Computational insights into orthotropic fracture: crack-tip fields in strain-limiting materials under non-uniform loads

Abstract A finite element framework is presented for analyzing crack-tip phenomena in transversely isotropic, strain-limiting elastic materials. Mechanical response is characterized by an algebraically nonlinear constitutive model, relating stress to linearized strain. Non-physical strain singularities at the crack apex are mitigated, ensuring bounded strain magnitudes. This methodology significantly advances boundary value problem (BVP) formulation, especially for first-order approximate theories. For a transversely isotropic elastic solid with a crack, the governing equilibrium equation, derived from linear momentum balance and the nonlinear constitutive model, is reduced to a second-order, vector-valued, quasi-linear elliptic BVP. This BVP is solved using a robust numerical scheme combining Picard-type linearization with a continuous Galerkin finite element method for spatial discretization. Numerical results are presented for various loading conditions, including uniform tension, non-uniform slope, and parabolic loading, with two distinct material fiber orientations. It is demonstrated that crack-tip strain growth is substantially lower than stress growth. Nevertheless, strain energy density is found to be concentrated at the crack tip, consistent with linear elastic fracture mechanics principles. The proposed framework provides a robust basis for formulating physically meaningful, rigorous BVPs, critical for investigating fundamental processes like crack propagation, damage, and nucleation in anisotropic, strain-limiting elastic solids under diverse loading conditions.