Two-Temperature Generalized Thermoelastic Interactions in an Infinite Body with a Spherical Cavity

Abstract

This paper is concerned with the determination of the thermoelastic displacement, stress, conductive temperature, and thermodynamic temperature in an infinite isotropic elastic body with a spherical cavity in the context of the two-temperature generalized thermoelasticity theory (2TT). The two-temperature Lord-Shulman (2TLS) model and two-temperature Green-Naghdi (2TGN) models of thermoelasticity are combined into a unified formulation introducing the unified parameters. The medium is assumed initially quiescent. The basic equations have been written in the form of a vector-matrix differential equation in the Laplace transform domain which is then solved by (a) the state-space approach and (b) the eigenvalue approach for any set of boundary conditions. The general solution obtained is applied to a specific problem when the boundary of the cavity is subjected to thermomechanical loading. The numerical inversion of the transform is carried out using Fourier-series expansion techniques. The computed results for thermoelastic stresses, conductive temperature, and thermodynamic temperature are shown graphically for the Lord Shulman model and for two models of Green-Naghdi and the effects of two temperatures are discussed. A comparative study of the two methods has also been carried out.

On the spatial behavior in two-temperature generalized thermoelastic theories

“…To simplify the notation, we will write K(φ) = (k ij φ ,i ) ,j . It will be useful to take into account the following identities: 2 . We note that the last equality comes from the relation (hθ + dθ)(θ + αθ) = 1 2…”

“…We note that (θ) 2 . We now consider an auxiliary function to control the expression involving the term K(φ)φ.…”

“…It is worth recalling that the combination of the theories proposed by Green and Lindsay and Lord and Shulman with the two temperatures theory was suggested by Youssef [33]. This theory has attracted a lot of attention in the past years ( [2], [19], [26]). The spatial behavior for the classical theory with two temperatures was considered by Awad [1].…”

Thermoelastic response in a symmetric spherical shell

“…A half space problem filled with an elastic material has been solved in the context of the two-temperature generalized thermoelasticity theory using the state-space approach by Youssef and Al-Lehaibi [55]. Banik and Kanoria [56,57] have studied two-temperature generalized thermoelastic interactions in an infinite body with a spherical cavity. Twotemperature generalized thermoelasticity has also been studied by many authors [58][59][60].…”

Fractional heat conduction with finite wave speed in a thermo-visco-elastic spherical shell

“…When Fourier conductivity is dominant the temperature equation reduces to classical Fourier's law of heat conduction and when the effect of conductivity is negligible, the equation has undamped thermal wave solutions without energy dissipation. Applying the above theories of generalized thermoelasticity, several problems have been solved by Mallik and Kanoria (2008), Kar and Kanoria (2009), Islam and Kanoria (2011), Ghosh and Kanoria (2010), Banik and Kanoria (2011).…”