Fourier and Laplace transforms in micropolar thermoelastic solid with rotation and Hall current in the case of energy di
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ORIGINAL PAPER
Fourier and Laplace transforms in micropolar thermoelastic solid with rotation and Hall current in the case of energy dissipation and thermal shock M I M Hilal1,2* 1
Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt 2
Department of Basic Sciences, Faculty of Engineering, Sinai University, Al-Arish, Egypt Received: 12 March 2019 / Accepted: 15 July 2019
Abstract: The present article discusses the problem of micropolar thermoelastic medium affected by the Hall current. The problem is investigated in the presence of rotation with Green–Naghdi (G–N) model of type III, using Laplace and Fourier transforms to obtain the considered physical quantities, which are calculated numerically and represented graphically. It is found that the rotation and time affected significantly in the calculated physical quantities. The used method predicts finite speeds of the thermal wave propagations. The present work may be applied on the field of steel and geomechanics, as well as in earthquakes engineering. Keywords: Hall current; Rotation; Micropolar thermoelastic; Fourier and Laplace transforms PACS Nos.: 62.20.Dc; 44.10.?i; 62.20.Dc; 80A17
1. Introduction The basic foundations of the classical theory of thermoelasticity laid in the nineteenth century by Duhamel, Neumann and Lord Kelvin are based on Fourier’s law of heat conduction. Combining these foundations with the basic laws of mechanics and thermodynamics, such as the geometrical relations, equations of motion, conservation of energy law, dissipation inequality and constitutive relations, Fourier’s law leads to the displacement-temperature field equations of hyperbolic-parabolic type that implies an infinite speed of propagation of thermoelastic waves. To correct this unrealistic feature, various modifications of the classical theory of thermoelasticity have been proposed. The first is introduced by Biot [1]; the heat equations for both theories are of the diffusion type, predicting infinite speeds of propagation for heat waves contrary to physical observations. Generalization of thermoelasticity theories is established by Lord and Shulman [2] with one relaxation time for the special case of an isotropic body. Green and Lindsay [3] developed the theory of thermoelasticity after taking two relaxation times. The last two models [2, 3] allow a finite speed for the wave
*Corresponding author, E-mail: [email protected]
propagations. Chandrasekharaiah [4] referred to this wavelike thermal disturbance as a ‘‘second sound’’. Green and Naghdi [5–7] introduced three new thermoelastic theories based on the entropy equality rather than the usual entropy inequality. The constitutive assumptions for the heat flux vector are different in each theory. Thus, they obtained three theories that are called thermoelasticity of type I, of type II and of type III. When the theory of type I is linearized, one can obtain the classical system of thermoelasticity. The theory of type II (a limiting case of type III) does not admit energy
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