Beam elements with frictional contact in the material point method

Abstract

This article presents a novel approach to modeling beam elements that accommodate large displacements/rotations in the context of the material point method (MPM). The MPM is a hybrid Lagrangian/Eulerian approach for solving solid mechanics problems involving extremely large deformations and rotations. A solid body is discretized into a set of Lagrangian material points, called particles in the MPM. The equations of motion are solved on a fixed Eulerian background grid. The beam particle developed herein consists of two end nodes, each possessing three translational and three rotational degrees of freedom (DOF) in 3D. The end nodes are tracked and define the evolving beam particle domain. As in the conventional MPM, the linear momentum is conserved on the background grid, but here we also use this grid to enforce C1 continuity between beam particles by mapping angular velocities and accelerations for the beam rotational DOF to/from the background grid. The contact interactions between the beam particles are treated using multiple velocity fields on the background grid. Spatial nodes representing the spatial extent of the beam particle are introduced, which allows for collision detection and associated frictional contact. The effectiveness of the proposed approach is demonstrated through a series of numerical examples. Beam structures subjected to large displacements and rotations are compared with analytical/numerical solutions, in which good agreement is obtained.