Phase field modeling of fracture in nonlinearly elastic solids via energy decomposition

Phase-field models for fracture problems have attracted considerable attention in recent years, which are capable of tracking the discontinuities numerically, and also produce complex crack patterns in many applications. In this paper, a phase-field model for a general nonlinearly elastic material is proposed using a novel additive decomposition of strain energy. This decomposition has two parts: one is principal stretch related and the other solely composed of volumetric deformation, which accounts for different behaviors of fracture in tension and compression. We construct the Lagrangian by integrating the split energies and the separation energy from phase-field approximation for discrete cracks. A coupled system of equations is also derived that governs the deformation of the body and the evolution of phase field. The capability and performance of the proposed model are demonstrated in several representative examples. Our results show that the predicted fracture surfaces are in good agreement with experimental observations. Compared with the previous models in which the energy is simply split into the isochoric and volumetric parts, the present model is numerically more robust and effective in simulating sharp cracks. The present model can also aid researchers to control the degree of tension–compression asymmetry in the nonlinear regime of deformation, which can be naturally extended to simulate the fracture of the rubber-like materials with tension–compression asymmetry.