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Energy analysis at the boundary interface of elastic and piezothermoelastic half-spaces

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Abstract

The present investigation analyzed the energy ratios at the interface of elastic half-space (ES) and nonlocal orthotropic piezothermoelastic half-space with dual-phase lag memory-dependent derivatives (NPS) in the context of different temperature models: hyperbolic two-temperature, classical two-temperature, and without-two-temperature in the presence and absence of memory-dependent derivatives and nonlocal effects. Plane waves involving P or SV type propagating through ES and striking at the interface result in two waves reflected and four waves transmitted. Amplitude ratios are determined in closed form, and these are used to obtain the energy ratios of various reflected and transmitted waves. The law of conservation of energy is justified.

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Acknowledgements

The author (Mr. Vipin Gupta) is thankful and acknowledges to Council of Scientific and Industrial Research (CSIR), New Delhi (File. No. 09/1293(0003)/2019-EMR-I) for the financial support to carry out this research work.

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Authors and Affiliations

  1. Department of Mathematics, Kurukshetra University, Kurukshetra, Haryana, 136119, India

    Rajneesh Kumar

  2. Department of Mathematics, Indira Gandhi University, Meerpur, Rewari, Haryana, 123401, India

    M. S. Barak, Rajesh Kumar & Vipin Gupta

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  1. M. S. Barak
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  2. Rajesh Kumar
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Correspondence to Vipin Gupta.

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Appendices

Appendix

$$ \begin{aligned} f_{{11}} = & m_{{12}} (m_{{22}} (m_{{34}} m_{{44}} + m_{{32}} m_{{46}} ) + m_{{31}} (m_{{24}} m_{{46}} - m_{{26}} m_{{44}} ) \\ & + \;m_{{42}} (m_{{24}} m_{{34}} + m_{{26}} m_{{32}} )) \\ \end{aligned} $$
$$ \begin{aligned} f_{12} = & m_{1} [m_{14} (m_{22} m_{34} - m_{26} m_{31} - m_{31} m_{46} - m_{34} m_{42} ) - m_{13} (m_{32} m_{46} + m_{24} m_{34} + m_{26} m_{32} - m_{34} m_{44} ) \\ & + \;m_{16} (m_{22} m_{32} + m_{24} m_{31} + m_{31} m_{44} - m_{32} m_{42} )] + m_{11} [m_{22} (m_{34} m_{44} + m_{32} m_{46} ) + m_{24} (m_{34} m_{42} + m_{31} m_{46} ) \\ & + m_{26} (m_{32} m_{42} - m_{31} m_{44} )] + m_{12} [m_{21} (m_{34} m_{44} + m_{32} m_{46} ) + m_{22} (m_{32} m_{45} + + m_{34} m_{43} + m_{33} m_{44} ) \\ & + m_{23} (m_{34} m_{42} + m_{31} m_{46} ) + m_{24} (m_{33} m_{42} + m_{31} m_{45} + m_{34} m_{41} ) + m_{25} (m_{32} m_{42} - m_{31} m_{44} ) + m_{26} (m_{32} m_{41} \\ & - m_{31} m_{43} )] + m_{1} [m_{14} (m_{22} m_{46} + m_{26} m_{42} ) + m_{13} (m_{26} m_{44} - m_{24} m_{46} ) - m_{16} (m_{22} m_{44} + m_{24} m_{42} )] \\ \end{aligned} $$
$$ \begin{aligned} f_{{13}} = & m_{1} [m_{{14}} (m_{{21}} m_{{46}} + m_{{22}} m_{{45}} + m_{{21}} m_{{34}} + m_{{22}} m_{{33}} \\ & + m_{{25}} m_{{42}} + m_{{26}} m_{{41}} - m_{{34}} m_{{41}} - m_{{31}} (m_{{25}} + m_{{42}} \\ & + \;m_{{45}} )) + m_{{15}} (m_{{22}} m_{{32}} + m_{{24}} m_{{31}} - m_{{22}} m_{{44}} - m_{{32}} m_{{42}} \\ & - m_{{24}} m_{{42}} m_{{31}} m_{{44}} ) + m_{{13}} (m_{{25}} m_{{44}} + m_{{26}} m_{{43}} \\ & + \;m_{{23}} m_{{34}} - m_{{32}} m_{{45}} - m_{{33}} m_{{44}} - m_{{34}} m_{{43}} - m_{{23}} m_{{46}} \\ & - \;m_{{24}} m_{{45}} - m_{{24}} m_{{33}} - m_{{25}} m_{{32}} ) + m_{{16}} (m_{{21}} m_{{32}} \\ & + \;m_{{23}} m_{{31}} - m_{{32}} m_{{41}} + m_{{31}} m_{{43}} - m_{{23}} m_{{42}} - m_{{24}} m_{{41}} \\ & - \;m_{{21}} m_{{44}} - m_{{22}} m_{{43}} )] + m_{{11}} [m_{{31}} (m_{{23}} m_{{46}} + m_{{24}} m_{{45}} \\ & - \;m_{{25}} m_{{44}} - m_{{26}} m_{{43}} ) + m_{{32}} (m_{{21}} m_{{46}} + m_{{25}} m_{{42}} \\ & + \;m_{{26}} m_{{41}} + m_{{22}} m_{{45}} ) + m_{{33}} (m_{{22}} m_{{44}} + m_{{24}} m_{{42}} ) \\ & + \;m_{{34}} (m_{{23}} m_{{42}} + m_{{24}} m_{{41}} + m_{{21}} m_{{44}} + m_{{22}} m_{{43}} )] \\ & + \;m_{{12}} [m_{{21}} ( + m_{{33}} m_{{44}} m_{{32}} m_{{45}} + m_{{34}} m_{{43}} ) + m_{{23}} (m_{{33}} m_{{42}} \\ & + \;m_{{34}} m_{{41}} + m_{{31}} m_{{45}} ) + m_{{25}} (m_{{32}} m_{{41}} - m_{{31}} m_{{43}} ) \\ & + \;m_{{33}} (m_{{24}} m_{{41}} + m_{{22}} m_{{43}} )] \\ \end{aligned} $$
$$ \begin{aligned} f_{14} = & m_{1} [m_{14} m_{33} (m_{21} - m_{41} ) + m_{13} (m_{25} m_{43} - m_{23} m_{45} ) - m_{13} m_{33} (m_{23} + m_{43} ) + m_{15} (m_{21} m_{32} + m_{23} m_{31} \\ & - \;m_{32} m_{41} + m_{31} m_{43} - m_{23} m_{42} - m_{24} m_{41} - m_{21} m_{44} - m_{22} m_{43} ) - m_{1} (m_{21} m_{43} + m_{23} m_{41} )] + m_{11} [m_{21} (m_{32} m_{45} \\ & + \;m_{33} m_{44} + m_{34} m_{43} ) + m_{23} (m_{34} m_{41} + m_{31} m_{45} + m_{33} m_{42} ) + m_{25} (m_{32} m_{41} - m_{31} m_{43} ) + m_{33} (m_{22} m_{43} \\ & + \;m_{24} m_{41} )] + m_{12} m_{33} (m_{21} m_{43} + m_{23} m_{41} ) + m_{14} m_{1} (m_{21} m_{45} + m_{25} m_{41} ) \\ \end{aligned} $$
$$ f_{15} = m_{11} m_{33} (m_{21} m_{43} + m_{23} m_{41} ) - m_{15} m_{1} (m_{21} m_{43} + m_{23} m_{41} ) $$

\(F_{1i} = - \left[ {\frac{{c_{13} }}{c} + \left( {c_{33} W_{i} + \frac{{\eta_{33} \beta_{11} T_{0} }}{{\eta_{31} }}\Phi_{i} } \right)q_{i} - \frac{{\beta_{33} c_{11} }}{{\beta_{11} \iota \omega }}p_{i} \Theta_{i} } \right]\), \(F_{2i} = - \frac{{c_{55} W_{i} }}{c} - c_{55} q_{i} - \frac{{\eta_{15} \beta_{11} T_{0} \Phi_{i} }}{{c\eta_{31} }}\) \(F_{3i} = 1\), \(F_{4i} = W_{i}\), \(F_{5i} = p_{i} q_{i} \Theta_{i}\), \(F_{6i} = - \frac{{\eta_{31} }}{c} - \eta_{33} q_{i} W_{i} + \frac{{\varepsilon_{33} T_{0} \beta_{11} }}{{\eta_{31} }}q_{i} \Phi_{i} - \frac{{\tau_{3} c_{11} }}{{\iota \omega \beta_{11} }}p_{i} \Theta_{i}\)\(p_{i} = \omega^{2} \frac{{\tau_{11}^{*} }}{{c^{2} }} + \tau_{33}^{*} \omega^{2} q_{i}^{2} - 1\;\;\left( {i = 1,\;2,\;3,\;4} \right)\).

For incident P wave

\(F_{15} = - \iota \omega \rho^{e} c_{1}^{2} \left[ {1 - \frac{{2\alpha_{2}^{{e^{2} }} \sin^{2} \theta_{0} }}{{\alpha_{1}^{{e^{2} }} }}} \right]\), \(F_{16} = \iota \omega \rho^{e} c_{1}^{2} \sin 2\theta_{2}\), \(F_{25} = \frac{{\iota \omega \alpha_{2}^{{e^{2} }} \rho^{e} c_{1}^{2} \sin 2\theta_{0} }}{{\alpha_{1}^{{e^{2} }} }}\).

\(F_{26} = \iota \omega \rho^{e} c_{1}^{2} \cos 2\theta_{2}\), \(F_{35} = \frac{{\iota \omega \sin \theta_{0} }}{{\alpha_{1}^{*} }}\), \(F_{36} = \frac{{\iota \omega \cos \theta_{2} }}{{\alpha_{2}^{*} }}\), \(F_{45} = - \frac{{\iota \omega \cos \theta_{0} }}{{\alpha_{1}^{*} }}\), \(F_{46} = \frac{{\iota \omega \sin \theta_{2} }}{{\alpha_{2}^{*} }}\).

\(X_{i} = \frac{{L_{i} }}{{A_{0}^{e} }},\,(i = 1,\;2,\;3,\;4)\), \(X_{5}^{e} = \frac{{A_{1}^{e} }}{{A_{0}^{e} }}\), \(X_{6}^{e} = \frac{{B_{1}^{e} }}{{A_{0}^{e} }}\), \(S_{1} = - F_{15}\), \(S_{2} = F_{25}\), \(S_{3} = - F_{35}\), \(S_{4} = F_{45}\).

For incident SV wave

\(F_{15} = - \iota \omega \rho^{e} c_{1}^{2} \left[ {1 - \frac{{2\alpha_{2}^{{e^{2} }} \sin^{2} \theta_{1} }}{{\alpha_{1}^{{e^{2} }} }}} \right]\), \(F_{16} = \iota \omega \rho^{e} c_{1}^{2} \sin 2\theta_{0}\), \(F_{25} = \frac{{\iota \omega \alpha_{2}^{{e^{2} }} \rho^{e} c_{1}^{2} \sin 2\theta_{1} }}{{\alpha_{1}^{{e^{2} }} }}\).

\(F_{26} = \iota \omega \rho^{e} c_{1}^{2} \cos 2\theta_{0}\), \(F_{35} = \frac{{\iota \omega \sin \theta_{1} }}{{\alpha_{1}^{*} }}\), \(F_{36} = \frac{{\iota \omega \cos \theta_{0} }}{{\alpha_{2}^{*} }}\), \(F_{45} = - \frac{{\iota \omega \cos \theta_{1} }}{{\alpha_{1}^{*} }}\), \(F_{46} = \frac{{\iota \omega \sin \theta_{0} }}{{\alpha_{2}^{*} }}\).

\(S_{1} = F_{16}\), \(S_{2} = - F_{26}\), \(S_{3} = F_{36}\), \(S_{4} = - F_{46}\).

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Barak, M.S., Kumar, R., Kumar, R. et al. Energy analysis at the boundary interface of elastic and piezothermoelastic half-spaces. Indian J Phys 97, 2369–2383 (2023). https://doi.org/10.1007/s12648-022-02568-w

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