Continuum Shape Sensitivity Analysis of Mode-I Fracture in Functionally Graded Materials

This paper presents a new method for continuum shape sensitivity analysis of a crack in an isotropic, linear-elastic functionally graded material. The method involves the material derivative concept of continuum mechanics, domain integral representation of the J-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 meshless method, finite element method, boundary element method, or others. In addition, since the J-integral is represented by domain integration, only the first-order sensitivity of the displacement field is needed. Numerical results show that first-order sensitivities of J-integral obtained by using the proposed method are in excellent agreement with the reference solutions obtained from finite-difference methods for the structural and crack geometries considered in this study.