Thermo-mechanical interactions in a fractional order generalized thermoelastic solid with diffusion

Abstract

The main objective of the present paper is to analyze the effects of fractional order parameter, diffusion, velocity of moving load and time on wave propagation in a two dimensional generalized thermoelastic half-space. Medium is assumed to be unstrained and unstressed initially and has uniform temperature. The governing equations of the fractional generalization of Lord-Shulman model for an isotropic and homogeneous medium are solved by means of the Laplace–Fourier transforms. By employing a numerical inversion technique, the distributions of different fields like displacement, stresses, temperature, concentration and chemical potential are computed numerically and displayed graphically for copper material. All the physicals fields are found to be significantly affected by the above mentioned parameters. The problem of generalized thermoelasticity with diffusion has been reduced as a special case of our problem.

Appendices

Appendix A

$Q = \frac{1}{E}\left[ {F - 3\xi^{2} E} \right] ,$ $N = \frac{1}{E}\left[ {G - 2\xi^{2} F + 3\xi^{4} E} \right],$ $I = \frac{1}{E}\left[ {H - \xi^{2} G + \xi^{4} F - \xi^{6} E} \right],$ where $E = 1 - \alpha_{3},$

$$\begin{aligned} F & = s\left\{ {1 + \frac{{\left( {\tau_{0} s} \right)^{\alpha } }}{{\varGamma \left( {\alpha + 1} \right)}}} \right\}\left[ {\alpha_{3} \left( {1 + \varepsilon } \right) + \varepsilon \alpha_{1} \left( {2 + \alpha_{1} } \right) - 1} \right] \\ & \quad + \alpha_{2} s\left\{ {1 + \frac{{\left( {\tau s} \right)^{\alpha } }}{{\varGamma \left( {\alpha + 1} \right)}}} \right\} + \alpha_{3} s^{2} ,\end{aligned} $$

$G = - s^{2} \left[ {s\left( {\alpha_{3} + \varepsilon \alpha_{1}^{2} } \right)\left\{ {1 + \frac{{\left( {\tau_{0} s} \right)^{\alpha } }}{{\varGamma \left( {\alpha + 1} \right)}}} \right\} + \alpha_{2} \left\{ {1 + \frac{{\left( {\tau s} \right)^{\alpha } }}{{\varGamma \left( {\alpha + 1} \right)}}} \right\}\left[ {s + \left( {1 + \varepsilon } \right)\left\{ {1 + \frac{{\left( {\tau_{0} s} \right)^{\alpha } }}{{\varGamma \left( {\alpha + 1} \right)}}} \right\}} \right]} \right]$,

$$ H = \alpha_{2} s^{4} \left\{ {1 + \frac{{(\tau_{0} s)^{\alpha } }}{{\varGamma \left( {\alpha + 1} \right)}}} \right\}\left\{ {1 + \frac{{(\tau s)^{\alpha } }}{{\varGamma \left( {\alpha + 1} \right)}}} \right\} $$

.

Appendix B

$\lambda_{2} = \sqrt {\frac{1}{3}\left[ {2n\sin (q) - Q} \right]},$ $\lambda_{3} = \sqrt {\frac{1}{3}\left[ { - Q - n\left( {\sqrt 3 \,\cos (q) + \sin (q)} \right)} \right]} ,$ $\lambda_{4} = \sqrt {\frac{1}{3}\left[ { - Q + n\left( {\sqrt 3 \,\cos (q) - \sin (q)} \right)} \right]},$ and $n = \sqrt {Q^{2} - 3N} ,\,\,\,q = \frac{{\sin^{ - 1} \left( r \right)}}{3},\,\,\,r = - \frac{{ - 2Q^{3} + 9QN - 27I}}{{2n^{3} }} .$

Appendix C

$R^{\prime}_{i} \left( {\xi ,s} \right) = d_{i} \,R_{i} \left( {\xi ,s} \right),\;R^{\prime\prime}_{i} \left( {\xi ,s} \right) = e_{i} R_{i} \left( {\xi ,s} \right)$,where

$$ d_{i} = \frac{{X_{1} \lambda_{i}^{2} - X_{2} }}{{X_{3} \lambda_{i}^{2} - X_{4} }},\,\,\,e_{i} = \frac{{X_{5} \lambda_{i}^{4} - X_{6} \lambda_{i}^{2} + X_{7} }}{{X_{8} \lambda_{i}^{2} - X_{9} }},\,\,\,\left( {i = 2,3,4} \right) $$

and

$$ X_{1} = \varepsilon \left( {1 + \alpha_{1} } \right),\,\,X_{2} = \varepsilon \left[ {\alpha_{1} \left( {\xi^{2} + s^{2} } \right) + \xi^{2} } \right] $$

$X_{3} = \frac{1}{{s\left\{ {1 + \frac{{\left( {\tau_{0} s} \right)^{\alpha } }}{{\varGamma \left( {\alpha + 1} \right)}}} \right\}}},\,\,\,X_{4} = \xi^{2} X_{3} - \varepsilon \alpha_{1} + 1$,

$$ X_{5} = \frac{{X_{1} }}{\varepsilon },\,\,\,X_{6} = \alpha_{1} \left( {s^{2} + 2\xi^{2} } \right) + 2\xi^{2} ,\,\,X_{7} = \alpha_{1} \xi^{2} \left( {\xi^{2} + s^{2} } \right) + \xi^{4}, $$

$$ X_{8} = \alpha_{1} + \alpha_{3} ,\,\,X_{9} = X_{8} \xi^{2} + \alpha_{2} s\left\{ {1 + \frac{{\left( {\tau s} \right)^{\alpha } }}{{\varGamma \left( {\alpha + 1} \right)}}} \right\}. $$