A generally variational phase field model of fracture

Although the phase field method has been widely used in the field of fracture, various models exist, such as the second-order and fourth-order phase field model in the field of fracture mechanics, and the phase field evolution model in the field of physics, and it is not yet possible to establish these models under a unified framework. In this paper, a thermodynamically generalized framework of variational phase field model of fracture is presented, which can effectively unify different models under this framework. Based on this variational framework, not only the current second-order and fourth-order phase field model can be unified, but also new second-order, fourth-order, sixth-order, and eighth-order phase field model can be obtained from this framework. In addition, a method is proposed to convert the lower-order model into a higher-order model, which can convert a phase field distribution function of the lower-order model into a high-order phase field distribution function, and then obtain a higher-order phase field model. Finally, the new model is verified by some representative examples, and the second-order and fourth-order phase field models are compared, which shows the effectiveness of building a new phase field model from the variational framework.