Two-dimensional mixed mode crack simulation using the material point method

The material point method (MPM) has demonstrated its capabilities in the simulation of impact/contact/penetration and interfacial crack growth problems. Because of the use of material points in the description of a continuum, consistent with the particle description ( atoms) using molecular dynamics ( MD), it is natural to couple MPM with MD for simulation from atomistic to continuum levels. However, in addressing plane stress/plane strain problems, the MPM algorithm and simulation examples available in literature use a regular grid mesh with uniform square cells and enforce velocity and displacement continuities through its background grid, resulting in limitations in dealing with stress concentration, inclined dislocations and inclined crack, etc. In this article, an irregular mesh is implemented in MPM to eliminate the limitations resulting from the use of a regular mesh. The ray-crossing algorithm is employed to determine which cell a material point belongs to after deformation for interpolation and extrapolation of variables between material points and grid nodes. As an example to demonstrate the capability of the MPM using irregular mesh, the stress field in a continuum with an inclined crack is determined using arbitrary quadrilateral cells in the background grid mesh. The use of irregular mesh in MPM was validated by comparing MPM results with ABAQUS/Explicit simulation. The proposed method of using irregular mesh will be an essential element in using MPM to couple with atomistic scale simulation so that MPM can address inclined dislocations and cracks emanating from the atomistic simulation.