A rigorous and efficient explicit algorithm for irreversibility enforcement in phase-field finite element modeling of brittle crack propagation

In the present work, a computationally efficient and explicit algorithm for the rigorous enforcement of the irreversibility constraint in the phase-field modeling of brittle fracture is presented. The proposed approach is staggered and relies on the alternate minimization of the total energy functional. The phase-field evolution turns out to be governed by a complementarity boundary-value problem, where the complementarity stems from the irreversibility, while the boundary-value problem is originated by the presence of the gradient term in the phase-field functional. Several different techniques have been proposed in the literature to account for damage irreversibility in a computationally effective way. Following a similar approach proposed in the past for a gradient-plasticity problem, a particularly simple and effective solution strategy based on the Projected Successive Over-Relaxation (PSOR) method for constrained optimization, where an iterative explicit scheme is used for the solution of symmetric linear complementarity problems, is presented. Even though the proposed method is restricted to linear complementarity problems, it can be applied to the numerical simulation of a wide range of problems discussed in the literature. The performance of the suggested solution algorithm on a number of test cases is compared with that of a recently proposed penalty approach for the approximated enforcement of irreversibility.