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.
Phase field method for quasi‐static brittle fracture: an adaptive algorithm based on the dual variable
