On the calculation of derivatives of stress intensity factors using fractal finite element method

Abstract

This paper presents a new fractal finite element based method for continuum-based shape sensitivity analysis for a crack in a homogeneous, isotropic, and two dimensional 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, 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 predict the first-order sensitivity of the stress-intensity factors, K╻ and K╻╻, more efficiently and accurately than the finite-difference method. Unlike the integral based methods such as J-integral or M-integral no special finite elements and post-processing are needed to determine the first-order sensitivity of the stress-intensity factors, K╻ and K╻╻. Also a parametric study is carried out to examine the effects of the similarity ratio, the number of transformation terms, and the integration order on the quality of the numerical solutions. One mixed mode numerical example is presented to calculate the first-order derivative of the stress-intensity factors. The results show that the first-order sensitivities of the stress intensity factors obtained using the proposed method are in excellent agreement with the reference solutions obtained using the finite-difference method.