Computation of 3-D stress singularities for multiple cracks and crack intersections by the scaled boundary finite element method

Abstract

Among the various possible ways of dealing with notch and crack situations, the scaled boundary finite element method [SBFEM, (Wolf and Song in Finite element modelling of unbounded structures. Wiley, Chichester, 1996; Wolf in The scaled boundary finite element method. Wiley, Chichester, 2003)] has been adopted in this work. This method has been proved to be versatile, much less time consuming than the finite element method and generates highly accurate numerical predictions in cases of structures with notches and cracks. The SBFEM gives the advantage of boundary element method by reducing one dimension in modelling the structures but the mathematical formulations are more related to conventional displacement based finite element method. This method requires a certain scalability of the given structure with respect to a point called similarity center. Like in the case of the boundary element method, the structure needs to be discretized only at the surface where standard displacement based isoparametric finite element formulations are adequate. Unlike in the boundary element method, however, no fundamental solution is required by the scaled boundary finite element method. The similarity or scalability of the method requires separation of coordinates such that in the radial direction (i.e. scaling direction) it yields simple differential equations that can be solved analytically. So this approach can be considered as a semi-analytical method. Several two-dimensional examples have been analysed for crack and notch situations that are well known cases in fracture mechanics. A number of three-dimensional cases have been considered for different crack configurations that yield high order of singularity. The results, according to the authors’ knowledge are up to now unpublished in the open literature. Parametric studies are conducted for structures with bi-material interfaces.