As a regularization method for modeling fracture phenomenon, the phase field method can not 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$\times$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.

作为建模断裂现象的正则化方法，相场法无法准确地捕获裂纹尖端的位置。J积分是量化裂纹尖端应力状态的重要物理参数。本研究基于Noether定理，提出了脆性断裂四阶相场模型的J-，M-和L-积分，并给出了相应的Lie群的无穷小生成器。证明了相场断裂系统中的J和L积分模型与路径无关。此外，J的数值实现-积分是通过使用域积分法进行的。为了计算J积分模型中的相场的三阶导数项，$\times$9构造了雅可比变换矩阵。此外，通过数值算例，比较了具有和不具有损伤相场影响的J积分与解析解。数值验证了相场模型J积分的路径无关性和Γ收敛性。所提出的积分模型可以克服断裂相场法的缺点，并提供断裂力学的特征参数来评估脆性材料的断裂行为。