This paper introduces a three-dimensional (3-D) mathematical and computational framework for the characterization of crack-tip fields in star-shaped cracks within porous elastic solids. A core emphasis of this model is its direct integration of density-dependent elastic moduli, offering a more physically realistic representation of engineering materials where intrinsic porosity and density profoundly influence mechanical behavior. The governing boundary value problem, formulated for the static equilibrium of a 3-D, homogeneous, and isotropic material, manifests as a system of second-order, quasilinear partial differential equations. This system is meticulously coupled with classical traction-free boundary conditions imposed at the complex crack surface. For the robust numerical solution of this intricate nonlinear problem, we employ a continuous trilinear Galerkin-type finite element discretization. The inherent strong nonlinearities arising within the discrete system are effectively managed through a powerful and stable Picard-type linearization scheme. The proposed model demonstrates a remarkable ability to accurately describe the full stress and strain states in a diverse range of materials, crucially recovering the well-established classical singularities observed in linearized elastic fracture mechanics. A comprehensive numerical examination of tensile stress, strain, and strain energy density fields consistently reveals that these quantities attain their peak values in the immediate vicinity of the crack tip, an observation that remarkably aligns with established findings in standard linearized elastic fracture mechanics.
Mallikarjunaiah et al. (Sat,) studied this question.