On the Choice of Integrity Base of Strain Invariants for Constitutive Equations of Isotropic Materials

Abstract

Using the well known representation theorem for isotropic tensor functions, the stress-strain law for a thermoelastic material can be written as:

$$\sigma ={{\phi }_{0}}1+{{\phi }_{1}}B+{{\phi }_{2}}{{B}^{2}},$$

(1)

where σ and B are appropriate tensors of stress and strain respectively. Although not yet necessary at the moment, we will define for later use σ as the Cauchy stress and B as the left Cauchy-Green tensor. The coefficients ø ₀, ø ₁, ø ₂ depend on three orthogonal invariants I _B, II _B, III _B of B forming an integrity base and on the temperature T:

$${{\phi }_{i}}+{{\phi }_{i}}\left( {{I}_{B}},I{{I}_{B}},II{{I}_{B}},T \right)i=0,1,2$$

(2)

In many cases the principal invariants of B are taken to be I _B, II _B, III _B and for the present we will do the same. However, any other integrity base of strain invariants is admissible, as long as no further restrictions are imposed on (1).