A mixed three‐field total Lagrangian material point method for phase‐field fracture modeling of nearly incompressible rubber‐like solids

The rubber‐like solids, as commonly observed materials in nature and engineering areas, often show high extensibility and weak compressibility, which lead to difficulties for modeling the fracture behavior of these materials. In this paper, we propose a mixed three‐field total Lagrangian material point method (TLMPM) to deal with the fracture of nearly incompressible rubber‐like solids. In this method, the phase‐field method combining with perturbed Lagrangian approach is developed to describe the fracture of nearly incompressible materials. Based on the Lagrangian equation, the governing equations of the displacement, pressure and phase fields are derived for the system considering dynamic effect. Then, a mixed three‐field TLMPM discrete formulation with a remapping strategy is then built to solve the problem involving large deformation. Besides, a staggered solving scheme with explicit time integration is developed for the three‐field problem. The accuracy and capabilities of the proposed method for dealing with the fracture of incompressible materials are demonstrated by modeling several quasi‐static and dynamic numerical problems and comparing the results with other numerical methods and experiments. Moreover, the applicability of the method is further proved by an out‐of‐plane tearing case.