Study of an arbitrarily oriented mode-III crack using gradient elasticity theory in a bidirectional functionally graded material

In this research, we conduct a thorough analysis of an arbitrarily oriented mode-III crack in a bidirectional functionally graded material (FGM) using strain gradient elasticity (SGE) theory. The focus is on understanding the growth and behavior of the crack when it is positioned at an angle counterclockwise to the x-axis. The material gradation in the bidirectional FGM is assumed to follow an exponential distribution within the xy-plane. By transforming the global coordinate system into a local system, the x1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_1$$\end{document}-axis is aligned with the crack’s direction, forming a specific angle with the x-axis. The SGE theory uses two material characteristic lengths, ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document} and ℓ′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^\prime $$\end{document}, to account for volumetric and surface strain gradient factors, respectively. To solve the crack boundary value problem, we utilize a methodology that combines Fourier transforms with an innovative hyper-singular integrodifferential equation approach. This methodological framework allows us to derive a comprehensive system of equations, which are then solved using Chebyshev polynomial expansion techniques and the selection of suitable collocation points. Our study includes a detailed examination of the crack surface displacement under various material parameter configurations. We also analyze the stress intensity factors and the energy release rate at the crack tips, providing critical insights into the mechanical behavior of cracks in bidirectional FGMs under the influence of strain gradient elasticity.