Mechanics modeling of sheet metal forming

Mechanics Modeling of Sheet Metal Forming Sing C. Tang Jwo Pan Warrendale, Pa. All rights reserved. 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 SAE. For permission and licensing requests, contact: SAE Permissions 400 Commonwealth Drive Warrendale, PA 15096-0001 USA E-mail: permissions@sae.org Tel: 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. (Jwo). II. Title. TS250.T335 2007 671.8'23011--dc22 2006039364 SAE International 400 Commonwealth Drive Warrendale, PA 15096-0001 USA E-mail: CustomerService@sae.org Tel: 877-606-7323 (inside USA and Canada) 724-776-4970 (outside USA) Fax: 724-776-1615 Copyright © 2007 SAE International ISBN 978-0-7680-0896-8 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 Pan Contents Preface . xi 1. Introduction to Typical Automotive Sheet Metal Forming Processes 1 1.1 Stretching 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 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 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 Stress 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 60 vi Mechanics Modeling of Sheet Metal Forming 3.4.3.2 A General Plane Stress Anisotropic Yield Condition .........65 3.5 Evolution of Yield Surface 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 Anisotropic 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 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 with 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 References 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 118 Contents 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 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 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 (CAE) Requirements for Shell Elements 150 6.1.2 Displacement Method ....150 6.2 Finite Element Method for the Classical Shell Theory—Total 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 Discretization 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 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 Deformation—Updated Lagrangian Formulation 173 6.3.1 Strain-Displacement Relationship in Incremental Form 173 6.3.2 Virtual Work Due to the Internal 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 181 viii Mechanics Modeling of Sheet Metal Forming 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 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-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 Dynamic 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 Newmark's Integration Scheme 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 Number 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 Integration to Sheet Metal Forming Analysis 224 Contents 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 Solutions 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 Rate-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 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 Quadrilateral 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. 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 Dimensional Space 280 8.2.4 Solution for Snap-Through Buckling 281 x 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 Methods 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 References 295 Index............. ..297 About the Authors 309 Preface 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 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. 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 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 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 simple 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 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 element 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 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 typical 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 experience 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. 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 infinitesimal deformation. Hooke's 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 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 simulation 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 of failure processes based on void mechanics are summarized. Chapter 4 introduces formulations for analyses of sheet metal forming processes, including binder closing, stretching/ drawing, trimming, flanging, and hemming. More attention is paid to the most basic analysis of the stretching/drawing xii Mechanics Modeling of Sheet Metal Forming process, which then can be extended to analyses of all other processes. The formulations include equations of motion, constitutive equations, tool surface modeling, surface contact forces, and draw-bead modeling. Chapter 5 discusses thin shell theories. Tensors with reference to the curvilinear coordinate system are used. Most sheet metal parts are made of thin sheets and can be modeled by thin shells for numerical efficiency and accuracy. Engineers may be tempted to use three-dimensional (3-D) solid elements, which are more general, to model a metal sheet under plastic deformation. However, the solid element model contains too many degrees of freedom to be solved using the current generation of digital computers. Even for the explicit time integration method, we cannot handle a finite element model with too many degrees of freedom for reasonable computation accuracy and time. The reason is that the dimension in the thickness direction of the sheet is very small compared to other dimensions. To satisfy the stability requirement for a numerical solution using the explicit time integration method, an extremely small time increment for a three-dimensional mesh must be used. However, it still is not practical at the present time, and the shell model is emphasized in this book. Chapter 6 presents formulations of two shell elements for finite element models appropriate for use in computation. The interpolation (shape) function for the C1 continuous shell element is complex but accurate, and it provides good convergence for the implicit integration method. The interpolation function for the C0 continuous element is simple, but it might have a shear locking problem for thin sheets. Chapter 7 presents solution methods for the equations of motion by the explicit time integration and implicit time integration methods. The contact forces are computed by the direct, Lagrangian multiplier, or penalty methods. If the dynamic effects are neglected, the equations of motion are reduced to the equations of equilibrium that are solved by the quasi-static method. Although the quasi-static method is more appropriate for analyses of sheet metal forming processes, it has convergence problems. Also, it would break down for a singular stiffness matrix when structural instability occurs. Structural stability problems also are discussed in Chapter 7. The radial return method is discussed to compute the stress increment from a given strain increment for more accurate numerical results. Computation of springback also is discussed briefly. For more efficient computations, adaptive meshing is introduced. Finally, various numerical examples for forming, springback, and flanging operations are given. Chapter 8 on buckling and wrinkling analyses briefly introduces Rik's approach to the solution of snap-through and bifurcation buckling. This type of instability may occur when the global stiffness matrix in the quasi-static method becomes singular. Because analyses of sheet metal forming processes mainly involve surface contact with friction, Rik's method cannot be applied directly without modification. Some methods are suggested to compute sheet defor- mation continuously to the post-buckling and wrinkling region. Numerical examples for buckling and wrinkling in production automotive panels are demonstrated at the end of Chapter 8. Recently, hydroforming processes have become popular in manufacturing automotive body panels and structural members. Although we do not specifically include simulations of hydroforming processes in this book, the principles and solution methods presented in this book can be applied to the simulation of hydroforming processes. In fact, one specifies the hydropressure instead of a punch movement in simulations of hydroforming processes. Therefore, the methods proposed in this book are ready to be applied to simulations of hydroforming processes with slight modifica- tions. We would like to thank Professor Pai-Chen Lin of the National Chung-Cheng University for preparing most of the figures in this book. We also want to thank Ms. Selina Pan of the University of Michigan for preparing some figures in this book. Sing C. Tang Jwo Pan Ann Arbor, Michigan June, 2006