A thermodynamically consistent phase field model for crack propagation is analyzed. The thermodynamic driving force for the crack propagation is derived based on the laws of thermodynamics. The Helmholtz free energy satisfies the thermodynamic equilibrium and instability conditions for the crack propagation. Analytical solutions for the Ginzburg–Landau equation including the surface profile and the estimation of the kinetic coefficient are found. It is shown how kinetic coefficient affects the local stress field. The local critical stress for the crack propagation is calibrated with the theoretical strength which gives the value of the crack surface width. The finite element method is utilized to solve the complete system of crack phase field and mechanics equations. The analytical expressions for the crack surface profile and the crack tip velocity are verified with the numerical solutions. It is shown that under mode I cracking and proper calibration of parameters, phase field models always agree with Griffith’s theory.

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裂纹扩展的相场建模及参数分析处理

分析了裂纹扩展的热力学一致相场模型。根据热力学定律推导出裂纹扩展的热力学驱动力。亥姆霍兹自由能满足裂纹扩展的热力学平衡和不稳定性条件。找到了Ginzburg-Landau方程的解析解，包括表面轮廓和动力学系数估计。它显示了动力学系数如何影响局部应力场。用理论强度校准裂纹扩展的局部临界应力，该理论强度给出裂纹表面宽度的值。利用有限元方法求解了完整的裂纹相场和力学方程组。用数值解验证了裂纹表面轮廓和裂纹尖端速度的解析表达式。结果表明，在模式I破裂和参数的正确校准下，相场模型始终与格里菲斯的理论一致。