Evaluation of fracture parameters for three-dimensional cracks by a hierarchical quadrature element method

This work integrates the hierarchical quadrature element method (HQEM), which is known to have p-convergence, into the virtual crack closure method (VCCM) to evaluate fracture parameters for three-dimensional (3D) crack configurations. The prerequisite of the VCCM when dealing with 3D crack problems is the orthogonality of mesh arrangement in the vicinity of the crack front, which cannot be strictly met when traditional h-version finite element methods are employed. Compared with the h-version methods, one of the distinguished advantages of the HQEM is its simplicity in pre-processing, which is helpful to solve the difficulty of orthogonal mesh generation. The technical details regarding the combination of HQEM and VCCM are illustrated in this work. Firstly, the method of generating higher-order mesh which strictly meets the orthogonality requirement is proposed. Then, a universal formula for crack closure integral is proposed for hexahedral hierarchical quadrature element, regardless of node arrangements and the number of nodes per element boundary. In addition, the subdomain integration technique is incorporated to estimate SIFs at a large number of subsegments along the crack front under a coarse mesh consisting of only a few elements. The effectiveness and accuracy of the present method are verified by several typical numerical examples, including through-the-thickness cracks, embedded elliptical cracks and semi-elliptical surface cracks. The results show that with only one or two elements arranged along the crack front, the present method is capable of easily and accurately obtaining the SIF distribution of 3D crack configurations with straight or curved crack fronts.