Preferential stiffness and the crack-tip fields of an elastic porous solid based on the density-dependent moduli model

In this paper, we study the preferential stiffness and the crack-tip fields for an elastic porous solid of which material properties are dependent upon the density. Such a description is necessary to describe the failure that can be caused by dam aged pores in many porous bodies such as ceramics, concrete and human bones. To that end, we revisit a new class of implicit constitutive relations under the assumption of small deformation. Although the constitutive relationship appears linear in both the Cauchy stress and linearized strain, the governing equation bestowed from the balance of linear momentum results in a quasi-linear partial differential equation (PDE) system. For the linearization and obtaining a sequence of elliptic PDEs, we propose the solution algorithm comprise a Newton’s method coupled with a bilinear continuous Galerkin-type finite elements for the discretization. Our algorithm exhibits an optimal rate of convergence for a manufactured solution. In the numerical experiments, we set the boundary value problems (BVPs) with edge crack under different modes of loading (i.e., the pure mode-I, II, and the mixed mode). From the numerical results, we find that the density-dependent moduli model describes diverse phenomena that are not captured within the framework of classical linearized elasticity. In particular, numerical solutions clearly indicate that the nonlinear modeling parameter depending on its sign and magnitude can control preferential mechanical stiffness along with the change of volumetric strain; larger the parameter is in the positive value, the responses are such that the strength of porous solid gets weaker against the tensile loading while stiffer against the in plane shear (or compressive) loading, which is vice versa for the negative value of it. The modeling framework of the density-dependent material moduli proposed in this study can provide a mathematical and computational foundation to further model the quasi-static and dynamic evolution of cracks and many other multi-physics applications such as the fluid flow or heat transfer in porous media