A computational model for crack-tip fields in a 3-D porous elastic solid with material moduli dependent on density

A mathematical model for crack-tip fields is proposed in this paper for the response of a three-dimensional (3-D) porous elastic solid whose material moduli are dependent on the density. Such a description wherein the generalized Lam\`e coefficients are nonlinear functions of material stiffness is more realistic because most engineering materials are porous, and their material properties depend on porosity and density. The governing boundary value problem for the static equilibrium state in a 3-D, homogeneous, isotropic material is obtained as a second-order, quasilinear partial-differential-equation system with a classical traction-free crack-surface boundary condition. The numerical solution is obtained from a continuous trilinear Galerkin-type finite element discretization. A Picard-type linearization is utilized to handle the nonlinearities in the discrete problem. The proposed model can describe the state of stress and strain in various materials, including recovering the classical singularities in the linearized model. The role of \textit{tensile stress}, \textit{stress intensity factor} (SIF), and \textit{strain energy density} are examined. The results indicate that the maximum values of all these quantities occur directly before the crack-tip, consistent with the observation made in the canonical problem for the linearized elastic fracture mechanics. One can use the same classical local fracture criterion, like the maximum of SIF, to study crack tips' quasi-static and dynamic evolution within the framework described in this article.