On fundamental solution of Moore–Gibson–Thompson (MGT) thermoelasticity theory

Abstract

In the theory of partial differential equations, the fundamental solutions have undoubtedly occupied serious attention, as they have a critical role to play in a wide range of problems under mathematical physics and continuum mechanics. The present work explores the Galerkin-type representation of the field equations based on the recently developed Moore–Gibson–Thompson thermoelasticity theory. With the help of the Laplace transform technique, we then establish the short-time approximated fundamental solutions for two separate cases: concentrated body force and concentrated heat source. For the time domain, the fundamental solutions are obtained by taking inverse Laplace transforms of displacement and temperature components. Lastly, we obtain the fundamental solution for steady vibration in terms of elementary functions.

Appendix

$Y_{11}=\frac{\left( n_{1}^{2}-n_{2}^{2}\right) \left( K-\tau K_{1}n_{2}^{2}\right) -\alpha \tau n_{2}^{2}n_{3}\varTheta _{0}}{\left( -1+\vartheta _{11}n_{2}^{2}\right) \left( -1+\vartheta _{21}n_{2}^{2}\right) },\,\,Y_{12}=\frac{-n_{2}^{2}\left( K_{1}\left( n_{1}^{2}-n_{2}^{2}\right) +\alpha n_{3}\varTheta _{0}\right) \left( K-\tau K^{*}\right) }{K\left( -1+\vartheta _{11}n_{2}^{2}\right) \left( -1+\vartheta _{21}n_{2}^{2}\right) },$

$Y_{13}=\frac{\left( n_{1}^{2}-n_{2}^{2}\right) \left( K\vartheta _{11}-\tau K_{1}\right) -\alpha \tau n_{3}\varTheta _{0}}{\left( \vartheta _{11}-\vartheta _{21}\right) \left( -1+\vartheta _{11}n_{2}^{2}\right) },\,\,Y_{14}=\frac{K\vartheta _{12}\left( n_{1}^{2}-n_{2}^{2}\right) -\left( K_{1}\left( n_{1}^{2}-n_{2}^{2}\right) +\alpha n_{3}\varTheta _{0}\right) \left( K-\tau K^{*}\right) }{\left( \vartheta _{11}-\vartheta _{21}\right) \left( -1+\vartheta _{11}n_{2}^{2}\right) },$

$Y_{15}=\frac{-\left( K\vartheta _{21}-\tau K_{1}\right) \left( n_{1}^{2}-n_{2}^{2}\right) +\alpha \tau n_{3}\varTheta _{0}}{\left( \vartheta _{11}-\vartheta _{21}\right) \left( -1+\vartheta _{21}n_{2}^{2}\right) },\,\,Y_{16}=\frac{K\vartheta _{22}\left( -n_{1}^{2}+n_{2}^{2}\right) +\left( K_{1}\left( n_{1}^{2}-n_{2}^{2}\right) +\alpha n_{3}\varTheta _{0}\right) \left( K-\tau K^{*}\right) }{\left( \vartheta _{11}-\vartheta _{21}\right) \left( -1+\vartheta _{21}n_{2}^{2}\right) },$

$Y_{17}=\frac{\alpha \tau \varTheta _{0}}{\left( \vartheta _{11}-\vartheta _{21}\right) },\,\,\,Y_{18}=\frac{\alpha \varTheta _{0}\left( -K\left( \tau \vartheta _{12}+\vartheta _{21}-\tau \vartheta _{22}\right) +\tau \vartheta _{21}K^{*}+\vartheta _{11}\left( K-\tau K^{*}\right) \right) }{K\left( \vartheta _{11}-\vartheta _{21}\right) ^{2}},$

$$\begin{aligned} Y_{19}= & {} \frac{\alpha \varTheta _{0}}{K^{2}\left( \vartheta _{11}-\vartheta _{21}\right) ^{3}}\left( K^{2}\left( \vartheta _{12}^{2}+\tau \vartheta _{13}\vartheta _{21}-\vartheta _{21}\vartheta _{22}+\tau \vartheta _{22}^{2}+\vartheta _{12}\left( \vartheta _{21}-2\tau \vartheta _{22}\right) -\tau \vartheta _{21}\vartheta _{23}\right. \right. \\{} & {} \left. \left. +\vartheta _{11}\left( -\vartheta _{12}-\tau \vartheta _{13}+\tau \vartheta _{22}+\tau \vartheta _{23}\right) \right) \right) \\{} & {} \left. \right. -K\left( \vartheta _{11}-\vartheta _{21}\right) \left( \vartheta _{11}-\tau \vartheta _{12}-\vartheta _{21} +\tau \vartheta _{22}\right) K^{*}+\tau \left( \vartheta _{11}-\vartheta _{21}\right) ^{2}\left( K^{*}\right) ^{2}\biggr ),\\ Z_{11}= & {} \frac{\tau n_{3}}{K\left( \vartheta _{11}-\vartheta _{21}\right) },\,\,\,\,\,\,Z_{12}=\frac{n_{3}\left( -K\left( \tau \vartheta _{12}+\vartheta _{21}-\tau \vartheta _{22}\right) +\tau \vartheta _{21}K^{*}+\vartheta _{11}\left( K-\tau K^{*}\right) \right) }{K^{2}\left( \vartheta _{11}-\vartheta _{21}\right) ^{2}},\\{} & {} Z_{13}=\frac{\tau \left( -1+\vartheta _{11}n_{1}^{2}\right) }{K\left( \vartheta _{11}-\vartheta _{21}\right) },\\ Z_{14}= & {} \frac{1}{K^{2}\left( \vartheta _{11}-\vartheta _{21}\right) ^{2}}\left( K\left( -\vartheta _{11}+\tau \vartheta _{12}+\vartheta _{21} -\tau \vartheta _{22}+\left( \vartheta _{11}^{2}-\tau \vartheta _{12}\vartheta _{21}+\vartheta _{11}\left( -\vartheta _{21}+\tau \vartheta _{22}\right) \right) n_{1}^{2}\right) \right. \\{} & {} \left. -\tau \left( \vartheta _{11}-\vartheta _{21}\right) \left( -1+\vartheta _{11}n_{1}^{2}\right) K^{*}\right) ,\\ Z_{15}= & {} \frac{1}{K^{3}\left( \vartheta _{11}-\vartheta _{21}\right) ^{2}}\biggl ( -K^{2}\tau \vartheta _{12}^{2}n_{1}^{2}+K\vartheta _{22}\left( -1+\vartheta _{11}n_{1}^{2}\right) \left( K-\tau K^{*}\right) +K\vartheta _{12}\left( K+K\tau \vartheta _{22}n_{1}^{2}-\tau K^{*}\right. \\{} & {} \left. +\vartheta _{21}n_{1}^{2}\left( -K+\tau K^{*}\right) \right) \\{} & {} +\left( \vartheta _{11}-\vartheta _{21}\right) \left( K^{2}\tau \vartheta _{13}n_{1}^{2}+\left( \vartheta _{11}n_{1}^{2}-1\right) K^{*}\left( -K+\tau K^{*}\right) \right) \biggr ),\\ Z_{16}= & {} \frac{\tau \left( -1+\vartheta _{21}n_{1}^{2}\right) }{K\left( \vartheta _{11}-\vartheta _{21}\right) },\\ Z_{17}= & {} \frac{1}{K^{2}\left( \vartheta _{11}-\vartheta _{21}\right) ^{2}} \left( K\left( \tau \vartheta _{12}+\vartheta _{21}-\tau \vartheta _{22}-\vartheta _{21}\left( \tau \vartheta _{12}+\vartheta _{21}\right) n_{1}^{2}+\vartheta _{11}\left( -1+\left( \vartheta _{21}+\tau \vartheta _{22}\right) n_{1}^{2}\right) \right) \right. \\{} & {} \left. -\tau \left( \vartheta _{11}-\vartheta _{21}\right) \left( -1+\vartheta _{21}n_{1}^{2}\right) K^{*}\right) ,\\ Z_{18}= & {} \frac{1}{K^{3}\left( \vartheta _{11}-\vartheta _{21}\right) ^{2}} \biggl (K^{2}\tau \vartheta _{22}^{2}n_{1}^{2}+K\vartheta _{22}\left( -1+\vartheta _{11}n_{1}^{2}\right) \left( K-\tau K^{*}\right) +K\vartheta _{12}\left( K-\tau \left( K\vartheta _{22}n_{1}^{2}+K^{*}\right) \right. \\{} & {} \left. +\vartheta _{21}n_{1}^{2}\left( -K+\tau K^{*}\right) \right) \\{} & {} +\left( \vartheta _{11}-\vartheta _{21}\right) \left( K^{2}\tau \vartheta _{23}n_{1}^{2}+\left( -1+\vartheta _{21}n_{1}^{2}\right) K^{*}\left( -K+\tau K^{*}\right) \right) \biggr ). \end{aligned}$$