Conservational integrals of the fourth-order phase field model for brittle fracture via Noether theorem

As a regularization method for modeling fracture phenomenon, the phase field method cannot accurately capture the location of crack tip. J-integral is an important physical parameter to quantify the stress state of crack tip. In this study, the J-, M- and L-integrals of the fourth-order phase field model for brittle fracture are proposed based on the Noether theorem, and the corresponding infinitesimal generator of Lie group are given. The J- and L-integral models in the phase field fracture system are proved to be path-independent. Besides, the numerical implementation of the J-integral is conducted by using the domain integral method. In order to calculate the third derivative term of phase field in the J-integral model, a 9×9 Jacobian transformation matrix is constructed. Moreover, the J-integrals with and without effect of damage phase field are compared with analytic solution by numerical examples. The path-independence and Γ-convergence of the J-integral of phase field model are numerically verified. The proposed integral models can overcome the drawback of the fracture phase field method and provide the characteristic parameters of fracture mechanics to evaluate fracture behavior of brittle material.