State-space approach for an infinite medium with a spherical cavity based upon two-temperature generalized thermoelasticity theory and fractional heat conduction

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. A general solution to the problem based on the two-temperature generalized thermoelasticity theory (2TT) is introduced. The theory of thermal stresses based on the heat conduction equation with Caputo’s time-fractional derivative of order α is used. Some special cases of coupled thermoelasticity and generalized thermoelasticity with one relaxation time are obtained. The general solution is provided by using Laplace’s transform and state-space techniques. It is applied to a specific problem when the boundary of the cavity is subjected to thermomechanical loading (thermal shock). Some numerical analyses are carried out using Fourier’s series expansion techniques. The computed results for thermoelastic stresses, conductive temperature, and thermodynamic temperature are shown graphically and the effects of two-temperature and fractional-order parameters are discussed.