An adaptive local algorithm for solving the phase-field evolution equation in the phase-field model for fracture

Abstract

In the phase-field model for fracture, material damage can be characterized by a variable called the crack phase-field. Usually-two sub-problems controlled by the equilibrium equation and the evolution equation, respectively, are needed to be solved in the whole domain. In this paper, an adaptive local algorithm is proposed for the unified phase-field model to solve the evolution equation in the local domain based on the fact that the value of phase-field is non-zero only around the crack. In the initial unified phase-field model, the boundedness and irreversibility conditions are not easy to be satisfied. In this paper, a history field is proposed to convert the inequality into equality in the evolution equation, and the boundedness and irreversibility conditions can be treated easily as in the classical phase-field model. To further obtain the local algorithm, some failure criteria are introduced to identify the damaged area, and the elements in the local domain are updated adaptively in each iteration step. The local algorithm can be implemented easily since the boundary conditions of the local domain are the same as in the non-local algorithm. Numerical examples have shown that the proposed local algorithm can obtain the same results as the non-local algorithm.