Memory response in quasi-static thermoelastic stress in a rod due to distributed time-dependent heat sources

Purpose To find the quasi-static thermoelastic stress and displacement, the proposed model looks at how the microstructures interact with each other and how the temperature changes inside a rod. It uses the fractional-order dual-phase-lag (FODPL) theory to derive analytical solutions for one-dimensional problems in nonsimple media within the MDD framework. The dimensionless equations are used to analyze a finite rod experiencing the heat sources continuously distributed over a finite portion of the rod which vary with time according to the ramp-type function with other sectional heat supplies kept at zero temperature. The study introduces a technique using integral transforms for exact solutions in the Laplace transform domain for different kernel functions. Design/methodology/approach A novel mathematical model incorporating dual-phase-lags, two-temperatures and Riesz space-fractional operators via memory-dependent derivatives has been established to analyze the effects of thermal stress and displacement in a finite rod. The model takes into account the continuous distribution of heat sources over a finite portion of the rod and their time variation according to the ramp-type function. It incorporates the finite Riesz fractional derivative in two-temperature thermoelasticity with dual-phase-lags via memory effect, and its solution is obtained using Laplace transform with respect to time and sine-Fourier transform with respect to spatial coordinates defined over finite domains. Findings In memory-dependent derivatives, thermal field variables are strongly influenced by the phase-lag heat flux and temperature gradient. The non-Fourier effects of memory-dependent derivatives substantially impact the distribution and history of the thermal field response, and energy dissipation may result in a reduction in temperature without heat transfer. The temperature, displacement and stress profile exhibit a reduced magnitude with the MDD effect compared to when the memory effect is absent (without MDD). To advance future research, a new categorization system for materials based on memory-dependent derivative parameters, in accordance with the principles of two-temperature thermoelasticity theory, must be constructed. Research limitations/implications The one-dimensional assumption introduces limitations. For example, local heating of a one-dimensional plate will not extend radially, and heating one side will not heat the surrounding sides. Furthermore, while estimating heat transfer, object shape limits may apply. Originality/value This paper aims to revise the classical Fourier law of heat conduction and develop analytical solutions for one-dimensional problems using fractional-order dual-phase-lag (FODPL) theory in nonsimple media in the context of MDD.