Mathematical Problems in Plasticity

This study of the problem of the equilibrium of a perfectly plastic body under specific conditions employs tools and methods that can be applied to other areas, including the mechanics of fracture and certain optimal control problems.
The three-part approach begins with an exploration of variational problems in plasticity theory, covering function spaces, concepts and results of convex analysis, formulation and duality of variational problems, limit analysis, and relaxation of the boundary condition. The second part examines the solution of variational problems in the finite-energy spaces; its topics include relaxation of the strain problem, duality between the generalized stresses and strains, and the existence of solutions to the generalized strain problem. The third and final part addresses asymptotic problems and problems in the theory of plates. The text includes a substantial bibliography and a new Preface and appendix by the author.

Roger Teman is Director of the Institute for Scientific Computing and Applied Mathematics at Indiana University. He was previously on the faculty of the University of Paris-Sud from 1967–2003. A member of the French Academy of Sciences he was also elected to the American Academy of Arts and Sciences in 2015. His many books include Mathematical Modelling in Continuum Mechanics and Navier-Stokes Equations.

Table of Contents

Preface 2018

Foreword

Chapter I Variational problems in plasticity theory
Introduction
1. Function spaces
2. Convex analysis--a review of some basic concepts and results
3. Formulation of the variational problems of plasticity theory
4. Duality of the variational problems
5. Limit Analysis
6. Relaxation of the boundary condition

Chapter II Solution of the variational problems in the finite-energy spaces
Introduction
1. Further results on the space LD (<<ohm symbol>>)
2. The space BD(<<ohm symbol>>) (I)
3. The space BD(<<ohm symbol>>) (II)
4. Convex functions of a measure
5. Convex functionals of a measure
6. Example of a convex function of a measure: relaxation of the strain problem
7. Duality between the generalised stresses and strains
8. Existence of solutions to the generalised strain problem

Chapter III Asymptotic problems and problems in the theory of plates
Introduction
1. Some asymptotic problems: problems of imperfectly plastic bodies
2. Some problems in the theory of plates

Principal Notations

Index

Bibliography

Appendix