A Material Point Method for Reissner-Mindlin Shells

This study proposes a novel material point method (MPM) for thin Reissner–Mindlin (RM) shells. The weak form of the RM shell is derived on 2D curved surfaces using Hamilton’s principle, and then the formulations in the 2D surface space are transformed to 3D Cartesian space in which the MPM background grid is defined. The proposed MPM shell method describes shell dynamics using a unified equation that is compatible with either the updated or total Lagrangian framework. The discretization of the weak form is performed with a general interpolation material point (GIMP) approach. Newmark-β time integration is adopted, which allows a much larger time step than the widely used update-stress-last (USL) scheme. For numerical stability, stress points are employed, and the gradients of shell field values are calculated on the curvilinear surface using the kernel function method. At stress points, stress tensors are updated incrementally, considering large deformations and rotations. Furthermore, an anti-hourglass term is proposed to further improve the numerical stability of computing pseudo-normals of the shell. Self- or multi-body contact of the shell can be treated easily, inheriting MPM’s natural handling of collisions and topological changes. A set of benchmark cases demonstrates the robustness and accuracy of the proposed method. Notably, the MPM shell method can produce highly accurate predictions with an MPM grid cell size much larger than the shell thickness, which eliminates the need for a very fine grid to resolve stress in the thickness direction, as in traditional MPM. To the authors’ knowledge, this is the first MPM work that uses thin shell theory and avoids relying on a meshed discretization of the shell.