| In the current probabilistic fracture-mechanics (PFM) analysis, the response-surface based methods are widely used to approximate crack-driving force, such as stress-intensity factor (SIF) or J-integral, and calculate the reliability of cracked structures. The response-surface approximations significantly reduce the complexity in calculating the derivatives of SIF or J-integral, thereby exploiting the first- and second-order reliability methods (FORM/SORM) effectively. However, their usefulness is limited, since it cannot be applied for general (linear-elastic or elastic-plastic) fracture-mechanics analysis. Because of the complexity in crack geometry, external loads, and material behavior, more advanced computational tools, such as finite element method (FEM) or boundary element method, have to be employed to provide the necessary computational framework for analyzing cracked structures. This raises complexity in the reliability analysis because the performance function for FORM/SORM is not available as an explicit, closed-form function of input random variables. Although, calculating SIF and J and other relevant fracture parameters by FEM is not unduly difficult, the evaluation of response derivatives or sensitivities is a challenging task. Clearly, new and efficient methods need to be developed for predicting sensitivities of SIF and J-integral and then applying these sensitivities for predicting reliability of cracked structures involving FEM.; This research presents new methods for continuum-based shape sensitivity and reliability analyses for cracks in a homogeneous, isotropic, linear-elastic or nonlinear-elastic body subject to plane loading conditions. The methods involve material derivative concept of continuum mechanics, domain integral representation of J-integral and M-integral, and direct differentiation. Unlike the virtual crack extension techniques, no mesh perturbation is needed in the proposed sensitivity analysis. Since the governing variational equation is differentiated before the process of discretization, the resulting sensitivity equations are independent of any approximate numerical techniques, such as the finite element method, boundary element method, and others. Based on continuum sensitivities, the first-order reliability method was formulated to conduct probabilistic fracture-mechanics analysis. Numerical examples are presented to illustrate the usefulness of the proposed sensitivity equations for probabilistic analysis. Since all of the gradients are calculated analytically, the reliability analysis of cracks can be performed efficiently. |
