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Effect of two-temperature on the energy ratios at the elastic–piezothermoelastic interface with phase lags

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Abstract

The phenomenon of reflection and transmission of plane waves between two half-spaces elastic and orthotropic piezothermoelastic with three-phase-lag and two-temperature is investigated. Amplitude ratios of the waves are obtained and used to compute energy ratios. The trend of energy ratios of different reflected and transmitted waves with angle of incidence is displayed graphically depicting the effect of two-temperature parameters with the consideration of three-phase-lag theory. The conservation of energy is justified. A particular case is also deduced from the study.

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All data generated or analysed during this study are included in this article.

Abbreviations

OPTHS:

Orthotropic piezothermoelastic half-space

ES:

Elastic half-space

WTT:

Energy ratio curves for orthotropic piezothermoelastic with phase lags and two-temperature

WOTT:

Energy ratio curves for orthotropic piezothermoelastic with phase lags and without two-temperature

\(c_{ijkl}\) :

Elastic parameters

\(\alpha_{ij}\) :

Tensor of thermal moduli

\(\rho\) :

Mass density

\(C_{e}\) :

Specific heat at constant strain

\(T_{0} ,\,T\) :

Reference temperature, absolute temperature

\(\sigma_{ij} {,}\varepsilon_{ij}\) :

Components of the stress and strain tensor in OPTHS

\(E_{i}\) :

Electric field intensity

\(D_{i}\) :

Electric displacement

\(\phi\) :

Electric potential

\(\tau_{i}\) :

Pyroelectric constants

\(e_{ijk} ,\xi_{ij}\) :

Tensors of piezothermal moduli

\(u_{i} ,\,u_{i}^{e}\) :

Components of displacement vectors in OPTHS and ES, respectively

\(\,\tau_{t} ,\,\tau_{\nu } ,\tau_{q}\) :

Phase lags of temperature gradient, thermal displacement gradient and heat flux, respectively, such that \(0 \le \tau_{\nu } < \tau_{t} < \tau_{q}\)

\(K_{ij} ,K_{ij}^{*}\) :

Components of thermal conductivity and material constant

\(\varphi\) :

Conductive temperature of the medium

\(\nu\) :

Thermal displacement

\(\rho^{e} ,\lambda^{e} ,\mu^{e}\) :

Mass density and the Lame’s constants in elastic media, respectively

\(\sigma_{ij}^{e} ,\,\,\varepsilon_{ij}^{e}\) :

Components of the stress and strain tensor in ES

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Funding

No funding was received for conducting this study.

Author information

Authors and Affiliations

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

    Rajneesh Kumar & Poonam Sharma

Authors
  1. Rajneesh Kumar
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  2. Poonam Sharma
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Correspondence to Poonam Sharma.

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Software application is stated in this article.

Appendices

Appendix A

$$ \begin{aligned} t_{11} & = T_{0} (i\omega - \omega^{2} \tau_{0} ),t_{12} = \frac{{\alpha_{11} T_{0} }}{{e_{31} }},t_{13} = - \frac{{i\rho c_{1}^{2} }}{\omega c},t_{14} = 1 + \omega^{2} \frac{{a_{11} }}{{c^{2} }},\\ t_{15} & = \omega^{2} a_{33} ,t_{16} = - \frac{{i\rho c_{1}^{2} \alpha_{33} }}{{\omega \alpha_{11} }}, t_{17} = - \frac{\rho }{{\alpha_{11} }},t_{18} = t_{17} \omega_{1} \omega^{2} ,\\ t_{19} & = - t_{11} c_{1}^{2} r,\,{\text{where}}\,r = \frac{{\rho C_{e} }}{{T_{0} }},t_{20} = \frac{{i\rho c_{1}^{2} \tau_{3} }}{{\omega \alpha_{11} }}\tau_{T} = (1 + i\omega \tau_{t} ), \\ \tau_{X} & = (1 + i\omega \tau_{\upsilon } ),_{Q} = (1 + i\omega \tau_{q} - \frac{{\omega^{2} }}{2}\tau_{q}^{2} ),\\ t_{11} & = \frac{{\alpha_{11} T_{0} }}{{e_{31} }},t_{12} = - \frac{{i\rho c_{1}^{2} }}{\omega c}, \\ t_{13} & = 1 + \omega^{2} \frac{{a_{11} }}{{c^{2} }}, t_{14} = \omega^{2} a_{33} ,t_{15} = - \frac{{i\rho c_{1}^{2} \alpha_{33} }}{{\omega \alpha_{11} }},t_{16} = \frac{{i\omega \rho \omega_{1} \tau_{T} }}{{\alpha_{11} }},t_{17} = \frac{{\rho \tau_{X} }}{{\alpha_{11} }}, \\ t_{18} & = - \frac{{\rho c_{1}^{2} \tau_{Q} T_{0} r}}{{\alpha_{11} }}\,{\text{where}}\,r = \frac{{\rho C_{e} }}{{T_{0} }},t_{19} = \frac{{i\rho c_{1}^{2} \tau_{3} }}{{\omega \alpha_{11} }}. \\ x_{11} & = \frac{{c_{11} }}{{c^{2} }} - \rho c_{1}^{2} ,x_{12} = \frac{{(c_{13} + c_{55} )}}{c} = x_{16} ,x_{13} = \frac{{(e_{31} + e_{15} )}}{c}t_{11} ,\\ x_{14} & = t_{12} t_{13} ,x_{15} = t_{12} t_{14} ,x_{17} = \frac{{c_{55} }}{{c^{2} }} - \rho c_{1}^{2} , \\ x_{18} & = \frac{{t_{11} e_{15} }}{{c^{2} }},x_{19} = t_{11} e_{33} ,x_{20} = t_{13} t_{15} ,x_{21} = t_{15} t_{14} ,x_{22} = \frac{{i\omega \alpha_{11} \tau_{Q} T_{0} }}{c},\\ x_{23} &= i\omega \alpha_{33} \tau_{Q} T_{0} , x_{24} = - \frac{{i\omega \tau_{Q} \left( {T_{0} } \right)^{2} \tau_{3} \alpha_{11} }}{{e_{31} }},x_{25} = \frac{{t_{16} K_{11} }}{{c^{2} }} + \frac{{t_{17} K_{{_{11} }}^{ * } }}{{c^{2} }} + t_{13} t_{18} ,\\ x_{26} & = t_{16} K_{33} + t_{17} K_{{_{33} }}^{ * } + t_{14} t_{18} , x_{27} = \frac{{(e_{31} + e_{15} )}}{c},x_{28} = \frac{{e_{15} }}{{c^{2} }},x_{29} = - \frac{{t_{11} \xi_{11} }}{{c^{2} }},\\ x_{30} & = - t_{11} \xi_{33} ,x_{31} = t_{13} t_{19} ,x_{32} = t_{14} t_{19} . b_{11} = x_{17} x_{33} - x_{18} x_{32} ,b_{12} = c_{33} x_{33} + x_{17} x_{34} ,\\ b_{13} & = - (x_{19} x_{32} + x_{18} e_{33} ),b_{14} = b_{12} + b_{13} ,b_{15} = x_{18} e_{33} - c_{55} x_{33} , \\ b_{16} & = x_{12} x_{33} - x_{18} x_{31} ,b_{17} = x_{12} x_{34} - x_{19} x_{31} ,\\ b_{18} &= x_{12} x_{32} - x_{17} x_{31} ,b_{19} = x_{12} e_{33} - c_{33} x_{31} , \\ b_{20} & = c_{55} b_{11} + x_{11} b_{14} - x_{12} b_{16} + x_{13} b_{18} ,b_{21} = \,c_{55} b_{14} + x_{11} b_{15} - x_{12} b_{17} + x_{13} b_{19} . \\ m_{11} & = - c_{55} x_{26} b_{19} ,m_{12} = - (b_{19} x_{25} c_{55} + x_{26} b_{21} ),\\ m_{13} & = - (b_{21} x_{25} + b_{20} x_{26} ),m_{14} = - (b_{20} x_{25} + x_{26} x_{11} b_{17} ), \\ m_{15} & = - x_{11} x_{25} b_{17} . \\ \end{aligned} $$

Appendix B

$$ \begin{aligned} D_{{1i}} & = - \frac{{c_{{13}} }}{c} - c_{{33}} q_{i} W_{i} - \frac{{\alpha _{{11}} T_{0} e_{{33}} }}{{e_{{31}} }}q_{i} \Phi _{i} - \frac{{\alpha _{{33}} \rho c_{1}^{2} }}{{i\omega \alpha _{{11}} }}\left( {1 + \omega ^{2} \frac{{a_{{11}} }}{{c^{2} }} + \omega ^{2} a_{{33}} q_{i}^{2} } \right)\Theta _{i} ,\,\,\, \\ D_{{2i}} & = - c_{{55}} \left( {\frac{{W_{i} }}{c} + q_{i} } \right) + \frac{{e_{{15}} c_{1} \alpha _{{11}} T_{0} }}{{ce_{{31}} \omega _{1} }}\Phi _{i} ,\,\,D_{{3i}} = 1\,\,,\\ D_{{4i}} & = W_{i} ,\,\,\,D_{{5i}} = \left( {1 + \omega ^{2} \frac{{a_{{11}} }}{{c^{2} }} + \omega ^{2} a_{{33}} q_{i}^{2} } \right)q_{i} \Theta _{i} ,\,\, \\ D_{{6i}} & = - \frac{{e_{{31}} }}{c}\, - e_{{33}} q_{i} W_{i} + \frac{{\xi _{{33}} T_{0} \alpha _{{11}} }}{{e_{{31}} }}q_{i} \Phi _{i}\\ &\quad + \frac{{\tau _{3} \rho c_{1}^{2} }}{{i\omega \alpha _{{11}} }}\left( {1 + \omega ^{2} \frac{{a_{{11}} }}{{c^{2} }} + \omega ^{2} a_{{33}} q_{i}^{2} } \right)\Theta _{i} \,,\,(i = 1,2,3,4). \\ \end{aligned} $$
  1. (i)

    For incident P wave

$$ \begin{aligned} D_{{15}} & = - i\omega \rho ^{e} c_{1}^{2} \left( {1 - \frac{{2\beta ^{{e^{2} }} \sin ^{2} \theta }}{{\alpha ^{{e^{2} }} }}} \right),\,\,\,D_{{16}} = i\omega \rho ^{e} c_{1}^{2} \sin 2\theta _{2} ,\,\,D_{{25}} = \frac{{i\omega \beta ^{{e^{2} }} \rho ^{e} c_{1}^{2} \sin 2\theta }}{{\alpha ^{{e^{2} }} }},\,\, \\ D_{{26}} & = i\omega \rho ^{e} c_{1}^{2} \cos 2\theta _{2} ,\,\,\,D_{{35}} = \frac{{i\omega c_{1} \sin \theta }}{{\alpha ^{e} }},\,\,\,D_{{36}} = \frac{{i\omega c_{1} \cos \theta _{2} }}{{\beta ^{e} }},\\ D_{{45}} & = - \frac{{i\omega c_{1} \cos \theta }}{{\alpha ^{e} }},\,\,D_{{46}} = \frac{{i\omega c_{1} \sin \theta _{2} }}{{\beta ^{e} }}, \\ X_{i} & = \frac{{U_{i} }}{{A_{0}^{e} }},\,(i = 1,2,3,4),\,\,X_{5}^{e} = \frac{{A_{1}^{e} }}{{A_{0}^{e} }},X_{6}^{e} = \frac{{B_{1}^{e} }}{{A_{0}^{e} }},\\ N_{1} &= - D_{{15}} ,\,\,N_{2} = D_{{25}} ,\,\,N_{3} = - D_{{35}} ,\,\,N_{4} = D_{{45}} . \\ \end{aligned} $$
  1. (ii)

    For incident SV wave

$$ \begin{aligned} D_{{15}} & = - i\omega \rho ^{e} c_{1}^{2} \left( {1 - \frac{{2\beta ^{{e^{2} }} \sin ^{2} \theta _{1} }}{{\alpha ^{{e^{2} }} }}} \right),\,\,\,D_{{16}} = i\omega \rho ^{e} c_{1}^{2} \sin 2\theta ,\,\,D_{{25}} = \frac{{i\omega \beta ^{{e^{2} }} \rho ^{e} c_{1}^{2} \sin 2\theta _{1} }}{{\alpha ^{{e^{2} }} }},\,\, \\ D_{{26}} & = i\omega \rho ^{e} c_{1}^{2} \cos 2\theta ,\,\,\,D_{{35}} = \frac{{i\omega c_{1} \sin \theta _{1} }}{{\alpha ^{e} }},\,\,\,D_{{36}} = \frac{{i\omega c_{1} \cos \theta }}{{\beta ^{e} }},\\ D_{{45}} & = - \frac{{i\omega c_{1} \cos \theta _{1} }}{{\alpha ^{e} }},\,\,D_{{46}} = \frac{{i\omega c_{1} \sin \theta }}{{\beta ^{e} }}, \\ X_{i} & = \frac{{U_{i} }}{{B_{0}^{e} }},\,(i = 1,2,3,4),\,\,X_{5}^{e} = \frac{{A_{1}^{e} }}{{B_{0}^{e} }},X_{6}^{e} = \frac{{B_{1}^{e} }}{{B_{0}^{e} }},\,\,N_{1} = D_{{16}} ,\\ N_{2}& = - D_{{26}} ,\,\,N_{3} = D_{{36}} ,\,\,N_{4} = - D_{{46}} . \\ \end{aligned} $$

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Kumar, R., Sharma, P. Effect of two-temperature on the energy ratios at the elastic–piezothermoelastic interface with phase lags. Eur. Phys. J. Plus 136, 1200 (2021). https://doi.org/10.1140/epjp/s13360-021-02042-2

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