An explicit phase field material point method for modeling dynamic fracture problems

A novel explicit phase field material point method (ex-PFMPM) is proposed for modeling dynamic fracture problems. The rate-dependent phase field governing equation is discretized by a set of particles, and the phase field is updated by the explicit forward-difference time integration. Furthermore, the stability of the ex-PFMPM is studied. A novel explicit critical time step formula is obtained based on the system eigenvalues in one dimension and then extended to two and three dimensions. The critical time step formula takes the effect of particle position and neighboring cell interaction into consideration, and can also be used in an explicit phase field finite element method. Several numerical examples, including a dynamic crack branching, a plate with pre-existing crack under velocity boundary conditions and a three point bending problem are studied to verify the proposed ex-PFMPM. The use of the history field in the explicit method is studied, which shows that it will lead to fake phase field update and overestimation of the fracture energy in the unloading case. All of the numerical results show that the proposed ex-PFMPM has the capacity of modeling the crack initiation and propagation problems with both accuracy and efficiency.