Study On Higher-Order Phase Field Theory Of Crack Propagation

It is a common phenomenon that crack initiation,propagation and even complete failure occur in materials under load.So far,there are many methods applied to the study of crack propagation,etc.However,the prediction of crack initiation and propagation is still an open problem.In recent years,it has been found that the phase field method has potential advantages in prediction of crack initiation,propagation and bifurcation.In view of this research status,this paper is based on the lower-order phase field theory to expand,established higher-order phase filed model to study the crack growth of uniform brittle materials and non-uniform materials.In this paper,the lower-order phase field theory is firstly extended,the surface density function of higher-order crack is proposed,and the governing equilibrium equation of higher-order phase field theory is derived by using variational principle.Under the same load,material parameters and geometric dimensioning,lower-order phase field and higher-order phase field are used to predict crack propagation in materials.The results show that the crack propagation and load displacement curves between the lower-order phase filed theory and the higher-order phase field theory are consistent.Secondly,the regionally independent integral for uniform brittle materials in phase field method is derived,and the expression and proof of the regionally independent integral are given.This integral expression is suitable for both lower-order phase field theory and higher-order phase field theory,and the regionally independent property of the integral expression is further verified by numerical simulation of edge crack propagation in tensile and shear models.Finally,using higher-order phase field theory to simulate the crack propagation of heterogeneous materials,for qualitative analysis,a functionally graded material composed of two materials is adopted.Crack propagation in functionally graded materials under tensile loading is studied by using higher-order phase field theory.The results show that the higher-order phase field theory is suitable for numerical simulation of crack propagation in heterogeneous materials.