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S P R I N G E R B R I E F S I N A P P L I E D S C I E N C E S A N D T E C H N O LO G Y CO N T I N U U M M E C H A N I C S

Sergei Alexandrov

Singular Solutions in Plasticity

SpringerBriefs in Applied Sciencesand Technology

Continuum Mechanics

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Sergei Alexandrov

Singular Solutionsin Plasticity

123

Sergei AlexandrovInstitute for Problems in MechanicsRussian Academy of SciencesMoscowRussia

ISSN 2191-530X ISSN 2191-5318 (electronic)SpringerBriefs in Applied Sciences and TechnologySpringerBriefs in Continuum MechanicsISBN 978-981-10-5226-2 ISBN 978-981-10-5227-9 (eBook)DOI 10.1007/978-981-10-5227-9

Library of Congress Control Number: 2017944291

© The Author(s) 2018This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made. The publisher remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.

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Preface

This monograph concerns with singular solutions in plasticity and their applica-tions. The source of these singular solutions is the maximum friction surface. Thepresentation of the introductory material and the theoretical developments appear ina text of six chapters. The topics chosen are primarily of interest to engineers aspostgraduates and practitioners but they should also serve to capture a readershipfrom among applied mathematicians. The monograph provides both a descriptionof general approaches to finding the asymptotic expansion of solutions in thevicinity of maximum friction surfaces and the specific asymptotic expansions forseveral rigid plastic material models. Numerical analysis is restricted to plane staindeformation of rigid perfectly plastic material. The potential of singular solutions todescribe the generation of narrow fine grain layers in the vicinity of frictionalinterfaces in metal forming processes is outlined. The possibility of using singularsolutions to build up kinematically admissible fields that account for the behavior ofreal velocity fields near maximum friction surfaces is discussed. Among the topicsthat are either new or presented in greater detail than would be found in similar textsare the following:

1. Expansions of singular velocity fields near maximum friction surfaces forseveral rigid plastic models (rigid perfectly plastic material, viscoplastic materialand anisotropic material)

2. An approach to using singular velocity fields for describing the generation offine grain layers in metal forming processes

3. An approach to using singular velocity fields for constructing accurate upperbound solutions.

Chapter 1 concerns with the constitutive equations for several rigid plastic models(rigid perfectly plastic material, viscoplastic material, and anisotropic material). Themodels for viscoplastic and anisotropic materials reduce to the rigid perfectlyplastic model at specific values of input parameters. Several definitions of themaximum friction law are provided. These definitions are connected to the con-stitutive equations. The constitutive equations introduced in this chapter and themaximum friction law are used in subsequent chapters.

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Chapters 2–4 deal with the asymptotic expansions of solutions in the vicinity ofmaximum friction surfaces for the material models introduced in Chap. 1. First, inChap. 2, planar and axisymmetric flows of the rigid perfectly plastic material areconsidered. The main result for the rigid perfectly plastic material model isextended to a class of viscoplastic models in Chap. 3. It is shown that the consti-tutive equations for viscoplastic material should satisfy certain conditions for thereducibility of viscoplastic solutions to rigid perfectly plastic solutions when theconstitutive equations for viscoplastic material reduce to the constitutive equationsfor rigid perfectly plastic material. Chapter 4 concerns with plane strain deformationof anisotropic materials. A simple analytic solution is provided in each chapter toillustrate the general theory.

In Chap. 5 a numerical method for calculating the strain rate intensity factor,which is the coefficient of the leading singular term in a series expansion of theequivalent strain rate in the vicinity of maximum friction surfaces, is introduced forplane strain deformation of rigid perfectly plastic material. The method is applied tofind the strain rate intensity factor in compression of a plastic layer between twoparallel plates.

Chapter 6 includes a brief discussion of two applied aspects of the theory ofsingular solutions. First, it is shown that there is a correlation between the theo-retical strain rate intensity factor and the experimental thickness of a fine grain layergenerated in the process of direct extrusion of an AZ31 alloy through a conical die.Second, a general kinematically admissible velocity field that accounts for thesingular behavior of the real velocity field near maximum friction surfaces isconstructed. This kinematically admissible velocity field is then used in conjunctionwith the upper bound theorem for analysis of forging inside a confined chamber.

Moscow, Russia Sergei AlexandrovMarch 2017

vi Preface

Acknowledgements

This work was made possible by the grant RSCF-14-11-00844.The figures in Sect. 6.1 (except Fig. 6.1) have been reproduced, courtesy of the

publisher of this author’s earlier article published in Journal of ManufacturingScience and Engineering, ASME.

Sections 3.1, 4.1, 4.2, Chap. 5, and Sect. 6.2 have been reproduced, courtesyof the publisher of this author’s earlier articles, from the following journals:

Meccanica, Springer-VerlagContinuum Mechanics and Thermodynamics, Springer-VerlagInternational Journal of Advanced Manufacturing Technology, Springer-Verlag

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Definition of the Subject. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Coordinate Systems and Fundamental Equations . . . . . . . . . . . . . . 21.3 Material Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Maximum Friction Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Rigid Perfectly Plastic Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Plane Strain Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Axisymmeric Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Compression of a Layer Between Rough Plates . . . . . . . . . . . . . . . 20

2.3.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.2 Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Rigid Viscoplastic Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1 Plane Strain Deformation [6–8] . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Axisymmeric Deformation [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3 Compression of a Layer Between Rough Plates . . . . . . . . . . . . . . . 46References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 Anisotropic Rigid Plastic Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.1 Plane Strain Deformation [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Compression of a Layer Between Rough Plates [2] . . . . . . . . . . . . 56References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 Numerical Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.1 The Strain Rate Intensity Factor in Characteristic Coordinates . . . . 615.2 Benchmark Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.3 Numerical Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.3.1 General Numerical Scheme for the Boundary ValueProblem of Section 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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5.3.2 Verification of the Accuracy of the Numerical Scheme. . . . 735.3.3 The Strain Rate Intensity Factor in Compression

of a Finite Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.1 Generation of Fine Grain Layers Near Frictional Interfaces . . . . . . 816.2 Upper Bound Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.2.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 936.2.2 General Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.2.3 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

x Contents

Symbols

The intention within the various theoretical developments given in this monographhas been to define each new symbol where it first appears in the text. In this regardseach chapter should be treated as self-contained in its symbol content. There are,however, certain symbols that re-appear consistently throughout the text. Thesesymbols are given in the following list.

q, s Local coordinate system in plane strain and axisymmetry (Figs. 1.1and 1.2)

uq, us q� and s� velocity components in plane strain and axisymmetryD Strain rate intensity factorH Scale factor for q coordinate line (Figs. 1.1 and 1.2)neq Equivalent strain rate defined in Eqs. (1.3) and (1.11))nqq, nss Normal strain rates in q; sð Þ coordinate system in plane strain and

axisymmetrynqs Shear strain rate in q; sð Þ coordinate system in plane strain and

axisymmetrynhh Circumferential strain rater Stress invariant defined in Eq. (1.7)rh Hydrostatic stressrqq, rss Normal stresses in q; sð Þ coordinate system in plane strain and

axisymmetryrqs Shear stress in q; sð Þ coordinate system in plane strain and axisymmetryrhh Circumferential stressr1 Major principal stressw Orientation of principal stress direction corresponding to r1 relative to

the q direction (Fig. 1.3)

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Chapter 1Introduction

1.1 Definition of the Subject

Singular solutions considered in the present monograph are restricted to the behaviorof rigid plastic solutions in the vicinity of maximum friction surfaces. The defini-tion for such surfaces depends on the material model chosen. Several definitions areprovided in Sect. 1.4. A common feature of many maximum friction surfaces is thatthe equivalent strain rate (the quadratic invariant of the strain rate tensor) approachesinfinity near such surfaces if the regime of sliding occurs. Most likely, the conditionsrequired by the maximum friction law are never satisfied in real processes. Neverthe-less, the asymptotic behavior of singular velocity fields in the vicinity of maximumfriction surfaces can be useful for applications. In particular, the stress intensity factoris one of basic concepts in linear elastic fracture mechanics. This parameter appearsin asymptotic analyses performed in the vicinity of a sharp crack-tip and is the coef-ficient of the leading singular term. In spite of the fact that the assumptions underwhich the stress intensity factor is determined are not satisfied in real materials (thecrack-tip is not sharp and a region of inelastic deformation exists in its vicinity), thisapproach is effective in structural design, for example [24]. In particular, a fracturecriterion is formulated in the form K = Kc where K is the stress intensity factor andKc is its critical value, a material parameter. The stress intensity factor controls themagnitude of stress in the vicinity of crack tips. Analogously, the strain rate intensityfactor introduced in [14] controls the magnitude of the equivalent strain rate in thevicinity of maximum friction surfaces and the asymptotic behavior of the equivalentstrain rate predicts a very high gradient of this quantity in the vicinity of maximumfriction surfaces. On the other hand, it is well known that plastic strain is one of themain contributory mechanisms responsible for high gradients in material propertiesnear frictional interfaces in machining and deformation processes [20]. Therefore, itis natural to assume, by analogy to linear elastic fracture mechanics, that a new typeof constitutive equations involving the strain rate intensity factor should be developedto describe the evolution of material properties in the vicinity of surfaces with highfriction in machining and deformation processes [8, 21]. To this end, it is necessary

© The Author(s) 2018S. Alexandrov, Singular Solutions in Plasticity, SpringerBriefsin Continuum Mechanics, DOI 10.1007/978-981-10-5227-9_1

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