BASIC ENGINEERING PLASTICITY
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BASIC ENGINEERING
PLASTICITY An Introduction with ...

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BASIC ENGINEERING PLASTICITY

This page intentionally left blank

BASIC ENGINEERING

PLASTICITY An Introduction with Engineering Manufacturing Applications and Manufacturing D. W. A. Rees

School of Engineering and Design, Brunei University, University, UK Brunel

AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD • PARIS SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

ELSEVIER

Butterworth-Heinemann is an imprint of Elsevier

Butterworth-Heinemann is an imprint of Elsevier Butterworth-Heinemann 0X2 8DP Linacre House, Jordan Hill, Oxford OX2 30 Corporate Drive, Suite 400, Burlington, MA 01803

First edition 2006 Copyright © 2006, D. W. A. Rees. Published by Elsevier Ltd. All rights asserted reserved identified as the author of this work has been The right of D. W. A. Rees to be identified 1988 asserted in accordance with the Copyright, Designs and Patents Act 1988

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier's Elsevier’s Science & Technology Rights Oxford, UK: phone (+44) (0) 1865 1865 843830; fax (+44) (0) 1865 1865 853333; Department in Oxford, email: [email protected] Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, http://elsevier.com/locate/permissions, and selecting Obtaining permission permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid verification of diagnoses and drug advances in the medical sciences, in particular, independent verification dosages should be made Cataloging in Publication Data British Library Cataloguing catalog record for this book is available from the British Library A catalogue Library of Congress Cataloging-in-Publication Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress

ISBN-13: 978-0-7506-8025-7 0-7506-8025-3 ISBN-10: 0-7506-8025-3 information on all Butterworth-Heinemann Butterworth-Heinemann publications For information http://books.elsevier.com visit our web site at http://books.elsevier.com

Printed and bound in the UK 10 10 10 9 8 7 6 5 4 3 2 1 06 07 08 09 10

Working together to grow libraries in developing countries www.elsevier.com | www.bookaid.org | www.sabre.org

ELSEVIER

?n°°?t£S

Sabre Foundation

CONTENTS

Preface

xi

Acknowledgements

xii

List of Symbols

xiii

CHAPTER 1 STRESS ANALYSIS

1.1 Introduction .,2 Cauchy Definition of Stress 1.3 Three Dimensional Stress Analysis 1.4 Principal Stresses and Invariants 1.5 Principal Stresses as Co-ordinates 1.6 Alternative Stress Definitions Bibliography Exercises

1 4 7 15 21 27 31 31

CHAPTER 2 STRAIN ANALYSIS

2.1 Introduction 2.2 Infinitesimal Strain Tensor 2.3 Large Strain Definitions 2.4 Finite Strain Tensors 2.5 Polar Decomposition 2.6 Strain Definitions References Exercises

33 33 40 47 58 62 62 63

vi

CONTENTS

CHAPTER 3 YIELD CRITERIA 3.1 Introduction 3.2 Yielding of Ductile Isotropie Materials 3.3 Experimental Verification 3.4 Anisotropic Yielding in Polyerystals 3.5 Choice of Yield Function References Exercises

65 65 71 83 90 91 93

CHAPTER 4 NON-HARDENING PLASTICITY 4.1 Introduction 4.2 Classical Theories of Plasticity 4.3 Application of Classical Theory to Uniform Stress States 4.4 Application of Classical Theory to Non-Uniform Stress Slates 4.5 Hencky versus Prandtl-Reuss References Exercises

95 95 98 111 123 124 124

CHAPTER 5 ELASTIC-PERFECT PLASTICITY 5.1 Introduction 5.2 Elastic-Plastic Bending of Beams 5.3 Elastic-Plastic Torsion 5.4 Thick-Walled, Pressurised Cylinder with Closed-Ends 5.5 Open-Ended Cylinder and Thin Disc Under Pressure 5.6 Rotating Disc References Exercises

127 127 137 144 149 154 159 159

CONTENTS

CHAPTER 6 SLIP LINK FIELDS

6.1 Introduction 6.2 Slip Line Field Theory 6.3 Frictionless Extrusion Through Parallel Dies 6.4 Frictionless Extrusion Through Inclined Dies 6.5 Extrusion With Friction Through Parallel Dies 6.6 Notched Bar in Tension 6.7 Die Indentation 6.8 Rough Die Indentation 6.9 Lubricated Die Indentation References Exercises

161 161 180 191 195 197 199 204 207 210 211

CHAPTER 7 LIMIT ANALYSIS 7.1 Introduction 7.2 Collapse of Beams 7.3 Collapse of Structures 7.4 Die Indentation 7.5 Extrusion 7.6 Strip Rolling 7.7 Transverse Loading of Circular Plates 7.8 Concluding Remarks References Exercises

213 213 215 221 225 230 234 238 239 239

vili

CONTENTS

CHAPTER 8 CRYSTAL PLASTICITY 8.1 Introduction 8.2 Resolved Shear Stress and Strain 8.3 Lattice Slip Systems 8.4 Hardening 8.5 Yield Surface 8.6 Flow Rule 8.7 Micro- to Macro-Plasticity 8.8 Subsequent Yield Surface 8.9 Summary References Exercises

241 242 246 248 250 255 257 262 266 267 268

CHAPTER 9 THE FLOW CURVE 9.1 Introduction 9.2 Equivalence in Plasticity 9.3 Uniaxial Tests 9.4 Torsion Tests 9.5 Uniaxial and Torsional Equivalence 9.6 Modified Compression Tests 9.7 Bulge Test 9.8 Equations to the Flow Curve 9.9 Strain and Work Hardening Hypotheses 9.10 Concluding Remarks References Exercises

269 269 274 280 283 286 290 294 298 304 304 305

C H A P T E R 10 PLASTICITY WITH HARDENING 10.1 Introduction 10.2 Conditions Associated with the Yield Surface 10.3 Isotropic Hardening 10.4 Validation of Levy Mises and Drucker Flow Rules 10.5 Non-Associated Flow Rules 10.6 Prandtl-Reuss Flow Theory 10.7 Kinematic Hardening 10.8 Concluding Remarks References Exercises

309 309 313 318 325 326 331 336 336 337

CONTENTS

ta

CHAPTER 11 ORTHOTROPIC PLASTICITY 11.1 Introduction 11.2 Ortnotropie Flow Potential 11.3 Qrtholropic How Curves 11.4 Planar Isotropy 11.5 Rolled Sheet Metals 11.6 Extruded Tubes 11.7 Non-Linear Strain Paths 11.8 Alternative Yield Criteria 11.9 Concluding Remarks References Exercises

339 339 343 348 351 357 362 365 366 367 368 C H A P T E R 12

PLASTIC INSTABILITY 12.1 Introduction 12.2 Inelastic Buckling of Struts 12.3 Buckling of Plates 12.4 Tensile Instability 12.5 Circular Bulge Instability 12.6 Ellipsoidal Bulging of Orthotropic Sheet 12.7 Plate Stretching 12.8 Concluding Remarks References Exercises

371 371 378 388 393 395 399 408 409 409

C H A P T E R 13 STRESS WAVES IN BARS 13.1 Introduction 13.2 The Wave Equation 13.3 Particle Velocity 13.4 Longitudinal Impact of Bars 13.5 Plastic Waves 13.6 Plastic Stress Levels 13.7 Concluding Remarks References Exercises

411 411 412 415 421 432 436 436 436

CONTENTS

CHAPTER 14 PRODUCTION PROCESSES 14.1 Introduction 14.2 Hot Forging 14.3 Cold Forging 14.4 Extrusion 14.5 Hot Rolling 14.6 Cold Rolling 14.7 Wire and Strip Drawing 14.8 Orthogonal Machining 14.9 Concluding Remarks References Exercises

439 439 442 444 448 454 457 461 475 475 475

C H A P T E R 15 APPLICATIONS OF FINITE ELEMENTS 15.1 Introduction 15.2 Elastic Stiffiiess Matrix 15.3 Energy Methods 15.4 Plane Triangular Element 15.5 Elastic-Plastic Stiffiiess Matrix 15.6 FE Simulations 15.7 Concluding Remarks References Exercises

479 479 482 484 490 496 502 503 503

Index

505

PREFACE

This book brings together the elements of the mechanics of plasticity most pertinent to engineers. The presentation of the introductory material, the theoretical developments and the use of appropriate experimental data appear within a text of 15 chapters. A textbook style has been adopted in which worked examples and exercises illustrate the application of the theoretical material. The latter is provided with appropriate references to journals and other published sources. The book thereby combines the reference material required of a researcher together with the detail in theory and application expected from a student. The topics chosen are primarily of interest to engineers as undergraduates, postgraduates and practitioners but they should also serve to capture a readership from among applied mathematicians, physicists and materials scientists. There is not a comparable text with a similar breath in the subject range. Within this, much new work has been drawn from the research literature. The package of topics presented is intended to complement, at a basic level, more advanced monographs on the theory of plasticity. The unique blend of topics given should serve to support syllabuses across a diversity of undergraduate courses including manufacturing, engineering and materials. The first two chapters are concerned with the stress and strain analyses that would normally accompany a plasticity theory. Both the matrix and tensor notations are employed to emphasise their equivalence when describing constitutive relations, co-ordinate transformations, strain gradients and decompositions for both large and small deformations. Chapter 3 outlines the formulation of yield criteria and their experimental confirmation for different initial conditions of material, e.g. annealed, rolled, extruded etc. Here the identity between the yield function and a plastic potential is made to provide flow rules for the ideal plastic solids examined in Chapters 4 and 5. Chapter 4 compares the predictions from the total and incremental theories of classical plasticity with experimental data. Differences between them have been attributed to a strain history dependence lying within non-radial loading paths. Chapter 5 compiles solutions to a number of elastic-perfect plastic structures. Ultimate loads, collapse mechanisms and residual stress are among the issues considered from a loading beyond the yield point. In Chapter 6 it is shown how large scale plasticity in a number of forming processes can be described with slip line fields. For this an ideal, rigid-plastic, material is assumed. The theory identifies the stress states and velocities within a critical deformation zone. The rolling Mohr's circle and hodograph constructions are particulary useful where a full field description of the deformation zone is required. Alternative upper and lower bound analyses of the forming loads for metal forming are given in Chapter 7. Bounding methods provide useful approximations and are more rapid in their application. Chapters 8-10 allow for material hardening behaviour and its influence upon practical plasticity problems. Firstly, in Chapter 8, a description of hardening on a micro-scale is given. It is shown from the operating slip processes and their directions upon closely packed atomic planes, that there must exist a yield criterion and a flow rule. There follows from this the concept of an initial and a subsequent yield surface, these being developed further in later chapters. The measurement and description of the flow curve (Chapter 9) becomes an essential requirement when the modelling the observed, macro-plasticity behaviour. The

xli

PREFACE

simplest isotropic hardening model is outlined in chapter 10. Also discussed here is the model of kinematic hardening for when a description of the Bauschinger effect is required. In Chapter 11 the theory of orthotropic plasticity for rolled sheet metals and extruded tubes is given. These two models of hardening behaviour are extended in Chapter 12 to provide predictions to plastic instability in structures and necking in sheet metal forming. A graphical analysis of the plasticity induced by longitudinal impact of bars is given in Chapter 13. The plasticity arising from high impact stresses is shown to be carried by a stress wave which interacts with an elastic wave to disfribute residual stress in the bar. Chapter 14 considers the control of plasticity arising in conventional produetion processes including: forging, extrusion, rolling and machining. Here, the detailed analyses of ram forces, roll torques and strain rates employ the principles of force equilibrium and strain compatibility. This approach recognises that there are alternatives to slip lines and bounding methods, all of which are complementary when describing plasticity in practice. Thanks are due to the author's past teachers, students and conference organisers who have kept him active in this area. The subject of plasticity continues to develop with many solutions provided these days by various numerical techniques. In this regard, the material presented here will serve to provide the essential mechanics required for any numerical implementation of a plasticity theory. Examples of this are illustrated within the final Chapter 15, where my collaborations with the University of Liege (Belgium) and the Warwick Manufacturing Centre (UK) are gratefully acknowledged.

ACKNOWLEDGEMENTS The figures listed below have been reproduced, courtesy of the publishers of this author's earlier articles, from the following journals: Acta Mechanica, Springer-Verlag (Figs 3.11, 3.13,11,6) Experimental Mechanics, Society for Experimental Mechanics (Figs 10.5,11.10,11.11) Journal of Materials Processing Technology, Elsevier (Fig. 12.19,12.23) Journal Physics IV, France, EDP Sciences (Fig. 11.15) Research Meccanica, Elsevier (Figs 8.13,10.9,10.13) Proceedings of the Institution of Mechanical Engineers, Council I. Mech. E. (Fig. 10.15) Proceedings ofthe Royal Society, RoyaL Society (3.7,3.14,4.1,4.5,9.20,10.7,10.8,10.12) ZeitschriftfurAngewandteMathematikundMechamk, Wiley VCH (Figs 5.15, 5.16) and from the following conference proceedings: Applied SolidMechanics 2 (eds A. S. Tooth andJ, Spence) Elsevier Applied Science, 1988, Chapter 17 (Figs 4,8,4.9,4.10).

sili

LIST OF SYMBOLS The intention within the various theoretical developments given in this book has been to define each new symbol where it first appears in the text. In this regard each chapter should be treated as self-contained in its symbol content. There are, however, certain symbols that re-appear consistently throughout the text, such as those representing force, stress and strain. These symbols are given in the following list along with others most commonly employed in plasticity theory. O.P

I,H,F

curvilinear co-ordinates (slip lines) kinematic hardening translations Schmidt's orientation factors friction and shear angles rolling draft normal and shear strains normal and shear rates of strain micro-plastic strain tensor equivalent plastic strain direct and shear stress principal stresses mean or hydrostatic stress micro-stress tensor transformed stress tensile and compressive strengths equivalent stress friction coefficient Lode's parameters scalar multipliers angular twist die angles hardening measure Poisson's ratio density extension (stretch) ratio hardening functions

a, h, I, z A b, t c C, T «?!, «j, e% eti ep E

lengths section or surface area breadth and thickness propagation velocity torque principal engineering strains distortions subscripts denoting elastic-plastic superscript denoting elastic

aJ, ft?

P>* 3

e, Y e a, T Of % °3

am *

i & M p., v

4>,Ji 0 d,m

f V

P A

MV

LIST OF SYMBOLS

E,G,K

elastic constants yield function (plastic potential) f F,G,H,L.. anisotropy parameters F,P force orthotropic tensors Hfj, Cm Hm, HyUma orthotropic tensors continued second moments of area I,J strain invariants ?uh>h stress invariants / „ J2, J% stress deviator invariants buckling coefficient K direction cosines l,m, n P r . anisotropy parameters continued half-waves in buckling m, n bending moment M hardening exponent n pressure P P superscript denoting plastic stress ratio Q shape and safety factors Q,S polar co-ordinates rB,z r r incremental strain ratios (r values) nt R extrusion ratio radii of curvature R UR 2 back tensions u,v,w displacements strain energy U linear and angular velocities v, a y volume work done W Cartesian co-ordinates x,y,z spacial co-ordinates x. material co-ordinates X( equivalence coefficients x,z tensile and shear yield stresses Y(=ao),k Considere's subtangent z Q (= 6^) B, C, G, L E(=£g) F, H m, n, u M (= IQ) S T(=er9) T (=00 U, V

rotation tensor/matrix deformation tensors infinitesimal strain tensor/matrix deformation gradients unit vectors rotation matrix nominal stress tensor stress tensor/matrix deviatoric stress tensor/matrix stretch tensors

CHAPTER 1 STRESS ANALYSIS

1,1 Introduction Before we can proceed to the study of flow in a deforming solid it is necessary to understand what is meant by the term stress. Various definitions of stress have been used so it is pertinent to begin with explanations as to how it arises and is quantified. Firstly, it is essential that the tensorial nature of stress is appreciated. It will be shown that stress is a symmetrical second order Cartesian tensor. Where deformation is small (infinitesimal) we can represent stress in both the tensor component and matrix notations. Stress is first introduced for simple uniaxial and shear loadings. A combination of these loadings gives both normal and shear stress, these eomprising two of the six independent components that are possible within a stress tensor. The transformation properties of stress are to be examined following a rotation in the orthogonal co-ordinates chosen to define the stress state at a point. Alternative stress definitions are given when it becomes necessary to distinguish between the initial and current areas for large (finite) deformations. Finite deformation will affect the definition of stress because the initial and current areas can differ appreciably. The chosen definition of stress becomes important when connecting the stress and strain tensors within a constitutive relationship for elastic and plastic deforming solids. The following analyses will alternate between the engineering and mathematical coordinate notations listed in Table 1.1. This will enable the reader to interchange between notations in recognition of the equivalence between them.

Table 1.1 Symbol Equivalence fa Engineering and Mathematical Notations

Quantity Material co-ordinates Spacial co-ordinates Material displacements Spacial displacements Unit co-ordinate vectors Direction cosines Unit normal equation Unit normal column matrix Normal stress Shear stress Normal strain (see Ch. 2) Shear strain (see Ch. 2) Stresses on oblique plane

Engineering Notation

Mathematical Notation

x,y,z

*H-^2!^"3

X,T,Z

u,v,w

«!»«a> «3

U,V,W

uu u2, u3

tti, U 25 U 3

/, m, n uB = lu» + wu^+«u a ffx, a,,

n = /tUj + l 2 u 2 + l3Uj

B={/, / 2 / , } T

fft

ex, eP ex'

^11» ^ 2 2 ' ^ 3 3

a, r

au',an',aM'

*U'

^ 1 3 ' ^23

BASIC ENGINEERING PLASTICITY

Note that a rotation matrix M employs the direction cosines in the above table for a co-ordinate transformation between Cartesian axes 1, 2 and 3, in each notation as follows:

hi hi M = hi hi

m

l

s

*31

This page intentionally left blank

BASIC ENGINEERING

PLASTICITY An Introduction with Engineering Manufacturing Applications and Manufacturing D. W. A. Rees

School of Engineering and Design, Brunei University, University, UK Brunel

AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD • PARIS SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

ELSEVIER

Butterworth-Heinemann is an imprint of Elsevier

Butterworth-Heinemann is an imprint of Elsevier Butterworth-Heinemann 0X2 8DP Linacre House, Jordan Hill, Oxford OX2 30 Corporate Drive, Suite 400, Burlington, MA 01803

First edition 2006 Copyright © 2006, D. W. A. Rees. Published by Elsevier Ltd. All rights asserted reserved identified as the author of this work has been The right of D. W. A. Rees to be identified 1988 asserted in accordance with the Copyright, Designs and Patents Act 1988

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier's Elsevier’s Science & Technology Rights Oxford, UK: phone (+44) (0) 1865 1865 843830; fax (+44) (0) 1865 1865 853333; Department in Oxford, email: [email protected] Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, http://elsevier.com/locate/permissions, and selecting Obtaining permission permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid verification of diagnoses and drug advances in the medical sciences, in particular, independent verification dosages should be made Cataloging in Publication Data British Library Cataloguing catalog record for this book is available from the British Library A catalogue Library of Congress Cataloging-in-Publication Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress

ISBN-13: 978-0-7506-8025-7 0-7506-8025-3 ISBN-10: 0-7506-8025-3 information on all Butterworth-Heinemann Butterworth-Heinemann publications For information http://books.elsevier.com visit our web site at http://books.elsevier.com

Printed and bound in the UK 10 10 10 9 8 7 6 5 4 3 2 1 06 07 08 09 10

Working together to grow libraries in developing countries www.elsevier.com | www.bookaid.org | www.sabre.org

ELSEVIER

?n°°?t£S

Sabre Foundation

CONTENTS

Preface

xi

Acknowledgements

xii

List of Symbols

xiii

CHAPTER 1 STRESS ANALYSIS

1.1 Introduction .,2 Cauchy Definition of Stress 1.3 Three Dimensional Stress Analysis 1.4 Principal Stresses and Invariants 1.5 Principal Stresses as Co-ordinates 1.6 Alternative Stress Definitions Bibliography Exercises

1 4 7 15 21 27 31 31

CHAPTER 2 STRAIN ANALYSIS

2.1 Introduction 2.2 Infinitesimal Strain Tensor 2.3 Large Strain Definitions 2.4 Finite Strain Tensors 2.5 Polar Decomposition 2.6 Strain Definitions References Exercises

33 33 40 47 58 62 62 63

vi

CONTENTS

CHAPTER 3 YIELD CRITERIA 3.1 Introduction 3.2 Yielding of Ductile Isotropie Materials 3.3 Experimental Verification 3.4 Anisotropic Yielding in Polyerystals 3.5 Choice of Yield Function References Exercises

65 65 71 83 90 91 93

CHAPTER 4 NON-HARDENING PLASTICITY 4.1 Introduction 4.2 Classical Theories of Plasticity 4.3 Application of Classical Theory to Uniform Stress States 4.4 Application of Classical Theory to Non-Uniform Stress Slates 4.5 Hencky versus Prandtl-Reuss References Exercises

95 95 98 111 123 124 124

CHAPTER 5 ELASTIC-PERFECT PLASTICITY 5.1 Introduction 5.2 Elastic-Plastic Bending of Beams 5.3 Elastic-Plastic Torsion 5.4 Thick-Walled, Pressurised Cylinder with Closed-Ends 5.5 Open-Ended Cylinder and Thin Disc Under Pressure 5.6 Rotating Disc References Exercises

127 127 137 144 149 154 159 159

CONTENTS

CHAPTER 6 SLIP LINK FIELDS

6.1 Introduction 6.2 Slip Line Field Theory 6.3 Frictionless Extrusion Through Parallel Dies 6.4 Frictionless Extrusion Through Inclined Dies 6.5 Extrusion With Friction Through Parallel Dies 6.6 Notched Bar in Tension 6.7 Die Indentation 6.8 Rough Die Indentation 6.9 Lubricated Die Indentation References Exercises

161 161 180 191 195 197 199 204 207 210 211

CHAPTER 7 LIMIT ANALYSIS 7.1 Introduction 7.2 Collapse of Beams 7.3 Collapse of Structures 7.4 Die Indentation 7.5 Extrusion 7.6 Strip Rolling 7.7 Transverse Loading of Circular Plates 7.8 Concluding Remarks References Exercises

213 213 215 221 225 230 234 238 239 239

vili

CONTENTS

CHAPTER 8 CRYSTAL PLASTICITY 8.1 Introduction 8.2 Resolved Shear Stress and Strain 8.3 Lattice Slip Systems 8.4 Hardening 8.5 Yield Surface 8.6 Flow Rule 8.7 Micro- to Macro-Plasticity 8.8 Subsequent Yield Surface 8.9 Summary References Exercises

241 242 246 248 250 255 257 262 266 267 268

CHAPTER 9 THE FLOW CURVE 9.1 Introduction 9.2 Equivalence in Plasticity 9.3 Uniaxial Tests 9.4 Torsion Tests 9.5 Uniaxial and Torsional Equivalence 9.6 Modified Compression Tests 9.7 Bulge Test 9.8 Equations to the Flow Curve 9.9 Strain and Work Hardening Hypotheses 9.10 Concluding Remarks References Exercises

269 269 274 280 283 286 290 294 298 304 304 305

C H A P T E R 10 PLASTICITY WITH HARDENING 10.1 Introduction 10.2 Conditions Associated with the Yield Surface 10.3 Isotropic Hardening 10.4 Validation of Levy Mises and Drucker Flow Rules 10.5 Non-Associated Flow Rules 10.6 Prandtl-Reuss Flow Theory 10.7 Kinematic Hardening 10.8 Concluding Remarks References Exercises

309 309 313 318 325 326 331 336 336 337

CONTENTS

ta

CHAPTER 11 ORTHOTROPIC PLASTICITY 11.1 Introduction 11.2 Ortnotropie Flow Potential 11.3 Qrtholropic How Curves 11.4 Planar Isotropy 11.5 Rolled Sheet Metals 11.6 Extruded Tubes 11.7 Non-Linear Strain Paths 11.8 Alternative Yield Criteria 11.9 Concluding Remarks References Exercises

339 339 343 348 351 357 362 365 366 367 368 C H A P T E R 12

PLASTIC INSTABILITY 12.1 Introduction 12.2 Inelastic Buckling of Struts 12.3 Buckling of Plates 12.4 Tensile Instability 12.5 Circular Bulge Instability 12.6 Ellipsoidal Bulging of Orthotropic Sheet 12.7 Plate Stretching 12.8 Concluding Remarks References Exercises

371 371 378 388 393 395 399 408 409 409

C H A P T E R 13 STRESS WAVES IN BARS 13.1 Introduction 13.2 The Wave Equation 13.3 Particle Velocity 13.4 Longitudinal Impact of Bars 13.5 Plastic Waves 13.6 Plastic Stress Levels 13.7 Concluding Remarks References Exercises

411 411 412 415 421 432 436 436 436

CONTENTS

CHAPTER 14 PRODUCTION PROCESSES 14.1 Introduction 14.2 Hot Forging 14.3 Cold Forging 14.4 Extrusion 14.5 Hot Rolling 14.6 Cold Rolling 14.7 Wire and Strip Drawing 14.8 Orthogonal Machining 14.9 Concluding Remarks References Exercises

439 439 442 444 448 454 457 461 475 475 475

C H A P T E R 15 APPLICATIONS OF FINITE ELEMENTS 15.1 Introduction 15.2 Elastic Stiffiiess Matrix 15.3 Energy Methods 15.4 Plane Triangular Element 15.5 Elastic-Plastic Stiffiiess Matrix 15.6 FE Simulations 15.7 Concluding Remarks References Exercises

479 479 482 484 490 496 502 503 503

Index

505

PREFACE

This book brings together the elements of the mechanics of plasticity most pertinent to engineers. The presentation of the introductory material, the theoretical developments and the use of appropriate experimental data appear within a text of 15 chapters. A textbook style has been adopted in which worked examples and exercises illustrate the application of the theoretical material. The latter is provided with appropriate references to journals and other published sources. The book thereby combines the reference material required of a researcher together with the detail in theory and application expected from a student. The topics chosen are primarily of interest to engineers as undergraduates, postgraduates and practitioners but they should also serve to capture a readership from among applied mathematicians, physicists and materials scientists. There is not a comparable text with a similar breath in the subject range. Within this, much new work has been drawn from the research literature. The package of topics presented is intended to complement, at a basic level, more advanced monographs on the theory of plasticity. The unique blend of topics given should serve to support syllabuses across a diversity of undergraduate courses including manufacturing, engineering and materials. The first two chapters are concerned with the stress and strain analyses that would normally accompany a plasticity theory. Both the matrix and tensor notations are employed to emphasise their equivalence when describing constitutive relations, co-ordinate transformations, strain gradients and decompositions for both large and small deformations. Chapter 3 outlines the formulation of yield criteria and their experimental confirmation for different initial conditions of material, e.g. annealed, rolled, extruded etc. Here the identity between the yield function and a plastic potential is made to provide flow rules for the ideal plastic solids examined in Chapters 4 and 5. Chapter 4 compares the predictions from the total and incremental theories of classical plasticity with experimental data. Differences between them have been attributed to a strain history dependence lying within non-radial loading paths. Chapter 5 compiles solutions to a number of elastic-perfect plastic structures. Ultimate loads, collapse mechanisms and residual stress are among the issues considered from a loading beyond the yield point. In Chapter 6 it is shown how large scale plasticity in a number of forming processes can be described with slip line fields. For this an ideal, rigid-plastic, material is assumed. The theory identifies the stress states and velocities within a critical deformation zone. The rolling Mohr's circle and hodograph constructions are particulary useful where a full field description of the deformation zone is required. Alternative upper and lower bound analyses of the forming loads for metal forming are given in Chapter 7. Bounding methods provide useful approximations and are more rapid in their application. Chapters 8-10 allow for material hardening behaviour and its influence upon practical plasticity problems. Firstly, in Chapter 8, a description of hardening on a micro-scale is given. It is shown from the operating slip processes and their directions upon closely packed atomic planes, that there must exist a yield criterion and a flow rule. There follows from this the concept of an initial and a subsequent yield surface, these being developed further in later chapters. The measurement and description of the flow curve (Chapter 9) becomes an essential requirement when the modelling the observed, macro-plasticity behaviour. The

xli

PREFACE

simplest isotropic hardening model is outlined in chapter 10. Also discussed here is the model of kinematic hardening for when a description of the Bauschinger effect is required. In Chapter 11 the theory of orthotropic plasticity for rolled sheet metals and extruded tubes is given. These two models of hardening behaviour are extended in Chapter 12 to provide predictions to plastic instability in structures and necking in sheet metal forming. A graphical analysis of the plasticity induced by longitudinal impact of bars is given in Chapter 13. The plasticity arising from high impact stresses is shown to be carried by a stress wave which interacts with an elastic wave to disfribute residual stress in the bar. Chapter 14 considers the control of plasticity arising in conventional produetion processes including: forging, extrusion, rolling and machining. Here, the detailed analyses of ram forces, roll torques and strain rates employ the principles of force equilibrium and strain compatibility. This approach recognises that there are alternatives to slip lines and bounding methods, all of which are complementary when describing plasticity in practice. Thanks are due to the author's past teachers, students and conference organisers who have kept him active in this area. The subject of plasticity continues to develop with many solutions provided these days by various numerical techniques. In this regard, the material presented here will serve to provide the essential mechanics required for any numerical implementation of a plasticity theory. Examples of this are illustrated within the final Chapter 15, where my collaborations with the University of Liege (Belgium) and the Warwick Manufacturing Centre (UK) are gratefully acknowledged.

ACKNOWLEDGEMENTS The figures listed below have been reproduced, courtesy of the publishers of this author's earlier articles, from the following journals: Acta Mechanica, Springer-Verlag (Figs 3.11, 3.13,11,6) Experimental Mechanics, Society for Experimental Mechanics (Figs 10.5,11.10,11.11) Journal of Materials Processing Technology, Elsevier (Fig. 12.19,12.23) Journal Physics IV, France, EDP Sciences (Fig. 11.15) Research Meccanica, Elsevier (Figs 8.13,10.9,10.13) Proceedings of the Institution of Mechanical Engineers, Council I. Mech. E. (Fig. 10.15) Proceedings ofthe Royal Society, RoyaL Society (3.7,3.14,4.1,4.5,9.20,10.7,10.8,10.12) ZeitschriftfurAngewandteMathematikundMechamk, Wiley VCH (Figs 5.15, 5.16) and from the following conference proceedings: Applied SolidMechanics 2 (eds A. S. Tooth andJ, Spence) Elsevier Applied Science, 1988, Chapter 17 (Figs 4,8,4.9,4.10).

sili

LIST OF SYMBOLS The intention within the various theoretical developments given in this book has been to define each new symbol where it first appears in the text. In this regard each chapter should be treated as self-contained in its symbol content. There are, however, certain symbols that re-appear consistently throughout the text, such as those representing force, stress and strain. These symbols are given in the following list along with others most commonly employed in plasticity theory. O.P

I,H,F

curvilinear co-ordinates (slip lines) kinematic hardening translations Schmidt's orientation factors friction and shear angles rolling draft normal and shear strains normal and shear rates of strain micro-plastic strain tensor equivalent plastic strain direct and shear stress principal stresses mean or hydrostatic stress micro-stress tensor transformed stress tensile and compressive strengths equivalent stress friction coefficient Lode's parameters scalar multipliers angular twist die angles hardening measure Poisson's ratio density extension (stretch) ratio hardening functions

a, h, I, z A b, t c C, T «?!, «j, e% eti ep E

lengths section or surface area breadth and thickness propagation velocity torque principal engineering strains distortions subscripts denoting elastic-plastic superscript denoting elastic

aJ, ft?

P>* 3

e, Y e a, T Of % °3

am *

i & M p., v

4>,Ji 0 d,m

f V

P A

MV

LIST OF SYMBOLS

E,G,K

elastic constants yield function (plastic potential) f F,G,H,L.. anisotropy parameters F,P force orthotropic tensors Hfj, Cm Hm, HyUma orthotropic tensors continued second moments of area I,J strain invariants ?uh>h stress invariants / „ J2, J% stress deviator invariants buckling coefficient K direction cosines l,m, n P r . anisotropy parameters continued half-waves in buckling m, n bending moment M hardening exponent n pressure P P superscript denoting plastic stress ratio Q shape and safety factors Q,S polar co-ordinates rB,z r r incremental strain ratios (r values) nt R extrusion ratio radii of curvature R UR 2 back tensions u,v,w displacements strain energy U linear and angular velocities v, a y volume work done W Cartesian co-ordinates x,y,z spacial co-ordinates x. material co-ordinates X( equivalence coefficients x,z tensile and shear yield stresses Y(=ao),k Considere's subtangent z Q (= 6^) B, C, G, L E(=£g) F, H m, n, u M (= IQ) S T(=er9) T (=00 U, V

rotation tensor/matrix deformation tensors infinitesimal strain tensor/matrix deformation gradients unit vectors rotation matrix nominal stress tensor stress tensor/matrix deviatoric stress tensor/matrix stretch tensors

CHAPTER 1 STRESS ANALYSIS

1,1 Introduction Before we can proceed to the study of flow in a deforming solid it is necessary to understand what is meant by the term stress. Various definitions of stress have been used so it is pertinent to begin with explanations as to how it arises and is quantified. Firstly, it is essential that the tensorial nature of stress is appreciated. It will be shown that stress is a symmetrical second order Cartesian tensor. Where deformation is small (infinitesimal) we can represent stress in both the tensor component and matrix notations. Stress is first introduced for simple uniaxial and shear loadings. A combination of these loadings gives both normal and shear stress, these eomprising two of the six independent components that are possible within a stress tensor. The transformation properties of stress are to be examined following a rotation in the orthogonal co-ordinates chosen to define the stress state at a point. Alternative stress definitions are given when it becomes necessary to distinguish between the initial and current areas for large (finite) deformations. Finite deformation will affect the definition of stress because the initial and current areas can differ appreciably. The chosen definition of stress becomes important when connecting the stress and strain tensors within a constitutive relationship for elastic and plastic deforming solids. The following analyses will alternate between the engineering and mathematical coordinate notations listed in Table 1.1. This will enable the reader to interchange between notations in recognition of the equivalence between them.

Table 1.1 Symbol Equivalence fa Engineering and Mathematical Notations

Quantity Material co-ordinates Spacial co-ordinates Material displacements Spacial displacements Unit co-ordinate vectors Direction cosines Unit normal equation Unit normal column matrix Normal stress Shear stress Normal strain (see Ch. 2) Shear strain (see Ch. 2) Stresses on oblique plane

Engineering Notation

Mathematical Notation

x,y,z

*H-^2!^"3

X,T,Z

u,v,w

«!»«a> «3

U,V,W

uu u2, u3

tti, U 25 U 3

/, m, n uB = lu» + wu^+«u a ffx, a,,

n = /tUj + l 2 u 2 + l3Uj

B={/, / 2 / , } T

fft

ex, eP ex'

^11» ^ 2 2 ' ^ 3 3

a, r

au',an',aM'

*U'

^ 1 3 ' ^23

BASIC ENGINEERING PLASTICITY

Note that a rotation matrix M employs the direction cosines in the above table for a co-ordinate transformation between Cartesian axes 1, 2 and 3, in each notation as follows:

hi hi M = hi hi

m

l

s

*31

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