Three-dimensional numerical modeling of a stationary v-notch in a porous material constituted by a non-linear response relationship

A novel three-dimensional (3D) mathematical model is presented in this paper to characterize crack-tip fields within porous elastic solids, specifically addressing materials where elastic moduli are intrinsically dependent on density. A more realistic portrayal of engineering materials is offered by this approach, as their mechanical properties are significantly influenced by their inherent porosity and density. The governing boundary value problem, describing the static equilibrium of a 3D, homogeneous, and isotropic material, is formulated as a system of second-order, quasilinear partial differential equations. This formulation is coupled with the classical traction-free boundary condition at the crack surface. To numerically solve this intricate problem, a continuous trilinear Galerkin-type finite element discretization is employed. The inherent nonlinearities within the discrete system are effectively managed through a robust Picard-type linearization scheme. By the proposed model, a remarkable ability to accurately describe the stress and strain states in a diverse range of materials is demonstrated, and the classical singularities observed in linearized models are crucially recovered. Through a comprehensive examination of tensile stress, strain, and stress-power, it is revealed that peak values for these quantities are consistently attained in the immediate vicinity of the crack tip. This observation is found to be in remarkable alignment with established findings in standard linearized elastic fracture mechanics. Furthermore, the framework presented in this article allows the quasi-static and dynamic evolution of crack tips to be rigorously investigated through the application of conventional local fracture criteria, providing a powerful tool for the prediction of material failure.