Explicit finite element material point method coupled with phase-field model for solid brittle fracture

Material fracture is one of the main causes of material instability and damage. To investigate the brittle fracture propagation behavior of solid materials, an explicit finite element material point coupled with phase-field model (PF-FEMPM) is proposed. This method discretizes the solid into finite elements and embeds them into Euler grids, effectively avoiding issues such as numerical errors between particles when two particles are separated by an empty grid, as well as difficulties in applying boundary conditions, which are common in the phase-field material point method (PF-MPM). Integration points within the elements serving as material points are calculated by finite element method to store internal variables, while the finite element mesh nodes are used to store external forces and apply boundary conditions. The Euler grid is employed to integrate the momentum conservation equation and the phase-field dynamic evolution equation, and then map back to the finite element mesh nodes to update the nodes velocity and displacement. Mapping functions among integration points, finite element nodes, and background grid nodes are derived, and an algorithm is established using a forward difference solution scheme. Through a series of numerical examples, including single-edge notched tension problem, dynamic crack branching, a plate with center prefabricated crack under dynamic velocity boundary conditions, and the fragmentation of equal-thickness cylinder, the rationality and validity of the algorithm are verified.