A constitutive model for an internally balanced compressible elastic material

Many large deformation constitutive models for the mechanical behavior of solid materials make use of the multiplicative decomposition [Formula: see text] as, for example, used by Kröner in the context of finite-strain plasticity. Then [Formula: see text] describes the elastic effect by letting the potential energy of the deformation depend upon [Formula: see text]. In this paper we allow the potential energy to depend upon both portions of the multiplicative decomposition. As in hyperelasticity, energy minimization with respect to displacement gives equilibrium field equations and traction boundary conditions. The new feature, minimization with respect to the decomposition itself, generates an additional mathematical requirement that is interpreted here in terms of a principle of internal mechanical balance. We specifically consider a Blatz–Ko-type solid suitably generalized to incorporate the notion of internal balance. Conventional results of hyperelasticity are retrieved for certain limiting forms of the energy density, whereas the general form of the energy density gives rise to an overall softening response, as is demonstrated in the context of pure pressure and uniaxial loading.