Memory-dependent derivative under generalized three-phase-lag thermoelasticity model with a heat source

Abstract

Purpose

This study aims to attempt to construct a new mathematical model of the generalized thermoelasiticity theory based on the memory-dependent derivative (MDD) considering three-phase-lag effects. The governing equations of the problem associated with kernel function and time delay are illustrated in the form of vector matrix differential equations. Implementing Laplace and Fourier transform tools, the problem is sorted out analytically by an eigenvalue approach method. The inversion of Laplace and Fourier transforms are executed, incorporating series expansion procedures. Displacement component, temperature and stress distributions are obtained numerically and illustrated graphically and compared with the existing literature.

Design/methodology/approach

This study is to analyze the influence of MDD of three-phase-lag heat conduction interaction in an isotropic semi-infinite medium. The current model has been connected to generalize two-dimensional (2D) thermoelasticity problem. The governing equations are shown in vector matrix form of differential equation concerning Laplace-transformed domain and solved by using the eigenvalue technique. The combined Laplace Fourier transform is applied to find the analytical interpretations of temperature, stresses, displacement for silicon material in a non-dimensional form. Inverse Laplace transform has been found by applying Fourier series expansion techniques introduced by Honig and Hirdes (1984) after performing the inverse Fourier transform.

Findings

The main conclusion of this current study is to demonstrate an innovative generalized concept for heat conducting Fourier’s law associated with moderation of time parameter, time delay variable and kernel function by applying the MDDs. However, an important role is played by the time delay parameter to characterize the behavioral patterns of the physical field variables. Further, a new categorization for materials may be created rendering to this new idea along MDD for the time delay variables to develop a new measure of its potential to regulate heat in the medium.

Originality/value

Generalized thermoelasticity is hastily undergoing modification day-by-day from basic thermoelasticity. It has been progressed to get over from the limitations of fundamental thermoelasticity, for instance, infinite velocity components of thermoelasticity interference, in the adequate thermoelastic response of a solid to short laser pulses and deprived illustrations of thermoelastic performance at low temperature. In the past few decades, the fractional calculus is used to change numerous existing models of physical procedure, and its applications are used in various fields of physics, continuum mechanics, fluid mechanics, biology, viscoelasticity, biophysics, signal and image processing, control theory, engineering fields, etc.

Citation

Singh, B., Pal (Sarkar), S. and Barman, K. (2020), "Memory-dependent derivative under generalized three-phase-lag thermoelasticity model with a heat source", Multidiscipline Modeling in Materials and Structures, Vol. 16 No. 6, pp. 1337-1356. https://doi.org/10.1108/MMMS-10-2019-0182