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Double porous thermoelastic waves in a homogeneous, isotropic solid with inviscid liquid

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Abstract

A two-dimensional model based on Lord–Shulman’s thermoelasticity theory has been constructed to examine the impact of various physical parameters of the medium in a double porous thermoelastic, isotropic, homogeneous half-space in contact with an inviscid liquid half-space. It has been discovered that four coupled longitudinal waves and one uncoupled transverse wave may exist in the solid medium and one mechanical wave in the liquid medium, all propagating at different speeds. It has been observed that linked longitudinal waves exhibit dispersion, attenuation, etc., and depend on both kinds of voids and the thermal characteristics of the medium. The simplest form of the complex secular equation has been derived. The amplitudes of solid and liquid displacements, volume fractional fields, and temperature change at the interface have also been obtained. The computer-simulated results for magnesium crystal material for different wave profiles have been graphically displayed.

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Acknowledgements

The third author is thankful to the Council of Scientific and Industrial Research (CSIR), New Delhi, India (File no. 09/1293(0003)/2019-EMR-I), for providing the junior research fellowship (JRF).

The authors declare that they have no conflict of interest.

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Authors and Affiliations

  1. Department of Mathematics, Himachal Pradesh University Regional Centre, Dharamshala, HP, India

    Vijayata Pathania

  2. Department of Mathematics, Indira Gandhi University, Meerpur, Rewari, HR, India

    Rajesh Kumar, Vipin Gupta & M. S. Barak

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  1. Vijayata Pathania
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  2. Rajesh Kumar
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  3. Vipin Gupta
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  4. M. S. Barak
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Appendices

Appendix A

$$ L_{1} = 1 + \tau_{0} (1 + \varepsilon ) + \frac{1}{{(a_{8} - a_{3} )}}\left\{ {\left( {\frac{1}{{\delta_{2}^{2} }} + \frac{{a_{8} }}{{\delta_{1}^{2} }}} \right) + \frac{1}{{\omega^{2} }}\left( {a_{6} - a_{11} + a_{2} a_{9} - a_{2} a_{4} + a_{3} a_{10} - a_{5} a_{8} + a_{1} a_{4} a_{8} - a_{1} a_{3} a_{9} } \right)} \right\} $$
$$ \begin{aligned} L_{2} = & \tau _{0} + \frac{1}{{(a_{8} - a_{3} )}}\left[\frac{1}{{\delta _{1}^{2} \delta _{2}^{2}}} + \left( {\frac{1}{{\delta _{2}^{2} }} + \frac{{a_{8} }}{{\delta _{1}^{1} }}} \right)\left\{ 1 + \tau _{0} (1 + \varepsilon )\right\} + \frac{1}{{\omega ^{2} }}\left\{ a_{6} - a_{{11}} + a_{3} a_{{10}} - a_{5} a_{8} + \frac{1}{{\delta _{1}^{2} }}(a_{2} a_{9} - a_{{11}} ) \right.\right.\\ & \quad + \frac{1}{{\delta _{2}^{2} }}(a_{1} a_{4} - a_{5} ) + \tau _{0} (a_{6} - a_{{11}} + a_{2} a_{9} - a_{2} a_{4} + a_{3} a_{{10}} - a_{5} a_{8} + a_{7} a_{{14}} - a_{4} a_{{14}} + a_{9} a_{{14}} - a_{{12}} a_{{14}} \\ & \quad + a_{4} a_{8} a_{{13}} - a_{1} a_{3} a_{9} + a_{1} a_{4} a_{8} - a_{3} a_{9} a_{{13}} + a_{3} a_{{12}} a_{{13}} - a_{7} a_{8} a_{{13}} + \varepsilon (a_{6} - a_{{11}} + a_{2} a_{{12}} - a_{2} a_{7} + a_{3} a_{{10}} \\ &\left. \left. \quad -\; a_{5} a_{8} + a_{1} a_{3} a_{{12}} + a_{1} a_{7} a_{8} ))\vphantom{\left\{ a_{6} - a_{{11}} + a_{3} a_{{10}} - a_{5} a_{8} + \frac{1}{{\delta _{1}^{2} }}(a_{2} a_{9} - a_{{11}} ) \right.}\right\} + \frac{1}{{\omega ^{4} }}(a_{5} a_{{11}} - a_{6} a_{{10}} + a_{1} a_{6} a_{9} - a_{1} a_{4} a_{{11}} + a_{2} a_{4} a_{{10}} - a_{2} a_{5} a_{9} )\right] \\ \end{aligned} $$
$$\begin{aligned} L_{3} & = \frac{1}{{(a_{8} - a_{3} )}}\left[\frac{1}{{\delta _{1}^{2} \delta _{2}^{2} }}\{ 1 + \tau _{0} (1 + \varepsilon )\} + \tau _{0} \left( {\frac{1}{{\delta _{2}^{2} }} + \frac{{a_{8} }}{{\delta _{1}^{2} }}} \right) - \frac{1}{{\omega ^{2} }}\left\{ \frac{{a_{{11}} }}{{\delta _{1}^{2} }} + \frac{{a_{5} }}{{\delta _{2}^{2} }} - \tau _{0} \left\{ a_{6} - a_{{11}} + a_{3} a_{{10}} - a_{5} a_{8} \right.\right.\right. \\ & \quad + a_{7} a_{{14}} - a_{{12}} a_{{14}} + a_{3} a_{{12}} a_{{13}} - a_{7} a_{8} a_{{13}} - \frac{1}{{\delta _{1}^{2} }}(a_{{11}} - a_{2} a_{9} + a_{{12}} a_{{14}} - a_{9} a_{{14}} ) - \frac{1}{{\delta _{2}^{2} }}(a_{5} - a_{1} a_{4} + a_{7} a_{{13}} \\ &\left.\left. \quad -\; a_{4} a_{{13}} ) - \varepsilon \left\{ \frac{1}{{\delta _{1}^{2} }}(a_{{11}} - a_{2} a_{{12}} ) + \frac{1}{{\delta _{2}^{2} }}(a_{5} - a_{1} a_{7} )\right\} \right\} \right\} - \frac{1}{{\omega ^{4} }}\{ a_{6} a_{{10}} - a_{5} a_{{11}} + \tau _{0} \{ a_{6} a_{{10}} - a_{1} a_{6} a_{9} \\ & \quad + a_{1} a_{4} a_{{11}} - a_{2} a_{4} a_{{10}} + a_{2} a_{5} a_{9} - a_{4} a_{{10}} a_{{14}} + a_{4} a_{{11}} a_{{13}} - a_{6} a_{9} a_{{13}} + a_{5} a_{9} .a_{{14}} - a_{5} a_{{12}} a_{{14}} + a_{6} a_{{12}} a_{{13}} \\ & \quad - a_{7} a_{{11}} a_{{13}} + a_{7} a_{{10}} a_{{14}} - a_{1} a_{7} a_{9} a_{{14}} + a_{1} a_{4} a_{{12}} a_{{14}} - a_{2} a_{4} a_{{12}} a_{{13}} + a_{2} a_{7} a_{9} a_{{13}} - \varepsilon (a_{5} a_{{11}} - a_{6} a_{{10}} \\ & \left. \quad +\; a_{1} a_{6} a_{{12}} - a_{1} a_{7} a_{{11}} + a_{2} a_{7} a_{{10}} - a_{2} a_{5} a_{{12}} )\} \}\right] \\ \end{aligned} $$
$$ \begin{aligned} L_{4} & \quad = \frac{{\tau_{0} }}{{a_{8} - a_{3} }}[\frac{1}{{\delta_{1}^{2} \delta_{2}^{2} }} - \frac{1}{{\omega^{2} }}\left\{ {\frac{1}{{\delta_{1}^{2} }}(a_{11} + a_{12} a_{14} ) + \frac{1}{{\delta_{2}^{2} }}(a_{5} + a_{7} a_{13} )} \right\} + \frac{1}{{\omega^{4} }}(a_{5} a_{11} - a_{6} a_{10} + a_{5} a_{12} a_{14} \\ & \quad \;- a_{6} .a_{12} a_{13} + a_{7} a_{11} a_{13} - a_{7} a_{10} a_{14} )] \\ \end{aligned} $$

Appendix B

$$ \begin{aligned} V_{q} & \quad = ((a_{3} a_{9} - a_{4} a_{8} - a_{3} a_{12} + a_{7} a_{8} )m_{q}^{4} + (a_{6} a_{12} - a_{6} a_{9} - a_{2} a_{4} a_{12} + a_{2} a_{7} a_{9} + \Gamma_{3}^{2} a_{4} a_{8} + \Gamma_{1}^{2} a_{3} a_{12} - \Gamma_{1}^{2} a_{7} a_{8} \\ & \quad - \Gamma_{3}^{2} a_{7} a_{8} - 2a_{3} a_{9} k^{2} + a_{4} a_{8} k^{2} + a_{3} a_{12} k^{2} )m_{q}^{2} + a_{3} a_{9} k^{4} - \Gamma_{1}^{2} a_{6} a_{12} + a_{6} a_{9} k^{2} + a_{2} a_{4} a_{12} k^{2} - a_{2} a_{7} a_{9} k^{2} \\ & \quad + \Gamma_{1}^{2} \Gamma_{3}^{2} a_{7} a_{8} - \Gamma_{3}^{2} a_{4} a_{8} k^{2} - \Gamma_{1}^{2} a_{3} a_{12} k^{2} )/\sigma^{*} \\ \end{aligned} $$
$$ \begin{aligned} W_{q} & = ((a_{4} - a_{7} + a_{12} - a_{9} )m_{q}^{4} + (a_{7} a_{10} - a_{4} a_{10} + \Gamma_{1}^{2} a_{7} - \Gamma_{1}^{2} a_{12} + \Gamma_{2}^{2} a_{9} - \Gamma_{2}^{2} a_{12} + a_{7} k^{2} - 2a_{4} k^{2} + a_{9} k^{2} - a_{1} a_{7} a_{9} \\ & \quad + a_{1} a_{4} a_{12} )m_{q}^{2} + a_{4} k^{4} - \Gamma_{1}^{2} a_{7} k^{2} + \Gamma_{1}^{2} \Gamma_{2}^{2} a_{12} - \Gamma_{2}^{2} a_{9} k^{2} + a_{4} a_{10} k^{2} - \Gamma_{1}^{2} a_{7} a_{10} + a_{1} a_{7} a_{9} k^{2} - a_{1} a_{4} a_{12} k^{2} )/\sigma^{*} \\ \end{aligned} $$
$$ \begin{aligned} S_{q} & = ((a_{8} - a_{3} )m_{q}^{6} + (a_{6} - a_{2} a_{4} + a_{2} a_{9} + a_{3} a_{10} + \Gamma_{1}^{2} a_{3} - \Gamma_{1}^{2} a_{8} - \Gamma_{2}^{2} a_{8} - \Gamma_{3}^{2} a_{8} + 2a_{3} k^{2} - a_{1} a_{3} a_{9} \\ & \quad + a_{1} a_{4} a_{8} )m_{q}^{4} + (\Gamma_{1}^{2} \Gamma_{2}^{2} a_{8} - \Gamma_{1}^{2} a_{6} - a_{3} k^{4} - a_{6} k^{2} - a_{6} a_{10} + \Gamma_{1}^{2} \Gamma_{3}^{2} a_{8} + \Gamma_{2}^{2} \Gamma_{3}^{2} a_{8} - 2\Gamma_{1}^{2} a_{3} k^{2} \\ & \quad + a_{1} a_{6} a_{9} + a_{2} a_{4} a_{10} - \Gamma_{2}^{2} a_{2} a_{9} - \Gamma_{1}^{2} a_{3} a_{10} + 2a_{2} a_{4} k^{2} - a_{2} a_{9} k^{2} - a_{3} a_{10} k^{2} - \Gamma_{3}^{2} a_{1} a_{4} a_{8} + 2a_{1} a_{3} a_{9} k^{2} \\ & \quad - a_{1} a_{4} a_{8} k^{2} )m_{q}^{2} + \Gamma_{1}^{2} a_{3} k^{4} + \Gamma_{1}^{2} a_{6} k^{2} + \Gamma_{1}^{2} a_{6} a_{10} - a_{2} a_{4} k^{4} - a_{1} \cdot a_{3} a_{9} k^{4} - a_{1} a_{6} a_{9} k^{2} - a_{2} a_{4} a_{10} k^{2} \\ & \quad + \Gamma_{2}^{2} a_{2} a_{9} k^{2} + \Gamma_{1}^{2} a_{3} a_{10} k^{2} - \Gamma_{1}^{2} \Gamma_{2}^{2} \Gamma_{3}^{2} a_{8} + \Gamma_{3}^{2} a_{1} a_{4} a_{8} \cdot k^{2} )/\sigma^{*} \\ \end{aligned} $$
$$ \begin{aligned} \sigma^{*} & = (a_{3} - a_{8} )m_{q}^{4} + (a_{8} \Gamma_{2}^{2} - 2a_{3} k^{2} + a_{8} \Gamma_{3}^{2} - a_{6} + a_{2} a_{7} - a_{3} a_{10} + a_{1} a_{3} a_{12} - a_{2} a_{12} - a_{1} a_{7} a_{8} )m_{q}^{2} + a_{6} a_{10} \\ & \quad - \Gamma_{2}^{2} \Gamma_{3}^{2} a_{8} + a_{3} k^{4} - a_{1} a_{6} a_{12} + a_{6} k^{2} - a_{2} a_{7} a_{10} + \Gamma_{2}^{2} a_{2} a_{12} - a_{2} a_{7} k^{2} + a_{3} a_{10} k^{2} - a_{1} a_{3} a_{12} k^{2} + \Gamma_{3}^{2} a_{1} a_{7} a_{8} \\ \end{aligned} $$

Appendix C

$$ \begin{aligned} U_{2} /U_{1} & = ((V_{3} W_{1} - V_{1} W_{3} )m_{1} m_{3} m_{6} + (V_{1} W_{4} - V_{4} W_{1} )m_{1} m_{4} m_{6} + (V_{4} W_{3} - V_{3} W_{4} )m_{3} m_{4} m_{6} + (V_{1} W_{3} - V_{3} W_{1} + V_{4} W_{1} \\ & \quad + V_{4} W_{1} - V_{1} W_{4} + V_{3} W_{4} - V_{4} W_{3} + p_{1} \iota k(V_{1} W_{3} - V_{3} W_{1} + V_{4} W_{1} - V_{1} W_{4} + V_{3} W_{4} - V_{4} W_{3} ))p_{2} m_{1} m_{3} m_{4} \\ & \quad + (V_{3} W_{1} - V_{1} W_{3} + V_{1} W_{4} - V_{4} W_{1} + V_{4} W_{3} - V_{3} W_{4} )p_{1}^{2} m_{1} m_{2} m_{4} m_{5} m_{6} )/\sigma^{**} \\ \end{aligned} $$
$$ \begin{aligned} U_{3} /U_{1} & = ((V_{1} W_{2} - V_{2} W_{1} )m_{1} m_{2} m_{6} + (V_{4} W_{1} - V_{1} W_{4} )m_{1} m_{4} m_{6} + (V_{2} W_{4} - V_{4} W_{2} )m_{2} m_{4} m_{6} + (V_{2} W_{1} - V_{1} W_{2} \\ & \quad + V_{1} W_{4} - V_{4} W_{1} + V_{4} W_{2} - V_{2} W_{4} + p_{1} \iota k(V_{2} W_{1} - V_{1} W_{2} + V_{1} W_{4} - V_{4} W_{1} + V_{4} W_{2} - V_{2} W_{4} ))p_{2} m_{1} m_{2} m_{4} \\ & \quad + (V_{1} W_{2} - V_{2} W_{1} + V_{4} W_{1} - V_{1} W_{4} + V_{2} W_{4} - V_{4} W_{2} )p_{1}^{2} m_{1} m_{2} m_{4} m_{5} m_{6} )/\sigma^{**} \\ \end{aligned} $$
$$ \begin{aligned} U_{4} /U_{1} & = ((V_{2} W_{1} - V_{1} W_{2} )m_{1} m_{2} m_{6} + (V_{1} W_{3} - V_{3} W_{1} )m_{1} m_{3} m_{6} + (V_{3} W_{2} - V_{2} W_{3} )m_{2} m_{3} m_{6} + (V_{1} W_{2} - V_{2} W_{1} \\ & \quad + V_{3} W_{1} - V_{1} W_{3} + V_{2} W_{3} - V_{3} W_{2} + p_{1} \iota k(V_{1} W_{2} - V_{2} W_{1} + V_{3} W_{1} - V_{1} W_{3} + V_{2} W_{3} - V_{3} W_{2} ))p_{2} m_{1} m_{2} m_{3} \\ & \quad + (V_{2} W_{1} - V_{1} W_{2} + V_{1} W_{3} - V_{3} W_{1} + V_{3} W_{2} - V_{2} W_{3} )p_{1}^{2} m_{1} m_{2} m_{3} m_{5} m_{6} )/\sigma^{**} \\ \end{aligned} $$
$$ \begin{gathered} U_{5} /U_{1} = p_{1} m_{6} ((V_{1} W_{2} - V_{2} W_{1} + V_{3} W_{1} - V_{1} W_{3} + V_{2} W_{3} - V_{3} W_{2} )m_{1} m_{2} m_{3} + (V_{2} W_{1} - V_{1} W_{2} + V_{1} W_{4} - V_{4} W_{1} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + V_{4} W_{2} - V_{2} W_{4} )m_{1} m_{2} m_{4} + (V_{1} W_{3} - V_{3} W_{1} + V_{4} W_{1} - V_{1} W_{4} + V_{3} W_{4} - V_{4} W_{3} )m_{1} m_{3} m_{4} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + (V_{3} W_{2} - V_{2} W_{3} + V_{2} W_{4} - V_{4} W_{2} + V_{3} W_{4} + V_{4} W_{3} ))/\sigma^{**} \hfill \\ \end{gathered} $$
$$ \begin{aligned} U_{6} /U_{1} & = (p_{1} \iota k + 1)((V_{2} W_{1} - V_{1} W_{2} + V_{1} W_{3} - V_{3} W_{1} + V_{3} W_{2} - V_{2} W_{3} )m_{1} m_{2} m_{3} + (V_{1} W_{2} - V_{2} W_{1} + V_{4} W_{1} - V_{1} W_{4} \\ & \quad + V_{2} W_{4} - V_{4} W_{2} )m_{1} m_{2} m_{4} + (V_{3} W_{1} - V_{1} W_{3} + V_{1} W_{4} - V_{4} W_{1} + V_{4} W_{3} - V_{3} W_{4} )m_{1} m_{3} m_{4} \\ & \quad + (V_{2} W_{3} - V_{3} W_{2} + V_{4} W_{2} - V_{2} W_{4} - V_{3} W_{4} - V_{4} W_{3} ))/\sigma^{**} \\ \end{aligned} $$
$$ \begin{gathered} \sigma^{**} = (V_{2} W_{3} - V_{3} W_{2} )m_{2} m_{3} m_{6} + (V_{4} W_{2} - V_{2} W_{4} )m_{2} m_{4} m_{6} + (V_{3} W_{4} - V_{4} W_{3} )m_{3} m_{4} m_{6} + (V_{3} W_{2} - V_{2} W_{3} + V_{2} W_{4} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - V_{4} W_{2} + V_{3} W_{4} - V_{4} W_{3} + p_{1} \iota k(V_{3} W_{2} - V_{2} W_{3} + V_{2} W_{4} - V_{4} W_{2} + V_{4} W_{3} - V_{3} W_{4} ))p_{2} m_{2} m_{3} m_{4} \,\, \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + (V_{2} W_{3} - V_{3} .W_{2} + V_{4} W_{2} - V_{2} W_{4} + V_{3} W_{4} - V_{4} W_{3} )p_{1}^{2} m_{1} m_{2} m_{3} m_{5} m_{6} \hfill \\ \end{gathered} $$

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Pathania, V., Kumar, R., Gupta, V. et al. Double porous thermoelastic waves in a homogeneous, isotropic solid with inviscid liquid. Arch Appl Mech 93, 1943–1962 (2023). https://doi.org/10.1007/s00419-023-02364-w

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