Mathematical Foundations of Elasticity

by Marsden, Jerrold E.; Hughes, Thomas J. R.

• ISBN13: 9780486678658
• ISBN10: 0486678652
• Edition: Reprint
• Format: Paperback
• Copyright: 1994-02-18
• Publisher: Dover Publications

Summary

This graduate-level study approaches mathematical foundations of three-dimensional elasticity using modern differential geometry and functional analysis. It is directed to mathematicians, engineers and physicists who wish to see this classical subject in a modern setting with examples of newer mathematical contributions. Relevant problems appear throughout the text. 1983 edition.

Author Biography

Jerrold E. Marsden is Professor of Mathematics, University of California, Berkeley. Thomas J. R. Hughes is Professor of Mechanical Engineering, Stanford University.

Table of Contents

  Preface

Brief glossary of conventions and notations

A point of departure

Kinematics

Balance laws

Elastic materials

Boundary value problems

Constitutive inequalities

The role of geometry and functional analysis

Geometry and kinematics of bodies

Motions of simple bodies

Vector fields, one-forms, and pull-backs

The deformation gradient

Tensors, two-point tensors, and the covariant derivative

Conservation of mass

Flows and lie derivatives

Differential forms and the Piola transformation

Balance principles

The master balance law

The stress tensor and balance of momentum

Balance of energy

Classical spacetimes, covariant balance of energy, and the principle of virtual work

Thermodynamics II; the second law

Constitutive theory

The constitutive hypothesis

Consequences of thermodynamics, locality, and material frame indifference

Covariant constitutive theory

The elasticity tensor and thermoelastic solids

Material symmetries and isotropic elasticity

Linearization

The implicit function theorem

Linearization of nonlinear elasticity

Linear elasticity

Linearization stability

Hamiltonian and variational principles

The formal variational structure of elasticity

Linear Hamiltonian systems and classical elasticity

Abstract Hamiltonian and Lagrangian systems

Lagrangian field theory and nonlinear elasticity

Conservation laws

Reciprocity

Relativistic elasticity

Methods of functional analysis in elasticity

Elliptic operators and linear elastostatics

Abstract semigroup theory

Linear elastodynamics

Nonlinear elastostatics

Nonlinear elastodynamics

The energy criterion

A control problem for a beam equation

Selected topics in bifurcation theory

Basic ideas of static bifurcation theory

A survey of some applications to elastostatics

The traction problem near a natural state (Signorini's problem)

Basic ideas of dynamic bifurcation theory

A survey of some applications to elastodynamics

Bifurcations in the forced oscillations of a beam

Bibliography

Index

Table of Contents provided by Publisher. All Rights Reserved.