Enriched partition-of-unity finite element method for stress intensity factors at crack tips

Abstract

The traditional finite element method (FEM) is very versatile but is not efficient in dealing with crack-tip problems, even within the linear elastic regime. It is due to the presence of singular stress field in the vicinity of a crack tip. Although many endeavors have been reported, no efficient and ‘one-for-all’ type of solution has emerged yet. This paper presents a unified approach, which is particularly efficient in solving ‘stress intensity factor’ at crack tips. The present approach is based on the partition-of-unity (PU) method, which is known as one of the meshless methods. Actually, the PU method encompasses the traditional FEM. In the present PU formulation, an analytical solution exactly describing the stress field around the crack tip is embedded in the FE shape function. It results in an enriched approximation, which is mathematically singular and in the form of local asymptotic expansion. It yields more accurate solutions. In comparison with the traditional FE analysis with p-version refinements, the present approach is much simpler, more robust and efficient. On the other hand, the notorious difficulty encountered in the meshless method in satisfying the essential boundary conditions is circumvented. Moreover, the present formulation has another merit that the stress intensity factors at crack tip can be obtained directly from numerical solution without any post-processing. This paper includes illustrative examples showing the rate of convergence and efficiency, benchmarked against other available solutions, in particular those obtained by p-version FEM. The efficiency and versatility of present formulation are demonstrated.