Phase-Field Material Point Method for dynamic brittle fracture with isotropic and anisotropic surface energy

Abstract

A novel phase field material point method is introduced for robust simulation of dynamic fracture in elastic media considering the most general case of anisotropic surface energy. Anisotropy is explicitly introduced through a properly defined crack density functional. The particular case of impact driven fracture is treated by employing a discrete field approach within the material point method setting. In this, the equations of motion and phase field governing equations are solved independently for each discrete field using a predictor-corrector algorithm. Contact at the interface is resolved through frictional contact conditions. The proposed method is verified using analytical predictions. The influence of surface energy anisotropy and loading conditions on the resulting crack paths is assessed through a set of benchmark problems. Comparisons are made with the standard Phase Field Finite Element Method and experimental observations. 65 3 (PF-MPM) has been successfully introduced by the authors in [44] for quasistatic brittle fracture problems while a variant accounting for anisotropy in the quasi-static regime has been developed in [45].Moving beyond the state-of-the-art, we present a phase field MPM method for the solution of dynamic fracture considering materials with anisotropic frac-70 ture energy; isotropy emerges as a special case of the proposed formulation.Following, the method is extended to also account for frictional contact fracture problems. We use as point of departure the MPM contact algorithm introduced in Bardenhagen et al. [46] where multiple fields, termed discrete fields, are introduced in the non-deforming Eulerian mesh so that each contact body 75 corresponds to a different field. We define the variational structure of our phase field implementation of impact driven fracture at each discrete field from which the coupled weak form of the contact problem naturally emerges. Finally, we develop a predictor corrector solution algorithm for the solution of the governing equations over time. 80This paper is organized as follows. In section 2 phase-field modelling is briefly described in both isotropic and anisotropic brittle fracture. The discrete field formulation for phase field fracture due to impact is presented in section 3. The Material Point Method implementation for frictional contact fracture is presented in section 4. Finally, in section 5, a set of benchmark problems are 85 examined to demonstrate the accuracy and robustness of the proposed method. Preliminaries Phase field modellingIn the following, the case of an arbitrary deformable domain Ω is considered, with an external boundary ∂Ω and a crack path Γ as shown in Fig. (1a). 90The deformable domain Ω with domain volume V , is subjected to body forces b = b 1 b 2 b 3 T . Furthermore, a set of traction/pressure loadst is applied on the boundary ∂Ωt ⊆ ∂Ω. A prescribed displacement field, denoted asū, is imposed on the boundary ∂Ωū ⊆ ∂Ω.

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