A study of thermoelastic interactions in thin and long radiating rods under Moore–Gibson–Thompson theory of thermoelasticity

Abstract

The current work attempts to look at the consequence of the external supply of heat source which has an effect of thermal radiation towards the surrounding in accordance with the linearized form of Stefan–Boltzmann law (SBL) on the propagation of waves within a thermoelastic medium. Formulation of the problem is accomplished by considering the recent thermoelasticity theory based on the Moore–Gibson–Thompson (MGT) heat conduction equation. Here, a deformable thermal conductor, particularly a long, thin, and solid rod of thermoelastic material, is considered to study the behavior of the suggested theory in detail. After formulation of the problem and employing the Laplace transform technique, the solutions for the displacement, temperature, and stress fields in the Laplace transform domain are obtained. Further, the analytical solutions for all the field variables in the case of short-time approximation are derived by incorporating Laplace inversion. The investigation concentrates on certain key findings and observations under this model and compares them to those predicted by previously proposed models in this direction.

Appendix

The constants $A_{1,2}$, $B_{1,2}$ and $C_{1,2}$ which are used in Eq. (24) are given as follows:

$$\begin{aligned} {A_{1,2}}=~&{\frac{K_{0}}{\tau _{0}}L_{0},\ \ \ L_{0}=R_{0}\pm S_{0},} \end{aligned}$$

(31)

$$\begin{aligned} {B_{1,2}}=~&K_{0}\left\{ \frac{-k^{*}}{k}L_{0}+H_{0}\pm \frac{-4k^{*}v^{2}\tau _{q}+2H_{0}R_{0}-4kv^{2}\chi }{2S_{0}}\right\} , \nonumber \\ R_{0}=~\,&kv^{2}+\tau _{q}+\epsilon \tau _{q}\ \ K_{0}=\frac{\tau _{0}}{2v^{2}k},\nonumber \\ S_{0}=~\,&\sqrt{R_{0}^{2}-4kv^{2}\tau _{q}}\ \ H_{0}=(1+\epsilon +k^{*}v^{2}),\nonumber \\ G_{1,2}=~\,&\frac{\mp \tau _{0}(-4k^{*}v^{2}\tau _{q}+2H_{0}R_{0}-4kv^{2}\chi )^{2}}{4S_{0}^{3}}\pm \frac{-4\tau _{0}k^{*}v^{2}+\tau _{0}H_{0}^{2}+2R_{0}\chi }{S_{0}},\nonumber \\ C_{1,2}=~\,&K_{0}\left[ \frac{k^{*2}\tau _{0}}{k^{2}}L_{0}+\chi +\frac{G_{1,2}}{2}-\frac{k^{*}\tau _{0}}{k}\left( H_{0}\pm \frac{-4kv^{2}\tau _{q}+2H_{0}R_{0}-4kv^{2}\chi }{2S_{0}}\right) \right] . \end{aligned}$$

(32)

For the Eqs. (25)–(30), the constants are given as follows:

$$\begin{aligned} M_{1,2}=~\,&\frac{\mp \sqrt{A_{1,2}}}{A_{1}-A_{2}},\ \ N_{1,2}=\mp \frac{\sqrt{A_{1,2}}(B_{2}-B_{1})}{(A_{2}-A_{1})^{2}}\mp \frac{B_{1,2}}{2\sqrt{A_{1,2}}(A_{1}-A_{2})},\\ Q_{1,2}=~\,&\frac{1-A_{1,2}}{A_{1}-A_{2}},\ \ R_{1,2}=\frac{(1-A_{1,2})(B_{2}-B_{1})}{(A_{2}-A_{1})^{2}}-\frac{B_{1,2}}{A_{1}-A_{2}},\\ W_{1,2}=~\,&\pm \frac{C_{1,2}}{A_{1}-A_{2}}\pm \frac{B_{1,2}(B_{2}-B_{1})}{(A_{2}-A_{1})^{2}}+\frac{D(1-A_{1,2})}{(A_{1}-A_{2})^{3}},\\ D=~\,&-B_{2}^{2}+2B_{1}B_{2}-B_{1}^{2}+A_{1}C_{1}-A_{1}C_{2}-A_{2}C_{1}+A_{2}C_{2},\\ Z_{1,2}=~\,&-L_{1}Q_{1,2},\ \ V_{1,2}=-R_{1,2}L_{1},\ \ O_{1,2}=-W_{1,2}L_{1}+Q_{1,2}Q_{1,2}^{\prime },\\ L_{1}=~\,&\frac{1+\epsilon }{\epsilon },\ \ Q_{1,2}^{\prime }=~F_{1}A_{1,2}-J_{1},\ \ J_{1}=~\frac{\tau _{0}\chi }{\epsilon \tau _{q}},F_{1}=\frac{kv^{2}}{\epsilon \tau _{q}}. \end{aligned}$$