The influence of bi-directional material gradation on a mode-III crack in functionally graded material via strain gradient elasticity theory

In contrast to classical mechanics, which primarily relies on continuum assumptions and neglects micro-structural effects, the strain gradient elasticity (SGE) theory represents a paradigm shift in understanding the mechanical behavior of materials at small length scales. In this article, the influence of the bi-directional material gradation on a mode-III crack in functionally graded material via SGE theory is studied. The SGE theory uses two material characteristic lengths, $\ell$ and ${\ell }^{\prime }$, to account for volumetric and surface strain-gradient factors, respectively. Our investigation is centered on a material gradation model assumed to vary exponentially, with the shear modulus represented as $G(x,y)=G_0{e}^{\beta x+\gamma y}$, where $\beta$ and $\gamma$ are material gradation constants. To address the crack boundary value problem under consideration, we employ a methodology combining Fourier transforms and an innovative hyper-singular integro-differential equation approach. Using this approach, we systematically formulate a system of equations, which can be solved by selecting suitable collocation points. The closed-form analytical expressions are derived for the standard fracture parameters such as crack surface displacement (CSD), stress intensity factor (SIF), and energy release rate (ERR). Numerical studies are illustrated for the derived standard fractures, and the influence of these parameters $\beta$, $\gamma$, $\ell$, ${\ell }^{\prime }$, and applied shear load is graphically presented. Through comprehensive analysis, our aim is to provide insights into the complex interplay between material parameters, loading conditions, and crack behavior in functionally graded materials.