Fractal theory of thermoelasticity in non-integer dimension space

Motivated by the abundance of fractals within the natural world, the present study aims to explore a generalization of vector calculus for space having a non-integer dimension. A novel concept of thermoelasticity for fractal media has been presented utilizing a continuum model with non-integer dimensions. Invoking the analytic continuation in dimension, the first- and second-order differential vector operators, including gradient, divergence, and the scalar and vector Laplacian operators, have been found for a non-integer dimension space. Rotationally covariant vector and scalar point functions that are independent of angles have been taken into consideration for simplicity's sake. By incorporating the memory dependence within a sliding interval, the Moore–Gibson–Thompson theory of generalized thermoelatsicity frames the heat conduction equation for the present problem. Prescribed thermal loading and mechanical shock are applied to the continuum's bounding plane. To find the solution to the governing equations, the Laplace transform technique is employed. After that, a technique based on Fourier series expansion is used to numerically invert the Laplace transform. The graphical representations and computational findings show that a number of parameters, including the delay duration and the importance of various fractal dimensions, have a substantial impact. It also looks at the effects of various kernel functions, showing that in this new theoretical framework, nonlinear kernels are better than linear kernels.