems," Paper presented at Japan-U.S. Seminaron Matrix Methods of Structural Analysis and Design, Tokyo, Japan.

3. Lee, C. H., and Kobayashi, S., (1973), "New Solutions to Rigid-PlasticDeformation Problems using a Matrix Method," Trans. ASME, J. Engr. Ind.,Vol. 95, p. 865.

4. Lee, S. H., and Kobayashi, S., (1975), "Rigid-Plastic Analysis of BoreExpanding and Flange Drawing with Anisotropic Sheet Metals by the MatrixMethod," Proc. 15th Int. Mach. Tool Des. Res. Conf. p. 561.

5. Wang, N. M., (1970), "A Variational Method for Problems of Large PlasticDeformation of Metal Sheets," General Motors Research Publication GMR-1038.

6. Kim, J. H., and Kobayashi, S., (1978), Deformation Analysis of AxisymmetricSheet Metal Forming Processes by the Rigid-Plastic Finite Element Method,in, "Mechanics of Sheet Metal Forming," (Edited by D. P. Koistinen, andN. M. Wang), Plenum Press, New York, p. 341.

7. Kim, J. H., Oh, S. I., and Kobayashi, S., (1978), "Analysis of Stretching ofSheet Metals with Hemispherical Punch," Int. J. Machine Tool Des. Res., Vol18, p. 209.

8. Win, S. A., (1976), "An Incremental Complete Solution of the Stretch-forming and Deep-drawing of a Circular Blank Using a Hemispherical Punch,"Int. J. Mech. ScL, Vol. 18, p. 23.


9. Wang, N. M., and Budiansky, B., (1978), "Analysis of Sheet Metal Stampingby a Finite-element Method," General Motors Research Publication GMR-2423.

10. Onate, E., and Zienkiewicz, O. C., (1983), "A Viscous Shell Formulation forthe Analysis of Thin Sheet Metal Forming," Int. J. Mech. ScL, Vol. 25, p. 305.

11. Toh, C. H., and Kobayashi, S., (1983), "Finite-element Process Modeling ofSheet Metal Forming of General Shapes," Grundlagen der Umformtechnik ISymposium, Stuttgart, p. 39.

12. Toh, C. H., and Kobayashi, S., (1985), "Deformation Analysis and BlankDesign in Square Cup Drawing," Int. J. Machine Tool Des. Res, Vol. 25, No.1, p. 15.

13. Hill, R., (1950), "Mathematical Theory of Plasticity," Oxford UniversityPress, London.

14. Rebelo, N., and Kobayashi, S., (1980), "Axisymmetric Punch Stretching ofStrain Rate Sensitive Sheet Metals," Proc. 8th NAMRC, University ofMissouri, Rolla, MO., p. 235.

15. Park, J. J., Oh, S. I., and Altan, T., (1987), "Analyses of Axisymmetric SheetForming Processes by Rigid-Viscoplastic Finite Element Method," Trans.ASME, J. Engr. Ind., Vol. 109, p. 347.

16. Kim, J. H., (1977), "Analysis of Sheet Metal Forming by the Finite-ElementMethod," Ph.D. Dissertation, University of California, Berkeley.

17. Kaftanoglu, B., and Alexander, J. M., (1970), "On Quasistatic Axisymmetri-cal Stretch Forming," Int. J. Mech. Sci. Vol. 12, p. 1065.

18. Wood, D. M., (1968), "On the Complete Solutions of the Deep DrawingProblem," Int. J. Mech. Sci. Vol. 10, p. 83.

19. Toh, C. H., (1983), "Process Modeling of Sheet Forming of General Shapesby the Finite Element Method Based on Large Strain formulation," Ph.D.Dissertation, University of California, Berkeley.

20. Thomson, T. R., (1975), "Influence of Material Properties in the Forming ofSquare Shells," /. Australian Inst. Metals, Vol. 20, No. 2, p. 106.

21. Yoshida, K., and Miyauchi, K., (1978), Experimental Studies of MaterialBehavior as Related to Sheet Metal Forming, in "Mechanics of Sheet MetalForming" (Edited by D. P. Koistinen, and N. M. Wang), Plenum Press, NewYork, p. 19.

22. El-Walkil, D., Kamal, M. N. E., and Darwish, A. H., (1980), "Mechanics ofthe Square Box Drawing Operation of Aluminum Blanks," Sheet MetalIndustry, August, p. 679.

23. Woodthorpe, J., and Pearce, R., (1970), "The Anomalous Behavior ofAluminum Sheet under Balanced Biaxial Tension." Int. J. Mech. Sci., Vol. 12,p. 341.

24. Hill, R., (1979), "Theoretical Plasticity of Textured Aggregates," Math. Proc.Camb. Phil. Soc., Vol. 85, p. 179.

25. Parmar, A., and Mellor, P. B., (1978), "Predictions of Limit Strains in SheetMetal Using a More General Yield Criterion," Int. J. Mech. Sci., Vol. 20, p.385.

26. Parmar, A., and Mellor, P. B., (1978), "Plastic Expansion of a Circular Holein Sheet Metal Subjected to Biaxial Tensile Stress," Int. J. Mech. Sci., Vol. 20,p. 707.

27. Kobayashi, S., Caddell, R. M., and Hosford, W. F., (1985), "Examination ofHill's Latest Yield Criterion Using Experimental Data for Various AnisotropicSheet Metals," Int. J. Mech. Sci., Vol. 27, p. 509.

28. Wang, N. M., (1984), A Rigid-Plastic Rate-sensitive Finite Element Methodfor Modeling Sheet Metal Forming Processes, in "Numerical Analysis ofForming Processes," (Edited by J. F. T. Pittman et al.), Wiley, New York, p.117.

12THERMO-VISCOPLASTIC ANALYSIS

12.1 IntroductionThe main concern here is the analysis of plastic deformation processes inthe warm and hot forming regimes. When deformation takes place at hightemperatures, material properties can vary considerably with temperature.Heat is generated during a metal-forming process, and if dies are at aconsiderably lower temperature than the workpiece, the heat loss byconduction to the dies and by radiation and convection to the environmentcan result in severe temperature gradients within the workpiece. Thus, theconsideration of temperature effects in the analysis of metal-formingproblems is very important. Furthermore, at elevated temperatures, plasticdeformation can induce phase transformations and alterations in grainstructures that, in turn, can modify the flow stress of the workpiecematerial as well as other mechanical properties.

Since materials at elevated temperatures are usually rate-sensitive, acomplete analysis of hot forming requires two considerations—the effect ofthe rate-sensitivity of materials and the coupling of the metal flow and heattransfer analyses.

A material behavior that exhibits rate sensitivity is called viscoplastic. Atheory that deals with viscoplasticity was described in Chap. 4. It wasshown that the governing equations for deformation of viscoplasticmaterials are formally identical to those of plastic materials, except thatthe effective stress is a function of strain, strain-rate, and temperature. Theapplication of the finite-element method to the analysis of metal-formingprocesses using rigid-plastic materials leads to a simple extension of themethod to rigid-viscoplastic materials [1].

The importance of temperature calculations during a metal-formingprocess has been recognized for a long time. Until recently, the majority ofthe work had been based on procedures that uncouple the problem of heattransfer from the metal deformation problem. Several researchers haveused the following approach. They determined the flow velocity fields inthe problem either experimentally or by calculations, and they then usedthese fields to calculate heat generation. Examples of this approach are theworks of Johnson and Kudo [2] on extrusion, and of Tay et al. [3] onmachining. Another approach [4] uses Bishop's numerical method in which

222

Thermo-Viscoplastic Analysis 223

heat generation and transportation are considered to occur instantaneouslyfor each time-step with conduction taking place during the time-step. Thistechnique was used by Altan and Kobayashi [5] on extrusion; by Lahotiand Altan [6] on compression and torsion; and by Nagpal, et al. [7] onforging. Usually the temperature calculations are done by using finitedifferences, or finite elements, and the upper-bound technique is the mostcommon method for determining flow patterns theoretically.

In order to handle a coupled thermo-viscoplastic deformation problem,it is necessary to solve simultaneously the material-flow problem for agiven temperature distribution and the heat transfer equations. Numericalsolutions of such forming problems were discussed by Zienkiewicz et al. [8]with examples of steady flow in extrusion, drawing, rolling, and sheet-metal forming. Zienkiewicz et al. [9] made a coupled thermal analysis ofsteady-state extrusion. Rebelo and Kobayashi [10,11] developed themethod for a coupled analysis of transient viscoplastic deformation andheat transfer. They applied the method to solid cylinder compression andring compression.

12.2 Viscoplastic Analysis of Compression of a Solid CylinderThere are a number of materials that exhibit viscoplastic behavior. Theyinclude most metals at high temperature, superplastic materials, heatedglass, and polymers. When the deformation is large, most of them can beconsidered to be rigid-viscoplastic.

Because of the importance of the application of viscoplastic behavior tothe metal-forming processes, the treatment of time-dependent materialbehavior within the framework of the theory of viscoplasticity is thesubject of several recent studies. Cristescu [12,13] applied the theory tothe upper-bound approach in drawing. Zienkiewicz and Godbole [14, 15]have shown the feasibility of the finite-element approach in the deforma-tion analysis of rigid-viscoplastic materials by treating them as non-Newtonian viscous fluids. Price and Alexander [16,17] have applied thisformulation to creep forming.

There are only a few references in the literature in which strain-rateeffects in non-uniform deformation are explicitly in evidence. Amongthem, the work by Klemz and Hashmi [18] on simple upsetting of leadcylinders at room temperatures provides good experimental results forcomparison with the finite-element solution, since lead is strain-ratesensitive at room temperatures. A cylinder having a diameter of 25.4mm(1.0 in.) and a height of 24.13mm (0.95 in.) was considered. The stress,strain, and strain-rate data for pure lead at room temperature were takenfrom Reference [18] and from experimental results by Loizou and Sims[19]. The static stress—strain curve was approximated by fitting experimen-tal data to curves of the form o = Y[l + (e/y)m], and the obtained valuesof strain-rate exponent m and ym were interpolated in the program forintermediate strain values. The friction factor m was taken as 0.06 (m = a


fraction of the static yield shear strength), in agreement with the assumedvalue in Reference [18].

For the finite-element formulation of viscoplastic deformation, a par-ticular attention should be paid to the strain-rate sensitivity of the flowstress of the material. The linearized stiffness equations (6.4) or (6.48) inChap. 6 contains the second derivative of the functional given by eq.(6.44), in which the effective stress a is now a function of the effectivestrain-rate e as well as the effective strain e.

The calculations simulate the deformation of the cylinder by a drop-hammer, with a tup mass of 35.5 Ib, hitting the specimen at a speed of30 ft/s. The assumption of a quasistatic process is justified for thesevelocities [20]. Accordingly, the calculated work done by the contact forcesbetween tup and workpiece in each step of deformation was subtractedfrom the kinetic energy of the tup, and the process ended when the tupcame to a stop. For comparison, the calculations were repeated withoutintroducing strain-rate effects (making y = °o in the formulation).

This nonsteady-state deformation problem was analyzed in a step-by-step manner by treating it quasilinearly during each incremental deforma-tion. The reduction in height at each step was 1%. The solution of thevelocity field for uniform compression was used as an initial guess. Thesolution obtained from the previous step was then used as an initial guessfor the subsequent step. Normally, seven iterations were required for thefirst step to reach an accuracy of ||Av||/||v|| < 0.000 05. For subsequentsteps, only two to four iterations were necessary to reach the sameaccuracy.

The calculated overall quantities are compared with experimental dataand with the numerical dynamic analysis of Klemz and Hashmi [18] inFigs. 12.1 and 12.2. Some observations can be made from the figures. Theresults for nonstrain-rate-sensitive analysis (Fig. 12.1) tend to follow thoseobtained by the analysis of Klemz and Hashmi, which did not take intoaccount strain-rate effects but was based on a dynamic analysis. Theagreement of the two analyses indicates that the dynamic effect is

FIG. 12.1 Comparison of the finite-element rigid-plastic analysis (•) with dynamic analysis( ) [18] aid experiment (—) [18].


FIG. 12.2 Comparison of the finite element rigid-viscoplastic analysis (•) with dynamicanalysis ( ) [18] and experiment (—) [18].

negligible. However, both solutions show poor agreement with experi-ments. The analysis that includes strain-rate sensitivity shows goodagreement with experiments (Fig. 12.2). No "rigid" zones were formed atsuch a low value of friction in either rate-sensitive or rate-insensitive cases.

12.3 Heat Transfer AnalysisBasic Equations

The basic equations and corresponding finite-element formulations for thedeformation analysis are described in previous chapters. For the heattransfer analysis, we begin with the energy balance equation, expressed by

where kl Ttii is the heat transfer rate, k^ denotes thermal conductivity, f isthe heat generation rate, and peris the internal energy-rate. In eq. (12.1),the notation TiU is used for T,,_,- , with the comma denoting differentiationand repeated subscript meaning summation (Laplace differential operatorapplied to temperature T). We consider that the heat generation in thedeforming body is due only to plastic deformation

where the heat generation efficiency K, represents the fraction of mechani-cal energy transformed into heat and is usually assumed to be 0.9. Thefraction of the remainder of the plastic deformation energy (1 - K) isexpended to cause changes in dislocation density, grain boundaries, andphases. This energy is usually recoverable by annealing.

Along the boundaries of the deforming material, either the temperatureT is prescribed or a heat flux is given. The energy balance, eq. (12.1), canbe written in the form


for arbitrary variation in temperature 6T. By using the divergencetheorem, eq. (12.3) becomes

where qn is the heat flux across the boundary surface 5g, n denotes the unitnormal to the boundary surface, and

Solutions to problems of this nature require the temperature field to satisfythe prescribed boundary temperatures and eq. (12.4) for arbitrary pertur-bation dT.

Finite-Element FormulationThe temperature field in eq. (12.4) is approximated by

where qa is the shape function and Ta is the temperature at ath node.With the quadrilateral element shown in Fig. 8.1,

and

where q\, . . . , q* are given by eq. (8.2b) in Chap. 8.Putting

and substituting eq. (12.6) into eq. (12.4),

Because of the arbitrariness of 6T, the following system of equations isobtained,


Equations (12.9) can be expressed in the form

where C is the heat capacity matrix, Kc is the heat conduction matrix, Q isthe heat flux vector, T is the vector of nodal point temperatures, and T thevector of nodal point temperature-rates. The heat flux vector Q in eq.(12.10) has several components and is expressed with the interpolationfunction N by

The first term on the right is the heat, generated by plastic deformationinside the deforming body. The second term defines the contribution of theheat radiated from the environment to the element, where a is theStefan-Boltzman constant, e is the emissivity, and Te and Ts are environ-ment and surface temperatures, respectively. The third term describes theheat convected from the body surface to the environment with heatconvection coefficient h. The fourth term represents the contribution of theheat transferred from the workpiece to the die through their interface. Td

and Tw are die and workpiece temperatures, respectively, and /ilub is theheat transfer coefficient for the lubricant. The last term is the contributionof the heat generated by friction along the die-workpiece interface, qf

being the surface heat generation rate due to friction.The theory necessary to integrate (12.10) can be found in numerical

analysis books; see for instance Dahlquist and Bjorck, [21]. The conver-gence of a scheme requires consistency and stability. Consistency issatisfied by an approximation of the type

where /3 is a parameter varying between 0 and 1, and t denotes time.For unconditional stability, /? should be greater than 0.5, and a value of

0.75 was chosen. Selection of a proper value of /? is an important factor insituations where it is desirable for the time step to be as large as possible,provided that the increments in strain are compatible with an infinitesimalanalysis.

12.4 Computational Procedures for Thermo-Yiscoplastic AnalysisWe treat the workpiece and the die separately, assuming that the dieproperties do not change. Thus we greatly reduce the number of equationsto be solved simultaneously, which, in turn, reduces the cost of thesolution. There is no internal heat generation in the die and thereforedeformation calculations are not necessary.


The heat generated through friction, qf, is calculated as

where fs is the friction stress, and \us\ the relative velocity between die andworkpiece. The heat qf is evenly distributed between the die anddeforming material.

It should be noted that the nodes on the die do not generally coincidewith those on the deforming material along the interface, and that thecalculation of nodal point temperatues requires interpolation. Whenboundary conditions such as the convection term in eq. (12.11) apply, theyare split into two parts, wherein one containing the unknown temperaturesis added to the matrix Kc. Boundary conditions such as the radiation termin eq. (12.11) are applied using previous iteration values for bodytemperatures.

The equations for the flow analysis and the temperature calculation arestrongly coupled, making a simultaneous solution of their finite-elementcounterparts necessary.

Considering Tt+&t as a primary dependent variable, we have, from eq.(12.12), with t = 0 initially,

The circumflex over T denotes

Substituting eq. (12.14) into eq. (12.10), gives

The coupling procedure makes use of eq. (12.15) through the followingsequence.

1. Assume the initial temperature field T0.2. Calculate the initial velocity field u corresponding to the tempera-

ture field T0.3. Calculate the initial temperature-rate field T0 from eq. (12.10) using

values from (1) and (2).4. Calculate the quantity T.5. Update the nodal point positions and the effective strain of elements

for the next step.6. Use the velocity field at the previous step to calculate the first

approximate temperature T such as

Thermo-Viscoplastic Analysis

7. Calculate a new velocity field with the solution of (6).8. Use the new velocity field to calculate the second temperature field

such as

9. Repeat steps for (7) and (8) until both have converged.10. Calculate the new temperature rate field TA,.11. Repeat steps from (5) to (10) until the desired deformation state is

reached.

The iteration process for temperature calculations is not likely to requiremuch computing time, because only the heat input vector Q is changedduring iterations and, as a result triangularization of the matrix isnecessary only once. Moreover, additional iterations necessary to obtain avelocity field after a new temperature field is obtained should be relativelyfew, because the velocity field does not show much sensitivity to smallvariations of the temperature field.

12.5 ApplicationsApplications of the thermo-viscoplastic analysis include compression of asolid steel cylinder [11,22], hot nosing of a steel shells [23], and forging oftitanium alloys [24, 25].

Compression of Steel Cylinder [11]

Pohl [26] conducted temperature measurements in order to test hisuncoupled analysis, in which approximate stream functions were used forthe deformation, and finite differences for the heat balance. A solidcylinder of a carbon steel AISI 1015 was compressed between flat dies atroom temperature. Thermocouples were inserted in the cylinder atdifferent locations. Upon deformation, their readings indicated the tem-perature increases due to the heat generation. Figure 12.3 shows thedimensions and locations of measuring points. The conditions used incomputations with the finite-element method [11] were as follows. Thedeformation took place at room temperature, until a reduction of 33% inheight was achieved. The finite-element grid was composed of 132four-node quadrilateral elements in the workpiece, and 119 in the die.Because of symmetry, only one-quarter of the cylinder needed to beanalyzed. The friction factor m was taken as 0.65.

The die velocity was changed at each time-step to simulate a mechanicalpress. Each step corresponded to 1% reduction in height, which wasequivalent to time-steps of up to 0.03 s. The flow stress was considered tobe independent of strain-rate and temperature and its values were given byPohl. The heat transfer characteristics, other than the thermal conductivityand the heat capacity of the AISI 1015, which were given by Pohl, weretaken from handbooks.

229

FIG. 12.3 Compression of a steel cylinder: specimen geometry and thermocouple locations, and comparison of temperature distribution between theory (—)and experiments [26].


The dimensions of the die were such that at the outside boundaries aconstant temperature was imposed. The temperature values measured byPohl are compared with the calculated values in Fig. 12.3. The agreementsare excellent for the internal points. For the three points near the billetsurface, however, the computed results indicate that the temperaturedifference for these points is small, while experiments show largerdifferences. This discrepancy may be attributed to inaccuracies in thematerial constants used in computations or to inaccuracies in experimentalmeasurements.

Hot Compression of Steel CylinderWu and Oh [22] developed an FEM-based computer program (ALPIDT)that is capable of simulating nonisothermal forming operations witharbitrarily shaped workpieces and dies. This program consists of twoindependent FEM programs, ALPID for viscoplastic deformation analysis[27] and the program for the heat transfer analysis. They are coupled in anefficient manner in the program ALPIDT for simulation of thermo-viscoplastic deformation. To demonstrate the capability of ALPIDT, a hotcompression process was simulated. The temperature changes during theinitial resting and dwelling periods were included in the simulation. Thesimulation also accounted for the changes in the heat transfer coefficientsbetween the workpiece and the die during the process.

A cylindrical billet of AISI 1020 steel was compressed between two flatdies. The initial billet temperature was 1232°C (2250°F) and the initial dietemperature was 204°C (400°F). In order to estimate the workpiecetemperature accurately, simulation was performed in the process consistingof three periods as follows:

1. The heated billet is placed on the lower die for 6 seconds withoutdeformation (free resting period).

2. The workpiece is compressed to 67% in height (deformation period),1.5 seconds).

3. The deformed workpiece stays on the lower die for 3 seconds afterthe upper die is retracted (dwelling period).

Detailed computational conditions are given in Reference [22].In Fig. 12.4 photographs of grid-distortions and temperature distribu-

tions are shown at various stages of the process. The temperature scale(°F) is also shown in the figure.

The predicted temperatures at the end of the resting period are shown inFig. 12.4a. Owing to relatively small loss of heat to the environment, thetemperatures of the upper die and of the upper portion of the workpieceremained almost unchanged. However, heat loss from the bottom portionof the billet to the lower die was considerable. The temperature at thebottom of the workpiece dropped by 280°C (536°F) during the free restingperiod of 6 s. At the same time, the surface of the lower die was heated tonearly 600°C (1112°F). It is to be noted that the flow stress of the


workpiece material was doubled when the temperature was reduced from1230° to 950°C (2246° to 1742°F).

The predicted temperature distributions and grid distortions duringthe deformation period are shown in Figs. 12.4b to e. At the beginning ofthe deformation period, the temperature of the workpiece at the upper dieinterface was higher than that at the lower die interface. However, thetemperature of the top surface dropped quickly after the upper diecontacted the workpiece. The bottom and top surface temperatures of theworkpiece become almost the same, reaching 700-800°C (1292-1472°F) at0.375 s after deformation started. However, it is noted that the tempera-ture gradient at the top was higher than at the bottom of the workpiece.These differences between the temperature distributions in the upper andlower regions of the workpiece became less as the deformation proceededfurther. The temperature distributions shown in Figs 12.4b to e suggestthat the flow stress of workpiece near the die workpiece interface couldhave been three times that in the mid-height region.

The predicted grid distortions reflect the effect of temperature on theflow stress of the workpiece, influencing metal flow. During the early stageof deformation (see Figs. 12.4b and c), the barreling near the top surfacewas more pronounced than near the bottom surface. It is also noticed thatthe upper surface moved radially more than the bottom surface, suggestingthat the average flow stress was higher near the bottom surface. It can bealso seen that, throughout the deformation period, the deformationoccurred mainly in the mid-height region, while the chilled regions nearthe die workpiece interfaces remained almost rigid (Figs. 12.4d and e).

Figure 12.4f shows the temperature distributions at 3 s after the upperdie was retracted from the workpiece at the end of deformation. It is seenfrom the figure that the top surface of the workpiece has warmed up again,while the average temperature of the workpiece had dropped. This wasdue to the redistribution of heat within the workpiece, without die chillingat the top surface of the workpiece. The predictions shown in Fig. 12.4agree with general observations made in nonisothermal forging of cylindri-cal billets.

Forging of Titanium Alloy 776242

The preform considered is a cylindrical composite material with a centralcore of (a + ft} phase and an outer ring of (/^-transformed phase, the twodiffusion-bonded together. Its dimensions are given in Fig. 12.5a. Thepreform was forged isothermally (dies and workpiece at the same initialtemperature) at 1227 K (1750°F) at a constant ram speed of 5.08mm/min.(0.2in./min). The total reduction in height was 60%. At these hightemperatures, a glass-type lubricant is very effective; a friction factor ofm = 0.2 was used in computer simulations.

In isothermal forging, very slow speeds are usually employed in order to(1) avoid increasing the flow stress of the material, which is strain-ratedependent, and (2) allow the heat generated during deformation to spread

FIG. 12.4 Predicted temperature distributions and grid distortions at various stages of hotcompression process [22], (a) At the end of free resting (elapsed time t ~ 6s); (b) 16.67%reduction in height (/ - 6,375s); (c) 33,34% reduction in height (/ - 6.750s); (d) 50.00%reduction in height (i = 7.125s); (e) 66.67% reduction in height (/ = 7.500s); (f) at the endof 3s free resting after deformation (/ - 10.5s).

FIG. 12.5 Die and workpiece temperatures in forging a titanium 6242 alloy composite billetat 60% reduction in height with different ram speeds: (a) preform dimensions; (b) ramspeed = 0.2 in./min.; (c) ram speed = 2.0 in./min.

uniformly throughout the workpiece, assuring uniformity in temperatureand deformation. If higher speeds can be used without adverse effects, theincrease in productivity is obvious. To investigate this aspect, the wholeprocess was simulated at 5.08 mm/min. (0.2 in./min.) and at 50.8 mm/min.(2.0 in./min).

The difference in speed of deformation did not produce any noticeabledifferences in the overall deformation process, or even in the localdeformation histories. The only changes found were in the temperaturefields, also shown in Figs. 12.5b and c. An average calculated temperatureincrease of 6.5 K in the slow-speed deformation increased to 12.5 K for thefaster deformation. However, the temperature gradients were such thatthe greatest difference between two points in the workpiece was roughly5 K at the end of deformation. This indicates that the speeds ofdeformation can be increased (at least to 2.0 in./min) without loss ofuniformity of properties, provided the average temperatures reached arenot critical.



The resulting microstructures at various locations after forging areshown in Fig. 12.6. The strain, strain-rate, and temperature variationsduring forging were found to be almost the same at all locations, andtypical variations for 5.08mm (0.2in./min.) ram speed are shown for theelements near the outer periphery and near the center in Fig. 12.6.

FIG. 12.6 Local strain, strain-rate, and temperature variations during (a + f})/(i compositeforging and corresponding microstructures after forging. (Microstructures courtesy of C. C.Chen, Chen Tech. Industries.)


The temperature histories indicate that no appreciable changes tookplace. Therefore, any microstructural modifications must be due tomaintaining the workpiece at an elevated temperature for some time underpressure and also to the amount of deformation imposed.

The information given in Fig. 12.6 allows further qualitative interpreta-tion of the relationships between local strain, strain-rate, and temperaturehistories and corresponding microstructures.

Isothermal and Hot-Die Disk ForgingUsing the program developed by Oh [27], coupled with the heat transferfinite-element calculation, the compressor disk forging of a (/J)-phase Ti6242 alloy was analyzed under two sets of conditions, isothermal forgingand hot-die forging [25].

In isothermal forging, the initial temperatures of both die and preformwere chosen to be 1171.9 K (1650°F) and the die velocity was5.08mm/min. (0.2in./min). Again, a glass-type lubricant was used at thedie-preform interface.

In addition to isothermal forging, hot-die forging was simulated for thesame die and preform geometries in order to determine the effects oftemperature and strain-rate on the details of deformation behavior. Theinitial temperature of the preform was 1171.9 K (1650°F), but the die andenvironmental temperatures were assumed to be 644 K (700°F) and 293 K(68°F), respectively. The die speed was 76.2mm/min. (3.0in./min).

The grid distortions at 70% reduction under the two forging conditionsare compared in Fig. 12.7a. The difference in metal flow, perhaps mainlydue to temperature effects, can be seen qualitatively. Distortion is moresevere in hot-die forging than in isothermal forging. Nonuniformity ofdeformation due to temperature gradients within the workpiece in hot-dieforging can be seen from the shape of the grid distortions. The contactarea at the rim region is different, being larger in isothermal forging. It isevident that the bulge at the outer surface is greater with hot-die forging.Although the observation is qualitative, it is clear that the effect oftemperature, possibly combined with the strain-rate effect, causes themetal flow to differ under the two forging conditions.

Figure 12.7b shows a comparison of temperature distributions under thetwo forging conditions. The temperature variation within the workpiecefor isothermal forging is only a few degrees, while a severe temperaturegradient can be seen near the contact region between the die and theworkpiece in hot-die forging.

The strain distributions, on the other hand, are almost the same for thetwo forging conditions, as shown in Fig. 12.7c. It appears that the strainsare predominantly determined by the preform geometry and the dieconfiguration.

Hot Nosing [23]The nosing process was described in Chap. 2. To study the nature of thenosing process of shells, Carlson [28] conducted hot nosing on a small

FIG. 12.7 (a) Grid-distortions; (b) temperature distributions; and (c) strain distributions inisothermal and hot-die forging. Material: Ti 6242 alloy; /J-phase. Isothermal, die andworkpiece temperature 1227 K; speed 0.2in./min., hot-die, die temperature 624 K, speed3.0in./min.

238


FIG. 12.7 (cont'd)

model of a 105-mm shell. Commercial open-hearth, cold-rolled steel wasused in making the specimens. Geometric details of the specimens aregiven in Fig. 12.8.

The die that was employed throughout the experiments had a curvatureradius of 300mm (11.85 in.) for an ogive profile. To simulate the amountof nosing, 36.8mm (1.45 in.) of the specimen was made to enter the die,giving a maximum reduction in the mean diameter of about 33%.

In preparation for the test, the die was heated for about 1 hour to bringit to a temperature of 811 K (1000°F). The shell temperature varied fromabout 1144 K (1600°F) to 1311K (1900°F) at the tip, depending upon

FIG. 12.8 Hot nosing specimen [28].


FIG. 12.9 Temperature distributions obtained by induction heating for a shell [28].

heating conditions. To measure the temperature distribution, six chromel-alumel thermocouples were placed along the shell specimen. Themeasured temperature distributions are shown in Fig. 12.9.

The heating of the shell was done in about 1 min by an induction coil.The length of time for nosing was in the order of 1 s. For the finite-elementsimulations, the flow stress expressions were obtained from experimentaldata of Altan and Boulger [29], and the heat transfer characteristics weretaken from standard handbooks.

The predicted load-displacement curves for hot nosing at variousfriction conditions, with initial tip temperature at 1269 K (1825°F) and diespeed of 41 mm/s (1.62 in./s), are given in Fig. 12.10a. It can be seen thatthe simulation results, obtained with a friction coefficient of ju = 0.1, arevery close to the experimental results. In Fig. 12.10b, the load-displacement curves are computed for initial tip temperatures of 1158K(1625°F), 1227 K (1750°F), and 1269 K (1825°F), respectively, at a nosingspeed of 41 mm/s and for /j = 0.1. All of the finite-element simulationresults showed good agreement with the experimental data.


Using the capabilities of a coupled thermo-viscoplastic analysis, attemptswere made to predict metal flow and forming loads, and to correlate themetallurgical changes with the information obtained through simulation.Gegel et al. [30] provided a detailed interpretation of the microstructure


FIG. 12.10 Load-displacement curves for (a) various friction conditions and (b) differentinitial tip temperatures, and comparison with experiment [28].

that develops during hot forging. Furthermore, Gegel et al. [31] proposeda new method of modeling the dynamic material behavior that explicitlydescribed the dynamic metallurgical processes occurring during hotdeformation.

Recently, Dawson [32] developed the numerical solution formulation for


simulation of hot or warm metal forming under steady-state conditions. Ofparticular importance was the incorporation of a methodology for usingparticle stress-temperature trajectories in conjunction with deformationmechanism maps. Thus, the assumptions made regarding the assumedconstitutive equations could be evaluated. The analysis of slab rolling ofaluminum was given as an example of possible applications. A significantadvance, made by Dewhurst and Dawson [33], is the development of afinite-element program that models steady-state viscoplastic flow and heattransfer in three dimensions.

References1. Oh, S. I., Rebelo, N. M., and Kobayashi, S., (1978), "Finite-Element

Formulation for the Analysis of Plastic Deformation of Rate-Sensitive Mate-rials in Metal Forming," IUTAM Symposium, Tutzing/Germany, p. 273.

2. Johnson, W., and Kudo, H., (1960), "The Use of Upper-Bound Solutions forthe Determination of Temperature Distributions in Fast Hot Rolling andAxisymmetric Extrusion Processes," Int. J. Mech. Sci., Vol. 1, p. 175.

3. Tay, A. O., Stevenson, M. G., and Davis, G. V., (1974), "Using theFinite-Element Method to Determine Temperature Distributions in Orthogo-nal Machining," Proc. Inst. Mech. Engr. Vol. 188, p. 627.

4. Bishop, J. F. W., (1956), "An Approximate Method for Determining theTemperature Reached in Steady-State Motion Problems of Plane PlasticStrain," Q. J. Mech. Appl. Math., Vol. 9, p. 236.

5. Altan, T., and Kobayashi, S., (1968), "A Numerical Method for Estimatingthe Temperature Distributions in Extrusion Through Conical Dies," Trans.ASME, J. Engr. Ind., Vol. 90, p. 107.

6. Lahoti, G., and Altan, T., (1975), "Prediction of Temperature Distributions inAxisymmetric Compression and Torsion," Trans. ASME, J. Engr. MaterialsTechnology, Vol. 97, p. 113.

7. Nagpal, V., Lahoti, G. D., and Altan, T., (1978), "A Numerical Method forSimultaneous Prediction of Metal Flow and Temperatures in Upset Forging ofRings," Trans. ASME, J. Engr. Ind. Vol. 109, p. 413.

8. Zienkiewicz, O. C., Jain, P. C., and Onate, E., (1978), "Flow of Solids DuringForming and Extrusion: Some aspects of numerical solutions," Int. J. SolidsStructures Vol. 14, p. 15.

9. Zienkiewicz, O. C., Onate, E., and Heinrich, J. C., (1978), "Plastic Flow inMetal Forming—I. Coupled Thermal Behavior in Extrusion—II. Thin SheetForming." Applications of Numerical Methods to Forming Processes, ASME,AMD, Vol. 28, p. 107.

10. Rebelo, N., and Kobayashi, S., (1980), "A Coupled Analysis of ViscoplasticDeformation and Heat Transfer—I. Theoretical Considerations," Int. J. Mech.Sci., Vol. 22, p. 699.

11. Rebelo, N., and Kobayashi, S., (1980), "A Coupled Analysis of ViscoplasticDeformation and Heat Transfer—II. Applications." Int. J. Mech. Sci., Vol.22, p. 707.

12. Cristescu, N., (1975), "Plastic Flow Through Conical Converging Dies, Usinga Viscoplastic Constitutive Equation," Int. J. Mech. Sci., Vol. 17, p. 425.

13. Cristescu, N., (1976), "Drawing Through Conical Dies—An Analysis Com-pared with Experiments," Int. J. Mech. Sci., Vol. 18, p. 45.

14. Zienkiewicz, O. C., and Godbole, P. N., (1975), Viscous, IncompressibleFlow with Special Reference to Non-Newtonian (plastic) Fluids, Chap. 2 in"Finite Elements in Fluids" (Edited by R. H. Gallagher et al), Wiley, NewYork.


15. Zienkiewicz, O. C., and Godbole, P. N., (1974), "Flow of Plastic andViscoplastic Solids with Special Reference to Extrusion and Forming Proc-esses," Int. J. Num. Meth. Eng., Vol. 8, p. 3.

16. Price, J. W. H., and Alexander, J. M., (1976), "A Study of IsothermalForming or Creep Forming of a Titanium Alloy," Proc. of the 4th NAMRC,Columbus, Ohio, p. 46.

17. Price, J. W. H., and Alexander, J. M., (1976), "The Finite-Element Analysisof Two High-Temperature Metal Deformation Processes," 2d Int. Symposiumon FEM in Flow Problems, p. 717.

18. Klemz, F. B., and Hashmi, S. J., (1977), "Simple Upsetting of CylindricalBillets: Experimental Investigation and Theoretical Prediction," 18th MTDRConference, London, p. 323.

19. Loizou, N., and Sims, R. B., (1953), "The Yield Stress of Pure Lead inCompression," J. Mech. Phys. Solids, Vol. 1, p. 234.

20. Lippman, H., (1966), "On the Dynamics of Forging," Proc. 7th Int. MTDRConference, Birmingham, England, p. 53.

21. Dahlquist, G., and Bjorck, A., (1974), "Numerical Methods," Prentice-Hall,Englewood Cliffs, NJ.

22. Wu, W. T., and Oh, S. I., (1985), "ALPIDT: A General Purpose FEM Codefor Simulation of Non-Isothermal Forming Processes," Proc. NAMRI—XIII,Berkeley, California, p. 449.

23. Tang, M.-C., and Kobayashi, S., (1982), "An Investigation of the Shell NosingProcess by the Finite Element Method. Part 2: Nosing at Elevated Tempera-tures," Trans. ASME, J. Engr. Ind., Vol. 104, p. 312.

24. Rebelo, N., and Kobayashi, S., (1981), "Thermo-Viscoplastic Analysis ofTitanium Alloy Forging," ASME Publications FED—Vol. 3, ManufacturingSolutions Based on Engineering Sciences, p. 151.

25. Oh, S. I., Park, J. J., Kobayashi, S., and Allan, T., (1983), "Application ofFEM Modeling to Simulate Metal Flow in Forging a Titanium Alloy EngineDisk," Trans. ASME, J. Engr. Ind., Vol. 105, p. 251.

26. Pohl, W., (1972), "A Method for Approximate Calculation of Heat Genera-tion and Transfer in Cold Upsetting of Metals," Doctoral Dissertation,University of Stuttgart.

27. Oh, S. I., (1982), "Finite Element Analysis of Metal Forming Process withArbitrarily Shaped Dies," Int. J. Mech. ScL, Vol. 24, p. 479.

28. Carlson, R. K., (1943), "An Experimental Investigation of the Nosing ofShells," Forging of Steel Shells, presented at the Winter Annual Meeting ofASME, New York.

29. Altan, T., and Boulger, F. W., (1973), "Flow Stress of Metals and itsApplications in Metal Forming Analysis," Trans. ASME J. Engr. Ind., Vol.95, No. 4, p. 1009.

30. Gegel, H. L., Nadiv, S., Malas, J. C., and Morgan, J. T., (1980), "Applicationof Process Modeling to Analysis of Microstructural Changes During the HotWorking of a Two-phase Titanium Alloy," Appendix K, AFWAL-TR-80-4162, p. 403.

31. Gegel, H. L., Prasad, Y. V. R. K., Malas, J. C., Morgan, J. T., and Lark, K.A., (1984), "Computer Simulations for Controlling Microstructure During HotWorking of Ti-6242," ASME, PVP, vol. 87, p. 101.

32. Dawson, P. R., (1984), "A Model for the Hot or Warm Forming of Metalswith Special Use of Deformation Mechanism Maps," Int. J. Mech. Sci., Vol.26, p. 227.

33. Dewhurst, T. B., and Dawson, P. R., (1984), "Analysis of Large PlasticDeformation at Elevated Temperatures Using State Variable Models forViscoplastic Flow," Proc. Symp. Constitutive Equations: Micro, Macro, andComputational Aspects, ASME., p. 149.

13COMPACTION AND FORGING

OF POROUS METALS

13.1 Introduction

Powder forming, once considered a laboratory curiosity, has evolved into amanufacturing technique for producing high-performance componentseconomically in the metal-working industry because of its low manufactur-ing cost compared with conventional metal-forming processes [1,2].Generally, the powder-forming process consists of three steps: (1) com-pacting a precise weight of metal powder into a "green" preform with10-30% porosity (defined by the ratio of void volume to total volume ofthe preform); (2) sintering the preform to reduce the metal oxides andform strong metallurgical structures; (3) forming the preform by repressingor upsetting in a closed die to less than 1% residual porosity.

Powder forming has disadvantages in that the preform exhibits porosity.Because of this porosity, the ductility of the sintered preform is low incomparison with wrought materials [3]. In forging compacted and sinteredpowdered-metal (P/M) preforms, where large amount of deformation andshear is involved, pores collapse and align in the direction perpendicular tothat of forging and result in anisotropy. However, repressing-type defor-mation, where very little deformation and shear are present, does not leadto marked anisotropy [4]. A low-density preform will result in more localflow and a higher degree of anisotropy than will a preform of high initialdensity [5]. These anisotropic structures can lead to nonuniform impactresistances of the forged P/M parts. Also, in forming of sintered preforms,materials are more susceptible to fracture than in forming of solidmaterials, and the analysis is of particular importance in producingdefect-free components by determining the effect of various parameters(preform and die geometries, sintering conditions, and the frictionconditions) on the detailed metal flow. In this chapter, the plasticity theoryfor solid materials is extended to porous materials, applicable to thedeformation analysis of sintered powdered-metal preforms.

In characterizing the mechanical response of porous materials, aphenomeno'ogical approach (introducing a homogeneous continuum mo-del) is employed. For the finite-element formulations of the equilibriumand energy equations based on the infinitesimal theory, the followingassumptions are made: the elastic portion of deformation is neglected

244

Compaction and Forging of Porous Metals 245

because the practical forming process involves very large amounts ofplastic deformation; the normality of the plastic strain-rates to the yieldsurface holds; anisotropy that occurs during deformation is negligible; andthermal properties of the porous materials are independent of thetemperatures.

13.2 Yield Criterion and Flow Rules

For porous metals, a number of plasticity theories have been proposedwith a yield function /(a/,-) of the following form [6,7]:

(13.1)

where 7X is the linear invariant of stress tensor, J2 is the quadratic invariantof deviatoric stress tensor, and A and B are functions of void ratio orrelative density. The invariants 1^ and J2 are denned as

Il = ax + ay + oz

and

Starting with eq. (13.1), Oyane and his colleagues [8,9,10] derived theplasticity equations. On the basis of this theory, they derived the slip-linefield equations and the upper-bound theorem applicable to porous metals.

The yield surface can be defined by

where YR is the apparent yield stress of porous materials determined byuniaxial tension or compression. It can then be shown that B = 1 — (A/3).The yield function f(oti) is now expressed by

With this yield function, flow rules are expressed by

where o,, and e,y are apparent stresses and strain-rates, respectively,considering a porous metal to be a continuum. The proportionality factor Ain the flow rule, eq. (13.3), is given by

where the apparent strain-rate eR is defined according to the equivalence


of the work-rate, namely, atje,j = YReK [11,12] and expressed by

where y,-,- is the engineering shear strain rate and ev is the volumetric strainrate.

The apparent yield stress YR depends on the property of the base metaland the relative density R (ratio of the volume of base metal to the totalvolume of porous metal) [13] according to

where Yb is the yield stress of the base metal and 17 is a function of relativedensity. The effects of strain, strain-rate, and temperature on the yieldstress are included in Yb = Yb(eb, eb, Tb), where eb, eb, and Tb are strain,strain-rate, and temperature of the base metal. The relationship betweenthe apparent strain and strain-rate and those of the base metal are given by

and

In order to complete the constitutive equations, A in eq. (13.2) and r\ ineq. (13.6) must be determined as functions of relative density byexperiment. Among the proposed constitutive equations, those proposedby Doraivelu et al. [13] and by Shima et al. [8], appear to agree withexperimental measurements quite well. The expressions for A and rj byDoraivelu et al. are

It is to be noted that for R = 0.707, r\ = 0, which implies that the apparentyield stress is zero according to eq. (13.6). For the analysis in this chapter,the constitutive equation (13.8) is used. Further discussion on the validityof the constitutive equations for porous materials can be found inReferences [14] and [15].

13.3 Finite-Element Modeling and Numerical Procedures

The variational form of the equilibrium equation, as a basis for discretiza-tion, is derived as

where 8 denotes variations.


DiscretizationThe element used for discretization is an isoparametric quadrilateralelement with bilinear shape function (see Fig. 8.1 and Fig. 9.1 for thetwo-dimensional and the axisymmetric deformations, respectively). Theelemental velocity field is approximated by

where v is the velocity vector of nodal-point value and N is the shapefunction matrix. Applying differentiation to eq. (13.10),

The matrices N and B have been defined for a quadrilateral element in theprevious chapters. Substitution of eq. (13.11) into eq. (13.5) leads to

and the variation 6eR becomes, because of symmetry of the matrix P,

Substituting eqs. (13.10) and (13.13) into eq. (13.9) at the elemental leveland assembling the element equations with global constraints, we obtain

where (;') indicates the y'th element. Because the variation <5v is arbitrary,eq. (13.14) results in the stiffness equations. The matrix P in eq. (13.12) isobtained as follows. The inversion of the flow rules (13.3) can be expressedby

where, for the axisymmetric case,

with the stress and strain-rate vectors are defined according to

and


respectively. Then, from the requirement that OTE= YRER,

Updating of Relative DensityThe volumetric strain-rate is related to the density-rate according to

where the relative density R is defined by

with Vb being volume of the base metal and Vv as the volume of void.Integrating eq. (13.16), we have

In eq. (13.17), R0 is the current relative density and Aey is the change ofvolumetric strain in one deformation step. The average relative density Ra

is defined by

where Rt and Vt are the relative density and the volume of an element,respectively.

Fully Dense MaterialsFor fully dense materials, R = 1.0 and A = 3. Then, the matrix D of eq.(13.15) becomes infinity. Consequently, convergence behavior for thesolution becomes erratic when the relative density in the elementapproaches unity.

A constraint was incorporated in the numerical procedures such that anelement with R = 0.9990 was considered as a fully dense element. Thisconstraint was helpful in obtaining well-behaved convergence for thesolution. Osakada et al. [16] and Mori et al. [17], also using this theory,developed the finite-element method and applied it to the rigid-plasticdeformation analysis of fully dense materials.

Volume IntegrationThe program has been tested by analyzing compression of cylinders andrings of porous materials [18]. During the test it was found that thereduced integration must be applied to the terms involving the volumetricstrain-rate. Such a reduced integration strategy is straightforward in theformulation of fully dense material, since the term of the volumetric strain


energy-rate is fully uncoupled from that of the distortional energy-rate. Inthe present formulation, the terms with volumetric strain-rate cannot bedecoupled from those with distortional strain-rate, as seen in eq. (13.14).Thus, the matrix P is decomposed into two components, Pj and P2. Here,P! contains the terms with distortional strain-rate and P2 involves the termswith volumetric strain-rate. The matrices P1 and P2 are obtained bydecomposing the matrix D into the two components D1 and D2 as

D = D! + D2

where

and

Then, P = P! + P2 = BTDjB + BTD2B. For evaluating the matrix P, thematrix P2 is evaluated at the reduced integration point, while Px isevaluated at the regular integration points. The evaluation of the matrix Pis then obtained at the regular integration points. It should be noted againthat for solid materials A = 3 in eq. (13.19), which gives

and l/[3(3 — A)] is replaced by K, the penalty constant.

13.4 Simple Compression [14]

In simple compression, a cylindrical sintered P/M preform is compressedbetween two flat dies. The preform was 50.4 mm (2 in.) in diameter and50.4mm (2 in.) in height. It was assumed that the preform has uniformrelative density of 0.8. The simulations were carried out with two differentfriction factors, m = 0.2 and 0.5. The flow stress of the matrix material wasassumed to be expressed by o = 1.0 + O.Ole and the constitutive eq. (13.8)was used.

Figure 13.1 shows the predicted grid distortions at 20, 50 and 70%reduction in height for the friction factor m = 0.5. It can be seen from the


FIG. 13.1 Simple compression of a cylindrical sintered P/M preform. The predicted griddistortions during simple compression at 20, 50, and 70% reductions in height. The dashedlines are the predicted boundaries with fully dense initial preform. Initial relative density =0.8; friction factor = 0.5.

figure that the predicted grid distortions resemble those expected in thecompression of a fully dense preform. The predicted boundaries of thedeformed workpiece for a fully dense initial preform are shown in dashedlines for comparison. It is seen that the porous preform changes its volumeduring the compression process. The average density change as a functionof height strain is shown in Fig. 13.2. The height strain is defined by\n(H/H0) where H0 is the height of the undeformed preform and H is thedeformed height. It is interesting to note that the average density variesvery little with differerent friction conditions.


FIG. 13.2 The predicted average relative density changes as a function of height strain insimple compression.

Although the average density changes are not affected by differentfrictions, the differences in the local density distributions are considerable.Figures 13.3a and b show the predicted relative density distributions at 20,50, and 70% reductions in height for the friction factors m = 0.2 and 0.5,respectively. It is seen from Fig. 13.3a that at 20% reduction the density islowest near the center of the die contact surface where the deformation isrestricted by friction. The highest density is observed at the center of theworkpiece and also near the outside radius of the die contact surface. At50% reduction in height, the density is lowest at the equator of the sidesurface, and at this stage, the side surface barrels considerably. The highermean stress at the equator results in the low density at this point. It is alsonoted that at 50% reduction in height, densification near the center of thedie contact surface has been accelerated. This can be explained by the factthat the pressure near the z axis increases as radius-to-thickness ratioincreases. At 70% reduction in height, a large portion of the workpiecenear the z axis becomes fully dense while the material near the side surfaceremains porous. In fact, the density of the workpiece near the free surfacedecreases as the deformation progresses from 50% to 70% in height.

Comparing Figs. 13.3a and b, it is seen that the overall densifications ofthe workpiece, with the two different interface frictions, are almost thesame. However, it is noted that at 20% reduction, the gradient of thedensity distribution is larger with the higher friction. This fact supports theabove argument that the density near the center of the die contact surfaceincreases because of the increase in mean pressure near the axis. It is wellknown that higher pressure near the axis can be achieved with higherfriction when a thin disk is compressed between two flat dies. It is also

Radius

FIG. 13.3 The predicted relative density distributions in simple compression at 20, 50, and70% reductions in height. Initial density = 0.8; (a) friction factor = 0.2 and (b) frictionfactor = 0.5.

252


observed that the density near the free surface is lower with higherfriction, owing to the larger barrelling. At 70% reduction in height, it isnoted that the size of the fully dense region near the axis is larger withhigher friction, while the density near the free surface decreases con-siderably as the workpiece undergoes deformation from 50 to 70%reduction in height. In practice, fracturing is often observed at theequatorial surface because of this density reduction.

Figure 13.4 shows the predicted load-displacement curves for twodifferent friction conditions. For comparison, the predicted load-displacement curve for the fully dense preform with friction factor m = 0.5is also shown. The load is lower with the porous preform than that for thefully dense preform. It can be seen from the figure that the differencebetween the forming loads with different frictions is very small during theearly stage of deformation. However, during the later stage, the loadbegins to increase faster in the case of higher friction.

13.5 Axisymmetric Forging of Flange-Hub Shapes [18]

In a closed-die forging, the stress state favors complete densification,because of compressive mean stress. Experimental studies on closed-die

FIG. 13.4 The predicted load versus die stroke curves in simple compression. Ra indicatesthe initial relative density and m indicates the friction factor.


forging of metal powders were presented by Downey and Kuhn [19].These studies allowed qualitative determination of the preform shapes forforging to full density with a sound metallurgical structure and withoutfracture. One of the forgings investigated had a flange-hub shape. Theforging dies and two preform specimens, with the initial mesh system usedfor the analysis, are shown in Fig. 13.5. Since the specimens areaxisymmetric, only a quarter of the workpiece is used for calculations.

Equation (13.8) was used with the stress-strain relationship given by

for the base metal (OFHC copper). The initial relative density wasR0 = 0.800. Y0 is the initial yield stress of the base metal and was given as42061 psi (290MN/m2). The frictional condition is given by mky (ky

represents the apparent yield stress in shear) and the friction factor m wasassumed to be 0.1.

The results, illustrating the deformation zone and the extent ofdensification, are given for two preforms in Fig. 13.6, where thedistributions of apparent effective strain and relative density are shown at24% reduction in height. In this figure, the pattern of equistrain anddensity contours are remarkably similar. This is because the effective straincontains not only the term of distortion but also the volume change that isassociated with the relative density. The effective strains are largest at thecenter of the forging and at the edge of the die-specimen interface, as arethe relative densities.

For preform I, the central region under the die shows no deformationand no change in relative density from the initial value (R0 = 0.800) for

FIG. 13.5 Preform shapes and mesh systems in flange-hub forging.


FIG. 13.6 Effective strain (right) and relative density (left) distributions at 24% reduction inheight in forging (a) preform I and (b) preform II.

small reduction. Another observation of importance is that the relativedensity near the bulged free surface increases at first, then begins todecrease as reduction increases. Thus, the equatorial free surface is apossible fracture site and this was indeed observed in an experiment [19].

For preform II, the workpiece is deforming radially as well as axiallyrelative to the die motion to fill the cavity. Characteristics of strongsingularity, where the gradients of strain and density are large, are seenalong the die-workpiece contact surface. This results in a distinctlydifferent deformation pattern from that obtained for preform I. It wasnoted for preform II that friction at the die-workpiece interface issensitive to the metal flow. It was also found that at a certain stage ofdeformation the severe distortions are localized around the die corners andremeshing was needed for continuing forging. A need for remeshingoccurred at 37% reduction in height for the case of preform II.

The technique for remeshing has been discussed in detail in Chap. 7.The area-weighted average method is adopted for the problem because ofits simplicity and computational efficiency. In this method, the nodal pointvalue of relative density or apparent strain is determined from the averageof the adjacent element values surrounding that nodal point weighted bythe associated element areas. Further details of the method can be foundin Reference [20].

Figure 13.7a shows the mesh system at 37% reduction in height beforeand after remeshing. The comparisons of relative density and effectivestrain fields before and after remeshing are shown in Fig. 13.7b. Forremeshing, no change was made in the number of elements and nodalpoints, and therefore the element connectivity was not changed.

In Fig. 13.7b close similarity between the relative density contoursbefore and after remeshing is found. Other comparisons of the apparentaverage relative densities before and after remeshing confirm the credibi-lity of the program as follows: At 37% reduction in height, Ra — 0.9378


FIG. 13.7 (a) Mesh systems and (b) relative density contours before (left) and after (right)remeshing in flange-hub forging.

and 0.9373 before and after remeshing, respectively; and at completefilling, at 46% reduction in height, Ra = 0.9933 in both cases. Relativedensity and effective strain contours at complete filling are shown in Fig.13.8, which shows that the relative density reaches full density over theentire forging for preform II.

13.6 Axisymmetric Forging of Pulley Blank

The finite-element program has been applied to the simulation ofclosed-die forging of a pulley blank for the two preforms shown in Fig.13.9 [20]. The sintered preforms used in the simulations were made of

FIG. 13.8 Distributions of relative density (left) and apparent strain (right) at the completionof forging the flange-hub shape.

FIG. 13.9 Preform shapes and mesh systems in axisymmetric closed-die forging of a pulleyblank.

257


aluminum powders. The yield stress of the preform was determined fromthat of the base metal of commercially pure aluminum and the constitutiveeq. (13.8). The rate-sensitivity of the material property and the tempera-ture effect on deformation were not included in this simulation. The initialrelative density was assumed to be uniform and to be 0.780 for bothpreforms. Two values of m were assumed for friction, m = 0-l andm = 0-5.

The nonsteady-state forging process was simulated by a step-by-stepmethod with increments of 2% of the initial height for preform A and 5%of the initial height for preform B. In order to complete the calculationsuccessfully, four and ten remeshings were required for preforms A and B,respectively.Preform A. The calculation was stopped at 38% and 43% reductions inheight for m = 0.1 and m = 0.5, respectively, in forging preform A. Theprogram is designed to automatically discontinue the calculation if therelative density of any element falls below the limiting value of the relativedensity for which the apparent yield stress becomes zero, because thisindicates the initiation of fracture.

In Figs. 13.10a and b, the distributions of relative density and hydro-static stress at the final stage of forging are shown for two frictionconditions. The patterns of densification and hydrostatic stress are similarto each other. For the two friction conditions, the relative densitydistributions differ somewhat in the rim area, as do the hydrostatic stressdistributions. It was found that the effect of friction on the densification ofthe materials was not evident at an early stage of forging. As deformationcontinues, however, the densification and the compressive hydrostaticstress are greater with higher friction. It can also be seen that the locationsof the large and small values of relative density are the same for bothfriction conditions and lie at the flange part around the die corners andnear the free surfaces, respectively.Preform B. Contrary to preform A, complete forging was possible withpreform B. This was due to the differences in material flow in the twopreforms.

The distributions of relative density and hydrostatic stress with lowfriction were compared with those with high friction at the final stage offorging in Fig. 13.11. The distribution patterns are similar to each other forthe two friction conditions at the same stage of forging except that forhigher friction, the densification and the compressive hydrostatic stresswere greater. Because of this effect of friction on densification, the finalforged part in preform B with high friction did not completely fill thecavity at 60% reduction in height due to the volume change, whilecomplete filling was achieved for low friction at the same reduction inheight (see Figs. 13.Ha and b).

As in forging of preform A, densification and the compressive hydro-static stress are concentrated at the flange part near the die corners. Thefigure also shows that the weakest mechanical properties will be near the


FIG. 13.10 Relative density (left) and hydrostatic stress (3am, units of ksi) (right)distributions at the final stage for the two friction conditions in forging preform A (darkenedareas indicate the possible fracture sites): (a)m = 0.1; (b)m=0.5.

tips of the rim section. However, the relative density in most of the forgedhub section reached full density, R = 0.999, in the final stage.

The experiments [19] also showed that preform B produced defect-freepulley blanks. This agreement allows the prediction of the probablelocations of fracture during forging using the existing finite-elementprogram.

13.7 Heat Transfer in Porous Materials

Heat transfer analysis for solid metals was presented in Chap. 12. Forporous materials the energy balance equation, corresponding to eq. (12.1)


FIG. 13.11 Relative density (left) and hydrostatic stress, (3am units of ksi) (right)distributions at the final stage for the two friction conditions in forging preform B: (a)m=Q.\, (b) m=0.5.

in Chap. 12, is expressed by

where the subscript R denotes apparent (or equivalent) quantities,representing a porous material as an equivalent continuum. Thus, usingthe apparent thermal properties, kR and CR derived from the base-metalproperties, the heat transfer analysis follows exactly the same proceduredescribed in Chap. 12.

In porous materials, heat transfer takes place in three ways: conductionthrough base metal, convection, and radiation through the pores. Whenthe size of the pores is sufficiently small, convection is negligible.


Radiation through pores can be neglected when the temperature is low.Therefore, conduction plays a crucial role in heat transfer in porousmaterials.

Change in thermal conductivity, as determined by porosity, has beenstudied extensively and numerous attempts have been made to correlatethe complex effects of the pores with experimental results in a simplifiedform [21]. Among published proposals, the most generally employed arethe expressions by Russel [22] and by Eucken [23]. Both expressions werederived from Maxwell's relation for conductors and resistors. Although theexpressions differ from each other in form, they lead to similar results.They assume that the pores are discontinuous, spherical, and evenlydistributed throughout a continuous base-metal matrix.

Im and Kobayashi [24] derived a simple linear relationship between thethermal conductivities of the powdered metal and the base metal. Underthe assumptions that the heat flow is unidirectional and that the volumepore fraction is equal to the linear pore fraction and to the cross-sectionalpore fraction, as well as that the thermal properties are homogeneous, theheat balance equation yields

where kR is the apparent thermal conductivity, kb is the thermal conduc-tivity of the base metal, and kv is the thermal conductivity of the voids.

According to eq. (13.21), kR depends on both the volume pore fractionand the ratio between the thermal conductivity of the base metal and air.In addition to the conduction of air, at high temperatures the radiationacross the pores contributes to heat transfer. In this case, the radiationportion of heat transfer can be added to the thermal conductivity of air andlowers the ratio between the thermal conductivity of the base metal and airin eq. (13.21). By neglecting kv/kb terms compared to 1, eq. (13.21)becomes

Since the effect of temperature on the yield stress is determined by thetemperature of the base metal, the relationship between the temperaturesof the base metal and the porous material should be determined. Byintroducing the apparent density pR and the specific heat, CR of porousmaterials, the total change in internal energy can be expressed by

where the subscripts R, b, and v denote the quantity related to the totalporous materials, base metals, and voids, respectively, t is the time, and p,c, and V are the density, the specific heat, and the volume, respectively.

The temperature-rate of voids is at most of the same order as the


temperature-rate of the base metal, since the heat generation due to plasticdeformation is limited to that of the base metal. The change in the internalenergy of voids can also be neglected, since the thermal capacity of thevoids is much smaller than that of the base metal. Therefore, eq. (13.23)can be reduced to

where the apparent specific heat of the porous material, CR, can bedetermined by

Combining this equation with eq. (13.24), noting that pRVR = pbVb, gives

Integration of eq. (13.26) leads to the expression of the temperature of thebase metal as,

13.8 Hot Pressing Under the Plane-Strain Condition

Much experimental work has been performed on powdered-metal hotforging. Preform ductility and transient cracking in forging were analyzedby Fischmeister et al. [25], who showed that large amounts of plasticdeformation are beneficial in obtaining good impact and fatigue properties.Malik [26] has published work on the producibility of titanium powdered-metal shapes for aerospace structural applications. Ferguson et al. [27]have analyzed the hot isostatic process, which involves a consolidation ofloose powders or sintered preforms.

Im and Kobayashi [24] applied the finite-element technique to theanalysis of plane-strain compression at elevated temperatures. The work-pieces, made of sintered iron powders, were assumed to have uniforminitial relative densities of 0.802 and 0.743. Computed results werecompared in terms of macroscopic densification and forging pressure withexperimental values published by Fischmeister et al. [28]. In upsetting longbars between flat dies, (see Fig. 13.12), the plane-strain condition (ex =0,txy — Txz = 0) was assumed at the central cross-section, owing to therestraint of longitudinal material flow.

The conditions used in the computation were as follows. The dimensionsof the workpieces were 10x10x100 mm. (0.39 x 0.39 x 3.94 in.) Twoinitial relative densities, R0 = 0.743 and R0 = 0.802, were selected fromfour experimental cases reported in Reference [28]. Since no lubricant wasused between the workpiece and dies in the experiments, two frictionfactors, 1.0 and 0.5 were assumed for both cases. Initially the temperaturesof the dies and workpieces were assumed to be 293 K (68°F), and 1433 K


FIG. 13.12 Schematic of plane-strain compression: (a) geometry of the die and workpiece;(b) boundary conditions. (Superscripts c, r, f on the heat flow qn refer to convection,radiation, and friction, respectively.)

(120°F), respectively. The material property of the base metal wasassumed to be rigid-plastic, and the yield stress was taken as 28717 psi(198 MN/m2) from the handbook. Again, the consitutive eq. (13.8) wasused.

The thermal properties necessary for computation were taken from thehandbook as follows:

kb = 55 N/(s K), kv = 0.045 N/(s K), kd = 19 N/(s K),

Cfcp6=3.6N/(mm2K), capu =0.4N/(mm2K), cdpd = 3.77N/(mm2K),

CT£ = 3.6xlON/(mm2sK4) ,/zc = 5.5N/(mmsK), /z00 = 0.01 N/(mms K),

K = 0.85


Because of geometrical symmetry, only a quarter-section of the centralcross-section of the bar was analyzed for deformation and temperature.The boundary conditions are also shown in Fig. 13.12. Deformation wasanalyzed by the step-by-step method with increments of 0.4% of the initialheight of the workpiece. Total reduction in height was 70% for R0 = 0.743with low friction. For R0 = 0.802 with low friction, the calculation stoppedat 67% owing to the severe distortion in the corner element of thedeformed free surface in contact with the die. For both /?„ = 0.743 andR0 = 0.802 with high friction, the calculations stopped at 58% and 64%,respectively, because the relative density of an element fell below thelimiting value.

The computed and the experimental results are compared in terms ofthe macroscopic deformation behavior. The changes of lateral flow duringthe plane-strain compression are given in Fig. 13.13. The average width Wa

of the deformed free surface is used to calculate the true strain in the ydirection, ey = ln(Wa/W0), where W0 is the original width. According to thefigure, the results show good agreement between computation andexperiment. The straight line in the figure is for the fully dense material(R = 1.0), where ez = — ey because of incompressibility. In a porousmaterial, volume change occurs during deformation and the increase inwidth becomes less than the decrease in height.

The macroscopic level of densification as function of axial compression isgiven in Fig. 13.14. As seen in Fig. 13.14 the predictions are excellent forall cases. It is also seen that the densification occurs mainly at an earlystage of upsetting. As deformation proceeds, the densification becomessaturated and lateral flow is enhanced.

Figure 13.15a reproduces the local hardness distribution in the homoge-

True S t ra in in y - d i r e c t i o n

FIG. 13.13 Comparison of variations in lateral flow between theory and experiment [28].

FIG. 13.14 Variations of average relative density during compression, and comparison withexperiment [28].

(b)FIG. 13.15 (a) Experimental local hardness distribution in the center cross-section of a bar[28]. (b) Relative density distributions at 40% reduction in height for Ra = 0.743 with twofriction conditions: left (m = 0.5); right (m = 1.0).

265


FIG. 13.16 Relative density distributions at the final stages for Ra = 0.743 with two frictionconditions: (left) 70% reduction and m = 0.5; (right) 58% reduction and m = 1.0.

neous center region, obtained from the experimental results given inReference [28]. This information can be correlated with the computedrelative density distributions, such as shown in Fig. 13.15b. These figuresshow that densification is effective near the edge (owing to folding duringcompression) and in the central region of the deformed workpieces.Excellent agreement of relative density and hardness distributions betweenexperiment and computation can be seen from these figures.

Figure 13.16 shows the relative density distributions at the final stages ofcompression for R0 = 0.743 with two friction conditions. It can be seen thatmost of the central region is fully dense and that less densification occursnear the free surface. In some elements at the free surface, the relativedensity reaches the limiting value and then tends to decrease. This isinterpreted as possible fracturing and the critical elements are shown asdarkened areas in the figures.

The temperature profiles in the workpiece and dies are presented inFigs. 13.17a and b for R0 = 0.743. Figure 13.17a shows that the effect offriction on the temperature distribution is negligible. It is also seen that theconductive heat flow from the workpiece to the die is dominant whencompared to heat generation due to plastic deformation at an early stage.Consequently, the isothermal lines in the workpiece are almost linear. Asdeformation increases, the isothermal lines encircle the center, becauseheat generation due to plastic deformation increases and is maximal at thecenter, as seen in the left-hand side of Fig. 13.17b. A transition betweenthese two types of temperature distributions can be observed at 58%reduction in height, shown in the right-hand side of Fig. 13.17b.

13.9 Compaction

In the process of compaction, a P/M preform is placed in a cylindricalcontainer and pressed in a double-action press, as shown in Fig. 13.18. Forthe simulation [14], the container was 50.8 mm (2.0 in.) in diameter and152.4 mm (6.0 in.) in height. Because of symmetry, the analysis wasperformed for the half-height. The flow stress behavior of the matrixmaterial was assumed to be Yb =0.1. The constant-friction factor law was

FIG. 13.17 (a) Temperature distributions for R0 = 0.743 at 40% reduction in height with twofriction conditions: (left) m=0.5; (right) m = 1.0. (b) Temperature distributions forR0 = 0.743 at the final stages with two friction conditions: (left) m = 0.5; (right) m = 1.0.

267


FIG. 13.18 Schematic diagram of simple die compaction.

assumed, and the friction factors used in simulation were m = 0.1, 0.2, and0.5. The punch velocities were 76.2 mm/s (3.0 in./s) for both the upperand lower punches. The initial relative density in the simulation was takento be 0.8. It was also assumed that the air and the lubricant trapped in thepores did not affect the compaction process.

Figures 13.19a and b show the predicted relative density distributions forthe friction facors m =0.2 and 0.5, respectively. It is seen that the densityis greatest near the outside radius of the moving punch and decreases inregions remote from the punch, particularly near the container die wall.As the deformation progresses towards full compaction, the densitydistribution becomes more uniform; that is, the difference between thehighest and the lowest density becomes smaller. It can be seen that thetrend of density distributions for both friction conditions are similar, butthe density is more uniform with lower friction.

From the predicted density distribution (which is a function of radial aswell as height location), the average density was calculated as a function ofheight. The average density was obtained by integrating the relativedensity distribution over the cross-sectional area and dividing by thatcross-sectional area at a given height.

Figures 13.20a and b show the predicted density variation as a functionof height for different friction factors, m = 0.2 and m = 0.5, respectively. Itcan be seen from the figures that the density is lowest at the mid-height,and that it increases toward the punch. It is also seen that the distributionbecomes more uniform with increasing deformation and lower friction.


FIG. 13.19 (a) Predicted relative density distribution during die pressing; initial relativedensity = 0.8; friction factor = 0.2. (b) Predicted relative density distribution during diepressing; initial relative density = 0.8; friction factor = 0.5.

In order to validate the prediction of the current model, experimentaldata for pressing of a powder preform were sought from the literature.None that were directly comparable could be found. However, densitymeasurements for powder compaction in a closed die were found in aGerman doctoral dissertation [29]. Figure 13.21 shows measured densitydistributions of P/M preforms that were prepared by die compaction andisostatic compaction. From this figure, it is apparent that the density


D E N S I T Y DIS. M=0.2

DENSITY DIS M=0.5

FIG. 13.20 (a) Predicted average density as function of height; initial relative density = 0.8;friction factor = 0.2. (b) Predicted average density as function of height; initial relativedensity = 0.8; friction factor = 0.5.

distributions of the preform obtained by die compaction are similar tothose predicted.

Figure 13.22 shows the predicted punch load vs. punch stroke curves forfriction factors m = 0.1, 0.2 and 0.5. The figure illustrates that the pressingload is higher for higher friction. The load is important in a die pressingoperation because the maximum allowable load determines the level ofachievable average density.

Billet Height

FIG. 13.21 Measured average relative densities obtained in compaction of powder in closeddies and by hydrostatic pressing [29].

271


Punch Stroke (in.)FIG. 13.22 The predicted punch load during die pressing.

References

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2. Jones, P. K., (1973), "New Perspectives in Powder Metallurgy," Vol. 6,Plenum Press, New York.

3. Kaufman, S. M., and Mocarski, S., (1971), "Effect of Small Amount ofResidual Porosity on the Mechanical Properties of P/M Forgings," Int. J.Powder Metal., Vol. 7, p. 19.

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Powder Metallurgy," (Edited by H. H. Hausner and W. E. Smith), Vol. 7,American Powder Metallurgy Institute, Princeton, NJ, p. 235.

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7. Green, R. J., (1972), "A Plasticity Theory for Porous Metals," Int. J. Mech.Sci., Vol. 14, p. 215.

8. Shima, S., and Oyane, M., (1976), "Plasticity Theory for Porous Metals," Int.J. Mech. Sci., Vol. 18, p. 285.

9. Tabata, T., and Masaki, S., (1975), "Plane-Strain Extrusion of PorousMaterials," Memoirs of the Osaka Institute of Technology, Series B., Vol. 19,No. 2.

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11. Hill, R., (1950), "The Mathematical Theory of Plasticity." Oxford UniversityPress, London.

12. Johnson, W., and Miller, P. B., (1980), "Engineering Plasticity." VanNostrand Reinhold, London.

13. Doraivelu, S. M., Gegel, H. L., Gunasekera, J. S., Malas, J. C., and Morgan,J. T., (1984), "A New Yield Function for Compressible P/M Materials," Int.J. Mech. Sci., Vol. 26, p. 527.

14. Oh, S. L, and Gegel, H. L., (1986), "ALPIDP—Modeling of P/M Forming bythe Finite Element Method," Proc. NAMRC XIV, Minneapolis, MN, p. 284.

15. Oh, S. L, Wu, W. T., and Park, J. J., (1987), "Application of the FiniteElement Method to P/M Forming Processes," Proc. 2nd ICPT, Stuttgart, WestGermany, p. 961.

16. Osakada, K., Nakano, J., and Mori, K., (1982), "Finite Element Method forRigid-Plastic Analysis of Metal Forming—Formulation for Finite Deforma-tion," Int. J. Mech. Sci., Vol. 24, p. 459.

17. Mori, K., Shima, S., and Osakada, K., (1980), "Finite Element Method forthe Analysis of Plastic Deformation of Porous Metals," Bull. JSME, Vol. 23,No. 178.

18. Im, Y. T., and Kobayashi, S., (1985), Finite Element Analysis of PlasticDeformation of Porous Materials, in "Metal Forming and Impact Mechanics,"(Edited by S. R. Reid), Pergamon Press, Oxford, p. 103.

19. Downey, C. L. Jr., and Kuhn, H. A., (1975), "Application of a Forming LimitConcept to the Design of Powder Preforms for Forging," /. Engr. Mat. Tech,Vol. 97, p. 121.

20. Im, Y. T., and Kobayashi, S., (1986), "Analysis of Axisymmetric Forging ofPorous Materials by the Finite Element Method," Advanced ManufacturingProcesses, Vol. 1, p. 473.

21. Austin, J. B., (1939), Factors Influencing Thermal Conductivity of Non-metallic Materials, in "Symposium on Thermal Insulating Materials," Ameri-can Society for Testing Materials, Philadelphia, p. 3.

22. Russel, H. W., (1935), "Principles of Heat Flow in Porous Insulators," /. Am.Ceram. Soc., Vol. 18, p. 1.

23. Eucken, A., (1932), VDI, Forschungsheft, No. 353 (in German), Forschungauf dem Gebiete des Ingenieurwesens Ausgabe B, Band 3, March-April.

24. Im, Y. T., and Kobayashi, S. (1986), "Coupled Thermo-Viscoplastic Analysisin Plane-Strain Compression of Porous Materials," Advanced ManufacturingProcesses, Vol. 1, p. 269.

25. Fischmeister, H., Sjoberg, G., Elfstrom, B. O., Hamberg, K., and Mironov,V., (1977), Preforms Ductility and Transient Cracking in Powder Forging, in"Modern Developments in Powder Metallurgy," (Edited by H. H. Hausner


and P. V. Taubenblat), Vol. 9, American Powder Metallurgy Institute,Princeton, NJ. p. 437.

26. Malik, R. K., (1974), "Hot Pressing of Titanium Aerospace Components,"Int. J. Powder Metal, and Powder Tech., Vol. 10, No. 2.

27. Ferguson, B., Kuhn, A., Smith, O. D., and Hofstatter, F., (1984), "HotConsolidation of Porous Preforms Using Soft Tooling," Int. J. Powder Metal,and Powder Tech., Vol. 20, No. 2, p. 131.

28. Fischmeister, H. F., Aren, B., and Eastering, K. E., (1971), "Deformationand Densification of Porous Preforms in Hot Forging," Powder Metal., Vol.14, No. 27, p. 144.

29. Schacher, H. D., (1978), Kaltmassivumformen von Sintermetall, Ph.D.Dissertation, Institute fur Umformtechnik, Universitat Stuttgart.

14THREE-DIMENSIONAL PROBLEMS

14.1 Introduction

A majority of the finished products made by metal forming are geometri-cally complex and the metal flow involved is of a three-dimensional nature.Thus, any analysis technique will become more useful in industrialapplications if it is capable of solving three-dimensional metal-flowproblems.

Nagpal and Altan [1,2] introduced dual-stream functions for describingmetal flow in three dimensions. This work showed that the properselection of a flow function makes the incompressibility requirementautomatically satisfied and provides general kinematically admissiblevelocity fields. Yang and Lee [3] utilized the conformal transformation of aunit circle onto a cross-section in the analysis of curved die extrusion. Theyderived the stream-line equation from which a kinematically admissiblevelocity field was determined. The upper-bound method was then appliedto determine the extrusion pressure for a rigid-perfectly plastic material.An important aspect of three-dimensional plastic deformation is theanalysis of spread in metal-forming operations, such as spread in rolling orin flat tool forging, and spread in compression of noncircular disks.Solutions to such problems have been obtained by the use of Hill's generalmethod [4] and the upper-bound method [5-7].

The extension of the finite-element method to solve three-dimensionalproblems is natural and not new, particularly in the area of elasticity [8].However, the simulation of three-dimensional forming operations by thefinite-element method is relatively recent. Park and Kobayashi [9] de-scribed the formulation for the three-dimensional rigid-plastic finite-element method and the implementation of the boundary conditions. Theyapplied this technique to the analysis of block compression between twoparallel flat platens. For certain forming problems, such as those involvinglateral spread, the use of a simplified three-dimensional element is efficientand some examples can be found for analysis of spread in rolling and flattool forging [10, 11].

275


14.2 Finite-Element Formulation

Element EquationsThe matrices for evaluation of elemental stiffness equations are defined fora three-dimensional brick element in Chap. 6 and some of them arerecapitulated in this section. A three-dimensional brick element used forthe analysis is an eight-node hexahedral isoparametric element. As shownin Fig. 14.1 the global coordinate system (x, y, z) is transformed into thenatural coordinate system (£, rj, £). The natural coordinate system isdefined such that £, r\, and £ vary from —1 to 1 within each element. Thevelocity field within an element is expressed as

where

with

For eqs. (14.1), (14.2), and (14.3) above, reference can be made to eqs.(6.17), (6.18), and (6.19) in Chap. 6.

FIG. 14.1 Three-dimensional hexahedral element.

Three-Dimensional Problems 277

The strain-rate vector ET= {ex, ey, ez, jxy, yyz, YZ*} is expressed by

by taking the space derivatives of velocities, where the matrix B is given by

where

and

The determinant of the Jacobian matrix of the coordinate transformationthat is necessary for integration of elemental stiffness equation is given byeq. (6.38) in Chap. 6.

Traction Prescribed Boundary

The boundary conditions along the interface between workpiece and dieare mixed, because the velocity is given in the direction normal to theinterface surface and the traction is prescribed by friction along the contactsurface in the direction opposite to the relative sliding of the workpiecewith respect to the die. The prescribed velocity is treated as an essentialboundary condition, and the prescribed traction, which is a suppressibleboundary condition, is imposed during the discretization process.

Suppose that the surface (1-2-3-4) of the element shown in Fig. 14.1 isin contact with the die (x, y plane). On that surface, the natural coordinate £is equal to unity and the shape functions become qt = \(l + ££,)(! + r/Jj,).Equation (14.2) reduces to

and the nodal point velocity vector is given by

he determinant of the Jacobian transformation matrix is given (see eq.(6.35b) in Chap. 6) by


where

The frictional stress is represented as velocity dependent (see eq. (5.30)in Chap. 5). For the Newton-Raphson procedure, the complete formula-tion requires the detailed manipulations for the derivatives of thefunctional and can be found elsewhere [12].

14.3 Block Compressions [9]Compression of Rectangular BlocksThe compression of rectangular blocks was investigated by Kanacri et al.[13] using annealed Al-1100. The same conditions of workpiece geometry,material property, and interface friction were used for simulations.

The workpiece dimensions were 1/1/0.5 and 2/1/0.5 (width W/breadthB/thickness H). The friction factors m — 0.1 and 0.2 were chosen for thelubricated conditions for the two geometries, while m = 0.5 was used forthe dry (nonlubricated) condition. The flow stress of annealed Al-1100 isapproximately represented by

with r0 = 9102 psi (62.74 MN/m2).The mesh systems used for computation consist of a total of 75 elements

interconnected at 144 nodal points for W0/B0 = 1, and 72 elements with 104nodal points for W0/B0 = 2. The computation was carried out up to 50%reduction in height with 2% height reduction as a step size. Theconvergence was rapid and steady. Only four iterations were necessary toreach the accuracy of the velocity error norm less than 0.00001 with adeceleration coefficient of 0.5. Each simulation needed about 600 CPseconds execution time on a CDC 7600 computer.

The results are summarized in terms of contact pressure distributions,compression loads, spread contours, and strain distributions. Comparisonsof the computed loads with the experimental results are shown in Fig.14.2. It is of interest to note that the computed results are closer to theinterrupted loadings for both friction conditions. Also, it is seen thatassuming m = 0.1 underestimates the actual friction for the lubricated casefor WQ/B0 = 1. However, m = 0.2 and m = 0.5 are good choices for overallestimations of the friction conditions for W0/B0 = 2.

Geometrical changes of the equatorial planes under both frictionconditions are compared in Fig. 14.3. Because the height of the workpieceis relatively short compared to the other two dimensions, geometricalchanges along the height direction are negligibly small. As the deformationcontinues the workpiece spreads out and the compression load increases.A discrepancy in the spread contour at the corner is shown for W0/B0 = 1(lubricated), which suggests that m = 0.1 used in simulation is less than the

and

FIG. 14.2 Comparisons of the computed compression loads (broken curves) with the experimental results (solid curves) [13] in rectangular blockKJ compression.

£o

FIG. 14.3 Comparison of the computed (broken curves) and experimental (solid curves) [13] spread contours at the equatorial plane in rectangular blockcompression.


actual friction value that is present in the experiment. On the other hand,in the simulation for WJB0 = I, m = 0.5 is a proper value to assume forthe dry condition. The simulation, for W0/B0 = 2 with m = 0.2 for thelubricated condition, gives an excellent agreement with the experimentalresult. Some discrepancy around the corner in simulation for the drycondition may indicate that friction is not uniform over the entiredie-workpiece interface.

Compression of Wedge-Shaped Blocks

Compression of a wedge-shaped block (Fig. 14.4) between two flat paralleldies is used in practice for workability and microstructural studies inforging. The boundary conditions for the analysis are more complex incomparison with those for rectangular block compression owing to thelower degree of symmetry in the geometry. The determination of theneutral regions at the contact surfaces, with both upper and lower dies,represents an important problem. In particular, along the die-workpiececontact surface, at the lower die the extent of the actual contact is notknown and varies during deformation. The finite-element code for thisproblem includes additional schemes in order to take into account theseboundary conditions, as well as folding, that occur in rectangular blockcompression.

The workpiece dimensions used in the simulations are shown in Fig.14.4. The workpiece geometry and deformation are symmetrical withrespect to the (x, z) plane, so that one-half of the workpiece is taken as thecontrol volume for simulation. The flow stress of the material, Al-2024(annealed), is represented by a = 225 (1 + e/1.6147) (MN/m2). Twosimulations were performed under different friction conditions: dry (m =0.4) and lubricated (m = 0.1). A step size of 2% in height reduction was

FIG. 14.4 Dimensions of the workpiece in compression of wedge-shaped block.


used up to 30% reduction in height, and a step size of 1%, thereafter.Sixty elements (126 nodes) were used for the lubricated condition, while 48elements (105 nodes) were used for the dry condition. The computationaltime (CP time) for the total deformation of 60% reduction in height was2300 s with 60 elements and 1700 s with 48 elements on a CDC 7600computer.

Experiments were performed, and measurement such as load-displacement curves and geometrical changes were compared with some ofthe computed results. Details of the experiments and comparison with thetheory are reported in Reference [14].

Grid distortions at three stages of height reduction are shown in Fig.14.5. Several observations about the mode of deformation can be madefrom this figure. The side surface of the block portion bulges outward,while the side surface of the wedge portion becomes concave. This modeof deformation is more pronounced with the dry interface condition. Thecontours of the die-workpiece contact area show more severe distortion inthe width direction for higher interface friction.

Effective strain distributions in the mid-plane (plane of symmetry) areshown in Fig. 14.6. Locations of the neutral zones are indicated by arrows.The results show that higher concentration of strain occurs around the

FIG. 14.5 Grid distortion of the wedge-shaped workpiece at several height reductions (20,40, and 60%).

NJ FIG. 14.6 Effective strain distributions and location of neutral zones (indicated by arrows) in the plane of symmetry at several height reductions ofw compression of a wedge-shaped block.


lower center of the block portion and that the strain decreases monotoni-cally toward the end of the wedge. The strains are, in general, uniform inthe height direction, particularly in the wedge portion and for low friction.Locations of the neutral zones are symmetrical and approximately posi-tioned at the mid-point of the contact surface at larger reductions inheight. However, at 20% reduction, the interfacial friction conditionappears to have a significant influence on the location of the neutral zones.

In using the wedge compression for a workability test, fracture is usuallyobserved on the side surface. The present analysis provides someinformation concerning the occurrence of fracture, such as the strain pathand the corresponding stress variations at critical sites.

14.4 Square-Ring Compression [15]

In square-ring compression, a cube shaped billet with a coaxial square holeis pressed between flat dies. A schematic diagram of the undeformedsquare-ring is shown in Fig. 14.7. It is expected, in this process, thatdifferent modes of deformation may result depending upon the aspect ratioof workpiece and friction. The three-dimensional finite-element methodwas used to simulate the process [15]. The undeformed preform was a cube(50.8 x 50.8 x 50.8 mm) with co-axial square hole (25.4 x 25.4 x 50.8 mm).The friction between the workpiece and the die was assumed to followthe constant shear factor friction law with m=0.5. The workpiecematerial was annealed Al-1100 and its flow stress was represented by o =62.74(1 + £/0.052)a3 (MN/m2). Owing to the symmetry, it was sufficient toinclude one-sixteenth of the workpiece in the analysis, as shown by thick

FIG. 14.7 Schematic diagram of the square ring used in simulation. The thick lines indicatethe volume included in the analysis.


lines in Fig. 14.7. A total of 250 elements with 396 nodes were usedfor the analysis. Simulation was done in a step-by-step manner and 2%of the undeformed height was used as the step size.

The results show that the buckling-type deformation mode takes place atthe beginning of deformation. That is, the mid-height of the side wallmoves outward while the die contact surface moves very little in thehorizontal direction. As a result, nodes at the die contact surface near theinner edge separate from the die at the initial stage of the deformation.The separated nodes touch the die shortly afterwards. Figure 14.8 showsthe perspective view of one-quarter of the workpiece with the predictedmesh distortions of the cross section on the YZ plane at steps of 20% inheight. It is seen in the figure that at 20% reduction in height, themid-plane moves outward while the top and bottom surfaces change theirpositions very little. This deformation mode causes buckling. At 40%reduction in height, the outside wall forms a considerable bulge, while theinner wall forms a dimple. The dimple at the inner wall becomes sharperas deformation progresses and, finally, it collapses, forming a fold at 54%reduction in height. This inner surface folding takes place first at the YZcross section and propagates towards the corner of the hole. The folding ofthe inner surface can be seen clearly from the predicted grid distortion at60% reduction in height.

In Fig. 14.8, locations of the neutral points are indicated by arrows. It isseen that the spatial position of the neutral point on the YZ cross sectionchanges very little with deformation. However, its location relative to amaterial point changes considerably.

Figure 14.9 shows the top view of the distorted mesh at steps of 20%reduction in height. In the figure, the neutral lines are indicated by brokenlines. It is seen from the figure that the material points at the mid-side ofthe inner surface move inward, while points at the corner of the innersurface move outward until 40% reduction in height is reached; the cornerpoints then start to move inward. Because of the different movements ofthe material points on the die contact surface, the inner surface, lookingfrom the top, becomes inwardly convex and the angle between the twoinner surfaces becomes less than 90°.

Aku et al. [16] carried out square-ring compression tests by usingplasticine as a model material. Though the detailed measurements of theprocess have not been reported, they include the sketch of the top view ofthe deformed workpiece at 30, 50 and 77% reductions in height. Bycomparing their sketch with Fig. 14.8, it is found that the presentpredictions agree qualitatively very well with the experimental observa-tions. Seen from the top, the inner hole (which is inwardly convex), thebulge at the inner surface (which is not noticeable), and the large amountof bulge at the outside surface all indicate reasonable agreement betweenthe predictions and experiment. These observations are only qualitativeand further critical comparison may be necessary to fully verify the resultsof the simulations.

FIG. 14.8 Perspective views of one-quarter of the workpiece for square-ring compression,with the predicted mesh distortions on the YZ cross section at 0, 20, 40 and 60% reductionsin height (units of mm). (Arrows indicate the position of the neutral surface.) [15].

286


FIG. 14.9 Top views of grid distortions predicted by the analysis at 0, 20, 40, and 60%reductions in height during square-ring compression. The broken lines indicate the neutrallines (units of mm) [15].

A total of 44 hours of CPU time was required for 412 iterations tocomplete the simulation on a VAX 750. About 70% of the total CPU timewas used after 40% reduction in height, owing to a large amount of freesurface folding.

14.5 Simplified Three-Dimensional Elements

The use of simplified three-dimensional elements can be reasonable inanalyzing some aspects of three-dimensional deformation. One such aspectis the analysis of lateral spread. Simplified hexahedral elements can bederived by taking one layer of elements in the direction normal to theplane of symmetry and assuming that the element-sides, which are initiallynormal to the plane, remain normal during deformation.

In the element shown in Fig. 14.1, suppose that the plane of symmetry is(XOY) and that the nodes 5, 6, 7, and 8 will remain on that plane


throughout the process. Then the assumption states that the velocitycomponents ux and uy are independent of the z coordinate. We then have

and

We also have

and

This assumption leads to a linear distribution of uz in the z direction,and the coordinate transformation and shape functions become

With this simplification, a three-dimensional eight-node element can bedescribed by the coordinates and the velocities of only four lateral nodesinstead of eight nodes.

For the simplified 8-node hexahedral element defined by eqs. (14.11),(14.12) and (14.13), the strain-rate matrix B and the Jacobian matrix oftransformation simplify accordingly.

In the three-dimensional treatment of friction, where the contact surfaceis inclined with respect to the global coordinates, the surface integrationmust be performed numerically. Therefore, special care should be taken inthe evaluation of the second-order derivative in the numerical integrationscheme. With reference to the element arrangement shown in Fig. 14.1, let(3-4-8-7) be the surface of an element on which the frictional stress isacting, in the example of the rolling process (Fig. 14.10). The relativevelocity components with respect to the die velocity in the tangential andthe transverse directions are u, and uz, respectively. According to thevelocity distribution in the simplified element, we have

In eq. (14.14), the directions of «,<3) and w,(4) are tangential to the contactsurface at points 4 and 3, respectively. Thus, the distribution of u, given ineq. (14.14) is the result of an approximation usually involved indiscretization.


FIG. 14.10 Roll-workpiece contact surface of an element.

For surface integration of these expressions, the Gaussian quadraturewas used in the z direction and the Simpson rule was used in the tangentialdirection. Further, the Gaussian quadrature at eight integration points wasused for volume integration over the element.

14.6 Analysis of Spread in Rolling and Flat-Tool Forging

Rolling [10]

In rolling wide strips or plates, where the ratio of width-to-length of thedeformation zone is large (say, more than 10), spread in width is usuallynegligible and such deformation problems can be considered to be inplane-strain. Thus, most of the early theories of rolling were concernedwith predicting the load and torque under the assumption of plane-straindeformation. However, in rolling flat sections for which the width-to-length ratio is small (say, less than 6), spread is appreciable and cannot beneglected.

Since there was no theory of rolling to deal with three-dimensionaldeformation, with few exceptions, studies on deformation in rolling havebeen largely experimental. Chitkara and Johnson [17] and Helmi andAlexander [18] conducted experiments of rolling of lead bars andmild-steel bars, respectively, and compared the experimentally determinedspreads with those predicted by empirical formulas.

Few attempts have been made to predict the spread theoretically.Gokyu et al. [19] derived the expression for the width spread, applying theminimum-work hypothesis to the rolling problem. Lahoti and Kobayashi


[20] performed the computation for spread neglecting friction between rolland strip, and using the general method proposed by Hill (see Chap. 5).Kummerling and Lippmann [21] analyzed the spread by using a simplifiedtwo-parameter theory of plasticity. Oh and Kobayashi [5] analyzedthree-dimensional deformation in rolling using an extremum principle forrigid, perfectly plastic materials.

Several researchers have tried to solve the rolling problem by the use ofthe finite-element technique [22-27]. All the studies, however, used theplane-strain assumption, and the only approach taking into account thelateral spread in rolling has been the analysis by Kanazawa and Marcal[28], using line elements. At the international conference on numericalmethods in industrial forming processes, held at the University College ofSwansea in 1982, two papers [10, 29] were presented on the analysis ofspread in rolling. Both used the rigid-plastic finite-element method withsimplified three-dimensional elements.

For the analysis of spread in rolling, two arrangements of elements arepossible. The first considers the plane of symmetry (XOY) and the secondrefers to the plane (XOZ) as a plane of symmetry, as shown in Figs.14.11a and b, respectively. The element arrangement of Fig. 14.11aprovides the bulge profile of the front and side surfaces of the rolled stock,in addition to the amount of spread, while the arrangement of Fig. 14.libresults in the spread contour of the front edge, but does not give bulging ofthe spread surfaces. Therefore, the choice of the element arrangementdepends on the information required and possibly on the rolling condition,such as the width-to-thickness ratio. Mori and Osakada [29] applied thefirst type of element arrangement for the analysis of edge rolling(WQ/H0«1), and the second type for the spread analysis in plate rolling(WQ/HQ a 1). The rolling problem can be analyzed either as a nonsteady-state deformation or as a steady-state deformation. Mori and Osakadaadopted the steady-state approach. For this case, a solution is obtained interms of velocities and strain-rates. Then the flow lines are constructed andthe effective strain distribution, and thus the flow stress distribution, iscomputed by integrating the effective strain-rates along the flow lines. Thecalculation is repeated until the solution converges (see Chap. 10).

For the nonsteady-state procedure, deformation of the workpiece issimulated in a step-by-step manner from the beginning of the bite,updating the coordinates of material points and the material property aftereach step. The steady state is assumed to be reached when the spreadcontour becomes stationary in space and the roll torque has reached asteady value.

The analysis of lateral spread was performed using the nonsteady-stateapproach with the simplified eight-node element and the element arrange-ment shown in Fig. 14.11a. The conditions under which simulation wereperformed were as follows: roll radius R = 203 mm (8 in.); initial thicknessof the plate 2H0-2.5mm (0.1 in.); width-to-thickness ratios WQ/H0 = land 3; reduction in thickness, 5, 12.5, and 20%; material, annealed AISIsteel 1018; friction factor m = 0.5.

FIG. 14.11 Flat section rolling: configuration and two arrangements of elements.

291


FIG. 14.12 The profiles of a rolled square bar for 20% reduction: W0/HQ= I .

These computational conditions and the experimental data were takenfrom Lahoti et al. [30]. Figure 14.12 shows the spread of the rolled stockfor 20% reduction with a width-to-thickness ratio of unity. The averagelateral spreads of the rolled bars, computed and measured, are comparedin Fig. 14.13. Also, the reduction in cross-sectional area is plotted as afunction of reduction in height, as an indirect measure of the amount ofspread. The figure shows that the agreement between the theoreticalpredictions and the experimental results is excellent for both W0/H0 = 1and 3. It is to be noted that in the rolling process, as the material iscompressed, its cross-sectional area also is reduced. In the case ofplane-strain, the reduction in height is equal to that in the cross-sectionalarea; however, for smaller W0/H0 ratios, the reduction in cross-sectionalareas is less than that in height. Figure 14.13 also shows the relationshipbetween these two reductions, i.e., in cross-sectional area and in height.


FIG. 14.13. (a) Comparison between computed and measured [30] lateral spread forroom-temperature rolling of mild-steel plates, (b) Reduction in cross-sectional area vs.reduction in height for room-temperature rolling of mild-steel plates.

Flat-Tool Forging [11]

In flat-tool forging the initial efforts to predict the lateral spread werebased on experiment [31-33]. Theoretical predictions of spread can bemade by the use of simplified three-dimensional elements. During conven-tional bar forging, the bar is moved longitudinally with respect to the dieseveral times between forging strokes in one pass (Fig. 14.14). Someresults of the analysis of a three-bite flat-tool forging operation in whichthe thickness of a bar of initial length L0 is reduced from 2H0 to 2H^ bythree strokes, are shown in Figs. 14.14 and 14.15. An example of the straindistributions after each bite is shown in Fig. 14.14 for B/2W0 = 0.5. It canbe seen that interaction of the deformation zones occurs near to the dieedges where strain concentrations are observed.

Experimental results of spread and elongation are compared with thecomputed results in Fig. 14.15. Comparisons of the changes in width andlength show that the agreement between theory and experiment is good interms of trends but not in terms of magnitudes. Several reasons can begiven for this discrepancy. In experiments the specimen dimensions werenot exact and the reductions in height were not exactly controlled.Furthermore, the measurements of the final specimen dimensions includedsome errors. In the theoretical calculations, besides the use of simplifiedelements, some uncertainties are obviously associated with the frictionconditions. It was shown, however, that the use of the simplifiedthree-dimensional elements in the finite-element analysis provides goodpredictions of spread in rolling and flat-tool forging.

Stroke 3

FIG. 14.14 Flat-bar forging and strain distributions in three-bite flat-bar forging.

294

Three-Dimensional Problems

Stroke

295

FIG. 14.15 Comparison between theory and experiment for spread (AWmax/W0) andelongation (AL/L0) in three-bite flat-bar forging (Bar and tool geometry are shown in Fig.14.14). [11]


It is concluded that for geometrically simple block compression, thefinite-element method provides reasonably good solutions for the detail ofthree-dimensional plastic deformation with relatively few elements. How-ever, it is evident that the economic constraint becomes more severe forthree-dimensional metal-flow analysis in comparison with two-dimensionalproblems. Therefore, in the development of analysis techniques for thethree-dimensional metal flow, special consideration must be given toachieving a balance between computational efficiency and solution ac-curacy. In this regard, the use of simplified three-dimensional elements canbe helpful. Although the problems for which simplified elements can beused are limited, the approach can be effectively applied to a class of


problems in which prediction of spread is important. Nevertheless, thedevelopment of the three-dimensional analysis technique will continue formore complex problems. This is indeed indicated by recent publications.These developments must deal with not only geometrical complexities, butalso with three-dimensional heat transfer and its coupling with flowanalysis. Also, complex numerical techniques, such as rezoning, must bedeveloped. In addition, the analysis program must be interfaced with thecapabilities of a computer-graphics and data-display system in order toevaluate the results quickly and effectively. Eventually, three-dimensionalprograms will become useful for process design; for example, for design ofroll-pass schedules, design of operational sequences in flat-tool forging,and design of preforms in hot and cold forging.

References

1. Nagpal, V., (1977), "On the Solution of Three-Dimensional Metal FormingProcesses," Trans. ASME, J. Engr. Ind., Vol. 99, p. 624.

2. Nagpal, V., and Allan, T., (1975), "Analysis of the Three-Dimensional MetalFlow in Extrusion of Shapes with the Use of Dual Stream Functions," Proc. 3dNAMRC., Pittsburgh, PA, p. 26.

3. Yang, D. Y., and Lee, C. H., (1978), "Analysis of Three-DimensionalExtrusion of Sections Through Curved Dies by Conformal Transformation,"Int. J. Mech. ScL, Vol. 20, p. 541.

4. Lahoti, G. D., and Kobayashi, S., (1974), "Flat-tool Forging," Proc. 2dNAMRC, University of Wisconsin, Madison, p. 73.

5. Oh, S. I., and Kobayashi, S., (1975), "An Approximate Method forThree-Dimensional Analysis of Rolling," Int. J. Mech. ScL, Vol. 17, p. 293.

6. Sagar, R., and Juneja, B. L., (1979), "An Upper Bound Solution for Flat ToolForging Taking into Account the Bulging of Sides," Int. J. Machine Tool. Des.Res., Vol. 19, p. 253.

7. Braun-Angott, P., and Berger, B., (1982), An Upper Bound Approximationfor Spread and Pressure in Flat Tool Forging, in "Numerical Method inIndustrial Forming Processes," (edited by J. F. T. Pittman, R. D. Wood, J. M.Alexander and O. C. Zienkiewicz) Pineridge Press, Swansea UK, p. 165.

8. Zienkiewicz, O. C., (1977), "The Finite Element Method," 3d Edn. McGraw-Hill, New York.

9. Park, J. J., and Kobayashi, S., (1984), "Three-Dimensional Finite-ElementAnalysis of Block Compression," Int. J. Mech. Sci., Vol. 26, p. 165.

10. Li, G.-J., and Kobayashi, S., (1982), Spread Analysis in Rolling by theRigid-Plastic Finite Element Method, in "Numerical Method in IndustrialForming Processes," (edited by J. F. T. Pittman, R. D. Wood, J. M.Alexander and O. C. Zienkiewicz) Pineridge Press, Swansea UK, p. 777.

11. Sun, Jie-Xian, Li, G.-J., and Kobayashi, S., (1983), "Analysis of Spread inFlat-Tool Forging by the Finite Element Method," Proc. llth North AmericanManufacturing Research Conf., May 1983, Madison, Wisconsin, p. 224.

12. Park, J. J., (1982), "Applications of the Finite Element Method to MetalForming Problems," Ph.D. Dissertation, Department of Mechanical Engi-neering, University of California, Berkeley.

13. Kanacri. F., Lee, C. H., Beck, L. R., and Kobayashi, S., (1972), "PlasticCompression of Rectangular Blocks Between Two Parallel Platens," Proc.13th. Int. Mach. Tool Des. Res. Conference, Birmingham, p. 481.


14. Sun, Jie-Xian, and Kobayashi, S., (1984), "Analysis of Block Compressionwith Simplified Three-Dimensional Element," Proc. 1st Int. Conf. on Tech-nology of Plasticity, Tokyo, p. 3.

15. Park, J. J., and Oh, S. I., (1987), "Application of Three-Dimensional FiniteElement Analysis to Metal Forming Processes," Proc. NAMRC XV, Beth-lehem, PA, p. 296.

16. Aku, S. T., Slater, A. C., and Johnson, W., (1967), "The Use of Plasticine toSimulate the Dynamic Compression of Prismatic Blocks of Hot Metal," Int. J.Mech. ScL, Vol. 9, p. 495.

17. Chitkara, N. R., and Johnson, W., (1966), "Some Experimental ResultsConcerning Spread in the Rolling of Lead," Trans. ASME, J. Basic Engr.,Vol. 88, p. 489.

18. Helmi, A., and Alexander, J. M., (1968), "Geometric Factors AffectingSpread in Hot Flat Rolling of Steel," J. Iron Steel Inst., Vol. 206, p. 1110.

19. Gokyu, I., Kiharai, J., and Mae, Y., (1970), "Study on the Width Spread inFlat Rolling," /. Japan Soc. Tech. Plast, Vol. 11, p. 11.

20. Lahoti, G. D., and Kobayashi, S., (1974), "On Hill's General Method ofAnalysis for Metal-Working Processes," Int. J. Mech. ScL, Vol. 16, p. 521.

21. Kummerling, R., and Lippmann, H., (1975), "On Spread in Rolling," Mech.Res. Commun., Vol. 2, p. 113.

22. Tamano, T., (1973), "Finite Element Analysis of Steady Flow in MetalProcessing," J. Japan Soc. Tech. Plast., Vol. 14, p. 766.

23. Zienkiewicz, O. C., Jain, P. C., and Onate, E. (1978), "Flow of Solids DuringForming and Extrusion: Some Aspects of Numerical Solutions," Int. J. SolidsStructures, Vol. 14, p. 15.

24. Zienkiewicz, O. C., Onate, E., and Heinrich, J. C., (1981), "A GeneralFormulation for Coupled Thermal Flow of Metals using Finite Elements," Int.J. Num. Meth. Eng., Vol. 17, p. 1497.

25. Dawson, P. R., (1978), "Viscoplastic Finite Element Analysis of Steady-stateForming Processes Including Strain History and Stress Flux Dependence,"Applications of Numerical Methods to Forming Processes, ASME, AMD,Vol. 28, p. 55.

26. Shima, S., Mori, K., Oda, T., and Osakada, K., (1980), "Rigid-Plastic FiniteElement Analysis of Strip Rolling," Proc. 4th Int. Conf. on Prod. Eng.,Tokyo, Japan, p. 82.

27. Li, G.-J., and Kobayashi, S., (1982), "Rigid-Plastic Finite Element Analysis ofPlane Strain Rolling," ASME Trans., J. Engr. Ind., Vol. 104, p. 55.

28. Kanazawa, K., and Marcal, P. V., (1982), "Finite Element Analysis of PlaneStrain Rolling," ASME Trans., J. Engr. Ind., Vol. 104, p. 55.

29. Mori, K., and Osakada, K., (1982), "Simulation of Three-Dimensional Rollingby the Rigid-Plastic Finite Element Method," Proc. Num. Methods Ind.Forming Processes, Swansea, p. 747.

30. Lahoti, G. D., Akgerman, N., Oh, S. I., and Altan, T., "Computer-aidedAnalysis of Metal Flow and Stresses in Plate Rolling," /. Mech. Work. Tech.,Vol. 4, p. 105.

31. Tomlinson, A., and Stringer, J. D., (1959), "SPread and Elongation inFlat-tool Forging," /. Iron Steel Inst., Vol. 193, p. 157.

32. Baraya, G. L., and Johnson, W., (1964), "Flat-bar Forging," Proc. 5th Int.Conf. MTDR, p. 449.

33. Kudo, H., and Nagahama, T., (1969), "Experimental Results for UpsettingPressure and Material Spread—Study in Transverse Upsetting Process ofCircular Rods, 1st Report," /. Japan Soc. Tech. Plast., Vol. 10, No. 106, p.827.

15PREFORM DESIGN IN METAL FORMING

15.1 Introduction

Preform design in metal forming refers to the design of an initial shape ofthe workpiece that, when it has undergone an associated forming process,forms the required product shape with desired property successfullywithout formation of defects and without excessive waste of materials. Acarefully selected preform can contribute significantly to the reduction ofthe production costs. Preform design problems are encountered in variousmetal-forming processes, such as closed-die forging, shell nosing, rolling,and sheet-metal forming. Design of an optimal preform shape requiressimultaneous determination of optimal process conditions. However, weare here concerned with the determination of the best preform shapeunder a given set of process conditions.

In this chapter, a new method of "backward tracing" is introduced as analternative approach to the solution of preform design [1], and theapplications of this method to some specific processes are discussed.

15.2 Method for Design

Similarly to the forward simulation technique, the backward tracingmethod uses the finite-element method. The forward simulation techniquehas been discussed in the previous chapters.

Backward tracing refers to the prediction of the part configuration at anystage in a deformation process, when the final part geometry and processconditions are given. The concept is illustrated in Fig. 15.1. At time t = ta,the geometrical configuration x0 of a deforming body is represented by apoint Q. The point Q is arrived at from the point P, whose configuration isgiven as XQ.J at t = f0-i, through the displacement field during a time-stepAt, namely, XQ = XQ_I + u0_! Af, where u0_i is the velocity field at t = f0-i.Therefore, the problem is to determine UG-I, based on the information (XQ)at point Q. The solution scheme is as follows: taking a loading solution u()

(forward) at Q, a first estimate of P can be made according toPm = XQ — u0 Af. Then, the loading solution u^, can be calculated fromthe configuration of P(1), with which the configuration x() at Q can becompared with P(1) + u^t Af = Q(l\ If Q and Q(1) are not sufficiently closeto each other, then P(2) can be estimated by P(2) = XQ- u(V_i Af. The

298

Preform Design in Metal Forming 299

FIG. 15.1 Concept of the backward tracing method (geometrical configuration only).

solution for loading at P(2) is then u,^, and the second estimate of theconfiguration Q(2) = F(2)+ u,(,221 Af can be made. The iteration is carriedout until Q(n) = P(n> + v$2i Af becomes sufficiently close to Q.

The formulation of the concept described above will further clarify thebackward tracing technique. After finite-element discretization, the solu-tion of the boundary value problem associated with isothermal viscoplasticdeformation satisfies a system of nonlinear, coupled algebraic equations,expressed in the form

with the boundary conditions in terms of velocity and traction, where K isthe stiffness matrix; f is the force vector; x is the nodal point coordinates; vis the nodal point velocities; e is the integration point effective strains; e isthe integration point strain rates; and T is the given temperature(constant).

In eq. (15.1), x and E at time t are supposed to be known, and e is afunction of v. In forward simulation (t & t0), the solution v() of eq. (15.1) att = tn is used for the solution at the next step t = t(>+ { = t(} + Af, according tothe explicit scheme as

and


LOADING PROGRAM BACKWARD TRACING PROGRAM

FIG. 15.2 Flow chart for the backward tracing procedure implemented into the finite-element flow analysis.

In backward tracing (t<t0), at time t = t0_1 = t0 — At, XQ_I, e0-i shouldsatisfy the following conditions, in addition to eq. (15.1),

For a given time-increment, the substitution of eq. (15.3) into eq. (15.1) attime t0-i results in


where XQ, §o are known and EO_I is a function of v0_!. It is to be noted thatCQ is needed only for work-hardening materials. Equations (15.4) aresolvable for v0_i if the boundary conditions can be specified at time t0_i.Repeated calculations of the solutions for times £0_2, t0_3, . . . will lead tobackward tracing of the entire deformation process. A flow chart for thebackward tracing procedure implemented into the finite-element deforma-tion analysis is shown in Fig. 15.2. It is to be noted that the application offorward simulation (loading) and backward tracing becomes straightfor-ward if the change of the boundary conditions during a process is known.In many cases the change of the boundary conditions during loadingdepends on the shape of the preform itself. Then it becomes an importantissue in the design methodology to develop appropriate design criteria forcontrolling the change of the boundary conditions during deformation.This aspect of design, together with some others, such as consideration ofwork-hardening and temperature, are discussed in the following specificapplications of the method.

15.3 Shell Nosing at Room Temperature

Shell nosing is a process of forming an ogive nose at an open end ofcylindrical shell by forcing the shell into a contoured die, as described inChap. 2. In all modern methods of shell manufacturing, the cavity isformed to a finish shape and machining is restricted to the outer surface ofthe nosed shell. Thus, in this process it is essential to determine a preformshape that forms precisely the required final configuration with a specifiedwall thickness distribution after nosing. The metal flow in nosing iscomplex. Some large shells are manufactured at elevated temperaturesowing to the severe deformation required at the open end of the shell.

Until recently, there was very little quantitative information availableand, in most shell manufacturing plants, extensive experience and expen-sive trial-and-error techniques were necessary in order to design the nosingprocess. Pioneering work in this field is attributed to Nadai [2], whooutlined an approach for the strain analysis in the shell nose. UsingNadai's approach, Carlson [3] suggested a method for determining theoriginal shell profile, knowing the final shape. Based on Carlson's method,Lahoti et al. [4] developed a computer program for the preform designprocedure. Kobayashi [5] derived approximate solutions for determiningthe shell profile from the tangential velocity distributions along the diecontour. The approaches mentioned above, however, give only approxim-ate answers.

Park et al. [1] applied the backward tracing method to the determinationof preforms that result in a nosed shell with uniform wall thickness. For diegeometry and nosed shell configurations shown in Fig. 15.3, two preformconfigurations were determined. The wall thickness distributions requiredin the preform can be accommodated by varying either the inner or theouter diameters of the preform. This criterion was imposed for controllingthe boundary condition during the backward tracing process.


FIG. 15.3 Die dimensions and nosed shell configurations. L0 = 47.26, Lf = 48.84, R =21.526, Rt = 14.648, Bf = 14.47, Hf = 6.88 (uniform), a0 = 237, b0 = 18.75 (units of mm).

For the preform shape with constant outer diameter (Type-0 preform),RO = R0 for all n, where RQ is the radius of a nodal point located along theouter surface of the shell and R0 is the radius at the die entrance. Theboundary condition for this type of preform is controlled in such a way thatnodes that are in contact with the die (7?o < RO) become force free whenthe condition that RQ = R0 is reached during backward tracing.

For the preform shape with constant inner diameter (Type-I preform),the criterion for controlling the boundary condition is that as soon asR? = Rf for the nodes Rp

t < Rh the condition of the corresponding outsidenode is changed from that of die-contact to the force free condition, whereRp

t is the radius of a nodal point along the inner surface of the shell.Backward tracing was carried out computationally by taking 2 mm

penetration (in loading) as one step. Thus, a total of 22 steps wasnecessary to complete the calculation. The solution convergence neededabout six to seven iterations. For completing one step of backward tracing,it was necessary to have one to two iterations for the case where theboundary condition does not change, and two to three iterations wererequired for the case where a change in the boundary condition occurs.Convergence for backward tracing was determined by the closeness of thesolution (between Q and <2(M) described in Section 15.2) within a limit.The two types of preforms determined by this method are compared withthe final nose configuration in Fig. 15.4. In the figure, the lengths £° and£o, for Tpye-0 and Type-I preforms, respectively, indicate the portion ofthe preform that becomes the nosed shell length Lf after nosing. Because


FIG. 15.4 Nosed shell configuration and two types of preform shapes for nonwork-hardeningmaterials [1].

the shell elongates during nosing, the preforms are shorter than the nosedshell. The shell material was assumed to be rigid-perfectly plastic(nonwork-hardening).

In order to include the work-hardening effect in preform design, it isnecessary to know the strain distributions in the nosed shell, in addition tothe geometrical configurations. However, the strain distributions in thenosed shell depend on the preform shapes and are not known (or cannotbe specified). In Fig. 15.5 the procedure used to take into account thework-hardening effect is illustrated.

By applying the backward tracing scheme to nonwork-hardening mate-rials, the two types of preforms can be determined. With these preformsthe solutions for loading are obtained for a work-hardening material. Theresults provide strain (effective strain) distributions and the nosed shellconfigurations that are very close to but not identical to the specifiedgeometries. These configurations are then corrected exactly according tothe specifications, maintaining the same effective strain distributions. With

FIG. 15.5 Schematic illustration of the procedure to include the work-hardening effect in prefr»rn design.


these strain distributions, the application of the backward tracing schemeresults in the preform shapes for a work-hardening material. At each step,the strain and corresponding stress are also traced backward following thestress-strain curve of the material. It should be mentioned that the firstcomputational procedure given in Fig. 15.5 may be eliminated by usingapproximate solutions for preform shapes of nonwork-hardening materials.

15.4 Plane-Strain Rolling [7]

In rolling a flat-ended ingot, shapes with defective ends may be formed.These shapes have an overhang at the front and rear ends, and formoverlaps after several rolling passes. In addition, the so-called "fishtail" isformed at the rear end if edge-rolling is applied. These defective endregions, shown in Fig. 15.6, are cut off and lost as scrap. Johnson [6] hasremarked that the end shape changes found in rolled slabs will be difficultto predict, since they occur in the context of nonsteady-state conditionsand are of a complicated three-dimensional character. However, theoreti-cal predictions of the defective shapes and end shape designs to eliminatethem are possible if the plane-strain condition is assumed. Hwang andKobayashi [7] tested an approximate design of preformed end shapes fornonwork-hardening materials [8] by simulating plate rolling by thefinite-element method.

Figure 15.7 shows the contours of end shapes used in rolling nonwork-hardening materials. Figures 15.7a and b show the changes in the front andthe back end shapes, respectively. After rolling, the end shapes are notperfectly flat, but they are fairly flat. This demonstrates that the designbased on approximate solutions is very good. When the required endflatness is of the degree shown in the figure, the approximate approach willgive various preform end shapes under various process variables, such asthe roll diameter-to-plate-thickness ratio and reduction in thickness.

The backward tracing scheme was used to determine the preform endshapes that are perfectly flat after rolling. For the front end, the shapeobtained with an approximate solution was modified to a flat end and theloading pass that would result in this desired end shape was tracedbackward. The front end shapes obtained at several locations from the exit

FIG. 15.6 Defective end shapes in flat and ingot rolling.

FIG. 15.7 (a) Front and (b) back end contours at several stages of rolling of a plate with apreform design based on approximate solution (reduction in thickness = 0.2; roll diameter =4.0; thickness of rolled plate = 1.0; material, nonwork-hardening; friction, sticking).

306

FIG. 15.8 Backward tracing applied to a preform design of (a) the front end and (b) the backend in rolling (conditions are the same as those of Fig. 15.7).

307


are shown in Fig. 15.8a. The preform front end obtained by this methodhas a circular convex contour. The figure also shows that during rolling thecurvature of the front end contour is reduced and becomes slightly concavebefore the front end becomes flat at the exit. Similarly, in order to applythe backward tracing scheme to the determination of back end preformshapes, the back end contour obtained by the approximate solution wasmodified to a flat end, and the rolling process was then traced backward.Variations of the back end contour at several stages are shown in Fig.15.8b. The resulting preform shape is a circular arc, similar to the frontend preform, but with a smaller curvature.

For work-hardening materials, the procedure illustrated in Fig. 15.5 wasused. With preform shapes of a plate obtained for nonwork-hardeningmaterials (Fig. 15.8), the rolling simulation is performed for a work-hardening material. The results provide the strain distributions and the endcontours. The end shapes are not exactly flat. The ends are then correctedgeometrically to become flat ends, maintaining the same strain distribu-tions. With these strain distributions, the application of the backwardtracing scheme results in preform shapes for a work-hardening material.The results for an aluminum alloy are shown in Fig. 15.9. Comparing thepreform shapes for a work-hardening material with those for nonwork-hardening materials, it is seen in Fig. 15.9 that at both front and back ends

FIG. 15.9 Comparison of preform end shapes for nonwork-hardening and work-hardeningmaterials in rolling.


the deviations of the contours from the flat ends are smaller for awork-hardening material. The end shapes obtained by the backwardtracing scheme for a work-hardening material are more complex thanthose for nonwork-hardening materials. This indicates that duringnonsteady-state rolling, the deformation is influenced by the materialproperty.

15.5 Axially Symmetric Forging [14]

In forging, preform design involves the determination of the number ofpreforms and the design of the shapes and dimensions of each preform.According to Biswas and Knight [9], the bulk of the work conducted todetermine the preform shapes for complex forging components has beencarried out in Germany and the Soviet Union, as reported by Spies[10]. Chamouard [11] in France and Akgerman et al. [12] in the U.S. dealtwith some practical problems of preform design for certain forgingcross-sections. Yu and Dean [13] recently reviewed existing guidelines forthe design of dies and discussed previous computer-aided approaches.

Examples of preform design for steel forgings of various H-shapes areshown in Fig. 15.10. The present practice in handling this complexproblem is to computerize the design calculations required for thedesigner's decisions on basis of qualitative guidelines that have beenderived mainly from experience or experimental studies. A preform isrequired to provide complete filling of the die cavity during finish-forging.In addition the preform must be such that it assists in obtaining desiredproduct properties. An example of such a design problem is encounteredin disk forging [14].

FIG. 15.10 Preforms for steel finish forgings of H cross sections.


Problem Statement

A simple compression of a cylindrical bar stock between two flat paralleldies produces a flat disk. If friction at the die-workpiece interface werenegligible, the forged disk would have uniform strain throughout thedeforming body. The problem, then, is how to produce a uniformlydeformed disk under the presence of friction at the die-workpieceinterface. In the present case we assume that friction at the die-workpieceinterface is equal to the shear strength of the deforming material (frictionalstress = mk, where k is the shear strength, and m = 1.0), and that thematerial is nonwork-hardening. The process conditions are: (i) the stock isof cylindrical shape and has a radius of 1.25 and a height of 1.0 and (ii) thestock is to be deformed to 50% reduction in height. Under theseconditions, it is necessary to determine the number of forming operationsand the preform shapes for producing a uniformly deformed disk. It shouldbe noted, however, that we have some flexibility in selecting the processconditions and that the design requirements are not very rigid. Forexample, the amount of final strain in the forged disk, and the stockgeometry, are not definitely prescribed. Note also that the requirement ofuniform straining of the disk throughout the body cannot be satisfied in astrict sense. All that we can hope for is to achieve strain uniformity asmuch as possible.

Preliminaries

For developing design procedures, it would be helpful if we could obtainsome knowledge of the metal flow involved in forging. Thus, twosimulations by the finite-element method were performed as part of apreliminary investigation.

One was the simulation of the simple compression of a cylinder betweentwo parallel flat dies (with m = 1.0). The grid patterns before and aftercompression are shown in Fig. 15.11. Also shown in the figure are theeffective strain distributions after 50% reduction.

From the results of Fig. 15.11, it is seen that in the core region, a nearlydead-metal zone is formed beneath the die, resulting in a large straingradient in the thickness direction. In the peripheral region, strains are lessthan the average strain value within the main body, but they are greaterthan average near the contact surface, because of distortion due to friction.

The strain gradient in the core region may be reduced if the size of thedead zone can be reduced. One may achieve this by applying indentationat the core region, followed by flat die compression. A symmetricindentation was simulated by a hemispherical punch with a radius of 1.5 tothe depth of z = 0.35 (the amount of punch displacement = 0.15). Simula-tion of flat-die compression was then performed until the final diskconfiguration was reached. The grid distortions in the workpiece afterindentation, and at the final stage, are shown in Fig. 15.12 along withresulting strain distributions. It is seen that the strain distribution in the


FIG. 15.11 The grid patterns before and after deformation and the effective straindistributions after 50% reduction in height in disk forging (black zone: e>l.l; white zone:1.1 >e>0.8; gray zone: e<0.8).

core region was altered and some improvement toward uniform strains, ingeneral, was obtained by indentation and compression.

Design Procedures

In the present problem, and in preform design in forging in general, theboundary conditions and their variations during the process depend on thepreform shape itself. During the actual forming operation, a change in theboundary conditions occurs when a nodal point on the free surface comesinto contact with the die. In backward tracing, nodal points along thedie-workpiece interface in the final configuration should be detached fromthe die at some stage of deformation and all the nodal points must becomefree at the completion of backward tracing.

In developing design procedures for these cases, it is necessary toconsider criteria for controlling when and where and in what sequence aboundary node should be detached from the die. Two criteria areconceivable at this point—one for controlling the workpiece geometry andthe other for controlling deformation.

For the first backward tracing step, the boundary conditions were


FIG. 15.12 The grid patterns after identation and in the finish-forged disk, and the effectivestrain distributions in the finished disk (black zone: £>!.!, white zone: 1.1>E>0.8; grayzone: e<0.8).

primarily controlled by geometrical considerations of the preform shape.The preliminary investigations show that the preform shape with a dimpleat the center improves the strain distributions in the core region.Therefore, the boundary condition was regulated in such a way that thepreform shape was smooth and convex, similar to the shape obtained byindentation. This was done by allowing a node to be detached from the dieat each time-step, sequentially from both the center and the periphery ofthe interface.

During the second tracing step, the residual strain distributions wereexamined after each time-increment and used for controlling the boundaryconditions. When the residual strains in the thickness direction reach thevalue within a given limit (say, ±0.1) in the main body (excluding a narrowregion near the interface), a corresponding boundary node was detachedfrom the die. In the peripheral region, however, the residual strains neverreach the desirable values (values close to zero), and separation of theboundary nodes was required mostly for the physical reason that the nodalforce must be compressive in order for a nodal point to remain in contactwith the die.


Final Design

Following the design procedures discussed above, the shapes of thepreform and the stock and the resulting strain distributions in the finisheddisk were obtained, and are shown in Fig. 15.13a. In Fig. 15.13a it is seenthat improvement has been achieved for the strain distributions in the coreregion, but not in the peripheral region. Further improvement in designappears to be possible only by increasing the amount of distortion in theperipheral region with an additional preforming operation. A radialforging operation with circular-contoured dies is considered as a processfor this purpose. In forward simulation as well as in backward tracing, theradial forging process is replaced by a fictitious process in which thediameter of a ring-die decreases or increases radially. During backwardtracing of this process, the boundary conditions were regulated in such away that the side-surface of the stock became as close to a circular cylinderas possible. The resulting shape of the stock and the corresponding straindistributions in the finished disk are shown in Fig. 15.13b. It can be seen

FIG. 15.13 Shapes of preforms and stocks and the effective strain distributions in the finisheddisk: (a) design I (black zone: e>l . l ; white zone: 1.1>£S0.8; gray zone: esO.8); (b)design II (black zone: e a 1.3, white zone: 1.3 > £ > 1.0; gray zone: e < 1.0).


that improvement in uniformity of strain distributions in the peripheralregion was achieved.

In the design procedures discussed above, no particular attention waspaid to irregular shapes of designed stocks and preforms. The reason isthat the shapes and contours can always be smoothed out or modified toobtain realistic and simple configurations. The final design can then beevaluated by simulating the operation sequence with modified stock andpreform shapes. The modified stock shape and the radial forging processesare shown in Fig. 15.14a. The initial stock is a cylindrical solid with ahemispherical dimple at the center. Each die for radial forging has acircular contour, and moves radially inward.

The second preforming operation is forging with a die that has the

FIG. 15.14 (a) Simulation of first preforming operation—radial forging, (b) Simulation ofsecond preforming operation—forging, (c) Simulation of finishing operation—flat-toolcompression.



smoothed contour of the preform 2 shown in Fig. 15.13b. The griddistortions before and after the operation are shown in Fig. 5.14b. Thesimulation of the finishing operation and the results are shown in Fig.15.14c. The strain distributions in the finished disk obtained for the finaldesign show that geometrical modification of stock and preforms did notchange the results appreciably.

15.6 Hot Forming [15]

For preform design problems in thermo-viscoplastic deformation, back-ward tracing of the process should also include temperature variations. Inaddition to the velocity equations given by eq. (15.1), the followingtemperature equations, along with the boundary conditions, must besatisfied:

at time t


Kc is the heat conduction matrix; C is the heat capacity matrix; Q is theheat flux vector; T is the vector of nodal point temperatures; T is thevector of nodal-point temperature rates.

For backward tracing (t < t0), at time t = t0_i = t0 — Af, x0_!, E0-i shouldsatisfy the conditions given by eq. (15.3) and T0^ should satisfy theconditions that

(see eq. (12.12) in Chap. 12). Substitution of eqs. (15.3) into eq. (15.5) attime t0-i, gives

In eq. (15.7), x0, £0> VQ-I, EO-I are supposed to be known or obtained fromthe solution of the velocity equations. The temperature Tis a function of T0_i according to eq. (15.6). Since T0 and T0 are known,eq. (15.7) can be solved for T0-i by assuming the boundary conditions attime t0_i. Once T0_! is obtained, then T0_! can be calculated andcompared with the assumed boundary values. Iterative procedures willresult in backward tracing of temperature calculations. Details of thecalculation procedure involved in backward tracing of a thermo-viscoplastic deformation process is given in Reference [15].

To illustrate the design method, a preform design in shell nosing atelevated temperatures is considered. The specifications described in thestudy by Lahoti et al. [16] and by Tang and Kobayashi [17] were used.They were: dimensions of the nosed shell and die; interface friction(frictionless); material (AISI 1045 steel); initial temperature distributions;and nosing speed (270 mm/s). The preform design procedure consists ofthe following three steps.Step (1)—Loading simulation. Backward tracing begins with the strain andthe temperature fields in the specified configuration and in the die. Inorder to know these fields, loading is simulated with a trial or "guessed"preform, using a given initial temperature distribution. A trial preformmay be designed by considering that the material volume of the preformmust be the same as that of the shell to be formed, and by assuming thatthe inner profile of the preform is a straight line. The geometry of the trialpreform is then determined by the requirement of straight outer profile.The trial preform shape and the initial temperature distributions in thepreform are shown in Fig. 15.15.Step (2)—Preparation for backward tracing. The final deformed shape,obtained from the simulation in Step (1) is not the same as that specified.The nodal point positions are adjusted so that the specified shellconfiguration results and strain and temperature distributions in thisconfiguration are obtained. Figure 15.16 shows these distributions.Step (3)—Backward tracing and control of boundary conditions. Startingfrom the specified shell configuration and corresponding strain andtemperature fields, backward tracing is performed. In order to obtain a


FIG. 15.15 (a) The trial preform shape and finite-element layout for loading simulation ofnosing, (b) The initial temperature distributions in the preform.

preform with straight outer wall surface, we control the boundaryconditions in such a way that each node that is in contact with the dieleaves the die at the die entrance and then moves axially. The finalpreform shape is shown in Fig. 15.17a. Figure 15.17b gives the initialtemperature distribution, obtained from backward tracing of temperaturechanges. The temperature distribution of Fig. 15.17b is very close to the


FIG. 15.16 (a) The strain distributions and (b) the temperature distributions in the specifiedshell configuration and in the die.

given initial temperature distribution shown in Fig. 15.15b, indicating thatthere is no need for iteration.


A new technique, the backward tracing scheme, was devised for preformdesign. This is a rather unique application of the finite-element method tosolving problems in metal forming. It was demonstrated that the technique


FIG. 15.17 (a) The preform shape and (b) the initial temperature distributions obtained atthe completion of backward tracing for shell nosing.

can be applied to preform design in shell nosing and rolling at roomtemperature. It was found that one of the aspects critical for furtherdevelopment of the technique is the treatment of the boundary conditions.In the disk-forging problem, where the boundary conditions change duringthe process depending on the preform shape, criteria were suggested forcontrolling the boundary conditions during the process. Further develop-ment of the method was presented for backward tracing of a thermo-viscoplastic deformation process, and the method was applied to preform


design in shell nosing at elevated temperature. In conclusion, it isimportant to note that in order to obtain the desired (optimal) preform ofa forming process, the path for backward tracing must be known. Thus,future research should concentrate on developing criteria for the selectionof optimal preforms and corresponding deformation path in generalforming processes.

References

1. Park, J. J., Rebelo, N., and Kobayashi, S., (1983), "A New Approach toPreform Design in Metal Forming with the Finite Element Method," Int. J.Machine Tool Des. Res., Vol. 23, p. 71.

2. Nadai, A., (1944), "Plastic State of Stress in Curved Shells: The ForcesRequired for Forging of the Nose of High-Explosive Shells," Forging of SteelShells, ASME Trans., p. 31.

3. Carlson, R. K., (1944), "An Experimental Investigation of the Nosing ofShells," Forging of Steel Shells, ASME Trans., p. 45.

4. Lahoti, G. D., Subramanian, T. L., and Allan, T., (1978), "Development of aComputerized Mathematical Model for the Hot/Cold Nosing of Shells,"Report ARSCD-CR-78019 to U.S. Army Research and DevelopmentCommand.

5. Kobayashi, S., (1983), "Approximate Solutions for Preform Design in ShellNosing," Int. J. Machine Tool Des. Res., Vol. 23, p. 111.

6. Johnson, W., (1980), The Mechanics of Some Industrial Pressing, Rolling, andForging Processes, in "Mechanics of Solids," (Edited by H. G. Hopkins andM. J. Sewell), Pergamon Press, Oxford, p. 303.

7. Hwang, S. M., and Kobayashi, S., (1984), "Preform Design in Plane-StrainRolling by the Finite Element Method." Int. J. Machine Tool Des. Res., Vol.24, p. 253.

8. Kobayashi, S., (1984), "Approximate Solutions for Preform Design in Roll-ing," Int. J. Machine Tool Des. Res., Vol. 24, p. 215.

9. Biswas, S. K., and Knight, W. A., (1974), "Computer-Aided Design ofAxisymmetric Hot Forging Dies," Proc. 15th Int. MTDR Conf., p. 135.

10. Spies, K., (1957), "The Preforms in Closed-Die Forging and Their Preparationby Reducer Rolling" (in German), Doctoral Dissertation Technical University,Hannover.

11. Chamouard, A., (1964), "Estampage et Forge." Dunod, Paris.12. Akgerman, N., Becker, J. R., and Allan, T., (1973), "Preform Design in

Closed Die Forging," Metal Forming, Vol. 40, p. 135.13. Yu, G. B., and Dean, T. A., (1985), "A Practical Computer-Aided Approach

to Mould Design for Axisymmetric Forging Die Cavities," Int. J. MachineTool Des. Res., Vol. 25, p. 1.

14. Hwang, S. M., and Kobayashi, S., (1986), "Preform Design in Disk Forging,"Int. J. Machine Tool Des. Res., Vol. 26, p. 231.

15. Hwang, S. M., and Kobayashi, S., (1987), "Preform Design in Shell Nosing alElevaled Temperatures," Int. J. Machine Tool Manufacture, Vol. 27, p. 1.

16. Lahoti, G. D., Oh, S. I., and Allan, T., (1981), "Development andConfirmation of a Series of Mathematical Models for the Blocking, Cabbaging,Piercing, and Nosing Operations Involved in Shell Manufacluring," ReportARSCD-CR-81010, to U.S. Army Research and Development Command.

17. Tang, M. C., and Kobayashi, S., (1982), "An Investigation of the Shell NosingProcess by the Finite Element Method, Part 2. Nosing at Elevated Tempera-tures (Hot Nosing)," ASME Trans., J. Engr. Ind., Vol. 104, p. 312.

16SOLID FORMULATION, COMPARISON

OF TWO FORMULATIONS, ANDCONCLUDING REMARKS

16.1 IntroductionIn the previous chapters we have discussed only the applications of flowformulation to the analysis of metal-forming processes. Lately, elastic-plastic (solid) formulations have evolved to produce techniques suitable formetal-forming analysis. This evolution is the result of developmentsachieved in large-strain formulation, beginning from the infinitesimalapproach based on the Prandtl-Reuss equation.

A question always arises as to the selection of the approach—"flow"approach or "solid" approach. A significant contribution to the solution ofthis question was made through a project in 1978, coordinated by Kudo[1], in which an attempt was made to examine the comparative merits ofvarious numerical methods. The results were compiled for upsetting ofcircular solid cylinders under specific conditions, and revealed the impor-tance of certain parameters used in computation, such as mesh systems andthe size of an increment in displacement. This project also showed that thesolid formulation needed improvement, particularly in terms of predictingthe phenomenon of folding. For elastic-plastic materials, the constitutiveequations relate strain-rate to stress—rates, instead of to stresses. Conse-quently, it is convenient to write the field equation in the boundary-valueproblem for elastic-plastic materials in terms of the equilibrium of stressrates.

In this chapter, the basic equations for the finite-element discretizationinvolved in solid formulations are outlined both for the infinitesimalapproach and for large-strain theory. Further, the solutions obtained bythe solid formulation are compared with those obtained by the flowformulation for the problems of plate bending and ring compression. Adiscussion is also given concerning the selection of the approach for theanalysis.

In conclusion, significant recent developments in the role of thefinite-element method in metal-forming technology are summarized.

16.2 Small-Strain Solid FormulationThe field equation for the boundary-value problem associated with thedeformation of elastic-plastic materials is the equilibrium equation of stress

321


rates. As stated in Chap. 1 (Section 1.3), the internal distribution of stress,in addition to the current states of the body, is supposed to be known, andthe boundary conditions are prescribed in terms of velocity and traction-rate.

The derivation of the basic equations for the finite-element discretizationfollows a procedure similar to that for rigid-plastic materials, (see eqs.(5.23-5.25)), using the stress-rate equilibrium equations.

The basic equation is

where the traction-rate Ft is prescribed on SP as <7,y«; = Fit where «,• is a unitnormal to the surface SF.

The independent variables in eq. (16.1) are velocity components, andthe Prandtl-Reuss equations provide the relationship between the stress-rate and velocity components. The Prandtl-Reuss equations are given byeq. (4.32). Substituting A given by eq. (4.38) and noting that a,y =dij - difidkk and 2G = £7(1 + v), eq. (4.32) becomes

where dtj is the Kronecker delta. The first term of eq. (16.2) can berewritten, using the strain-hardening relationship in the form of eq. (4.37),as

where H' is the slope of the stress-plastic strain curve H' — da/de. Inderiving eq. (16.3), note that

oa = \ o'mnd'mn = | o'mn(amn - &mn\akk)

where o'mn5mn = 0. Then, eq. (16.2) can be written in the form

For discretization of eq. (16.1), we need the inversion of eq. (16.4). Theinversion was obtained by Yamada et al. [2] as

For the last term in eq. (16.5), the coefficient a is added such that a = 0for elastic elements and a = 1 for plastic elements. After an incremental

Solid Formulation, Comparison of Two Formulations 323

deformation of time-step Af, the stress-state is updated by addingACT,-,- = btj A? to the current stress <7,y.

Early applications of the finite-element method have been based on thisinfinitesimal theory. The method was useful for the analysis of problems inwhich plastic strains are small and the effect of material rotation on theformulation is negligible.

16.3 Large Deformation: Rate Form

Various forms of the finite-element method dealing with large strain havebeen used by several investigators [3-6]. Among them, the most elegantforms are those given by Needleman [5] and by McMeeking and Rice [6].Needleman employed the Lagrangian formulation using convected coordi-nate systems, while McMeeking and Rice adopted the updated-Lagrangianformulation. Although their approaches to the problem are different, theirformulations are equivalent to and are based on that given by Hill [7, 8]. Inthis section the updated-Lagrangian formulation is outlined.

Consider a body with volume V0 and surface S0 at time t = t0 as areference state. After a certain increment of time Af, the body occupiesthe new position V and S. At the reference state, each particle of the bodyis labeled by a set of coordinates (Xt) that is embedded in the material.Another coordinate system (#,•) that is fixed in space and not moving withthe body will be chosen, as shown in Fig. 16.1. Then, at any time t > tn, wehave the following relations between the two coordinate systems.

The coordinate system xa is chosen to be rectangular Cartesian and theLagrangian description uses Xf and t as independent variables. SinceXj =Xj at t = t0, the strain-rate £,-, as a measure of deformation is given by

REFERENCE STATE

FIG. 16.1 Reference and current states, and coordinate systems.


and the rotation rate is

The three stress measures were introduced in Chap. 4 and discussed insome detail in Chap. 11 in conjunction with the considerations of largedeformation in the flow formulation. They are the Cauchy stress a,-,-,nominal stress (or first Piola—Kirchhoff stress) pijt and the secondPiola-Kirchhoff stress s^. They are related to each other according to

where / = det \dxil9Xa\. It can be noted that the second Piola-Kirchhoffstress sap is symmetric but that paj is not.

As a stress measure we introduce another stress Tfi, the Kirchhoff stress,which is defined by

where p0 and p are the material densities at the reference and thedeformed states, respectively. The time derivative of the stress measure isthe Jauman derivative (DTij/Dt), where time differentiation is carried outwith the coordinate system that rotates but does not deform with thematerial. It is expressed by

Because r,-, = a,-,- and Xt = xf at t = t0, and noting that t,y =Pij: + (dUj/dxk)okj, from eqs. (16.8) and (16.9), eq. (16.10) becomes

For the finite-element discretization, the variational form of the equi-librium equation of stress-rate is expressed as

Then

Equation (16.12) is the virtual work-rate principle in terms of stress-rate.Substituting pn of eq. (16.11) into eq. (16.12) and using the boundary

condition for the traction-rate F/ on 5n (and further imposing the restriction


that dUj — 0 on the part of S0 where the velocity is prescribed),

Using the symmetry of Dr^lDf, e,7 and a^, eq. (16.13) can be rewritten inthe form

Assuming that the Prandtl-Reuss equation holds for the Jauman deriva-tive of the Kirchhoff stress and strain rates, the constitutive relation isgiven by

Equations (16.14) and (16.15) are the bases for the finite-elementdiscretization.

In the updated Lagrangian description, the reference state is assumed tocoincide with the current state at time t. The procedures used to integratethe constitutive equations for a time-step Af are based on the mean normalmethod of Rice and Tracey [9] and the radial return method of Krieg andKrieg [10]. These two methods are further explained in Reference [28].

The formulations described above have been used for the analyses ofvarious forming processes. A complete stress and deformation analysis hasbeen given for plane-strain and axisymmetric extrusion until the steadystate is reached [11,12]. In applying the finite-element method tocan-extrusion the main problem was to simulate the metal flow around asharp corner [13]. The use of a special element to overcome this problemwas discussed.

Complete solutions of stretch-forming and deep drawing problems,taking into account the contact problem at blank holder, die, die profile,and punch head, were obtained by Wifi [14]. On the basis of the nonlineartheory of membrane shells, Wang and Budiansky [15] developed theprocedure for calculating the deformations in the stamping of sheet metalby arbitrarily shaped punches and dies.

Plane-strain rolling was analyzed by Rao and Kumar [16]. Key et al. [17]obtained the solutions for rolling, extrusion, and sheet stretching. Hartleyet al. [18] developed a method for introducing friction into the finite-element analysis [19].

Hartley et al. [20] also included the effects of strain-rate and tempera-ture variations within the billet in the analysis of axial compression of asolid cylinder. The fixed-mesh updated-Lagrangian technique was pro-


posed for the study of steady-state metal-forming processes by Derbailanet al. [21]. Analytical formulations of computational sheet-metal formingwere given by Wang [22]. Plane-strain plate bending was analyzed by Ohand Kobayashi [23] and the results were compared with those based on theflow approach. Some of the comparison of the two solutions are shown inChap. 8 and a more detailed comparison is discussed in Section 16.5 of thischapter.

16.4 Large Deformation: Incremental FormIn Section 16.3, the updated-Lagrangian approach was described in termsof rate equations. An alternative formulation is the derivation of thegoverning equations directly in incremental form. This formulation wasdeveloped by Nagtegaal and De Jong [24]. Implementation of theformulation into numerical models and applications of the finite-elementcode to metal-forming analyses are discussed in a series of publications byNagtegaal and his group [25-28]. The outline of the formulation is given inthe following.

At a generic time t, the virtual work-rate principle is expressed in termsof the first Piola-Kirchhoff stress p,y in the reference state as

where wy is the displacementt and Fj = Pijnoi is the traction prescribed onSQ. Using the relationship given in eq. (16.8), eq. (16.16a) is written interms of the second Piola-Kirchhoff stress s,y as

where a comma denotes a spatial In the updated-Lagrangian description, the reference state is the current

state at the beginning of the increment, and various stress measures aremomentarily equal to each other. Therefore, eq. (16.16b) becomes

At the end of the increment At, the spatial position of the body x, isdescribed by

(16.18)

with Aw, being the incremental displacement. Then eq. (16.16b) impliesthat the virtual work-rate principle, applicable at the end of the increment,

t Note that Uj has been used for velocity throughout this book but is used for displacementin Sections 11.4 through 11.7 of Chap. 11 and in this section.

derivative


is written as

Subtracting eq. (16.17) from eq. (16.19), we obtain the incrementalvirtual-work equation as

The incremental constitutive equations are expressed in the form

where &Ekl is the increment of Lagrangian strain and <£ikjl are theconstitutive moduli. The increment of the Lagrangian strain A£w is relatedto the displacement increment according to

The moduli J£ijk, are not the small-strain moduli Lijkl given by eq. (16.5),and are given by

The derivation of eq. (16.23) is given in Reference [24].The presence of the last term in eq. (16.23) causes the constitutive

equations to be nonsymmetric. However, if the deformations are nearlyincompressible, the last term can be neglected and the constitutiveequations remain symmetric.

At the end of the increment, the state variables need to be updated.Stresses are updated, following the relationships given by eq. (16.8), by

where / is the Jacobian of the deformation increment, and will be equal tounity if the material is incompressible.

The applications of this formulation to metal-forming processes includesample problems, such as tension, compression, extrusion, and forging ofcomplex shapes, for the purpose of demonstrating the capability of theformulations. A study undertaken by Rebelo et al. [29] is noteworthy fromthe application point of view. It is a comparative study of severalalgorithms implemented according to elastic-plastic (solid) and rigid-plastic(flow) formulations. The results of the investigation are illustrated in thesimulation of ring compression and are given in the next section.

16.5 Comparison with Rigid-Plastic (Flow) SolutionsPlate Bending [23]

The analysis of plane-strain plate bending using the rigid-plastic in-finitesimal deformation theory was presented in chap. 8 (Section 8.5). Thesame process conditions are used for the elastic-plastic analysis.


FIG. 16.2 Finite-element mesh system of the undeformed workpiece for (a) rigid-plastic and(b) elastic-plastic analyses in plate bending.

With reference to Fig. 8.9 in Chap. 8, the boundary conditions fortraction in the flow formulation are changed to traction-rate. In Fig. 16.2the undeformed grid systems are shown for both the rigid-plastic andelastic-plastic analyses. These two mesh systems are exactly the sameexcept that, for the elastic-plastic analysis, the rectangular mesh is furtherdivided into four triangular elements where plastic strain is expected to belarge, and into two triangular elements where only elastic strain isexpected.

For elastic-plastic calculations, each incremental step solution is deter-mined by solving a system of linear equations. This linear system assumesthat the nonlinearity of material behavior and of geometry can beapproximated by first-order linear equations. Therefore, the step size mustbe kept small to obtain accurate solutions. Also, the assumption that theboundary conditions remain unchanged during an increment limits the stepsize. Changes in boundary conditions include touching of a free nodalpoint to the punch, or separation of nodal points from the punch. The stepsize is also restricted by limiting the element rotation to within 1° for eachincrement.

In the computation the maximum punch displacement per step was setto 0.05-0.06 (sheet thickness is unity). However, the actual step size waslimited to 0.01-0.012 after considerable yielding took place, because of therestrictions described above. The total punch stroke to achieve a 90°bending angle is about 3.2.

Some of the results obtained by elastic-plastic analysis were comparedwith those obtained by rigid-plastic analysis in Chap. 8. Comparisons ofgrid distortions, bend angle-punch displacement relationship, and stressand strain distributions revealed that the results for both analyses agreewith each other very well with only minor differences.


In order to compare the stress and strain distributions quantitatively,bending stresses and effective strains at two stages of deformation areplotted against undeformed thickness positions at several cross-sections ofthe workpiece in Figs. 16.3 and 16.4. For identification, the numbers of thecross-sections correspond to those given in Fig. 16.2 (shown as elementnumber).

Figure 16.3 shows the excellent agreement between the bending stressesthat are computed from both formulations. The agreement is almostperfect at the outer fiber of the workpiece. However, there appear to besome discrepancies near the mid-surface of the plate thickness.

The effective strain distributions in Fig. 16.4 show a nearly uniformgradient across the cross-section. The figure also shows that, for bothstages of deformation, the rigid-plastic analysis predicts higher effective

FIG. 16.3 Variations of normalized bending stress along the thickness directions at variouscross sections (as indicated in Fig. 16.2), for punch displacements of (a) 2.25 and (b) 3.2.


ELASTO-PLASTIC R I G I D - P L A S T I C

FIG. 16.4 Variation of effective strain along the thickness direction at various cross sections(as indicated in Fig. 16.2), for punch displacements of (a) 2.25 and (b) 3.2.

strains near the outer fiber and lower values near mid-surface, than thecorresponding values predicted by the elastic-plastic analysis.

One of the main disadvantages of the rigid-plastic formulation is that itcannot predict unloading behavior. However, in the case where theunloading conditions can be well-defined, such as in sheet bending, it ispossible to calculate springback and the residual stress distribution byperforming the elastic-plastic unloading calculations using the results of therigid-plastic loading solution.

The springback angle calculated with the elastic-plastic solution is 3.03°,and 2.94° when obtained from the other solution. Figure 16.5 shows thedistributions of the bending component of residual stress along thethickness of several selected cross-sectional areas for both cases. Theresults show that the residual stresses obtained from the elastic-plasticsolution are somewhat higher along the cross sections near the bendingaxis. Comparison of the two solutions shows excellent agreement betweenthem, not only in overall quantities but also in detail.

The amount of springback was obtained by performing unloadingcalculations. Two methods were used, namely, elastic-plastic loading andunloading, and rigid-plastic loading and elastic-plastic unloading. Theresults show that the springback angle, as well as the residual stressdistributions, calculated by both methods are in excellent agreement.


FIG. 16.5 Normalized residual stress distributions in plate bending: (a) elastic-plastic loadingand unloading; (b) rigid-plastic loading and elastic-plastic unloading.

From an economic point of view, the elastic-plastic finite-elementmethod requires 430 steps to reach a 90° bend angle, while the rigid-plasticprogram needs a total of 498 iterations for the same deformation. The totalcomputing time was 350 s by the rigid-plastic program and about 1000 s bythe elastic-plastic program, using the CDC 7600 computer.

Ring compression

Rebelo et al. [29] presented a comparative study of several algorithms usedin the simulation of metal-forming processes. The same forming problemwas solved with all algorithms and the results were compared. The resultsof the simulation of ring compression are given here.

The same finite-element mesh was used in all tests, consisting of 100bilinear quadrilateral elements, representing half of a ring of height20 mm, outer diameter 40 mm, and inner diameter 20 mm. The upper diemoves at a velocity of 1 mm/s. The material being deformed was assumedto be commercially pure aluminum. The primary set of test runscompressed the workpiece to a 50% reduction in height in 100 increments.In all cases, a constant friction factor m = 0.7 along the interface was used.In the following, the test results obtained by the solid formulation(elastic-plastic; mean normal method of integrating plasticity equations;full Newton-Raphson algorithm; 0.5% step-size) and by the flow formula-tion (rigid-plastic; direct substitution algorithm; 0.5% step-size) arecompared. The relative execution times on a PRIME 9955 computer were1 and 0.26, respectively. In Fig. 16.6 the deformed meshes of the twosolutions at 50% reduction in height appear to be almost identical,although some differences in bulge contours may be noticed. In a morequantitative way, a comparison is shown in Fig. 16.7, displaying totalplastic strain contours at the same reduction in height as that given in Fig.


FIG. 16.6 Deformed mesh in ring compression (shown for upper half of the ringcross-section) after 50% reduction in height [29]: (a) elastic-plastic analysis; (b) rigid-plasticanalysis.

16.6. It is clear that the two solutions give almost identical results in detail.Figure 16.8 shows plots of nodal-point loads at the die-workpieceinterface. The closeness of the two distributions confirms that bothformulations are physically consistent.

Comments on ComparisonThe comparison shown above indicates that both the flow and solidformulations are capable of predicting certain physical phenomena inmetal forming. A variety of formulations and algorithms are now availablefor simulation of forming processes. The following remarks may be helpfulfor the selection of an approach to a specific forming problem.

When we consider simulation of forming processes, it is important for anengineer to be aware of the objective of simulation. Specific informationwill be sought by simulation, depending on the objective. Each problemhas its peculiarities and specific requirements, and the selected methodmust have the capability of delivering the required information. The

FIG. 16.7 Equivalent plastic strain contours in ring compression at 50% reduction in height[29]: (a) elastic-plastic analysis; (b) rigid-plastic analysis.


FIG. 16.8 Axial nodal force distributions along the die-workpiece contact surface in ringcompression [29]: (a) elastic-plastic analysis: (b) rigid-plastic analysis.

method should be computationally efficient. A measure of efficiency isexecution time, but true efficiency of the method must be considered in thecontext of the complete system of simulation. The solution accuracydepends on the mesh system and the step size, and influences efficiency.The schemes for data preparation, remeshing, and postprocessing areimportant factors that must be considered for total efficiency. Ease ofinterfacing a simulation method with those peripheral activities greatlyenhances the overall efficiency of simulation.

Finally, solution reliability is a major concern. Reliability of thecomputed solutions can only be determined by comparing predictions withexperimental observations. Since experimental observations are limited tocertain aspects of deformation, such as geometry and hardness distribu-tion, the relative reliability of computed solutions is difficult to determine.


The determination of reliability depends on how closely a boundary valueproblem is defined for a physical problem. In this regard, formulationsconcerning materials behavior and friction at the die-workpiece interfacemust include parameters that reflect the real physical conditions. Inparticular, the influence of the change of the constitutive equationparameters upon deformation behavior was investigated for rigid-plasticmaterials by Tomita [30]. Eggert and Dawson [31] used internal variableconstitutive equations for characterizing the changing state of a materialduring a deformation process.


The applications of the finite-element method in metal-forming are beingcontinuously improved and expanded. Many improvements are presentedin the book "NUMIFORM 86", Numerical Methods in Industrial FormingProcesses [32]. In concluding, some of the studies discussed in this bookare described here. One improvement is to expand the analysis capabilitiesof the method to three-dimensional deformation. Using the flow formula-tion, Chenot and his co-workers have developed a three-dimensionalviscoplastic finite-element code and analyzed shape rolling [33]. Further,they used explicit and implicit Euler schemes for time integration inconnection with three-dimensional forging analysis [34].

Three-dimensional analysis capability has also been developed with thesolid formulation. Pillinger et al. [35] discussed various applications toforging, rolling, and ring compression. In particular, the analysis of ringcompression shows the capability of the method, including predictions oftemperature and strain-rate distributions.

Deformation analyses of porous materials were discussed in Chap. 13 inthe framework of thermo-viscoplasticity. Constitutive equations of elasto-viscoplastic type were used for the analysis of hot isostatic pressing ofmetal powders [36]. A major concern is for improvement in computationalefficiency. Zienkiewicz et al. [37] suggested some new directions towardthis objective. These are mixed formulations in the iterative procedure,and mesh regeneration—in particular, automatic mesh regeneration.

Thompson offered an approach for transient analyses of metal-formingoperations—the method of pseudoconcentrations [38]. He concluded thatthe method is easy to implement and requires a minimum of userinterventions during an analysis. It was mentioned in the previous sectionthat improvements in solution reliability are mainly influenced by realisticformulations of constitutive equations and friction boundary conditions. Aconstruction of frictional constitutive equations was presented for finite-strain elastic-plastic material behavior by Baaijens et al. [39]. Theusefulness of the formulation, however, requires verification by ex-perimental investigations. With respect to constitutive equations, a generalformulation of elastic-plastic theory that is valid for finite deformations,involving strain-induced anisotropy, was developed by Lee and Agah-


Tehran! [40]. Lush and Anand [41] proposed implicit time integrationprocedures for internal variables of constitutive equations for large-deformation elasto-viscoplasticity.

Pecherski [42] presented the phenomenological model for large plasticstrains and the onset of deformation instability. For calculation of theflatness defect of rolled strip, Yukawa et al. [43] solved the eigenvalueproblem for the buckling limit and determined the shape after buckling bythe large-displacement finite-element method. Mori et al. [44], in thecoupled analysis of a forming process with elastic tools, treated the plasticdeformation of the workpiece by the rigid-plastic finite-element methodand the elastic deflection by the boundary-element method.

These investigations provide some indication for ever-increasing ap-plications of the finite element method in metal forming.

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25. Nagtegaal, J. C., (1982), "On the Implementation of Inelastic ConstitutiveEquations with Special Reference to Large Deformation Problems," ComputerMethods Appl. Mech. Engr., Vol. 33, p. 469.

26. Nagtegaal, J. C., and Veldpaus, F. E., (1984), On the Implementation ofFinite Strain Plasticity Equations in a Numerical Model, in "NumericalAnalysis of Forming Processes", (Edited by J. F. T. Pittman, O. C.Zienkiewicz, R. D. Wood and J. M. Alexander), Wiley, New York, p. 351.

27. Rebelo, N., and Wertheimer, T. B., (1986), "General Purpose Procedures forElastic-Plastic Analysis of Metal Forming Processes," Proc. 14th NAMRC,Minneapolis, MN., p. 414.

28. Nagtegaal, J. C., and Rebelo, N., (1986), On the Development of a GeneralPurpose Finite Element Program for Analysis of Forming Processes, in"NUMIFORM 86," Balkema, Rotterdam, p. 41 [see Ref. 32].

29. Rebelo, N., Nakazawa, S., Wertheimer, T. B., and Nagtegaal, J. C., (1986),"A Comparative Study of Algorithms Applied in Finite Element Analysis ofMetal Forming Problems," Special Publication ASME, PED-Vol. 20, p. 17.

30. Tomita, Y., (1982), "A Rigid Plastic Finite Element Method for the EffectivePrediction of the Influence of the Change in Parameters in the ConstitutiveEquation upon Deformation Behavior," Int. J. Mech. ScL, Vol. 24, p. 711.

31. Eggert, G. M., and Dawson, P. R., (1987), "On the Use of Internal VariableConstitutive Equations in Transient Forming Processes," Int. J. Mech. Sci.,Vol. 29, p. 95.

32. NUMIFORM 86 (1986), "Numerical Methods in Industrial Forming Proc-


esses," (Edited by K. Mattiasson, A. Samuelson, R. D. Wood and O. C.Zienkiewicz), A. A. Balkema, Rotterdam.

33. David, C., Bertrand, C., Chenot, L. J., and Buessler, P., (1986), A TransientThree Dimensional Finite Element Analysis of Hot Rolling of Thick Slabs, in"NUMIFORM 86," p. 219 [see Ref. 32].

34. Surdon, G., and Chenot, J. L., (1986), Finite Element Calculation of ThreeDimensional Hot Forging, in "NUMIFORM 86," p. 287 [see Ref. 32].

35. Pillinger, I., Hartley, P., Sturgess, C. E. N., and Rowe, G. W., (1986), FiniteElement Modeling of Metal Flow in Three-Dimensional and Temperature-Dependent Forming, in "NUMIFORM 86," p. 151 [see Ref. 32].

36. Abouaf, M., Chenot, J. L., Raisson, G., and Baudin, P., (1986), FiniteElement Simulation of Hot Isostatic Pressing of Metal Powders, in"NUMIFORM 86," p. 79 [see Ref. 32].

37. Zienkiewicz, O. C., Liu, Y. C., Zhee, J. Z., and Toyoshima, S., (1986), Flowformulation for Numerical Solution of Forming Processes II, in "NUMIFORM86," p. 3 [see Ref. 32].

38. Thompson, E., (1986), Transient Analysis of Metal Forming operations UsingPseudo-Concentrations, in "NUMIFORM 86," p. 65 [see Ref. 32].

39. Baaijens, F. P. T., Brekelmans, W. A. M., Veldpaus, F. E., and Starmans, F.J. M., (1986), A Constitutive Equation for Frictional Phenomena IncludingHistory Dependency, in "NUMIFORM 86," p. 91 [see Ref. 32].

40. Lee, E. H., and Agah-Tehrani, A., (1986), The Structure of ConstitutiveEquations for Finite Deformation of Elastic-Plastic Materials with Strain-Induced Anisotropy, in "NUMIFORM 86," p. 29 [see Ref. 32].

41. Lush, A., and Anand, L., (1986), Implicit Time-Integration Procedures for aSet of Internal Variable Constitutive Equations for Hot-working, in"NUMIFORM 86," p. 131 [see Ref. 32].

42. Pecherski, R. B., (1986), The Disturbed Spin Concept and its Consequences inPlastic Instability, in "NUMIFORM 86," p. 145 [see Ref. 32].

43. Yukawa, N., Ishikawa, T., and Tozawa, T., (1986), Numerical Analysis of theShape of Rolled Strip, in "NUMIFORM 86," p. 249 [see Ref. 32].

44. Mori, K., Osakada, K., Nakadori, K., and Fukuda, M., (1986), CoupledAnalysis of Steady State Forming Processes with Elastic Tools, in"NUMIFORM 86," p. 237 [see Ref. 32].

AppendixTHE FEM CODE, SPID

(SIMPLE PLASTIC INCREMENTALDEFORMATION)

A.I Introduction

The development and testing of the computer code is an important part ofthe finite-element method. In programming a practical finite-elementmethod code, one should consider various factors, such as generality,computational efficiency, and pre- and postprocessing. However, suchconsiderations are beyond the scope of this book. Instead, a simpletwo-dimensional FEM code, called SPID, written for metal-formingsimulation is presented as an example and discussed in this appendix. Themain purpose of the code is to illustrate the programming of key stepsexplained in this book.

The program SPID, Simple Plastic /ncremental Deformation, is basedon rigid viscoplastic finite-element formulation. SPID is capable ofhandling only simple forming processes, such as simple compression andring compression. Some important features and limitations of SPID aresummarized below.

• SPID is valid for rigid plastic as well as rigid viscoplastic materials.• A constant shear factor friction law is used.• No heat transfer simulation is included.• It can handle one flat die only.• It cannot handle free surface folding to the die.• The initial guess is generated automatically.• A banded matrix solver is used.• SPID can handle a finite-element model with up to 100 nodes.

SPID is written in FORTRAN 77 standard, with special consideration ofportability. The program was tested on two computer systems. One wasVAX 750 with VMS version 4.4 operating system. The other was IBMPC/AT with a math coprocessor, where the professional FORTRAN compilerof Ryan-McFarland Corp. and DOS 3.10 operating system of Micro SoftInc. were used. The required minimum memory size for the IBM PC torun SPID is 256 K Bytes.

338

Appendix The FEM Code, SPID 339

A.2 Program StructureThe program structure of SPID in terms of subroutine calling sequence isshown schematically in Fig. A.I. Brief descriptions of each subroutine aregiven as comment lines in the program listing in Section A.6. Thefunctional procedure of SPID is described below. Subroutine namescorresponding to each logical step are also given in parenthesis.

1. Program starts. (SPID)2. Read input from input file SPID.DAT. (INPRED)3. Print input information. (PRTINP)4. Determine the maximum half-bandwidth and number of stiffness

equations. (BAND)5. If initial guess is required, then set up for the direct iteration

procedure. If not, then use Newton-Raphson iteration method.(NONLIN)

6. Evaluate elemental strain-rate matrix. (STRMTX)7. Evaluate elemental stiffness matrix. (ELSHLF, VSPLON,

VSPLST)

FIG. A.I Subroutine calling sequence of SPID. The names enclosed in boxes are thesubroutine names and the double-line box indicates the continuation of the diagram.


8. Calculate nodal force. (NFORCE)9. Apply ffictional boundary conditions. (FRCBDY, FRCINT)

10. Assemble global stiffness matrix. (ADDBAN)11. Apply displacement boundary condition. (DISBDY)12. Solve stiffness equations. (BANSOL)13. Adjust the velocity solution based on the result of step 12.

(NONLIN)14. Determine the error norm. (NORM)15. If converged, go to Step 16. Otherwise, go to Step 5. (NONLIN)16. Update the workpiece geometry. (POTSOL)17. Stress and strain evaluation. (POTSOL)18. Write the results on file SPID.OUT. (PRTSOL)19. Generate restart file, SPID.RST. (RSTFIL)20. If desired step reached, then go to Step 21. Otherwise, go to Step 5.

(SPID)21. Terminate the program. (SPID)

A.3 Input and Output Files

SPID requires one input file, SPID.DAT, to run a simulation and itgenerates three output files, SPID.MSG, SPID.OUT, and SPID.RST atthe end of each run. These files are described below.

SPID.DAT Name of the input file for initial run and continuing run.FORTRAN file units are indicated by INPT = 6

SPID.MSG SPID.MSG is one of SPID output files and it containsmessages generated by SPID. This message file clearlycontains

(a) information on iterations for each step solution(b) error messages from SPID

SPID.OUT SPID.OUT is output file of SPID and it contains(a) initial input summary(b) simulation results for each step solution

SPID.RST SPID.RST is one of SPID output files to be used as inputfile for a continuing run. SPID.RST has the same formatas that of SPID.DAT and it is ready to be used as an inputfile to SPID for a continuing run. However, the usershould rename this to SPID.DAT and change data inputfor NSEND (refer data input description of Line 2 in A.4)before continuing the run.

A.4 Input Preparations

The initial input file, SPID.DAT, to be prepared by the user, is describedbelow. Input is read by FORTRAN list directed FORMAT.


1. Master Control DataLine 1. TITLE

TITLE Heading up to 70 characters

Line 2. MINI, NSENDNINI Starting step numberNSEND Ending step number

Line 3. ALPH, DIATALPH Limiting strain-rateDIAT Penalty constant to enforce volume constancy

Line 4. IPLNAXIPLNAX If = 1, deformation is axisymmetric

If = 2, deformation is plane-strain

Line 5. FRCFACFRCFAC Constant shear friction factor

2. Node Coordinates DataLine 6. NUMNP

NUMNP Total number of nodal points

Line 7. N, (RZ(I,N), I = 1,2)*N Node numberRZ(1,N) x or r coordinate of node NRZ(2,N) y or z coordinate of node N

* This line should be repeated from node 1 to node NUMNP.

3. Element DataLine 8. NUMEL

NUMEL Total number of elementsLine 9. N, (NOD(I,N), I = 1,4)*

N Element numberNOD(I,N) Element connectivity of element N

* Element connectivity should be given counterclockwise.

4. Boundary Condition Code DataLine 10. N, (NBCD(I,N), 1 = 1,2)*

N Node numberNBCD(1,N) Boundary condition code in * or r direction

= 0, nodal force is specified.= 1, nodal velocity is specified.

* If the boundary condition codes for both directions are zero, then thecorresponding line can be skipped. However, data corresponding the lastnode (NUMNP) cannot be skipped.


NBCD(2, N) Boundary condition code in y or z direction= 0, nodal force is specified.= 1, nodal velocity is specified.= 3, node is in contact with the die.

5. Node Velocity DataLine 11. N, (URZ(I,N), 1 = 1, 2)*

N Node numberURZ(1,N) x or r velocity of node TVURZ(2,N) y or z velocity of node TV

* This program assumes that both velocity components are zero for theskipped node. Last node velocity data cannot be skipped.

6. Flow Stress of Workpiece MaterialThe flow stress of the workpiece material is given by the user suppliedsubroutine, FLWSTS. The program listing in Section A.6 shows anexample for subroutine FLWSTS.

A.5 Description of the Major VariablesMost of the major variables used in the program SPID are included inlabeled COMMONS. The descriptions of these variables are given below.

COMMON/INOT/INPT, MSSG, IUNIT, IUNI2INPT Input file unit number, INPUT = 5MSSG Message file unit number, MSSG = 6IUNIT Output file unit number, IUNIT = 3IUNI2 Restart file unit number, IUNI2 = 4

COMMON/MSTR/NUMNP,NUMEL,IPLNAXNUMNP Total number of nodal pointsNUMEL Total number of elementsIPLNAX Deformation mode indicator

If = 1, axisymmetric modeIf = 2, plane-strain mode

COMMON/CNEQ/NEQ,MBANDNEQ Total number of stiffness equationsMBAND Half-bandwidth

COMMON/RIGD/RTOL,ALPH,DIATRTOL Error limit to check the convergenceALPH Limiting strain-rateDIAT Penalty constant for volume constancy

COMMON/TSTP/NINI,NCUR,NSEND,NITR,DTMAXNINI Starting step numberNCUR Current step number


NSEND Ending step numberNITR Current iteration numberDTMAX Step size in time unit

COMMON/DIES/FRCFACFRCFAC Constant shear friction factor

COMMON/ITRC/ITYP,ICONVITYP Type of iteration process

If = 1, Newton-Raphson iterationIf = 2, direction iteration

ICONV Convergence indicatorIf = 1, solution is convergedIf = 2, solution is not converged

COMMON/RVA1/RZ(2,100),URZ(2,100),FRZ(2,100)RZ(I,N) Coordinates of node NURZ(I,N) Velocity of node NFRZ(I,N) Nodal force of node N

COMMON/RAV2/EPS(5,100) ,STS(5,100) ,TEPS(100)EPS(I,N) Strain rate components of element N

1 = 1, ex or er

= 2, ey or ez

= 3, e2 or ee

= 4, jxy or jrz

= 5, effective strain rate eSTS(I,N) Stress components of element N

1=1, ax or or

= 1, oy or az

= 3, crz or aH

= 4, Txy or xrz

= 5, effective stress aTEPS(N) Total effective strain of element TV

COMMON/INVR/NOD(4,100),NBCD(2,100)NOD(I,N) Element connectivity of node TVNBCD(I,N) Boundary condition code of node N

COMMON A(5000),B(200)*A Array for stiffness matrixB Array for load vector

* Variables are set up to handle up to 100-node FEM model.

A.6 Program Listing

A FORTRAN program listing of SPID is given here with the main program,SPID, listed at the beginning and all other subroutines listed in alphabeti-cal order by the subroutine names. Chapter and Equation numbers givenin some subroutines refer to those in this book.


PROGRAM SPIDC

c*************************************************

C C METAL FORMING AND THE FINITE ELEMENT METHOD C C BY SHIRO KOBAYASHI, SOO-IK OH AND TAYLAN ALTAN C C OXFORD UNIVERSITY PRESS, 1988

CC THIS IS MAIN PROGRAM OF RIGID VISCOPLASTIC FINITEC ELEMENT METHOD FOR SIMPLE FORMING PROCESSES.C

IMPLICIT INTEGERS (I-N), REAL*8 (A-H, 0-Z)CHARACTER TITLE*70COMMON /TITL/ TITLECOMMON /RVA2/ EPS(5,100), STS(5,100), TEPS(IOO)COMMON /INOT/ INPT,MSSG,IUNIT,IUNI2COMMON /MSTR/ NUMNP.NUMEL.IPLNAXCOMMON /INVR/ NOD(4,100),NBCD(2,100)COMMON /TSTP/ NINI,NCUR,NSEND,NITR,DTMAXCOMMON /ITRC/ ITYP.ICONVINPT - 5MSSG - 6IUNIT = 3IUNI2 = 4

CC READ INPUTC

CALL INPREDOPEN(IUNIT,FILE='SPID.OUT',STATUS='UNKNOWN',FORM='FORMATTED')OPEN(MSSG, FILE='SPID.MSG',STATUS='UNKNOWN',FORM='FORMATTED')WRITE(MSSG,1020) TITLECALL PRTINPCALL BAND(NOD,NUMEL,NUMNP)

CC STEP SOLUTIONSC

NINI = NINI + 1C

DO 300 N = NINI, NSENDNCUR - N

CWRITE(MSSG,1050) NIF(N .NE. NINI) GO TO 80ICOUNT = 0

50 ITYP = 2CALL NONLINICOUNT - ICOUNT + 1

80 ITYP - 1CALL NONLINIF(ICONV .EQ. 2 .AND. ICOUNT .GT. 3) GO TO 900


IF(ICONV .EQ. 2) GO TO 50C

CALL POTSOLCALL PRTSOLCALL RSTFIL

300 CONTINUEC

CLOSE(IUNIT)CLOSE(MSSG)STOP

C900 CONTINUE

WRITE(MSSG,1070)C

STOP1020 FORMAT(1H1,//,5X,'OUTPUT OF S P I D',//,

+ 5X,' MESSAGE FILE FOR '/,5X,A,//)1050 FORMAT(///,' ITERATION PROCESS FOR STEP ',I5,//)1070 FORMAT{/,' STOP BECAUSE SOLUTION DOES NOT CONVERGE. ')

END

SUBROUTINE ADDBAN(B,A,NQ,LM,Qq,PP)CC CHAPTER 7.2,C EQUATION (7.8)CC ASSEMBLE GLOBAL STIFFNESS MATRIX FROM ELEMENTAL STIFFNESS MATRIXC

IMPLICIT REAL*8 (A-H, 0-Z), INTEGERM (I-N)DIMENSION B(l), A(NQ,1), QQ(1), PP(8,8), LM(1)

CDO 100 I = 1, 8II = LM(I)DO 50 J - 1, 8JJ = LM(J) - LM(I) + 1IF(JJ .LE. 0) GO TO 50A(II.JJ) = A(II,JJ) + PP(I.J)

50 CONTINUEB(II) = B(II) + QQ(I)

100 CONTINUERETURNEND

SUBROUTINE BAND(NOD,NUMEL,NUMNP)CC CHAPTER 7.2C DETERMINE MAXIMUM HALF BANDWIDTH, MBAND ANDC TOTAL NUMBER OF EQUATIONS, NEQC

IMPLICIT REAL*8 (A-H,0-Z), INTEGERM (I-N)COMMON /CNEQ/ NEQ,MBANDDIMENSION NOD(4,1)

CMBAND = 0

c

c


DO 100 N - 1, NUMELNMIN - NOD(1,N)NHAX - NOD(l.N)DO 50 I - 2,4IF(NMIN .GT. NOD(I.N)) NMIN - NOD(I.N)IF(NHAX .LT. NOD(I,N)) NMAX = NOD(I.N)

50 CONTINUEMB - (NMAX - NMIN + 1) * 2IF(MBAND .LT. MB) MBAND = MB

100 CONTINUENEQ - NUHNP * 2RETURNEND

SUBROUTINE BANSOL(B,A,NQ,MM)CC CHAPTER 7.2C THIS SUBROUTINE SOLVES THE BANDED SYMMETRIC MATRIX EQUATIONS BYC THE GAUSSIAN ELIMINATION.C B LOAD VECTORC A SYMMETRIC MATRIX IN BANDED FORMC MM HALF BANDWIDTHC NQ NUMBER OF EQUATIONSC

IMPLICIT REAL*8 (A-H, 0-Z), INTEGERS (I-N)COMMON /INOT/ INOT,MSSG,IUNIT,IUNI2DIMENSION B(l), A(NQ,1)

CDO 200 N = 1, NQIF(A(N,1) .LE. 0.) GO TO 800DO 150 L = 2, MMIF(A(N,L) .EQ. 0.) GO TO 150C = A(N,L) / A(N,1)I = N + L - 1J - 0DO 100 K = L, MMJ - J + 1

100 A(I,J) - A(I,J) - C*A(N,K)A(N,L) = C

150 CONTINUE200 CONTINUE

CC LOAD VECTOR REDUCTIONC

DO 300 N - 1, NQDO 250 L = 2, MMI = N + L - 1IF(I .GT. NQ) GO TO 250B(I) = B(I) - A(N,L) * B(N)

250 CONTINUEB(N) = B(N) / A(N,1)

300 CONTINUECC BACK SUBSTITUTION

c


rDO 400 M - 1,NQN = NQ + 1 - MDO 350 K - 2, MML - N + K - 1

350 B(N) - B(N) - A(N,K)*B(L)400 CONTINUE

RETURNC800 CONTINUE

WRITE(MSSG,1020) NSTOP

1020 FORMAT(/,' NEGATIVE PIVOT AT EQUATION NO. ',15)END

SUBROUTINE DISBDY(URZ,NBCD,B,A,NEQ,MBAND,ITYP)CC CHAPTER 7.3C APPLY DISPLACEMENT BOUNDARY CONDITIONC

IMPLICIT REAL*8 (A-H.O-Z), INTEGERM (I-N)DIMENSION B(1),A(NEQ,1),NBCD(1),URZ(1)

CIF(ITYP .EQ. 2) GO TO 120DO 100 N > 1, NEQIF(NBCD(N) .EQ. 0) GO TO 100DO 70 I = 2, MBANDII - N - I + 1IF(II .LE. 0) GO TO 50A(II,I) = 0.

50 CONTINUEII - N + I - 1IF(II .GT. NEQ) GO TO 70A(N,I) - 0.

70 CONTINUEB(N) - 0.A(N,1) = 1.

100 CONTINUERETURN

C120 CONTINUE

DO 200 N = 1, NEQIF(NBCD(N) .EQ. 0) GO TO 200DO 170 I = 2, MBANDII - N - I + 1IF(II .LE. 0) GO TO 150B(II) = B(II) - A(II,I) * URZ(N)A(II,I) - 0.

150 CONTINUEII - N + I - 1IF(II .GT. NEQ) GO TO 170B(II) = B(II) - A(N,I) * URZ(N)A(N,I) = 0.

170 CONTINUE

c


B(N) * URZ(N)A(N,1) - 1.

200 CONTINUEEND

SUBROUTINE ELSHLFfPP.QQ.RZ.URZ.EPS.TEPS.IPLNAX.IDREC.NEL)CC CHAPTER 6.5C EQUATIONS (6.43) AND (6.44)C EVALUATION OF ELEMENTAL STIFFNESS MATRIXCC IDREC : IF = 1, NEWTON-RAPHSON ITERATIONC = 2, DIRECT ITERATIONC

IMPLICIT REAL*8 (A-H, 0-Z), INTEGER** (I-N)DIMENSION RZ(2,1), URZ(2,1), B(4,8), EPS(l), TEPS(l)DIMENSION QQ(1),PP(8,8),S2(2),W2(2)DATA S2/-0.57735026918963DO,0.57735026918963DO/,+ W2/2*1.0DO/

CDO 10 I - 1, 8QQ(I) - 0.DO 10 J = 1, 8PP(I,J) = 0.

10 CONTINUECC CARRY OUT ONE POINT INTEGRATIONC

S = 0.T = 0.CALL STRMTXfRZ.B.WDXJ.SJ.IPLNAX.NEL)WDXJ = WDXJ * 4CALL VSPLON(QQ,PP,B,URZ,EPS,WDXJ,IDREC)

CC REGULAR INTEGRATIONC

DO 100 I - 1, 2S - S2(I)DO 50 J - 1, 2T - S2(J)CALL STRMTXfRZ.B.WDXJ.S.T.IPLNAX.NEL)WDXJ - WDXJ * W2(I)*W2(J)CALL VSPLST(QQ,PP,B,URZ,TEPS,WDXJ,IDREC)


RETURNEND

CSUBROUTINE FLWSTS(YS,FIP,STRAN,STRRT)

CC USER SUPPLIED SUBROUTINE TO DESCRIBE THE MATERIALC FLOW STRESS.C THIS SUBROUTINE SHOWS THE EXAMPLE OFC YS - 10. * (STRAIN RATE) ** 0.1

c


CIMPLICIT INTEGER*4 (I.J.K.L.M.N), REAL*8 (A-H.O-Z)COMMON /RIGD/ RTOL, ALPH, DIAT

CIF(STRRT .LT. ALPH) GO TO 100YS = 10. * STRRT ** 0.1FIP = STRRT ** (-0.9)RETURN

C100 YO = 10. * ALPH ** 0.1

FIP - YO / ALPHYS = FIP * STRRTRETURNEND

SUBROUTINE FRCBDY(RZ,URZ,NBCD,TEPS,EPS,QQ,PP,IPLNAX)CC APPLY FRICTION BOUNDARY CONDITIONC

IMPLICIT INTEGERM (I-N), REAL*8 (A-H, 0-Z)COMMON /DIES/ FRCFACCOMMON /INOT/ INPT, MSSG, IUNIT, IUNI2DIMENSION RZ(2,1),URZ(2,1),NBCD(2,1),EPS(5),QQ(1},PP(8,1),+ ER(2,2),FR(2),XY(2,2),VXY(2,2)

CDO 100 N = 1,411 = N + 112 = NIF(N .EQ. 4) II = 1IF(NBCD(2,I1) .NE. 3 .OR. NBCD(2,I2) .NE. 3) GO TO 100CALL FLWSTS(FLOW,DUM,TEPS,EPS(5))XY(1,1) - RZ(1,I1)XY(2,1) = RZ(2,I1)XY(1,2) - RZ(1,I2)XY(2,2) = RZ(2,I2)VXY(l.l) - URZ(1,I1)VXY(2,1) = URZ(2,I1)VXY{1,2) - URZ(1,I2)VXY(2,2) - URZ(2,I2)CALL FRCINT(XY,VXY,FLOW,FR.ER,FRCFAC,IPLNAX)Jl = II * 2 - 1J2 - 12 * 2 - 1QQ(J1) - QQ(J1) + FR(1)QQ(J2) = QQ(J2) + FR(2)PP(J1,J1) = PP(J1,J1) + ER(1,1)PP(J2,J2) - PP(J2,J2) + ER(2,2)PP(J1,J2) - PP(J1,J2) + ER(1,2)PP(J2,J1) = PP(J2,J1) + ER(2,1)


SUBROUTINE FRCINT(RZ,URZ,FLOW,FR.ER,FRCFAC,IPLNAX)C

c

c


C CHAPTER 7.3C EQUATIONS (7.14) AND (7.15)C INTEGRATION METHOD : SIMPSON'S FORMULAC THIS ROUTINE CALCULATES THE FRICTION MATRIXC USED FOR BOTH TYPES OF ITERATION SCHEMEC

IMPLICIT INTEGERM (I-N), REAL*8 (A-H, 0-Z)COMMON /INOT/ INPT,MSSG,IUNIT,IUNI2COMMON /ITRC/ ITYP.ICONVDIMENSION RZ(2,1),URZ(1),QQ(1),PP(8,1)DIMENSION SLIV(2),ER(2,2),FR(2)DATA PI/3.1415926535898DO/DATA UA/0.0005DO/

CC INITIALIZE FR AND ER ARRAYC

DO 10 I « 1, 2FR(I) = 0.DO 10 0 = 1, 2ER(I,J) - 0.

10 CONTINUEMINT = 5FAC = DSQRT((RZ(1,2)-RZ(1,1))**2 + (RZ(2,2FK FLOW * FRCFAC / SQRT(3.)DH 2. / (NINT - 1)S -1. - DHCON 2. / PI * FKWD DH / 3. * FAC * 0.5 * CON

CDO 300 N - 1, NINTS = S + DHHI = 0.5 * (1. - S)H2 = 0.5 * (I. + S)WDXJ = WDIF(IPLNAX .NE. 1) GO TO 90RR = H1*RZ(1,1) + H2*RZ(1,2)WDXJ - RR * WDXJ

90 CONTINUEC

IF(N .EQ. 1 .OR. N .EQ. NINT) GO TO 100NMOD = N - N/2*2IF(NMOD .EQ, 0) WDXJ = WDXJ * 4IF(NMOD .EQ. 1) WDXJ = WDXJ * 2

100 CONTINUEC

US * H1*URZ(1) + H2*URZ(3)AT = DATAN(US/UA)IF(ITYP .EQ. 2) GO TO 200US2 = US * USUSA - US2 + UA*UACT1 = AT*WDXJCT2 - UA/USA*WDXJGO TO 250

C


C FOR D-ITERATION CASEC200 CONTINUE

IF(DABS(US) .LE. l.OD-5) SLOP - UA/(UA*UA+US*US)IF(DABS(US) .GT. l.OD-5) SLOP = AT / USCT1 - 0.CT2 = SLOP*WDXJ

CC CALCULATE CONTRIBUTION TO STIFFNESSC250 CONTINUE

FR(1) - FR(1) - H1*CT1FR(2) - FR(2) - H2*CT1ER(1,1) = ER(1,1) + H1*H1*CT2ER(1,2) = ER(1,2) + H1*H2*CT2ER(2,2) = ER(2,2) + H2*H2*CT2ER(2,1) - ER(1,2)


SUBROUTINE INPREDCC READ INPUT FROM INPUT FILEC

IMPLICIT REAL*8 (A-H, 0-Z), INTEGERM (I-N)CHARACTER TITLE*70COMMON /TITL/ TITLECOMMON /TSTP/ NINI,NCUR,NSEND,NITR,DTMAXCOMMON /RVA1/ RZ(2,100), URZ(2,100), FRZ(2,100)COMMON /RVA2/ EPS(5,100), STS(5,100), TEPS(IOO)COMMON /INVR/ NOD(4,100), NBCO(2,100)COMMON /DIES/ FRCFACCOMMON /RIGD/ RTOL, ALPH, DIATCOMMON /MSTR/ NUMNP, NUMEL, IPLNAXCOMMON /INOT/ INPT, MSSG, IUNIT, IUNI2

CC READ MASTER CONTROL DATAC

OPEN(INPT,FILE='SPID.DAT',FORM='FORMATTED',STATUS-'OLD')READ (INPT,1000) TITLEREAD (INPT,*) NINI.NSEND.DTMAXREAD (INPT,*) ALPH, DIATREAD (INPT,*) IPLNAX

CC READ DIE DATAC

READ (INPT,*) FRCFACCC READ FEM NODE INFORMATIONC

READ (INPT,*) NUMNPIF(NUMNP .GT. 300) GO TO 500DO 20 I = 1, NUMNP


READ (INPT,*) N, (RZ{J,N), J - 1, 2)20 CONTINUE

CC READ ELEMENT INFORMATIONC

READ (INPT,*) NUMELIF(NUMEL .GT. 300) GO TO 500DO 40 I - 1, NUMELREAD (INPT,*) N, (NOD(J,N), J-1,4)

40 CONTINUECC READ BOUNDARY CONDITION DATAC

DO 60 N = 1, NUMNPDO 60 I = 1, 2NBCD(I,N) = 0

60 CONTINUEC

DO 80 N - 1, NUMNPREAD (INPT,*) M, (NBCD(I,M), I = 1,2)IF (M .GE. NUMNP) GO TO 100


CC READ NODE VELOCITY DATAC

DO 120 N = 1, NUMNPDO 120 1 = 1,2URZ(I.N) = 0.

120 CONTINUEDO 140 N = 1, NUMNPREAD (INPT,*) M, (URZ(I.M), 1=1,2)IF(M. GE. NUMNP) GO TO 160

140 CONTINUE160 CONTINUE220 CONTINUE

CC READ STRAIN DATAC

IF(NINI .EQ. 0) GO TO 300DO 240 N = 1, NUMELREAD (INPT,*) M, TEPS(M)

240 CONTINUE300 CLOSE(INPT)

RETURNC500 CONTINUE

WRITE(MSSG,1010)STOP

C1000 FORMAT(A)1010 FORMAT(/,' SORRY, THIS PROGRAM CANNOT HANDLE MORE THAN 100'

+ ' NODES OR ELEMENTS ')END


SUBROUTINE NFORCE(QQ,FRZ,LM)CC ADD NODAL POINT FORCEC

IMPLICIT REAL*8 (A-H.O-Z), INTEGERM (I-N)DIMENSION QQ{1),FRZ(1),LM(1)

CDO 100 I = 1, 8N - LM(I)FRZ(N) - FRZ(N) - QQ(I)


SUBROUTINE NONLINCC THIS ROUTINE CONTROLS THE ITERATIONSC

IMPLICIT INTEGERM (I-N), REAL*8 (A-H, 0-Z)COMMON /INOT/ INPT.MSSG,IUNIT.IUNI2COMMON /MSTR/ NUMNP.NUMEL.IPLNAXCOMMON /TSTP/ NINI,NCUR,NSEND,NITR,DTMAXCOMMON /ITRC/ ITYP, ICONVCOMMON /CNEQ/ NEQ,MBANDCOMMON /RVA1/ RZ(2,100), URZ(2,100), FRZ(2,100)DIMENSION UNORM(2),ENORM(2),FNORM(2)COMMON A(5000),B(200)

CRTOL •= 0.00001IF(ITYP .EQ. 2) RTOL - 0.0005ACOEF - 1.NSTEL - NEQ * MBANDIF(NSTEL.LE.5000 .AND. NEQ.LE.200) GO TO 10HRITE(MSSG,1010)STOP

C10 CONTINUE

DO 30 N - 1,2UNORM(N) = 0.ENORM(N) - 0.FNORM(N) - 0.

30 CONTINUEC

ITRMAX - 20IF(ITYP .EQ. 2) ITRMAX - 200DO 200 N = 1, ITRMAXNITR = NCALL STIFF(B,A,NEQ,MBAND,ITYP)IDREC = 1CALL NORM(FRZ,B,FDUM,DFN,NEQ,IDREC)IF(ITYP .EQ. 2) DFN = 0.CALL BANSOL(B,A,NEQ,MBAND)IDREC - ITYP

c

c


CALL NORM(URZ,B,UC,EC,NEQ,IDREC)IF(ITYP .EQ. 1) WRITE(MSSG,1030) NIF(ITYP .EQ. 2) WRITE(MSSG,1050) NWRITE(HSSG,1070) UC.EC.DFNIF(N .EQ. 1) GO TO 130IF(EC .LT. RTOL .AND. DFN .LT. RTOL) GO TO 300IF(ITYP .EQ. 2) GO TO 130IF(EC .LT. ENORH(2)) GO TO 100

CC ADJUST THE ACOEFC

ACOEF = ACOEF * 0.7GO TO 130

100 CONTINUEIF(ENORM(1) .GT. ENORM(2) .AND. ENORM(Z) .GT. EC)

+ ACOEF - ACOEF * 1.3IF(ACOEF .GT. 1.) ACOEF = 1.0

CC VELOCITY UPDATEC

130 CONTINUENB - 0DO 150 I - 1, NUMNPDO 150 J - 1,2NB - NB + 1IF(ITYP .EQ. 1) URZ(J.I) - URZ(J.I) + ACOEF * B(NB)IF(ITYP .EQ. 2) URZ(J,I) = B(NB)

150 CONTINUEC

170 CONTINUEUNORM(l) UNORM(2)ENORM(l) ENORH(2)FNORH(l) FNORM(2)UNORM(2) UCENORM(2) ECFNORM(2) DFN

200 CONTINUECC SET FLAGC

ICONV - 2RETURN

C300 CONTINUE

C CONVERGED CASEC SET FLAGC

ICONV - 1C

RETURN1010 FORHAT(/,' YOU NEED MORE SPACE IN THE BLANK COMMON ')1030 FORMAT(/,' N-R ITERATION NO. ',I5,/)1050 FORMAT(/,' DRT ITERATION NO. ', 15,/)1070 FORMAT( ' VELOCITY NORM = '.F15.7,/,


+ ' REL. ERROR NORM - '.F15.7,/,+ ' REL. FORCE ERROR NORM = '.F15.7,/)END

SUBROUTINE NORM(URZ,V,UC,EROR,NEQ,ITYP)CC CALCULATE THE ERROR NORM FOR LINEAR AND NONLINEAR CASEC

IMPLICIT INTEGERM (I-N), REAL*8 (A-H.O-Z)DIMENSION URZ(1),V(1)

CUC = 0.EROR = 0.DO 100 N = 1, NEQUC = UC + URZ(N) * URZ(N)IF(ITYP .EQ. 1) EROR = EROR + V(N) * V(N)IF(ITYP .EQ. 2) EROR - EROR + (URZ(N)-V(N))**2

100 CONTINUEC

UC - DSQRT(UC)EROR - DSQRT(EROR)IF(UC .NE. 0.) EROR - EROR / UCRETURNEND

SUBROUTINE POTSOLCC THIS SUBROUTINE HANDLES THE POST SOLUTION PROCEDURES, IE,C GEOMETRY UPDATESC STRESS EVALUATIONC TOTAL STRAIN EVALUATIONC

IMPLICIT REAL*8 (A-H.O-Z), INTEGER*4 (I-N)COMMON /TSTP/ NINI,NCUR,NSEND,NITR,DTMAXCOMMON /MSTR/ NUMNP.NUMEL.IPLNAXCOMMON /RIGD/ RTOL.ALPH.DIATCOMMON /RVA1/ RZ(2,100), URZ(2,100), FRZ(2,100)COMMON /RVA2/ EPS(5,100), STS(5,100),TEPS(100)COMMON /DIES/ FRCFAC

CC GEOMETRY UPDATESC

DO 100 N = 1, NUMNPRZ(1,N) - RZ(1,N) + DTMAX * URZ(l.N)RZ(2,N) - RZ(2,N) + DTMAX * URZ(2,N)

100 CONTINUECC STRESS EVALUATIONC

DO 200 N - 1, NUMELAL - EPS(5,N)IF(AL .LT. ALPH) AL = ALPHCALL FLWSTS(EFSTS, STRT, TEPS(N), AL)EM = (EPS(l.N) + EPS(2,N) + EPS(3,N)) / 3.

c


DO 150 I = 1 , 3STS(I.N) = 2./3. * EFSTS * (EPS(I.N)-EM) / AL + DIAT * EM * 3.

150 CONTINUESTS(4,N) = EFSTS * EPS(4,N) / AL / 3.STS(5,N) = EFSTS

200 CONTINUECC UPDATE TOTAL EFFECTIVE STRAINC

DO 300 N = 1, NUMELTEPS(N) = TEPS(N) + EPS(5,N) * DTMAX


SUBROUTINE PRTINPCC THIS SUBROUTINE PRINTS THE INPUT DATAC

IMPLICIT REAL*8 (A-H, 0-Z), INTEGERM (I-N)CHARACTER TITLE*70COMMON /TITL/ TITLECOMMON /TSTP/ NINI,NCUR,NSEND,NITR,DTMAXCOMMON /RVA1/ RZ(2,100), URZ(2,100), FRZ(2,100)COMMON /RVA2/ EPS(5,100), STS(5.100), TEPS(IOO)COMMON /INVR/ NOD(4,100), NBCD(2,100)COMMON /DIES/ FRCFACCOMMON /RIGD/ RTOL, ALPH, DIATCOMMON /MSTR/ NUMNP, NUMEL, IPLNAXCOMMON /INOT/ INPT, MSSG, IUNIT, IUNI2

CC INPUT SUMMARYC

WRITE(IUNIT,1010) TITLEWRITE(IUNIT,1020)WRITE(IUNIT,1030) NINI.NSEND,DTMAXWRITE(IUNIT,1050) ALPH,DIATWRITE(IUNIT,1070) IPLNAXWRITE(IUNIT,1110) FRCFACWRITE(IUNIT,1130) NUMNPWRITE(IUNIT,1150)WRITE(IUNIT,1180) (N,(RZ(I,N), 1=1,2), N = 1, NUMNP)

CC PRINT NODE VELOCITYC

WRITE(IUNIT,1220)WRITE(IUNIT,1180) (N,(URZ(I,N), 1=1,2), N = 1, NUMNP)

CC ELEMENT INFORMATIONC

WRITE(IUNIT,1270) NUMELWRITE(IUNIT,1330)WRITE(IUNIT,1350) (N, (NOD(I,N),1=1,4), N=l,NUMEL)

c

c


C BOUNDARY CONDITIONC

WRITE(IUNIT,1400)WRITE(IUNIT,1430) (N, (NBCD(I,N),1-1,2), N=1,NUMNP)

CC WRITE STRAIN DISTRIBUTION AT INPUT STAGEC

WRITE(IUNIT,1500)WRITE(IUNIT,1550) (N,TEPS(N), N=1,NUMEL)RETURN

C1010 FORHAT(1H1,///,5X,'OUTPUT OF S P I D ',//,

+ 5X.A,///)1020 FORMAT(5X,'INITIAL INPUT SUMMARY'///)1030 FORMATC INITIAL STEP NUMBER 15,/,

+ ' FINAL STEP NUMBER 15,/,+ ' STEP SIZE IN TIME UNIT F10.5)

1050 FORMATC LIMITING STRAIN RATE F15.7,/,+ ' PENALTY CONSTANT F15.7)

1070 FORMATC DEFORMATION CODE 15,/,+ ' IF - 1, AXISYMMETRIC ',/+ ' = 2, PLANE STRAIN ')

1110 FORMATC FRICTION FACTOR = ', F15.7,/)1130 FORMAT{///' NUMBER OF NODAL POINTS = ', 15,/)1150 FORMAT(//,' NODE COORDINATES ',//,

+ ' NODE NO X-COORD Y-COORD',/)1180 FORMAT(5X,I5,5X,2F15.7)1220 FORMAT(///,' NODE VELOCITY ',//,

+ ' NODE NO X-VELOCITY Y-VELOCITY'/)1270 FORMAT(//, NUMBER OF ELEMENTS = ', 15,/)1330 FORMAT(//, ELEMENT CONNECTIVITY ',//,

+ ELE NO. I J K L ',/)1350 FORMAT(5I71400 FORMAT(//, BOUNDARY CONDITION CODE ',//,

+ NODE NO Xl-CODE X2-CODE ',/)1430 FORMAT(3I7)1500 FORMAT(///,' STRAIN DISTRIBUTION AT INPUT STAGE ',//,

+ ' NODE NO. STRAIN ',/)1550 FORMAT(I5,5X,F15.7)

END

SUBROUTINE PRTSOLCC THIS SUBROUTINE PRINT THE SOLUTION RESULTSC

IMPLICIT REAL*8 (A-H.O-Z), INTEGERM (I-N)CHARACTER TITLE*70COMMON /TITL/ TITLECOMMON /INOT/ INPT,MSSG,IUNIT,IUNI2COMMON /TSTP/ NINI,NCUR,NSEND,NITR,DTMAXCOMMON /MSTR/ NUMNP,NUMEL,IPLNAXCOMMON /RVA1/ RZ(Z.IOO), URZ(2,100), FRZ(2,100)COMMON /RVA2/ EPS(5,100), STS(5,100), TEPS(IOO)COMMON /INVR/ NOD(4,100), NBCD(2,100)

c

358

CC PRINT NODE COORDINATESC

WRITE(IUNIT,1010) TITLE, NCURWRITE(IUNIT,1020)WRITE(IUNIT,1040) (N,(RZ(I,N), 1=1,2), N -1, NUHNP)

CC PRINT NODE VELOCITY, NODAL FORCEC

WRITE(IUNIT,1080)WRITE(IUNIT,1100) (N,(URZ(I,N), 1-1,2),

+ (FRZ(I,N), 1=1,2), N = 1, NUMNP)

C STRAIN RATE, STRESS, TOTAL EFFECTIVE STRAINC

WRITE(IUNIT,1130)WRITE(IUNIT,1180) (N,(EPS(I,N),1-1,5), N - 1, NUMEL)WRITE(IUNIT,1230)WRITE(IUNIT,1180) (N,(STS(I,N),1=1,5), N = 1, NUMEL)WRITE(IUNIT,1330)WRITE(IUNIT,1360) (N, TEPS(N), N = 1, NUMEL)RETURN

C1010 FORMAT(1H1,///,5X,'OUTPUT OF S P I D',//,5X,A,//,

+ 10X, SOLUTION AT STEP NUMBER - ',I5,///)1020 FORMAT( /, NODE COORDINATES',//,

+ NODE NO X-COORD Y-COORD',/)1040 FORMAT(5X,I5,5X,2F15.7)1080 FORMAT(///, NODAL VELOCITY AND FORCE',//,

+ NODE NO X-VELOCITY Y-VELOCITY',+ ' X-FORCE Y-FORCE ',//)

1100 FORMAT(3X,I5,3X,4F15.7)1130 FORMAT(///,' STRAIN RATE COMPONENTS ',//,

+ ELE. NO. Ell E22 E33'+ E12 EBAR',//)

1180 FORMAT(I5,5F15.7)1230 FORMAT(//, STRESS COMPONENTS',//,

+ ELE. NO. Sll S22 S33'+ S12 SBAR',//)

1330 FORMAT(///,' TOTAL EFFECTIVE STRAIN ',//,+ ' ELE. NO. EFFECTIVE STRAIN '//)

1360 FORMAT(5X,I5,5X,F15.7)END

c

SUBROUTINE RSTFILCC GENERATE RESTART FILEC

IMPLICIT REAL*8 (A-H, 0-Z), INTEGERM (I-N)CHARACTER TITLE*70COMMON /TITL/ TITLECOMMON /TSTP/ NINI,NCUR,NSEND,NITR,DTMAXCOMMON /RVA1/ RZ(2,100), URZ(2,100), FRZ(2,100)COMMON /RVA2/ EPS(5,100), STS(5,100), TEPS(IOO)

cc



COMMON /INVR/ NOD(4,100), NBCD(2,100)COMMON /DIES/ FRCFACCOMMON /RIGD/ RTOL, ALPH, DIATCOMMON /MSTR/ NUMNP, NUMEL, IPLNAXCOMMON /INOT/ INPT, MSSG, IUNIT, IUNI2

CNN - NCUR + 1OPEN(IUNI2,FILE='SPID.RSr,STATUS*'UNKNOWN',FORM='FORMATTED')WRITE(IUNI2,1010) TITLEWRITE(IUNI2,1040) NCUR.NN.DTMAXWRITE(IUNI2,1060) ALPH.DIATHRITE(IUNI2,1080) IPLNAXWRITE(IUNI2,1060) FRCFACWRITE(IUNI2,1080) NUMNPWRITE(IUNI2,1120) (N,(RZ(I,N),1=1,2),N=l,NUMNP)WRITE(IUNI2,1080) NUMELWRITE(IUNI2,1080) (N,(NOD(I.N),1-1,4), N=l,NUMEL)WRITE(IUN12,1160) (N,(NBCD(I,N),1-1,2),N=1,NUMNP)WRITE(IUNI2,1120) (N,(URZ(I,N),I-=1,2),N-1,NUMNP)WRITE(IUNI2,1200) (N,TEPS(N), N-l,NUMEL)CLOSE(IUNI2)RETURN

C1010 FORMAT(1X,A)1040 FORMAT(2I10,F20.7)1060 FORMAT(3F20.10)1080 FORMAT(517)1120 FORMAT(I5,2F20.10)1160 FORMAT(3I7)1200 FORMAT(I7,F20.10)

END

SUBROUTINE STIFF(B,A,NEQ,MBAND,ITYP)CC STIFFNESS MATRIX GENERATIONC ITYP - 1, NEWTON-RAPHSON ITERATIONC ITYP - 2, DIRECT ITERATIONC

IMPLICIT REAL*8 (A-H, 0-Z), INTEGER*4 (I-N)COMMON /INOT/ INPT,MSSG,IUNIT.IUNI2COMMON /RVA1/ RZ(2,100), URZ(2,100), FRZ(2,100)COMMON /RVA2/ EPS(5,100), STS(5,100), TEPS(IOO)COMMON /INVR/ NOD(4,100), NBCD(2,100)COMMON /DIES/ FRCFACCOMMON /MSTR/ NUMNP, NUMEL, IPLNAXDIMENSION A(NEQ,1),B(1)DIMENSION RZE(2,4), URZE(2,4), NBCDE(2,4),PP(8,8), QQ(8),+ LM(8)

CC INITIALIZE LOAD VECTOR, STIFFNESS MATRIX, ANDC NODAL POINT FORCE ARRAYC

DO 20 N = 1, NEQB(N) = 0.

c


DO 20 I = 1, MBANDA(N,I) = 0.

20 CONTINUEDO 50 N - 1, NUHNPDO 50 I - 1, 2

50 FRZ(I.N) - 0.C

DO 200 N - 1, NUMELCC CHANGE RZ, URZ, AND NBCD FROM GLOBAL ARRANGEMENT TO ELEMENTALC ARRANGEMENTC

DO 100 I - 1, 412 - 1 * 2II = 1 2 - 1NE = NOD(I,N)RZE(1,I) - RZ(1,NE)RZE(2,I) - RZ(2,NE)URZE(l.I) - URZ(l.NE)URZE(2,I) - URZ(2,NE)NBCDE(l.I) = NBCD(l.NE)NBCDE(2,I) - NBCD(2,NE)LM(I2) - NOD(I,N)*2LH(I1) - LM(I2) - 1

100 CONTINUEC

CALL ELSHLF(PP,QQ,RZE,URZE,EPS{1,N),TEPS(N),IPLNAX,ITYP,N)IF(ITYP .EQ. 1) CALL NFORCE(QQ,FRZ,LM)IF(FRCFAC.NE. 0.)

+ CALL FRCBDY(RZE,URZE,NBCDE,TEPS(N),EPS(1,N),QQ,PP,IPLNAX)CALL ADDBAN(B,A,NEQ,LM,QQ,PP)

200 CONTINUECC APPLY DISPLACEMENT BOUNDARY CONDITIONC

CALL DISBDY(URZ,NBCD,B,A,NEQ,MBAND,ITYP)RETURNEND

SUBROUTINE STRMTX(RZ,B,WDXJ,S,T,IPLNAX,NEL)CC CHAPTER 6.4C EQUATIONS (6.25), (6.27) AND (6.35)C EVALUATE STRAIN RATE MATRIX OF QUADRILATERAL ELEMENTCC B(4,8) : STRAIN RATE MATRIXC RZ(2,4) : NODE COORDINATESC (S,T) : NATURAL COORDINATEC

IMPLICIT REAL*8 (A-H, 0-Z), INTEGERM (I-N)COMMON /INOT/ INPT,MSSG,IUNIT,IUNI2DIMENSION RZ(2,1), B(4,l)

CR12 = RZ(1,1) - RZ(1,2)

c


R13 RZ(1,1) - RZ(1,3)R14 RZ(1,1) - RZ(1,4)R23 RZ(1,2) - RZ(1,3)R24 RZ(1,2) - RZ(1,4)R34 RZ(1,3) - RZ(1,4)

CZ12 RZ(2,1) - RZ(2,2)Z13 RZ(2,1) - RZ(2,3)Z14 RZ(2,1) - RZ(2,4)Z23 RZ(2,2) - RZ(2,3)Z24 RZ(2,2) - RZ(2,4)Z34 RZ(2,3) - RZ(2,4)

CC REFER EQUATION (6.34)C

DX08 = ( (R13*Z24 - R24*Z13) + (R34*Z12 - R12*Z34)*S ++ (R23*Z14 - R14*Z23)*T )DXJ = DXJ8 / 8.IF(DXO .GT. 0.) GO TO 10WRITE(MSSG,1010) NELWRITE(HSSG,1030) DXJ.SJSTOP

10 CONTINUEC

XI ( Z24 - Z34*S - Z23H) / DXJ8X2 (-Z13 + Z34*S + Z14*T) / DXJ8X3 (-Z24 + Z12*S - Z14*T) / DXJ8X4 ( Z13 - Z12*S + Z23*T) / DXJ8

CYl (-R24 + R34*S + R23*T) / DX08Y2 ( R13 - R34*S - R14*T) / DXJ8Y3 ( R24 - R12*S + R14*T) / DXJ8Y4 (-R13 + R12*S - R23*T) / DXJ8

CDO 20 I = 1, 4DO 20 J = 1,8B(I,J) - 0.

20 CONTINUEC

B(l,l) XI8(1,3) X2B(l,5) X3B(l,7) X4B(2,2) YlB(2,4) Y2B(2,6) Y3B(2,8) Y4

CWDXJ = DXJIF(IPLNAX .NE. 1) GO TO 40Ql = (l.-S) * (l.-T) * 0.25Q2 - (l.+S) * (l.-T) * 0.25Q3 - (l.+S) * (l.+T) * 0.25Q4 = (l.-S) * (l.+T) * 0.25


CR - Q1*RZ(1,1) + Q2*RZ(1,2) + Q3*RZ(1,3) + Q4*RZ(1,4)B(3,l) - Ql / RB(3,3) - QZ / R8(3,5} - Q3 / R8(3,7) - Q4 / RWDXJ - WDXJ * R

C40 CONTINUE

B(4,l) - YlB(4,3) •= Y28(4,5) = Y3B(4,7) - Y4B(4,2) - XIB(4,4) - X2B(4,6) = X3B(4,8) - X4RETURN

C1010 FORMAT(/,' SORRY, NEGATIVE JACOBIAN DETECTED AT ELEMENT NO.

+ 15)1030 FORMAT(' DXJ.S.T = \3F15.7)

END

SUBROUTINE VSPLON(QQ,PP, B,URZ,EPS,WDXJ,IDREC)CC CHAPTERS 6.5 AND 7.1C EQUATIONS (6.43) AND (6.44)C REDUCED INTEGRATION OF VOLUME STRAIN RATECC PP = ELEMENTAL STIFFNESS MATRIXC QQ = ELEMENTAL LOAD VECTORC 8 = STRAIN RATE MATRIXC

IMPLICIT REAL*8 (A-H, 0-Z), INTEGER*4 (I-N)COMMON /RIGD/ RTOL.ALPH.DIATDIMENSION PP(8,8), QQ(8), B(4,8), URZ(l), EPS(l)DIMENSION D(6),XX(8)DATA 0/3*0.666666666666666700, 3*0.3333333333333333DO/

CC GENERATE DILATATIONAL STRAIN RATE MATRIXC

DO 20 I = 1, 8XX(I) = B(1,I) + 8(2,1) + 8(3,1)

20 CONTINUECC CALCULATE STRAIN RATE COMPONENTSC

DO 40 I = 1, 5EPS(I) = 0.

40 CONTINUEXVOL = 0.DO 60 J = 1, 8XVOL = XVOL + XX(J) * URZ(J)

c


DO 60 I = 1, 4EPS(I) - EPS(I) + B(I,J) * URZ(J)

60 CONTINUEEB2 = (EPS(1)**2 + EPS(2)**2 + EPS(3)**2) * D(l) ++ EPS(4)**2 * 0(4)EPS(5) - DSQRT(EB2)

CC EVALUATE VOLUMETRIC CONTRIBUTION Of STIFFNESS MATRIXC

DO 80 I - 1, 8IF(IDREC .EQ. 1)

+ QQ(I) = QQ(I) - DIAT * WDXJ * XVOL * XX(I)TEN - DIAT * WDXJ * XX(I)DO 80 J = I, 8PP(I,J) = PP(I,J) + TEM * XX(J)PP(J,I) = PP(I,J)


c SUBROUTINE VSPLSTtQQ.PP.B.URZ.TEPS.WDXJ.IDREC)

CC CHAPTERS 6.5C EQUATIONS (6.43), (6.44) AND (6.46)C FOUR POINTS INTEGRATION OF VOLUME STRAIN RATECC PP = ELEMENTAL STIFFNESS MATRIXC QQ = ELEMENTAL LOAD VECTORC B = STRAIN RATE MATRIXC

IMPLICIT REAL*8 (A-H, 0-Z), INTEGER*4 (I-N)COMMON /TSTP/ NINI,NCUR,NSEND,NITR,DTMAXCOMMON /RIGD/ RTOL,ALPH,DIATDIMENSION PP(8,8), QQ(8), 8(4,8), URZ(l)DIMENSION D(6),FDV(8),E(4),XX(8)DATA 0/3*0.6666666666666700, 3*0.333333333333333DO/

CC ELIMINATE DIALATATIONAL COMPONENT FROM STRAIN RATE MATRIXC

DO 20 I = 1, 8XX(I) = (B(1,I) + B(2,I) + B(3,I)) / 3.

20 CONTINUEDO 40 I = 1, 8DO 40 J = 1,3B(J,I) = B(J,I) - XX(I)

40 CONTINUECC CALCULATE STRAIN RATEC

DO 60 J = 1, 4E(J) = 0.DO 60 I = 1, 8E(J) = E(J) + B(J,I) * URZ(I)

60 CONTINUE


EFSR2 - D(1)*E(1)*E(1) + D(2)*E(2)*E(2) + D(3)*E(3)*E(3) ++ D(4)*E(4)*E(4)IF(NITR.EQ.l .AND. NCUR.EQ.NINI .AND. IDREC.EQ.2)

+ EFSR2 - (ALPH*100.) ** 2ALPH2 - ALPH **2IF(EFSR2 .LT. ALPH2) EFSR2 - ALPH2EFSR - DSQRT(EFSR2)CALL FLWSTS(EFSTS, STRAT, TEPS, EFSR)

CC CALCULATE FIRST DERIVATE OF EFSR **2C

DO 80 I - 1, 8FDV(I) - 0.DO 80 J - 1,4FDV(I) - FDV(I) + D(J)*E(J)*B(J,I)

80 CONTINUECC ADD POINT CONTRIBUTION TO STIFNESS MATRIXC

Fl - EFSTS / EFSR * WDXJIF(IDREC .EQ. 2) GO TO 200F2 - STRAT / EFSR2 * WDXJ - Fl / EFSR2DO 120 I - 1,8QQ(I) - QQ(I) - FDV(I) * FlDO 110 0 - 1,8TEM - 0.DO 100 K - 1, 4TEM - TEM + D(K)*B(K,I)*B(K,J)

100 CONTINUEPP(I,J) - PP(I,J) + TEM*F1IF(EFSR2 .LT. ALPH2) GO TO 105PP(I,J) - PP(I,J) + FDV(I)*FDV(J)*F2

105 PP(J,I) - PP(I,J)110 CONTINUE120 CONTINUE

RETURNC200 CONTINUE

DO 300 1 - 1 , 8DO 280 J - 1,8TEM - 0.DO 250 K = 1,4TEM - TEM + D(K)*B(K,I)*B(K,J)

250 CONTINUEPP(I,J) = PP(I.J) + TEM*F1PP(J,I) - PP(I,J)


RETURNEND

A.7 Example Solution1. Simulation ConditionsA simple compression of circular cylinder was simulated by using SPID.The computational conditions used for the simulation are as follows.

Initial billet dimension 1.0 (in.) diameter1.0 (in.) height

Die velocity -1.0(in./s)Friction factor 0.5Flow stress 10.0(£)° ' (Ksi)Total reduction in height 40%Incremental step size 2% of initial height


Total number of nodesTotal number of elements

2516

It may be mentioned that the degrees of freedom, 25 nodes, used in thesimulation are hardly enough to obtain accurate results. However, a smallnumber of nodes were used for demonstration purposes.

2. Input FileThe input file, SPID.DAT, for the simple compression is given in thefollowing.

SIMPLE UPSETTING, M - 0.50 20 0.02

0.01 10000.10.5251 0.00 1.002 0.25 1.003 0.50 1.004 0.75 1.005 1.00 1.006 0.00 0.757 0.25 0.758 0.50 0.759 0.75 0.7510 1.00 0.7511 0.00 0.5012 0.25 0.5013 0.50 0.5014 0.75 0.5015 1.00 0.5016 0.00 0.2517 0.25 0.2518 0.50 0.2519 0.75 0.2520 1.00 0.2521 0.00 0.0022 0.25 0.0023 0.50 0.0024 0.75 0.0025 1.00 0.00161 1 6 7 22 2 7 8 33 3 8 9 44 4 9 10 55 6 11 12 76 7 12 13 87 8 13 14 98 9 14 15 109 11 16 17 1210 12 17 18 1311 13 18 19 1412 14 19 20 1513 16 21 22 1714 17 22 23 1815 18 23 24 1916 19 24 25 20

1 1 32 0 33 0 34 0 35 0 36 1 0

AXISYMMETRIC DEFORMATIONFRICTION FACTORNUMBER OF NODESNODE COORDINATES

NUMBER OF ELEMENTSELEMENT CONNECTIVITY

BOUNDARY CONDITION CODE


1116212223242512345

25

11100000.0.0.0.0.0.

0011111

-1.-1.-1.-1.-1.0.

3. Output FileThe output file, SPID.OUT, contains the results of 20 solution steps. The20th step (40% reduction in height) solution is printed here as an exampleof the output file.

OUTPUT OF S P I D

SIMPLE UPSETTING, M = 0.5

SOLUTION AT STEP NUMBER 20

NODE NO

12345678910111213141516171819202122232425

NODE COORDINATES

X-COORD

0.00000000.25393330.51832400.77394611.14175720.00000000.28694000.56913760.89179921.23117680.00000000.33352090.66375721.00268701.28073030.00000000.37524390.74035351.05378571.32468270.00000000.39508730.76223121.07127741.3397564

Y-COORD

0.60000000.60000000.60000000.60000000.60000000.39247660.38021430.40232070.40777250.52101010.22491960.22484860.23732090.29191730.36399480.10971410.09517480.12297320.14840540.18588720.00000000.00000000.00000000.00000000.0000000

VELOCITY BOUNDARY CONDITONS


NODAL VELOCITY AND FORCE

NODE NO X-VELOCITY Y-VELOCITY X-FORCE Y-FORCE

12345678910111213141516171819202122232425

STRAIN RATE COMPONENTS

ELE. NO. Ell E22 E33 E12 EBAR

0.00000000.04379790.15195690.17296270.54164250.00000000.22296440.32619110.59320880.79500780.00000000.33078260.58511400.93857450.92882040.00000000.38936060.84055101.05065901.04898230.00000000.43470560.91179621.08340721.0878362

-1.0000000-1.0000000-1.0000000-1.0000000-1.0000000-0.7072642-0.8703353-0.7340251-0.9273866-0.4385331-0.4869862-0.4963833-0.6170831-0.4774093-0.2414312-0.1617441-0.3021957-0.2709927-0.2204407-0.10395320.00000000.00000000.00000000.00000000.0000000

-0.0206875-0.1886367-0.3876530-0.7469869-0.55317700.02231280.00000040.0000003-0.0000002-0.00000040.0064280-0.0000002-0.00000010.00000010.00000000.0011846-0.00000030.00000000.00000020.00000000.0176562-0.00000010.00000000.00000010.0000000

-0.1677586-0.8241910-1.3597889-2.9630664-3.52695400.0000000-0.00000050.00000040.00000120.0000004-0.00000010.00000000.00000020.00000020.00000000.00000000.00000000.00000010.00000000.00000000.48829122.07010902.93865262.42786990.9161880

12345678910111213141516

0.49167740.41672930.52399171.06487860.90580190.63898591.08311570.67991371.03607871.08243540.98861680.10537801.09138101.31152020.63220190.0186890

-0.9905694-0.8789797-0.9806039-1.5922324-1.8161495-1.4431107-1.8779439-1.4301980-2.0749139-2.1205662-1.9961498-0.9711997-2.1862521-2.4730094-1.7276470-0.9266512

0.49802160.46161600.45598970.52605080.90873890.80321430.79361880.74994461.03716291.03667811.00669850.86570711.09317311.15986111.09482330.9079304

-0.6989617-0.4731238-1.6138913-0.7080532-0.5110853-0.7441022-1.25699370.2619431-0.3153663-0.7567893-0.38672070.1472329-0.3355728-0.1449076-0.13512490.0679924

1.06935300.92060891.35312001.67263661.83943591.50839182.01984491.43862732.08233232.16479932.00834851.06914942.19425662.47543101.74970681.0600883

STRESS COMPONENTS

ELE. NO. Sll S22 S33 S12 SBAR

1 -5.1711015 -14.4740431 -5.1312835 -2.1934270 10.06727912 -3.0252520 -12.3309426 -2.7028793 -1.6989706 9.91762123 -3.2440441 -10.8846159 -3.5893688 -4.0977962 10.30703224 -7.8939030 -19.0434449 -10.1548860 -1.4855362 10.5278614


5678910111213141516

-11.7723892-5.6950626-7.63872120.0451580

-12.2963810-10.1809774-4.3976851-0.4260495-12.5221537-11.5784259-3.3509895-0.1836357

-22.2574977-15.2834789-18.1237511-10.0953735-23.0143553-20.8368751-15.0210956-7.1840704-23.2945029-22.7376738-12.8597561-6.1634682

-11.7610756-4.9387623-8.66382140.3817044

-12.2926460-10.3332052-4.33332864.3467787

-12.5162635-12.0256153-1.48690425.4413395

-0.9843645-1.7133599-2.22548980.6294081-0.5432490-1.2588613-0.68821000.4621138-0.5514508-0.2136407-0.27223600.2150461

10.628414110.419608910.728322110.370383810.761058810.802935010.722200110.067087510.817545810.948763710.575393310.0585228

TOTAL EFFECTIVE STRAIN

ELE. NO. EFFECTIVE STRAIN

12345678910111213141516

0.16624180.19018510.32061230.56115100.44000650.44839660.56765230.49736810.70787420.69814890.61998240.41766140.87540110.80861280.60549530.4199022

~Z


FIG. A.2 FEM grid distortions of simple compression predicted by SPID: (a) undeformed;(b) 40% reduction in height.

/Figure A.2 shows (a) the initial FEM mesh and (b) the deformed FEM

mesh at 40% reduction in he/ght predicted by SPID.The computations were performed on an IBM PC/AT with Professional

FORTRAN compiler of Ryarj/McFarland Corp. and also on VAX 750 withVMS operating system. Tar>le A.I summarizes the CPU time requirementsto run this simulation on/VAX-750 and IBM PC/AT.

TABLE A/1 CPU Time Summary for Simple Compression

VMS version 4.4 operating system was used.

* IBM PC/AT equipped with a hard disk and math co-processor. Professional FORTRAN compiler byRyan-McFarland Corp. and DOS operating system, Version 3.10, were used.

INDEX

Accuracy of simulation, 333Air bending, 21, 141-147Air rounding, 21ALPID program, 136Analysis in metal forming, 26-52

closed-die forging, 34, 35-36, 37cold extrusion, 39-41cold forging, 39-41drawing of rod, wire, shapes, and

tubes, 45-47equations for, 73flow stress of metals, 28-30friction in metal forming, 30-33hot extrusion of rods and shapes,

36-39hot forming, 28-29, 34-35impression, 35-36, 37methods of. See Methods of

analysisobjectives of, 26rolling of strip, plate, and shapes,

41-45sheet-metal forming, 47-52temperatures, 28, 29, 33-35variables in, 26-28

Anisotropy, 189plastic, 190-191

Area-weighted averaging, 126-127,129, 255

Assemblage, element, 115-116Automobile wheel center, 51-52Axisymmetric forging

of flange-hub shapes, 253-256, 257in preform design, 309-315of pulley blank, 256-259See also Axisymmetric isothermal

forgingAxisymmetric isothermal forging,

151-172cabbaging, 165-168compression of cylinders, 153-159

compressor disk forging, 168-169finite-element formulation, 151-153flashless forging, 170, 171, 172friction, evaluation of, 163-165, 166heading of cylindrical bars, 153-159ring compression, 159-163, 163-165spike forging, 163, 165

Backward extrusion forging, 14Backward tracing, 298-301, 301-305

in hot forming, 316-317in plane-strain rolling, 305-308in shell nosing, 301-305in symmetric forging, 311-312

Bar drawing, 178-183multipass, 183-186

Bar extrusion, 176-178multipass, 183-184

Beverage cans, 46Block compression, 278-284

rectangular blocks, 278-281wedge-shaped blocks, 281-284

Blocker operation, 35Bore-expanding, 194-195, 196Boundary conditions, 86-87, 117-121

with hemispheric punch, 201-203Boundary lubrication, 31Brake-bending, 21

Cabbaging, 165-168CAD. See Computer-aided designCan ironing press, 46-47Cauchy stress, 56, 57, 58, 208, 324Closed-die forging

analysis, 35-36, 37load vs displacement curves in, 34metal forming system using, 9with flash, 14, 133-136without flash, 12

Coining operation, 13Cold extrusion, 39-41

371

372 Index

Cold forging, 39-41Compaction, 266-270Compatibility conditions, 73Compression

block, 278-284of cylinder, 75-78, 80-83, 153-159,

223-225, 229-234, 364-369flat-tool, 314of porous metals, 249-253ring, 159-163, 163-165, 331-332,

333square-ring, 284-287See also Hot compression; Plane-

strain compressionCompression test, 29Compressor disk forging, 168-169Computer-aided design (CAD), 35-36,

93-94Computer-aided techniques, 1, 186.

See also Extrusion dies,computer-aided design of

Connectivity, element, 115Constitutive equations, 73Cup drawing, 48

square, 210-217Cylinders

compression of, 75-78, 80-83,153-159, 223-225, 229-234,364-369

steel, compression of, 229-234Cylindrical bars, heading of, 153-159

Deep-drawing processes, 201-206Deformation mechanics, 10. See also

PlasticityDeformation speed, 26-28Die bending, 21Die rounding, 21Direct iteration method, 121-122Disk forging, 311, 312Distortion energy criterion, 59Drawbead, 50, 51Drawing, 20

bar, 178-183bar, multipass, 183-186cup, defects in, 48deep, 23, 201-206flange, 194-195, 197of rod, 45-47of shapes, 45-47

square-cup, 210-217of tubes, 45-47of wire, 45-47

Dry conditions, 30

Effective strain, 66-68interpolation of, 126-127

Effective strain-rate matrix, 107Effective stress, 66-68Efficiency of simulation, 333Element assemblage, 115-116Element connectivity, 115Element strain-rate matrix,

101-107Equilibrium, with tractions, 62Equilibrium equations, 61-62, 73Equipment, 10Equivalent stress, 66Euler equation, 86Extremum principles, 68-70, 71-72,

74Extrusion, 19

bar, 176-178bar, multipass, 183-184cold, 39-41direct and indirect, 19hot, 20, 36-39of rods, 36-39of shapes, 36-39

Extrusion dies, computer-aided designof, 186

FEM. See Finite-element methodFEM code. See SPIDFinite deformation, 56Finite-element method (FEM), 83-87,

88, 90-110, 111-129admissibility requirement for

velocity field, 91-92advantages of, 4alternative approach, 84assemblage, element, 115-116axisymmetric isothermal forging,

151-153axisymmetric out-of-plane

deformation, 199-201basic concept of, 3basic equations, comments on,

85-86basic equations, derivation of, 83

Index 373

basic principles and concepts in,73-74

basis of, 4boundary conditions, 86-87,

117-121computer-aided, 93-94construction of model, 3-4direct iteration method, 121-122discretization of problem, 90element strain-rate matrix, 101-107elemental stiffness equation,

108-110geometry updating, 123-125history of, 90in-plane deformation processes,

192-194introduction to, 3-4Lagrange multiplier method, 110linear matrix solver, 115-117linearization, 92metal forming and, 5-6nodal point velocities, 91notation, 101numerical integrations, 111-114penalty function method, 110plane-strain problems, 131-133porous metals, 246-249procedures, 90-94rectangular element family, 97-100,

105-106rezoning, 126-129rigid region, treatment of, 87ring compression, 160-163sheet-metal forming of general

shapes, 209-210stiffness equations, 92three-dimensional brick element,

100-101, 106-107three-dimensional problems,

276-278time-increment, 123-125triangular element family, 95-97,

104-105, 113variational approach, 83-84, 86

Flange drawing, 194-195, 197Flange-hub shapes, axisymmetric

forging of, 253-256, 257Flashless forging, 170, 171, 172Flat-face die, 36, 38Flat-tool compression, 314

Flat-tool forging, 293, 294, 295Flow formulation, 4-5Flow rule, 63-66, 245-246

plasticity and, 63-66porous metals, 245-246

Flow stress, 28-30, 66Folding, 153Forging. See also Axisymmetric

forging; Closed-die forgingclassification of, 151disk, 311, 312flashless, 170, 171, 172flat-tool, 293, 294, 295forward extrusion, 13hot-die disk, 237, 238, 239isothermal, 237, 238, 239. See also

Axisymmetric isothermal forgingopen-die, 15orbital, 16process design objective, 35process design steps, 35, 36radial, 314titanium alloy Ti6242, 234-237

Forward extrusion forging, 13Friction, 10, 30-33.

at tool-workpiece interface,163-165, 166

Friction coefficient, 206Friction hill, 138-139Friction shear stress, 32Frictional stress, 119-120

Gaussian elimination, 116-117Gaussian quadrature formula,

111-113Gear blank forging, 172Generalized stress, 66Geometry updating, 123-125

H cross sections, preforms for, 309Hasek method, 50Heading of cylindrical bars, 153-159Heat transfer

in porous metals, 259-262in thermo-viscoplastic analysis,

225-227Hexahedral element, 276Hill's general method, 3, 78-83, 88Hobbing, 14Hooke's law, 63

374 Index

Hot compression, of steel cylinder,231-234

Hot-die disk forging, 237, 238, 239Hot extrusion, 20

of rods and shapes, 36-39Hot forming, 28-29, 34-35

preform design, 315-318Hot nosing, 237-240Hot pressing, under plane-strain

compression, 262-266Huber-Mises criterion, 59Hydrodynamic conditions, 31

Impression, 35-36, 37Infinitesimal deformation theory,

55-56Infinitesimal plastic strain, 63-64In-plane deformation processes,

192-195Integration points ,111Ironing, 21Ironing press, 46-47Isoparametric elements, 96, 98, 99Isothermal forging, 237, 238, 239. See

also Axisymmetric isothermalforging

Jauman derivative, 324

Kirchhoff stress, definition of, 324

Lagrange multiplier method, 110Lagrangian description, 55-57Lagrangian family, 98Lagrangian strain, 208Levy-Mises equations, 66Limiting drawing ratio, 205Lubrication, basic types of, 30-31Lubricity, 32-33

Maximum plastic work principle,64-65

Maxwell-Heuber-Mises criterion, 59Mechanics of deformation, 10. See

also PlasticityMetal flow, 27

in non-steady-state upset forging, 27in rolling, 42in steady-state extrusion, 27

Metal forming and finite-elementmethod, 5-6

Metal-forming processes, 8-24. Seealso specific processes

backward extrusion forging, 14brake-bending, 21classification of, 11-12closed-die forging, 9, 12, 13closed-die forging, without flash, 12coining operation, 13deep drawing, 23deformation mechanics, 10. See

also Plasticitydescription of, 11-12drawing, 20extrusion process, 19, 20forward extrusion forging, 13friction, 10hobbing, 14ironing, 21material variables, 9-10nosing, 15open-die forging, 15orbital forging, 16product properties, 10radial forging of shaft, 16ring rolling, 18roll bending, 22roll forming, 22rolling, 17rotary tube piercing, 18rubber-diaphragm hydroforming, 24rubber-pad forming, 24shear forming from plate, 19sheet-metal, 11, 12spinning, 23systems approach in, 8-10tooling and equipment, 10upsetting with flat-heading tool, 17

Metal powders. See Porous metalsMethods of analysis, 73-88

equations for, 73finite-element method, 83-87, 88.

See also Finite-element methodHill's general method, 3, 78-83, 88for non-steady-state processes,

175-176for steady-state processes, 174-176upper-bound method, 3, 74-78, 88von Mises criterion, 59

Index 375

Modeling, process, 1-3Multipass bar drawing and extrusion,

183-186

Nakajima method, 50Necking, 47Newton-Raphson method, 92, 93, 121Nominal stress, 324Nonquadratic yield criterion, 217-220Non-steady-state flow, 27Nosing, 15

hot, 237-240shell, 301-305

Open-die forging, 15Orbital forging, 16

Parent element, 97Penalty function method, 110Piola-Kirchhoff stress, 208, 324Piola-Kirchhoff stress tensor, 57Plane plastic flow, 131. See also

Plane-strain problemsPlane-strain compression, hot pressing

under, 262-266Plane-strain problems, 131-149

closed-die forging with flash,133-136

finite-element formulation, 131-133plate bending, 141-147sheet rolling, 137-141side pressing, 148-149

Plane-strain rolling, 305-309Plastic anisotropy, 190-191Plastic strain-rate, 63-64Plasticity, 54-70

effective strain, 66-68effective stress, 66-68equilibrium equations, 61-62extremum principles, 68-70, 71-72flow rule, 63-66infinitesimal deformation theory,

55-56Lagrangian description, 55—57maximum plastic work principle,

64-65plastic potential, 63-66strain, 54-58strain-hardening, 66-68strain-rate, 54-58

stress, 54-58virtual work-rate principle, 62-63viscoplasticity, 70-72yield criterion, 58-61

Plastometers, 29Plate

bending of, 141-147, 327-331rolling of, 41-45

Porous metals, 244-270axisymmetric forging of flange-hub

shapes, 253-256, 257axisymmetric forging of pulley

blank, 256-259compaction, 266-270discretization, 247-248finite-element modeling, 246-249flow rules, 245-246fully dense materials, 248heat transfer in, 259-262hot pressing under plane-strain

condition, 262-266numerical procedures, 246-249simple compression, 249-253updating relative density, 248volume integration, 248-249yield criterion, 245-246

Powder compaction, 266-270Powder forming, 244. See also Porous

metalsPrandtl-Reuss equations, 65-66Preform design, 298-320

axially symmetric forging, 309-315backward tracing, 298-301,

301-305, 305-308, 311-312,316-317, 318-320. See alsoBackward tracing

definition of, 298H cross sections, 309hot forming, 315-318method for design, 298-301plane-strain rolling, 305-309shell nosing, 301-305

Pressing, hot, 262-266Process modeling, 1-3Product properties, 10Pseudoconcentrations, 334Pulley blank, axisymmetric forging of,

256-259Punch loads/displacements, 144-145,

146

376 Index

Punch-stretching, 201-206

Radial forging, 314of shaft, 16

Rectangular block compression,278-281

Rectangular element family, 97-100strain-rate matrix, 105-106

Reliability of simulation, 333-334Rezoning, 126-129Rigid-plastic formulation, 327-334

disadvantages of, 330solid formulation vs., 327-334

Ring compression, 159-165, 331-332,333

axisymmetric isothermal forging,159-163

friction and, 163-165solid formulation, 331-332, 333

Ring rolling, 18Ring test, 33Rods, 36-39, 45-47

drawing of, 45-47hot extrusion of, 36-39

Roll bending, 22Roll forming, 22Rolling, 17

analysis of spread in, 289-292metal flow in, 42plane-strain, 305-309plate, 17, 43-44ring, 18of shapes, 44-45sheet, 17, 137-141strip, 42-43, 45

Rotary tube piercing, 18Rubber-diaphragm hydroforming, 24Rubber-pad forming, 24

Shaft, radial forging of, 16Shapes

drawing of, 45-47flange-hub, axisymmetric forging of,

253-256, 257hot extrusion of, 36-39rolling of, 44-45sheet-metal forming of, 206-210

Shear forming from plate, 19Shear strength of deforming material,

32

Shear stress criterion, 59Sheet rolling, 137-141Sheet-metal forming, 47-52, 189-220

axisymmetric out-of-planedeformation, 195-201

axisymmetric punch-stretching,201-206

bore expanding, 194-195, 196classification of processes, 11, 12deep-drawing processes, 201-206flange drawing, 194-195, 197general shapes, 206-210in-plane deformation processes,

192-195nonquadratic yield criterion,

217-220plastic anisotropy, 190-191square-cup drawing, 210-217

Shell nosing, 301-305Side pressing, 148-149Simple compression, 249-253,

364-369of cylinder, 364-369of porous metals, 249-253

Simple plastic incrementaldeformation. See SPID

Simpson's formula, 114Simulation of forming processes,

332-334Slab method, 1Slip-line field method, 1-3, 138Small-strain solid formulation,

321-323Smooth entry dies, 38Solid formulation, 4-5, 321-335

large deformation: incrementalform, 326-327

large deformation: rate form,323-326

plate bending, 327-334rigid-plastic (flow) solutions vs.,

327-334ring compression, 331-332, 333small-strain, 321-323three-dimensional analysis with, 334

SPID (simple plastic incrementaldeformation), 338-369

description of major variables,342-343

example solution, 364-369

Index 377

input and output files, 340input preparations, 340-342program listings, 343-364program structure, 339-340simulation conditions, 364-369

Spike forging, 163, 165Spinning, 23Springback angle, 330Square-cup drawing, 210-217Square-ring compression, 284-287Steady-state flow, 27Steady-state processes, 174-187

applications to process design,186-187

bar drawing, 178-183bar drawing, multipass, 183-186bar extrusion, 176-178bar extrusion, multipass, 183-184method of analysis, 174-176

Steel cylinder, compression of,229-234

Stiffness equations, 120assembled, 116-117elemental, 108-110

Strain, 54-58effective, 66-68

Strain-hardening, 66-68Strain-rate, 54-58

in Cartesian coordinate system, 101Strain-rate matrix, element, 101-107Strain-rate vector, 103, 153Stress, 54-58

effective, 66-68Stress measures, 324Strip, rolling of, 42-43, 45Swift test, 205

Temperature in metal forming, 28, 29,33-35. See also Thermo-viscoplastic analysis

Thermal conductivity. See Heattransfer

Thermo-viscoplastic analysis, 222-242applications of, 229-240compression of cylinder, 223-225,

229-234computational procedures for,

227-229forging titanium alloy Ti6242,

234-237

heat transfer analysis, 225-227hot nosing, 237-240hot-die disk forging, 237, 238, 239isothermal forging, 237, 238, 239

Three-dimensional brick element,100-101

strain-rate matrix, 106-107Three-dimensional problems, 275-296

block compressions, 278-284brick element, 100-101, 106-107finite-element formulation, 276-278flat-tool forging, 293, 294, 295hexahedral element, 276rolling, analysis of spread in,

289-292simplified elements, 287-289solid formulation, 334square-ring compression, 284-287

Time increment, 123-125Titanium alloy Ti6242, 234-237Tooling, 10Tresca criterion (shear stress

criterion), 59Triangular elements, 95-97, 104-105,

113strain-rate matrix of, 104-105

True stress, 56, 57, 58Tubes, drawing of, 45-47

Upper-bound method, 3, 74-78, 88Upsetting, with flat-heading tool, 17

Virtual work-rate principle, 62-63Viscoplasticity, 70-72. See also

Thermo-viscoplastic analysisdefinition of, 222

Visioplasticity method, 3Volume strain-rate matrix, 107von Mises criterion, 59

Wedge-shaped blocks, 281-284Wheels, automobile, 51-52Wire, drawing of, 45-47Wrinkling, 47, 50

Yield criterionnonquadratic, 217-220plasticity, 58-61porous metals, 245-246

metal forming and the finite element method

Education