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Effect of rotation and gravity on the reflection of P-waves from thermo-magneto-microstretch medium in the context of three phase lag model with initial stress

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Abstract

In this paper the linear theory of the thermoelasticity has been employed to study the effect the reflection of plane harmonic waves from a semi-infinite elastic solid under the effect of the magnetic field , rotation, initial stress and gravity. The medium under consideration is traction free, homogeneous, isotropic, as well as with three-phase-lag. The normal mode analysis is used to solve the resulting non-dimensional coupled equations. The expressions for the reflection coefficients, which are the relations of the amplitudes of the reflected waves to the amplitude of the incident waves, are obtained similarly, the reflection coefficient ratio variations with the angle of incident under different conditions are shown graphically. Comparisons are made with the results predicted by different theories Lord-Shulman theory (L-S), the Green-Naghdi theory of type III (G-N III) and the three-phase-lag model in the absence and presence of a magnetic field, rotation, initial stress and gravity. The results indicate that the effect of rotation, magnetic field, initial stress and gravity field are very pronounced.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, South Valley University, Qena, 83523, Egypt

    S. M. Abo-Dahab

  2. Department of Mathematics, Faculty of Science, Taif University, Taif, Saudi Arabia

    S. M. Abo-Dahab

  3. Department of Mathematics, Faculty of Science, Sohag University, Sohag, 82524, Egypt

    A. M. Abd-Alla, A. A. Kilany & M. Elsagheer

Authors
  1. S. M. Abo-Dahab
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  2. A. M. Abd-Alla
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  3. A. A. Kilany
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  4. M. Elsagheer
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Corresponding author

Correspondence to A. M. Abd-Alla.

Appendix

Appendix

We represent \(A,\,B,\,C,\,D\) and \(E\) in terms \(L = C_{k} - \iota C_{\nu } \omega - C_{T} \omega^{2} ,\quad M = \zeta_{0} + \zeta_{1} \omega^{2} ,\) \(N = \zeta_{8} + \zeta_{9} \omega^{2} ,\quad R = - \frac{{C_{2}^{2} }}{{\omega^{*2} }} + \omega^{2} ,\quad S = \frac{{2kc_{0}^{2} }}{{\gamma \omega^{*2} }} - \frac{{j\rho c_{0}^{2} \omega^{2} }}{\gamma }\) as follow:

$$A = \frac{{C_{1}^{2} }}{{c_{0}^{2} }}L,$$
$$\begin{aligned} B & = - LR - \zeta_{5} \frac{{C_{3}^{2} }}{{\omega^{*2} }}L - \frac{{\zeta_{10} kC_{1}^{2} }}{{\gamma \omega^{*2} }}L + \frac{{C_{1}^{2} }}{{c_{0}^{2} }}(LS - \omega^{2} \tau_{q}^{*} - \zeta_{4} \varepsilon_{1} \omega^{2} \tau_{q}^{*} - LM - LN - \zeta_{2} \zeta_{6} \sin^{2} \theta L), \\ C & = \omega^{2} \tau_{q}^{*} R - a_{0} \varepsilon_{2} \omega^{2} \tau_{q}^{*} + \varepsilon_{2} \omega^{2} \tau_{q}^{*} \zeta_{4} \frac{{C_{3}^{2} }}{{\omega^{*2} }} + \, \varepsilon_{1} \omega^{2} \tau_{q}^{*} \zeta_{4} R + a_{0} \varepsilon_{1} \omega^{2} \tau_{q}^{*} \zeta_{5} + LMR + \omega^{2} \tau_{q}^{*} \frac{{C_{1}^{2} }}{{c_{0}^{2} }}M + LNR \\ & \quad + \omega^{2} \tau_{q}^{*} \frac{{C_{1}^{2} }}{{c_{0}^{2} }}N - LRS - \omega^{2} \tau_{q}^{*} \frac{{C_{1}^{2} }}{{c_{0}^{2} }}S + \omega^{2} \tau_{q}^{*} \zeta_{5} \frac{{C_{3}^{2} }}{{\omega^{*2} }} + \zeta_{2} \zeta_{6} \sin^{2} \theta LR + \zeta_{2} \zeta_{6} \omega^{2} \tau_{q}^{*} \sin^{2} \theta \frac{{C_{1}^{2} }}{{c_{0}^{2} }} \\ & \quad + \zeta_{5} \frac{{C_{3}^{2} }}{{\omega^{*2} }}LN + \varepsilon_{1} \omega^{2} \tau_{q}^{*} \zeta_{4} \frac{{C_{1}^{2} }}{{c_{0}^{2} }}N - \zeta_{5} \frac{{C_{3}^{2} }}{{\omega^{*2} }}LS - \varepsilon_{1} \omega^{2} \tau_{q}^{*} \zeta_{4} \frac{{C_{1}^{2} }}{{c_{0}^{2} }}S + \zeta_{10} \frac{{kc_{0}^{2} }}{{\gamma \omega^{*2} }}LR + \zeta_{10} \omega^{2} \tau_{q}^{*} \frac{{kC_{1}^{2} }}{{\gamma \omega^{*2} }} \\ & \quad - \zeta_{3} \zeta_{7} \omega^{2} \frac{{C_{1}^{2} }}{{c_{0}^{2} }}L + \frac{{C_{1}^{2} }}{{c_{0}^{2} }}LMN - \frac{{C_{1}^{2} }}{{c_{0}^{2} }}LMS - \frac{{C_{1}^{2} }}{{c_{0}^{2} }}LNS + \zeta_{5} \zeta_{10} \frac{{C_{3}^{2} kc_{0}^{2} }}{{\gamma \omega^{*4} }}L + \varepsilon_{1} \omega^{2} \tau_{q}^{*} \zeta_{4} \zeta_{10} \frac{{kC_{1}^{2} }}{{\gamma \omega^{*2} }} \\ & \quad - \zeta_{2} \zeta_{6} \sin^{2} \theta \frac{{C_{1}^{2} }}{{c_{0}^{2} }}LS + \zeta_{10} \frac{{kC_{1}^{2} }}{{\gamma \omega^{*2} }}LM, \\ \end{aligned}$$
$$\begin{aligned} D & = a_{0} \varepsilon_{2} \omega^{2} \tau_{q}^{*} M + a_{0} \varepsilon_{2} \omega^{2} \tau_{q}^{*} N - a_{0} \varepsilon_{2} \omega^{2} \tau_{q}^{*} S - \omega^{2} \tau_{q}^{*} MR - \omega^{2} \tau_{q}^{*} NR + \omega^{2} \tau_{q}^{*} RS - \varepsilon_{2} \omega^{2} \tau_{q}^{*} \zeta_{4} \frac{{C_{3}^{2} }}{{\omega^{*2} }}N \\ & \quad + \varepsilon_{2} \omega^{2} \tau_{q}^{*} \zeta_{4} \frac{{C_{3}^{2} }}{{\omega^{*2} }}S + a_{0} \varepsilon_{2} \omega^{2} \tau_{q}^{*} \zeta_{2} \zeta_{6} \sin^{2} \theta + a_{0} \varepsilon_{2} \omega^{2} \tau_{q}^{*} \zeta_{10} \frac{{kc_{0}^{2} }}{{\gamma \omega^{*2} }} + \zeta_{3} \zeta_{7} \omega^{2} LR + \zeta_{3} \zeta_{7} \omega^{4} \tau_{q}^{*} \frac{{C_{1}^{2} }}{{c_{0}^{2} }} \\ & \quad - \zeta_{2} \zeta_{6} \omega^{2} \tau_{q}^{*} \sin^{2} \theta R\, - \omega^{2} \tau_{q}^{*} \zeta_{5} \frac{{C_{3}^{2} }}{{\omega^{*2} }}N - \varepsilon_{1} \omega^{2} \tau_{q}^{*} \zeta_{4} NR + \omega^{2} \tau_{q}^{*} \zeta_{5} \frac{{C_{3}^{2} }}{{\omega^{*2} }}S + \varepsilon_{1} \omega^{2} \tau_{q}^{*} \zeta_{4} RS \\ & \quad - \omega^{2} \tau_{q}^{*} \zeta_{10} \frac{{kc_{0}^{2} }}{{\gamma \omega^{*2} }}R - a_{0} \varepsilon_{1} \omega^{2} \tau_{q}^{*} \zeta_{5} N + a_{0} \varepsilon_{1} \omega^{2} \tau_{q}^{*} \zeta_{5} S - LMNR - \omega^{2} \tau_{q}^{*} \frac{{C_{1}^{2} }}{{c_{0}^{2} }}MN + LMRS \\ & \quad + \omega^{2} \tau_{q}^{*} \frac{{C_{1}^{2} }}{{c_{0}^{2} }}MS + LNRS + \omega^{2} \tau_{q}^{*} \frac{{C_{1}^{2} }}{{c_{0}^{2} }}NS - \omega^{2} \tau_{q}^{*} \zeta_{5} \zeta_{10} \frac{{C_{3}^{2} }}{{\omega^{*2} }}\frac{{kc_{0}^{2} }}{{\gamma \omega^{*2} }} + \omega^{2} \tau_{q}^{*} \zeta_{2} \zeta_{6} \sin^{2} \theta \frac{{C_{1}^{2} }}{{c_{0}^{2} }}S \\ & \quad - \varepsilon_{1} \omega^{2} \tau_{q}^{*} \zeta_{4} \zeta_{10} \frac{{kc_{0}^{2} }}{{\gamma \omega^{*2} }}R - a_{0} \varepsilon_{1} \omega^{2} \tau_{q}^{*} \zeta_{5} \zeta_{10} \frac{{kc_{0}^{2} }}{{\gamma \omega^{*2} }} + \zeta_{2} \zeta_{6} \sin^{2} \theta LRS\, - \zeta_{3} \zeta_{7} \omega^{2} \frac{{C_{1}^{2} }}{{c_{0}^{2} }}LS \\ & \quad + \zeta_{5} \frac{{C_{3}^{2} }}{{\omega^{*2} }}LNS + \varepsilon_{1} \omega^{2} \tau_{q}^{*} \zeta_{4} \frac{{C_{1}^{2} }}{{c_{0}^{2} }}NS - \zeta_{10} \frac{{kc_{0}^{2} }}{{\gamma \omega^{*2} }}LMR - \zeta_{10} \frac{{kC_{1}^{2} }}{{\gamma \omega^{*2} }}M + \frac{{C_{1}^{2} }}{{c_{0}^{2} }}LMNS \\ & \quad - \varepsilon_{2} \omega^{2} \tau_{q}^{*} \zeta_{4} \zeta_{10} \frac{{C_{3}^{2} kc_{0}^{2} }}{{\gamma \omega^{*4} }}, \\ \end{aligned}$$
$$\begin{aligned} E & = a_{0} \varepsilon_{2} \omega^{4} \tau_{q}^{*} \zeta_{3} \zeta_{7} - a_{0} \varepsilon_{2} \omega^{2} \tau_{q}^{*} MN + a_{0} \varepsilon_{2} \omega^{2} \tau_{q}^{*} MS + a_{0} \varepsilon_{2} \omega^{2} \tau_{q}^{*} NS + \omega^{4} \tau_{q}^{*} \zeta_{3} \zeta_{7} R + \omega^{2} \tau_{q}^{*} MNR \\ & \quad - \omega^{2} \tau_{q}^{*} MRS + \zeta_{3} \zeta_{7} \omega^{2} LRS + \omega^{4} \tau_{q}^{*} \zeta_{3} \zeta_{7} \frac{{C_{1}^{2} }}{{c_{0}^{2} }}S - \omega^{2} \tau_{q}^{*} \zeta_{2} \zeta_{6} \sin^{2} \theta RS - a_{0} \varepsilon_{2} \omega^{2} \tau_{q}^{*} \zeta_{10} \frac{{kc_{0}^{2} }}{{\gamma \omega^{*2} }}M\, \\ & \quad - \omega^{2} \tau_{q}^{*} \zeta_{5} \frac{{C_{3}^{2} }}{{\omega^{*2} }}NS - \varepsilon_{1} \omega^{2} \tau_{q}^{*} \zeta_{4} NRS + \omega^{2} \tau_{q}^{*} \zeta_{10} \frac{{kc_{0}^{2} }}{{\gamma \omega^{*2} }}MR - a_{0} \varepsilon_{1} \omega^{2} \tau_{q}^{*} \zeta_{5} NS - LMNRS \\ & \quad - \omega^{2} \tau_{q}^{*} \frac{{C_{1}^{2} }}{{c_{0}^{2} }}MNS - \varepsilon_{2} \omega^{2} \tau_{q}^{*} \zeta_{4} \frac{{C_{3}^{2} }}{{\omega^{*2} }}NS + a_{0} \varepsilon_{2} \omega^{2} \tau_{q}^{*} \zeta_{2} \zeta_{6} \sin^{2} \theta S - \omega^{2} \tau_{q}^{*} NRS, \\ \end{aligned}$$
$$F = a_{0} \varepsilon_{2} \omega^{4} \tau_{q}^{*} \zeta_{3} \zeta_{7} S - a_{0} \varepsilon_{2} \omega^{2} \tau_{q}^{*} MNS - \omega^{2} \tau_{q}^{*} \zeta_{3} \zeta_{7} \omega^{2} RS + \omega^{2} \tau_{q}^{*} MNRS.$$

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Abo-Dahab, S.M., Abd-Alla, A.M., Kilany, A.A. et al. Effect of rotation and gravity on the reflection of P-waves from thermo-magneto-microstretch medium in the context of three phase lag model with initial stress. Microsyst Technol 24, 3357–3369 (2018). https://doi.org/10.1007/s00542-017-3697-x

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