On an Implicit Model Linear in Both Stress and Strain to Describe the Response of Porous Solids

We study some mathematical properties of a novel implicit constitutive relation wherein the stress and the linearized strain appear linearly that has been recently put into place to describe elastic response of porous metals as well as materials such as rocks and concrete. In the corresponding mixed variational formulation the displacement, the deviatoric and spherical stress are three independent fields. To treat well-posedness of the quasi-linear elliptic problem, we rely on the one-parameter dependence, regularization of the linear-fractional singularity by thresholding, and applying the Browder-Minty existence theorem for the regularized problem. An analytical solution to the nonlinear problem under constant compression/extension is presented.