Fourth-order phase field model with spectral decomposition for simulating fracture in hyperelastic material

We present a fourth-order phase field model for fracture behavior simulations of hyperelastic material undergoing finite deformation. Governing equations of the fourth-order phase field model consist of the biharmonic operator of the phase field, which requires the second-order derivatives of shape function. Therefore, a 5 x 5 Jacobian matrix of isoparametric transformation is constructed. Neo-Hooken model and Hencky model are adopted as the material constitutive models. The spectral decomposition of stored strain energy is used to distinguish the contributions of tension and compression, and the corresponding stress tensor and constitutive tensors are derived, and subsequently, the numerical framework of modeling fracture with the fourth-order phase field model is implemented in details. Several typical numerical examples are conducted to demonstrate the robustness and effectiveness of the fourth-order phase field model in simulating the fracture phenomenon of rubber-like materials.