Mathematical Foundations of Elasticity

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.

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

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.

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Mathematical Foundations of Elasticity

Summary

Author Biography

Table of Contents

Supplemental Materials