A Blended Variationally Consistent Phase Field Material Point Method for Material Fragmentation Problems

Abstract

The material point method (MPM) can effectively addresses material fragmentation issues over mesh-based method due to its meshfree nature but faces numerical inaccuracies such as cell-crossing instability and Galerkin inexactness inherent in its numerical framework. The utilization of previously developed variational consistent (VC) corrections with smooth reproducing kernel (RK) approximations has proven effective in preventing cell-crossing and ensuring Galerkin exactness, albeit limited to continuum bodies. In this study, we present a blended variationally consistent phase field material point method for modeling material fragmentation process. The phase field damage method is integrated into MPM to facilitate the tracking of damage/cracks during the fragmentation process. The incorporation of VC correction alongside smooth RK approximation offers benefits not only for non-oscillatory stress modes but also for robust phase field evolution. To address the complexities of VC in handling domain boundary tracking during fragmentation, we introduce a deformation-driven blending scheme to seamlessly blend non-VC for fragmented regions and VC for continuous domains with a smooth transition zone in the Galerkin MPM. This approach enhances the stability and robustness of the formulation for simulating fragmentation problems. Benchmark examples involving extreme deformation scenarios illustrate the effectiveness of the proposed MPM framework, confirming its consistency and stability.