A strain gradient elasticity analysis of exponentially graded FGM containing multiple cracks under anti-plane shear load

In this study, we extend the investigation of mechanical behavior in functionally graded materials (FGMs) to include bi-directional exponential variations in material properties, addressing the case of two unequal collinear mode-III cracks weakened FGM plate. Employing the strain gradient elasticity theory, which incorporates two characteristic lengths, $$\ell$$ and $$\ell ^{\prime }$$, to capture volumetric and surface strain-gradient effects, respectively, we analyze the shear modulus as a function of both spatial coordinates, represented as $$\mathcal {G}(x,y)=\mathcal {G}_0e^{\beta x + \gamma y}$$. This bi-directional gradation introduces new complexities into the system, necessitating advanced numerical techniques. We construct a system of hyper-singular integral-differential equations to derive numerical solutions for various fracture parameters. Our analysis covers different variations of bi-directional material gradation parameters, $$\beta$$ and $$\gamma$$, and strain gradient parameters, $$\ell$$ and $$\ell ^{\prime }$$ to explore their influence on stress intensity factor, strain distributions, stresses, and crack surface displacement. Additionally, we examine the effect of varying inter-crack distances, providing a comprehensive understanding of the interplay between crack geometry and bi-directional material gradation. The findings demonstrate significant variations in fracture behavior and highlight the importance of considering bidirectional material gradation in FGMs.