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Dynamic response of imperfect interfaces on the reflection and transmission of the waves in context of generalised thermo-elasticity

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Abstract

The imperfection at the interface among two materials exists to various reasons such as thin inter phase, chemical action or interface damage. The interfaces such as mechanical, thermal and electrical significantly affect the performance of structural component that are utilized in various fields. The present mathematical study analyzes the reflection and transmission phenomenon of thermo-elastic wave being incident at the distinct types of separating interfaces in an initially stressed rotating structure with two different type half-spaces comprised of piezo-thermo-elastic (PTE) materials. The mathematical model of this study is performed under three-phase-lag model of generalised thermo-elasticity for normal stiffness interface (NSI), transverse stiffness interface (TSI), thermal conductance interface (TCI), electric imperfect interface (EII), and welded contact interface (WCI). The mathematical expressions for the amplitude and energy ratios are derived for the reflected and transmitted qP, qSV, qT and EA waves. The effects of initial stresses, rotation, phase lag and bonding parameters on amplitude ratios are performed, and energy balance law is established to validate the model. The validation of the result has been made with an existing problem for the correctness of the results. Numerical computations have been executed to demonstrate the mathematical findings by the graphical mean.

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Acknowledgements

Authors deeply acknowledge the valuable contribution of Reviewer in light of useful suggestions and valuable comments provided for the improvement of this manuscript. The First author is grateful to Guru Ghasidas Vishwavidyalaya Bilaspur, Chhattisgarh, India, for approving the proposal under "Research Seed Money Grant Scheme" and is also thankful to the Department of Science and Technology, New Delhi, India, for approving FIST program (Ref. No. SR/FST/MS-I/2022/122 dated 19 December 2022).

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Authors and Affiliations

  1. Department of Mathematics, Guru Ghasidas Vishwavidyalaya, Bilaspur, Chhattisgarh, 495009, India

    Brijendra Paswan

  2. Department of Mathematics, Buddha Post-Graduate College, Kushinagar, Uttar Pradesh, 274403, India

    Pooja Singh

  3. Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand, 826004, India

    Sanjeev A. Sahu & Amares Chattopadhyay

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  1. Brijendra Paswan
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  2. Pooja Singh
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  3. Sanjeev A. Sahu
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  4. Amares Chattopadhyay
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Correspondence to Pooja Singh.

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Appendix 1

Appendix 1

$$\begin{aligned} \varepsilon^{\left( i \right)} & = \frac{{T_{0}^{\left( i \right)} \left( {\beta_{11}^{\left( i \right)} } \right)^{2} }}{{\rho^{\left( i \right)} c_{E}^{\left( i \right)} c_{11}^{\left( i \right)} }},\,\,\overline{\beta }^{\left( i \right)} = \frac{{\beta_{33}^{\left( i \right)} }}{{\beta_{11}^{\left( i \right)} }},\,\,\overline{K}^{\left( i \right)} = \frac{{K_{33}^{\left( i \right)} }}{{K_{11}^{\left( i \right)} }},\,\,\overline{\varepsilon }^{\left( i \right)} = \frac{{\varepsilon_{11}^{\left( i \right)} }}{{\varepsilon_{33}^{\left( i \right)} }},\,\,p^{\left( i \right)} = \frac{{p_{3}^{\left( i \right)} c_{11}^{\left( i \right)} }}{{\beta_{11}^{\left( i \right)} e_{33}^{\left( i \right)} }},\,\,\eta^{\left( i \right)} = \frac{{\varepsilon_{33}^{\left( i \right)} c_{11}^{\left( i \right)} }}{{\left( {e_{33}^{\left( i \right)} } \right)^{2} }},\,\,c_{1}^{\left( i \right)} = \frac{{c_{33}^{\left( i \right)} }}{{c_{11}^{\left( i \right)} }}, \\ c_{2}^{\left( i \right)} & = \frac{{c_{44}^{\left( i \right)} }}{{c_{11}^{\left( i \right)} }},c_{3}^{\left( i \right)} = \frac{{c_{13}^{\left( i \right)} + c_{44}^{\left( i \right)} }}{{c_{11}^{\left( i \right)} }},\,\,e_{1}^{\left( i \right)} = \frac{{e_{15}^{\left( i \right)} + e_{31}^{\left( i \right)} }}{{e_{33}^{\left( i \right)} }},e_{2}^{\left( i \right)} = \frac{{e_{15}^{\left( i \right)} }}{{e_{33}^{\left( i \right)} }},\,\,e_{3}^{\left( i \right)} = \frac{{e_{31}^{\left( i \right)} }}{{e_{33}^{\left( i \right)} }},\,\,\overline{K}_{1}^{\left( i \right)} = \frac{{K_{11}^{ * \left( i \right)} }}{{\omega^{ * } K_{11}^{\left( i \right)} }},\,\,\overline{K}_{2}^{\left( i \right)} = \frac{{K_{33}^{ * \left( i \right)} }}{{\omega^{ * } K_{11}^{\left( i \right)} }}, \\ \end{aligned}$$
$$\begin{aligned} q_{11}^{\left( i \right)} & = \left( {1 + \tau_{xx}^{0} } \right) + \left( {c_{2}^{\left( i \right)} + \tau_{zz}^{0} } \right)\left( {m^{\left( i \right)} } \right)^{2} + 2\tau_{xz}^{0} m^{\left( i \right)} - \chi_{1}^{\left( i \right)} ,\,\,q_{12}^{\left( i \right)} = c_{3}^{\left( i \right)} m^{\left( i \right)} - \chi_{2}^{\left( i \right)} ,\,\,q_{13}^{\left( i \right)} = e_{1}^{\left( i \right)} m^{\left( i \right)} ,\,\, \\ q_{14}^{\left( i \right)} & = i\frac{c}{\omega },\,\,q_{21}^{\left( i \right)} = c_{3}^{\left( i \right)} m^{\left( i \right)} + \chi_{2}^{\left( i \right)} ,\,\,q_{22}^{\left( i \right)} = \left( {c_{2}^{\left( i \right)} + \tau_{xx}^{0} } \right) + \left( {c_{1}^{\left( i \right)} + \tau_{zz}^{0} } \right)\left( {m^{\left( i \right)} } \right)^{2} + 2\tau_{xz}^{0} m^{\left( i \right)} - \chi_{1}^{\left( i \right)} , \\ q_{23}^{\left( i \right)} & = e_{2}^{\left( i \right)} + \left( {m^{\left( i \right)} } \right)^{2} ,\,\,q_{24}^{\left( i \right)} = i\overline{\beta }^{\left( i \right)} m^{\left( i \right)} \frac{c}{\omega },\,\,q_{31}^{\left( i \right)} = e_{1}^{\left( i \right)} m^{\left( i \right)} ,\,\,q_{32}^{\left( i \right)} = e_{2}^{\left( i \right)} + \left( {m^{\left( i \right)} } \right)^{2} ,\,\, \\ q_{33}^{\left( i \right)} & = - \eta^{\left( i \right)} \left( {\overline{\varepsilon }^{\left( i \right)} + \left( {m^{\left( i \right)} } \right)^{2} } \right),\,\,q_{34}^{\left( i \right)} = - ip^{\left( i \right)} m^{\left( i \right)} \frac{c}{\omega },\,\,q_{41}^{\left( i \right)} = \varepsilon^{\left( i \right)} ,\,\,q_{42}^{\left( i \right)} = \overline{\beta }^{\left( i \right)} m^{\left( i \right)} \varepsilon^{\left( i \right)} ,\,q_{43}^{\left( i \right)} = - p^{\left( i \right)} m^{\left( i \right)} \varepsilon^{\left( i \right)} , \\ q_{44}^{\left( i \right)} & = - i\frac{c}{\omega }\left( {1 - \frac{i}{{c^{2} \omega }}\tau_{2} \left( {\overline{K}_{1}^{\left( i \right)} + \left( {m^{\left( i \right)} } \right)^{2} \overline{K}_{2}^{\left( i \right)} } \right) - \frac{1}{{c^{2} }}\tau_{1} \left( {1 + \left( {m^{\left( i \right)} } \right)^{2} \overline{K}^{\left( i \right)} } \right)} \right), \\ \chi_{1}^{\left( i \right)} & = c^{2} + \frac{{\Omega^{2} c^{2} }}{{\omega^{2} }},\,\,\chi_{2}^{\left( i \right)} = 2i\frac{\Omega }{\omega }c^{2} ,\,\,\tau_{1} = \frac{{\tau_{T} + \frac{i}{\omega }}}{{\frac{{\tau_{q}^{2} }}{2} + i\frac{{\tau_{q} }}{\omega } - \frac{1}{{\omega^{2} }}}},\,\,\tau_{2} = \frac{{\tau_{\upsilon } + \frac{i}{\omega }}}{{\frac{{\tau_{q}^{2} }}{2} + i\frac{{\tau_{q} }}{\omega } - \frac{1}{{\omega^{2} }}}}, \\ \end{aligned}$$
$$\begin{aligned} H_{1\zeta }^{\left( 1 \right)} & = c_{4}^{\left( 1 \right)} + c_{1}^{\left( 1 \right)} m_{\zeta }^{\left( 1 \right)} W_{\zeta }^{\left( 1 \right)} + m_{\zeta }^{\left( 1 \right)} \Phi_{\zeta }^{\left( 1 \right)} + i\frac{c}{\omega }\overline{\beta }^{\left( 1 \right)} E_{\zeta }^{\left( 1 \right)} ,\,\,H_{2\zeta }^{\left( 1 \right)} = c_{2}^{\left( 1 \right)} m_{\zeta }^{\left( 1 \right)} + c_{2}^{\left( 1 \right)} W_{\zeta }^{\left( 1 \right)} + e_{2}^{\left( 1 \right)} \Phi_{\zeta }^{\left( 1 \right)} , \\ H_{3\zeta }^{\left( 1 \right)} & = e_{3}^{\left( 1 \right)} + m_{\zeta }^{\left( 1 \right)} W_{\zeta }^{\left( 1 \right)} - \eta^{\left( 1 \right)} m_{\zeta }^{\left( 1 \right)} \Phi_{\zeta }^{\left( 1 \right)} - i\frac{c}{\omega }p^{\left( 1 \right)} E_{\zeta }^{\left( 1 \right)} ,\,\,H_{1\xi }^{\left( 1 \right)} = c_{4}^{\left( 1 \right)} + c_{1}^{\left( 1 \right)} m_{\xi }^{\left( 1 \right)} W_{\xi }^{\left( 1 \right)} + m_{\xi }^{\left( 1 \right)} \Phi_{\xi }^{\left( 1 \right)} + i\frac{c}{\omega }\overline{\beta }^{\left( 1 \right)} E_{\xi }^{\left( 1 \right)} , \\ H_{2\xi }^{\left( 1 \right)} & = c_{2}^{\left( 1 \right)} m_{\xi }^{\left( 1 \right)} + c_{2}^{\left( 1 \right)} W_{\xi }^{\left( 1 \right)} + e_{2}^{\left( 1 \right)} \Phi_{\xi }^{\left( 1 \right)} ,\,\,H_{3\xi }^{\left( 1 \right)} = e_{3}^{\left( 1 \right)} + m_{\xi }^{\left( 1 \right)} W_{\xi }^{\left( 1 \right)} - \eta^{\left( 1 \right)} m_{\xi }^{\left( 1 \right)} \Phi_{\xi }^{\left( 1 \right)} - i\frac{c}{\omega }p^{\left( 1 \right)} E_{\xi }^{\left( 1 \right)} , \\ H_{1\xi }^{\left( 2 \right)} & = c_{4}^{\left( 2 \right)} + c_{1}^{\left( 2 \right)} m_{\xi }^{\left( 2 \right)} W_{\xi }^{\left( 2 \right)} + m_{\xi }^{\left( 2 \right)} \Phi_{\xi }^{\left( 2 \right)} + i\frac{c}{\omega }\overline{\beta }^{\left( 2 \right)} E_{\xi }^{\left( 2 \right)} ,\,\,H_{2\xi }^{\left( 2 \right)} = c_{2}^{\left( 2 \right)} m_{\xi }^{\left( 2 \right)} + c_{2}^{\left( 2 \right)} W_{\xi }^{\left( 2 \right)} + e_{2}^{\left( 2 \right)} \Phi_{\xi }^{\left( 2 \right)} , \\ H_{3\xi }^{\left( 2 \right)} & = e_{3}^{\left( 2 \right)} + m_{\xi }^{\left( 2 \right)} W_{\xi }^{\left( 2 \right)} - \eta^{\left( 2 \right)} m_{\xi }^{\left( 2 \right)} \Phi_{\xi }^{\left( 2 \right)} - i\frac{c}{\omega }p^{\left( 2 \right)} E_{\xi }^{\left( 2 \right)} \\ \end{aligned}$$
$$\begin{aligned} L^{\left( i \right)} \left( {m_{\xi }^{\left( i \right)} } \right) & = i\frac{c}{\omega }\left[ {\left( {m^{\left( i \right)} } \right)^{4} \left\{ {c_{3}^{\left( i \right)} \left( {\overline{\beta }^{\left( i \right)} \eta^{\left( i \right)} - p^{\left( i \right)} } \right) - 1 - \left( {c_{1}^{\left( i \right)} + \tau_{zz}^{0} } \right)\eta^{\left( i \right)} + \left( {c_{1}^{\left( i \right)} + \tau_{zz}^{0} } \right)p^{\left( i \right)} e_{1}^{\left( i \right)} + e_{1}^{\left( i \right)} \overline{\beta }^{\left( i \right)} } \right\}} \right. \\ & \quad + \left( {m^{\left( i \right)} } \right)^{3} \left\{ {2\tau_{xz}^{0} p^{\left( i \right)} e_{1}^{\left( i \right)} - \left( {\overline{\beta }^{\left( i \right)} \eta^{\left( i \right)} - p^{\left( i \right)} } \right)\chi_{2}^{\left( i \right)} - 2\tau_{xz}^{0} \eta^{\left( i \right)} } \right\} + \left( {m^{\left( i \right)} } \right)^{2} \left\{ {c_{3}^{\left( i \right)} \left( {\eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} \overline{\beta }^{\left( i \right)} - p^{\left( i \right)} e_{2}^{\left( i \right)} } \right)} \right. \\ & \quad \left. { + \left( {c_{2}^{\left( i \right)} + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right)e_{1}^{\left( i \right)} p^{\left( i \right)} + \overline{\beta }^{\left( i \right)} e_{1}^{\left( i \right)} e_{2}^{\left( i \right)} - 2e_{2}^{\left( i \right)} - \left( {c_{2}^{\left( i \right)} + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right)\eta^{\left( i \right)} - \left( {c_{1}^{\left( i \right)} + \tau_{zz}^{0} } \right)\eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} } \right\} \\ & \quad \left. { + m^{\left( i \right)} \left\{ { - \chi_{2}^{\left( i \right)} \left( {\eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} \overline{\beta }^{\left( i \right)} - p^{\left( i \right)} e_{2}^{\left( i \right)} } \right) - 2\tau_{xz}^{0} \eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} } \right\} - \left\{ {\left( {c_{2}^{\left( i \right)} + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right)\eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} + \left( {e_{2}^{\left( i \right)} } \right)^{2} } \right\}} \right], \\ \end{aligned}$$
$$\begin{aligned} L_{1}^{\left( i \right)} \left( {m_{\xi }^{\left( i \right)} } \right) & = i\frac{c}{\omega }\left[ { - \left( {m^{\left( i \right)} } \right)^{5} \left( {c_{2}^{\left( i \right)} + \tau_{zz}^{0} } \right)\left( {\overline{\beta }^{\left( i \right)} \eta^{\left( i \right)} - p^{\left( i \right)} } \right) - \left( {m^{\left( i \right)} } \right)^{4} 2\tau_{xz}^{0} \left( {\overline{\beta }^{\left( i \right)} \eta^{\left( i \right)} - p^{\left( i \right)} } \right)} \right. \\ & \quad - \left( {m^{\left( i \right)} } \right)^{3} \left\{ {\left( {c_{2}^{\left( i \right)} + \tau_{zz}^{0} } \right)} \right.\left( {\eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} \overline{\beta }^{\left( i \right)} - p^{\left( i \right)} e_{2}^{\left( i \right)} } \right)\\ &\quad - \left( {c_{3}^{\left( i \right)} \eta^{\left( i \right)} + e_{1}^{\left( i \right)} } \right) + e_{1}^{\left( i \right)} c_{3}^{\left( i \right)} p^{\left( i \right)} + \left( {e_{1}^{\left( i \right)} } \right)^{2} \overline{\beta }^{\left( i \right)} + \left( {1 + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right) \\ & \quad \left. {\left( {\overline{\beta }^{\left( i \right)} \eta^{\left( i \right)} - p^{\left( i \right)} } \right)} \right\} - \left( {m^{\left( i \right)} } \right)^{2} \left\{ {e_{1}^{\left( i \right)} \chi_{2}^{\left( i \right)} p^{\left( i \right)} + 2\tau_{xz}^{0} \left( {\eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} \overline{\beta }^{\left( i \right)} - p^{\left( i \right)} e_{2}^{\left( i \right)} } \right) - \chi_{2}^{\left( i \right)} \eta^{\left( i \right)} } \right\} \\ & \quad \left. { - m^{\left( i \right)} \left\{ {\left( {1 + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right)\left( {\eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} \overline{\beta }^{\left( i \right)} - p^{\left( i \right)} e_{2}^{\left( i \right)} } \right) - \left( {c_{3}^{\left( i \right)} \eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} + e_{1}^{\left( i \right)} e_{2}^{\left( i \right)} } \right)} \right\} + \chi_{2}^{\left( i \right)} \eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} } \right], \\ \end{aligned}$$
$$\begin{aligned} L_{2}^{\left( i \right)} \left( {m_{\xi }^{\left( i \right)} } \right) & = i\frac{c}{\omega }\left[ { - \left( {m^{\left( i \right)} } \right)^{5} \left( {c_{2}^{\left( i \right)} + \tau_{zz}^{0} } \right)\left\{ {\left( {c_{1}^{\left( i \right)} + \tau_{zz}^{0} } \right)p^{\left( i \right)} + \overline{\beta }^{\left( i \right)} } \right\} - \left( {m^{\left( i \right)} } \right)^{4} \left\{ {2\tau_{xz}^{0} p^{\left( i \right)} \left( {c_{2}^{\left( i \right)} + \tau_{zz}^{0} } \right)} \right.} \right. \\ & \quad \left. { + 2\tau_{xz}^{0} \left( {\left( {c_{1}^{\left( i \right)} + \tau_{zz}^{0} } \right)p^{\left( i \right)} + \overline{\beta }^{\left( i \right)} } \right)} \right\} + \left( {m^{\left( i \right)} } \right)^{3} \left\{ {\left( {p^{\left( i \right)} c_{3}^{\left( i \right)} + \overline{\beta }^{\left( i \right)} e_{1}^{\left( i \right)} } \right)c_{3}^{\left( i \right)} } \right.\\ &\quad - \left( {1 + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right)\left( {\left( {c_{1}^{\left( i \right)} + \tau_{zz}^{0} } \right)p^{\left( i \right)} + \overline{\beta }^{\left( i \right)} } \right) \\ & \quad \left. { - \left( {c_{2}^{\left( i \right)} + \tau_{zz}^{0} } \right)\left( {\left( {c_{2}^{\left( i \right)} + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right)p^{\left( i \right)} + \overline{\beta }^{\left( i \right)} e_{2}^{\left( i \right)} } \right) - 4\left( {\tau_{xz}^{0} } \right)^{2} p^{\left( i \right)} - \left( {\left( {c_{1}^{\left( i \right)} + \tau_{zz}^{0} } \right)e_{1}^{\left( i \right)} - c_{3}^{\left( i \right)} } \right)} \right\} \\ & \quad + \left( {m^{\left( i \right)} } \right)^{2} \left\{ {p^{\left( i \right)} c_{3}^{\left( i \right)} \chi_{2}^{\left( i \right)} } \right. - \chi_{2}^{\left( i \right)} \left( {p^{\left( i \right)} c_{3}^{\left( i \right)} + \overline{\beta }^{\left( i \right)} e_{1}^{\left( i \right)} } \right) \\ &\quad- 2\tau_{xz}^{0} p^{\left( i \right)} \left( {1 + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right) - 2\tau_{xz}^{0} \left( {\left( {c_{2}^{\left( i \right)} + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right)p^{\left( i \right)} + \overline{\beta }^{\left( i \right)} e_{2}^{\left( i \right)} } \right) \\ & \quad \left. { - \left( {2\tau_{xz}^{0} e_{1}^{\left( i \right)} - \chi_{2}^{\left( i \right)} } \right)} \right\} + m^{\left( i \right)} \left\{ { - p^{\left( i \right)} \left( {\chi_{2}^{\left( i \right)} } \right)^{2} - \left( {1 + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right)\left( {\left( {c_{2}^{\left( i \right)} + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right)p^{\left( i \right)} + \overline{\beta }^{\left( i \right)} e_{2}^{\left( i \right)} } \right)} \right. \\ & \quad \left. { - \left. {\left( {c_{2}^{\left( i \right)} + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right)e_{1}^{\left( i \right)} - e_{2}^{\left( i \right)} c_{3}^{\left( i \right)} } \right\} + e_{2}^{\left( i \right)} \chi_{2}^{\left( i \right)} } \right], \\ \end{aligned}$$
$$\begin{aligned} L_{3}^{\left( i \right)} \left( {m_{\xi }^{\left( i \right)} } \right) & = \left( {m^{\left( i \right)} } \right)^{6} \left\{ {\left( {c_{2}^{\left( i \right)} + \tau_{zz}^{0} } \right)\left( {1 + \eta^{\left( i \right)} \left( {c_{1}^{\left( i \right)} + \tau_{zz}^{0} } \right)} \right)} \right\} + \left( {m^{\left( i \right)} } \right)^{5} \left\{ {2\tau_{xz}^{0} \eta^{\left( i \right)} \left( {c_{2}^{\left( i \right)} + \tau_{zz}^{0} } \right)} \right. \\ & \quad + 2\tau_{xz}^{0} \left. {\left( {1 + \eta^{\left( i \right)} \left( {c_{1}^{\left( i \right)} + \tau_{zz}^{0} } \right)} \right)} \right\} + \left( {m^{\left( i \right)} } \right)^{4} \\ &\quad\left\{ { - c_{3}^{\left( i \right)} \left( {\eta^{\left( i \right)} c_{3}^{\left( i \right)} + e_{1}^{\left( i \right)} } \right) + e_{1}^{\left( i \right)} \left( {e_{1}^{\left( i \right)} \left( {c_{1}^{\left( i \right)} + \tau_{zz}^{0} } \right) - c_{3}^{\left( i \right)} } \right) + \left( {1 + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right)} \right. \\ & \quad \times \left( {1 + \eta^{\left( i \right)} \left( {c_{1}^{\left( i \right)} + \tau_{zz}^{0} } \right)} \right) + \left( {c_{2}^{\left( i \right)} + \tau_{zz}^{0} } \right)\left( \eta^{\left( i \right)} \left( {c_{2}^{\left( i \right)} + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right) \right.\\ &\left. \left.\quad+ \eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} \left( {c_{1}^{\left( i \right)} + \tau_{zz}^{0} } \right) + 2e_{2}^{\left( i \right)} \right) + 4\left( {\tau_{xz}^{0} } \right)^{2} \eta^{\left( i \right)} \right\} \\ & \quad + \left( {m^{\left( i \right)} } \right)^{3} \left\{ {\chi_{2}^{\left( i \right)} \left( {\eta^{\left( i \right)} c_{3}^{\left( i \right)} + e_{1}^{\left( i \right)} } \right) - c_{3}^{\left( i \right)} \eta^{\left( i \right)} \chi_{2}^{\left( i \right)} + \left( {2\tau_{xz}^{0} e_{1}^{\left( i \right)} - \chi_{2}^{\left( i \right)} } \right)e_{1}^{\left( i \right)} + 2\tau_{xz}^{0} \eta^{\left( i \right)} \left( {1 + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right)} \right. \\ & \quad \left. { + 2\tau_{xz}^{0} \eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} \left( {c_{2}^{\left( i \right)} + \tau_{zz}^{0} } \right) + 2\tau_{xz}^{0} \left( {\eta^{\left( i \right)} \left( {c_{2}^{\left( i \right)} + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right) + \eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} \left( {c_{1}^{\left( i \right)} + \tau_{zz}^{0} } \right) + 2e_{2}^{\left( i \right)} } \right)} \right\} + \left( {m^{\left( i \right)} } \right)^{2} \\ \end{aligned}$$
$$\begin{aligned} & \quad \left\{ {\eta^{\left( i \right)} \left( {\chi_{2}^{\left( i \right)} } \right)^{2} - \left( {e_{1}^{\left( i \right)} e_{2}^{\left( i \right)} + \eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} c_{3}^{\left( i \right)} } \right)c_{3}^{\left( i \right)} + e_{1}^{\left( i \right)} \left( {e_{1}^{\left( i \right)} \left( {c_{2}^{\left( i \right)} + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right) - e_{2}^{\left( i \right)} c_{3}^{\left( i \right)} } \right) + \left( {1 + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right)} \right. \\ & \quad \times \left( {\eta^{\left( i \right)} \left( {c_{2}^{\left( i \right)} + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right) + \eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} \left( {c_{1}^{\left( i \right)} + \tau_{zz}^{0} } \right) + 2e_{2}^{\left( i \right)} } \right) + \left( {c_{2}^{\left( i \right)} + \tau_{zz}^{0} } \right)\left( {\eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} \left( {c_{2}^{\left( i \right)} + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right) + \left( {e_{2}^{\left( i \right)} } \right)^{2} } \right) \\ & \quad \left. { + 4\left( {\tau_{xz}^{0} } \right)^{2} \eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} } \right\} + m^{\left( i \right)} \left\{ {\left( {e_{1}^{\left( i \right)} e_{2}^{\left( i \right)} + \eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} c_{3}^{\left( i \right)} } \right)\chi_{2}^{\left( i \right)} - \eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} c_{3}^{\left( i \right)} \chi_{2}^{\left( i \right)} - e_{1}^{\left( i \right)} e_{2}^{\left( i \right)} \chi_{2}^{\left( i \right)} } \right. \\ & \quad \left. { + 2\tau_{xz}^{0} \eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} \left( {1 + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right) + 2\tau_{xz}^{0} \left( {\eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} \left( {c_{2}^{\left( i \right)} + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right) + \left( {e_{2}^{\left( i \right)} } \right)^{2} } \right)} \right\} + \eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} \left( {\chi_{2}^{\left( i \right)} } \right)^{2} \\ & \quad + \left( {1 + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right)\left( {\eta^{\left( i \right)} \overline{\varepsilon }^{\left( i \right)} \left( {c_{2}^{\left( i \right)} + \tau_{xx}^{0} - \chi_{1}^{\left( i \right)} } \right) + \left( {e_{2}^{\left( i \right)} } \right)^{2} } \right), \\ \end{aligned}$$
$$\begin{aligned} G_{11} & = - k_{n} W_{1}^{\left( 1 \right)} ,\,\,G_{12} = - k_{n} W_{3}^{\left( 1 \right)} ,\,\,G_{13} = - k_{n} W_{5}^{\left( 1 \right)} ,\,\,G_{14} = - k_{n} W_{7}^{\left( 1 \right)} , \\ G_{15} & = {\text{i}}k\left( {H_{12}^{\left( 2 \right)} + \tau_{zx}^{0} W_{2}^{\left( 2 \right)} + \tau_{zz}^{0} m_{2}^{\left( 2 \right)} W_{2}^{\left( 2 \right)} } \right) + k_{n} W_{2}^{\left( 2 \right)} , \\ G_{16} & = {\text{i}}k\left( {H_{14}^{\left( 2 \right)} + \tau_{zx}^{0} W_{4}^{\left( 2 \right)} + \tau_{zz}^{0} m_{4}^{\left( 2 \right)} W_{4}^{\left( 2 \right)} } \right) + k_{n} W_{4}^{\left( 2 \right)} , \\ G_{17} & = {\text{i}}k\left( {H_{16}^{\left( 2 \right)} + \tau_{zx}^{0} W_{6}^{\left( 2 \right)} + \tau_{zz}^{0} m_{6}^{\left( 2 \right)} W_{6}^{\left( 2 \right)} } \right) + k_{n} W_{6}^{\left( 2 \right)} , \\ G_{18} & = {\text{i}}k\left( {H_{18}^{\left( 2 \right)} + \tau_{zx}^{0} W_{8}^{\left( 2 \right)} + \tau_{zz}^{0} m_{8}^{\left( 2 \right)} W_{8}^{\left( 2 \right)} } \right) + k_{n} W_{8}^{\left( 2 \right)} , \\ G_{21} & = - k_{t} ,\,\,G_{22} = - k_{t} ,\,\,G_{23} = - k_{t} ,\,\,G_{24} = - k_{t} ,\,\,G_{25} = {\text{i}}k\left( {H_{22}^{\left( 2 \right)} + \tau_{zx}^{0} + \tau_{zz}^{0} m_{2}^{\left( 2 \right)} } \right) + k_{t} , \\ \end{aligned}$$
$$\begin{aligned} G_{26} & = {\text{i}}k\left( {H_{24}^{\left( 2 \right)} + \tau_{zx}^{0} + \tau_{zz}^{0} m_{4}^{\left( 2 \right)} } \right) + k_{t} ,\,\,G_{27} = {\text{i}}k\left( {H_{26}^{\left( 2 \right)} + \tau_{zx}^{0} + \tau_{zz}^{0} m_{6}^{\left( 2 \right)} } \right) + k_{t} , \\ G_{28} & = {\text{i}}k\left( {H_{28}^{\left( 2 \right)} + \tau_{zx}^{0} + \tau_{zz}^{0} m_{8}^{\left( 2 \right)} } \right) + k_{t} ,\,\,G_{31} = - \left( {H_{11}^{\left( 1 \right)} + \tau_{zx}^{0} W_{1}^{\left( 1 \right)} + \tau_{zz}^{0} m_{1}^{\left( 1 \right)} W_{1}^{\left( 1 \right)} } \right), \\ G_{32} & = - \left( {H_{13}^{\left( 1 \right)} + \tau_{zx}^{0} W_{3}^{\left( 1 \right)} + \tau_{zz}^{0} m_{3}^{\left( 1 \right)} W_{3}^{\left( 1 \right)} } \right),\,\,G_{33} = - \left( {H_{15}^{\left( 1 \right)} + \tau_{zx}^{0} W_{5}^{\left( 1 \right)} + \tau_{zz}^{0} m_{5}^{\left( 1 \right)} W_{5}^{\left( 1 \right)} } \right), \\ G_{34} & = - \left( {H_{17}^{\left( 1 \right)} + \tau_{zx}^{0} W_{7}^{\left( 1 \right)} + \tau_{zz}^{0} m_{7}^{\left( 1 \right)} W_{7}^{\left( 1 \right)} } \right),\,\,G_{35} = \left( {H_{12}^{\left( 2 \right)} + \tau_{zx}^{0} W_{2}^{\left( 2 \right)} + \tau_{zz}^{0} m_{2}^{\left( 2 \right)} W_{2}^{\left( 2 \right)} } \right), \\ G_{36} & = \left( {H_{14}^{\left( 2 \right)} + \tau_{zx}^{0} W_{4}^{\left( 2 \right)} + \tau_{zz}^{0} m_{4}^{\left( 2 \right)} W_{4}^{\left( 2 \right)} } \right),\,\,G_{37} = \left( {H_{16}^{\left( 2 \right)} + \tau_{zx}^{0} W_{6}^{\left( 2 \right)} + \tau_{zz}^{0} m_{6}^{\left( 2 \right)} W_{6}^{\left( 2 \right)} } \right), \\ G_{38} & = \left( {H_{18}^{\left( 2 \right)} + \tau_{zx}^{0} W_{8}^{\left( 2 \right)} + \tau_{zz}^{0} m_{8}^{\left( 2 \right)} W_{8}^{\left( 2 \right)} } \right),\,\,G_{30} = \left( {H_{1\zeta }^{\left( 1 \right)} + \tau_{zx}^{0} W_{\zeta }^{\left( 1 \right)} + \tau_{zz}^{0} m_{\zeta }^{\left( 1 \right)} W_{\zeta }^{\left( 1 \right)} } \right), \\ \end{aligned}$$
$$\begin{aligned} G_{41} & = - \left( {H_{21}^{\left( 1 \right)} + \tau_{zx}^{0} + \tau_{zz}^{0} m_{1}^{\left( 1 \right)} } \right),\,\,G_{42} = - \left( {H_{23}^{\left( 1 \right)} + \tau_{zx}^{0} + \tau_{zz}^{0} m_{3}^{\left( 1 \right)} } \right), \\ G_{43} & = - \left( {H_{25}^{\left( 1 \right)} + \tau_{zx}^{0} + \tau_{zz}^{0} m_{5}^{\left( 1 \right)} } \right),\,\,G_{44} = - \left( {H_{27}^{\left( 1 \right)} + \tau_{zx}^{0} + \tau_{zz}^{0} m_{7}^{\left( 1 \right)} } \right), \\ G_{45} & = \left( {H_{22}^{\left( 2 \right)} + \tau_{zx}^{0} + \tau_{zz}^{0} m_{2}^{\left( 2 \right)} } \right),\,\,G_{46} = \left( {H_{24}^{\left( 2 \right)} + \tau_{zx}^{0} + \tau_{zz}^{0} m_{4}^{\left( 2 \right)} } \right), \\ G_{47} & = \left( {H_{26}^{\left( 2 \right)} + \tau_{zx}^{0} + \tau_{zz}^{0} m_{6}^{\left( 2 \right)} } \right),\,\,G_{48} = \left( {H_{28}^{\left( 2 \right)} + \tau_{zx}^{0} + \tau_{zz}^{0} m_{8}^{\left( 2 \right)} } \right), \\ G_{40} & = \left( {H_{2\zeta }^{\left( 1 \right)} + \tau_{zx}^{0} + \tau_{zz}^{0} m_{\zeta }^{\left( 1 \right)} } \right),\,\,G_{51} = - k_{e} \Phi_{1}^{\left( 1 \right)} ,\,\,G_{52} = - k_{e} \Phi_{3}^{\left( 1 \right)} ,\,\,G_{53} = - k_{e} \Phi_{5}^{\left( 1 \right)} , \\ G_{54} & = - k_{e} \Phi_{7}^{\left( 1 \right)} ,\,\,G_{55} = {\text{i}}kH_{32}^{\left( 2 \right)} + k_{e} \Phi_{2}^{\left( 2 \right)} ,\,\,G_{56} = {\text{i}}kH_{34}^{\left( 2 \right)} + k_{e} \Phi_{4}^{\left( 2 \right)} ,\,\,G_{57} = {\text{i}}kH_{36}^{\left( 2 \right)} + k_{e} \Phi_{6}^{\left( 2 \right)} , \\ \end{aligned}$$
$$\begin{aligned} G_{58} & = {\text{i}}kH_{38}^{\left( 2 \right)} + k_{e} \Phi_{8}^{\left( 2 \right)} ,\,\,G_{61} = - H_{31}^{\left( 1 \right)} ,\,\,G_{62} = - H_{33}^{\left( 1 \right)} ,\,\,G_{63} = - H_{35}^{\left( 1 \right)} ,\,\,G_{64} = - H_{37}^{\left( 1 \right)} , \\ G_{65} & = H_{32}^{\left( 2 \right)} ,\,\,G_{66} = H_{34}^{\left( 2 \right)} ,\,\,G_{67} = H_{36}^{\left( 2 \right)} ,\,\,G_{68} = H_{38}^{\left( 2 \right)} ,\,\,G_{71} = - k_{c} E_{1}^{\left( 1 \right)} ,\,\,G_{72} = - k_{c} E_{3}^{\left( 1 \right)} , \\ G_{73} & = - k_{c} E_{5}^{\left( 1 \right)} ,\,\,G_{74} = - k_{c} E_{7}^{\left( 1 \right)} ,\,\,G_{75} = \left( {{\text{i}}k\psi_{2} m_{2}^{\left( 2 \right)} + k_{c} } \right)E_{2}^{\left( 2 \right)} ,\,\,G_{76} = \left( {{\text{i}}k\psi_{2} m_{4}^{\left( 2 \right)} + k_{c} } \right)E_{4}^{\left( 2 \right)} , \\ G_{77} & = \left( {{\text{i}}k\psi_{2} m_{6}^{\left( 2 \right)} + k_{c} } \right)E_{6}^{\left( 2 \right)} ,\,\,G_{78} = \left( {{\text{i}}k\psi_{2} m_{8}^{\left( 2 \right)} + k_{c} } \right)E_{8}^{\left( 2 \right)} ,\,\,G_{81} = - m_{1}^{\left( 1 \right)} E_{1}^{\left( 1 \right)} , \\ G_{82} & = - m_{3}^{\left( 1 \right)} E_{3}^{\left( 1 \right)} ,\,\,G_{83} = - m_{5}^{\left( 1 \right)} E_{5}^{\left( 1 \right)} ,\,\,G_{84} = - m_{7}^{\left( 1 \right)} E_{7}^{\left( 1 \right)} ,\,\,G_{85} = \frac{{\psi_{2} }}{{\psi_{1} }}m_{2}^{\left( 2 \right)} E_{2}^{\left( 2 \right)} , \\ G_{86} & = \frac{{\psi_{2} }}{{\psi_{1} }}m_{4}^{\left( 2 \right)} E_{4}^{\left( 2 \right)} ,\,\,G_{87} = \frac{{\psi_{2} }}{{\psi_{1} }}m_{6}^{\left( 2 \right)} E_{6}^{\left( 2 \right)} ,\,\,G_{88} = \frac{{\psi_{2} }}{{\psi_{1} }}m_{8}^{\left( 2 \right)} E_{8}^{\left( 2 \right)} . \\ \end{aligned}$$

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Paswan, B., Singh, P., Sahu, S.A. et al. Dynamic response of imperfect interfaces on the reflection and transmission of the waves in context of generalised thermo-elasticity. Acta Mech 234, 6041–6068 (2023). https://doi.org/10.1007/s00707-023-03695-z

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  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-023-03695-z

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