Photothermoelastic behavior of fractal semiconductor media in noninteger-dimensional space via memory and nonlocal effects

Abstract

In this research, we investigate the photothermoelastic properties of fractal semiconductor materials by extending vector calculus principles to fractional-dimensional spaces. We develop generalized photothermoelastic equations for regions with noninteger dimensions and provide solutions for achieving the equilibrium state of these materials. Our approach involves defining both basic and advanced mathematical tools, such as gradient, divergence, and Laplacian, by extending their use to different dimensions. To streamline our analysis, we focus on scalar point functions and rotationally invariant vectors that are independent of angles. We derive the heat conduction equation in accordance with the Moore–Gibson–Thompson (MGT) framework of generalized photothermoelasticity, which incorporates memory effects over a sliding interval. The governing equations are solved using the Laplace transform method. The continuum boundary is exposed to mechanical shock and specified thermal loading. After obtaining the Laplace transform solution, numerically invert it using the Gaver–Stehfest algorithm. Graphical representation of the numerical results validates our new theory, demonstrating that certain parameters significantly influence thermoelastic behavior. These findings are critical for accurately predicting thermoelastic responses in the design and processing of nanostructures. Fractal-based modeling improves semiconductor technology, NEMS resonators, energy materials, optoelectronics, structural engineering, and environmental remedies by improving lenses and filters and supporting resistant composites.