A new weight function approach to determine the stress intensity factor(SIF) using indirect boundary integral equation has been presented. The crack surface displacement field was approximated using a single-layer potential in the indirect boundary integral equation method for the displacement based weight function approach. The single-layer potential for the crack surface displacement was represented by one boundary integral term whose kernel was modified from Kelvin's fundamental solution.
A closed-form stress intensity factor solution was derived analytically. The proposed method enables the calculation of the SIF using only one SIF solution without any modification for the various crack geometries symmetric in two-dimensional plane such as a center crack in a plate with or without an internal hole, double edge cracks, and circumferential crack or radial cracks in a pipe. The application procedure to those variety of crack geometries is very simple and straightforward with the closed-form solution, where hand calculation of the SIF is possible. In most cases, a linear approximation with two reference load cases for the determination of the SIF showed good results. To improve the solution behavior, a quadratic approximation with three reference load cases was also applied to the crack geometries with internal hole such as radial and circumferential cracks in a hollow cylinder. However, the linear approximation of the SIFs with two reference SIFs showed reasonable accuracies in most cases. The analysis results using several examples verified that the present closed-form solutions were in good agreement with those of the literature and applicable to various crack geometries.