On the computation of the J-integral for three-dimensional geometries in inhomogeneous materials

An analytical expression of the J-integral for inhomogeneous materials is presented in a form suitable for the numerical analysis of arbitrary three-dimensional (3-D) mode-I crack configurations. This integral is given using the principle of virtual work and Eshelby's energy moment tenser. The virtual crack extension method was developed from finite element considerations and was based on the calculations of the released energy when a crack in a finite element model was extended by a small amount Delta a. For numerical calculations, the 3-D analogue of the volume integral appears to be an attractive approach for obtaining pointwise values of the stress intensity factors along a crack front. The formulation is easily incorporated into a finite element program, but can also conveniently be used as part of a post-processing program, which uses stress and displacement data from a finite element analysis to calculate the stress intensity factors. Numerical examples are presented, using twenty node, isoparametric, quarter-point elements, for the compact tension specimen, homogeneous and inhomogeneous materials. Stress intensity factors are calculated for various moduli of the inclusion and distances of the crack from the interface. Numerical results are compared with results obtained by two-dimensional models.