Evaluation of mechanical and energy properties for the phase field modeling of failure

In recent years, various phase field models have been developed in variational methods to simulate the failure of brittle solids. However, there is a lack of objective evaluation of the existing results, and in particular, there are few studies on model nonhomogeneous resolution, stress-strain linear elastic properties, and failure stress estimation. To compensate for the above gaps, the commonly used variational phase field model is systematically analyzed to solve the problem of evaluating the mechanics and energy properties of the model in this paper. The unified expression of the analytical solution and the nonhomogeneous solution under specific boundary conditions is analyzed and verified. Additionally, we theoretically analyze the energy properties of the phase field model and study the influence of the critical strain energy on the damage field, stress and strain of different models. Finally, the effect of different effective material parameters on the material failure stress is analyzed, the stress estimation formula for the failure of mode I of various phase field models is given, and the accuracy of the theory is verified by some examples.