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
Download
http://s18.alxa.net/s18/srvs2/01/Con...Elasticity.rar
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
Download
http://s18.alxa.net/s18/srvs2/01/Con...Elasticity.rar
