Fracture analysis of unequal arbitrarily orientated cracks in bi-directional functionally graded materials via strain gradient elasticity theory

Abstract

In this article, we have analyzed the interaction of arbitrarily oriented cracks under anti-plane deformation in a bi-directional functionally graded material (bi-FGM) using strain gradient elasticity (SGE) theory. The first crack is aligned along the $x_1$ -axis, and the second crack is aligned along the $x_2$ -axis, each obtained by rotating the global $xy$ -coordinate system to their respective local systems. The material gradation within the bi-FGM follows an exponential distribution in the $xy$ -plane. Using SGE theory, which incorporates two characteristic lengths of the material, $\ell$ and ${\ell }^{\mathrm{\prime }}$ , to account for the effects of volumetric and surface strain gradients, we adopt a robust methodological framework. This involves the application of Fourier transforms and a novel approach of hypersingular integro-differential equations. By solving the resulting system of equations with Chebyshev polynomial expansion techniques and appropriate collocation points, analytical expressions are obtained for stress intensity factor (SIF), strain distributions, stresses, and crack surface displacement (CSD) profiles for both cracks. An illustrative numerical case study is presented to show the influence of the orientation angle on the various fracture parameters. In addition, we explore the impact of intercrack distances, providing a thorough understanding of the interaction between crack geometry and bi-directional material gradation. Moreover, the variation in the CSD profile under linear and quadratic loading conditions is examined to highlight the influence of loading patterns on crack behavior. These findings offer a deeper understanding of the fracture mechanics in bi-FGMs under the influence of strain gradient elasticity.