On the calculation of derivatives of stress intensity factors for multiple cracks

In this paper, the work of Lin and Abel [Lin SC, Abel JF. Variational approach for a new direct-integration form of the virtual crack extension method. Int J Fract 1988;38:217-35] is further extended to the general case of multiple crack systems under mixed-mode loading. Analytical expressions are presented for stress intensity factors and their derivatives for a multiply cracked body using the mode decomposition technique. The salient feature of this method is that the stress intensity factors and their derivatives for the multiple crack system are computed in a single analysis. It is shown through two-dimensional numerical examples that the proposed method gives very accurate results for the stress intensity factors and their derivatives. It is also shown that the variation of mode I and II displacements at one crack-tip influence the mode I and II stress intensity factors at any other crack. The computed errors were about 0.4-3% for stress intensity factors, and 2-4% for their first order derivatives for the mesh density used in the examples.