Accurate determination of mode I and II leading coefficients of the Williams expansion by finite element analysis

Leading coefficients of the Williams expansion are evaluated by using the fractal finite-element method (FFEM). By means of the self-similarity principle, an infinite number of elements is generated at the vicinity of the crack tip to model the crack tip singularity. The Williams expansion series with higher-degree coefficients is used to capture the singular and non-singular stress behaviour around the crack tip and to condense the large amount of nodal displacements at the crack tip to a small set of unknown coefficients. New sets of coefficients up to the sixth degree for mode I and fourth degree for mode II problems are solved. The important fracture parameters such as stress intensity factors and T-stress can be obtained directly from the coefficients without employing any path independent integrals. Convergence study reveals that the present method is simple and very coarse finite element meshes with 12 leading terms in the William expansion can yield very accurate solutions. The effects of the influence of crack length on the higher-degree coefficients of some common plane crack problems are studied in detail.