On unimodular constitutive expressions

We consider constitutive expressions which the stress σ(X, t) at a particle X at time t is given by σ (X, t) = $\text{F}$FX, τ)] where $\text{F}$F(X, τ)] denotes a functional of the history of the deformation gradient matrix F(X, τ)] from time τ = 0 unti τ = t. This expression is restricted by the requirement of invariance under a superposed rotation of the physical system and by the further requirement that the constitutive expression shall be invariant under the group of unimodular transformations, i.e. $\text{F}$F(X, τ)] = $\text{F}$F(X, τ) H] must hold for all matrices H such that det H - 1. We employ results from the classical theory of invariants in order to determine the general form of the expression $\text{F}$F(X, τ)] which is consistent with these restrictions. Special cases are considered where the functional is replaced by a function of the strain, rate of strain, ? matrices. The case of shear flow is briefly discussed.