Wave Propagation of Generalized Magneto-Thermoelastic Interactions in an Elastic Medium Under Influence of Initial Stres
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RESEARCH PAPER
Wave Propagation of Generalized Magneto‑Thermoelastic Interactions in an Elastic Medium Under Influence of Initial Stress Kh. Lotfy1,2 · A. A. El‑Bary3 Received: 31 May 2017 / Accepted: 30 August 2019 © Shiraz University 2019
Abstract Magneto-thermoelastic interactions in an isotropic homogeneous elastic half-space with two temperatures are studied using mathematical methods under the purview of the Lord–Şhulman (LS) and Green–Lindsay (GL) theories, as well as the classical dynamical coupled theory (CD). The medium is considered to be permeated by a uniform magnetic field. The general solution obtained is applied to a specific problem of a half-space and the interaction between them under the influence of magnetic field subjected to one type of heating the thermal shock type. The normal mode analysis is used to obtain the exact expressions for the displacement components, force stresses, temperature and couple stress distribution. The variations in the considered variables through the horizontal distance are illustrated graphically. Comparisons are made with the results between the three theories. Numerical work is also performed for a suitable material with the aim of illustrating the results. Keywords Generalized thermoelasticity · Thermal shock · Two temperatures · Lord–Şhulman theory · Magnetic field List of Symbols 𝜆, 𝜇 Counterparts of Lame’s parameters p Initial pressure η Initial stress parameter a Two-temperature parameter 𝛼t Coefficient of linear thermal expansion 𝜃 = T − T0 Thermodynamic temperature 𝜙 = 𝜙0 − T Conductive heat temperature (thermal temperature) T Absolute temperature T0 Temperature of the medium in its natural | T−T | state assumed to be | T 0 | < 1 | 0 | 𝜎ij Components of the stress tensor ui Components of the displacement vector 𝜌 Density of the medium * Kh. Lotfy [email protected] A. A. El‑Bary [email protected] 1
Department of Mathematics, Faculty of Science, Taibah University, Madina, Kingdom of Saudi Arabia
2
Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt
3
Basic and Applied Science Department, Arab Academy of Science, Technology and Maritime Transport, P.O. Box 1029, Alexandria, Egypt
eij Components of the strain tensor e Cubical dilatation CE Specific heat at constant strain K Thermal conductivity 𝜏0 Thermal relaxation time 𝜇0 Magnetic permeability 𝜀0 Electric permittivity Fi Lorentz force 𝛿ij Kronecker delta function
1 Introduction The classical coupled thermoelasticity theory proposed by Biot (1956) with the introduction of the strain rate term in the Fourier heat conduction equation leads to a parabolictype heat conduction equation, called the diffusion equation. This theory predicts finite propagation speed for elastic waves but an infinite speed for thermal disturbance. This is physically unrealistic. To overcome such an absurdity, generalized thermoelasticity theories have been propounded by Lord and Shulman (1967) as well as Green and Lindsay (1972) advocating the exist
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