Preface
Brief glossary of conventions and notations
A point of departure
1. Kinematics
2. Balance laws
3. Elastic materials
4. Boundary value problems
5. Constitutive inequalities
6. The role of geometry and functional analysis
1. Geometry and kinematics of bodies
1.1 Motions of simple bodies
1.2 Vector fields, one-forms, and pull-backs
1.3 The deformation gradient
1.4 Tensors, two-point tensors, and the covariant derivative
1.5 Conservation of mass
1.6 Flows and lie derivatives
1.7 Differential forms and the Piola transformation
2. Balance principles
2.1 The master balance law
2.2 The stress tensor and balance of momentum
2.3 Balance of energy
2.4 Classical spacetimes, covariant balance of energy, and the principle of virtual work
2.5 Thermodynamics II; the second law
3. Constitutive theory
3.1 The constitutive hypothesis
3.2 Consequences of thermodynamics, locality, and material frame indifference
3.3 Covariant constitutive theory
3.4 The elasticity tensor and thermoelastic solids
3.5 Material symmetries and isotropic elasticity
4. Linearization
4.1 The implicit function theorem
4.2 Linearization of nonlinear elasticity
4.3 Linear elasticity
4.4 Linearization stability
5. Hamiltonian and variational principles
5.1 The formal variational structure of elasticity
5.2 Linear Hamiltonian systems and classical elasticity
5.3 Abstract Hamiltonian and Lagrangian systems
5.4 Lagrangian field theory and nonlinear elasticity
5.5 Conservation laws
5.6 Reciprocity
5.7 Relativistic elasticity
6. Methods of functional analysis in elasticity
6.1 Elliptic operators and linear elastostatics
6.2 Abstract semigroup theory
6.3 Linear elastodynamics
6.4 Nonlinear elastostatics
6.5 Nonlinear elastodynamics
6.6 The energy criterion
6.7 A control problem for a beam equation
7. Selected topics in bifurcation theory
7.1 Basic ideas of static bifurcation theory
7.2 A survey of some applications to elastostatics
7.3 The traction problem near a natural state (Signorini's problem)
7.4 Basic ideas of dynamic bifurcation theory
7.5 A survey of some applications to elastodynamics
7.6 Bifurcations in the forced oscillations of a beam
Bibliography, Index

This advanced-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. Prerequisites include a solid background in advanced calculus and the basics of geometry and functional analysis.
The first two chapters cover the background geometry ― developed as needed ― and use this discussion to obtain the basic results on kinematics and dynamics of continuous media. Subsequent chapters deal with elastic materials, linearization, variational principles, the use of functional analysis in elasticity, and bifurcation theory. Carefully selected problems are interspersed throughout, while a large bibliography rounds out the text.

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