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Thermal effect on the transient waves in piezoelectric half-space subjected to dynamic loading

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Abstract

Propagation of transient waves in the piezoelectric half-space under anti-plane dynamic force and in-plane electrical displacement loading is studied theoretically when thermal effect considered. One-sided and two-sided Laplace transforms are firstly employed in obtaining the solutions of mechanical displacement, electrical potential, shear stress and electrical displacement in Laplace space. Cagniard-de Hoop method is then adopted for inverse Laplace transform in determining the analytical transient solutions in time domain. Transient response of the mechanical displacement, shear stress, electric potential and displacement are finally evaluated numerically, and the effect of thermal stress on the transient waves propagating in the piezoelectric half-space is discussed in details. Furthermore, two different electro-mechanic boundary conditions are considered for the propagation of transient waves in the piezoelectric half-space.

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Acknowledgements

Support from the National Natural Science Foundation of China (No. 12272036) is greatly appreciated.

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Authors and Affiliations

  1. Department of Mechanics, Beijing Jiaotong University, Beijing, 100044, China

    Fangzhi Wu, Xiang Zhou & Guoshuang Shui

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  1. Fangzhi Wu
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  2. Xiang Zhou
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Correspondence to Xiang Zhou or Guoshuang Shui.

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Appendices

Appendix 1

The transient response of the displacement field can be expressed as:

$$ \frac{{\pi \tilde{c}_{44} L}}{{c_{s} P}}w^{\prime}\left( {x,y,t} \right) = \frac{1}{{{1} - k_{e}^{4} }}\left\{ {{\text{Re}} \left( {{\Omega }_{{\text{a}}}^{\prime } } \right)H\left( {\tau - \tau_{a} } \right) - {\text{Im}} \left( {{\Omega }_{{{\text{ae}}}}^{\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ae}}}} } \right) - H\left( {\tau - \tau_{{\text{a}}} } \right)} \right]} \right\} $$
(104)

The transient response of electrical potential, stress and electric displacement can be expressed as:

$$ \begin{gathered} \frac{{\pi \tilde{c}_{44} \in_{{{11}}} L}}{{\tau_{0} c_{s} e_{15} }}\phi^{\prime}\left( {x,y,t} \right) = \frac{{C_{f} }}{{{1} - k_{e}^{4} }}\left\{ {{\text{Re}} \left( {{\Omega }_{{\text{a}}}^{\prime } } \right)H\left( {\tau - \tau_{{\text{a}}} } \right)} \right. \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. { - {\text{Im}} \left( {{\Omega }_{{{\text{ae}}}}^{\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ae}}}} } \right) - H\left( {\tau - \tau_{{\text{a}}} } \right)} \right] - {\text{Re}} \left( {{\Omega }_{{\text{e}}}^{\prime } } \right)H\left( {\tau - \tau_{{\text{e}}} } \right)} \right\} \hfill \\ \end{gathered} $$
(105)
$$ \begin{gathered} \frac{\pi L}{{\tau_{0} }}\tau_{{{\text{yz}}}} \left( {x,y,t} \right) = \frac{1}{{{1} - k_{e}^{4} }}\left\{ {{\text{Re}} \left( {q\sqrt {1 - \vartheta_{{\text{a}}}^{\prime + 2} } {\Omega }_{a}^{\prime } } \right)H\left( {\tau - \tau_{a} } \right)} \right. \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \quad- {\text{Im}} \left( {q\sqrt {1 - \vartheta_{{{\text{ae}}}}^{\prime + 2} } {\Omega }_{ae}^{\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ae}}}} } \right) - H\left( {\tau - \tau_{{\text{a}}} } \right)} \right] \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\;\;\;\;\quad + k_{e}^{2} {\text{Re}} \left( {\sqrt {\tau_{l}^{2} - \vartheta_{{\text{e}}}^{\prime + 2} } {\Omega }_{{\text{e}}}^{\prime } } \right)H\left( {\tau - \tau_{{\text{e}}} } \right)} \right\} \hfill \\ \end{gathered} $$
(106)
$$ \begin{gathered} \frac{\pi L}{{\tau_{0} }}\tau_{{{\text{xz}}}} \left( {x,y,t} \right){ = } - \frac{1}{{{1} - k_{{\text{e}}}^{4} }}\left\{ {{\text{Re}} \left( {\vartheta_{{\text{a}}}^{\prime + } {\Omega }_{{\text{a}}}^{\prime } } \right)H\left( {\tau - \tau_{a} } \right)} \right. \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \quad- {\text{Im}} \left( {\vartheta_{ae}^{\prime + } {\Omega }_{{{\text{ae}}}}^{\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ae}}}} } \right) - H\left( {\tau - \tau_{{\text{a}}} } \right)} \right] \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. { \quad+ k_{{\text{e}}}^{2} {\text{Re}} \left( {\vartheta_{{\text{e}}}^{\prime + } {\Omega }_{e}^{\prime } } \right)H\left( {\tau - \tau_{{\text{e}}} } \right)} \right\} \hfill \\ \end{gathered} $$
(107)
$$ \begin{gathered} \frac{{\pi \tilde{c}_{44} L}}{{\tau_{0} e_{15} }}D_{y} \left( {x,y,t} \right) = \frac{1}{{{1} - k_{{\text{e}}}^{4} }}\left\{ {\left( {1 - C_{f} } \right){\text{Re}} \left( {q\sqrt {1 - \vartheta_{{\text{a}}}^{\prime + 2} } {\Omega }_{{\text{a}}}^{\prime } } \right)H\left( {\tau - \tau_{{\text{a}}} } \right)} \right. \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\quad - \left( {1 - C_{{\text{f}}} } \right){\text{Im}} \left( {q\sqrt {1 - \vartheta_{{{\text{ae}}}}^{\prime + 2} } {\Omega }_{{{\text{ae}}}}^{\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ae}}}} } \right) - H\left( {\tau - \tau_{{\text{a}}} } \right)} \right] \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\quad + C_{{\text{f}}} {\text{Re}} \left( {\sqrt {\tau_{{\text{l}}}^{2} - \vartheta_{{\text{e}}}^{\prime + 2} } {\Omega }_{{\text{e}}}^{\prime } } \right)H\left( {\tau - \tau_{{\text{e}}} } \right)} \right\} \hfill \\ \end{gathered} $$
(108)
$$ \begin{gathered} \frac{{\pi \tilde{c}_{44} L}}{{\tau_{0} e_{15} }}D_{{\text{x}}} \left( {x,y,t} \right) = - \frac{1}{{{1} - k_{e}^{4} }}\left\{ {{\text{Re}} \left( {\vartheta_{{\text{a}}}^{\prime + } {\Omega }_{{\text{a}}}^{\prime } } \right)H\left( {\tau - \tau_{{\text{a}}} } \right)} \right. \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \quad- \left( {1 - C_{{\text{f}}} } \right){\text{Im}} \left( {\vartheta_{{{\text{ae}}}}^{\prime + } {\Omega }_{{{\text{ae}}}}^{\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ae}}}} } \right) - H\left( {\tau - \tau_{{\text{a}}} } \right)} \right] \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \quad+ C_{{\text{f}}} \left. {{\text{Re}} \left( {\vartheta_{{\text{e}}}^{\prime + } {\Omega }_{{\text{e}}}^{\prime } } \right)H\left( {\tau - \tau_{{\text{e}}} } \right)} \right\} \hfill \\ \end{gathered} $$
(109)

where

$$ \begin{gathered} {\Omega }_{{\text{a}}}^{\prime } = \frac{{q\sqrt {1 - \vartheta_{{\text{a}}}^{\prime + 2} } + k_{{\text{e}}}^{2} \sqrt {\tau_{l}^{2} - \vartheta_{a}^{\prime + 2} } }}{{\tau_{{{\text{bge}}}}^{2} - \vartheta_{{\text{a}}}^{\prime + 2} }}\frac{{q\sqrt {1 - \vartheta_{a}^{\prime + 2} } }}{{\sqrt {\tau^{2} - \tau_{a}^{2} } }} \hfill \\ {\Omega }_{{{\text{ae}}}}^{\prime } = \frac{{q\sqrt {1 - \vartheta_{{{\text{ae}}}}^{\prime + 2} } + k_{e}^{2} \sqrt {\tau_{l}^{2} - \vartheta_{ae}^{\prime + 2} } }}{{\tau_{{{\text{bge}}}}^{2} - \vartheta_{{{\text{ae}}}}^{\prime + 2} }}\frac{{q\sqrt {1 - \vartheta_{{{\text{ae}}}}^{\prime + 2} } }}{{\sqrt {\tau_{a}^{2} - \tau^{2} } }} \hfill \\ {\Omega }_{{\text{e}}}^{\prime } = \frac{{q\sqrt {1 - \vartheta_{a}^{\prime + 2} } + k_{e}^{2} \sqrt {\tau_{l}^{2} - \vartheta_{a}^{\prime + 2} } }}{{\tau_{{{\text{bge}}}}^{2} - \vartheta_{e}^{\prime + 2} }}\frac{{\sqrt {\tau_{l}^{2} - \vartheta_{e}^{\prime + 2} } }}{{\sqrt {\tau^{2} - \left( {x^{\prime 2} + q^{2} y^{\prime 2} } \right)\tau_{l}^{2} } }} \hfill \\ \end{gathered} $$
(110)
$$ \begin{gathered} \vartheta_{a}^{\prime \left( \pm \right)} = - \frac{{x^{\prime}\tau }}{{x^{\prime 2} + q^{2} y^{\prime 2} }} \pm {\text{i}}\frac{{qy^{\prime}\sqrt {\tau^{2} - \left( {x^{\prime 2} + q^{2} y^{\prime 2} } \right)} }}{{x^{\prime 2} + q^{2} y^{\prime 2} }} \hfill \\ \vartheta_{ae}^{\prime \left( \pm \right)} = - \frac{{x^{\prime}\tau }}{{x^{\prime 2} + q^{2} y^{\prime 2} }} \pm \frac{{qy^{\prime}\sqrt {\left( {x^{\prime 2} + q^{2} y^{{\prime}{2}} } \right) - \tau^{2} } }}{{x^{\prime 2} + q^{2} y^{\prime 2} }} \hfill \\ \vartheta_{e}^{\prime \left( \pm \right)} = - \frac{{x^{\prime}\tau }}{{x^{\prime 2} + y^{\prime 2} }} \pm {\text{i}}\frac{{y^{\prime}\sqrt {\tau^{2} - \left( {x^{\prime 2} + y^{\prime 2} } \right)\tau_{l}^{2} } }}{{x^{\prime 2} + y^{\prime 2} }} \hfill \\ \end{gathered} $$
(111)

Appendix 2

After dimensionless processing, each physical quantity can be expressed as:

$$ \frac{{\pi c_{44}^{v} L}}{{c_{s} \tau^{v} }}w^{\prime}\left( {x,y,t} \right){ = }\frac{1}{{1 - k_{e}^{v4} }}\left\{ {{\text{Re}} \left( {{\Omega }_{a}^{v\prime } } \right)H\left( {\tau - \tau_{{\text{a}}} } \right) - {\text{Im}} \left( {{\Omega }_{{{\text{ae}}}}^{v\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ae}}}} } \right) - H\left( {\tau - \tau_{{\text{a}}} } \right)} \right]} \right\} $$
(112)
$$ \begin{gathered} \frac{{\pi c_{44}^{v} \in_{{{11}}} L}}{{\tau^{v} c_{{\text{s}}} e_{15} }}\phi^{\prime}\left( {x,y,t} \right) = \frac{{C_{f} }}{{1 - k_{e}^{v4} }}\left\{ {{\text{Re}} \left( {\Omega_{a}^{v\prime } } \right)H\left( {\tau - \tau_{{\text{a}}} } \right)} \right. \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. { - {\text{Im}} \left( {\Omega_{ae}^{v\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ae}}}} } \right) - H\left( {\tau - \tau_{{\text{a}}} } \right)} \right]} \right\} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \frac{{ \in_{{{11}}} }}{{ \in_{0} }}\frac{1}{{1 - k_{e}^{v4} }}\left\{ {{\text{Re}} \left( {\Omega_{e1}^{v\prime } } \right)H\left( {\tau - \tau_{e} } \right)} \right. \hfill \\ \left. {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - {\text{Im}} \left( {\Omega_{{{\text{ef}}1}}^{v\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ef}}}} } \right) - H\left( {\tau - \tau_{{\text{e}}} } \right)} \right]} \right\} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - \frac{{ \in_{{{11}}} }}{{ \in_{0} }}\frac{{c_{44}^{v} }}{{\frac{{e_{15}^{2} }}{{E_{cf} \in_{11} }} + \frac{{e_{15}^{{2}} }}{{ \in_{11} + \in_{0} }}}}\left\{ {{\text{Re}} \left( {\Omega_{e2}^{v\prime } } \right)H\left( {\tau - \tau_{{\text{e}}} } \right)} \right. \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. { - {\text{Im}} \left( {\Omega_{{{\text{ef}}2}}^{v\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ef}}}} } \right) - H\left( {\tau - \tau_{{\text{e}}} } \right)} \right]} \right\} \hfill \\ \end{gathered} $$
(113)
$$ \begin{gathered} \frac{{\pi c_{44}^{v} L}}{{\tau^{v} \tilde{c}_{44} }}\tau_{yz} \left( {x,y,t} \right){ = }\frac{1}{{1 - k_{e}^{v4} }}\left\{ {{\text{Re}} \left( {q\sqrt {1 - \vartheta_{a}^{\prime + 2} } {\Omega }_{a}^{v\prime } } \right)\left( {\tau - \tau_{{\text{a}}} } \right)} \right. \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. { - {\text{Im}} \left( {q\sqrt {1 - \vartheta_{{{\text{ae}}}}^{\prime + 2} } {\Omega }_{{{\text{ae}}}}^{v\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ae}}}} } \right) - H\left( {\tau - \tau_{a} } \right)} \right]} \right\} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \frac{{e_{15}^{2} }}{{\tilde{c}_{44} \in_{0} }}\frac{1}{{1 - k_{e}^{v4} }}\left\{ {{\text{Re}} \left( {\sqrt {\tau_{l}^{2} - \vartheta_{e}^{\prime + 2} } {\Omega }_{e1}^{v\prime } } \right)H\left( {\tau - \tau_{e} } \right)} \right. \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. { - {\text{Im}} \left( {\sqrt {\tau_{l}^{2} - \vartheta_{ef}^{\prime + 2} } {\Omega }_{ef1}^{v\prime } } \right)\left[ {H\left( {\tau - \tau_{ef} } \right) - H\left( {\tau - \tau_{e} } \right)} \right]} \right\} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - \frac{{e_{15} }}{{\frac{{e_{15} }}{{E_{cf} \in_{11} }} + \frac{{e_{15} }}{{ \in_{11} + \in_{0} }}}}\frac{{c_{44}^{v} }}{{\tilde{c}_{44} \in_{0} }}\left\{ {{\text{Re}} \left( {\sqrt {\tau_{l}^{2} - \vartheta_{e}^{\prime + 2} } {\Omega }_{e2}^{v\prime } } \right)H\left( {\tau - \tau_{{\text{e}}} } \right)} \right. \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. { - {\text{Im}} \left( {\sqrt {\tau_{l}^{2} - \vartheta_{{{\text{ef}}}}^{\prime + 2} } {\Omega }_{ef2}^{v\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ef}}}} } \right) - H\left( {\tau - \tau_{{\text{e}}} } \right)} \right]} \right\} \hfill \\ \end{gathered} $$
(114)
$$ \begin{gathered} \frac{{\pi c_{44}^{v} L}}{{\tau^{v} \tilde{c}_{44} }}\tau_{{{\text{xz}}}} \left( {x,y,t} \right) = - \frac{1}{{1 - k_{{\text{e}}}^{v4} }}\left\{ {{\text{Re}} \left( {\vartheta_{{\text{a}}}^{\prime + } \Omega_{{\text{a}}}^{v\prime } } \right)H\left( {\tau - \tau_{{\text{a}}} } \right)} \right. \hfill \\ \left. {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - {\text{Im}} \left( {\vartheta_{{{\text{ae}}}}^{\prime + } \Omega_{{{\text{ae}}}}^{v\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ae}}}} } \right) - H\left( {\tau - \tau_{{\text{a}}} } \right)} \right]} \right\} \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - \frac{{e_{15}^{2} }}{{\tilde{c}_{44} \in_{0} }}\frac{1}{{1 - k_{{\text{e}}}^{v4} }}\left\{ {{\text{Re}} \left( {\vartheta_{{\text{e}}}^{\prime + } \Omega_{{{\text{e}}1}}^{v\prime } } \right)H\left( {\tau - \tau_{{\text{e}}} } \right)} \right. \hfill \\ \left. {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - {\text{Im}} \left( {\vartheta_{{{\text{ef}}}}^{\prime + } \Omega_{{{\text{ef}}1}}^{v\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ef}}}} } \right) - H\left( {\tau - \tau_{{\text{e}}} } \right)} \right]} \right\} \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + \frac{{e_{15} }}{{\frac{{e_{15} }}{{E_{{{\text{cf}}}} \in_{11} }} + \frac{{e_{15} }}{{ \in_{11} + \in_{0} }}}}\frac{{c_{44}^{v} }}{{\tilde{c}_{44} \in_{0} }}\left\{ {{\text{Re}} \left( {\vartheta_{{\text{e}}}^{\prime + } \Omega_{{{\text{e}}2}}^{v\prime } } \right)H\left( {\tau - \tau_{{\text{e}}} } \right)} \right. \hfill \\ \left. {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - {\text{Im}} \left( {\vartheta_{{{\text{ef}}}}^{\prime + } \Omega_{ef2}^{v\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ef}}}} } \right) - H\left( {\tau - \tau_{{\text{e}}} } \right)} \right]} \right\} \hfill \\ \end{gathered} $$
(115)
$$ \begin{gathered} \frac{{\pi c_{44}^{v} L}}{{\tau^{v} e_{15} }}D_{{\text{y}}} \left( {x,y,t} \right) = \frac{{1 - C_{{\text{f}}} }}{{1 - k_{{\text{e}}}^{v4} }}\left\{ {{\text{Re}} \left( {q\sqrt {1 - \vartheta_{{\text{a}}}^{\prime + 2} } \Omega_{{\text{a}}}^{v\prime } } \right)H\left( {\tau - \tau_{{\text{a}}} } \right)} \right. \hfill \\ \left. {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - {\text{Im}} \left( {q\sqrt {1 - \vartheta_{{{\text{ae}}}}^{\prime + 2} } \Omega_{{{\text{ae}}}}^{v\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ae}}}} } \right) - H\left( {\tau - \tau_{{\text{a}}} } \right)} \right]} \right\} \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + \frac{{ \in_{11} }}{{ \in_{0} \left( {1 - k_{v}^{4} } \right)}}\left\{ {{\text{Re}} \left( {\sqrt {\tau_{l}^{2} - \vartheta_{{\text{e}}}^{\prime + 2} } \Omega_{{{\text{e}}1}}^{v\prime } } \right)H\left( {\tau - \tau_{{\text{e}}} } \right)} \right. \hfill \\ \left. {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - {\text{Im}} \left( {\sqrt {\tau_{l}^{2} - \vartheta_{{{\text{ef}}}}^{\prime + 2} } \Omega_{{{\text{ef}}1}}^{v\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ef}}}} } \right) - H\left( {\tau - \tau_{{\text{e}}} } \right)} \right]} \right\} \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\;\; - \frac{{c_{44}^{v} }}{{\frac{{e_{15}^{2} }}{{E_{{{\text{cf}}}} \in_{11} }} + \frac{{e_{15}^{2} }}{{ \in_{11} + \in_{0} }}}}\frac{{ \in_{11} }}{{ \in_{0} }}\left\{ {{\text{Re}} \left( {\sqrt {\tau_{l}^{2} - \vartheta_{{\text{e}}}^{\prime + 2} } \Omega_{{{\text{e}}2}}^{v\prime } } \right)H\left( {\tau - \tau_{{\text{e}}} } \right)} \right. \hfill \\ \left. {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\; - {\text{Im}} \left( {\sqrt {\tau_{l}^{2} - \vartheta_{{{\text{ef}}}}^{\prime + 2} } \Omega_{{{\text{ef}}2}}^{v\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ef}}}} } \right) - H\left( {\tau - \tau_{{\text{e}}} } \right)} \right]} \right\} \hfill \\ \end{gathered} $$
(116)
$$ \begin{gathered} \frac{{\pi c_{44}^{{\text{v}}} L}}{{\tau^{{\text{v}}} e_{15} }}D_{{\text{x}}} \left( {x,y,t} \right){ = } - \frac{{1 - C_{{\text{f}}} }}{{1 - k_{{\text{e}}}^{v4} }}\left\{ {{\text{Re}} \left( {\vartheta_{{\text{a}}}^{\prime + } {\Omega }_{{\text{a}}}^{v\prime } } \right)H\left( {\tau - \tau_{{\text{a}}} } \right)} \right. \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \left. { - {\text{Im}} \left( {\vartheta_{{{\text{ae}}}}^{\prime + } {\Omega }_{{{\text{ae}}}}^{v\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ae}}}} } \right) - H\left( {\tau - \tau_{{\text{a}}} } \right)} \right]} \right\} \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - \frac{{ \in_{11} }}{{ \in_{0} \left( {1 - k_{v}^{4} } \right)}}\left\{ {{\text{Re}} \left( {\vartheta_{e}^{\prime + } {\Omega }_{e1}^{v\prime } } \right)H\left( {\tau - \tau_{{\text{e}}} } \right)} \right. \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \left. { - {\text{Im}} \left( {\vartheta_{{{\text{ef}}}}^{\prime + } {\Omega }_{{{\text{ef}}1}}^{v\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ef}}}} } \right) - H\left( {\tau - \tau_{{\text{e}}} } \right)} \right]} \right\} \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + \frac{{c_{44}^{v} }}{{\frac{{e_{15}^{2} }}{{E_{{{\text{cf}}}} \in_{11} }} + \frac{{e_{15}^{2} }}{{ \in_{11} + \in_{0} }}}}\frac{{ \in_{11} }}{{ \in_{0} }}\left\{ {{\text{Re}} \left( {\vartheta_{{\text{e}}}^{\prime + } {\Omega }_{e2}^{v\prime } } \right)H\left( {\tau - \tau_{{\text{e}}} } \right)} \right. \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \left. {\quad \quad \quad - {\text{Im}} \left( {\vartheta_{{{\text{ef}}}}^{\prime + } {\Omega }_{{{\text{ef}}2}}^{v\prime } } \right)\left[ {H\left( {\tau - \tau_{{{\text{ef}}}} } \right) - H\left( {\tau - \tau_{{\text{e}}} } \right)} \right]} \right\} \hfill \\ \end{gathered} $$
(117)
$$ \begin{gathered} \Omega_{{\text{a}}}^{v\prime } = \frac{{q\sqrt {1 - \vartheta_{{\text{a}}}^{\prime + 2} } + k_{{\text{e}}}^{v2} \sqrt {\tau_{{\text{l}}}^{2} - \vartheta_{{\text{a}}}^{\prime + 2} } }}{{\tau_{{{\text{bge}}}}^{v2} - \vartheta_{{\text{a}}}^{\prime + 2} }}\frac{{q\sqrt {1 - \vartheta_{{\text{a}}}^{\prime + 2} } }}{{\sqrt {\tau^{2} - \tau_{{\text{a}}}^{2} } }} \hfill \\ \Omega_{{{\text{ae}}}}^{v\prime } = \frac{{q\sqrt {1 - \vartheta_{{{\text{ae}}}}^{\prime + 2} } + k_{e}^{v2} \sqrt {\tau_{l}^{2} - \vartheta_{{{\text{ae}}}}^{\prime + 2} } }}{{\tau_{{{\text{bge}}}}^{v2} - \vartheta_{{{\text{ae}}}}^{\prime + 2} }}\frac{{q\sqrt {1 - \vartheta_{{{\text{ae}}}}^{\prime + 2} } }}{{\sqrt {\tau_{{\text{a}}}^{2} - \tau^{2} } }} \hfill \\ \end{gathered} $$
(118)
$$ \begin{gathered} \Omega_{e1}^{v\prime } = \frac{{q\sqrt {1 - \vartheta_{{\text{e}}}^{\prime + 2} } + k_{{\text{e}}}^{v2} \sqrt {\tau_{{\text{l}}}^{2} - \vartheta_{{\text{e}}}^{\prime + 2} } }}{{\tau_{{{\text{bge}}}}^{v2} - \vartheta_{{\text{e}}}^{\prime + 2} }} \cdot \hfill \\ \frac{{\sqrt {\tau_{l}^{2} - \vartheta_{{\text{e}}}^{\prime + 2} } \left[ {\left( {1 - C_{{\text{f}}} } \right)q\sqrt {1 - \vartheta_{{\text{e}}}^{\prime + 2} } - \frac{{ \in_{0} }}{{ \in_{11} }}C_{{\text{f}}} \sqrt {\tau_{0}^{2} - \vartheta_{{\text{e}}}^{\prime + 2} } } \right]}}{{\left( { \in \sqrt {\tau_{{\text{l}}}^{2} - \vartheta_{{\text{e}}}^{\prime + 2} } + \sqrt {\tau_{0}^{2} - \vartheta_{{\text{e}}}^{\prime + 2} } } \right)\sqrt {\tau^{2} - \tau_{{\text{e}}}^{2} } }} \hfill \\ \Omega_{ef1}^{v\prime } = \frac{{q\sqrt {1 - \vartheta_{{{\text{ef}}}}^{\prime + 2} } + k_{e}^{v2} \sqrt {\tau_{l}^{2} - \vartheta_{ef}^{\prime + 2} } }}{{\tau_{{{\text{bge}}}}^{{v{2}}} - \vartheta_{{{\text{ef}}}}^{\prime + 2} }} \cdot \hfill \\ \frac{{\sqrt {\tau_{{\text{l}}}^{2} - \vartheta_{{{\text{ef}}}}^{\prime + 2} } \left[ {\left( {1 - C_{{\text{f}}} } \right)q\sqrt {1 - \vartheta_{{{\text{ef}}}}^{\prime + 2} } - \frac{{ \in_{0} }}{{ \in_{11} }}C_{{\text{f}}} \sqrt {\tau_{0}^{2} - \vartheta_{{{\text{ef}}}}^{\prime + 2} } } \right]}}{{\left[ { \in \sqrt {\tau_{{\text{l}}}^{2} - \vartheta_{{{\text{ef}}}}^{\prime + 2} } + \sqrt {\tau_{0}^{2} - \vartheta_{{{\text{ef}}}}^{\prime + 2} } } \right]\sqrt {\tau_{{\text{e}}}^{2} - \tau^{2} } }} \hfill \\ \end{gathered} $$
(119)
$$ \begin{gathered} \Omega_{e2}^{v\prime } = \frac{{\sqrt {\tau_{{\text{l}}}^{2} - \vartheta_{{\text{e}}}^{\prime + 2} } }}{{\left[ { \in \sqrt {\tau_{{\text{l}}}^{2} - \vartheta_{{\text{e}}}^{\prime + 2} } + \sqrt {\tau_{0}^{2} - \vartheta_{{\text{e}}}^{\prime + 2} } } \right]\sqrt {\tau^{2} - \tau_{{\text{e}}}^{2} } }} \hfill \\ \Omega_{ef2}^{v\prime } = \frac{{\sqrt {\tau_{l}^{2} - \vartheta_{ef}^{\prime + 2} } }}{{\left[ { \in \sqrt {\tau_{{\text{l}}}^{2} - \vartheta_{{{\text{ef}}}}^{\prime + 2} } + \sqrt {\tau_{0}^{2} - \vartheta_{{{\text{ef}}}}^{\prime + 2} } } \right]\sqrt {\tau_{{\text{e}}}^{2} - \tau^{2} } }} \hfill \\ \end{gathered} $$
(120)

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Wu, F., Zhou, X. & Shui, G. Thermal effect on the transient waves in piezoelectric half-space subjected to dynamic loading. Arch Appl Mech 93, 1647–1669 (2023). https://doi.org/10.1007/s00419-022-02351-7

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