Nonlinear Elasticity Background

Abstract

In this chapter we summarize the main ingredients of continuum mechanics and their application to nonlinear elasticity with a view to the subsequent development of the nonlinear theories of electroelastic and magnetoelastic interactions. The setting is purely mechanical without reference to any electromagnetic effects. In particular, we review the necessary kinematics of deformation and motion; some required integral theorems involving tensors; the balance equations of mass, linear and angular momentum; and the associated stress tensors. Balance of energy is then used to introduce the strain-energy function of a hyperelastic material, following which we discuss the notions of objectivity and material symmetry as applied to the constitutive equations, and we provide some simple examples of constitutive equations for isotropic and transversely isotropic materials. We then give a general formulation of boundary-value problems in nonlinear elasticity, which is applied to some representative problems involving non-homogeneous deformations that will also feature in later chapters dealing with electroelastic and magnetoelastic boundary-value problems.