On the direct numerical simulation of plane-strain fracture in a class of strain-limiting anisotropic elastic bodies

In this paper, using a special class of nonlinear response relations between linearized strain and Cauchy stress tensors, we study a quasi-static mixed-mode (combination of mode-I and mode-II) fracture boundary value problem. A special subclass of such relations are strain-limiting in which the strains are guaranteed to be uniformly bounded in the neighborhood of crack-tip. In this article, we consider a finite element method based direct numerical simulation (DNS) of plane-strain mixed-mode fracture in a class of strain-limiting, transversely isotropic elastic bodies. We study the numerical solution of a nonlinear fracture boundary value problem with displacement as the primary unknown. The linearized version of the strong form was derived using damped Newton’s method and the corresponding weak formulation was solved using a conforming finite element method. The results of this DNS indicate that even very near the crack tip, both stress and strain remain much smaller in magnitude than the corresponding predictions from the linearized elastic fracture mechanics (LEFM). While away from strain concentrating crack-tip region, the solution to the nonlinear, strain-limiting model agrees closely with the LEFM solution. This supports recent asymptotic theoretical studies of anti-plane and plane-strain fracture boundary value problems. Further, we also study the behavior of cleavage stress in the neighborhood of crack-tip. The numerical results indicate that the cleavage stress is largest along a line directly ahead of the crack-tip in agreement with the classical LEFM solution for pure mode I loading.