Implicit nonlinear elastic bodies with density dependent material moduli and its linearization

We develop an implicit constitutive relation to describe the response of a compressible elastic solid, based on physical considerations, that captures all the characteristics exhibited by the popular Blatz–Ko model, but in addition presents some interesting novel features. The fact that the Cauchy stress appears linearly in the implicit constitutive relation between the stress and the left Cauchy–Green strain with the material moduli depending nonlinearly on the deformation gradient, allows us to capture several characteristic features of the response of rubber-like elastic solids. Interestingly, in the nonlinear implicit model that we develop, we find that it is possible to have the normal stress components of the stress influence the shearing motion at second order, when considering weakly nonlinear waves, that only occurs at third order within the case of the classical nonlinear Cauchy elasticity theory. Linearization of the constitutive relation under the assumption of small displacement gradient reduces the constitutive relation to one whose material moduli can depend on the trace of the linearized strain and hence the density in virtue of the balance of mass, such a feature is not possible within the context of the Blatz–Ko constitutive relation, or for that matter any Cauchy elastic body, as linearization leads to the classical linearized elastic constitutive relation that has constant material moduli.