Extracting edge stress intensity function for isotropic 3D cracked domains having stochastic material properties

Abstract

The Stress Intensity Factors (SIFs) are fundamental and valuable parameters in fracture mechanics since many failure criteria involve them. In the case of a 3D crack, the SIFs extraction, based on a post-processing solution using the Finite-Element Method (FEM), is performed repeatably at several points along the singular edge (as a point-wise method) to evaluate the functional representation of the SIF along the crack edge. The Quasi Dual Function Method (QDFM) provides a polynomial approximation of the SIFs along the edge and eliminates the need for point-wise repeated calculations along the edge. Herein, we extend the QDFM to provide a broader 3D extraction method that includes stochasticity in the material properties in the case of an isotropic cracked domain having a straight edge, using the generalized Polynomial chaos (gPC). We apply the 2D gPC SIF extraction method over the QDFM to provide a stochastic Edge Stress Intensity Function (ESIF) polynomial approximation for 3D cracked domains with stochastic material properties (stochastic variables). The obtained ESIF is expressed using three families of orthogonal polynomials: Jacobi polynomials which approximate the deterministic solution of the ESIF (as part of the QDFM) and two families of orthogonal polynomials, selected by the probability distribution function of the material properties. The proposed method provides a functional polynomial presentation of the stochastic ESIF along the crack edge. Numerical example problems are provided where the stochastic approximation of the ESIF is computed. The stochastic properties of the ESIF: Mean, Variance, Skewness and Kurtosis are calculated, and the convergence of these properties is examined. The stochastic approximation of the ESIF is compared with results obtained using the 2D point-wise gPC SIFs extraction method. The proposed method is very efficient compared with the known point-wise methods since it provides a continuous function along the singular edge, unlike the point methods that require a new calculation for each point separately. The numerical examples demonstrate the robustness and high accuracy of the proposed method.