Mechanics of Sheet Metal Forming Mechanics of Sheet Metal Forming Z. Marciniak The Technical University of Warsaw, Poland J.L. Duncan The University of Auckland, New Zealand S.J. Hu The University of Michigan, USA OXFORD AMSTERDAM BOSTON LONDON NEW YORK PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO Butterworth-Heinemann An imprint of Elsevier Science Linacre House, Jordan Hill, Oxford OX2 8DP 225 Wildwood Avenue, Woburn, MA 01801-2041 First published by Edward Arnold, London, 1992 Second edition published by Butterworth-Heinemann 2002 Copyright  2002 S.J. Hu, Z. Marciniak, J.L. Duncan All rights reserved The right of S.J. Hu, Z. Marciniak and J.L. Duncan to be identified as the authors of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 All rights reserved. No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England W1T 4LP. Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to the publishers British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress ISBN 0 7506 5300 0 For information on all Butterworth-Heinemann publications visit our website at www.bh.com Typeset by Laserwords Private Limited, Chennai, India Printed and bound in Great Britain Contents Preface to the second edition ix Preface to the first edition xi Disclaimer xii Introduction xi 1 Material properties 1 1.1 Tensile test 1 1.2 Effect of properties on forming 10 1.3 Other mechanical tests 12 1.4 Exercises 13 2 Sheet deformation processes 14 2.1 Introduction 14 2.2 Uniaxial tension 14 2.3 General sheet processes (plane stress) 16 2.4 Yielding in plane stress 17 2.5 The flow rule 22 2.6 Work of plastic deformation 24 2.7 Work hardening hypothesis 25 2.8 Effective stress and strain functions 26 2.9 Summary 27 2.10 Exercises 27 3 Deformation of sheet in plane stress 30 3.1 Uniform sheet deformation processes 30 3.2 Strain distributions 31 3.3 Strain diagram 31 3.4 Modes of deformation 33 3.5 Effective stress–strain laws 36 3.6 The stress diagram 39 3.7 Principal tensions or tractions 41 3.8 Summary 43 3.9 Exercises 43 v 4 Simplified stamping analysis 45 4.1 Introduction 45 4.2 Two-dimensional model of stamping 46 4.3 Stretch and draw ratios in a stamping 57 4.4 Three-dimensional stamping model 57 4.5 Exercises 59 5 Load instability and tearing 61 5.1 Introduction 61 5.2 Uniaxial tension of a perfect strip 62 5.3 Tension of an imperfect strip 64 5.4 Tensile instability in stretching continuous sheet 67 5.5 Factors affecting the forming limit curve 75 5.6 The forming window 79 5.7 Exercises 80 6 Bending of sheet 82 6.1 Introduction 82 6.2 Variables in bending a continuous sheet 82 6.3 Equilibrium conditions 84 6.4 Choice of material model 85 6.5 Bending without tension 86 6.6 Elastic unloading and springback 92 6.7 Small radius bends 96 6.8 The bending line 100 6.9 Bending a sheet in a vee-die 104 6.10 Exercises 106 7 Simplified analysis of circular shells 108 7.1 Introduction 108 7.2 The shell element 108 7.3 Equilibrium equations 110 7.4 Approximate models of forming axisymmetric shells 111 7.5 Applications of the simple theory 112 7.6 Summary 115 7.7 Exercises 116 8 Cylindrical deep drawing 117 8.1 Introduction 117 8.2 Drawing the flange 117 8.3 Cup height 123 8.4 Redrawing cylindrical cups 123 8.5 Wall ironing of deep-drawn cups 125 8.6 Exercises 127 vi Contents 9 Stretching circular shells 129 9.1 Bulging with fluid pressure 129 9.2 Stretching over a hemispherical punch 132 9.3 Effect of punch shape and friction 134 9.4 Exercises 135 10 Combined bending and tension of sheet 136 10.1 Introduction 136 10.2 Stretching and bending an elastic, perfectly plastic sheet 136 10.3 Bending and stretching a strain-hardening sheet 142 10.4 Bending a rigid, perfectly plastic sheet under tension 144 10.5 Bending and unbending under tension 145 10.6 Draw-beads 150 10.7 Exercises 151 11 Hydroforming 152 11.1 Introduction 152 11.2 Free expansion of a cylinder by internal pressure 153 11.3 Forming a cylinder to a square section 155 11.4 Constant thickness forming 159 11.5 Low-pressure or sequential hydroforming 161 11.6 Summary 163 11.7 Exercises 163 Appendix A1 Yielding in three-dimensional stress state 165 Appendix A2 Large strains: an alternative definition 168 Solutions to exercises 176 Index 205 Contents vii Preface to the second edition The first edition of this book was published a decade ago; the Preface stated the objective in the following way. In this book, the theory of engineering plasticity is applied to the elements of common sheet metal forming processes. Bending, stretching and drawing of simple shapes are analysed, as are certain processes for forming thin-walled tubing. Where possible, the limits governing each process are identified and this entails a detailed study of tensile instability in thin sheet. To the authors’ knowledge, this is the first text in English to gather together the mechanics of sheet forming in this manner. It does, however, draw on the earlier work of, for example, Swift, Sachs, Fukui, Johnson, Mellor and Backofen although it is not intended as a research monograph nor does it indicate the sources of the models. It is intended for the student and the practitioner although it is hoped that it will also be of interest to the researcher. This second edition keeps to the original aim, but the book has been entirely rewritten to accommodate changes in the field and to overcome some earlier deficiencies. Professor Hu joined the authors and assisted in this revision. Worked examples and new problems (with sample solutions) have been added as well as new sections including one on hydro- forming. Some of the original topics have been omitted or given in an abbreviated form in appendices. In recent years, enormous progress has been made in the analysis of forming of complex shapes using finite element methods; many engineers are now using these systems to analyse forming of intricate sheet metal parts. There is, however, a wide gulf between the statement of the basic laws governing deformation in sheet metal and the application of large modelling packages. This book is aimed directly at this middle ground. At the one end, it assumes a knowledge of statics, stress, strain and models of elastic deformation as contained in the usual strength of materials courses in an engineering degree program. At the other end, it stops short of finite element analysis and develops what may be called ‘mechanics models’ of the basic sheet forming operations. These models are in many respects similar to the familiar strength of materials models for bending, torsion etc., in that they are applied to simple shapes, are approximate and often contain simplifying assumptions that have been shown by experience to be reasonable. This approach has proved helpful to engineers entering the sheet metal field. They are confronted with an ix industry that appears to be based entirely on rules and practical experience and they require some assistance to see how their engineering training can be applied to the design of tooling and to the solution of problems in the stamping plant. Experienced sheet metal engineers also find the approach useful in conceptual design, in making quick calculations in the course of more extensive design work, and in interpreting and understanding the finite element simulation results. Nevertheless, users of these models should be aware of the assumptions and limitations of these approximate models as real sheet metal designs can be much more complex than what is captured by the models. The order in which topics are presented has been revised. It now follows a pattern developed by the authors for courses given at graduate level in the universities and to sheet metal engineers and mechanical metallurgists in industry and particularly in the automotive field. The aim is to bring students as quickly as possible to the point where they can analyse simple cases of common processes such as the forming of a section in a typical stamping. To assist in tutorial work in these courses, worked examples are given in the text as well as exercises at the end of each chapter. Detailed solutions of the exercises are given at the end of the text. The possibility of setting interesting problems is greatly increased by the familiarity of students with computer tools such as spread sheets. Although not part of this book, it is possible to go further and develop animated models of processes such as bending, drawing and stamping in which students can investigate the effect of changing variables such as friction or material properties. At least one package of this kind is available through Professor Duncan and Professor Hu. Many students and colleagues have assisted the authors in this effort to develop a sound and uncomplicated base for education and the application of engineering in sheet metal forming. It is impossible to list all of these, but it is hoped that they will be aware of the authors’ appreciation. The authors do, however, express particular thanks to sev- eral who have given invaluable help and advice, namely, A.G. Atkins, W.F. Hosford, F. Wang, J. Camelio and the late R. Sowerby. In addition, others have provided com- ment and encouragement in the final preparation of the manuscript, particularly M. Dingle and R. Andersson; the authors thank them and also the editorial staff at Butterworth- Heinemann. J.L. Duncan Auckland S.J. Hu Ann Arbor 2002 x Preface to the second edition Preface to the first edition In this book, the theory of engineering plasticity is applied to the elements of common sheet metal forming processes. Bending, stretching and drawing of simple shapes are analysed, as are certain processes for forming thin-walled tubing. Where possible, the limits governing each process are identified and this entails a detailed study of tensile instability in thin sheet. To the author’s knowledge, this is the first text in English to gather together the mechan- ics of sheet forming in this manner. It does, however, draw on the earlier work of, for example, Swift, Sachs Fukui, Johnson, Mellor and Backofen although it is not intended as a research monograph nor does it indicate the sources of the models. It is intended for the student and the practitioner although it is hoped that it will also be of interest to the researcher. In the first two chapters, the flow theory of plasticity and the analysis of proportional large strain processes are introduced. It is assumed that the reader is familiar with stress and strain and the mathematical manipulations presented in standard texts on the basic mechanics of solids. These chapters are followed by a detailed study of tensile instability following the Marciniak–Kuczynski theory. The deformation in large and small radius bends is studied and an approximate but useful approach to the analysis of axisymmetric shells is introduced and applied to a variety of stretching and drawing processes. Finally, simple tube drawing processes are analysed along with energy methods used in some models. A number of exercises are presented at the end of the book and while the book is aimed at the engineer in the sheet metal industry (which is a large industry encompassing automotive, appliance and aircraft manufacture) it is also suitable as a teaching text and has evolved from courses presented in many countries. Very many people have helped with the book and it is not possible to acknowledge each by name but their contributions are nevertheless greatly appreciated. One author (J.L.D.) would like to thank especially his teacher W. Johnson, his good friend and guide over many years R. Sowerby, the illustrator S. Stephenson and, by no means least, Mrs Joy Wallace who typed the final manuscript. Z. Marciniak, Warsaw J.L. Duncan, Auckland 1991 xi Disclaimer The purpose of this book is to assist students in understanding the mechanics of sheet metal forming processes. Many of the relationships are of an approximate nature and may be unsuitable for engineering design calculations. While reasonable care has been taken, it is possible that errors exist in the material contained and neither the authors nor the publisher can accept responsibility for any results arising from use of information in this book. xii [...]... strain when the load has been removed as shown in Figure 1 .3( b) Material properties 3 Engineering stress, MPa 30 0 Eu 250 TS 200 E Tot 150 (sf)0 100 50 0 0 5 10 15 20 25 30 35 Engineering strain, % 40 45 50 (a) Engineering stress seng ey (sf)0 E eeng 0 Engineering strain (b) seng Engineering stress sproof 0.2% 0 eeng Engineering strain (c) Figure 1 .3 (a) Engineering stress–strain curve for the test of drawing... extension of 13. 55 mm The test-piece fails at an extension of 22.69 mm Determine the following: tensile strength, = 12.5 × 0.80 = 10 mm2 = 10−5 m2 1.791 × 1 03 = = 179 × 106 P a = 179 MPa 10−5 = 2.94 × 1 03 ÷ 10−5 = 294 MPa total elongation, = (22.69/50) × 100% = 45.4% initial cross-sectional area, initial yield stress, true stress at maximum load, = 294(50 + 13. 55)/50 = 37 4 MPa 50 + 13. 55 = 0.24 maximum... condition will show the characteristics of this diagram At low strains in the elastic range, the curve is approximately linear with a slope of unity; this corresponds to an equation for the elastic regime of σ = Eε or log σ = log E + log ε (1.15) 3. 5 3 Log stress log K 2.5 n 2 1 1 1.5 Elastic 1 3. 5 3 Plastic −2.5 −2 −1.5 Log strain −1 −0.5 0 Figure 1.6 True stress–strain from the above diagram plotted... similar models for the deformation of sheet In this way, the engineer can apply a familiar approach to problem solving in sheet metal engineering Application to design The objective in studying the basic mechanics of sheet metal forming is to apply this to part and tool design and the diagnosis of plant problems It is important to appreciate that analysis is only one part of the design process The first... strain-hardening index, n [Ans: 156 MPa, 294 MPa, 0.24, 45%, 530 MPa, 0.24] Material properties 13 2 Sheet deformation processes 2.1 Introduction In Chapter 1, the appropriate definitions for stress and strain in tensile deformation were introduced The purpose now is to indicate how the true stress–strain curve derived from a tensile test can be applied to other deformation processes that may occur in typical... Metal Forming Tensile test s3 = 0, e3 = −e1/2 Plane stress s3 = 0, e3 = − (1 + b)e1 s2 = 0, e2 = −e1/2 s1, e1 s2 = as1 e2 = be1 s1, e1 (a) (b) Figure 2.2 Principal stresses and strains for elements deforming in (a) uniaxial tension and (b) a general plane stress sheet process principal directions so that σ1 > σ2 and the third direction is perpendicular to the surface where 3 = 0 The deformation mode... we may assume from symmetry that the strains in the width and thickness directions will be equal in magnitude and hence, from Equation 2 .3, 1 dε2 = d 3 = − dε1 2 (In the previous chapter we considered the case in which the material was anisotropic where dε2 = Rd 3 and the R-value was not unity We can develop a general theory for anisotropic deformation, but this is not necessary at this stage.) Sheet... does not rotate with respect to the principal directions If, and only if, these conditions apply, we may safely use the integrated or large strains defined in Chapter 1 For uniaxial deformation of an isotropic material, these strains are ε1 = ln l ; l0 ε2 = ln w 1 = − ε1 ; w0 2 3 = ln t 1 = − ε1 to 2 (2.5) 2 .3 General sheet processes (plane stress) In contrast with the tensile test in which two of... 1.22 2.059 × 1 03 × true stress at A, = = 211 MPa −5 10 50 50 + 1.22 true strain at A, = ln = 0.024 50 By fitting a power law to two points, point A, and the maximum load point, determine an approximate value of the strain-hardening index and the value of K n= log 37 4 − log 211 log σmax − log σA = = 0.25 log εu − log εA log 0.24 − log 0.024 By substitution, 211 = K × 0.0240.25 , ∴K = 536 MPa (Note that... surface where 3 = 0 The deformation mode is thus: ε1 ; ε2 = βε1 ; 3 = −(1 + β)ε1 σ1 ; σ2 = ασ1 ; 3 = 0 (2.6) The constant volume condition is used to obtain the third principal strain Integrating the strain increments in Equation 2 .3 shows that this condition can be expressed in terms of the true or natural strains: ε1 + ε2 + 3 = 0 (2.7) i.e the sum of the natural strains is zero For uniaxial tension, . 27 2.10 Exercises 27 3 Deformation of sheet in plane stress 30 3. 1 Uniform sheet deformation processes 30 3. 2 Strain distributions 31 3. 3 Strain diagram 31 3. 4 Modes of deformation 33 3. 5 Effective. deformation 33 3. 5 Effective stress–strain laws 36 3. 6 The stress diagram 39 3. 7 Principal tensions or tractions 41 3. 8 Summary 43 3.9 Exercises 43 v 4 Simplified stamping analysis 45 4.1 Introduction. Stretching over a hemispherical punch 132 9 .3 Effect of punch shape and friction 134 9.4 Exercises 135 10 Combined bending and tension of sheet 136 10.1 Introduction 136 10.2 Stretching and bending an