Circumventing volumetric locking in explicit material point methods: A simple, efficient, and general approach

The material point method (MPM) is frequently used to simulate large deformations of nearly incompressible materials such as water, rubber, and undrained porous media. However, MPM solutions to nearly incompressible materials are susceptible to volumetric locking, that is, overly stiff behavior with erroneous strain and stress fields. While several approaches have been devised to mitigate volumetric locking in MPM, none of them lends itself to a straightforward application to standard explicit MPM formulations. In this work, we propose a locking-mitigation approach that features an unprecedented combination of simplicity, efficacy, and generality for a family of explicit MPM formulations. The approach combines the assumed deformation gradient ($\bar{\boldsymbol{F}}$) method with a volume-averaging operation built on standard particle-grid transfer schemes in MPM. Upon explicit time integration, this combination yields a new and simple algorithm for updating the deformation gradient, preserving all other MPM procedures. The proposed approach is thus easy to implement, low-cost, and compatible with the existing machinery in MPM. Through various types of nearly incompressible problems in solid and fluid mechanics, we verify that the proposed approach efficiently circumvents volumetric locking in explicit MPM, regardless of the basis functions and material types.