On Efficient Evaluation of Derivatives of Fracture Parameters using Fractal Finite Element Method

Probabilistic fracture mechanics (PFM) that blends the theory of fracture mechanics and the probability theory provides a more rational means to describe the actual behavior and reliability of structures. However in PFM, the fracture parameters and their derivatives are often required to predict the probability of fracture initiation and/or instability in cracked structures. The calculation of the derivatives of fracture parameters with respect to load and material parameters, which constitutes size-sensitivity analysis, is not unduly difficult. 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 finite-difference method to calculate the shape sensitivities is often computationally expensive, in that numerous repetitions of deterministic finite element analysis may be required for a complete reliability analysis. Therefore, an essential need of probabilistic fracture-mechanics is to evaluate the sensitivity of fracture parameters accurately and efficiently. In this paper new continuum shape sensitivity based methods for evaluation of the fracture parameters and their derivatives for cracks in an isotropic, linear-elastic materials. These methods involve the fractal finite-element method which has been proved to be an accurate and efficient method to solve planar crack problems, the material derivative concept of continuum mechanics, domain integral representation of a J-integral or 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. Numerical results show that J-integral or stress intensity factors and first-order sensitivities of J-integral or stress intensity factors obtained by using the proposed methods are in excellent agreement with the reference solutions for the structural and crack geometries considered in this study.