  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 damaged 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 \textit{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 \textit{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 \textit{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.
