Mechanics Modeling of Sheet Metal Forming
|Publication Date:||1 January 2007|
Beverage cans and many parts in aircraft, appliances, and automobiles are made of thin sheet metals formed by stamping 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, development of an analytical method based on the mechanics principles to predict formability is difficult, if not impossible. Because of modem computer technology, the numerical finite element method at the present time is fesible 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 modem 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 deformation 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 modem 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
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 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 Inetal 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 deformation 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, hydro forming processes have become popular in manufacturing automotive body panels and structural members. Although we do not specifically include simulations of hy dro forming processes in this book, the principles and solution methods presented in this book can be applied to the simulation of hydro forming processes. In fact, one specifies the hydropressure instead of a punch movement in simulations of hydro forming processes. Therefore, the methods proposed in this book are ready to be applied to simulations of hydro forming processes with slight modifications.
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.