Shape sensitivity and reliability analyses of linear-elastic cracked structures

This paper presents a new method for continuum-based shape sensitivity and reliability analyses of a crack in a homogeneous, isotropic, and linear-elastic body subject to mode-I loading. 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 sensitivity analysis method. Since the governing variational equation is differentiated prior to the process of discretization, the resulting sensitivity equations are independent of any approximate numerical techniques, such as the finite element method, boundary element method, or others. Numerical results show that the maximum error in calculating the sensitivity of J using the proposed method is less than three percent. Based on continuum sensitivities, the first-order reliability method was formulated to conduct probabilistic fracture-mechanics analysis. A numerical example is presented to illustrate the usefulness of the proposed sensitivity equations for probabilistic analysis. Since all gradients are calculated analytically, the reliability analysis of cracks can be performed efficiently.