Introduction of (1) material frame indifference, (2) indifference with respect to rigid-body rotations of the reference configuration, and (3) material symmetry classification guides the development of non-linear (both Cauchy and hyper-) elastic constitutive laws for isotropic and anisotropic behaviour (for materials with a micro-structure). Incremental elasticity is introduced with reference to the relative Lagrangean description, and the Biot framework for incompressible elasticity is derived. Hypo-elasticity is also briefly included.
Broadly speaking, constitutive laws set a bridge between strain and stress, so they have to keep into account the behaviour of the specific material under consideration. This behaviour can be fully reversible and described by elasticity (e.g., in the case of rubber) or can be partially irreversible and described by elastoplasticity (e.g., in the case of mild steel), or it may be time-dependent and described by viscous laws (e.g., in the case of a Newtonian fluid). We exclude for the moment the elastoplastic behaviour, and we present constitutive laws in this chapter for anisotropic elastic materials and for incrementally deformed solids (elastoplasticity is deferred to Chapter 8).
The introduction of constitutive laws for materials subject to large strains requires preliminary statements of general principles, such as the so-called material frame indifference and indifference with respect to rigid-body rotations of the reference configuration. When these concepts are specified, the development of constitutive equations to model different materials is greatly simplified. After introduction of these general principles, we conclude the present chapter with the formulation of elastic constitutive laws, generalising concepts proposed in Chapter 4, now to includemicro-structural anisotropy.