A small deformations effective stress model of gradient plasticity phase-field fracture

A variational formulation of small strain ductile fracture, based on a phase-field modeling of crack propagation, is proposed. The formulation is based on an effective stress description of gradient plasticity, combined with an AT1 phase-field model. Starting from established variational statements of finite-step elastoplasticity for generalized standard materials, a mixed variational statement is consistently derived, incorporating in a rigorous way a variational finite-step update for both the elastoplastic and the phase-field dissipations. The complex interaction between ductile and brittle dissipation mechanisms is modeled by assuming a plasticity driven crack propagation model. A non-variational function of the equivalent plastic strain is then introduced to modulate the phase-field dissipation based on the developed plastic strains. Particular care has been devoted to the formulation of a consistent Newton–Raphson scheme for the case of Mises plasticity, with a global return mapping and relative tangent matrix, supplemented by a line-search scheme, for the solution of the gradient elastoplasticity problem for fixed phase field. The resulting algorithm has proved to be very robust and computationally effective. Application to several benchmark tests show the robustness and accuracy of the proposed model.