Shape sensitivity analysis in mixed-mode fracture mechanics

This paper presents a new method for continuum-based shape sensitivity analysis for a crack in a homogeneous, isotropic, and linear-elastic body subject to mixed-mode (modes I and II) loading conditions. The method is based on the material derivative concept of continuum mechanics, domain integral representation of an interaction integral, and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is needed in the proposed method to calculate the sensitivity of stress-intensity factors. Since the governing variational equation is differentiated prior to the process of discretization, the resulting sensitivity equations are independent of approximate numerical techniques, such as the finite element method, boundary element method, meshless methods, or others. In addition, since the interaction integral is represented by domain integration, only the first-order sensitivity of the displacement field is needed. Two numerical examples are presented to illustrate the proposed method. The results show that the maximum difference in the sensitivity of stress-intensity factors calculated using the proposed method and reference solutions obtained by analytical or finite-difference methods is less than four percent.