A consistent isotropic spectral constitutive equation: The infinitesimal strain depends nonlinearly on the stress

Recently, there have been several developments in nonlinear infinitesimal-strain stress constitutive relations to describe the response of elastic bodies. However, some of the nonlinear constitutive equations presented in the literature are unable to model both compressible and incompressible bodies. In this paper, we develop a spectral constitutive equation, where the infinitesimal strain depends nonlinearly on the stresses, and the incompressible behaviour is simply described as a special case obtained from the compressible elastic model, by just letting the value of the Poisson's ratio equal to a half. The constitutive equation satisfies the Baker-Ericksen inequality and a specific expression for the constitutive equation is proposed to fit experimental data for the compression of a sample of rock. Several boundary value problems with homogeneous deformations and stresses are analysed. Spectral components for the fourth order tensor that is required for the analysis of the propagation of P- and S-waves are given.