One-dimensional problem of a fractional order two-temperature generalized thermo-piezoelasticity

Abstract

This paper is concerned with the determination of the thermoelastic stress, strain and conductive temperature in a piezoelastic half-space body in which the boundary is stress free and subjected to thermal loading in the context of the fractional order two-temperature generalized thermoelasticity theory (2TT). The two-temperature three-phase-lag (2T3P) model, two-temperature Green-Naghdi model III (2TGNIII) and two-temperature Lord-Shulman (2TLS) model of thermoelasticity are combined into a unified formulation introducing unified parameters. The basic equations have been written in the form of a vector-matrix differential equation in the Laplace transform domain that is then solved by the state-space approach. The numerical inversion of the transform is carried out by a method based on Fourier series expansion techniques. The numerical estimates of the quantities of physical interest are obtained and depicted graphically. The effect of the fractional order parameter, two-temperature and electric field on the solutions has been studied and comparisons among different thermoelastic models are made.

Generalized Thermopiezoelectricity with Memory‐Dependent Derivative and Transient Thermoelectromechanical Responses Analysis

“…As a powerful mathematical tool, the methodology of fractional calculus has been widely employed in micro-/nanoscale heating problems [30][31][32][33]. Inspired by fractional calculus's resounding success, the generalized thermopiezoelectricity has been further extended into fractional ones [34][35][36]. Generally speaking, fractional calculus contains various types of definitions, the most widely adopted ones of which are Riemann-Liouville and Caputo fractional derivatives.…”

“…It is clear that these theories [34][35][36] may be totally referred to as time-fractional-order generalized thermopiezoelectricity, which can be applied to depict memorydependent behaviors. Nevertheless, it is easily seen that different formulas are proposed by different authors; that is, the aforementioned fractional-order generalized thermopiezoelectricity theories are not unique in the form.…”

Memory response in a two‐dimensional transversely isotropic thick plate with varying heat source

“…The stability in the three‐phase‐lag heat conduction equation and the relations among the three material parameters are discussed by Quintanilla and Racke [12]. Problems concerning three‐phase‐lag thermoelastic model have been studied by many authors [13–20].…”

An investigation on responses of thermoelastic interactions in a generalized thermoelasticity with memory-dependent derivatives inside a thick plate

“…In the last few decades, fractional calculus has been found to have applications to several phenomena in the areas of physics and engineering. Recently, many researchers have tried to modify the classical Fourier law of heat conduction equation using fractional calculus [15][16][17][18][19][20][21][22][23]. Diethelm [24] has explained that the Caputo [25] fractional derivative can be defined as (m) (s) ds, (…”