The spectral decomposition of the strain tensor is an essential technique to deal with the fracture problems via phase field method, and some incorrect results may be obtained without it. A novel phase field model for brittle fracture is developed based on cell-based smooth finite element (CS-FEM) and the spectral decomposition is taken into account. In order to describe the nonlinearity behaviors which contain the varied stress and elastic constitutive response caused by spectral decomposition. A second-order stress tensor and a fourth-order constitutive tensor based on decomposition of strain tensor are derived. A fundamental framework of CS-FEM is established to solve the phase field fracture problems, implemented by user-defined element (UEL) subroutine of ABAQUS software. The proposed model is validated by a typical Mode II crack, and the results show that the derived tensors are effective. Phase field parameter, CS-FEM parameter and mesh inhomogeneity are investigated to provide some useful suggestion for further development. Some classical numerical examples are solved by using the present model. The studies demonstrate that the proposed method can successfully overcome mesh distortion; the number of smooth cell does not show influences on the accuracy. Moreover, some results show that this method has the advantage over the standard FEM in convergence and computing efficiency.

中文翻译：

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考虑谱分解的基于单元光滑有限元的脆性断裂相场建模

应变张量的谱分解是用相场法处理断裂问题的必要技术，没有它可能会得到一些不正确的结果。基于单元光滑有限元（CS-FEM）建立了一种新的脆性断裂相场模型，并考虑了谱分解。为了描述包含由光谱分解引起的变化的应力和弹性本构响应的非线性行为。导出了基于应变张量分解的二阶应力张量和四阶本构张量。建立了解决相场断裂问题的CS-FEM基本框架，由ABAQUS软件的用户定义单元（UEL）子程序实现。所提出的模型通过典型的模式 II 裂纹进行验证，结果表明导出的张量是有效的。研究了相场参数、CS-FEM参数和网格不均匀性，为进一步的开发提供了一些有益的建议。使用本模型求解了一些经典的数值例子。研究表明，所提出的方法可以成功地克服网格变形；平滑单元的数量对精度没有影响。此外，一些结果表明，该方法在收敛性和计算效率方面优于标准有限元法。研究表明，所提出的方法可以成功地克服网格变形；平滑单元的数量对精度没有影响。此外，一些结果表明，该方法在收敛性和计算效率方面优于标准有限元法。研究表明，所提出的方法可以成功地克服网格变形；平滑单元的数量对精度没有影响。此外，一些结果表明，该方法在收敛性和计算效率方面优于标准有限元法。