# Finite-Element Plasticity and Metalforming Analysis

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Finite-element plasticity and metalforming analysis

Finite-element plasticity andmetalforming analysis

G.W. ROWEProfessor of Mechanical EngineeringUniversity of Birmingham

C.E.N. STURGESSJaguar Professor of Automotive EngineeringUniversity of Birmingham

P. HARTLEYSenior Lecturer, Department of Manufacturing EngineeringUniversity of Birmingham

I. PILLINGERSenior Computer Officer, Department of Mechanical EngineeringUniversity of Birmingham

The right of theUniversity of Cambridge

to print and sellall manner of books

was granted byHenry Vlll in 1534.

The University has printedand published continuously

since 1584.

CAMBRIDGE UNIVERSITY PRESSCambridgeNew York Port ChesterMelbourne Sydney

CAMBRIDGE UNIVERSITY PRESSCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo

Cambridge University Press

The Edinburgh Building, Cambridge CB2 2RU, UK

Published in the United States of America by Cambridge University Press, New York

www. Cambridge. org

Information on this title: www.cambridge.org/9780521383622

Cambridge University Press 1991

This book is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place withoutthe written permission of Cambridge University Press.First published 1991This digitally printed first paperback version 2005A catalogue record for this publication is available from the British Library

ISBN-13 978-0-521-38362-2 hardbackISBN-10 0-521-38362-5 hardback

ISBN-13 978-0-521-01731-2 paperbackISBN-10 0-521-01731-9 paperback

Contents

Preface xiAcknowledgements xiiiNomenclature xv

1 General introduction to the finite-element method 11.1 Introduction 11.2 Earlier theoretical methods for metalforming analysis 31.3 Basic finite-element approach 41.4 General procedure for structural finite-element analysis 51.5 Simple application of elastic FE analysis: a tensile test bar 7

1.5.1 Discretisation 71.5.2 Interpolation 71.5.3 Stiffness matrices of the elements 71.5.4 Assembly of element stiffness matrices 101.5.5 Boundary conditions 101.5.6 Numerical solution for the displacements 101.5.7 Strains and stresses in the elements 12

References 12

2 Basic formulation for elastic deformation 142.1 Types of elements 14

2.1.1 Linear elements 152.1.2 Plane-strain triangular elements 162.1.3 Linear quadrilateral elements 162.1.4 Higher-order (quadratic) elements 162.1.5 Three-dimensional elements 172.1.6 Size of elements 172.1.7 Shape and configuration of elements: aspect ratio 18

2.2 Continuity and equilibrium 182.3 Interpolation functions 19

Contents

2.3.1 Polynomials 192.3.2 Convergence 19

2.4 Displacement vector u and matrix [B] 202.5 The [D] matrix of elastic constants 22

2.5.1 General expression 222.5.2 Plane elastic stress 232.5.3 Plane elastic strain 242.5.4 Axial symmetry 24

2.6 Formulation of the element stiffness matrix [K{] 252.7 Formulation of the global stiffness matrix [K] 26

2.7.1 Assembly of element matrices 262.7.2 Properties of [K] 27

2.8 General solution methods for the matrix equations 312.8.1 Direct solution methods 312.8.2 Indirect solution methods 332.8.3 Iteration to improve a direct solution 35

2.9 Boundary conditions 352.10 Variational methods 36

2.10.1 Variational method of solution in continuum theory 362.10.2 Approximate solutions by the Rayleigh-Ritz method 372.10.3 Weighted residuals 372.10.4 Galerkin's method 38

2.11 The finite-difference method 38References 39

3 Small-deformation elastic-plastic analysis 403.1 Introduction 403.2 Elements of plasticity theory 40

3.2.1 Yielding 403.2.2 Deviatoric and generalised stresses 423.2.3 Constancy of volume in plastic deformation 423.2.4 Decomposition of incremental strain 433.2.5 Generalised plastic strain 433.2.6 The relationship of stress to strain increment: the Prandtl-Reuss equations 443.2.7 Elastic-plastic constitutive relationship 453.2.8 Strain hardening in FE solutions 453.2.9 Force/displacement relationship: the stiffness matrix [K{] 46

3.3 Example analysis using the small-deformation formulation 46References 48

4 Finite-element plasticity on microcomputers 494.1 Microcomputers in engineering 494.2 Non-linear plasticity demonstration programs on a

microcomputer using BASIC 50

Contents

4.2.1 Introduction 504.2.2 Structure of the finite-element program in BASIC 514.2.3 Simple application and comparison to mainframe results 52

4.3 A 'large' FORTRAN-based system for non-linearfinite-element plasticity 56

4.3.1 Hardware selection 564.3.2 Program transfer 584.3.3 Applications of the non-linear system 584.3.3.1 Axi-symmetric upsetting 584.3.3.2 Cold heading 604.3.4 Overcoming the FORTRAN compiler limitations 614.3.5 Introducing an improved FORTRAN compiler and the 8087 maths

co-processor 624.3.5.1 System improvements 624.3.5.2 Analysis of upsetting and heading with refined meshes 63

4.4 Summary 64References 65

5 Finite-strain formulation for metalforming analysis 665.1 Introduction 665.2 Governing rate equations 67

5.2.1 Rate of potential energy for updated-Lagrangian increment 675.2.2 Minimisation of rate of potential energy 685.2.3 Incorporation of strain rate 685.2.4 Importance of correct choice of stress rate 69

5.3 Governing incremental equations 705.3.1 Modification of rate expression 705.3.2 Effect of rotation 715.3.3 LCR expression for strain increment 72

5.4 Elastic-plastic formulation 735.4.1 Yield criterion 735.4.2 Elastic-plastic flow rule 745.4.3 Elastic-plastic constitutive relationship 745.4.4 Effect of plastic incompressibility 745.4.5 Element-dilatation technique 75

5.5 Element expressions 775.5.1 Interpolation of nodal displacement 775.5.2 Incremental element-stiffness equations 78

References 78

6 Implementation of the finite-strain formulation 806.1 Introduction 806.2 Performing an FE analysis - an overview 806.3 Pre-processing 82

6.3.1 Mesh generation 82

Contents

6.3.2 Boundary conditions 836.3.3 Material properties 856.3.4 Deformation 856.3.5 Initial state parameters 87

6.4 FE Calculation 876.4.1 Input of data 87

6.4.2 Displacement of surface nodes 886.4.3 Assembly and solution of the stiffness equations 906.4.3.1 Incorporation of frictional restraint 906.4.3.2 Solution techniques 926.4.3.3 Yield-transition increments 956.4.4 Updating of workpiece parameters 956.4.4.1 Strain components and rotational values 95

9699

100100101107108108108111112114

116116117117122122125131135137145149149153153155159159163167

6.4.46.4.46.4.46.4.46.4.46.4.5

6.56.6

6.6.16.6.26.6.3

77.17.2

7.2.17.2.27.2.37.2.47.2.57.2.67.2.77.2.8

7.37.3.1

7.47.4.17.4.2

7.57.5.17.5.2

.2 Deviatonc stress

.3 Hydrostatic stress

.4 External forces

.5 Strain rate

.6 TemperatureOutput of resultsPost-processingSpecial techniquesProcesses involving severe deformationSteady-state processesThree-dimensional analysesReferences

Practical applicationsIntroductionForgingSimple upsettingUpset forgingHeadingPlane-strain side-pressingRim-disc forgingExtrusion-forging'H'-section forgingForging of a connecting rodExtrusionForward extrusionRollingStrip rollingSlab rollingMulti-stage processes - forging sequence designAutomobile spigotShort hollow tube (gudgeon pin)References

Contents

8 Future developments 1698.1 Introduction 1698.2 Developments in software 1708.3 Advances in modelling 1708.4 Material properties 1728.5 Post-processing 1738.6 Expert systems 1748.7 Hardware 1748.8 Applications in the future 175

References 176

Appendices 1781 Derivation of small-strain [B] matrix for 2-D triangular element 1782 Derivation of elastic [D] matrix 1813 Derivation of elastic-plastic [D] matrix 1844 Derivation of small-strain stiffness matrix [K] for plane-stress

triangular element 1885 Solution of stiffness equations by Gaussian elimination and

back-substitution 1916 Imposition of boundary conditions 1967 Relationship between elastic moduli E, G and K 2028 Vectors and tensors 2059 Stress in a deforming body 20910 Stress rates 21611 Listing of BASIC program for small-deformation elastic-plastic

FE analysis 218

Bibliography 246Index 294

Preface

More and more sectors of the metalforming industry are beginning to recognisethe benefits that finite-element analysis of metal-deformation processes can bringin reducing the lead time and development costs associated with the manufactureof new components. Finite-element analyses of non-linear problems, such asmetal deformation, require powerful computing facilities and large amounts ofcomputer running time but advances in computer technology and the fallingprice of hardware are bringing these techniques within the reach of even themost modest R & D department.

Finite-element programs specially designed for the analysis of metalformingprocesses (such as the program EPFEP3*, which is used for the majority ofexamples in this monograph) are now commercially available. As a result, thereis an urgent need for a text that explains the principles underlying finite-elementmetalforming analysis to the people who are starting to use these techniquesindustrially, and to those undertaking the undergraduate and postgraduateengineering courses in the subject that industry is beginning to demand. One ofthe aims of this monograph is to fulfil that need.

The first chapters provide an introduction to the application of finite-element analysis to metalforming problems, starting from the basic ideas of thefinite-element method, and developing these