1、Mechanics Modeling of Sheet Metal Forming Sing C. Tang and Jwo Pan AMechanics Modeling of Sheet Metal Forming Sing C. Tang Jwo Pan Warrendale, Pa. Copyright 2007 SAE International eISBN: 978-0-7680-5097-4All rights reserved. No part of this publication may be reproduced, stored in a retrieval system
2、, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of SAE. For permission and licensing requests, contact: SAE Permissions 400 Commonwealth Drive Warrendale, PA 15096-0001 USA E-mail: permissionssae.org T
3、el: 724-772-4028 Fax: 724-772-4891 Library of Congress Cataloging-in-Publication Data Tang, Sing C. Mechanics modeling of sheet metal forming / Sing C. Tang, Jwo Pan. p. cm. Includes bibliographical references and index. ISBN 978-0-7680-0896-8 1. Sheet-metal work. 2. Continuum mechanics. I. Pan, J.
4、(Jwo). II. Title. TS250.T335 2007 671.823011-dc22 2006039364 SAE International 400 Commonwealth Drive Warrendale, PA 15096-0001 USA E-mail: CustomerServicesae.org Tel: 877-606-7323 (inside USA and Canada) 724-776-4970 (outside USA) Fax: 724-776-1615 Copyright 2007 SAE International ISBN 978-0-7680-0
5、896-8 SAE Order No. R-321 Printed in the United States of America.Thanks to our families for their support and patience. To my wife Kin Ling Sing C. Tang To my mom Mei-Chin and my wife Michelle Jwo PanContents Preface . xi 1. Introduction to Typical Automotive Sheet Metal Forming Processes 1 1.1 Str
6、etching and Drawing 2 1.2 Trimming 7 1.3 Flanging and Hemming 7 1.4 References 9 2. Tensor, Stress, and Strain 11 2.1 Transformation of Vectors and Tensors in Cartesian Coordinate Systems 11 2.2 Transformation of Vectors and Tensors in General Coordinate Systems 15 2.3 Stress and Equilibrium 19 2.4
7、Principal Stresses and Stress Invariants 23 2.5 Finite Deformation Kinematics 25 2.6 Small Strain Theory 28 2.7 Different Stress Tensors 32 2.8 Stresses and Strains from Tensile Tests 36 2.9 Reference 37 3. Constitutive Laws 39 3.1 Linear Elastic Isotropic Materials 40 3.2 Linear Elastic Anisotropic
8、 Materials 44 3.3 Different Models for Uniaxial Stress-Strain Curves 47 3.4 Yield Functions Under Multiaxial Stresses 52 3.4.1 Maximum Plastic Work Inequality 52 3.4.2 Yield Functions for Isotropic Materials 53 3.4.2.1 von Mises Yield Condition 55 3.4.2.2 Tresca Yield Condition 56 3.4.2.3 Plane Stre
9、ss Yield Conditions for Isotropic Materials 57 3.4.3 Yield Functions for Anisotropic Materials 59 3.4.3.1 Hill Quadratic Yield Condition for Orthotropic Materials 60vi Mechanics Modeling of Sheet Metal Forming 3.4.3.2 A General Plane Stress Anisotropic Yield Condition .65 3.5 Evolution of Yield Surf
10、ace 67 3.6 Isotropic Hardening Based on the von Mises Yield Condition 71 3.7 Anisotropie Hardening Based on the von Mises Yield Condition 76 3.8 Isotropic Hardening Based on the von Mises Yield Condition with Rate Sensitivity 79 3.9 Isotropic and Anisotropic Hardening Based on the Hill Quadratic Ani
11、sotropic Yield Condition 83 3.10 Plastic Localization and Forming Limit Diagram 86 3.11 Modeling of Failure Processes 88 3.12 References .92 4. Mathematical Models for Sheet Metal Forming Processes 95 4.1 Governing Equations for Simulation of Sheet Metal Forming Processes 95 4.2 Equations of Motion
12、for Continua .95 4.3 Equations of Motion in Discrete Form 96 4.3.1 Internal Nodal Force Vector 97 4.3.2 External Nodal Force Vector 97 4.3.3 Contact Nodal Force Vector 97 4.3.4 Mass and Damping Matrices 98 4.3.5 Equations of Motion in Matrix Form 99 4.4 Tool Surface Models 99 4.5 Surface Contact wit
13、h Friction 100 4.5.1 Formulation for the Direct Method .102 4.5.2 Formulation for the Lagrangian Multiplier Method 103 4.5.3 Formulation for the Penalty Method 107 4.6 Draw-Bead Model 109 4.6.1 Draw-Bead Restraint Force by Computation. 113 4.6.2 Draw-Bead Restraint Force by Measurement 113 4.7 Refer
14、ences 115 5. Thin Plate and Shell Analyses 117 5.1 Plates and General Shells 117 5.2 Assumptions and Approximations 117 5.3 Base Vectors and Metric Tensors 118Contents vii 5.4 Lagrangian Strains 125 5.5 Classical Shell Theory 126 5.5.1 Strain-Displacement Relationship 126 5.5.2 Principle of Virtual
15、Work 131 5.5.3 Constitutive Equation for the Classical Shell Theory 131 5.5.4 Yield Function and Flow Rule for the Classical Shell Theory 132 5.5.5 Consistent Material Tangent Stiffness Tensor 134 5.5.6 Stress Resultant Constitutive Relationship 140 5.6 Shell Theory with Transverse Shear Deformation
16、 141 5.6.1 Constitutive Equation for the Shell Theory with Transverse Shear Deformation 142 5.6.2 Consistent Material Tangent Stiffness Tensor with Transverse Shear Deformation 143 5.7 References .147 6. Finite Element Methods for Thin Shells 149 6.1 Introduction 149 6.1.1 Computer-Aided Engineering
17、 (CAE) Requirements for Shell Elements 150 6.1.2 Displacement Method 150 6.2 Finite Element Method for the Classical Shell TheoryTotal Lagrangian Formulation 151 6.2.1 Strain-Displacement Relationship in Incremental Forms 151 6.2.2 Virtual Work Due to the Internal Nodal Force Vector 152 6.2.3 Discre
18、tization of Spatial Variables in a Curved Triangular Shell Element 154 6.2.4 Increments of the Strain Field in Terms of Nodal Displacement Increments .156 6.2.5 Element Tangent Stiffness Matrix and Nodal Force Vector 160 6.2.6 Basic and Shape (Interpolation) Functions 162 6.2.7 Numerical Integration
19、 for a Curved Triangular Shell Element 167 6.2.8 Updating Configurations, Strains, and Stresses 171 6.3 Finite Element Method for a Shell with Transverse Shear DeformationUpdated Lagrangian Formulation 173 6.3.1 Strain-Displacement Relationship in Incremental Form 173 6.3.2 Virtual Work Due to the I
20、nternal Nodal Force Vector .177 6.3.3 Discretization of Spatial Variables in a Quadrilateral Shell Element. 179 6.3.4 Increment of the Strain Field in Terms of Nodal Displacement Increments 180 6.3.5 Element Tangent Stiffness Matrix and Nodal Force Vector 181viii Mechanics Modeling of Sheet Metal Fo
21、rming 6.3.6 Shape (Interpolation) Functions 186 6.3.7 Numerical Integration for a Quadrilateral Shell Element 187 6.3.8 Five to Six Degrees of Freedom per Node 189 6.3.9 Updating Configurations, Strains, and Stresses 189 6.3.10 Shear Lock and Membrane Lock 197 6.4 Discussion of C1 and C0 Continuous
22、Elements 199 6.5 References 200 7. Methods of Solution and Numerical Examples 201 7.1 Introduction to Methods for Solving Equations of Motion 201 7.1.1 Equations of Motion and Constraint Conditions 201 7.1.2 Boundary and Initial Conditions 204 7.1.3 Explicit and Implicit Integration 205 7.1.4 Quasi-
23、Static Equations 205 7.2 Explicit Integration of Equations of Motion with Constraint Conditions 206 7.2.1 Discretization and Solutions 206 7.2.2 Numerical Instability 208 7.2.3 Computing Contact Nodal Forces 209 7.2.4 Updating Variables for Dynamic Explicit Integration 209 7.2.5 Summary of the Dynam
24、ic Explicit Integration Method with Contact Nodal Forces Computed by the Penalty Method 210 7.2.6 Application of the Dynamic Explicit Integration Method to Sheet Metal Forming Analysis 210 7.3 Implicit Integration of Equations of Motion with Constraint Conditions 210 7.3.1 Newmarks Integration Schem
25、e 212 7.3.2 Newton-Raphson Iteration 212 7.3.3 Computing the Contact Nodal Force Vector by the Direct Method 213 7.3.4 Computing the Contact Nodal Force Vector by the Lagrangian Multiplier Method 216 7.3.5 Computing the Contact Nodal Force Vector by the Penalty Method 218 7.3.6 Solving a Large Numbe
26、r of Simultaneous Equations 220 7.3.7 Convergence of the Newton-Raphson Iteration 221 7.3.8 Updating Variables for Dynamic Implicit Integration 222 7.3.9 Summary of the Implicit Integration Method with Contact Nodal Forces Computed by the Penalty Method 223 7.3.10 Application of Dynamic Implicit Int
27、egration to Sheet Metal Forming Analysis 224Contents ix 7.4 Quasi-Static Solutions 224 7.4.1 Equations of Equilibrium and Constraint Conditions 225 7.4.2 Boundary and Initial Conditions for Quasi-Static Analysis 226 7.4.3 Quasi-Static Solutions Without an Equilibrium Check 226 7.4.4 Quasi-Static Sol
28、utions with an Equilibrium Check 227 7.4.5 Summary of the Quasi-Static Method with the Contact Nodal Force Vector Computed by the Penalty Method 230 7.4.6 Application of the Quasi-Static Method to Sheet Metal Forming Analysis 231 7.5 Integration of Constitutive Equations 232 7.5.1 Integration of Rat
29、e-Insensitive Plane Stress Constitutive Equations with Isotropic Hardening 236 7.5.2 Integration of Rate-Insensitive Plane Stress Constitutive Equations with Anisotropic Hardening 240 7.5.3 Integration of Rate-Insensitive Constitutive Equations with Transverse Shear Strains and Anisotropic Hardening
30、 244 7.6 Computing Springback 246 7.6.1 Approximate Method for Computing Springback 247 7.6.2 Constitutive Equations for Springback Analysis 248 7.7 Remeshing and Adaptive Meshing 250 7.7.1 Refinement and Restoration for Triangular Shell Elements 252 7.7.2 Refinement and Restoration for Quadrilatera
31、l Shell Elements 257 7.8 Numerical Examples of Various Forming Operations 258 7.8.1 Numerical Examples of Sheets During Binder Wrap 258 7.8.2 Numerical Examples of Sheets During Stretching or Drawing 258 7.8.3 Numerical Examples of Springback After Various Forming Operations 260 7.9 References 268 8
32、. Buckling and Wrinkling Analyses 271 8.1 Introduction 271 8.2 Riks Approach for Solution of Snap-Through and Bifurcation Buckling 273 8.2.1 Critical Points 274 8.2.2 Establishment of Governing Equations in the N + 1 Dimensional Space 278 8.2.3 Characteristics of Governing Equations in the N + 1 Dim
33、ensional Space 280 8.2.4 Solution for Snap-Through Buckling 281x Mechanics Modeling of Sheet Metal Forming 8.2.5 Methods to Locate the Secondary Path for Bifurcation Buckling 281 8.2.6 Method to Locate Critical Points and the Tangent Vector to the Primary Path for Bifurcation Buckling . 285 8.3 Meth
34、ods to Treat Snap-Through and Bifurcation Buckling in Forming Analyses 286 8.3.1 Introduction of Artificial Springs at Selected Nodes .286 8.3.2 Forming Analyses of Snap-Through Buckling and Numerical Examples .287 8.3.3 Forming Analyses of Bifurcation Buckling and Numerical Examples 290 8.4 Referen
35、ces 295 Index. 297 About the Authors 309Preface Beverage cans and many parts in aircraft, appliances, and automobiles are made of thin sheet metals formed by stamp- ing operations at room temperature. Thus, sheet metal forming processes play an important role in mass production. Conventionally, the
36、forming process and tool designs are based on the trial-and-error method or the pure geometric method of surface fitting that requires an actual hardware tryout that is called a die tryout. This design process often is expensive and time consuming because forming tools must be built for each trial.
37、Significant savings are possible if a designer can use simulation tools based on the principles of mechanics to predict formability before building forming tools for tryout. Due to the geometric complexity of sheet metal parts, especially automotive body panels, develop- ment of an analytical method
38、 based on the mechanics principles to predict formability is difficult, if not impossible. Because of modern computer technology, the numerical finite element method at the present time is feasible for such a highly nonlinear analysis using a digital computer, especially one equipped with vector and
39、 parallel processors. Although simulation of sheet metal forming processes using a modern digital computer is an important technology, a comprehensive book on this subject seems to be lacking in the literature. Fundamental principles are discussed in some books for forming sheet metal parts with sim
40、ple geometry such as plane strain or axisymmetry. In contrast, detailed theoretically sound formulations based on the principles of continuum mechanics for finite or large deforma- tion are presented in this book for implementation into simulation codes. The contents of this book represent proof of
41、the usefulness of advanced continuum mechanics, plasticity theories, and shell theories to practicing engineers. The governing equations are presented with specified boundary and initial conditions, and these equations are solved using a modern digital computer (engineering workstation) via finite e
42、lement methods. Therefore, the forming of any complex part such as an automotive inner panel can be simulated. We hope that simulation engineers who read this book will then be able to use simulation software wisely and better understand the output of the simulation software. Therefore, this book is
43、 not only a textbook but also a reference book for practicing engineers. Because advanced topics are discussed in the book, readers should have some basic knowledge of mechanics, constitutive laws, finite element methods, and matrix and tensor analyses. Chapter 1 gives a brief introduction to typica
44、l automotive sheet metal forming processes. Basic mechanics, vectors and tensors, and constitutive laws for elastic and plastic materials are reviewed in Chapters 2 and 3, based on course material taught at the University of Michigan by Dr. Jwo Pan. The remaining chapters are drawn from the experien
45、ce of Dr. Sing C. Tang, who had been working on simulations of real automotive sheet metal parts at Ford Motor Company for more than 15 years. Chapter 2 presents the fundamental concepts of tensors, stress, and strain. The definitions of the stresses and strains in tensile tests then are discussed.
46、Readers should pay special attention to the kinematics of finite deformation and the definitions of different stress tensors due to finite deformation because extremely large deformation occurs in sheet metal forming processes. Chapter 3 reviews the linear elastic constitutive laws for small or infi
47、nitesimal deformation. Hookes law for isotropic linear elastic materials, which is widely used in many mechanics analyses, is discussed first. Anisotropic linear elastic behavior also is discussed in detail. Then, deviatoric stresses and deviatoric strains are introduced. These concepts are used as
48、the basis for development of pressure-independent incompressible anisotropic plasticity theory. Chapter 3 also discusses fundamentals of mathematical plasticity theories. In sheet metal forming processes, most of the deformation is plastic. Therefore, knowledge of plasticity is essential in using si
49、mulation software and in understanding simulation results. Different mathematical models for uniaxial tensile stress-strain relations are introduced first. Then the yield conditions for isotropic incompressible materials under multiaxial stress states are presented. Because sheet metals generally are plastically anisotropic, the anisotropic yield conditions are discussed in detail. The basic concepts of the formation of constitutive laws with consideration of plastic hardening behavior of materials also are presented. Finally, the principles of plastic localization and modeling
