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 vari-ational equation is differentiated prior to the process of discretization, the resulting sensitivity equations are independent of approximate numerical techniques, such as the ®nite element method, boundary element method, meshless methods, or others. In addition, since the interaction integral is represented by domain integration, only the ®rst-order sensitivity of the displacement ®eld 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 ®nite-difference methods is less than four percent. 1 Introduction Sensitivity analysis of a crack-driving force plays an important role in many fracture-mechanics applications involving the stability and arrest of crack propagation, reliability analysis, parameter identi®cation, or other considerations. For example, the derivatives of the stress-intensity factor (SIF) or other fracture parameters are often required to predict the probability of fracture initiation and/or instability in cracked structures. The ®rst-and second-order reliability methods [1], frequently used in probabilistic fracture mechanics [2±8], require the gradient and Hessian of the performance function with respect to random parameters. In linear-elastic fracture mechanics (LEFM), the performance function is built on SIF. Hence, both ®rst-and/or second-order derivatives of SIF are needed for probabilistic analysis. The calculation of these derivatives with respect to load and material parameters, which constitutes size-sensitivity analysis, is not unduly dif®cult. However, the evaluation of response derivatives with respect to crack size is a challenging task, since it requires shape sensitivity analysis. Using a brute-force type ®nite-difference method to calculate the shape sensitivities is often computationally expensive, in that numerous repetitions of deterministic ®nite element analysis may be required for a complete reliability analysis. Furthermore, if the ®nite-difference perturbations are too large relative to ®nite element meshes, the approximations can be inaccurate, whereas if the perturbations are too small, numerical truncation errors may become signi®cant. Therefore, an important requirement of some fracture-mechanics applications is to evaluate the …