Quasi-static anti-plane shear crack propagation in nonlinear strain-limiting elastic solids using phase-field approach

Abstract

We study a quasi-static evolution of anti-plane crack with the nonlinear strain-limiting model using the phase-field approach. The nonlinear strain-limiting models, a subclass of the implicit constitutive relations, allow the linearized strain value to remain small even if the stress value tends to infinity. To compute the quasi-static crack, we solve the constrained energy minimization for the nonlinear bulk energy coupled with diffusive crack energy. An iterative staggered scheme (L-scheme) is employed for coupling the nonlinear mechanics and phase-field, and augmented Lagrangian is utilized to accommodate crack irreversibility. Several numerical experiments of the proposed framework substantiate the bounded strain in the neighborhood of the crack-tip for both static and quasi-static cracks. The comparisons of bulk and the crack energies and the discrete crack-tip speed between the LEFM and our model are presented.