Material point method with enriched shape function for crack problems

A material point method (MPM)/generalized interpolation material point method (GIMP) with enriched shape function (EMPM/EGIMP for short) is proposed for modelling crack problems in the MPM/GIMP framework. The EMPM/EGIMP enriches the nodal degrees of freedom based on the idea of the extended finite element method (XFEM). This improvement allows the crack problem, whose displacement and velocity are discontinuous, to be simulated by only one set of background grid meshes, so multigrid and multiple velocity fields are unnecessary. In addition, the technique we developed to lump the mass matrix makes the EMPM/EGIMP can be implemented easily in a conventional MPM/GIMP code. If there is no crack, the EMPM/EGIMP degenerates to the conventional MPM/GIMP. The Level Set Method (LSM) is employed in the EMPM/EGIMP for easily tracking crack surfaces. The crack moves with the material points physically, so that the LSM function is carried by particles. In each time step, the LSM function value is mapped to the grid nodes from the particles, and the nodes to be enriched can be identified conveniently from the grid nodal LSM function values. Numerical experiments for stress fields distribution, fracture parameters calculation and crack propagation are provided to validate the proposed EMPM/EGIMP.