Effectiveness of nonlinear kernel with memory for a functionally graded solid with size dependency

Structures made of graded composites play an important role in various industrial fields, such as aerospace and biomechanics. By incorporating nonlocal stress theory the internal length scale parameter of the nonlocal model provides detailed information on long-range forces of atoms or molecules. This paper investigates the size-dependent modeling of a functionally graded unbounded medium influenced by a heat source and an induced magnetic field on the bounding plane. The heat transport equation is governed by a unified formulation that integrates both the three-phase-lag model and Moore–Gibson–Thompson theory of generalized thermoelasticity, incorporating a memory-dependent derivative with nonlinear and linear kernels. Using nonlocal stress theory, the constitutive equations are addressed. The basic equations are simplified in the transformed domain through the Laplace and Fourier integral transforms. To obtain solutions in the real space-time domain, the Fourier transforms are analytically inverted using residue calculus, with poles of the integrand numerically determined in the complex domain via Laguerre’s method. Subsequently, the numerical inversion of the Laplace transform is performed using a method based on Fourier series expansion. The computational results and corresponding graphical representations reveal significant effects of parameters such as the nonlocality parameter, time-delay parameter, and the influence of the magnetic field. Furthermore, the impact of different kernel functions is examined, demonstrating the superiority of nonlinear kernels over linear kernels within this new theoretical framework.