Cracked plate analysis with the dual boundary element method and Williams׳ eigenexpansion

This paper provides a numerical verification that the singular term of Williams׳ series eigenexpansion can be used as a singular solution, valid in the neighborhood of each crack tip, in a single-region dual boundary element analysis of two-dimensional piece-wise flat multi-cracked plates, either with edge or internal cracks, in mixed-mode deformation, as an intermediate and necessary research step towards the implementation of the singularity subtraction technique.The dual equations are the displacement and traction boundary integral equations which allow the solution of general mixed-mode crack problems in a single-region boundary-element analysis.The singularity subtraction technique is a regularization procedure that uses a singular particular solution of the crack problem to introduce the stress intensity factors as additional primary unknowns in the dual boundary element method. Its implementation depends on the availability of closed-form singular solutions relative to a single-region of a general multi-cracked plate.In this paper, Williams׳ series eigenexpansion, which is valid for a semi-infinite edge crack, is used to compute the stress intensity factors, for both cases of edge and internal cracks, for each deformation mode. The singular term of the expansion is used as a singular particular solution in the neighborhood of each edge and internal crack tip. Collocation of this term, at a single internal point near the crack tip, is carried out to compute the stress intensity factors in post-processing.Several cracked plates were analyzed with this technique in order to assess the validity of using the singular term of Williams׳ series eigenexpansion for the regularization of the elastic field in a single-region dual boundary element analysis of a general piece-wise multi-cracked plate. The results obtained in this work are in perfect agreement with those obtained with the dual boundary element method, through the J-integral technique, and other published results for both cases of the edge and internal piecewise-flat cracks. Hence, it can be concluded that, in the singularity subtraction technique of the dual boundary element analysis of general edge and internal piecewise-flat multi-cracked plates under mixed-mode deformation, the singular term of Williams׳ series can be used as a closed-form particular solution, valid in the neighborhood of each crack tip.