Contact Problems in Elasticity





Contact Problems in Elasticity

SIAM Studies in Applied and Numerical Mathematics

This series of monographs focuses on mathematics and its applications to problems

of current concern to industry, government, and society. These monographs will be of

interest to applied mathematicians, numerical analysts, statisticians, engineers, and

scientists who have an active need to learn useful methodology.

Series List

Vol. 1 Lie-Bdcklund Transformations in Applications

Robert L. Anderson and Nail H. Ibragimov

Vol. 2 Methods and Applications of Interval Analysis

Ramon E. Moore

Vol. 3 Ill-Posed Problems for Integrodifferential Equations in Mechanics and

Electromagnetic Theory

Frederick Bloom

Vol. 4 Solitons and the Inverse Scattering Transform

Mark J. Ablowitz and Harvey Segur

Vol. 5 Fourier Analysis of Numerical Approximations of Hyperbolic Equations

Robert Vichnevetsky and John B. Bowles

Vol. 6 Numerical Solution of Elliptic Problems

Garrett Birkhoff and Robert E. Lynch

Vol. 7 Analytical and Numerical Methods for Volterra Equations

Peter Linz

Vol. 8 Contact Problems in Elasticity: A Study of Variational Inequalities and Finite

Element Methods

N. Kikuchi and J. T. Oden

Vol. 9 Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics

Roland Glowinski and P. Le Tallec

Vol. 10 Boundary Stabilization of Thin Plate Splines

John E. Lagnese

Vol.11 Electro-Diffusion of Ions

Isaak Rubinstein

Vol. 12 Mathematical Problems in Linear Viscoelasticity

Mauro Fabrizio and Angelo Morro

Vol. 13 Interior-Point Polynomial Algorithms in Convex Programming

Yurii Nesterov and Arkadii Nemirovskii

Vol. 14 The Boundary Function Method for Singular Perturbation Problems

Adelaida B. Vasil'eva, Valentin F. Butuzov, and Leonid V. Kalachev

Vol. 15 Linear Matrix Inequalities in System and Control Theory

Stephen Boyd, Laurent El Ghaoui, Eric Feron, and Venkataramanan Balakrishnan

Vol. 16 Indefinite-Quadratic Estimation and Control: A Unified Approach to H2 and

H"" Theories

Babak Hassibi, Ali H. Sayed, and Thomas Kailath



Preface

Webster's Ninth New Collegiate Dictionary defines "contact" as "a touching

or meeting of bodies." Even a nontechnical definition of the term thus brings

to mind a mechanical phenomenon involving solid bodies. It is not surprising,

therefore, that contact problems have always occupied a position of special

importance in the mechanics of solids. In this volume, we consider those

contact problems in the theory of elasticity that can be formulated as variational

inequalities. A multitude of results obtained through the use of this method

give rise to new techniques for solving this class of problems, techniques that

not only expose properties of solutions that were obscured by classical methods,

but also provide a basis for the development of powerful numerical schemes.

We believe this is the first truly comprehensive treatment of the problem of

unilateral contact that attempts to unify the physical problems of contact with

the mathematical modeling of the phenomena and numerical implementation

of the models. A detailed study of the qualitative features of the mathematical

model is presented, as well as approximation of the governing equations by

modern numerical methods, a study of the properties of the approximation

including numerical stability, accuracy, and convergence, the development of

algorithms to study the numerical approximations, and the application of the

algorithms to real-world problems.

The first nine chapters of the book discuss the mathematical formulation of

classical contact problems on linearly elastic bodies in which no friction is

present, along with finite element approximations and numerical algorithms

for solving problems of this type. In chapters 10 through 14, we discuss

generalizations of the theory, the complications of friction contact, various

models of dry friction, and applications to static, quasi-static, and dynamic

contact problems, including problems of large deformation, rolling contact,

and inelastic materials. Much of this latter section represents work still very

much in development and hence not fully explored from mathematical or

numerical points of view. Some concluding comments and opinions are

collected at the end of the volume.



A decade will have passed from when we commenced writing this volume

until its appearance in print. Our original plan was to produce a volume on

contact problems in elasticity that focused on variational inequalities in elastostatics

with an emphasis on approximation theory, finite elements, and numeri-

cal analysis. The first eight chapters of the final product still contain much of

the content of the original draft. Many copies of this draft circulated worldwide,

and the final, expanded scope of the work reflects our response to the various

readers, reviewers, and editors encountered along the way. Much of the added

material is drawn from more recent papers and results we have developed

with our students on frictional models and more general contact problems.

Despite the regrettable delay of years of rewriting, this final version still

contains much material published here for the first time.

Several colleagues and students read portions of the manuscript and made

suggestions that improved our presentation of this subject. We wish to thank

Professor Patrick Rabier and Drs. Y. J. Song, L. Demkowicz, L. Campos,

E. Pires, and J. A. C. Martins.

We have had the help of excellent technical typists during the preparation

of various drafts of this work over a period of ten years. An early draft was

typed by Mrs. Bernadette Ashman, another draft by Mrs. Rita Bunstock, and

various revisions by Mdmse. Nancy Webster, Dorothy Baker, Ruth Dye, and

Linda Manifold. To these patient and skillful friends, we owe our sincere thanks.

We gratefully acknowledge that our studies of contact problems, particularly

the use of finite element methods to study contact problems in elasticity, were

primarily supported by the United States Air Force Office of Scientific Research.

N. KIKUCHI

The University of Michigan

J. T. ODEN

The University of Texas



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