Computational investigation of crack-tip fields in a compressed nonlinear strain-limiting material

A finite element framework is presented for the analysis of crack-tip phenomena in an elastic material containing a single edge crack under compressive loading. The mechanical response of the material is modeled by a nonlinear constitutive relationship that algebraically relates stress to linearized strain. This approach serves to mitigate non-physical strain singularities and ensures that the crack-tip strains don't grow, unlike singular stresses. A significant advancement is thus achieved in the formulation of boundary value problems (BVPs) for such complex scenarios. The governing equilibrium equation, derived from the balance of linear momentum and the nonlinear constitutive model, is formulated as a second-order, vector-valued, quasilinear elliptic BVP. A classical traction-free boundary condition is imposed on the crack face. The problem is solved using a robust numerical scheme in which a Picard-type linearization is combined with a continuous Galerkin finite element method for the discretization. Analyses are performed for both an isotropic and a transversely isotropic elastic solid containing a crack subjected to compressive loads. The primary crack-tip variables**-stress, strain, and strain energy density-**are examined in detail. It is demonstrated that while high concentrations of compressive stress and strain energy density are observed at the crack tip, the growth of strain is substantially lower than that of stress. These findings are shown to be consistent with the predictions of linear elastic fracture mechanics, but a more physically meaningful representation of the crack-tip field is provided by the nonlinear approach. A rigorous basis is thus established for investigating fundamental processes like crack propagation and damage in anisotropic, strain-limiting solids under various loading conditions, including compression.