analysis of guillotining and slitting · analysis of guillotining and slitting, finite element...

Analysis of

Guillotining and SlittingFinite Element Simulations

Harm Wisselink

De promotie commissie is als volgt samengesteld:

Voorzitter en secretaris:Prof.dr.ir. H.J. Grootenboer Universiteit Twente (WB)Promotor:Prof.dr.ir. J. Huetink Universiteit Twente (WB)Leden:Prof.dr.ir. F.P.T. Baaijens Technische Universiteit EindhovenDr.ir. P.J. Bolt TNO IndustrieDr. R.M.J. van Damme Universiteit Twente (TW)Prof.dr.ir. F.J.A.M. van Houten Universiteit Twente (WB)Ir. W.E ten Napel Universiteit Twente (WB)Prof.dr.ir. J.J.W. van der Vegt Universiteit Twente (TW)

This research is financed by the Dutch IOP (Innovative Research Programme) as projectC94.704.UT.WB: ”EEM model van het knippen en slitten”.

Title:Analysis of Guillotining and Slitting, Finite Element SimulationsPh.D-Thesis, University of Twente, The NetherlandsJanuary 2000Author: H.H.WisselinkISBN: 90365.13995Subject headings: Guillotining, Slitting, Finite Element Method

Copyright c�

2000 by H.H.Wisselink, Enschede, the NetherlandsPrinted by Ponsen & Looijen, Wageningen

ANALYSIS OFGUILLOTINING AND SLITTING,

FINITE ELEMENT SIMULATIONS

PROEFSCHRIFT

ter verkrijging vande graad van doctor aan de Universiteit Twente,

op gezag van de rector Magnificus,prof.dr. F.A. van Vught,

volgens besluit van het College voor Promotiesin het openbaar te verdedigen

op donderdag 13 januari 2000 te 13:15 uur.

door

Hendrik Herman Wisselinkgeboren op 22 juni 1969

te Wisch

Dit proefschrift is goedgekeurd door de promotorProf.dr.ir. J. Huetink

Summary

Guillotining and slitting are two sheet metal cutting processes, which produce long straightcuts and can be seen as stationary processes. The analysis of these two processes is thesubject of this thesis. This research project is part of a cluster of IOP projects on sheetmetal cutting, originating from industrial requests for more fundamental knowledge, assheet metal cutting is still mainly based on experience and trial and error. The objective isto develop models of guillotining and slitting to increase the insight and to contribute to thequality and productivity of these shearing processes.

The finite element method is used to model guillotining and slitting. This model should beable to describe the phenomena occurring during shearing, such as the large elasto-plasticdeformations, the contact between the sheet and the knives and the ductile fracture of thesheet. The simulations are carried out with the finite element code DiekA, developed at theUniversity of Twente. DiekA contains an Arbitrary Lagrangian Eulerian (ALE) formula-tion, which combines the properties of the Lagrangian and Eulerian formulations. The ALEmethod is used for all the simulations in this thesis, and therefore large part of the thesisis dedicated to ALE formulations, mesh management procedures and the transfer of statevariables. Furthermore, attention is paid to material models for the description of plasticdeformation and ductile fracture, and to the applied contact model.

Two different types of finite element models of shearing processes are developed.

2D transient models of orthogonal shearingA plane strain model of orthogonal shearingis developed. This model is a 2D approximation of guillotining and slitting in whichall 3D effects are neglected. The ALE method is used in these simulations to avoidtoo much element distortion. The simulations can be continued to knife penetrationsof about half the sheet thickness. For larger punch penetrations or the initiation andpropagation of cracks the ALE method alone no longer suffices and remeshing be-comes necessary.

3D stationary models of guillotining and slitting Guillotining and slitting can be mod-elled as a 3D flow problem, because these processes can be seen as stationary. Thesteady state of guillotining and slitting is calculated from an initial estimation of the

vi

steady state geometry including a crack front. This initial geometry evolves during thesimulation, in which the material flows through the finite element mesh, to a steadystate geometry. The ALE method is now used to follow the free surfaces during thecomputation. The correctness of the assumed initial crack front, which position is notadapted during the simulation, is assessed by means of a fracture criterion.

Results of simulations with the shearing models described in this thesis are presented. Acomparison is made between the results of the 2D and 3D models. The results of guillotiningsimulations are compared with some experimental data of guillotining. Furthermore, a smallparameter study is performed.The conclusion is that the developed shearing models are able to qualitatively describe theobserved phenomena correctly. Hence it can be stated that these models contribute to abetter understanding of guillotining and slitting.

Contents

Summary v

1 Introduction 11.1 Guillotining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Slitting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Objectives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Outline of this thesis . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Continuum mechanics 112.1 Kinematics . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Deformation and strain . .. . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Stress and stress rate . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Conservation laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Constitutive relations 193.1 Introduction . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Material behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 Tensile test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.2 Ductile fracture . .. . . . . . . . . . . . . . . . . . . . . . . . . . 213.2.3 High-speed shearing. . . . . . . . . . . . . . . . . . . . . . . . . 213.2.4 Hydrostatic pressure. . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Material models .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3.1 Elasto-plastic material model .. . . . . . . . . . . . . . . . . . . . 233.3.2 Damage models . .. . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Finite element formulation 334.1 Introduction . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Types of FEM formulation. . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2.1 Lagrangian formulation . . .. . . . . . . . . . . . . . . . . . . . 33

viii Contents

4.2.2 Lagrangian formulation with remeshing. . . . . . . . . . . . . . . 344.2.3 Eulerian formulation. . . . . . . . . . . . . . . . . . . . . . . . . 354.2.4 ALE formulation .. . . . . . . . . . . . . . . . . . . . . . . . . . 354.2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.3 Problem definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3.1 Strong formulation. . . . . . . . . . . . . . . . . . . . . . . . . . 384.3.2 Weak formulation .. . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.4 ALE formulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.4.1 Coupled ALE . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 414.4.2 Decoupled ALE .. . . . . . . . . . . . . . . . . . . . . . . . . . 444.4.3 Semi-coupled ALE. . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.5 Material increment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.6 Convective increment . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.6.1 Weighed local and global smoothing . .. . . . . . . . . . . . . . . 484.7 Solving the equations . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.7.1 Steady state solution. . . . . . . . . . . . . . . . . . . . . . . . . 564.7.2 Proportional increment. . . . . . . . . . . . . . . . . . . . . . . . 564.7.3 Iterative solvers . .. . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Contact and friction 595.1 Introduction . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.1.1 Kinematics and stresses . . .. . . . . . . . . . . . . . . . . . . . 605.2 Constitutive relations for contact . . .. . . . . . . . . . . . . . . . . . . . 605.3 Weak form and FE discretisation . . .. . . . . . . . . . . . . . . . . . . . 635.4 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.5 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.5.1 Tool description .. . . . . . . . . . . . . . . . . . . . . . . . . . 655.5.2 Contact search . .. . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.6 Contact and ALE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.7 Concluding remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6 Shearing models 676.1 Introduction . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.2 Orthogonal shearing model. . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.2.1 2D quadrilateral mesh generation . . .. . . . . . . . . . . . . . . 686.2.2 2D mesh management. . . . . . . . . . . . . . . . . . . . . . . . 696.2.3 Plane strain shearing model .. . . . . . . . . . . . . . . . . . . . 74

6.3 3D models of guillotining and slitting. . . . . . . . . . . . . . . . . . . . 776.3.1 3D stationary crack growth . .. . . . . . . . . . . . . . . . . . . . 786.3.2 3D hexahedral mesh generation. . . . . . . . . . . . . . . . . . . 796.3.3 3D mesh management. . . . . . . . . . . . . . . . . . . . . . . . 80

Contents ix

6.3.4 3D boundary conditions and contact . .. . . . . . . . . . . . . . . 846.3.5 Preprocessing . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.4 Conclusions . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7 Simulation results 877.1 Comparison 2D and 3D models. . . . . . . . . . . . . . . . . . . . . . . . 877.2 Guillotining experiment 1 .. . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.2.1 Experimental results. . . . . . . . . . . . . . . . . . . . . . . . . 997.2.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . 1027.2.3 Comparison between experiment and simulation .. . . . . . . . . 103

7.3 Guillotining experiment 2 .. . . . . . . . . . . . . . . . . . . . . . . . . . 1067.3.1 Experimental results. . . . . . . . . . . . . . . . . . . . . . . . . 1067.3.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . 1097.3.3 Variations of the simulation .. . . . . . . . . . . . . . . . . . . . 113

7.4 Results of slitting simulations. . . . . . . . . . . . . . . . . . . . . . . . . 1187.5 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

8 Conclusions and recommendations 123

A Shearing geometry 125A.1 Slitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125A.2 Guillotining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

B The shape of the crack front 129

Bibliography 134

List of symbols 143

Samenvatting 147

Dankwoord 149

Chapter 1

Introduction

Shearing, in many variants, is a very frequently used separation process in the sheet metalforming industry. It is combined with other forming processes such as rolling, deep-drawingor bending, in order to manufacture a sheet metal product. Mechanical shearing is stillthe most economical process for mass production, although other cutting techniques as forinstance laser beam cutting are available.Because nowadays industry requires a higher output together with a higher and more con-stant quality, there is a need for a more fundamental knowledge of the shearing process.Therefore, originating from industrial requests, ”IOP-metalen” (the Innovative ResearchProgramme financed by the Dutch Ministry of Economic Affairs) started a cluster of projectson sheet metal separation techniques, one of which led to this thesis.

The research in this project is restricted to two sheet metal shearing processes, which pro-duce long straight cuts : guillotining and slitting. Contrary to punching these processes canbe seen as stationary processes. The guillotining and slitting processes are discussed in thenext sections, followed by the objectives of this research, a short overview of other researchin this field and the outline of this thesis.

1.1 Guillotining

The term ”guillotining” describes shearing with two straight knives, of which the movableupper knife is inclined, with a rake angleα � 0, compared to the fixed lower knife. (Fig-ure 1.1). The sheet is clamped on the lower knife by a down-holder. When the inclinedupper knife is forced downwards, with a small clearance (1%-25% of the sheet thickness)between the knives, the sheet is cut progressively from one end to the other. Ignoring thestart and the end of the cut, this can be seen as a stationary process. This becomes clearwhen the dotted box in Figure 1.1 is followed during the process.During the downward motion of the upper knife several phases can be distinguished, which

2 Introduction

A B C

A B C D

D

clearance

sheet

lower knife

down-holder upper knifeα

Figure 1.1: Guillotining.

are illustrated in pictures A to D in Figure 1.2. These phases are found next to each otherat the same time in guillotining (lines A to D in Figure 1.1). The assumption of a stationaryprocess becomes invalid in the case of orthogonal shearing (shearing with parallel knives,α=0), when the sheet is cut all at once and the phases A to D are sequentially passed. Thementioned phases of shearing can be recognised in the schematic view of the shape of asheared edge in Figure 1.3 and are explained next.

1.1 Guillotining 3

lower knife

upper knifedown-holder

D

B

C

horizontal clearance

A

Figure 1.2: Different phases in shearing.

thickness

fractured

sheared/burnished

tilted surface

burr

draw-in/roll-over

clearance

Figure 1.3: Sheared edge.

A. A bending moment is applied to the sheet as the upper knife moves down. Bendingof the sheet is prevented on one side by a down-holder. On the other side (under themoving upper knife) it is more difficult to prevent bending of the sheet and thereforesuch a device is generally omitted (compare left and right sides in Figure 1.2). Thesheet is first elastically deformed, and with further downward displacement of theknife the deformation becomes plastic.

4 Introduction

B. The complete cross-section is plastically deformed. The material is drawn into theclearance area, forming a rounded draw-in (or roll-over), growing with increasingknife penetration. At larger knife penetrations the draw-in is accompanied by asmooth, burnished part, making contact with the knife. Due to bending, a tilted sur-face develops under the knife on the top face of the cut-off sheet.

C. After some amount of plastic deformation cracks initiate at the edges of the lower orupper knife. The plastic deformation is now combined with the propagation of cracksin the material.

D. The sheet is completely separated as the cracks eventually meet, which generallytakes place before the knife has penetrated the complete thickness of the sheet. Be-sides a rough fractured part, burrs and slivers can be formed by the separation. Thesheet will springback elastically due to the relaxation of internal stresses, when thetools are removed. Residual stresses can remain in the cut sheet after elastic spring-back.

The shearing deformation in guillotining is accompanied by a bending deformation, as thesheet has to conform to the inclination of the knife. Both deformations influence each other,leading to the development of shape defects (Figure 1.4), which increase with increasingrake angle and decreasing width of the cut-off strip.

Figure 1.4: Shape defects in guillotining.

A typical force-displacement curve of the shearing process is given in Figure 1.5 for orthog-onal shearing and guillotining. The force in orthogonal shearing (α � 0) increases fast asthe sheet is elastically deformed, and keeps increasing (although more slowly) due to strainhardening when the sheet becomes plastic, until the force reaches a maximum (Fmax). Atthis point the force increase due to strain hardening is in equilibrium with the cross-sectionalarea reduction. This is a geometrical instability similar to necking in a tensile test. Usu-ally after the maximum, real material softening, due to the development of internal damageor temperature increase by plastic dissipation, will lead to localisation and crack initiation.The force will drop rapidly due to the propagating cracks that finally separate the parts.

1.2 Slitting 5

100%

D

C

A

B

Displacement [% of sheet thickness]

Total Force

guillotining α � 0

orthogonal α � 0Fmax

Fstat

Figure 1.5: Typical force-displacement curves for orthogonal shearing and guillotining.

The maximum force in orthogonal shearing is proportional to the length of the cut, whichleads to very high forces for long cuts. In guillotining (α � 0) only a part of the sheet iscut at one time. Therefore, the stationary cutting forceFstat is independent of the length ofthe cut. For a given material the cutting force depends mainly on the rake angle and sheetthickness. An increasing rake angle leads to lower forcesFstat over larger displacements.The objective of a guillotining process is to cut the sheet with a certain edge quality (burr)and flatness with a limited force. Therefore, in practice the rake angleα varies between 0� 5�

and 3� as a compromise between the quality of the sheet and the required force.

1.2 Slitting

Slitting is a sheet metal cutting process with circular knives, which is used to split widecoiled sheet metal into narrower widths or for edge trimming of rolled sheet. The slittingprocess is schematically shown in Figure 1.6. The slitter knives are mounted on two arborstogether with spacers . The spacers determine the width of the cut and the horizontal clear-ance. Sometimes also a vertical clearance (overlap, Appendix A) is used, e.g. to correctfor bending of the arbors. Rubber stripper rings are used to eject the strip from betweenthe knives and to support the strip during slitting. Just like guillotining, slitting combinesshearing with a bending deformation and the stages A, B, C and D from Figure 1.2 arefound next to each other at the same time.A slitting line consists of an uncoiler, a slitter and a recoiler. The uncoiler, slitter and recoilerare driven by separate motors. Depending on the way the uncoiler, slitter and recoiler aredriven, different modes to drive the material through the slitter knives can be distinguished(Madachy 1980). Which mode is chosen depends on the material, the sheet thickness and

6 Introduction

BC

DA

spacers

stripper rings

knife

arbor

horizontal clearance

clearancevertical

Figure 1.6: Slitting process.

the number of slits. These modes are depicted in Figure 1.7.

� Straight mode slittingIn this mode the driven recoiler pulls the material from the uncoiler through the slit-ters. It is calledpull-through slitting, when the uncoiler and slitter are driven only tofeed the material to the recoiler. Indriven slittingthe slitter is also driven during theprocess. The motors must now be synchronised to maintain a constant speed of thematerial as it moves through the line. The advantages of driven slitting are the abilityto slit thin sheets and an improved edge quality for all thicknesses.

� Free loop mode slittingIn free loop slitting, the material is allowed to form a free loop between the slitter andthe recoiler. A tensioning device in front of the recoiler is needed now, to producewell wound coils. This method makes it possible to process poorly shaped coils.

Shape defects (length variations such as buckles, twists and thickness variation across thewidth) and internal stresses in the incoming coil influence the result of the slitting process.Therefore, a leveller or a temper mill can be added to the slitting line, to correct defects inthe incoming coil, which ensures a better quality of the slitted strips.

1.3 Objectives

As is seen in the previous sections, there are many parameters that influence or can be usedto influence the result of the shearing operations. The quality of the cutted sheet dependson the combination of:

1.3 Objectives 7

recoiler

slitter

uncoiler

(a) Straight mode

uncoiler

tension device

recoiler

loop

slitter

leveling/temper mill

(b) Free loop mode

Figure 1.7: Slitting modes.

Material properties The formability and ductility of the sheet are important aspects in theshearing process;

Incoming material conditions Shape defects and internal stresses in the incoming mate-rial, which are not always known, can have a large influence on the process:

Tool setup The tooling can be adjusted to reach the optimal result for a given material. Themost important parameters are the horizontal and vertical clearances, the clamping ofthe sheet, rake angle or slitter diameter, the cutting speed and the sharpness of theknives. Due to wear, sharp knives will become blunt, leading to poor results, andparticularly larger burrs.

The knowledge of the process is mainly empirical and it is not yet fully understood how theaforementioned parameters affect the shearing process. Therefore it is difficult to controlthe guillotining and in particular the slitting process (Pennington 1998).

In practice the right setup for the knives is mostly found by trial and error combined withexperience. This can result in lengthy procedures to find a good setup in case of “new”materials (high strength steels or aluminium) or closer tolerances. Therefore models areneeded to shorten the time of trial and error, leading to better quality and higher productiv-

8 Introduction

ity.

The objective of this project is to build a (finite element) model for stationary shearingprocesses. With this model it should be possible to investigate the influence of some impor-tant parameters on guillotining and slitting. The results of the model should provide moreinsight into these process, and hence a better product quality and process control.

1.4 Literature review

General directives for setting up shearing processes can be found in handbooks such as(Spur and St¨oferle 1994), (Iliescu 1990) (Lange 1990) or (VDI 1994). These books presentrules of thumb that are based on experimental data ( e.g. from (Chang and Swift 1950) or(Johnson and Slater 1967)). Most of the literature found relates to punching, but a numberof articles deal mainly with guillotining (Sperling 1968), (Atkins 1981), (Atkins 1990) orslitting (Reilly 1971) and (FMA 1999).A number of analytical formulae are derived for the shearing process by (Atkins 1980),(Sauer 1980) and (Zhou and Wierzbicki 1996). These analytical models are used to es-timate the forces during shearing. More detailed information cannot be expected, as thedeformation pattern in shearing is too complex for analytical models.

A more powerful technique is the Finite Element Method (FEM), which is used to modelall kinds of forming process. During the last few years FEM has also often been used tomodel shearing processes. In contrast to most forming processes, where fracture shouldbe avoided, ductile fracture is an essential part of the shearing process. The lack of robustnumerical algorithms for remeshing, contact and damage, required to model the large lo-calised deformation and fracture in shearing, has obstructed earlier attempts to use FEM forshearing simulations. In the literature only 2D simulations, either plane strain or axisym-metric, of shearing processes are reported. To the authors’ knowledge no 3D simulations ofguillotining or slitting have been performed up to now. An overview of recent publicationsof shearing simulations is given in Table 1.1.The papers listed in Table 1.1 emphasise different aspects of the shearing process. Onlya limited number of FEM models are able to describe the complete shearing process, in-cluding the complete separation of the parts by ductile fracture. Most of the ideas usedfor the 2D simulation of shearing can be extended to 3D and applied to the simulation ofguillotining and slitting.Another forming process with large deformations and fracture is machining. Dependingon whether continuous or segmented chips are formed, machining can be seen as a station-ary process, like guillotining and slitting. Therefore use can be made of the work done inthe modelling of machining. A very comprehensive survey of all the literature about FEMmodelling of machining is given by (Mackerle 1999). Again, the majority of the models

1.5 Outline of this thesis 9

mentioned in the literature is 2D. (Marusich and Ortiz 1995) and (Ceretti et al. 1999) pro-vide some good 2D examples of the simulation of machining with remeshing and fracture.Another interesting piece of work is the 3D simulation of oblique cutting of (Pantal´e et al.1998) using the ALE method, which shows some similarities with guillotining.

Shape aberrations due to shearing processes, especially the sheet flatness, are investigatedby (Jimma and Sekine 1990), (Bolt and Sillekens 1998) and (Moesdijk 1999). They alsoexamined the effect of shearing operations on subsequent forming processes.

1.5 Outline of this thesis

The finite element code DiekA, developed by (Hu´etink 1986) and co-workers, is used tomodel guillotining and slitting. This code has been applied during the last fifteen years todifferent forming processes such as rolling (Lugt 1988), deep-drawing (Vreede 1992)(Car-leer 1997) and aluminium extrusion (Mooi 1996). The parts of the FEM code needed for thesimulations of guillotining and slitting are explained in this thesis, some extensively, othersbriefly.

First, some basic definitions and equations needed in the rest of the thesis are treatedin Chapter 2. The constitutive relations that describe the behaviour of a material during(un)loading are given in Chapter 3. Models are presented for elasto-plastic deformation andductile fracture. The finite element method, the numerical tool to solve the relations derivedin the previous chapters is elaborated in Chapter 4. A large part of this chapter deals withthe ALE formulation. The contact and friction between the sheet and the knives is handledin Chapter 5. Details of the finite element model of guillotining and slitting are treated inChapter 6. The subjects discussed here are preprocessing, meshing and mesh management,the boundary- and inflow conditions and the incorporation of fracture in the finite elementmodel. In Chapter 7 the results of the 3D stationary models of guillotining and slitting arecompared with the outcome of plane strain (2D) shearing models. Furthermore, the guillo-tining model is validated with experimental data. The influence of a range of parameters onthe guillotining and slitting processes is illustrated by several examples. The conclusions ofthis research and some recommendations for further research are given in the last chapter.

10Introduction

Reference Process axisymm. /

plane strain

FEM code Remeshing Crack propaga-

tion

Material model

(Popat et al. 1989) Blanking pl. strain Own code No No El. pl + Fract. crit.

(Chen et al. 1999) Fine blanking axisymm. Own code No No El. plastic

(Klocke et al. 1999) Fine blanking axisymm. DEFORM Yes El. elimination Fract. crit.

(Pyttel and Hoogen 1998) Shearing axisymm. MARC/Autof. Yes No El. plastic

(Behrens and Westhoff 1999) Blanking pl. strain MARC/Autof. Yes El. elimination El. Pl.+ Fract. crit.

(Golovashchenko 1999) Trimming pl. strain Own code No El. elimination El. pl. + Fract. crit.

(Taupin et al. 1996) Blanking axisymm. DEFORM Yes El. elimination Fract. crit.

(Quinlan and Monaghan 1998) Blanking axisymm. DEFORM Yes El. elimination Rigid pl. + Fract. crit.

(Ko et al. 1997) Shearing pl. strain DEFORM Yes El. elimination Therm. viscoplastic +

Fract. crit

(Brokken 1999) Blanking axisymm. MARC + user Yes Delete and fill El.pl. + Fract. crit. or

(Stegeman et al. 1999) subroutines Therm-viscopl.

(Goijaerts 1999)

(Faura et al. 1998) Blanking pl. strain ANSYS No No El. pl. + Fract. crit.

(Abdali et al. 1996) Cutting pl. strain Own code No No El. pl. with damage

(Bezzina and Saanouni 1997) Cutting axisymm. SIC No No El. pl. with damage

(Hambli and Potiron 1999) Punching axisymm. ABAQUS + user

routines

No El. elimination El. pl. with damage

(Post and Voncken 1996) Blanking axisymm. MARC Yes No Gurson

(Morancay et al. 1997) Punching axisymm. SIC Yes No El. plastic + Gurson

(Samuel 1998) Blanking axisymm. MARC Yes No Gurson

(Komori 1999) Shearing axisymm. Own code Yes Nodal Release Rigid-pl + Gurson

(Yoshida et al. 1999) Shearing axisymm. RIPAD-2D Yes No Rigid pl. + Gurson

Table 1.1: Overview of FEM simulation of shearing processes with some characteristics.

Chapter 2

Continuum mechanics

The deformation behaviour of the sheet during shearing is modelled with the continuummechanics theory. This is a macroscopic approach in which it is assumed that the materialbehaviour can be described using continuous functions. Continuum mechanics can be splitinto two parts:

� General principles, assumptions and consequences applicable for all media.

� Constitutive equations, describing the behaviour of a particular material.

The general principles cover the kinematics and conservation laws, irrespective of the actualmaterial. The material behaviour is taken into account by the constitutive relations, in whichmicrostructural effects can be incorporated.A brief description of the applied continuum mechanics is presented, based on (Malvern1969), (Huetink 1986) and (Besseling and Van der Giessen 1994). The general principlesare treated in this chapter, the constitutive relations in Chapter 3.

2.1 Kinematics

The motion and deformation of a continuum can be described in different ways.

� The Lagrangian or material description, in which the state variables are written in thematerial coordinatesX.

� The Eulerian or spatial description, in which the state variables are written in thecurrent coordinatesx.

� The Referential description, in which the state variables are written in referentialcoordinatesχχχ.

12 Continuum mechanics

χχχVχχχ

Vx

VX

X � ΨΨΨ χχχ t �

X

xΨΨΨ � 1 Φ

x � Φ χχχ t �

Figure 2.1: Relations between the domains

In Figure 2.1 the three domains are drawn. One-to-one mappingsΦΦΦ andΨΨΨ exist betweenthe different domains and are functions of time. Therefore, a referential point correspondsto one material point and one spatial point, but not always to the same points. The referen-tial description is used for the derivation of the ALE method in Chapter 4. The Lagrangianand Eulerian coordinates are special cases of the referential coordinates, which are obtainedwhenχχχ � X or χχχ � x. The domains coincide at timet � 0.

The material displacementum is the difference between the current position and the initialposition of a particle. The referential displacementug is not related to the position of aparticle.

um � X � t � � x � X � t � � x � X � 0� (2.1)

ug � χχχ � t � � x � χχχ � t � � x � χχχ � 0� (2.2)

The material velocityvm and the referential velocityvg are defined as

vm � ∂x∂t � X � x (2.3)

vg � ∂x∂t � χχχ � �x (2.4)

2.2 Deformation and strain 13

dxx � dx

dX

X

x

X � dX

F

Figure 2.2: Deformation gradientF

The time derivative with constant material coordinateX is denoted as ˙� � and the time

derivative with constant referential coordinateχχχ as �� The material time derivative of an arbitrary quantityξ depends on the coordinates in whichit is written.

ddt

ξ � X � t � � ∂∂t

ξ � X � t � � X � ξ (2.5)

ddt

ξ � x � t � � ∂∂t

ξ � x � t � � x � vm � ��ξ � x � t � (2.6)

ddt

ξ � χχχ � t � � ∂∂t

ξ � χχχ � t � � χχχ � vc � ��ξ � χχχ � t � � �ξ � vc � ��

ξ � χχχ � t � (2.7)

A convective term appears whenξ is not written as a function of the material coordinates.The convective velocityvc is the difference between the material velocity and the grid ve-locity.

vc � vm � vg (2.8)

The spatial time derivative vanishes for stationary processes.

ddt

ξ � x � t � � vm � ��ξ � x � t � (2.9)

2.2 Deformation and strain

The deformation gradient tensorF maps the current configuration of a line elementdx onthe initial configurationdX of the same line element (Figure 2.2).


dx � F � dX F � ∂x∂X

(2.10)

dX � F 1 � dx F 1 � ∂X∂x

(2.11)

The determinant ofF is called the JacobianJ and is a measure of the volumetric deforma-tion.

J � det� F � (2.12)

F can be decomposed into an orthogonal and a symmetric part:

F � R � U � V � R (2.13)

with:

R rotation tensor ( orthogonal,R 1 � RT)U right stretch tensorV left stretch tensor (U,V symmetric, positive definite)

From this, the right Cauchy-Green tensorC can be defined which is invariant under rotationof the frame of reference, and the objective left Cauchy-Green tensorB.

C � FT � F � U2 (2.14)

B � F � FT � V2 (2.15)

The relative velocity of two particles on a line element is:

dvm � ∂vm

∂x � dx � Ldx (2.16)

L � vm !�(2.17)

whereL is the spatial gradient of the velocity.L is decomposed into the symmetric rate ofdeformation tensorD and the skew-symmetric spin tensorW.

L � D � W (2.18)

D � 12 � vm !� � ��

vm � � 12 � L � LT � (2.19)

W � 12 � vm !� � ��

vm � � 12 � L � LT � (2.20)

F can be expressed inL andF

F � L � F˙� F 1 � � � F 1 � L (2.21)

2.3 Stress and stress rate 15

Strain is defined as the change in the squared length of the material line elementdX fromthe initial to the current statedx.

#dx

# 2 � #dX

# 2 � 2dX � E � dX � 2dx � e � dx (2.22)

The Green-Lagrange strain tensorE is expressed in material coordinates and the Euler-Almansi strain tensore in the current coordinates.

E � 12 � FT � F � I � � 1

2 � C � I � (2.23)

e � 12 � I � � FT � F � 1 � � 1

2 � I � B 1 � (2.24)

E � FT � e � F & e � F T � E � F 1 (2.25)

The finite strainsE andecan be expressed in terms of the displacementsu, giving geomet-rically non-linear terms.

E � 12 � ��

0u � u !�0 � � ��

0u � � � u !�0 � � (2.26)

e � 12 � ��

u � u !� � � ��u � � � u !� � � (2.27)

The difference between the Lagrangian and Eulerian strain measures disappears for smalldisplacement gradients. In that caseE andeare linearised to the small strain tensorεεε.

εεε � 12 � ��

u � u !� � ( e ( E (2.28)

2.3 Stress and stress rate

Stress is the force per unit area. Just as the strain, it can be defined in the deformed orthe undeformed state. In Figure 2.3 the force vectordP is shown for an (internal) surfaceelementdS, with normaln.The Cauchy stressσσσ is defined by the actual force on the deformed surface. Hence it is alsocalled the true stress.

dP � � n � σσσ � dS (2.29)

It is an objective tensor, which is symmetric when distributed moments or couple stressesare absent.

σσσ � σσσT (2.30)

The Piola-Kirchhoff stress tensors are defined with respect to the reference configuration.The first Piola-Kirchhoff stress tensorT1 gives the actual force on the undeformed surface.

dP � � n0 � T1 � dS0 (2.31)


current configurationinitial configuration

dS0 dS

n

dPdP

n0

Figure 2.3: Force vectors and surfaces with normals in the initial and current configuration

This stress tensor is not symmetric, therefore this tensor is modified. Instead of the actualforcedP, the forcedP is used which is related to the actual force in the same way that thematerial vector is related to the spatial vector (Equation 2.11).

dP � F 1dP � � n0 � T � dS0 (2.32)

The second Piola-Kirchhoff stress tensorT is symmetric and invariant under rigid bodymotions. The relation between the Cauchy stress and the Piola-Kirchhoff stresses are

σσσ � J 1F � T1 & T1 � J � F 1 � σσσ (2.33)

σσσ � J 1F � T � FT & T � J � F 1 � σσσ � F T

For small deformations all stress measures coincide with the Cauchy stress.The stress tensor can be split into a deviatoric and hydrostatic part:

σσσ � s � p1 (2.34)

wheres is the deviatoric stress and the hydrostatic pressurep is given by:

p � � 13tr � σσσ � � � σh (2.35)

The rate of the Cauchy stress is not objective, and hence objective rates are defined. One of

them is the Jaumann rate�σσσ, which reads for the Cauchy stress:

�σσσ � σσσ � W � σσσ � σσσ � W (2.36)

2.4 Conservation laws 17

2.4 Conservation laws

With the kinematics and the stress and strain definitions from the preceeding sections thebasic equations of continuum mechanics can be formulated.

Theconservation of massis given by

ρ � ρ �� v � 0 (2.37)

which can also be expressed as:

ρρ

� � JJ

� � trD � � �� v (2.38)

The initial and current densities are termedρ0 andρ respectively. Equation 2.38 can besimplified to � ρJ � ρ0 � if the variables are written in material coordinates.

The mechanical equilibrium of an arbitrary body is described by theconservation of mo-mentum, given in the current configuration by:

σσσ � !� � ρb � ρv onV (2.39)

n � σσσ � t � 0 onΓ (2.40)

whereb is the body force per unit mass andt the surface traction per unit surface. Asrelatively slow processes without gravitational forces are considered, the inertia terms (ρv)and body forces can be neglected. Equation 2.39 then reduces to

σσσ � !� � 0 on V (2.41)

Hence we have three first order partial differential equations with six unknown stresses. Tosolve the equilibrium equations extra equations have to be found. These equations are theconstitutive relations, which give the relation between the stress and strain for a particularmaterial (Chapter 3).

Chapter 3

Constitutive relations

3.1 Introduction

Constitutive relations describe the relation between the stresses and strains. For shearingprocesses not only the elastic and plastic behaviour but also the ductility of the material areof interest. These phenomena are described qualitatively in Section 3.2. Material models,which take into account (some of) the observed phenomena are given in Section 3.3.

3.2 Material behaviour

3.2.1 Tensile test

The tensile test is used here to illustrate the behaviour of a material when it is deformed.Although the deformation pattern in shearing is different from the pattern in the tensiletest, it gives an indication of the material behaviour in shearing. The shape of the force-displacement diagram of a tensile test is similar to the force-displacement diagram of or-thogonal shearing (Figure 1.5).The following stress and strain definitions are used in a tensile test. The engineering stressT and straineare given by:

T �

FA0

; e � , l

l0

dllo

� l � l0l0

(3.1)

and the true stressσ and the natural or logarithmic strainε by:

σ � FA

; ε � , l

l0

dll

� ln � ll0

� � ln � 1 � e� (3.2)

wherel0 andl are the initial and actual lengths of the specimen respectively andA0 andAthe initial and current areas of a cross section.

20 Constitutive relations

A

O

BC

D

e

T

necking

fracture

(a) Engineering stress-strain curve

F’

εp εe

F

A

O

E

ε

σy0

BC

D

D’σ

F”

(b) True stress-strain curve

Figure 3.1: Typical stress-strain curves for metals.

In Figure 3.1 the typical stress-strain curves obtained for metals are depicted. The engineer-ing stress-strain curve is almost equal to the true stress-strain curve for small strains, forlarger strains the difference between the curves becomes significant.When the applied stress is lower than the initial yield stressσy0, the behaviour is reversibleor elastic. The initial state is recovered when the load is removed. The stress-strain relationfor metals is linear with stiffnessE � σ . ε.If the the applied stress exceeds the initial yield strength, the initial state will no longer berecovered. The material is now plastically deformed and the total strain can be divided inan elastic and a plastic part

ε � εe� εp (3.3)

The phenomenon that an increasing stress is required for further plastic deformation (AC inFigure 3.1(b)) is called strain hardening.If the material is unloaded from point F it follows the same slope as in the elastic case topoint F’. If the material is loaded again, it will generally deform plastically at F” before thepreviously attained stressσF at F is reached. With further loading the curve approaches thecurve OAB, that would have been found without intermediate unloading.The stress and strain state is uniform until the strip necks at the maximum load (dF � 0)at B. This is a geometrical instability at which the increase in yield stress due to strainhardening levels the decrease in load carrying cross sectional area. The maximum at dF � 0is only present in the engineering stress-strain curve. The true stress-strain curve can also

3.2 Material behaviour 21

reach a maximum after which the yield stress decreases, this is called true softening (branchCD’). True softening is an instability due to damage of the material and is a precursor tolocalisation and fracture. The elastic properties of the material are also affected by thedevelopment of damage.Finally, at point D the material loses its load carrying capacity and fractures. Ductile fractureis described in the next section.

3.2.2 Ductile fracture

The process of ductile fracture consists of three sequential stages:

� Nucleation of micro-voidsVoids are nucleated at inclusions, second-phase particles or grain boundaries, by de-cohesion at the interface between a particle and the matrix or by particle crackingwhen the material is deformed. Which of the two mechanisms initiates a void, de-pends on the type of bonding between the particle and the surrounding matrix and onthe difference in deformation behaviour between the particle and the matrix. There-fore the presence and type of particles in the microstructure is an important factor forthe ductility of a material. The greater the volume fraction of inclusions and second-phase particles the smaller the ductility.

� Growth of the micro-voidsVoids can grow under the influence of the applied stress and strain fields to a size thatexceeds that of the original particles. Both the hole volume and shape are changing.

� Coalescence of voidsAs a critical void volume fraction or a critical distance between the voids is reached,the matrix between the voids fails, resulting in microcracks. Cracks propagate bycoalescence with voids in front of the crack tip.

The mechanism of void growth is shown in Figure 3.2. The resulting fracture surface con-sists of shallow holes, called dimples. More information about ductile fracture can be foundin (Dodd and Bai 1987), (Thomason 1998) and (G¨anser et al. 1998).Another form of ductile fracture is rupture. In that case the nucleation and growth of voidsis suppressed and the parts are separated without the preceding formation of cracks.

3.2.3 High-speed shearing

From blanking experiments reported in the literature it is known that blanking at high speedyields different results than at low speeds (Johnson and Slater 1967), (Jana and Ong 1989),(Svahn 1996). This means that for high punch velocities (1-10 m/s) or high slitting linespeeds (300 m/min) the strain rate and temperature sensitivity of the flow stress becomesimportant. The yield stress of metals decreases as the temperature increases (Figure 3.3(a))


Nucleation Growth Coalescence

Figure 3.2: Mechanism of ductile failure by voids.

T / T0

T 0 T0

T 1 T0

ε

σ

(a) Temperature effect.

ε

σ

ε 0 ε0

ε / ε0

(b) Strain rate effect.

Figure 3.3: Temperature and strain rate effects on the stress-strain curve.

3.3 Material models 23

and increases for higher strain rates (Figure 3.3(b)). About 90% of the work of plasticdeformation is converted into heat, which leads to an increase in the temperature at thedeformation zone. Especially for large strains and high strain rates, when the producedheat has no time to be conducted away from the deformation zone, thermally assisted oradiabatic shear can be observed.

3.2.4 Hydrostatic pressure

A superimposed hydrostatic pressure has only a small effect on the flow stress and maximumload in a tensile test, but significantly increases the strain at fracture (Dodd and Bai 1987).It is observed that the difference in strain between the maximum load� dF � 0� and themaximum true stress� dσ � 0� increases when the hydrostatic pressure increases. Thiseffect is used in fine blanking to prevent fracture by imposing extra pressure during blanking(Lange 1990).

3.3 Material models

The uni-axial yield stress has to be translated to a yield criterion to describe multi-axialstress states. This yield criterion is given by:

φ � σσσ � � � � � � � � � � 0 (3.4)

It depends on the stressσσσ and can also contain parameters that describe the influence ofstrain, strain rate, temperature and damage on the yield stress. Damage is a measure for thedegradation of mechanical properties due to the development of voids. The yield criterioncan be interpreted as a surface in the three-dimensional principal stress space. The materialdeforms elastically if the stress state is inside the yield surface, i.e:

� φ 2 0� or � φ � 0 and φ 2 0� (3.5)

Plastic deformation occurs for stresses on the yield surface:

φ � 0 and φ � 0 (3.6)

Stress states outside the yield surface are not possible.In the following sections an elasto-plastic material model and examples of damage modelsare presented.

3.3.1 Elasto-plastic material model

In this section an elasto-plastic material model is given for finite deformations. It is assumedthat the rate of deformation tensor can be additively decomposed into an elastic and a plastic


part.

D � De � Dp (3.7)

For small elastic strains Hooke’s law can be written as:

�σσσ � E : De � E : � D � Dp � (3.8)

The fourth order elasticity tensorE for isotropic material is given by:

E � 2GH � � Cb � 23

G� 11 (3.9)

with G the shear modulus andCb the bulk modulus, which can be expressed in the elasticitymodulusE and Poisson’s ratioν.

G � E2 � 1 � ν � ; Cb � E

3 � 1 � 2ν � (3.10)

For isotropic materials a yield surface can be expressed asφ � σσσ � εp � , which is a function ofthe stress tensor and a scalar history parameterεp, the equivalent plastic strain, which isdefined as

˙εp � 3 23

Dp : Dp (3.11)

εp � , ˙εp dt (3.12)

Becauseεp is a scalar, the yield surface can only expand or shrink. This type of hardeningis called isotropic hardening. A tensor is needed to describe the loading history for othertypes of hardening.According to the postulate ofDrucker , which is a valid approximation for metals, the yieldsurface is convex and the plastic rate of deformation tensor is orthogonal to the yield surface(flow rule)

Dp � λ∂φ∂σσσ

; λ 4 0 (3.13)

The scalarλ is positive for plastic deformation and set to zero for elastic deformation,according to the Kuhn-Tucker conditions:

φ 5 0 � λ 4 0 � φλ � 0 (3.14)

Equation 3.8 together with Equation 3.13 yields:

�σσσ � E : � D � λ∂φ∂σσσ

� (3.15)


The consistency condition (φ � 0) gives:

φ � σσσ � εp � � ∂φ∂σσσ

: σσσ � ∂φ∂εp

˙εp � 0 (3.16)

Substitution of the Jaumann stress rate (Equation 2.36) into Equation 3.16 gives:


: �σσσ � ∂φ∂σσσ

: � W � σσσ � σσσ � W � � ∂φ∂εp

˙εp � 0 (3.17)

The second term in Equation 3.17 must vanish, because the yield surface must be invariantfor rigid body motions. This requirement is fulfilled for isotropic materials (Mooi 1996),which leads to:


: �σσσ � ∂φ∂εp

˙εp � ∂φ∂σσσ

: �σσσ � hλ � 0 (3.18)

h � 1

λ∂φ∂εp

˙εp (3.19)

An expression forλ can be derived from the equations above:

λ � � ∂φ∂σσσ : E : D

h � ∂φ∂σσσ : E : ∂φ

∂σσσ

(3.20)

leading to:

�σσσ � � E � Y � : D (3.21)

with the yield tensorY given by:

Y � E :∂φ∂σσσ

∂φ∂σσσ

: E

∂φ∂σσσ

: E :∂φ∂σσσ

� h(3.22)

To determineY, the yield surface must be specified. Here theVon Mises yield criterionin quadratic form is used. This yield surface can be visualised by a cylinder around thehydrostatic axis in principal stress space:

φ � σσσ � εp � � 32

s : s � � σy � εp � � 2 � 0 (3.23)

The equivalent Von Mises stress is defined as:

σeq � 3 32

s : s (3.24)


The terms ofY can be calculated now:

∂φ∂σσσ

�

∂φ∂s

∂s∂σσσ

� 3s (3.25)

Combining the flow rule (Equation 3.13) with Equation 3.25 gives:

Dp� 3λs (3.26)

The equivalent plastic strain rate can now be expressed as:

˙εp� 2σyλ (3.27)

from which an expression forh is determined:

h � 2σy∂φ∂εp � 4σ2

y∂σy

∂εp (3.28)

For isotropic material the following holds:

E : s � s : E � 2Gs (3.29)

The preceding equations lead to the following expression forY:

Y �

2Gss

23σ2

y � 2σ2y

9Gdσydεp

(3.30)

Determination of the flow curve To calculate the yield tensorY, the flow curveσy � εp �must be determined. This can be done by a curve fit of the true stress-true strain diagramobtained from a tensile test. Frequently, functions are used from the (modified) Ludwik-Nadai model:

σy � εp � � CN � ε0 � εp � n � σy0 � σy � 0� � CN � ε0 � n (3.31)

and the Voce model:

σy � εp � � σy0 � CV1 7 1 � e8 CV2εp 9(3.32)

whereσy0 is the initial yield stress.CN, ε0, n, CV1 andCV2 are constants. Of course otherfunctions can also be used to fit the experimental data.

The uniform strain reached in a tensile test is in a range of 0.2 to 0.4. This is much lower thanthe maximum strains found in shearing, which can reach values of up to 4. Therefore, theresults of a standard tensile test must be extrapolated to the large strains found in shearing.This is shown on the basis of the results of a tensile test on AISI316 stainless steel, which


0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 0.5 1 1.5 2 2.5 3

Flo

w s

tres

s [M

Pa]

:

Equivalent plastic strain

Data from tensile testNadai curve fitVoce curve fit

Figure 3.4: Tensile test data plus extrapolated curve fits.

will be used for the simulations described in Section 7.3. In Figure 3.4 the measured stress-strain curve is compared with two curve fits, using the Nadai and Voce models. It is clearthat two different fits, that both fit well to the measured data, can produce very differentresults when extrapolated to large strains. Therefore the extrapolated values are not veryreliable.To obtain more reliable results for large strains, tensile tests on pre-rolled specimens werecarried out by (Vegter 1991) and (Goijaerts 1999). A curve-fit is made between the maximaof the stress-strain curves of specimens with a different initial deformation due to rolling.Another possibility was proposed by (Zhang et al. 1999). They used a correction factor fortensile tests on rectangular specimens, comparable to the correction factor of Bridgeman forround tensile bars, to enlarge the range of strain for which reliable data can be found.Information for large strains can also be obtained from the results of plane strain com-pression tests, with appropriate corrections for friction and inhomogeneous deformation(Rietman 1999).


3.3.2 Damage models

Damage models are used to describe the degradation of the material properties which leadto fracture initiation. These models can be divided into two types.

� Coupled damage models:The damage is incorporated in the constitutive relations, which leads to a redistribu-tion of the stress and strain fields when damage develops.

� Uncoupled damage models:The damage is computed from the stress and strain fields, but does not modify thesefields.

Examples of both types of damage models will be given in the rest of this section.

Coupled damage models

There are several coupled damage models proposed in the literature, e.g. the ContinuumDamage Mechanics (CDM) models developed by (Kachanov 1986), (Lemaitre 1985) and(Gelin 1995).The damage model of (Tvergaard 1990), which is based on (Gurson 1977) is briefly treatedhere. The total volume of a ductile material is divided into the volume of the voids and thematrix material.f is the volume fraction of voids.

Vtotal � Vvoids � Vmatrix (3.33)

f � Vvoids

Vtotal� 1 � Vmatrix

Vtotal(3.34)

The yield condition developed by (Gurson 1977) is modified by (Tvergaard 1990) to:

φ � σσσ � f � � ; σeq

σy =2 � 2 f > q1 cosh ; σh

2σy = � 7 1 � � q1 f > � 2 9 � 0 (3.35)

in whichσσσ is the macroscopic equivalent stress of a porous medium andσy the actual flowstress of the matrix. The yield surface shrinks for increasingf > and the stress carryingcapacity vanishes forf > � 1. q1, when the yield surface shrinks to a point. The originalflow rule of (Gurson 1977) is obtained forq1 � 1 and f > � f . Equation 3.35 degenerates tothe Von Mises flow criterion (Equation 3.23), when the void volume fraction is zero (f � 0).The final material failure at a realistic void volume fraction is accounted for by takingq1 � 1 � 5 and the functionf > � f � introduced by (Tvergaard and Needleman 1984).fc isthe critical void volume fraction for coalescence of voids andf f the void volume fraction atfinal fracture.

f > � f for f 5 fc (3.36)

f > � fc � f >u � fcf f � fc

� f � fc � for f � fc (3.37)


The ultimate value off >u is equal to 1. q1 according to (Equation 3.35). Based on experi-mental and numerical result the valuesfc A B 0 � 15� 0 � 2D and f f � 0 � 25 are determined.The volume fraction of voids changes during the process, due to the nucleation and growthof voids.

f � fnucleation � fgrowth (3.38)

The nucleation of voids can be stress-controlled, strain-controlled or a combination of bothtypes of nucleation.

fnucleation � A˙εp� B � σeq � σh � (3.39)

Considering a random distribution of inclusions, the material parametersA andB are oftendefined as a normal distribution around a mean nucleation stress or strain. The growth ofthe voids follows from the incompressibility of the matrix material:

fgrowth �VmatrixVtotal

V2total

� � 1 � f � tr � Dp � (3.40)

The last column of Table 1.1 provides references for shearing simulations which use a ma-terial model derived from (Gurson 1977).

Uncoupled damage models

Ductility curve An uncoupled approach is the use of ductility curves. A ductility curvegives the relation between the equivalent plastic strain at fractureεp

f and the triaxiality. Thetriaxiality ratio is defined as the hydrostatic stress divided by the equivalent Von Misesstress:

triaxiality ratio �σh

σeq(3.41)

The typical shape of such a curve is shown in Figure 3.5. The fracture strain is large fornegative triaxialities and decreases with increasing triaxiality. A ductility curve can be ob-tained by experiments with different loading conditions (Bolt 1989), (Peeters and Donkers1996). An example of the use of ductility curves is given by (Marusich and Ortiz 1995) forthe FEM simulation of machining. They used a formula based on (Rice and Tracey 1969):

εpf � 2 � 48e8

32

σhσeq (3.42)

A drawback of ductility curves is that the stress and strain path of the process is usuallydifferent from the stress and stain path of the experiments used to determine the ductilitycurve.


Tension

Compression

Triaxiality ratio

εpf

Figure 3.5: Ductility curve.

Fracture criteria Fracture criteria are based on the assumption that during plastic defor-mation damage develops, which leads to crack initiation. The damageD is calculated as anintegral of a function of the stress and strain history. Crack initiation occurs as the damagereaches a critical levelDc.

D � , εp

0f � σσσ � � � � � dεp

D � Dc for εp � εpf (3.43)

Some of the proposed functionsf are given in Table 3.1. The parametersC are materialconstants;σ1 � σ2 � σ3 are the principal stresses andn is a hardening parameter. Thedamage functions are based on void growth theories or found empirically. Some functionscontain the triaxiality ratio (Equation 3.41), which describes the influence of the hydrostaticpressure on the fracture behaviour. The damage grows slowly under hydrostatic pressureand fast under hydrostatic tension.A comparison of the different fracture criteria was made by (Clift et al. 1990), (Zhu andZacharia 1995), (Wifi et al. 1995), (Gouveia et al. 1996) and (Jan´ıcek and Petruˇska 1999).They combined experiments with FEM simulations of the same experiments, to calculatethe damage value at crack initiation. From their results can be concluded that the criticaldamage valueDc, which should be a material constant, can also be dependent on the process.Therefore one should take care when using critical damage values measured under differentloading conditions than the loading in the investigated process.The critical damage value in blanking can be determined by comparing the damage valuein an FEM simulation of blanking with the initiation of cracks in a blanking experiment.It was shown by (Brokken 1999) and (Taupin et al. 1996) that a critical damage leveldetermined by one clearance can be used to predict crack initiation at other clearances. To

3.4 Discussion 31

Reference Fracture criterion

(Freudenthal 1950) D � E εp

0 σeqdεp

(Cockcroft and Latham 1968) D � E εp

0 σ1 dεp

(McClintock 1968) D � E εp

0 F 2G3 H 1 8 nI sinh F G

3 H 1 8 nI2 J σ1 K σ2

σeq J dεp

(Rice and Tracey 1969) D � E εp

0 ; eCR

σhσeq = dεp

(Oyane et al. 1980) D � E εp

0 � 1 � COσhσeq

� dεp

(Goijaerts 1999) D � E εp

0 L 1 � CG1σhσeq M � εp � CG2 dεp

Table 3.1: Some proposed fracture criteria in the literature.

avoid decreasing damage (Brokken 1999) modified the fracture criterion of Oyane to:

D � , εp

02 1 � 3

σh

σeq� dεp (3.44)

2 x � � 0 for x 5 0

x for x � 0

A more attractive method to measure the critical value of the damage is to use a tensiletest. A correlation between the critical damage in a tensile test and the critical damage inblanking has been found by (Goijaerts 1999).

3.4 Discussion

A number of models were presented in this chapter to describe the plastic deformation andductile failure of a material. During the development of a 3D shearing model, it is preferredto use relatively simple material models. Hence an elasto-plastic material model combinedwith a fracture criterion was chosen for the shearing simulations.

The effects of strain rate and temperature dependent material properties are neglected, be-cause it is expected that three-dimensional thermo-mechanically coupled FEM simulationswill take too much calculation time. Therefore the applicability of the material models islimited to relatively low shearing speeds.

The modified fracture criterion of Oyane (Equation 3.44) is chosen, because such a criterioncan easily be incorporated in an FEM code and can give good results (Brokken 1999).


Coupled damage models, however, are more accurate in the deformation phase just beforefracture, as the degradation of material properties due to damage is accounted for. Theidentification of damage parameters is not always easy either for coupled or uncoupleddamage models.

Chapter 4

Finite element formulation

4.1 Introduction

The equilibrium equations with the accompanying boundary conditions, together with theconstitutive relations described in Chapter 2 and Chapter 3, have to be solved in order tosimulate guillotining and slitting. The solution of these partial differential equations can beapproximated with the finite element method (FEM). The finite element formulation usedfor the calculations in this thesis will be described in this chapter.

4.2 Types of FEM formulation

In this section some different types of finite element formulations are treated and assessedon their suitability for modelling guillotining and slitting. The formulation should be able todescribe 3D stationary processes with large deformations, free surfaces and ductile fracture.

4.2.1 Lagrangian formulation

In forming processes a Lagrangian formulation is often used. In such a formulation theframe of reference is fixed to the initial geometry (Total Lagrangian) or fixed to the geometryat the beginning of the time step, and thus moving with the material (Updated Lagrangian).With a Lagrangian formulation free surfaces are followed automatically and history depen-dent behaviour can easily be taken into account. A drawback of the Lagrangian formulationis that the shape of the elements can become distorted in the case of large deformations,leading to less accurate results or even to a premature end of the calculation. Therefore it isnot possible to model guillotining and slitting solely with the Updated Lagrangian formula-tion.

34 Finite element formulation

4.2.2 Lagrangian formulation with remeshing

The Updated Lagrangian (UL) formulation can be extended with a remeshing procedure,in which the old grid is replaced by a completely new grid, and the information on the oldgrid is transferred to this new grid. Remeshing is a flexible method, as during the wholesimulation the mesh can be adapted to best fit the current situation. It can be used to avoidtoo much grid distortion, for mesh refinement in critical areas or to model crack initiationand propagation.Some possibilities to use remeshing for the simulation of crack initiation and propagationare:

Nodal releaseWith the nodal release method cracks are assumed to initiate or propagatealong element edges. The node at the crack tip is separated into two nodes and thecrack is extended by one element length. A drawback of this method is that if thecrack trajectory is unknown, a very fine mesh must be used to describe the trajec-tory accurately. An advantage is the minimal modification of the mesh during thesimulation. Examples can be found in (Dodds and Tang 1993) and (Obikawa et al.1997).

Element elimination Another method is to remove elements (or give a zero stiffness) thatexceed a certain fracture criterion. The results of this procedure depend strongly onthe element size, as material is lost as the crack propagates. This technique is appliedto blanking and machining by (Taupin et al. 1996) and (Ceretti et al. 1999). Inaddition they used remeshing to keep elements small in front of the crack tip, whichminimizes the material loss.

Delete and fill The delete and fill technique consists of deleting the mesh around a cracktip, extending the crack and generating a new mesh around the new crack tip. Thisprocedure, that requires a good remeshing algorithm, is among others used by (K¨onke1995), (Marusich and Ortiz 1995), (Brokken 1999) and (Bouchard et al. 1999). Itoffers the possibility of using special types of elements (with 1. N r crack tip sin-gularity) or special mesh topologies around the crack tip and allows cracks to growindependently of the finite element mesh.

Important issues in the use of remeshing are the automatic generation of the new finiteelement mesh and the accuracy of the transfer of information between the old and newmeshes. Remeshing is widely used for the simulation of forming processes and also forfracture problems in two dimensions (all references mentioned above concern 2D models).The use of remeshing for three-dimensional problems with hexahedral elements, however,is not yet well developed. Some recent examples of the use of remeshing for 3D formingprocesses are (Tekkaya 1998) and (Gelten 1998).

4.2 Types of FEM formulation 35

Figure 4.1: Simulation of an upsetting process: left Updated Lagrange, right ALE.

4.2.3 Eulerian formulation

In a Eulerian formulation the material flows through a frame of reference which is fixedin space. Therefore, problems with grid distortion due to large deformation do not exist insuch a formulation, but in general material boundaries are not equal to the element edges, sospecial procedures are required to follow free surfaces, cracks or boundaries between dif-ferent materials, including contact (Dvorkin and Pet¨ocz 1993), (Benson 1997). To handlehistory dependent material properties convection must be taken into account.

4.2.4 ALE formulation

The Arbitrary Lagrangian Eulerian (ALE) formulation is a combination of the Lagrangianand the Eulerian descriptions. In such a formulation the frame of reference is not necessar-ily equal to the material displacement (Lagrangian formulation) nor equal to zero (Eulerianformulation), but can be chosen independently from the material displacement. This for-mulation was first applied in the finite element method by (Hughes et al. 1981), (Doneaet al. 1982) to calculate fluid-structure interactions and later also to forming processes by(Huetink 1986), (Schreurs et al. 1986). The ALE formulation can be used to solve the prob-lem with free surfaces in a Eulerian formulation or to avoid grid distortion in a Lagrangianformulation. As in the Eulerian formulation, convection of history dependent variables mustbe taken into account.

The difference between a simulation with an ALE formulation and a UL formulation isshown in Figure 4.1 for an upsetting process. The Updated Lagrange simulation on the leftcannot be continued because the elements are too distorted, also, the boundary conditionsare not described accurately (punch enlargement). When the same simulation is carried outwith an ALE formulation, (right half of Figure 4.1) it is possible to preserve well shapedelements and to describe the boundary conditions better. The ALE simulation can also becontinued from this stage.

A characteristic of the ALE formulation is that the topology of the mesh (number of ele-


ments and their connectivity) is constant during the entire simulation. This means that theinitial domain as well as the final domain must be described with the same topology, whichlimits the applicability of the ALE formulation.The transfer of information between the old mesh and the new mesh is in general moreaccurate for an ALE formulation (where convection techniques can be used) than for aremeshing step in the UL formulation. Therefore both methods can be combined in onesimulation to reduce the number of remeshings (Brokken 1999).

The ALE method can also be used to model crack propagation. The growth of an initiallypresent crack can be followed to a certain extent without changing the mesh topology, butit is not possible to describe a complete fracture process from initiation till complete sepa-ration. A coupled ALE formulation (Section 4.4.1), in which the crack tip position is takenas an unknown in the finite element equations, is used by (DeBruyne 1995) and (Koh et al.1995) for 2D crack propagation problems.

4.2.5 Discussion

An ALE formulation is chosen for the calculation of the steady state of guillotining and slit-ting. Stationary processes, like guillotining and slitting, can be seen as flow problems withsurfaces coming into contact with the tools, free surfaces, large localised deformations andductile fracture. The steady state of a stationary process can be calculated by continuing atransient calculation until a steady state is reached. An ALE formulation is very suitable forthe calculation of the steady state of such processes. When the calculation is started with avolume sufficiently close to the steady state volume and including the crack, grid distortioncan be avoided and the free surfaces are described correctly. This means that an estimationof the steady state geometry has to be made for the initial mesh (the exact shape of the freesurfaces and the crack are not known beforehand).

The ALE method is also used for plane strain shearing simulations (2D) to preserve regularelements and to keep the refinements in the areas with the largest gradients. This simulationcan be continued to penetrations of about fifty percent of the sheet thickness using the ALEmethod. For larger penetrations or crack initiation remeshing is required, as the shape ofthe domain becomes too different from the initial one.

4.3 Problem definition

To be able to follow the deformation history, the total load is applied in a number of timesteps. In the case of an elasto-plastic material model these are not real time steps, but morea sequence of events.

4.3 Problem definition 37

VnP 1m

VnP 1g

xnP 1g

xnP 1m

Constantχre f

ConstantXre f

χre f � Xre f � xre f

Vnm � Vn

g

Figure 4.2: Spatial domains ontn andtnK 1

It is assumed that the state att � tn is known on the material domainVnm, which is equal to

the computational domainVng . The displacements of a material pointxm and a grid pointxg

during a time step fromt � tn to t � tn � ∆t � tnK 1 are:

xnK 1m � xn

m � x∆t � xnm � vm∆t � xn

m � ∆um (4.1)

xnK 1g � xn

g � �x∆t � xng � vg∆t � xn

g � ∆ug (4.2)

with the material velocityvm and the grid velocityvg. In a Lagrangian formulation thecomputational domain is equal to the material domainVnK 1

g � VnK 1m . In a Eulerian formula-

tion the computational domain is fixed:VnK 1g � Vn

g . The new computational domainVnK 1g

formed by the displacement of all grid points is in an ALE formulation not necessarily equalto the material domainVnK 1

m formed by all material particles inVnm. This means that material

can flow in and out of the computational domain during a time step.The deformation and the values of the state variables at timetnK 1 have to be calculated onthe new computational domainVnK 1

g . The state variables are described in terms ofχχχre f ,which is equal to the spatial coordinatex at the beginning of the time step. The referentialcoordinateχχχ is connected to the grid points during a time step, which move with the gridvelocityvg.The values of the state variables in the new grid points attnK 1 are calculated using the grid

time derivative (�ξ) of the state variableξ, which consists of the material time derivative (ξ)


and a convective part.

ξnK 1g � ξn

g � �ξ∆t

� ξng � F ξ � � vm � vg � � ��

ξ J ∆t (4.3)

� ξng � ∆ξg

4.3.1 Strong formulation

The objective is to find the solution forσσσ andum that fulfils the equilibrium equations andboundary conditions attnK 1. The surface is split in a partSu, where the displacements areprescribed and a partSt , where the forces are prescribed, such thatS � St Q Su R St S Su � T .

σσσnK 1 � !� � 0 in VnK 1g (4.4)

u � u0 on Su (4.5)

σσσnK 1 � n � tnK 1 on St (4.6)

wheren is the outward normal on the boundary,u0 the prescribed displacements andtthe prescribed loads. Inflow conditions are set for the state variables and displacements atboundariesSf where material flows into the domain (vc � n 2 0):

u � u f on Sf (4.7)

ξ � ξ f on Sf (4.8)

4.3.2 Weak formulation

In the finite element method the equilibrium equations are only weakly enforced. Whenbody forces are neglected, the so-called weak form can be written as:

,VnW 1

� w !� � : σσσnK 1dV � ,SnW 1

w � tnK 1dSt (4.9)

with the weight functionw which is equal to zero onSu and integration over the spatialcoordinates. This weak form is equal to the virtual power equation if the weighting functionw is equal to the virtual velocityδv:

δWin � δWext (4.10)

Because the elastic-plastic material model is given as the material rate of the stress-strainrelation (Equation 3.21), the rate of the weak form is used. The increase in virtual workfrom Vn

g at tn toVnK 1g at tnK 1 can be approximated by:

δWnK 1in X δWn

in � δ �Win∆t (4.11)

δWnK 1ext X δWn

ext � δ �Wext∆t (4.12)

4.3 Problem definition 39

From this, the rate of weak form reads for the ALE formulation:

δ �Win � δ �Wext (4.13)

which can degenerate to an Updated Lagrangian formulation:

δWin � δWext (4.14)

or a Eulerian formulation:

∂∂t

δWin � ∂∂t

δWext (4.15)

Based on (Liu et al. 1988) and (Wang and Gadala 1997) the derivation of the rate of weakform for the ALE formulation is given. The weak form (Equation 4.9) is first transformedinto referential coordinates:

,Vχχχ

∂w∂χχχ

∂χχχ∂x

: σσσJχχχ dVχχχ � ,Sχχχ

w � tJSdSχχχ (4.16)

with the surface JacobianJS and:

Jχχχ � det ; ∂x∂χχχ = (4.17)

Using the following definitions forFg and the gradient of the grid velocityLg:

Fg � ∂x∂χχχ

� xg !�χχχ (4.18)

Lg � vg !�(4.19)

the grid time derivative of the internal virtual work becomes:

δ �Win � ,Vχχχ

∂w∂χχχ

: ; �Fg 1 � σσσJχχχ � Fg 1 � �σσσJχχχ � Fg 1 � σσσ �Jχχχ = dVχχχ (4.20)

The rates of the weight functionw vanish ((Huetink 1986)). Equation 4.20 contains threegrid time derivatives which can be rewritten as (Liu et al. 1988):

�Fg 1 � � Fg 1 � Lg (4.21)

�Jχχχ � Jχχχtr � Lg � (4.22)


�σσσ � σσσ � � vm � vg � � ��σσσ � �σσσ � σσσ � W � W � σσσ � � vm � vg � � ��

σσσ (4.23)

This can be seen when Equation 4.21 is compared with Equation 2.21 and Equation 4.22with Equation 2.38. The grid time derivative of the stress is found using the Jaumannderivative (Equation 2.36). Substitution of the derivatives and transforming back to thespatial coordinates gives the final form of the internal work:

δ �Win � ,V

w !�: Y � Lg � σσσ � σσσ � � vm � vg � � ��

σσσ � σσσtr � Lg � [ dV (4.24)

The virtual external work written in referential coordinates is:

δWext � ,St

w � t dSt � ,Sχχχ

w � tJSdSχχχ � ,Sχχχ

w � σσσ � nJSdSχ (4.25)

The grid derivative of the external virtual work is:

δ �Wext � ,Sχχχ

w � \ �σσσJS � σσσ �JS] ndSχχχ (4.26)

with Equation 4.23:

δWext _ `Sχχχ

w a \ b σσσ d e vm d vg g a hiσσσ j JS k σσσJS] ndSχχχ (4.27)

Transforming back to spatial coordinates and usingσσσ a n _ t gives:

δWext _ `Sw a t dS d `

Sw a l e vm d vg g a hi

σσσ n ndS k `Sw a tJSdS (4.28)

The complete rate of the weak form of the ALE equations is now:

`V

w oi: l d Lg a σσσ k σσσ d e vm d vg g a hi

σσσ k σσσtr e Lg g n dV _`

Sw a t dS d `

Sw a l e vm d vg g a hi

σσσ n ndS k `Sw a tJSdS (4.29)

This weak form degenerates to a UL formulation whenvg _ vm:

`V

w oi: r d L a σσσ k σσσ k σσσtr e L g t dV _ `

Sw a t dS k `

Sw a tJSdS (4.30)

or to a Eulerian formulation forvg _ 0:

`V

w oi: l σσσ d vm a hi

σσσ n dV _ `Sw a t dS k `

Sw a l vm a hi

σσσ n ndS (4.31)

4.4 ALE formulations 41

4.4 ALE formulations

Different approaches are possible to solve the ALE equations of Section 4.3. Some methodsare worked out in the following sections. First, the coupled ALE formulation is discussedin Section 4.4.1, next, the decoupled ALE formulation in Section 4.4.2 and last, in Sec-tion 4.4.3 the ALE formulation as it is used for the simulations in this work, which hassimilarities with both of the preceding approaches.

4.4.1 Coupled ALE

Equation 4.29 is discretised in the coupled ALE method. This rate of the weak form con-tains both material and convective terms which are solved simultaneously, and is thereforethe best possible solution of the ALE problem. It is used for simulations of forming pro-cesses by (Liu et al. 1988), (Bayoumi et al. 1998) and (Gadala and Wang 1999).

In the finite element method the total volume is discretised into a finite number of elements.In this study linear, isoparametric quadrilateral or hexahedral elements are used. This meansthat the same linear interpolation is used for the geometry and the state variables:

x _ ∑k

Nk e r v zv hg xk (4.32)

ξ _ ∑k

Nk e r v zv hg ξk (4.33)

in whichNk are the shape functions,e r v zv hg the local coordinates andξk the nodal values ofthe state variableξ. The (virtual) velocities can be expressed in nodal velocities as:

vm _ ∑k

Nkvmk; δvm _ ∑

k

Nkδvmk (4.34)

vg _ ∑k

Nkvgk; δvg _ ∑

k

Nkδvgk (4.35)

(4.36)

The gradient of the velocity and the rate of deformation tensor are then:

L _ ∑k

vmk hi

Nk (4.37)

D _ ∑k

Bk a vmk (4.38)

When the Jaumann derivative (Equation 2.36) of the stress and the constitutive relation(Equation 3.21) are substituted into Equation 4.29 (usingw _ δv), the semi-discretizedform is obtained (Equation 4.39). The stiffness matricesr K t can be derived by numericalintegration (using the gauss points) of the integrals of the weak form. These steps are not


worked out here. The semi-discretized form contains two unknown velocities, the materialvelocityvm and the grid velocityvg:

r Kmt w vm y k r Kg t w vg y _ w Fext y (4.39)

The velocity is assumed constant during a time step and equal to the average valuev:

vm _ ∆um

∆t; vg _ ∆ug

∆t(4.40)

Substitution into Equation 4.39 and multiplying by∆t leads to:

r Kmt w ∆um y k r Kgt w ∆ug y _ w ∆Fext y (4.41)

To solve this expression, extra equations are needed which describe the relation betweenw ∆ug y and w ∆um y :

r At w ∆um y k r Bt w ∆ug y _ w C y (4.42)

The grid displacementsw ∆ug y are taken as additional degrees of freedom, giving:

\ r Kmt r Kg tr At r Bt | ~ ∆um

∆ug � _ ~ ∆Fext

C � (4.43)

w ∆ug y can be eliminated from Equation 4.41 if it is expressed explicitly inw ∆um y :

w ∆ug y _ r D t w ∆um y (4.44)

leading to:

r r Kmt k r Kgt r D t t w ∆um y _ r K t w ∆um y _ w ∆Fext y (4.45)

Predictor/Corrector

To solve the nonlinear set of equations an iterative scheme is used, which consists of apredictor part and a corrector part (Figure 4.3).The first approximation for the incremental displacementw ∆um y is found by linearisingEquation 4.45 with the known state at the beginning of the time step. The iteration numberis denoted with the superscripti . The superscript0 refers to the beginning of the step.

r K0 t w ∆uim y _ w ∆Fext y for i _ 1 (4.46)

Assembling and solving this equation is called the predictor. In the corrector part the statevariables are integrated in time, using the results of the predictorw ∆u1

m y and w ∆u1g y (calcu-

lated from Equation 4.44):

ξi _ ξ0 k ` t � ∆t

tξdt

_ ξ0 k ` t � ∆t

tξdt d ` t � ∆t

te vm d vg g a hi

ξdt (4.47)


iteration loop

Convergence ?

steploop

Update

� ∆um � ∆ug

ALE predictor

ALE corrector � ξn� 1

Figure 4.3: Coupled ALE.

This integral consists of two parts, a material part and a convective part, which will bedescribed in Sections 4.5 and 4.6 respectively.From the stresses in the integration points the nodal reaction forcesw Fint y can be calculated:

w Fiint y _ `

ViBT a σσσi dV (4.48)

These internal forces should be in equilibrium with the prescribed forcesw Fext y , which aretaken constant during a time step:

w Ri y _ w Fext y d w Fiint y (4.49)

When the mechanical unbalance ratioεi , the norm of the unbalance forcew Ri y divided bythe norm of the internal force, is smaller thanεF , the iteration process is stopped and a newstep can be started.

εi _ � � w Ri y � �� w Fiint y � � �

εF (4.50)

If the convergence criterion is not fulfilled a new iteration will be performed. A new stiff-ness matrix is calculated with the values of the state variables from the last iteration. Theunbalance force of the previous iteration is carried on to the next iteration:

r Ki � 1 t w ∆∆uim y _ w Ri � 1 y for i � 2 (4.51)


The adaptationw ∆∆uim y is added to the displacement increment of the last iteration:

w ∆uim y _ w ∆ui � 1

m y k w ∆∆uim y (4.52)

The displacement ratio, defined as:

� � w ∆∆uim y � �

� � w ∆uim y � � �

εd (4.53)

can also be used as a convergence criterion. This ratio is equal to one for the first iteration.

In the coupled ALE method both material and grid displacements are calculated simulta-neously. Therefore, the solution of the iteration process is the best possible solution of theALE problem. But the disadvantages of this method are:

� In the coupled ALE, extra equations are required to couple the grid displacement tothe material displacement. This expression is added to the stiffness matrix, givingtwice as many degrees of freedom (Equation 4.43) and thus larger systems to solve.

� If the convective displacements are expressed as an explicit function of the materialdisplacements (Equation 4.45), they can be eliminated from the stiffness matrix. Inpractice this is limited to relatively simple functions (Bayoumi et al. 1998).

� In the case of convective terms the usual Galerkin finite element method leads to spu-rious oscillations at moderate mesh velocities (Liu et al. 1988), which can be over-come using an upwind-type finite element formulation (Brooks and Hughes 1982).The stiffness matrix then also becomes non-symmetric.

� The coupled method is time-consuming since convection must be taken into accountat each iteration.

4.4.2 Decoupled ALE

In the decoupled ALE method, also called operator split ALE, each time step is subdividedinto two parts which are solved separately. Examples can be found in (Benson 1989) and(Stoker 1999).First a normal Updated Lagrangian step is carried out, discretising and solving Equation 4.30.In the predictor part the material displacement is calculated from:

r K0mt w ∆ui

m y _ w ∆Fext y for i _ 1 (4.54)

r Kimt w ∆∆ui

m y _ w Ri y for i � 2


and in the corrector part the state variables are integrated in time (only the material incre-ment):

ξL _ ξ0 k `t � ∆t

tξdt (4.55)

step loop

iteration loop

Update

Convergence ?

Interpolation� ξn� 1

Meshing � ∆ug

UL corrector � ξL

UL predictor � ∆uim

Figure 4.4: Decoupled ALE.

The predictor and corrector are repeated until equilibrium is satisfied (Figure 4.4).

Now the material displacement and the state variablesξL after the Lagrangian step areknown, the grid displacement is determined using the material displacement and the newvalues of the state variables.Finally, the state variables at the integration points of the new gridξn� 1 at tn� 1 have to becalculated from the valuesξL in the integration points of the old grid, also attn� 1. Thisis an interpolation problem as no time change is involved, but it can also be written as aconvection problem:

ξn� 1 e xn� 1g g _ ξL e xn� 1

m d ∆uc g (4.56)

An advantage of the decoupled ALE method compared to the coupled ALE method is theflexibility in defining the new grid, as the latter is determined after the Lagrangian step iscompleted. The decoupled method is also more efficient since the grid displacement andthe convective increment are calculated only once per time step.


A drawback of the decoupled method is that equilibrium is not checked at the end of thestep. Equilibrium can be violated if the convection is not performed accurately. This leads tounbalance forces, which are carried as a load correction onto the next step. Large unbalanceforces can lead to possible divergence. Comparing the coupled and decoupled formulations(Baaijens 1993) found similar results, but the decoupled formulation was more efficient.

4.4.3 Semi-coupled ALE

The ALE method used in this thesis differs from the two methods described before. It isbased on the ALE implementation of (Hu´etink 1986). The formulation can be seen as acoupled formulation, in which the ALE predictor is replaced by the UL predictor by setting

w ∆ug y _ w ∆um y in Equation 4.41.

step loopiteration loop

Convergence ?

Update

UL predictor � ∆uim

Meshing � ∆ug

ALE corrector � ξn� 1

� ξC

Figure 4.5: Semi-coupled ALE.

With the UL predictor the material displacement increment is calculated as in the decoupledmethod (Figure 4.5). The meshing step is also the same as in the decoupled method, butis now performed at every iteration. In the corrector part first the convective increment iscalculated using the values of the state variables at the beginning of the step,

ξC e xn� 1g g _ ξ0 e xn

m d ∆uc g (4.57)

followed by the integration of the material increment:

ξn� 1 _ ξC k `t � ∆t

tξdt (4.58)

4.5 Material increment 47

Equilibrium is checked at the end of the time step and the iteration loop is repeated until theconvergence criterion is fulfilled.The advantages of the semi-coupled ALE method are:

� The flexibility in meshing of the decoupled method is preserved.

� Equilibrium is fulfilled at the end of the step, leading to a more robust algorithm,compared to the decoupled method, which is especially important in contact prob-lems.

Drawbacks are:� As in the coupled method, convection and meshing are performed at each iteration,

which is less efficient than the decoupled method.

� As the convective terms are neglected in the predictor part, a slower rate of con-vergence is expected compared to the coupled method. A Eulerian stiffness matrix(setting w ∆ug y _ w 0 y in the predictor part) may yield better results in the case of flowproblems.

4.5 Material increment

At each iteration a material increment is calculated in the corrector part. The mean normalmethod is used in this work, a simple and stable algorithm. More about integration methodscan be found in e.g. (Ortiz and Popov 1985).From the incremental material displacements the total strain increment can be calculated:

∆εεε _ B a ∆um (4.59)

Next a fictitious elastic stress increment is calculated:

∆σσσe _ E : ∆εεε (4.60)

Then, the mean fictive elastic stress is determined for points that are on the flow surface atthe beginning of the step:

σσσ � _ σσσn k 12∆σσσe (4.61)

The new stress is then:

σσσn� 1 _ σσσn k r E d Y e σσσ � g t : ∆εεε (4.62)

The yield tensorY is calculated using the mean fictive elastic stress. In the case of hard-ening, this stress will in general not fulfil the flow criterion, therefore the stress is adaptedby scaling of the deviatoric stresss� . For large rotations, corrections must be made to theintegration scheme to preserve incremental objectivity for the tensorial quantities (Hu´etinket al. 1999).


4.6 Convective increment

In the ALE formulation a convective increment has to be calculated at each step or iteration,depending on the ALE method used. This convective increment can be written as a con-vection or as an interpolation problem. The convective velocityvc is taken to be constantduring a time step:

∆ξg _ `t � ∆t

td vc a hi

ξdt � d ∆uc a hiξ (4.63)

In the case of an elasto-plastic material model the history dependent state variablesξ, theequivalent plastic strain, stress and damage parameters, which are known in the integrationpoints, have to be calculated at the new position of the integration points. Contrary to theincremental displacements, the total displacements must also be convected. Although thetotal displacements are not necessary for the simulation, they can be very useful for post-processing.

The convective increment is calculated with theWeighed local and global smoothingschemeof (Huetink 1986), which is abbreviated as WLGS. This is the only scheme implemented inthe finite element code DiekA for convection in 2D and 3D. Some other schemes in 2D arepresented by (Rekers 1995), (Helm et al. 1998), (Stoker 1999) and (Brokken 1999). Theyshowed the accuracy of their schemes by using the Molenkamp test, which makes it possi-ble to compare the different schemes. From their results can be concluded that the WLGSscheme is not the most accurate scheme in 2D (Rekers 1995),(Helm et al. 1998). Hence,the development of 3D versions of schemes other than WLGS would be an interesting topicfor further research.

4.6.1 Weighed local and global smoothing

The Weighed local and global smoothing scheme was first described in (Hu´etink 1986) and(Huetink, Vreede, and Lugt 1990). The main characteristics of this scheme are summarisedin this section. Some changes were made to the original scheme to avoid cross-wind diffu-sion. It was shown by (Helm, Hu´etink, and Akkerman 1998) that this method has similari-ties with the Lax-Wendroff schemes with added artificial diffusion used in finite difference.The WLGS scheme is an interpolation approach, which uses a smoothed continuous fieldas an intermediate step. It consists of the following steps:

1. At the end of a time step a continuous field is constructed from the integration pointvalues. The continuous field at the end of the last step is of course equal to the fieldat the beginning of the current step:

(a) Extrapolation of the integration point values to the nodes, with alocal smoothingfactor, which may depend on the Courant number.

4.6 Convective increment 49

(b) Calculation of the nodal average from the contributions of the surrounding ele-ments.

(c) Interpolation of the averaged nodal values.

This continuous field is used for convection in the next step and can also be used forpost-processing purposes.

2. In the iteration loop:

(a) Calculate the local Courant number of the current step.

(b) Calculate theglobal smoothingfactor.

(c) Calculate the integration point values in the new integration points using thecontinuous fields.

The different steps in this scheme are now elaborated for trilinear hexahedrals with shapefunctionsNk (Equation 4.33):

N1 _ 18

e 1 d r g e 1 d zg e 1 d hgN2 _ 1

8e 1 k r g e 1 d zg e 1 d hg

� � �N8 _ 1

8e 1 d r g e 1 k zg e 1 k hg (4.64)

with the local coordinatesr v zv h � r d 1 v 1t . The integration points are located at the localcoordinatese � 1

3 � 3 v � 13 � 3 v � 1

3 � 3g .

Courant number The Courant number is defined as the difference in local coordinatesbetween the old and new integration points, divided by the element size (2x2x2 in localcoordinates). Three Courant numbersCr ,Cz andCh are calculated for the different directionsat every integration point. The integration point values are then averaged to one value forthe whole element:

Cr _ 18

8

∑ip � 1

� ∆rip �2(4.65)

The total Courant number is then

C _ � C2r

k C2z

k C2h (4.66)


Extrapolation The integration point values are extrapolated from the integration pointsto the nodes using the shape functions (Equation 4.64):

ξnode _8

∑k� 1

Nk e ripβr v zipβz v hipβh g ξkip (4.67)

β � 0

β � � 3

nodeip value

Figure 4.6: Extrapolation from integration point to nodes.

Thelocal smoothingparameterβ � r 0 v � 3t is introduced to influence the extrapolation. Thesmoothing effect ofβ is illustrated for 1D in Figure 4.6. A value ofβ _ � 3 means normalextrapolation. According to (Hinton and Campbell 1974), (Hu´etink 1986) uses least squaresmoothing, which implies for linear elements that the mean value of the integration pointsis used as a nodal contributione β _ 0g . β can be a fixed number or a function of the Courantnumbers in the different local directions:

βr _ βz _ βh _ constant

βr _ f e Cr v Cz v Ch g (4.68)

Nodal averaging A nodal value is determined by averaging the contributions of the sur-rounding elements:

ξ �node _1ne

ne

∑k� 1

ξknode (4.69)

The average nodal value can be overruled by a predefined value, e.g. for inflow boundaries.

Interpolation By interpolation of the average nodal values using the shape functions, acontinuous field is obtained. An example of the construction of a continuous field is givenin 1D in Figure 4.7 for the minimum and maximum values ofβ. It can be seen that forβ _ 0a much smoother field is obtained than forβ _ � 3.


extrapolationcontinuous field

nodal averagenodeintegration point value

β � � 3

β � 0

Figure 4.7: Construction of continuous field, differentβ.

Convection The values of the state variables in the new integration points can be calcu-lated using the constructed continuous field. The local coordinates of the new integrationpoint in the old element are:

e r v zv hg new _ e r v zv hg old d e Cr v Ch v Cz g (4.70)

By interpolation of the continuous field the state variables at new integration points can becalculated:

ξnewip _

8

∑k � 1

Nk e e r v zv hg newg ξ �nodek (4.71)

Another possibility is to use the continuous field to calculate the convective increment andadd this to the old integration point value.

ξnewip _ ξold

ipk 8

∑k� 1

Nk e e r v zv hg newg ξ �nodek d

8

∑k � 1

Nk e e r v zv hg old g ξ �nodek (4.72)

The first method (Equation 4.71) is very diffusive and the second method (Equation 4.72)shows spurious oscillations (Hu´etink 1986). Therefore, a weighed combination of thesetwo methods was proposed by (Hu´etink 1986) with a weighting factor,α � r 0 v 1t , which isa function of the Courant number. The application of this weighting factor is calledglobalsmoothing.


ξnewip _ 8

∑k � 1

Nk e e r v zv hg newg ξ �nodek k

e 1 d α g ξoldip d 8

∑k � 1

Nk e e r v zv hg old g ξ �nodek ¢ (4.73)

Global smoothing From the results of some numerical experiments a good choice for theweight factorα for the global smoothing proved to be:

α _ min e 1 � 64C1 £ 2 v 1g (4.74)

It is clear from Equation 4.74 that the larger the Courant number is, the more smoothing isadded. The integration point values are not changed if no convection takes place:

ξnewip _ ξold

ip if C _ 0 ¤ α _ 0 (4.75)

Total displacements The total material displacements in an ALE simulation cannot di-rectly be seen from the element mesh. Although it is not necessary to calculate the total dis-placements for the simulation, it gives insight into the material flow during the simulation.Also, an impression of the accuracy of the calculation can be obtained if the initial meshis compared to the current mesh minus the total displacement. Contrary to the incrementaldisplacements the total material displacements also must be convected. The procedure toconvect the total displacement is a little different from what was described earlier, as thetotal displacements are stored per element in the nodes and the state variables are known atthe integration points.The element values of the total displacement are “extrapolated” to the nodes with a localsmoothing factor:

denode _ N

∑k� 1

Nk e βr r v βzzv βhhg dknode (4.76)

with βr _ 1 d Cr v � � �The nodal total displacement is calculated by averaging the contributions of the elements:

d �node _ 1ne

ne

∑k � 1

denode

k (4.77)

Interpolation of the smoothed continuous field gives the element value of the total displace-ment:

dnewnode _ 8

∑k� 1

Nk e e r v zv hg old d e Cr v Cz v Ch g g d �nodek (4.78)


Figure 4.8: Geometry convection test.

The displacements are in principle continuous at the element boundaries, but the gradientsare not continuous. Hence, when convection is taken into account, discontinuous fields areobtained.

Test problem The influence of the global and local smoothing parametersα and β isillustrated with a test problem. A block, with dimensions 1x2x5, is given a displacement of0.1 per step in the direction of the longest edge, see Figure 4.8. The grid displacement iszero, so the material flows through the block.An initial distribution of a state variableξ is applied to the block. This initial distribution isnow convected with the material.In Figure 4.9 the results are depicted for different values ofα andβ. The figures on theleft side are obtained withβ _ 0, the figures on the right withβ _ � 3. The figures onthe top show the initial distribution of the state variable. The shown average nodal valuesare different for the differentβ. The other figures in Figure 4.9 show the results after 25convection steps. In Figure 4.9(d) no local or global smoothing is applied, which leadsto oscillations. Adding global smoothing without local smoothing will not improve theresults (Figure 4.9(f)). Also, local smoothing without global smoothing (Figure 4.9(c))gives unstable behaviour. Only acombination of local smoothing and global smoothingleads to stable, but also diffusive results (Figure 4.9(e)). In (Hu´etink 1986) is shown thatsmaller steps and smaller elements will improve the results.

Cross-wind diffusion Another problem which can occur during convection is cross-winddiffusion: Gradients normal to the streamlines level out in regions where only convectiontakes place. This is illustrated with the same block as in the previous test problem, but nowan inflow condition for the state variable is used. A value of one is assigned to the statevariable at the nodes of the element in the left upper corner and zero to the other nodes inthe inflow surface. The initial distribution is given in Figure 4.10(a).


¥ ¦ ¨ ª ¬ ¯ °¥ ² ´ ª ¬ ¬ ¬

(a) initial, β ¶ 0

· ¸ ¹ º ¼ ½ ¾ ¿· À Á º ¼ ½ ¼ ¼

(b) initial, β ¶ Ã 3

Å Æ Ç È É Ê Ë ÌÅ Í Î È Ï É Ê Ñ Ó

(c) α ¶ 0 Ô β ¶ 0

Õ Ö × Ø Ù Ú Û ÜÕ Ý Þ Ø ß Ù Ú à Û

(d) α ¶ 0 Ô β ¶ Ã 3

â ã ä å æ ç è êâ ë ì å í æ ç æ ï

(e)α ¶ 1 ð 64C1 ñ 2 Ô β ¶ 0

ò ó ô õ ö ÷ ø ùò ú û õ ü ö ÷ ý ø

(f) α ¶ 1 ð 64C1 ñ 2 Ô β ¶ Ã 3

Figure 4.9: Results for different values ofα andβ.


þ ÿ � � � � � �

(a) initial

� � � � �

(b) β ¶ 0

� � � � � � � �

(c) β ¶ Ã 3

� � � � � �

(d) β from Equation 4.79

Figure 4.10: Results for three different values ofβ, α _ 1 � 64C1 £ 2

The distribution after 75 convection steps is illustrated in the other three pictures of Fig-ure 4.10. From Figure 4.10(b) can be concluded thatβ _ 0 leads to much cross-winddiffusion. Choosingβ _ � 3 solves the cross-wind diffusion problem, but also gives un-stable results (Figure 4.10(c)). As was seen in the previous test problem local smoothing isneeded for a stable convection scheme. Thereforeβ is made dependent on the flow direc-tion, so that the local smoothing is only applied in the flow direction, but not perpendicularto it (Equation 4.79). The factorGLF, added to make it possible to keep a little smoothingperpendicular to the flow direction, is generally chosen between 0� 9 and 1.

βr _ e 1 d Cr g e 1 d!!!!

Cr

C

!!!! g a � GLF � a � 3; βz _ � � � (4.79)

The result of thisorthotropic local smoothing is shown in Figure 4.10(d). No cross-winddiffusion or instabilities can be seen in this figure. Hence can be concluded that with or-thotropic smoothing in grid aligned flows, cross-wind diffusion is effectively suppressedand enough smoothing is applied in the flow direction to keep the solution stable.


4.7 Solving the equations

4.7.1 Steady state solution

In the simulations of guillotining and shearing a transient calculation is carried out until thesolution is steady. A steady state is reached when:

� The velocity of surface points is tangential to the surface, i.e. the free surface move-ment is zero:

vg a n _ 0 (4.80)

� The spatial time derivatives of the state variables are zero:

∂ξ∂t _ 0 (4.81)

If vg _ 0 this is equal to:

ξ d vm a hiξ _ 0 (4.82)

The conditions above are not checked during the simulation, but the simulation is continueduntil the material at the outflow has undergone the complete deformation process. Thisproved to be sufficient to satisfy the steady state conditions.

4.7.2 Proportional increment

The solutions for different time steps in transient calculations that converge to a steady stateare very similar. Therefore, the calculations can be speeded up if the predictor part of thefirst iteration is skipped in subsequent steps. As an initial guess the material displacementincrement of the converged solution of the last step is used:

w ∆u1m y n� 1 _ w ∆um y n w ∆un� 1

presyw ∆un

presy (4.83)

When the prescribed displacementsw ∆upresy are changed, the initial guess is also changedproportional to the change in the prescribed displacements.This method proved to be unstable when the solution converged in one iteration per step. Inthat case only the corrector part is carried out, which is very fast, but the Newton-Raphsoniterations diverges in one of the subsequent steps. Therefore, at least two iterations per stepare required, which means that the solution will always be adapted. This method can giveproblems when the material displacement increment changes abruptly from one step to theother, e.g. when points come into contact with the tools.

4.7 Solving the equations 57

4.7.3 Iterative solvers

In the predictor/corrector mechanism a linearised set of equations has to be solved at eachiteration. This is normally carried out using a direct solver, which factorises the stiffnessmatrix (e.g. Cholesky decomposition followed by back-substitution). Also, iterative solverscan be used, which approximate the solution in an iterative process (Barrett et al. 1994).These solvers introduce an extra error, which may worsen the Newton-Raphson iterationprocess. However, for large 3D systems iterative solvers can become attractive when theyprovide acceptable answers, using less time or memory. Examples of the use of iterativesolvers in forming processes are given in (Boogaard et al. 1998) and (Boman et al. 1998).

The memory used for a direct solver is in 2D proportional ton32 and in 3D proportional to

n52 , wheren is the number of degrees of freedom. As iterative solvers need memory which

is only proportional ton, memory can be saved using iterative solvers.The difference in speed between a direct and iterative solver is less clear. For a direct solverthe Cholesky decomposition takesn2

b a n and the substitutionnb a n operations, wherenb isthe bandwidth of the matrix. Using bandwidth-optimisation the solution speed can be in-creased. The speed of an iterative solver is determined by the number of iterations it needs toreach the convergence criterion. The convergence of an iterative solver depends on the con-dition of the matrix. Large condition numbers, caused by plasticity, stiff contact elementsor connectivity require more iterations. To speed up convergence, a preconditioner can beused. Preconditioning of the system of equations should decrease the condition number,which improves the convergence behaviour. The costs of preconditioning should be lowerthan the gain in the solution time.

The convergence criterion of the iterative solver is made dependent of the error in theNewton-Raphson iterations (Boogaard et al. 1998). The error in the i-th Newton-Raphsoniteration in the solution of the non-linear equations is (Equation 4.49):

w Ri y _ w Fext y d w Fiint y

The internal force vector at iterationi k 1 can be written as the internal force vector atiterationi, a linearised increment and an errorw ei � 1 y :

w Fi � 1int y _ w Fi

int y k r Ki t w ∆∆ui � 1 y k w ei � 1 y (4.84)

With:

w ∆∆ui � 1 y _ r Ki t � 1 w Ri y (4.85)

and some rewriting one finds that the norm of the error is equal to the norm of the unbalanceforce:

� � w ei � 1 y � � _ � � w Ri � 1 y � � (4.86)


The Newton-Raphson iterations are stopped when the mechanical unbalance is small enough.

εi _ � � w Ri y � �� w Fi

int y � �" εF (4.87)

The approximationw ∆∆u � y of w ∆∆u y of the linearised set of equations calculated by theiterative solver gives an errorw e� y :

r Ki t w ∆∆u � i � 1 y _ w Ri y k w e� y (4.88)

The iterative solver is stopped as:

� � w e� y � � �δ � � w Ri y � � (4.89)

or if the maximum number of iterations is exceeded.If w ∆∆u � y is used instead ofw ∆∆u y , Equation 4.84 becomes:

w Fi � 1int y _ w Fi

int y k r Ki t w ∆∆u � i � 1 y k w e� y k w ei � 1 y (4.90)

The total error now consists of two parts:

� � w Ri � 1 y � � _ � � w ei � 1 y k w e� y � � � � � w ei � 1 y � � k � � w e� y � � (4.91)

The global convergence norm is then:

� � w Ri � 1 y � �� w Fi � 1

int y � �_ εi � 1

�� w ei � 1 y � �

� � w Fi � 1int y � �

k δεi (4.92)

If the linear set of equations is solved exactly, thenδεi vanishes and the left part of the equa-tion above would be the unbalance ratio. Ifδεi # εF then the Newton-Raphson iterationprocess is hardly influenced by the error of the iterative solver. Thereforeδ is chosen as:

δ _ ηεF

εi (4.93)

with a small numberη. In the first Newton-Raphson iterationδ is a fixed number (δ _0 � 001), asεi is not known. In the subsequent iterationsδ is related to the convergence of theNewton-Raphson procesεi.

Chapter 5

Contact and friction

5.1 Introduction

The contact between the tools and sheet is an important factor in shearing. The result of theshearing operation depends on the shape of the tools and their motion. It has to be identifiedwhere the sheet and tools make contact and what the contact conditions (contact pressure,friction) are. The contact model and its consequences for the shearing models is describedin this chapter.

A contact model should satisfy the following mathematical constraints:

g � 0

τN

�0 (5.1)

τN a g _ 0

Which states that the product of the normal contact stressτN and the gap widthg, thenormal distance between the sheet and the tool, is always zero. The gap width is zero andthe contact stress is negative (pressure) if there is contact between the bodies. When thebodies do not make contact the gap width is positive and the contact stress is zero. Thecontact inequalities of Equation 5.1 have to be added to the weak form for the deformationof the continuum (Chapter 4).The contact model applied here is based on the contact layer approach of (Lugt 1988)and (Vreede 1992), in which the non-penetration condition is approximated by a penaltymethod. In this method a penalty, which can be seen as a contact stiffness, is applied as thepenetration of two contacting bodies occurs.Friction is modelled in a similar manner as elasto-plastic material behaviour. It is assumedto have a reversible (elastic) part where the bodies stick to each other and a irreversible(plastic) part in which the bodies slip with respect to each other. The shear stress at whichthe contact changes from stick to slip is determined by the normal contact pressure and the

60 Contact and friction

Coulomb friction coefficientµ.

The tools are assumed rigid. This means that the influence of the deformation of the toolson the process is neglected. But it will be possible to extend the shearing models withdeformable knives. It was shown by (Lugt 1988) that, for stationary processes (rolling), thedeformation of the tools could be calculated using this contact model.

5.1.1 Kinematics and stresses

The contact between two bodies is made at the many asperities on the surface. Becausethe height of an asperity is small compared to the dimensions of the total contact area, thecontact region can be modelled as a smooth layer with a small but finite thickness∆x.

∆x∆x

Figure 5.1: Contact layer with thickness∆x.

A local coordinate systeme e1 v e2 v e3 g can be defined in the reference surfaceSR having equalnormal distance to the surfacesSA andSB of the contacting bodies A and B. The orientationof the local basis is a function of space and time:

ei _ ei e x v t g (5.2)

e3 is taken normal toSR; e1 is perpendicular toe3 and a reference vector. These localcoordinates are used to define the contact stresses and the relative surface velocityd, thedifference in velocity of two contact pointsA andB on the surfacesSA andSB. The contactstresses are shown in Figure 5.2 with the normal contact stressτN _ τ3 and the shear stressτT with componentsτ1 andτ2.The definitions ofd and the rate of the contact stressτττ are elaborated in (Hu´etink, Vreede,and Lugt 1989). Corrections are made for the rotation and curvature of the reference surface

by introducing a Jaumann derivative of the contact stress$

τττ.

5.2 Constitutive relations for contact

The relation between the gap widthg, the relative surface velocityd and the contact stressτττmust now be determined. This relation has the same function as the constitutive equationsin continuum mechanics and is given in this section.

5.2 Constitutive relations for contact 61

SB

SR

SAτ3

τ1

e1

τ3

A

e2

e3

τ2

B

τ1

τ2

Figure 5.2: The contact stresses.

It is assumed that the normal component ofd is completely elastic and that the tangentialcomponent of the relative velocity can be decomposed into a reversible and an irreversiblepart:

d _ drev k dir (5.3)

The linear elastic contact is then written as:$

τττ _ EC a drev _ EC a e d d dir g (5.4)

The diagonal contact stiffness tensorEC, in matrix form, is:

r EC t _

%&E11 0 00 E22 00 0 E33

'((5.5)

The contact stiffnessE33 in normal direction is the penalty, which prevents the penetration ofone body into another body. The tangential stiffnesses are equal (E11 _ E22) and dependenton µ.The transition from stick to slip is governed by the contact functionφC. This function,comparable with the yield function in plasticity, divides the contact stress space into a regionφC " 0 where stick occurs and a surfaceφC _ 0 where slip takes place:

φC _ τ21

k τ22 d mine µ2τ2

3 v τ2maxg (5.6)

The slip surface, drawn in Figure 5.3, is a combination of a cone with top angleµ and acylinder with radiusτmax, the maximum possible shear stressτT .The contact stresses must remain on the slip surfaceφC _ 0 for sliding, which gives, for aconstant coefficient of friction:

φC _∂φC

∂τττa τττ _

∂φC

∂τττa

$τττ _ 0 (5.7)


τ2

τ3

τ1

φC

µ

Figure 5.3: The slipsurface in stress-space.

According to (Lugt 1988) a Jaumann rate$

τττ can be substituted forτττ in Equation 5.7.The irreversible relative contact velocity can be related to a potentialϕ using a normalityrule:

dir _ κ∂ϕ∂τττ

(5.8)

with the Kuhn-Tucker relations:

κ _ 0 if e φC " 0g or e φC _ 0 and φC " 0g (5.9)

κ � 0 if e φC _ 0 and φC _ 0g (5.10)

This potential is only a function of the shear stresses, as the normal component ofdir isassumed to be equal to zero:

ϕ _ τ21

k τ22 (5.11)

Combining Equations 5.3-5.11 leads to:$τττ _ ) EC d YC + a d _ C a d (5.12)

with the slip tensorYC

YC _EC :

∂ϕ∂τττ

∂φC

∂τττ: EC

∂φC

∂τττ: EC :

∂ϕ∂τττ

(5.13)

5.3 Weak form and FE discretisation 63

As ϕ ,_ φC, we have a non-associated flow rule andYC becomes a non symmetric tensor.Written in matrix form:

r YC t _ 1

E11τ21

k E22τ22

%&E2

11τ21 E11E22τ1τ2 d µE11E33τ1τ3

E11E22τ1τ2 E222τ2

2 d µE22E33τ2τ3

0 0 0

'((5.14)

The tensorC of Equation 5.12 is set to zero if there is no contact or the bodies lose contact.For contact without slip,YC _ 0 andC _ EC.

5.3 Weak form and FE discretisation

The contribution of the contact stresses to the weak form is:

δWC _ `VC

hiδv : σσσdV (5.15)

whereVC is the contact volume andδv the virtual velocity.With the assumption of a thin contact layer with a linear velocity distribution over the thick-ness between the contacting bodies A and B:

hiv _ ∆v

� ∆x �(5.16)

and some rewriting ((Lugt 1988) and (Vreede 1992)) this weak form becomes:

δWC _ `SC

δvA d δvB

� ∆x �a τττ � ∆x � dS _

`SC

e δvA d δvB g a τττdS _ `SC

δ e ∆v g a τττdS (5.17)

The rate of the weak form is then:

δWC _ `SC

δ e ∆v g a τττ k δ e ∆v g a τττJSdS (5.18)

which can be worked out using the definition of Jaumann derivative of the contact stressand Equation 5.12 (Lugt 1988).The contact volume is now divided into hexahedral contact elements, with one face con-nected to the rigid tool and the opposite face connected to the bulk material (Figure 5.4).dandδ e ∆v g can be expressed in the nodal velocitiesvk (Equation 4.36) using the assumptionof a linear velocity distribution in the contact layer (Equation 5.16). When the nodal ve-locities are transformed from the local to the global coordinates, the stiffness matrix of thecontinuum plus contact can be assembled.


rigid tool

contact node

bulk+contact node connected

bulk node

bulk elements

contact elements

Figure 5.4: Contact elements.

In an incremental formulation a step consists of a predictor part and a corrector part. In thepredictor part the displacement increments are calculated using the linearised weak form.When these displacements are known, the new gap width is calculated (Section 5.5.2). Inthe corrector part, corrections are made for non-linear effects, such as opening/closure andstick/slip. The normal contact stress is calculated directly from the gap width function:

τN _ E33 a g for g

�0 (5.19)

The shear stresses are adapted incrementally. They are increased using an elastic trial stress:

e τi g n� 1 _ e τi g n k Eii ∆di i _ 1 v 2 (5.20)

If the tangential stressτT is outside the slip surface of Equation 5.6, then the shear stressesare scaled back to the slip surface at constant normal stress and the contact is set to slip forthe next iteration or increment.

5.4 Damping

To stabilise the calculation when points come into contact or lose contact some dampingis added in the normal direction. The stiffness in the normal direction is increased in thepredictor part. Even for small positive values of the gap widthg " gdamp, some stiffness isadded. The damping valueCd is generally much smaller thanE33.The normal contact stress calculated in the corrector part becomes dependent on the damp-ing valueCd and the change in gap width∆g:

τN _ 0

τN _ b 1 d ggdamp

j Cd a ∆g

τN _ E33 a g k Cd a ∆g

forg - gdamp

0 "g

�gdamp

g

�0

(5.21)

5.5 Tools 65

This means that the contact pressure is already built up before the points come into contactand that a negative contact pressure (tension) prevents losing contact too fast. The dampingin the normal direction has also an indirect stabilising influence on the shear stresses. Aschematic view of the contact model with damping is given in Figure 5.5:

E11E22

E33Cd

SA

SB

Figure 5.5: Schematic view of the contact model.

5.5 Tools

5.5.1 Tool description

Rigid tools are used for the simulations in this thesis. It is assumed that the edges of thetools have a circular shape, and hence the shape of these idealised tools can be described bya limited number of standard shapes. These standard shapes can be written in rather simpleanalytical functions as lines and circles in 2D and (parts of) planes, cylinders and tori in 3D.More complicated tools can be described using a finite element mesh, which means thatthe shape of the tools is approximated by an assembly of many simple standard shapes (theelements).

5.5.2 Contact search

The gap widthg from the contact model has to be calculated at each Newton-Raphsoniteration of the finite element simulation. This gap width is determined using a closest point


projection algorithm. A projection is made from a contact node to some selected standardshapes, describing the tools. The closest projection becomes the contact point.In the shearing simulations the number of standard shapes is limited (� 10), but when manystandard shapes (elements) have to be searched for possible contact, efficiency in the contactsearch becomes very important. In that case, algorithms are used that limit the number ofshapes to be searched (Carleer 1997).When the contact points are found, the nodes of the contact elements located on the tools aremoved to these contact points. The gap width is now calculated from the nodal coordinatesof the contact element.

5.6 Contact and ALE

In the simulations of guillotining and slitting, the contact model is combined with an ALEformulation. The nodes of the contact element are displaced convectively in accordancewith the nodes on the surface of the bulk material and the tools.A consequence of this convective displacement is that convection has to be taken into ac-count for the history-dependent variables. The normal contact stress is only weakly historydependent when damping is applied, as it depends not only on the current gap width, butalso on the change in gap width. The shear stresses in the contact element are incrementallycalculated (Equation 5.20), and hence a convection increment of the shear stress should becalculated.The convection in the contact elements is neglected however. This appears not to influencethe results of the simulations too much. As a semi-coupled ALE formulation (Chapter 4) isused any unbalance in the contact is reduced during the iteration process.

5.7 Concluding remarks

For the shearing simulations a high penalty (E33 � 105d 106) is needed to keep the pene-

tration of the knives into the sheet sufficiently small. Increasing the penalty decreases thecondition of the system of equations, leading to more Newton-Raphson iterations, iterativesolver iterations or even non-converging solutions. This makes the choice of contact param-eters (stiffness, damping) often very critical. Therefore, it might be interesting to look ataugmented Lagrangian methods (Simo and Laursen 1992), in which a much lower contactstiffness can be used and the resulting larger penetration is reduced in some iterations.

Chapter 6

Shearing models

6.1 Introduction

In the preceding chapters the finite element method with an ALE formulation and a contactmodel were treated. In this chapter we explain how that theory can be applied to modelshearing processes. The domain must be subdivided into elements for the FEM method.In an ALE formulation the initial and the final geometries are described using the samemesh topology. Therefore, already in the definition of the initial mesh, the expected finalgeometry must be anticipated. The meshing and the mesh management for the shearingsimulations are treated in this chapter.

The mesh management is carried out within the iteration loop of the Newton-Raphson itera-tion in the semi-coupled ALE formulation (Section 4.4.3) that is used. The grid coordinatesafter the predictor part, i.e. the coordinates at the beginning of the step plus the materialdisplacement increments, are denoted byxm. The new grid coordinates, determined usingthe mesh management procedures, are written asxg. The convective displacement is thengiven byxm d xg. The mesh management must fulfil the following requirements:

� The domain must be described accurately. No material should be gained or lost duringthe simulation, except through in- or outflow boundaries.

� The element shape must be kept regular.

� The mesh management must be efficient, as it is carried out at each Newton-Raphsoniteration.

� It must be stable and robust.

Furthermore, it is desirable that the mesh management procedure be able to maintain or toproduce mesh refinements in critical areas.

68 Shearing models

Two types of shearing models are discussed. A 2D model of orthogonal shearing in Sec-tion 6.2 and 3D models of guillotining and slitting in Section 6.3. Results of simulationswith these shearing models can be found in Chapter 7.

6.2 Orthogonal shearing model

Orthogonal shearing is modelled with a plane strain finite element model. The choice ofthe initial mesh and the mesh management procedures will be explained here. 4-node linearquadrilateral elements were used for the simulation. Mesh management procedures weredeveloped to reach the maximum penetration possible, while keeping reasonably shapedelements. With this 2D model, the shearing process can be simulated up to a knife pene-tration of about half the sheet thickness. For further continuation of the simulation an ALEformulation alone is not sufficient. Some mesh management techniques used for this 2Dmodel are also used for the 3D models of guillotining and slitting in the subsequent section.

6.2.1 2D quadrilateral mesh generation

The initial, rectangular domain of the sheet to be cut is divided into a number of regions bymaster lines connected by poles. An example is given in Figure 6.1 with two regions andstraight and curved master lines.

P1

P2P3

P4

P5P6

P7R1

R2

ML6

ML1

ML2

ML5

ML4

ML3

ML8

ML7

Figure 6.1: Poles (Pi), Master lines (MLi) and regions (Ri).

The number of nodes and a refinement factor must be specified for each master line, whichdivides the master lines into segments. Then, the regions can be meshed separately usingthe segments on the master lines. The transfinite mapping method (Haber et al. 1981) isused here to subdivide the regions into elements, which leads to a structured mesh. Otherpossibilities leading to unstructured meshes are the paving method (Blacker and Stephenson

6.2 Orthogonal shearing model 69

1991), the conversion of a triangular mesh to quadrilaterals (Johnston et al. 1991) or theapproach of (Petersen and Martins 1997).

Transfinite mapping

Each region is bounded by four master lines, discretised into a number of nodes. Oppositemaster lines must be discretised with the same number of nodes.

10

1v

u .0 / 00

.1 / 00P1

.v0

P2

.v0

Q2

.u0

.1 / 10.

0 / 10

Q1

.u0

Figure 6.2: Transfinite mapping.

The discretisation of the master lines is given byP1 e ui g and P2 e ui g with 1

�i

�n and

Q1 e vj g andQ2 e vj g with 1

�j

�m, in which ui andvj are the coordinates of a point in a

unit square. The coordinates of nodesx in the interior are now given by a bilinear operator(Haber et al. 1981), which maps a unit square onto the true surface:

x e ui v vj g _ e 1 d vj g P1 e ui g k vjP2 e ui g k e 1 d ui g Q1 e vj g kuiQ2 e vj g d uivjx e 1 v 1g d e 1 d ui g e 1 d vj g x e 0 v 0g d (6.1)

e 1 d ui g vjx e 0 v 1g d ui e 1 d vj g x e 1 v 0g6.2.2 2D mesh management

When determining the new positions of the nodes, two different kind of nodes are distin-guished:

� surface nodes, which must remain on the surface;

� internal nodes, which can be moved freely in the material as long as a regular elementshape is preserved.

For the 2D simulations the mesh management procedures as described by (Ponthot 1989)are used. The new nodal coordinates are calculated in the following order:

1. The starting point is the mesh after the predictor partxm.

70 Shearing models

2. First, the new coordinates of the poles are calculated. The poles are preferably definedas Lagrangian or Eulerian points, having a grid displacement equal to the materialdisplacement or equal to zero respectively. If this is not possible, a suitable functionmust be formulated for the grid displacement of the pole, which depends on the gridor material displacement of other nodes. This choice is problem dependent and oftencrucial for a succesful mesh management.

3. Next, the nodes on a master line are redistributed on this master line. Master linesin the interior of the mesh are kept straight. Master lines describing a free boundaryshould follow that boundary. This redistribution is explained below.

4. When the new boundaries of a region are known, the nodal coordinates of nodes inthe interior of a region are calculated again with the transfinite mapping method ofEquation 6.2 or with some smoothing method.

5. Finally, a contact search is performed for the contact elements (Chapter 5) accordingto the new boundariesxg of the bulk material.

If the mesh management is finished the corrector part of the Newton-Raphson iteration canbe performed (Figure 4.5).

Redistribution of nodes on a master line

The nodes of a curve are repositioned with the help of an estimator functionE e sg , suchthat this estimator function is equally distributed over the segments (the line between twoneighbouring nodes) of the master line (Ponthot 1989), (Ponthot 1992).s is the coordinatealong the master line, which consists ofn nodes andn d 1 segments.The estimator function is defined as:

E e si g _i � 1

∑k � 1

wk∆sk; i _ 2 : n; E e s1 g _ 0 (6.2)

wherewk is the weight function and∆sk _ sk� 1 d sk is the length of thekth segment. Thetotal length of the master line isL. The functionE e sg is calculated using the coordinatesxm. The requirement that the estimator function is equally distributed gives:

E e sk� 1 g d E e sk g _ Constant (6.3)

Therefore, the new positionsi � of nodei is calculated by:

E e si � g _e i d 1g

e n d 1gE e sn g for i _ 2 : n d 1; s1 � _ 0; sn � _ L (6.4)

so that the grid will be fine where the the slope of the estimator function is large and coarseif the slope is small, as is illustrated in Figure 6.3.


E.s0

s

Figure 6.3: Determination of the new position on a curve, using the weight function.

The weight functionw e sg is not yet determined. The simplest form of the weight functionis wk _ 1, which results in an uniform element distribution along the master line. An initialrefinement of the form∆sn � 1 _ α∆s1 can be maintained during the simulation by choosingthe following weight function:

wk _ 1

α 1 k 2 1 31 n 2 13 (6.5)

These simple weight functions can already be very effective.More sophisticated weight functions can be composed by incorporating geometrical infor-mation, such as the angle between segmentsθk, or error indicators as the derivatives of thestate variablesS.

wk _ g1

α 1 k 2 131 n 2 13 L k

P e ∆sk g 4 g2 � S�6 L0 � S� ds

k g3 � S7 s �6 L0 � S7 s � ds

k g4 � S7 ss�6 L0 � S7 ss� ds

k g5 � θ �6 L0 � θ � ds8 (6.6)

The functionP e ∆sk g is added to avoid extremely small elements in the case of localiseddeformations:

P e ∆sk g _ 1 if ∆sk � ∆smean

P e ∆sk g _ exp 9 d g6∆smean: ∆sk d 1

n d 1 ; (6.7)

The optimum weight factorsg1 to g6 can be determined by trial and error.

Calculation of new coordinates on a master line

Now the new positions of the nodes along the master linesi � are known, the nodal coordi-nates can be calculated.

72 Shearing models

If the master line has to be kept straight, then the new coordinates can be found by a simpleinterpolation on a straight line through the new position of the poles. Straight master linesare used for internal master lines or for master lines that are suppressed in one direction.The determination of the new nodal positions of points on a curved master line is an inter-polation problem, which has the same characteristics as a convection problem. Thereforeconvection schemes can be used for the calculation of the new nodal coordinates, as shownbelow.

s

ygi ?

ymi < 1yi = 2

m

yimym

i = 1

si = 1 si > si

Figure 6.4: Interpolation of nodal coordinates.

The new position of the nodexig can be calculated by a linear interpolation between the

coordinates of the nodesxi � 1m andxi

m or xim andxi � 1

m depending on whethersi � is larger orsmaller thansi . Linear interpolation is the same as a first order upwind convection scheme(Equation 6.8). The componentse yv zg of xi

g are calculated independently from each otherwith this scheme, that for they component reads:

yig _ yi

m d C ) yim d yi � 1

m+ if si � " si (6.8)

WhereC is the local Courant number:

C _

!!!!si � d si

si d si � 1

!!!! (6.9)

If the Courant number is larger than the maximum valueCmax somewhere on the masterline, then all Courant numbers are limited. The maximum is usually chosen lower than orequal to 0� 5. This means that the error will not be equidistributed in this step, but that canbe corrected in subsequent steps.

First-order schemes are known to be diffusive. A more accurate description of the boundarycan be obtained with higher order interpolation or convection schemes. Hence, a Lax-Wendroff scheme with van Leer limiters is used here, which shows no spurious oscillations(Akkerman 1993). The newy-component of the new coordinatesxi

g calculated with this


scheme, forsi � � si , is given by:

yig _ yi


m+

d 12C e 1 d C g ? ψ e ri � 1A 2 g e yi � 1

m d yim g d ψ e ri � 1A 2 g e yi

m d yi � 1m g B (6.10)

The van Leer limiterψ e r g stabilises the Lax-Wendroff scheme. In the case of large gradients(ψ e r g D 0) the limited Lax-Wendroff scheme degenerates to the Upwind scheme. Theinfluence of the limiter is also small for small gradients, asψ e r g D 1 and the original Lax-Wendroff scheme is retained:

ri � 1A 2 _ yim d yi � 1

m

yi � 1m d yi

m

; ri � 1A 2 _ yi � 1m d yi � 2

m

yim d yi � 1

m; ψ e r g _ r k � r �1 k � r � (6.11)

It is assumed that the grid is locally equidistant, which is a reasonable approximation whenneighbouring elements do not differ too much in size. At the ends of a master line, whennot enough points are available for the limited Lax-Wendroff scheme, an Upwind schemeis used.

Smoothing

A smoothing algorithm can be used for the calculation of the new nodal positions in theinterior of a region.

xg

xm

xCm

xCg

Figure 6.5: Smoothing of internal mesh.

In a loop over the nodes in the interior, the new coordinates of a node are calculated as themean value of all coordinates of the neighbouring nodes in the interior. If the new coor-dinates of the neighbour node are already known (e.g. nodes on a master line), these newvalues are used (Figure 6.5). This is a simple form of the Laplacian smoothing technique.

xCg _ 1

N N F∑k � 1

xkg

k N

∑k � N F � 1

xkm

¢ (6.12)

74 Shearing models

One loop over all interior nodes proved to be sufficient to reach an accurate solution, as thisloop is repeated at each calculation step and the shape of the domain does not change muchin one step. This method can depend on the sequence in which the nodes are treated, butthat effect was not apparent in the simulations.This simple algorithm already gives good results for convex domains, but problems canarise near concavities, where the elements can be inverted. A proper subdivision of thedomain into convex subdomains solves this problem.A drawback of smoothing is the limited influence of the mesh on the master lines on themesh in the interior of the region. In spite of refinements on the master line, the smoothingalgorithm will produce equally sized elements in the interior of a region.Extensions of the smoothing algorithm (Equation 6.12) are possible by taking into accountthe element shape (angle, size ..) or an error indicator in the weighting of the contributionof each node in the averaging function (Joun and Lee 1997), (Canann et al. 1998), (Askeset al. 1999). These methods are more time consuming and therefore not always attractivefor ALE formulations, since the smoothing is performed at each iteration.

Transfinite mapping If the transfinite mapping method is used to create the initial mesh,it can also be used to adapt the coordinates of nodes in the interior of a region. An advantageof this method is that refinements in the boundary have an influence throughout the completeregion and therefore leads to better results than smoothing in the case of structured meshes.

6.2.3 Plane strain shearing model

In this section some details are presented of the application of the meshing and mesh man-agement techniques for the orthogonal shearing model. This model is illustrated in Fig-ure 6.6 and will be explained in the following paragraphs.

Initial mesh and mesh management The initial mesh is shown in Figure 6.7. By subdi-vision of the sheet into a number of regions a mesh refinement is achieved in the shear zone.The regions are meshed with the transfinite mapping method.

Boundary conditions and contact The knives are described by the contoursC1 andC2,which consist of two lines and a rounded corner with a small radius. Contact elements areplaced on the top and bottom of the sheet (the dash-dotted lines in Figure 6.6). The contoursC3 andC4 are auxiliary contours used for an easier projection of the contact nodes. Thesecontours are moved in such a way that they never make contact with the sheet.The clamping of the sheet is not modelled with a real downholder, but with boundary con-ditions applied directly at the nodes. Two cases are studied. In the first case both sidesare clamped by suppressing the displacement in thez-direction of the nodes onML1 andML2. In the other case only one side is clamped by suppressing thez-displacement ofML2.


P3

ML2

ML3

y

C1

C3

P4C2

P2

ML1

P1

C4

zx

ML4

ML5P6

ML6P5

Figure 6.6: Schematic view of 2D shearing model.

Figure 6.7: Initial mesh.

The model can be extended for a downholder by replacing the auxiliary contourC3 with acontour describing the downholder.

Mesh management The poles on the boundary are all Lagrangian, except forP5 andP6.These poles can only have a grid displacement in they-direction to keep them at the cuttingedge of the tool. The grid displacement of the inner poles is calculated by an interpolation

76 Shearing models

of the grid displacement of the other poles. All master lines keep their initial refinementduring the simulation, exceptML3 andML4. For these master lines, the angle between theelements is taken into account in the weighting function, as these boundaries will becomecurved. The grid displacement of the nodes in a region is determined using the transfinitemapping algorithm.

In Figures 6.8 and 6.9 the finite element meshes are given for a knife penetration of 40%of the sheet thickness for both types of boundary conditions. In these figures it can be seenthat the mesh management procedures succeeded in preserving a regular element shape.The refinements are kept at the cutting edge and develop at the transition from the draw-inzone to the shear zone, due to the weighting factor for the angle between the segments onthe master line.

Figure 6.8: Meshes, 40% penetration; both sides clamped.

Figure 6.9: Meshes, 40% penetration; one side clamped.

6.3 3D models of guillotining and slitting 77

The simulation with one side clamped (Figure 6.9) is carried out with three different meshmanagement options forML3 andML4. The differences in geometry between the simula-tions are shown in Figure 6.10 for the contour of the sheet at the edge of the lower knife andthe transition from the draw-in zone to the shear zone at the lower knife. A comparison ismade between master lines with a weight function containing the influence of the angle be-tween elements and master lines with a weight function containing only the influence of theelement length and also between convection with the Upwind or the limited Lax-Wendroffschemes. The differences are small, but it can be seen that the limited Lax-Wendroff schemewith a weight function containing the angle between the elements describes the geometrybest.

-0.2

-0.15

-0.1

-0.05

0

5.05 5.06 5.07 5.08 5.09 5.1 5.11 5.12

y-co

ordi

nate

[mm

]

G

z-coordinate [mm]

Contour sheet at the lower knife

Angle, UpwindAngle, Lax-W

No Angle, Lax-W

Figure 6.10: Results of different mesh management schemes.

6.3 3D models of guillotining and slitting

In this section the finite element models of guillotining and slitting are treated. These mod-els are almost identical, differences are found only in the boundary conditions and the shapeof the tools, which are described in Section 6.3.4. The implementation of ductile fracture inthe model is presented followed by the definition of the initial mesh and the mesh manage-ment procedures. This section ends with some notes about preprocessing.

78 Shearing models

Deformed

Crack front

Cracked

Upper knife

(a) guillotining

Crack front

Deformed

Cracked

Lower knife

(b) slitting

Figure 6.11: Estimate of the shape of the crack front.

6.3.1 3D stationary crack growth

The objective for the 3D stationary simulations of guillotining and slitting is to calculate thesteady state. In this steady state a 3D crack front is present, which grows with a constant(known) speed in the flow direction. The shape and position of the crack front, however, arenot known beforehand. A number of experiments were carried out by (Atkins 1981) andTNO (Appendix B) to determine the shape and position of the crack front. In these studiesthe guillotining process was interrupted to get an approximation of the steady state. The dif-ferent phases of the shearing process (Figure 1.2) can be recognised from the cross-sectionsof this incomplete guillotine cut. The theory of (Atkins 1981) is followed for the estimationof the shape of the crack front, although not clearly confirmed by the experiment presentedin Appendix B. More experiments are needed to extract general rules for the shape of thecrack front.

Atkins suggested the possible existence of vanishingly small cracks, keeping pace with theknives (and therefore hard to recognise), before cracks running ahead of the tools finallyseparate the parts. Therefore, it is assumed that cracks initiate at the edges of the toolsat some defined penetration. The cracks grow, first at the same speed as the knives, lateron faster until the parts are separated before 100% knife penetration. A symmetric shapeis chosen, although it can be non-symmetric due to non-symmetric shearing conditions(clamping, one knife sharper than the other). The estimate of the crack front is drawn inFigure 6.11.An ALE formulation is used for the calculation of the steady state in guillotining and slit-ting. The initial mesh for these simulations is some estimate of the steady state shape, inwhich a crack is present. Therefore a crack front is modelled in the initial mesh according


to the estimated shape of Figure 6.11. The crack can be seen as a special kind of free sur-face, producing new free surfaces. With a suitable mesh management procedure it shouldbe possible to adapt the position and shape of the crack front according to one of the frac-ture criteria of Section 3.3.2. The fracture criterion should have a critical value along thecrack front. A condition for this is that the initial chosen crack front is close enough to thesteady state crack front, as the shape of the domain may not change too much in an ALEcalculation. A combination of the ALE method with a nodal release procedure can increasethe flexibility of the method, without the need for a complete remesh.In the simulations shown in this thesis, the position and shape of the crack front was fixedduring the simulation. The grid displacement of nodes on the crack front was zero, whichmeans that the crack was propagating with constant speed but was probably not in thecorrect position. The correctness of the estimated position and shape of the crack frontwas assessed using a fracture criterion calculated during the simulation, but the initial crackfront has not yet been automatically adapted according to that fracture criterion. Results fordifferent positions and shapes of the crack front are given in Section 7.4.

6.3.2 3D hexahedral mesh generation

For the simulation of metal forming problems the use of hexahedral elements is to befavoured above other types of elements. A drawback of the use of hexahedrals is thatthe generation of a hexahedrals-only mesh can be difficult, as a robust and flexible meshgenerator for an arbitrary domain still does not exist.Some methods to create a hexahedral mesh are:

� The domain can be divided into simple subdomains, which can be meshed with amapping method or by sweeping, i.e. the extrusion of a 2D mesh into a third di-mension. This can be done manually, but also some algorithms are proposed ((Priceand Armstrong 1997), (Calvo and Idelsohn 1998)) to automatise the subdivision intosimple shapes.

� It is also possible to use the better developed tetrahedral mesh generators. The tetrahe-dral mesh can be split up into hexahedrals, but this yields highly unstructured meshesof poor quality. This method is used for automatic remeshing by (Zuo 1999).

� Another method is the combination of a quadrilateral surface mesh and a simple struc-tured hexahedral mesh in the interior of the volume. The gap between the boundarymesh and the hexahedrals in the interior is filled up with topologically correct hexa-hedral elements (Schneiders 1996). Improvement of the shape of boundary mesh ispossible with smoothing techniques (Lee and Yang 1999), (Tekkaya 1998). Coarsen-ing of the interior mesh is possible by the coalescence of elements and the additionof tyings (Gelten 1998), the opposite of the h-type of mesh refinement.

80 Shearing models

For the generation of the initial mesh it is chosen to subdivide the domain manually intosimple subdomains. Since all considered initial domains have a similar shape, this has to becarried out only once. An advantage is that with the mapping of simple shapes a structuredmesh can be obtained, which simplifies the mesh management and allows the use of moreaccurate convection schemes (orthotropic smoothing, Section 4.6.1).

Initial mesh The initial volume is divided into 4 blocks (Figure 6.12), which each consistof 8 mappable sub domains. For each block a refinement in thex-direction can be given.The cross-section is subdivided into some regions to create a mesh refinement in the shearzone. The element topology for a cross-section perpendicular to thex-direction is equal forall blocks and shows some similarities with the 2D mesh. All cross-sections are in a planewith constantx-coordinate.

I. The first block is rectangular and positioned before the penetration of the knives intothe sheet.

II. The shear deformation takes place in block II until crack initiation.

III. The crack front is modelled in block III by doubling some nodes. The crack frontused in the simulations in this thesis is given in Figure 6.13. The area to the right ofthe thick line is cracked.

IV. In the last block the parts are completely separated.

A drawback of the equal topology for every cross-section is that the cross-sections in block I,where almost no deformation occurs, contain as many elements as the other blocks, leadingto an unnecessary increase in calculation time. A solution would be to coarsen (refine) themesh by coalescence (subdivision) of elements and the addition of some constraints or themismatching refinement technique proposed by (Yang and Lee 1999), which allows changesin element size between different subdomains.

6.3.3 3D mesh management

The determination of the new mesh is carried out in the following order:

� First, the new position of the crack front has to be determined.

� Next, all surface nodes are put back in thex-direction, keeping cross-sections in oneyz-plane. The newy- and z-coordinates are calculated with a convection scheme,using the coordinates of neighbouring nodes on a grid line in thex-direction.

� Then, the surface nodes are repositioned on the surface in theyz-plane, to keep theinitial refinements around the tool tips, where the largest deformations are found.


HIJ KLM

N

O OQ Q Q

R SFigure 6.12: Initial mesh for slitting simulation.

U V V Y [ \ ^ ` a Yc d f g h i j l m g n o q r s t o w x y

Figure 6.13: Crack front in finite element mesh.

� When the new positions of all the surface nodes are known, the new positions of theinternal nodes can be calculated.

� Finally, a new contact search is performed.

The mesh management procedures will now be explained in more detail.

Crack front adaptation The fracture criterion should have a critical value along the crackfront. If this is not the case, then the assumed crack front was not correct and the shape orposition of the crack must be adapted. As no automatic crack adaptation procedures have

82 Shearing models

yet been implemented, the nodes on the crack front are spatially fixed. This means that thecrack propagates at a constant speed.

Mesh management in thex-direction The geometry is meshed in such a way that the ini-tial mesh on the surfaces is regular, with grid lines that almost coincide with thex-direction.During the simulation this surface mesh is kept regular. As no automatic crack adaptationprocedures are yet implemented, the complete mesh is kept fixed in thex-direction (themain flow direction). The grid follows the free surfaces in theyz-plane.

x

yi = 2m

yi = 1m

yiint?

yim

yi < 1m

xi = 1m xi

mxig

Figure 6.14: Convection of nodal coordinates in thex-direction.

The limited Lax-Wendroff scheme, used for convection along a master line in Section 6.2.2,is now applied to convection along a grid line in thex-direction. This is illustrated in Fig-ure 6.14. The Courant number is not determined from a weight function, but directly fromthe material displacement in thex-direction. The newx-coordinatexi

g is equal to the initialx-coordinate.

C _xi

g d xim

xim d xi � 1

m(6.13)

An intermediate y-coordinateyiint is now calculated with the Lax-Wendroff convection

scheme in thex-direction, in whichym are the knowny-coordinates after the predictor part.The same procedure is applied to calculate the newz-coordinate:

yiint _ yi


m+

d12C e 1 d C g ? ψ e ri � 1A 2 g e yi � 1

m d yim g d ψ e ri � 1A 2 g e yi

m d yi � 1m g B (6.14)

The application of this scheme in thex-direction is illustrated with a test problem. The initialmesh is shown in Figure 6.15. A “bubble” is moving in thex-direction with a velocityV.The nodes on the upper surface are adapted using the described scheme. All other nodes arespatially fixed.In Figure 6.16 the results are shown after a number of steps. From this can be concludedthat the scheme used is stable but also shows some diffusion. As the curvatures in the


{

Figure 6.15: Initial mesh of test problem.

flow direction in this test problem are larger than in the shearing simulations, the scheme isaccurate enough to be used for these calculations.

Figure 6.16: Geometry of test problem after 0/25/50/100 steps.

Mesh management in theyz-plane When the convection in the flow direction for allsurface nodes is completed, a second convection step in theyz-plane (perpendicular to theflow direction) will be performed. This convection is needed to keep the initial refinementsaround the edges of the knives in blocks II and III. For the small deformations in blocks Iand IV, convection in thex-direction only is enough. As, after convection in thex-direction,the mesh consists of 2D cross-sections in theyz-plane, the mesh management proceduresdescribed for the 2D shearing model can be used again in the 3D model.The mesh in theyz-plane is, as in the plane strain shearing model, subdivided into poles,master lines and regions. For block II it is required that the polesP3 andP7 can only movein the y-direction and thatP2 andP6 move in thee 0 v 1 v 1g direction, to keep them at theedges of the tools. The other poles do not get an extra convective displacement.The nodes on masterlinesML1 till ML4 are repositioned along these master lines keepingthe initial refinements. Extra weight factors to describe the boundary more accurately arenot applied. Contrary to the 2D case,ym is not used, butyint instead. The grid displacementof the nodes onML5 is zero.

84 Shearing models

P2P4

P5

P6P7P8

R1 R2

R4

P1

R3

P3

ML1

ML4

ML3

ML2

ML5

xz

y

ML6

ML7

Figure 6.17: Mesh management yz-plane block II.

Two types of mesh management are used for the interior nodes. The nodes in the processzone are repositioned with the smoothing procedure of Section 6.2.2. The process zone isdivided into two convex regionsR2 andR3 by ML5, to avoid problems with concave re-gions. For internal nodes outside the process zone (the regionsR1 andR4) where almostno deformation takes place, the grid displacement is taken proportional to the grid displace-ment of the surfaces as smoothing will not maintain the applied refinements.Some steady state meshes of a slitting simulation, of which the initial mesh is given inFigure 6.12, are shown in Figure 6.18 and Figure 6.19 to illustrate the result of the meshmanagement. The crack can be seen in the lower left picture of Figure 6.19.

6.3.4 3D boundary conditions and contact

Guillotining

Because guillotining and slitting are modelled as flow problems, inflow conditions areneeded for the displacements and the state variables. The material displacement at the in-flow boundary is prescribed in thex-direction and suppressed in they andz-directions. Thismeans that an extra clamping is applied at the inflow, which is not correct. However, whenthe inflow is far enough from the start of the cut, it is shown to have only a small influenceon the shearing process (Section 7.3). The grid displacement at the inflow becomes simplyzero, as the complete motion is prescribed. All state variables are set to zero at the inflow,


X

Y

Z X

Y

Z

Figure 6.18: Steady state mesh for a slitting simulation.

X

Y

Z X

Y

Z X

Y

Z X

Y

Z

X

Y

Z X

Y

Z X

Y

Z X

Y

Z

Figure 6.19: Some magnifications of cross-sections of the steady state mesh.

but it is also possible to assign them a value according to deformations and residual stressesof preceding processes.No real downholder is modelled, but the sheet is clamped by suppressing the displacementin thez-direction ofML7. The nodes onML7 are also given a prescribed displacement inthex-direction. The other side, under the upper knife, is free.

86 Shearing models

The upper and lower knives are moving with the same velocity in thex-direction as the ma-terial at the inflow, the upper knife also has a velocity in the negativey-direction accordingto the rake angle.No boundary conditions are prescribed at the outflow.

Slitting

The boundary conditions used for slitting depend on the slitting mode (Section 1.2). Forpull-through slitting, a displacement is prescribed at the outflow and the slitter rotates witha prescribed angle according to the displacement at the outflow. In the case of driven slit-ting, only the rotation of the slitters is prescribed. However, to build up the friction forcesbetween the sheet and slitter, in the first steps of the simulation a prescribed displacement atthe outflow is also needed. As in the guillotining simulations, only motion in thex-directionis possible at the inflow. A single cut of a multi-cut slitting line is modelled, which meansthat symmetry conditions can be applied at the boundaries. Therefore the nodes onML6andML7 are suppressed in thez-direction. The stripper rings are not modelled.

6.3.5 Preprocessing

Simple preprocessing is important for complex 3D models, as in the models for guillotin-ing and slitting. The commercial pre- and post processing package MSC/PATRAN wasused for that purpose. A parametric preprocessing program was written using PCL (PatranCommand Language), to generate an input file for the finite element code DiekA, from thegeometrical data of the shearing process. The input file contains all the data of the initialmesh, mesh management, contact, boundary conditions etc., and can be used directly to runa simulation. This makes it relatively easy to perform parameter studies.

6.4 Conclusions

It was shown in this chapter how the ALE method can be used for shearing processes. Themesh management procedures are able to preserve a sufficient element quality during thesimulations. By taking into account some adaptability, as was done with the weighting func-tion on a master line, also refinements are kept or created. Nevertheless mesh managementis problem dependent, and often trial and error is the only way to find good solutions.

Chapter 7

Simulation results

Results of simulations with the shearing models described in Chapter 6 are presented here.First, in Section 7.1 a comparison is made between the 2D shearing models and the 3Dmodels of guillotining and slitting. Next, in Section 7.2 and Section 7.3 the results of guil-lotining simulations are compared with two experiments. These experiments were carriedout at Hoogovens R&D and TNO Industry. The emphasis in the first comparison is on theshearing force and the shape of the sheared edge. The shape of the cut-off strip is studiedin the second comparison. Furthermore, the influence of a range of parameters on the pro-cess is investigated. Finally a number of slitting simulations are presented in Section 7.4, inwhich the emphasis is mainly on the fracture model used.

7.1 Comparison 2D and 3D models

In this section the results of four simulations of different shearing types are compared. Thesefour shearing types are ordered according to increasing constraints (the boundary conditionsused have been treated in Sections 6.2.3 and 6.3.4).

Orthogonal1 Plane strain model of orthogonal shearing with one side of the sheet clamped.

Guillotining 3D model for the calculation of the steady state of guillotining.

Slitting 3D model for the calculation of the steady state of slitting.

Orthogonal2 Plane strain model of orthogonal shearing with both sides of the sheet clamped.

The conditions for these four shearing simulations are kept as equal as possible. The sim-ulation data are given in Table 7.1. More information about the slitting simulation can befound in Section 7.4. The main difference is that the 2D models are transient calculationsand that the 3D models calculate a steady state. Another difference between the 2D and3D models is the mesh density. Although the total number of elements for the 3D models

88 Simulation results

Orthogonal1/2 Guillotining SlittingMaterial stainless steel;σy | 1228} 0 ~ 023 � ε � 0 � 4

Sheet thickness 1 mmSheet width 10 mmClearance 15%Rake angle 0� 4.67� 4.67� (mean

value)Radii knives 0.01 mmFriction coefficient 0.1 0.0Number of bulk elements (to-tal/per cross-section)

1291/- 13200/240 7540/240

Number of contact elements(total/per cross-section)

214/- 3500/60 1860/60

ALE convection scheme(Section 4.6)

Isotropicsmoothing

Orthotropic smoothing

Calculation times hours daysSolver Direct solver Preconditioned iterative solver

SSOR+GMRES(m)

Table 7.1: Simulation data of the four shearing models.

is much larger than for the 2D models, the number of elements per cross-section is muchlower. The 2D calculations are carried out with a direct solver and take several hours on aCOMPAQ Alpha DS20 server. An iterative solver is used for the 3D simulations, which lastfrom two till seven days on the same machine, dependent on the dimensions of the problem.The GMRES(m) solver with a SSOR preconditioner proved to be best in the simulationsof guillotining and slitting. The iteration numberm at which the solver is restarted is cho-sen 50 or 100, the maximum number of iterations is 500. The convergence criteria for theguillotining example areεF | 0 ~ 001 andη | 0 ~ 005 according to Section 4.7.3.

The deformation of the sheet for these four shearing types is compared at knife penetrationsof 25% and 50% of the sheet thickness. The results of the 3D models are given for cross-sections of the steady state meshes of guillotining and slitting. These steady state meshesare shown in Figure 7.1. The guillotining model is taken longer than the slitting model toavoid the influence of the inflow conditions.

Looking at the deformations of the sheets in Figure 7.2, it can be observed that the de-formation inOrthogonal1 andGuillotining is non-symmetric, due to the non-symmetricboundary conditions. The part of the sheet that is not clamped bends away from the upperknife. This bending is more severe inOrthogonal1 than inGuillotining as the bending ofthe sheet inGuillotining is restricted by other cross-sections, that are in a different stageof the process. The deformations inSlitting andOrthogonal2 are very similar and in both

7.1 Comparison 2D and 3D models 89

Figure 7.1: Steady state geometry of guillotining (left) and slitting (right).

cases symmetric.In Figures 7.3 to 7.8, the equivalent plastic strain, the hydrostatic stress and the damage inthe shearing zone for the meshes of Figure 7.2 are shown. In Figure 7.9 a magnification ispresented of the damage around the edges of the knives.The much higher extreme values of the state variables around the edges of the knives inthe 2D models compared to the 3D models is partially explained by the difference in meshdensity. The radii of the knives and the extreme values of the state variables are betterdescribed by the finer 2D models than the 3D models.When these four cases are compared, it can be seen that the stress and strain distributionsof Slitting andOrthogonal2 are similar: both are symmetric, and only the maximum val-ues differ. The stress and strain distributions ofGuillotining andOrthogonal1 are bothnon-symmetric, which is in agreement with the imposed non-symmetric deformation. Thedistribution of stress and strain inGuillotining is somewhere in between the distributionsfound inSlitting andOrthogonal1.From this comparison it can be concluded that the boundary conditions, i.e. the clampingof the sheet, have much influence on the stress and strain distributions in the sheet. Non-symmetric deformations lead to non-symmetric stress and strain distributions. The damageis also influenced by the boundary conditions as it is a function of the stress and strainhistory. This means that the predicted shape of the sheared edge is different for the fourshearing types studied. Therefore, special attention should be given to the correct modellingof the clamping of the sheet. This also means that the 3D models cannot be simply replacedby 2D approximations.


(a) Orthogonal1

(b) Guillotining

(c) Slitting

(d) Orthogonal2

Figure 7.2: Deformation at 25% and 50% penetration.


� � � � � � � � � � � � � � � �

(a) Orthogonal1

� � � � � � � ¡ ¢ £ ¤ ¥ § © ª

(b) Guillotining

Figure 7.3: Equivalent plastic strain at 25% and 50% penetration.


« ¬ ® ¯ ± ² ³ ´ µ ¶ · ¸ ¹ º º

(a) Slitting

» ¼ ½ ¾ ¿ À Á Â Ã Ä Å Æ Ç È É Ê

(b) Orthogonal2

Figure 7.4: Equivalent plastic strain at 25% and 50% penetration.


Ë Ì Í Î Ï Ð Ñ ÒË Ó Õ Î Ö Ð Ù Ú Û Ü Ý Þ ß à á â ã

Û ä æ Þ ç è à é ê ã

(a) Orthogonal1

ë ì í î ï ð ò ó ôë õ ö î ÷ ï ð ø ô ù ú û ü ý þ ÿ �

� �ú

� �ý � þ ÿ

� �

(b) Guillotining

Figure 7.5: Hydrostatic pressure [GPa] at 25% and 50% penetration.


� � � � � � � �

� � � � � � ! " �

$ % & ' ( ) * + +

$ , . ' / ( ) 1 3 *

(a) Slitting

5 6 7 8 9 ; 9 = ?

5 @ A 8 B C ; D E F

G H J L M N O P R

G T U L V X N Y Z O

(b) Orthogonal2

Figure 7.6: Hydrostatic pressure [GPa] at 25% and 50% penetration.


[ \ ] ^ _ ` a b c d e f g h i i

(a) Orthogonal1

j k l m n o p q r s t u v w x y

(b) Guillotining

Figure 7.7: Damage (Equation 3.44) at 25% and 50% penetration.


z { | } ~ � � � � � � � � � � �

(a) Slitting

� � � � � � � � � � � � � � � �

(b) Orthogonal2

Figure 7.8: Damage (Equation 3.44) at 25% and 50% penetration.


(a) Orthogonal1 (b) Guillotining

(c) Slitting (d) Orthogonal2

Figure 7.9: Damage (Equation 3.44) at 50% penetration.


In Figure 7.10 the stress in thex-direction is given forGuillotining andSlitting . Stressesin the shear zone, exceeding� 400 MPa, are not shown. It can be seen that the stresses inthex-direction form some bending moments: the stress is positive at the top and negativeat the bottom of the sheet or vice versa. The sheet is bent twice to conform to the shapeof the knives. These stresses are typical for guillotining and slitting and are not found inorthogonal shearing.

��

�

� � ¡ ¢ ¢ ¤ ¦ §

Figure 7.10: σxx between -400 and 400 MPa for guillotining (top) and slitting (bottom).

7.2 Guillotining experiment 1 99

7.2 Guillotining experiment 1

A number of guillotining experiments were carried out at Hoogovens R&D. The resultsof the experiments, the stationary shearing force and the shape of the sheared edge, arecompared to the results of the guillotining simulations.

7.2.1 Experimental results

At Hoogovens R&D a special cutting tool has been designed for guillotining experimentsand has been placed on MTS fatigue equipment. In this way a very well controlled stroke ofthe knife can be obtained. The knife is moved downwards with a speed of 1 mm/s. Duringthe experiment the distance travelled by the knife and the cutting force are measured.

Experiment Simulation

Sheet thickness 1.2 mmRake angle 2.5�

Clearance 2%, 5%, 10%, 15%Cutting width 20 mmClamped width 100 mm 5 mmCut length 150 mm —

Table 7.2: Geometrical data for experiment and simulation.

A strip of 20 mm by 150 mm was cut from an aluminium sheet of 120 mm by 150 mm witha rake angle of 2~ 5� (See also Table 7.2) for four different clearances in the rolling directionas well as perpendicular to the rolling direction.

Shearing force An example of a measured force-displacement diagram is given in Fig-ure 7.11. In this figure can be seen that the force during guillotining is almost constant. Afit is made from these data to obtain the stationary shearing force.

Shape of the sheared edgeThe shape of the sheared edge is recorded when the sheetis completely cut. Photographs of the shape of the sheared edge of the clamped part arepresented in Figure 7.12 for the four clearances. The cross-sections are etched to show thedeformation of the grains near the cut surface. The sheared edges of the clamped part andthe cut-off strip are not completely equal, due to the non-symmetric clamping conditions inguillotining. The tilted surface found at the sheared edge (Figure 7.13) gives an indicationof the clamping conditions. Almost no tilted surface is seen at the clamped part, whichindicates a good clamping. The cut-off strip is allowed to bend away from the upper knife,so that a tilted surface is found, which makes an angle of about 3� with the rest of the sheet.


0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8

For

ce [k

N]

¨

Downward displacement upper knife [mm]

MeasuredFitted

Figure 7.11: Example force measurement and fit (Hoogovens R&D).

From the photographs of the sheared edges an estimate can be made of the shape of theinitial crack front needed for the simulations of guillotining. Therefore the fracture heightis measured (Figure 7.13), which is different for the two parts and also shows some variationalong the cut line. The measured knife penetration at the start of the fracture, which is equalto the sheet thickness minus the fracture height, is given in Table 7.3.

Measured SimulationClearance Clamped Cut-off Clamped Cut-off

2% 31% 36% 34%5% 30% 28% 29%10% 30% 24% 27%15% 35% 31% 33%

Table 7.3: Knife penetrations at start of fracture in % of the sheet thickness.

Material model Tensile tests on the used aluminium (AA5182) were performed in differ-ent directions. A mean value was calculated, as an isotropic material model was used in thesimulations. The flow stress is described by:

σy } εp � | 488 © } 0 ~ 0072496� εp � 0 � 252 (7.1)

σy } 0� | Rp | 141


(a) 2% (b) 5%

(c) 10% (d) 15%

Figure 7.12: Photos of the sheared edge, clamped part (Hoogovens R&D).

thickness

sheared/burnished

tilted surfacetilt angle

fracture height

draw-in/roll-over

Figure 7.13: Schematic view of sheared edge.


E-modulus 70.000 MPa ν 0.33Rp 141 MPa Rm 266 MPaAg 19.5% A80 23.2 %C 488 n 0.252r 0.680

Table 7.4: Material data AA5182, mean values.

7.2.2 Simulation results

The dimensions used in the simulation differ a little from the dimensions in the experiment.In the simulation the down-holder was replaced by a boundary condition directly on thenodes (Section 6.3.4). It was expected that a boundary condition close to the process zonewould correspond best with the clamping of the experiment, as the tilt angle is almost zerofor the clamped part of the sheet. Therefore, the clamping width (5 mm) is chosen smallerthan in the experiment (100 mm). As, in the simulation, only the steady state is modelled,the length of the cut is not of interest. The total length of the modelled sheet was 96.2 mm,which is 55 mm in front of and 41.2 mm behind the point where the upper knife first makescontact with the sheet. The initial position of the crack front used in the simulations wasdetermined from the experimental results. A symmetrical shaped crack front was chosen(Section 6.3.1), which started at the knife penetration obtained by averaging the measuredfracture heights of the clamped and the cut off part (Table 7.3). The sheet was assumed tobe completely separated at a 10 percent further knife penetration.

The calculated shapes of the sheared edge are presented in Figure 7.14 for the four clear-ances. The calculated shapes of the sheared edges (Figure 7.14) are similar to the shearededges of the experiment (Figure 7.12). The shape of the crack front, measured from theexperiment, does not change during the simulation, which means that the resulting crack isalready defined before the simulation. Contrary to the shape of the crack, the shape of thedraw-in and the height of the shear zone are calculated in the simulation for a given crackfront. The transition between the shear zone and the draw-in is more rounded in the sim-ulation than in the experiment, due to the coarse finite element mesh. This transition willbecome sharper for finer element meshes, as can be seen in the comparison between the 2Dand 3D simulations in Section 7.1.Figure 7.15 gives an example of the development of the shearing forces of the upper knife inthree directions during the simulation. The forces increase until a stationary shearing forceis reached. It can be seen that the force perpendicular to the plane of the sheet (the shearingforce) is much greater than the forces in the plane of the sheet. The stationary value of forceFz is reached last, this is due to the development of the free surfaces during the simulation.Although the forces are already stationary, the simulation is continued until the stress andstrain distribution at the outflow is steady.


(a) 2% (b) 5%

(c) 10% (d) 15%

Figure 7.14: Shape of the sheared edge at clamped part, for different clearances.

7.2.3 Comparison between experiment and simulation

In Figure 7.16 the measured and calculated stationary shearing forces are compared. In thisfigure can be seen that the calculated cutting forces are about 25% larger than the measuredvalues. The measured forces for cutting in the rolling direction are a little below the forcesfor shearing perpendicular to the rolling direction, for all clearances.Handbooks (e.g. (Lange 1990)) give the following formula for the approximation of theshearing force in guillotining:

Fstat | k © Rmt2

2tan} α �(7.2)

wherek is a factor depending on the cutting conditions and the material, which should bechosen equal to 0~ 43 to fit to the measured forces.


0

0.5

1

1.5

2

2.5

3

0 20 40 60 80 100 120

For

ce [k

N]

¨

Prescribed displacement x-direction [mm]

Fx, c=5%, upper knifeFy, c=5%, upper knifeFz, c=5%, upper knife

Figure 7.15: Example of the calculated forces during the simulation.

0

0.5

1

1.5

2

2.5

2 4 6 8 10 12 14 16

For

ce [k

N]

¨

Clearance [%]

Exp. rolling dir.Exp. perp. to rolling dir.

Simulation resultsEquation 7.2 with k=0.43

Figure 7.16: Comparison of measured and calculated stationary shearing forces.


Possible reasons for the overestimate of the forces in the simulation are:

ª The flow curve is based on a standard tensile test, so that data aboveAg (the elongationat necking) are found by pure extrapolation, which can lead to inaccurate simulationresults (see page 27).

ª It can be shown that the initial estimate of the crack front has a large influence onthe calculated shearing force in the simulation. This initial estimate of the positionof the crack front is based on the sheared edges found in the experiment, hence thisestimate is rather good. The shape of the crack front, however, deviates from thereal crack front, as only symmetrical crack fronts are used in the simulations and theexperimentally found sheared edges show some asymmetry.

ª The friction between the sheet and the knives is neglected in the simulations. Whenfriction is taken into account the shearing force will probably increase, which in-creases the difference between the simulations and the experiments.

ª The mesh in these 3D guillotining simulations is rather coarse, especially around theradii of the knives. Mesh refinement would lead to lower shearing forces as largeelements behave too stiffly.

ª Because an uncoupled damage model is used (Section 3.3.2) the softening of thematerial before fracturing is neglected. It would be expected that a coupled damagemodel will lead to lower shearing forces.


7.3 Guillotining experiment 2

In the guillotining simulations the shape of the sheet in the steady state was calculated. Thedeformation of the sheet in the steady state can be determined experimentally by interrupt-ing the guillotining process. This second experiment was carried out at TNO Industry.

7.3.1 Experimental results

A sheet was partly cut on an industrial guillotine shear with a clearance of 10% of the sheetthickness and a rake angle of 1~ 5� . The dimensions of the sheet are given in Figure 7.17and Table 7.5. A coordinate frame is defined with the x- and z-axis in the plane of the sheetand the y-axis perpendicular to the sheet. The x-axis is parallel to the cutting direction. Theorigin is located at the lower surface of the sheet where the penetration of the upper knife iszero.

total length

length cutlength before cut

x

zcutting width

clamped width

Figure 7.17: Geometry of experiment 2.

The contour of the upper surface of the partly cut sheet was determined with a Zeiss UMC550S 3D coordinate measurement machine. The results are presented in Figure 7.18 forthree different views.The shape of the sheared edge was also recorded. A photograph of the sheared edge isshown in Figure 7.19. From this photograph can be concluded that the fractured part ofthe sheared edge is about 40% of the sheet thickness for both parts. This means that thefracture starts at a knife penetration of 60% of the sheet thickness. The clamping conditionscan be assessed by looking at the tilted surfaces. The tilt angle (Figure 7.13) is about 5� forthe cut-off part and 4� for the clamped part of the sheet. This means that some bending isallowed by the clamping device.


-15-10

-50

510

1520

z-15 -10 -5 0 5 10 15 20 25 30 35 40

x

-0.8-0.6-0.4-0.2

00.20.40.60.8

11.2

y

(a)

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-15 -10 -5 0 5 10 15 20

y«

z

(x=38)(x=33)(x=28)(x=22)(x=18)(x=13)(x=8)(x=3)

(x=-2)(x=-7)

(x=-12)

(b) lines x=constant

-1

-0.5

0

0.5

1

1.5

2

-20 -10 0 10 20 30 40

y«

x

z=-12z=-7z=-2z=3z=5z=8

z=13z=18

upper knife

(c) lines z=constant

Figure 7.18: Contour of the upper surface of experiment 2 (TNO).


SimulationExperiment 1 2 3

total length 100 mm 114.6 mmlength cut 40 mm 57.3 mmclamped width 80 mm 5 mmcutting width 20 mm 20 mmrake angle 1.5 � 1.5 �

clearance 10% 10%thickness 1 mm 1 mmmaterial model — Nadai Voce Nadaistart fracture 60% 50-60% 70-80%

Table 7.5: Geometrical data experiment and simulation

Figure 7.19: Photograph of the sheared edge of experiment 2 (TNO).

Material model

The experiment was carried out with an AISI 316 stainless steel sheet. A tensile test is usedto determine the flow curve of the material. As only isotropic material was used for thesimulations, a curve-fit was made from the mean value of the tests in different directions.Two curve fits of the experimental data were used in the simulations. One using a Nadaimodel, which yields:

σy } εp � | 1228} 0 ~ 023 � εp � 0 � 4; σy } 0� | 272 (7.3)


and also a Voce model:

σy } εp � | 272 � 850

¬1 e ® ¯ εp

0 ~ 3 ° ± (7.4)

The measured stress-strain curve and the curve-fits have already been given in Figure 3.4.It is clear from that figure that two different fits of the same test data can give very differentresults when they are extrapolated to larger strains than are recorded in a standard tensiletest. This means that the flow curve for the large strains occurring in the shearing processcan be unreliable. The Young’s modulus is taken equal to 210.000 N/mm2 and the Poisson’sratio ν is 0.3.

7.3.2 Simulation results

Three different simulations were carried out, in which the used material model and the po-sition of the initial crack front were changed. The dimensions of the sheet used for thesimulations are equal for all simulations and given in Table 7.5. The length between theinflow and the start of the cut (length before cut Figure 7.17) is taken slightly smaller thanin the experiment, which is expected to have no influence on the results of the simulation.The length of the cut is increased to obtain a longer, completely separated cut-off strip. Theclamping width is reduced from 80 mm to 5 mm to avoid excessive bending of the clampedpart of the sheet. The clamping of the sheet is modelled as a boundary condition atz | 5.The shape of the initial crack front is modelled according the experimentally found shearededge (Figure 7.19). The crack front starts at 50% penetration of the upper knife and is com-pletely separated at 60% knife penetration. In Simulation 3 is an other initial crack frontchosen, to investigate the influence of the position of the crack front on the deformationof the sheet. The Nadai material model of Equation 7.3 is used for the Simulations 1 and3. The influence of the different flow curves (Equations 7.3 and 7.4) is investigated in theSimulations 1 and 2.

The results of Simulation 1 are compared to the experiment in Figure 7.20. From thiscomparison it can be concluded that the shape of the contours are similar in the clampedpart, but differ in the cut-off part of the sheet.Another representation of the measured and calculated contours is given in Figure 7.21.In this figure is the twist of the sheet along thex-coordinate plotted for the simulations1 to 3 and the experiment. The twist is calculated from the coordinates of points of twoneighbouring grid lines on the upper surface, according to:

α } x | x1 � | arctan³ ∆y∆z ´ between z µ 2 and z µ 20

twist } x1 � 12 } x2 x1 � � |

α } x | x2 � α } x | x1 �

x2 x1© 1000 (7.5)


ExperimentSimulation 1

-15-10

-50

510

1520

z-20

-100

1020

3040

x

-2.5-2

-1.5-1

-0.50

0.51

1.5

y

(a)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-15 -10 -5 0 5 10 15 20

y·

z

Sim: x=-13Exp: x=-12

Sim: x=13.3Exp: x=13

Sim: x=37.9Exp: x=38

(b) lines x=constant

-4

-3

-2

-1

0

1

2

3

-60 -40 -20 0 20 40 60

y·

x

Exp: z=3Sim: z=2.7Exp: z=18

Sim: z=17.9upper knife

(c) lines z=constant

Figure 7.20: Contour of the upper surface; comparison between experiment and simula-tion 1.


-20

0

20

40

60

80

100

120

140

160

180

-60 -40 -20 0 20 40 60

twis

t [de

g./m

]

¸

x-coordinate

Simulation 1Simulation 2Simulation 3Experiment

Figure 7.21: Twist of experiment and simulations 1 to 3.

In this figure can be seen that the three calculated curves have a similar shape. The twistincreases from zero at the inflow to a maximum value at the beginning of the cut (x

|0).

The twist is constant during shearing and decreases when the sheet is fractured to a loweralmost constant value in the completely separated strip.The maximum of the measured twist is lower than the maxima from the simulations, butis also positioned atx

|0. Forx-coordinates larger than zero the calculated twist is larger

than the twist measured in the experiment, which decreases almost linearly to zero.Possible reasons for the deviation between the measured and calculated shape of the sheetare:

ª The penetration of the knife at (x|

40) is only 100% of the sheet thickness. Thereforeit is expected that the length of the sheet used for the experiment is too short torepresent the steady state of guillotining correctly. This means that entry phenomenain the experiment disturb the comparison. Performing the same experiment with asheet of larger length would give clarification.

ª In Section 7.1 it was shown that the boundary conditions influence the results of thesimulations. The boundary conditions used in the simulation are different from theclamping in the experiment, as no real downholder is used and inflow conditions aregiven. The differences in boundary conditions however cannot explain the differencebetween the experiment and the simulation.


-20

0

20

40

60

80

100

120

-60 -40 -20 0 20 40 60

twis

t [de

g./m

]

¸

x-coordinate

Simulation 2, before springbackSimulation 2, after springback

Experiment

Figure 7.22: Twist of experiment and simulation 2, before and after springback.

ª From the comparison of Sim. 1 and Sim. 2 in Figure 7.21, it becomes clear that theused flow curve has a considerable influence on the twisting of the strip. This showsagain that more accurate data are needed for the flow stress at large strains.

ª The position of the crack front also influences the twist of the strip. The results ofsimulations with different positions of the crack front (Sim. 1 and Sim.3) show thatpostponing fracture leads to a larger twist.

ª The contour of the partly cut sheet from the experiment was measured after the pro-cess. The contours of the simulation give the shape of the sheet during the process.This means that for a correct comparison the tools should be removed in the simula-tion, in order to determine the springback of the sheet. The influence of the spring-back on the twist is presented in Figure 7.22 for Simulation 2. Differences in twistlevel out during elastic springback, but the differences are much smaller than the dif-ferences found between experiment and simulation. Moreover the twist in the cut-offstrip was hardly changed by springback, as the stresses were already relaxed duringthe process.

ª The simulations were performed with no friction (µ|

0). The influence of friction onthe twist was not investigated.

It is concluded that the material model used and the position of the crack front have a largeinfluence on the calculated twist of the sheet, as is shown by a small parameter study. Other


reasons for the deviations between the experiment and the simulation are springback andthe too short length of the cut in the experiment.

7.3.3 Variations of the simulation

In the remainder of this section a number of simulations are presented showing the influenceof the cutting width, rake angle and the inflow conditions on the shearing process of theprevious subsections.

Influence of cutting width on the twist

It is a well known effect that the deformation of the cut-off sheet increases with decreasingcutting width. Some experiments are presented by (Murakawa and Yan 1997), which showan increasing twist by decreasing cutting width. Simulations with three different cuttingwidths were carried out at a constant rake angle of 1.5� . The trend in the simulations agreeswith the reported experiments: the twist, determined at the part of the cut-off strip that iscompletely cut, increases with decreasing width (Table 7.6).

Cutting width [mm] Twist angle [� /m]10 32820 4840 18

Table 7.6: Twist of the cut-off sheet.

The bending stressesσxx are given in Figure 7.23. This figure shows that the bendingstresses decrease with increasing distance from the shear zone. For a larger cutting width, alarger part of the sheet is only elastically deformed, which explains the decreasing twist.

Influence of rake angle

Simulations were carried out with three different rake angles: 1~ 5� , 3� and 6� . Again,the bending stressesσxx are shown in Figure 7.24. The bending stresses increase withincreasing rake angle as the sheet is bent over a larger angle. Not only the values, butalso the distribution of the stresses changes for increasing rake angles, as the process zonebecomes shorter for larger rake angles.The deformation of the cut-off strip changes for the different rake angles. This is illus-trated by Figure 7.25 in which the contour of the upper surface is given at the outflow (at aknife penetration of 150%). For the smallest rake angle of 1~ 5� the cut-off bends away, butremains straight. The strip curves at larger rake angles.


¹ º» ¼ ½ ½ ¾ À Â Ã Ä Å

Figure 7.23: σxx between -250 MPa and 250 MPa for cutting width 10, 20 and 40 mm (topto bottom).


Ç ÈÉ Ê Ë Ë Í Î Ï Ð Ñ Ò

Figure 7.24: σxx between -350 MPa and 350 MPa for rake angles 1~ 5� , 3� and 6� (bottomto top).


-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-5 0 5 10 15 20 25

yÔ

z

6 deg.3 deg.

1.5 deg.

Figure 7.25: Contour upper surface at outflow for different rake angles.

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-5 0 5 10 15 20 25 30

yÔ

z

1L, x=-9.42L, x=-9.43L, x=-9.21L, x=0.92L, x=0.93L, x=0.91L, x=5.42L, x=5.43L, x=5.4

1L, x=14.32L, x=14.33L, x=14.3

Figure 7.26: Contour of the upper surface at differentx-values; the distance between inflowand cut is L, 2L and 3L.


Õ Ö× Ø Ù Ù Û Ý ß à á â

Figure 7.27: σxx between -350 MPa and 350 MPa for different distances (L, 2L and 3L, topto bottom) between the inflow and the start of the cut.


Influence of inflow conditions

The influence of the inflow conditions on the results of the simulations was studied byvarying the distance between the inflow and the start of the cut (knife penetration of 0%).The rake angle was 6� and the distance 1, 2 or 3 timesL | 19mm.In Figure 7.26 and Figure 7.27 theσxx distribution and the contours of the upper surface fora range ofx-coordinates are given. From these figures can be concluded that the distancebetween the inflow and the start of the cut has only a small influence on the results of thesimulations. The distance between the inflow and the start of the cut should be taken largerfor decreasing rake angle and increasing cutting width.One thing should be noted: the sheet tends to move upward in front of the upper knife,when the length of the model is increased. This upward motion is prevented by the inflowcondition, which clamps the sheet.

7.4 Results of slitting simulations

Some results of slitting simulations are described in this section. These results are also pre-sented in (Wisselink and Hu´etink 1999). The geometry of the sheet and tools is given inTable 7.4. The material model applied is the same as the model used in Section 7.1. Sim-ulations were carried out for four different initial shapes of the crack, for three horizontalclearances and also for two different slitting modes (See Section 1.2 for an explanation ofthe slitting process). The simulation with the horizontal clearance of 15% has already beenpresented in Section 7.1.

sheet thickness 1 mmsheet width 10 mm

diameter knives 300 mmvert. clearance 0%hor. clearance 5%, 10%, 15%slitting mode driven, free

shape of the crack 40%-50%, 50%, 50%-60%, 60%-70%

Table 7.7: Slitting geometry.

Four different choices of the initial shape of the crack front are evaluated. Three curvedcrack fronts, as depicted in Figure 6.13, and one straight crack front are assessed for theirability to yield a constant damage value (Equation 3.44) on the fractured part of the shearededge.

In Figure 7.28 the damage along the fractured part of the sheared edge is given for a cross-section just before the material flows out of the mesh. The damage for the straight crack

7.4 Results of slitting simulations 119

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Cra

ck le

ngth

[mm

]

ã

Damage

straight; 50%curved;40%-50%curved;50%-60%curved;60%-70%

Figure 7.28: Damage along fractured part sheared edge.

front is largest at the edges and lowest in the middle, which means that if the material at theedges reaches a critical value, the material in the middle does not and should therefore notbe cracked.

The three curved crack fronts show a more constant damage value along the fractured partof the sheared edge. So the assumptions made in Section 6.3.1 about the shape of thecrack front are confirmed by these slitting simulations. A curved crack front means thatsome cross-sections exist that are partly cracked. In Figure 7.29 can be seen that for suchpartly cracked cross-sections very high hydrostatic stresses (=negative hydrostatic pressure)develop in front of the crack tips, leading to damage growth in the rest of the cross-section.When the damage values for the three curved crack fronts are compared, it can be seen thatthe damage increases when the crack front is moved downstream. This means that a criticalvalue of the damageDc can be used to asses the correctness of the position of the crack.This critical value must be determined by experiment and taken as an input parameter in thesimulation (page 30).

From Figure 7.30 and Figure 7.31 it can be concluded that the horizontal clearance and theslitting mode have a minor influence on the calculated damage.


(a) Hydrostatic pressure

(b) Damage

Figure 7.29: Partly cracked cross-sections at 53% and 58% knife penetration. Crack startat 50%, separation complete at 60%.

7.4 Results of slitting simulations 121

0

0.1

0.2

0.3

0.4

0.5

0.6

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Cra

ck le

ngth

[mm

]

ã

Damage

c=15%c=10%c=5%

Figure 7.30: Damage along crack for different horizontal clearances.

0

0.1

0.2

0.3

0.4

0.5

0.6

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Cra

ck le

ngth

[mm

]

ã

Damage

pull throughdriven

Figure 7.31: Damage along crack for different slitting modes.


7.5 Discussion

From the simulation results and the comparison with experimental results can be concludedthat most trends are well predicted by the simulations, but that quantitative differences be-tween simulations and experiments are found.

The main reasons for the differences between simulations and experiments are the coarsefinite element mesh in the 3D simulations, the poor material data for large strains (extrapo-lated from a tensile test) and the position and shape of the crack front.The correct position of the crack is important, as it was shown in this chapter that boththe shearing force and the twist of the cut strip depend on the position of the crack. Theinitial crack shape used in the simulations is determined from the experiments, but thedetermination of the shape of the crack during the simulation, by automatic adaptation ofthe crack front according to a critical damage value, should be preferred.

Chapter 8

Conclusions and recommendations

The objective of this research project was to develop a model of guillotining and slitting.This model should lead to more insight into these processes, which can be used to improvethe quality and productivity of the guillotining and slitting processes.Two- and three-dimensional finite element models were developed for the simulation of or-thogonal shearing and the calculation of the steady state of guillotining and slitting. Someconclusions are drawn from the results of simulations of shearing processes with these mod-els and recommendations for further research are given.

The main conclusion is that most trends in shearing processes are qualitatively well de-scribed by the developed shearing models. Hence it can be said that these models increasethe insight into guillotining and slitting and can be used together with experimental researchor as a support to the solution of practical problems. Quantitatively, however, differencesare found between numerical and experimental results.

The models developed here describe the deformation of the sheet (bending, twisting) due toshearing. Also the stress and strain distributions in the sheet during and after cutting (resid-ual stresses) are given by the model. The influence of the position and shape of the crackfront on the process can be investigated. From the performed simulations it is concludedthat the results (e.g. shearing force and twist) are sensitive for the following.

Material model The shape of the flow curve used in the elasto-plastic material model de-termines, to a large extent, the quantitative results. Therefore, more attention shouldbe paid to the accurate measurement of flow stresses at large strains.

Fracture model The position of the crack front is fixed during the simulations in this the-sis. Its position has a large influence on the results of the simulations. The assumedposition was verified by a fracture criterion. The model can be improved using a pro-cedure which automatically adapts the position and shape of the crack front accordingto the fracture criterion.

124 Conclusions and recommendations

Boundary conditions It was shown in this thesis that the clamping of the sheet (symmet-ric / non-symmetric) influences the shearing process. The clamping is adequatelydescribed in the current model by a geometric boundary condition to prevent bendingof the sheet. However, the modelling of a real downholder could refine the shear-ing model. The inflow conditions, required for numerical reasons, do not affect thesimulations, if applied sufficiently far from the process zone.

Mesh density/Calculation time Mesh refinement is needed around the edges of the knivesand the crack front to obtain more accurate solutions. The maximum possible meshdensity for the 3D simulations is limited by the required memory and the calculationtime. The 3D simulations shown in this thesis take from one day to one week, whichis a practical limit. More flexible hexahedral meshing methods however can improvethe accuracy of the calculations without increasing the number of elements too much.The mesh is fine enough to model the bending and twisting of the sheet, as theseeffects extend over a much larger area as the shearing of the sheet. However, thebending and twisting is dependent on the correct position of the crack front.

Iterative solvers were used for the 3D simulations. The error of the iterative solvershould be small enough to avoid deterioration of the convergence of the Newton-Raphson iterations. The speed (=the number of iterations needed to reach the con-vergence criterion) of the iterative solver is dependent on the dimensions of the sheetand the applied boundary conditions.

ALE method Theweighed local and global smoothingmethod used for the transfer of thestate variables is satisfactorily accurate at the moment. Cross-wind diffusion is ef-fectively suppressed by applying orthotropic local smoothing. Diffusion in the flowdirection is retained, to avoid oscillations due to the used convection scheme, but isless significant as the gradients in the flow direction are much smaller than perpen-dicular to the flow direction.

With the developed mesh management procedures it is possible to follow free surfacesand to preserve a reasonable element quality. The mesh management can be extendedto model the propagation of a crack, that is already present, which will be a majorimprovement to the model.

The results of the guillotining simulations are validated with two experiments. Consideringthe limitations of the model, the agreement is satisfying. However, more experiments areneeded to clarify the differences.

The final conclusion is that the developed model can be used to gain more insight in guil-lotining and slitting and can serve as a basis for further research on the 3D modelling ofguillotining and slitting, which should lead to more accurate quantitative results.

Appendix A

Shearing geometry

A.1 Slitting

For real overlap (o ä 0, Figure A.1):

d1 | å R2 ç R 12 } 1 � o� t é 2

(A.1)

d2 | å R2 ç R 12ot é 2

(A.2)

d3 | d1 d2 (A.3)

The shearing angles in slitting are, for a positive overlap (o ä 0):

sin} αinitial � |2d1

R(A.4)

sin} α f inal � |2d2

R(A.5)

12 tan} αmean� |

12t

d3 ê tan} αmean� |td3

(A.6)

and in the case of negative overlap (o ë 0, Figure A.2):

sin} αinitial � |2d1

R(A.7)

sin} α f inal � | 0 (A.8)

12 tan} αmean� |

12 } t o�

d1 ê tan} αmean� |t od1

(A.9)

126 Shearing geometry

t

R

R

12 ì t í oî

d2d3

d1

R í 12t ì 1 ï oî

R=radius slitter

t=thicknesso=overlap (in % of t)

o12αmean

Figure A.1: Positive overlap.

t

R

R

12 ì t í oî

d1

R í 12t ì 1 ï oî

R=radius slitter

t=thicknesso=overlap (in % of t)

o

12αmean

Figure A.2: Negative overlap.

A.2 Guillotining 127

t

$d_4$

t=thickness

αguil

αguil=rake angle

Figure A.3: Guillotining.

A.2 Guillotining

For guillotining the rake angleαguil is given by:

tan} αguil � |td4

(A.10)

The same shearing angle for slitting and guillotining is obtained when:

tan} αguil � | tan} αmean� (A.11)td4

|td3 ê

d4 | d3

Appendix B

The shape of the crack front

The experimental results that are presented here were obtained atTNO Industry . To mea-sure to shape of the crack front in the steady state, an incomplete cut is made with a guillo-tine shear. The stainless steel sheet of 1mmthickness is cut with a shearing angleα

|0 ~ 5�

and a clearance of 15%.By polishing back the incomplete cut, the cross-sections can be viewed. Photographs ofsome cross-sections are shown in this appendix. The photographs are taken at penetrationsof the upper knife ranging from 47% till 85% (given in a percentage of the sheet thickness)at an irregular interval. The penetrationp of the upper blade is given under the photograph.Three different magnifications were usedM | 32,M | 128 orM | 800.

(a) p ð 47% (b) p ð 72%

Figure B.1: M | 32

In the first picture (Figure B.1(a)) no cracks can be seen. In Figure B.1(b) and Figure B.2

130 The shape of the crack front

(a) M ñ 32 (b) M ñ 800

Figure B.2: p µ 83%

it can be seen that a crack has initiated at the upper knife atp µ 62%. This crack does notpropagate towards the tool tip of the lower knife and in fact propagates more slowly thanthe penetration of the knife, therefore we can state that fracture is combined with furtherplastic deformation.Although no cracks are visible in Figure B.2 atp µ 83% at the lower knife, deviations fromthe straight shear zone in Figure B.4 and Figure B.5 suggest that a crack has initiated at thelower knife at an estimated knife penetration ofp µ 75%.

131

(a) M ñ 800 (b) M ñ 128

Figure B.3: p µ 83~ 25%

132 The shape of the crack front

(a) p ð 83ò 65% (b) p ð 84ò 3%

Figure B.4: M|

128

(a) M ñ 128 (b) M ñ 32

Figure B.5: p µ 84~ 7%

133

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120

heig

ht [m

m]

ó

Distance from start cut [mm]

Crack at lower knifeCrack at upper knife

upper knife

Figure B.6: Measured shape of the crack front

The shape of the crack front determined from this experiment is given in Figure B.6. Thisshape is somewhat different from the shape resulting from the assumptions made in Sec-tion 6.3.1.

Some tilted surfaces can be seen on the side of the sheet, which makes contact with the tool,indicating that the sheet has bent. The tilt angle is 5� for the clamped part and 10� for thecut-off part.The clearance measured from the photographs is about 18%, which is larger then the pre-scribed clearance of 15%. This could have been caused by deformation of the tools duringcutting.

Bibliography

Abdali, A., K. Benkrid, and P. Bussy (1996). Simulation of sheet cutting by the large time incre-ment method.J. Mat. Proc. Tech. 60, 255–260.

Akkerman, R. (1993).Euler-Lagrange simulation of nonisothermal viscoelastic flows. Ph. D.thesis, University of Twente.

Askes, H., A. Rodriquez-Ferran, and A. Huerta (1999). Adaptive analysis of yield line patterns inplates with the Arbitrary Lagrangian-Eulerian method.Computers and Structures 70, 257–272.

Atkins, A. G. (1980). On cropping and related processes.Int. J. Mech. Sci. 22, 215–231.

Atkins, A. G. (1981). Surfaces produced by guillotining.Phil. Mag. A 43(3), 627–641.

Atkins, A. G. (1990). On the mechanics of guillotining ductile metals.J. Mat. Proc. Tech. 24,245–257.

Baaijens, F. P. T. (1993). An U-ALE formulation of 3-D unsteady viscoelastic flow.Int. J. Num.Meth. Eng. 36, 1115–1143.

Barrett, R., M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,C. Romine, and H. V. der Vorst (1994).Templates for the Solution of Linear Systems: BuildingBlocks for Iterative Methods. Philadelphia, PA: SIAM.

Bayoumi, H., M. Gadala, and J. Wang (1998). Numerical simulation of metal forming processes.In J. Huetink and F. P. T. Baaijens (Eds.),Simulation of Materials Processing: Theory, Methodsand Applications, Proc. NUMIFORM ’98, pp. 103–108. Balkema, Rotterdam.

Behrens, A. and B. Westhoff (1999). Material deformation in combination with material separation.In M. Geiger (Ed.),Advanced Technology of Plasticity, Vol. III: Proc. of the 6th ICTP, pp. 2251–2254.

Benson, D. (1989). An efficient, accurate, simple ALE method for nonlinear finite element pro-grams.Comput. Methods Appl. Mech. Eng. 72, 305–350.

Benson, D. (1997). A mixture theory for contact in multi-materials in Eulerian formulations.Com-put. Methods Appl. Mech. Eng. 140, 59–86.

Besseling, J. and E. Van der Giessen (1994).Mathematical Modelling of Inelastic Deformation.Chapman and Hall.

Bezzina, S. and K. Saanouni (1997). Computational procedures for finite strain elastoplasticity withdamage: application for sheet cutting. In D. Owen, E. Onate, and E. Hinton (Eds.),ComputationalPlasticity, Fundamentals and Applications: Proc.of the 5th Int. Conf., pp. 611–618.

136 Bibliography

Blacker, T. D. and M. B. Stephenson (1991). Paving: A new approach to automated quadrilateralelements.Int. J. Num. Meth. Eng. 32, 811–847.

Bolt, P. (1989).Prediction of Ductile Failure. Ph. D. thesis, Eindhoven University of Technology.

Bolt, P. and W. Sillekens (1998). Prediction of shape aberrations due to punching, shearing andslitting. In H. Kals (Ed.),Sheet Metal 98: Proc. of the 6th Int. Conf.

Boman, R., L. Colatonio, and J. P. Ponthot (1998). Iterative solvers for large deformation pressuredependent problems. In J. Hu´etink and F. P. T. Baaijens (Eds.),Simulation of Materials Processing:Theory, Methods and Applications, Proc. NUMIFORM ’98, pp. 225–231. Balkema, Rotterdam.

Boogaard, A. H. v. d., J. Hu´etink, and A. D. Rietman (1998). Iterative solvers in forming processes.In J. Huetink and F. P. T. Baaijens (Eds.),Simulation of Materials Processing: Theory, Methodsand Applications, Proc. NUMIFORM ’98, pp. 219–224. Balkema, Rotterdam.

Bouchard, P., F. Bay, Y. Chastel, and I. Tovena (1999). Experimental and numerical analysis offracture of zircaloy 4 tubes. In J. Covas (Ed.),Proc. 2nd ESAFORM Conference on MaterialForming, pp. 151–154.

Brokken, D. (1999).Numerical Modelling of Ductile Fracture in Blanking. Ph. D. thesis, Eind-hoven University of Technology.

Brooks, A. and T. Hughes (1982). Streamline Upwind/Petrov-Galerkin formulations for convectiondominated flows with particular emphasis on the incompressible Navier-Stokes equations.Comput.Methods Appl. Mech. Eng. 32, 199–259.

Calvo, N. A. and S. R. Idelsohn (1998). All-hexahedral meshing by generating the dual mesh. InS. Idelsohn, E. Onate, and E. Dvorkin (Eds.),Computational Mechanics, New Trends and Applica-tions. CIMNE, Barcelona, Spain.

Canann, S., J. R. Tristano, and M. L. Staten (1998). An approach to combined laplacian andoptimization-based smoothing for triangular, quadrilateral and quad-dominant meshes. InMeshingRoundtable ’98.

Carleer, B. D. (1997).Finite Element Analysis of Deepdrawing. Ph. D. thesis, University of Twente.

Ceretti, E., M. Lucchi, and T. Altan (1999). FEM simulation of orthogonal cutting: Serrated chipformation.J. Mat. Proc. Tech. 95, 17–26.

Chang, T. and H. Swift (1950). Shearing of metal bars.J. Inst. Metals 78, 119.

Chen, Z., C. Tang, T. Lee, and L. Chan (1999). A study of strain localization in the fine blankingprocess using the large deformation finite element method.J. Mat. Proc. Tech. 86, 163–167.

Clift, S., P. Hartley, C. E. N. Sturges, and G. Rowe (1990). Fracture prediction in plastic deforma-tion processes.Int. J. Mech. Sci. 32(1), 1–17.

Cockcroft, M. and D. Latham (1968). Ductility and the workability of metals.J. Inst. Metals 96,33–39.

DeBruyne, G. (1995). An arbitrary lagrangian description of 2D and 3D cracked structures.Nucl.Eng. Des. 158, 285–294.

Bibliography 137

Dodd, B. and Y. Bai (1987).Ductile Fracture and Ductility, Applications to Metalworking. Aca-demic press , London.

Dodds, R. and M. Tang (1993). Numerical procedures to model ductile crack extension.Eng.Fract. Mech. 46(2), 253–264.

Donea, J., S. Guiliani, and J. Halleux (1982). An Arbitrary Lagrangian Eulerian finite elementmethod for transient dynamic fluid structure interaction.Comput. Methods Appl. Mech. Eng. 33,689–723.

Dvorkin, E. N. and E. G. Pet¨ocz (1993). An effective technique for modelling 2D metal formingprocesses using a Eulerian formulation.Engineering Computations 10(4), 323–336.

Faura, F., A. Garcia, and M. Esterns (1998). Finite element analysis of optimum clearance in theblanking process.J. Mat. Proc. Tech. 80, 121–125.

FMA (Ed.) (1999).Coil Processing Conference. Fabricators & Manufacturers Association, Int.

Freudenthal, A. M. (1950).The Inelastic Behaviour of Engineering Materials and Structures.Wiley, New York.

Gadala, M. and J. Wang (1999). Simulation of metal forming processes with the finite elementmethod.Int. J. Num. Meth. Eng. 44, 1397–1428.

Ganser, H.-P., E. Werner, and F. Fisher (1998). Plasticity and ductile fracture of IF steels: Experi-ments and micromechanical modeling.Int. J. Plast. 14(8), 789–803.

Gelin, J. (1995). Theoretical and numerical modelling of isotropic and anisotropic ductile damagein metal forming processes. In S. Ghosh and M. Predeleanu (Eds.),Materials Processing Defects,pp. 123–140. Elsevier Science B.V.

Gelten, C. J. M. (1998). Application of automatic remeshing and contact to 3D forging simulations.In J. Huetink and F. P. T. Baaijens (Eds.),Simulation of Materials Processing: Theory, Methodsand Applications, Proc. NUMIFORM ’98, pp. 535–540. Balkema, Rotterdam.

Goijaerts, A. M. (1999).Prediction of Ductile Fracture in Metal Blanking. Ph. D. thesis, EindhovenUniversity of Technology.

Golovashchenko, S. F. (1999). A study on trimming of al alloy parts. In M. Geiger (Ed.),AdvancedTechnology of Plasticity, Vol. III: Proc. of the 6th ICTP, pp. 2261–2266.

Gouveia, B., J. Rodriques, and P. Martins (1996). Fracture predicting in bulk metal forming.Int.J. Mech. Sci. 38(4), 361–372.

Gurson, A. (1977). Continuum theory of ductile rupture by void nucleation and growth: Part I-yield criteria and flow rules for porous ductile media.J. Eng. Mat. Tech. 99, 2–15.

Haber, R., M. S. Shephard, J. F. Abel, R. H. Gallagher, and D. P. Greenberg (1981). A generaltwo-dimensional, graphical finite element preprocessor utilizing discrete transfinite mappings.Int.J. Num. Meth. Eng. 17, 1015–1044.

Hambli, R. and A. Potiron (1999). Finite element analysis of sheet metal punching processes. InM. Geiger (Ed.),Sheet Metal 99: Proc. of the 7th Int. Conf., pp. 153–160.

138 Bibliography

Helm, P. N. v. d., J. Hu´etink, and R. Akkerman (1998). Comparison of artificial dissipation andlimited flux schemes in Arbitrary Lagrangian Eulerian finite element formulations.Int. J. Num.Meth. Eng. 41(6), 1057–1076.

Hinton, E. and J. Campbell (1974). Local and global smoothing of discontinuous functions usinga least squares method.Int. J. Num. Meth. Eng. 8, 461–480.

Huetink, J. (1986). On the simulation of thermomechanical forming processes. Ph. D. thesis,University of Twente.

Huetink, J., D. G. Roddeman, H. J. M. Geijselaers, and A. H. v. d. Boogaard (1999). Mechanics offorming processes. Lecture notes Engineering Mechanics graduate course.

Huetink, J., P. T. Vreede, and J. v. d. Lugt (1989). The simulation of contact problems in formingprocesses using a mixed Eulerian-Lagrangian finite element method. In E. G. Thompson, R. D.Wood, O. C. Zienkiewicz, and A. Samuelsson (Eds.),Num. Methods in Ind. Form. Processes,Proc. NUMIFORM ’89, pp. 549–554. A.A.Balkema, Rotterdam.

Huetink, J., P. T. Vreede, and J. v. d. Lugt (1990). Progress in mixed Eulerian–Lagrangian finiteelement simulation of forming processes.Int. J. Num. Meth. Eng. 30, 1441–1457.

Hughes, T., W. Liu, and T. Zimmermann (1981). Lagrangian-Eulerian finite element formulationfor incompressible viscous flows.Comput. Methods Appl. Mech. Eng. 29, 329–349.

Iliescu, C. (1990).Cold-Pressing Technology. Elsevier.

Jana, S. and N. Ong (1989). Effect of punch clearance in the high speed blanking of thick metalsusing an accelerator designed for a mechanical press.J. Mech. Work. Tech. 19, 55–72.

Jan´ıcek, L. and J. Petruˇska (1999). Ductile fracture criteria in compression and cracking of notchedcylindrical specimens. In M. Geiger (Ed.),Advanced Technology of Plasticity, Vol. III: Proc. of the6th ICTP, pp. 2287–2292.

Jimma, T. and F. Sekine (1990). On high speed precision blanking of IC lead-frames using aprogressive die.J. Mat. Proc. Tech. 22, 291–305.

Johnson, W. and R. A. C. Slater (1967). A survey of the slow and fast blanking of metals at ambientand high temperatures. InProc. CIRP-ASTME, pp. 825–851.

Johnston, B. P., J. M. Sullivan, and A. Kwasnik (1991). Automatic conversion of triangular finiteelement meshes to quadrilateral elements.Int. J. Num. Meth. Eng. 31, 67–84.

Joun, M. S. and M. C. Lee (1997). Quadrilateral finite element generation and mesh quality controlfor metal forming simulation.Int. J. Num. Meth. Eng. 40, 4059–4075.

Kachanov, L. (1986).Introduction to Continuum Damage Mechanics. Martinus Nijhof Publishers.

Klocke, F., K. Sweeney, and H.-W. Raedt (1999). Improved tool design for fine blanking processthrough the application of numerical modelling techniques. In M. Geiger (Ed.),Sheet Metal 99:Proc. of the 7th Int. Conf., pp. 135–142.

Ko, D. C., B. M. Kim, and J. C. Choi (1997). Finite element simulation of the shear process usingthe element-kill method.J. Mat. Proc. Tech. 72, 129–140.

Bibliography 139

Koh, H., H. Lee, and U. Jeong (1995). An incremental formulation of the moving-grid finiteelement method for the prediction of dynamic crack-propagation.Nucl. Eng. Des. 158(2-3), 295–309.

Komori, K. (1999). Simulation of shearing by node separation method: Effect of parameter infracture criterion. In J. C. Gelin (Ed.),NUMISHEET 1999, pp. 607–612.

Konke, C. (1995). Damage evolution in ductile materials: From micro- to macro damage.Comp.Mech. 15, 497–510.

Lange, K. (1990).Umformtechnik, Handbuch for Ind. Und Wissenschaft, Volume 3 Blechbear-beitung. Springer-Verlag. in German.

Lee, Y. K. and D. Y. Yang (1999). Development of a grid-based mesh generation technique andits application to remeshing during the finite element simulation of a metal forming process.Engi-neering Computations 16(3), 316–336.

Lemaitre, J. (1985). A continuous damage mechanics model for ductile fracture.J. Eng. Mat.Tech. 107, 83–89.

Liu, W., H. Chang, J.-S. Chen, and T. Belytschko (1988). Arbitrary Lagrangian-Eulerian Petrov-Galerkin finite elements for non-linear continua.Comput. Methods Appl. Mech. Eng. 68, 259–310.

Lugt, J. v. d. (1988).A Finite Element Method for the Simulation of Thermo-Mechanical ContactProblems in Forming Processes. Ph. D. thesis, University of Twente.

Mackerle, J. (1999). Finite-element analysis and simulation of machining: A bibliography (1976-1996).J. Mat. Proc. Tech. 86(1), 17–44.

Madachy, J. (1980).Theory & Techniques of Coil Slitting. Paper No. 5060. Fabricators & Manu-facturers Association Int., Illinois USA.

Malvern, L. E. (1969).Introduction to the Mechanics of a Continuous Medium. Prentice-Hall.

Marusich, T. and M. Ortiz (1995). Modelling and simulation of high-speed machining.Int. J. Num.Meth. Eng. 38, 3675–3694.

McClintock, F. A. (1968). A criterion for ductile fracture by the growth of holes subjected tomulti-axial stress states.J. of Appl. Mech. 35, 363–371.

Moesdijk, R. D. v. d. (1999).Numerical modelling of shape aberrations due to blanking. Ph. D.thesis, University of Twente.

Mooi, H. G. (1996).Finite Element Simulations of Aluminium Extrusion. Ph. D. thesis, Universityof Twente.

Morancay, L., H. Homsi, and J. Roelandt (1997). Application of remeshing technics to the simu-lation of metal cutting by punching. In D. Owen, E. Onate, and E. Hinton (Eds.),ComputationalPlasticity, Fund. and Appl., pp. 1065–1070.

Murakawa, M. and L. Yan (1997). Precision cutting of sheets by means of a new shear based onrolling motion. J. Mat. Proc. Tech. 66, 232–239.

Obikawa, T., H. Sasahara, T. Shirakashi, and E. Usui (1997). Application of computational ma-chining method to discontinuous chip formation.J. Man. Sci. Eng. 119, 667–674.

140 Bibliography

Ortiz, M. and E. Popov (1985). Accuracy and stability of integration algorithms for elastoplasticconstitutive relations.Int. J. Num. Meth. Eng. 21, 1561–1576.

Oyane, M., T. Sato, K. Okimoto, and S. Shima (1980). Criteria for ductile fracture and theirapplications.J. Mech. Work. Tech. 4, 65–81.

Pantale, O., R. Rakotomalala, and M. Touratier (1998). An ALE three-dimensional model oforthogonal and oblique metal cutting processes.Int. J. Form. Proc. 1(3), 371–389.

Peeters, M. and K. Donkers (1996). The determination of a ductility curve through a combinedtensile/upsetting test. In H. Kals (Ed.),Sheet Metal 96: Proc. of the 4th Int. Conf., pp. 239–247.

Pennington, J. N. (1998, March). Slitting and shearing: Improving the practice.Modern Met-als 54(2), 50–53.

Petersen, S. and P. Martins (1997). Finite element remeshing: A metal forming approach forquadrilateral mesh generation and refinement.Int. J. Num. Meth. Eng. 40, 1449–1464.

Ponthot, J. P. (1989). Efficient mesh management in Eulerian–Lagrangian method for large defor-mation analysis. In E. G. Thompson, R. D. Wood, O. C. Zienkiewicz, and A. Samuelsson (Eds.),Num. Methods in Ind. Form. Processes, Proc. NUMIFORM ’89, pp. 203–210. Balkema, Rotterdam.

Ponthot, J. P. (1992). The use of the Eulerian–Lagrangian FEM with adaptive mesh. applications tometal forming simulation. InComputational Plasticity, Fundamentals and Applications: Proc.ofthe 3th Int. Conf., pp. 2269–2280.

Popat, P. B., A. Ghosh, and N. N. Kishore (1989). Theoretical investigation of optimum clearancein blanking.J. Mech. Work. Tech. 19, 251–259.

Post, J. and R. Voncken (1996). FEM analysis of the punching proces. In H. Kals (Ed.),SheetMetal 96: Proc. of the 4th Int. Conf., pp. 159–169.

Price, M. A. and C. G. Armstrong (1997). Hexahedral mesh generation by medial surface subdivi-sion: Part II. solids with flat and concave edges.Int. J. Num. Meth. Eng. 40, 111–136.

Pyttel, T. and M. Hoogen (1998). Simulation des Scherschneidens von Aluminiumblech.Blech,Rohre, Profile 45(9), 68–70. in German.

Quinlan, G. and J. Monaghan (1998). Finite element analysis of blanking. In H. Kals (Ed.),SheetMetal 98: Proc. of the 6th Int. Conf.

Reilly, D. (1971, March). Rotary slitting of sheet and strip.Sheet Metal Industries, 213–226.

Rekers, G. (1995).Numerical Simulation of Unsteady Viscoelastic Flow of Polymers. Ph. D. thesis,University of Twente.

Rice, J. R. and D. Tracey (1969). On the ductile enlargement of voids in triaxial stress fields.J.Mech. Phys. Solids 17, 201–217.

Rietman, A. D. (1999).Numerical analysis of inhomogeneous deformation in plane strain com-pression. Ph. D. thesis, University of Twente.

Samuel, M. (1998). FEM simulations and experimental analysis of parameters of influence in theblanking proces.J. Mat. Proc. Tech. 84, 97–106.

Bibliography 141

Sauer, R. (1980).Untersuchung Zur Mechanik Des Kaltscheren. Ph. D. thesis, TH Aachen. inGerman.

Schneiders, R. (1996). A grid based algorithm for the generation of hexahedral element meshes.Engineering with Computers 12(3-4), 168–177.

Schreurs, P., F. Veldpaus, and W. Brekelmans (1986). Simulation of forming processes, using thearbitrary eulerian-lagrangian formulation.Comput. Methods Appl. Mech. Eng. 58, 19–36.

Simo, J. and T. Laursen (1992). An augmented lagrangian treatment of contact problems involvingfriction. Computers and Structures 42, 97–116.

Sperling, H. (1968). Die schnittqualität bei Tafelscheren.Blech 15(4), 164–169. in German.

Spur, G. and T. Stöferle (1981-1994). Handbuch der Fertigungstechnik, Band 2.3. Hanser,Munchen. in German.

Stegeman, Y. W., A. Goijaerts, D. Brokken, W. Brekelmans, L. Govaert, and F. Baaijens (1999). Anexperimental and numerical study of a planar blanking process.J. Mat. Proc. Tech. 87, 266–276.

Stoker, H. C. (1999).Developments of the Arbitrary Lagrangian-Eulerian Method in Non LinearSolid Mechanics, Applications to Forming Processes. Ph. D. thesis, University of Twente.

Svahn, O. (1996). Stamping and blanking. In H. Kals (Ed.),Sheet Metal 96: Proc. of the 4th Int.Conf., pp. 15–22.

Taupin, E., J. Breitling, W. Wu, and T. Altan (1996). Material fracture and burr formation inblanking, results of fem simulations and comparison with experiments.J. Mat. Proc. Tech. 59,68–78.

Tekkaya, A. E. (1998). Fully automatic simulation of bulk metal forming processes. In J. Huétinkand F. P. T. Baaijens (Eds.),Simulation of Materials Processing: Theory, Methods and Applica-tions, Proc. NUMIFORM ’98, pp. 529–534. Balkema, Rotterdam.

Thomason, P. F. (1998). A view on ductile-fracture modelling.Fatigue and Fracture of EngineeringMaterials and Structures 21, 1105–1122.

Tvergaard, V. (1990). Material failure by void growth to coalescence.Adv. in Applied Mechan-ics 27, 83–151.

Tvergaard, V. and A. Needleman (1984). Analysis of cup-cone fracture in a round tensile bar.ActaMetall. 32(1), 157–169.

VDI (1994). VDI-Handbuch Betriebstechnik, Volume VDI 2906. Verein Deutsche Ingenieure,Dusseldorf. in German.

Vegter, H. (1991).On the Plastic Behaviour of Steel During Sheet Forming. Ph. D. thesis, Univer-sity of Twente.

Vreede, P. T. (1992).A Finite Element Method for Simulations of 3-Dimensional Sheet MetalForming. Ph. D. thesis, University of Twente.

Wang, J. and M. Gadala (1997). Formulation and survey of ALE method in nonlinear solid me-chanics.Finite Elements in Analysis and Design 24, 253–269.

142 Bibliography

Wifi, A., N. El-Abassi, and A. Abdel-Hamid (1995). A study of workability criteria in bulk formingprocesses. In S. Ghosh and M. Predeleanu (Eds.),Materials Processing Defects, pp. 333–357.

Wisselink, H. and J. Hu´etink (1999). Simulation of the slitting process with the finite elementmethod. In M. Geiger (Ed.),Sheet Metal 99: Proc. of the 7th Int. Conf., pp. 143–152.

Yang, D. Y. and Y. K. Lee (1999). Automatic mesh generation and remeshing for finite elementanalyses of 3D bulk forming processes. In M. Geiger (Ed.),Advanced Technology of Plasticity,Vol. I: Proc. of the 6th ICTP, pp. 521–532.

Yoshida, Y., N. Yukawa, and T. Ishikawa (1999). Deformation analysis of shearing process usingrigid plastic finite element method considering fracture phenomenon. In M. Geiger (Ed.),AdvancedTechnology of Plasticity, Vol. III: Proc. of the 6th ICTP, pp. 2245–2250.

Zhang, Z. L., M. Hauge, J. Ødeg˚ard, and C. Thaulow (1999). Determining material true stress-strain curve from tensile specimens with rectangular cross-section.Int. J. Solids Struct. 36, 3497–3516.

Zhou, Q. and T. Wierzbicki (1996). A tension zone model of blanking and tearing of ductile metalplates.Int. J. Mech. Sci. 38(3), 303–324.

Zhu, Y. and T. Zacharia (1995). Application of damage models in metal forming. In S. F. Shenand P. R. Dawson (Eds.),Simulation of Materials Processing: Theory, Methods and Applications,Proc. NUMIFORM ’95, pp. 375–380. Balkema, Rotterdam.

Zuo, X. (1999). 3D FEM simulation of multi-stage forging process using solid modeling of forgingtools. J. Mat. Proc. Tech. 91(1-3), 191–195.

List of symbols

Scalars:

c Horizontal clearancee Engineering strainf Damage function, Volume fraction of voidsg gap functionh Local coordinateh Hardenings coefficientip Integration pointl Lengthn Hardening parameter flow curvene Number of elementso Overlap or vertical clearancep Penetration knife in % thicknessp Hydrostatic pressurer Anisotropy factorr Local coordinates Coordinate along masterlinet Timez Local coordinateA Cross sectional areaAg Percentage elongation before reductionA80 Percentage elongation after fracture(l ô 80)C õ Cr Courant number (inr-direction)CN õ CV1 õ CV2 Parameters flow curveCO õ CG1 õ CG2 Parameters damage functionCb Bulk modulusD DamageE Youngs’ modulus, Estimator functionG Shear modulusJ Jacobian

144 List of symbols

N Shape functionR Radius slitterRp Proportional elongationRm Tensile strengthT Engineering stress, TemperatureS SurfaceV VolumeWõ δW Work, virtual workα Global smoothings parameter, Rake angleβ Local smoothings parameterε Logarithmic/natural strainεp Equivalent plastic strainεp

f Equivalent plastic strain at fractureε Mechanical unbalance ratioεF Treshold for convergence mechanical unbalanceεd Treshold for convergence displacement ratioκ Slip multiplierλ Plastic multiplierµ Friction coefficientν Poisson’s constantξ State variableρ Mass densityσ True stressσh Hydrostatic stressσeq Equivalent von Mises stressσy Flow stressτN Normal contact stressτT Shear contact stressφ Yield fuctionφC Contact fuctionϕ Potential fuction for slip

Vectors:

b Body forcesn Normal vectort Surface tractionu Displacementv õ δv Velocity, virtual velocityw Weight functionx Position in the current configuration

List of symbols 145

F Force vectorP Force vectorP Unbalance force vectorX Position in the initial configurationχχχ Position in the reference configuration

Second order tensors:

1 Unity tensore Euler-Almansi strains Deviatoric stressB Left Cauchy-Green tensorC Right Cauchy-Green tensorD Rate of deformationE Green-Lagrange strainF Deformation gradientL Spatial gradient of velocityR Rotation tensorT1 First Piola-Kirchhoff stressT Second Piola-Kirchhoff stressU Right stretch tensorV Left stretch tensorW Spin tensorεεε Small deformation strainσσσ Cauchy stressτττ Contact stress

Third order tensors:

B Tensor relatingD to v

Fourth order Tensors:

C Constitutive tensorE Elasticity tensorEC Contact elasticity tensorH Unity tensorY Yield tensorYC Contact slip tensor

Operators:

146 List of symbols

öx Gradient of scalar gives vectoröx Gradient of vector gives tensor÷ö

Pre-gradientøöPost-gradientö ù

x ô divx Divergence of vector gives scalarö ûx Curl of vector gives tensor

˙ý þMaterial derivative

�ý þGrid derivative

�ý þJaumann derivative

det Determinanttr Trace

ˆý þAverage

Subscripts:

0 Initial value

c Convective

c Critical value

g Grid

m Material

max Maximum value

stat Stationary value

Superscripts:

0 Initiale Elastici Iteration numberi Node number along masterlinek Node indexn Step numberp PlasticC ContactC Value after Eulerian part stepL Value after Lagrangian part step

Samenvatting

Knippen en slitten zijn twee mechanische scheidingsprocessen voor dunne metaal plaat,die lange rechte snedes produceren. Het onderwerp van dit proefschrift is het analyserenvan deze twee processen, die als stationaire processen beschouwd kunnen worden. Ditonderzoeksproject maakt deel uit van een cluster van IOP (Innovatiegericht OnderzoeksProgramma) projecten op het gebied van mechanisch scheiden van dunne plaat. Deze pro-jecten zijn opgezet n.a.v. vragen uit de industrie naar meer fundamentele kennis van hetmechanisch scheiden. Het ontwerpen en instellen van deze scheidings processen is nogsteeds gebaseerd op ervaringsregels en trial en error. Het doel van het onderzoek is het ont-wikkelen van modellen van het knippen en slitten, die meer inzicht moeten geven in dezeprocessen en daardoor bijdragen aan een hogere kwaliteit en produktiviteit.

Het knippen en slitten is gemodelleerd met behulp van de eindige elementen methode(EEM). Dit EEM model moet in staat zijn om de fenomenen die optreden gedurende hetknippen te beschrijven, zoals de grote elastisch-plastische vervormingen, het contact tussende plaat en de messen en de taaie breuk van de plaat. De simulaties zijn uitgevoerd methet eindige elementen pakket DiekA, dat ontwikkeld is aan de Universiteit Twente. DiekAbevat een Arbitrary Lagrangian Eulerian (ALE) formulering, die de eigenschappen van deLagrangiaanse en Eulerse formulering combineert. Deze ALE methode is gebruikt voor allesimulaties in dit proefschrift, daarom is een groot deel van dit proefschrift gewijd aan ALEformuleringen, mesh management en het overzetten van de toestand variabelen. Verder isaandacht besteed aan materiaal modellen voor plasticiteit en taaie breuk en de beschrijvingvan het contact tussen plaat en gereedschap.

Twee typen eindige elementen modellen van het mechanisch scheiden zijn ontwikkeld:

2D modellen van het mechanisch scheidenEr is een 2D model van het mechanisch schei-den ontwikkeld. Dit vlakke vervormings model is een benadering van knippen enslitten, waarin alle 3D effecten zijn verwaarloosd. In deze simulaties wordt de ALEmethode gebruikt om te voorkomen dat de elementen te veel vervormen. Hiermeekunnen de simulaties worden voortgezet tot penetraties van ongeveer 50% van deplaatdikte. Voor grotere penetraties of de initiatie en groei van scheuren is de ALE

148 Samenvatting

methode niet meer voldoende en wordt remeshing noodzakelijk.

3D modellen voor de berekening van de stationaire toestand van het knippen en slittenOmdat knippen en slitten beschouwd kunnen worden als stationaire processen, kun-nen deze processen gemodelleerd worden als een stromings probleem. De stationairetoestand van knippen en slitten wordt berekend van uit een begin benadering vande stationaire geometrie, compleet met een scheurfront. Deze initi¨ele geometrie on-twikkelt zich gedurende de simulatie, waarin het materiaal door de eindige elementenmesh stroomt, tot de stationaire geometrie. De ALE methode wordt nu gebruikt omde vrije oppervlakken te volgen. De juistheid van het gekozen initi¨ele scheurfront,dat niet wordt aangepast aan de resultaten van de berekening, wordt beoordeeld aande hand van een breuk criterium.

Resultaten van de simulaties van het knippen en slitten worden gepresenteerd. Er wordteen vergelijking gemaakt tussen de 2D en 3D modellen, die beschreven zijn in dit proef-schrift. De resultaten van de knip simulaties worden ook vergeleken met enkele experi-mentele gegevens. Verder is er een kleine parameter studie uitgevoerd.

De conclusie is dat de ontwikkelde modellen de waargenomen fenomenen kwalitatief goedbeschrijven. Daardoor kunnen deze modellen gebruikt worden om het inzicht in knippen enslitten te vergroten.

Dankwoord

Hierbij wil ik een aantal mensen bedanken voor hun bijdrage aan het tot stand komen vandit proefschrift.Ten eerste Han Hu´etink voor het in mij gestelde vertrouwen, de begeleiding in de afgelopenjaren en de vrijheid die ik heb genoten om dit onderzoek naar eigen inzicht in te vullen.Verder wil ik mijn collega’s en ex-collega’s van de leerstoel Mechanica van Vormgevings-processen bedanken voor hun tips en steun en ook voor de zin en onzin gedurende dekoffiepauzes.Bert Geijselaers bedankt voor het becommentari¨eren van de eerste versies van dit proef-schrift. Katrina Emmett heeft zeer zorgvuldig mijn Engels gecorrigeerd.De partners van het IOP-mechanisch scheiden hebben bijgedragen aan dit project door dediscussies en de resultaten van de andere projecten. In het bijzonder wil ik Pieter Jan Boltand Robert Werkhoven van TNO Industrie bedanken voor het uitvoeren van enkele knipex-perimenten.Jack Verschelling (Outokumpu) en de medewerkers van de strokenschaar bij Hoogovens,hebben mij kennis laten maken met de praktijk van het slitten.Peter Ament (Hoogovens) leverde zeer bruikbare experimentele gegevens van het knippenvan aluminium.Ik wil de secretaresses bedanken voor het uitvoeren van allerlei regelwerk en de systeembe-heerders voor hun ondersteuning op computergebied.Familie en vrienden bedankt voor jullie steun, begrip voor mijn regelmatige afwezigheidgedurende het schrijven van dit proefschrift en het zorgen voor ontspanning.

Harm Wisselink, 2 december 1999

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