Microstretch thermoelastic solid with temperature-dependent elastic properties under the influence of magnetic and gravitational field

Abstract

In this article, propagation of plane waves in generalized microstretch homogeneous isotropic thermoelastic half-space is considered. The elastic modulus is considered as a function having linear relation with the initial temperature. The normal mode method is used to fine exact expression for field variables. The results predicted by both types of Green and Naghdi theory under the presence and absence of magnetic and gravitational field are presented graphically. Influence of temperature-dependent elastic constants is also considered on field variables.

Appendices

Appendix A

$$\begin{array}{*{20}l} {c^{2} = \frac{1}{{\mu_{1} \varepsilon_{0} }},\quad \beta^{2} = \frac{{V_{A}^{2} }}{{c^{2} }} + 1,\quad V_{A}^{2} = \frac{{\mu_{1} H_{0}^{2} }}{\rho },\quad R_{H} = \frac{{V_{A}^{2} }}{{c_{2}^{2} }},\quad \alpha^{2} = 1 + R_{H} a_{0} } \hfill \\ \quad {c_{1}^{2} = \frac{{\left( {\lambda_{2} + 2\mu_{0} + k_{0} } \right)}}{\rho },\quad c_{3}^{2} = \frac{{2\alpha_{0}^{*} \left( {1 - \beta^{*} T_{0} } \right)}}{3\rho j},\quad c_{4}^{2} = \frac{{2\lambda_{3} \left( {1 - \beta^{*} T_{0} } \right)}}{9\rho j},\quad c_{5}^{2} = \frac{{2\lambda_{0}^{*} \left( {1 - \beta^{*} T_{0} } \right)}}{9\rho j},} \hfill \\ \end{array}$$

(A1)

$$\begin{array}{*{20}l} {a_{0} = \frac{{c_{2}^{2} }}{{c_{1}^{2} (1 - \beta^{*} T_{0} )}},\quad a^{\prime}_{0} = \frac{{c_{2}^{2} }}{{c_{1}^{2} }},\quad a_{1} = \frac{{\lambda_{0}^{*} }}{{\lambda_{2} + 2\mu_{0} + k_{0} }},\quad a_{2} = \frac{{\rho c_{2}^{2} }}{{(\mu_{0} + k_{0} )(1 - \beta^{*} T_{0} )}},\quad a_{3} = \frac{{k_{0} }}{{(\mu_{0} + k_{0} )}},} \hfill \\ {a_{4} = \frac{{k_{0} c_{2}^{2} }}{{\gamma_{0} \omega^{*2} }},\quad a_{5} = \frac{{\rho jc_{2}^{2} }}{{\gamma_{0} (1 - \beta^{*} T_{0} )}},\quad a_{6} = \frac{{c_{3}^{2} }}{{c_{2}^{2} }},\quad a_{7} = \frac{{c_{4}^{2} }}{{\omega^{*2} }},\quad a_{8} = \frac{{c_{5}^{2} }}{{\omega^{*2} }},\quad a_{9} = \frac{{2\hat{\gamma }_{1} c_{2}^{2} (1 - \beta^{*} T_{0} )}}{{9\hat{\gamma }_{0} j\omega^{*2} }}.} \hfill \\ \end{array}$$

(A2)

Appendix B

$$\begin{array}{*{20}l} {D = \frac{\text{d}}{{{\text{d}}z}},\quad A_{1} = \alpha^{2} b^{2} + a_{0} \beta^{2} \omega^{2} ,\quad A_{2} = ga_{0} ib,\quad A_{3} = b^{2} + a_{2} \beta^{2} \omega^{2} ,\quad A_{4} = ga_{2} ib,} \hfill \\ {A_{5} = b^{2} + 2a_{4} + a_{5} \omega^{2} ,\quad A_{6} = b^{2} a_{6} + a_{7} + \omega^{2} ,\quad \varepsilon = (\varepsilon_{2} + \varepsilon_{3} \omega ).} \hfill \\ \end{array}$$

(B1)

$$\begin{array}{*{20}l} {H_{1n} = {{\left[ {k_{n}^{4} a_{6} \varepsilon_{1} \omega^{2} - k_{n}^{2} g_{15} - g_{16} } \right]} \mathord{\left/ {\vphantom {{\left[ {k_{n}^{4} a_{6} \varepsilon_{1} \omega^{2} - k_{n}^{2} g_{15} - g_{16} } \right]} {\left[ {k_{n}^{4} a_{6} \varepsilon - k_{n}^{2} g_{13} + g_{14} } \right]}}} \right. \kern-0pt} {\left[ {k_{n}^{4} a_{6} \varepsilon - k_{n}^{2} g_{13} + g_{14} } \right]}},\quad H_{2n} = {{\left[ { - A_{4} \left( {k_{n}^{2} - A_{5} } \right)} \right]} \mathord{\left/ {\vphantom {{\left[ { - A_{4} \left( {k_{n}^{2} - A_{5} } \right)} \right]} {\left[ {k_{n}^{4} - k_{n}^{2} g_{5} + g_{6} } \right]}}} \right. \kern-0pt} {\left[ {k_{n}^{4} - k_{n}^{2} g_{5} + g_{6} } \right]}},} \hfill \\ {H_{3n} = {{\left[ {a_{4} A_{4} \left( {k_{n}^{2} - b^{2} } \right)} \right]} \mathord{\left/ {\vphantom {{\left[ {a_{4} A_{4} \left( {k_{n}^{2} - b^{2} } \right)} \right]} {\left[ {k_{n}^{4} - k_{n}^{2} g_{5} + g_{6} } \right]}}} \right. \kern-0pt} {\left[ {k_{n}^{4} - k_{n}^{2} g_{5} + g_{6} } \right]}},\quad H_{4n} = {{\left[ {a_{8} \left( {k_{n}^{2} - b^{2} } \right) - a_{9} H_{1n} } \right]} \mathord{\left/ {\vphantom {{\left[ {a_{8} \left( {k_{n}^{2} - b^{2} } \right) - a_{9} H_{1n} } \right]} {\left[ {a_{6} k_{n}^{2} - A_{6} } \right]}}} \right. \kern-0pt} {\left[ {a_{6} k_{n}^{2} - A_{6} } \right]}}.} \hfill \\ {H_{5n} = \left( {1 - \beta^{*} T_{0} } \right)\left[ {a_{10} H_{4n} + iba_{11} (ib - k_{n} H_{2n} ) + k_{n} a_{12} (k_{n} + ibH_{2n} ) - H_{1n} } \right],} \hfill \\ {H_{6n} = \left( {1 - \beta^{*} T_{0} } \right)\left[ {a_{10} H_{4n} + k_{n} a_{11} (k_{n} + ibH_{2n} ) + iba_{12} (ib - k_{n} H_{2n} ) - H_{1n} } \right],} \hfill \\ {H_{7n} = \left( {1 - \beta^{*} T_{0} } \right)\left[ { - k_{n} (ib - k_{n} H_{2n} ) - a_{13} ib(k_{n} + ibH_{2n} ) + a_{14} H_{3n} } \right],} \hfill \\ {H_{8n} = \left( {1 - \beta^{*} T_{0} } \right)\left[ { - ib(k_{n} + ibH_{2n} ) - k_{n} a_{13} (ib - k_{n} H_{2n} ) + a_{14} H_{3n} } \right].} \hfill \\ \end{array}$$

(B2)

$$\begin{array}{*{20}l} {A = g_{18} /g_{17} ,\quad B = g_{19} /g_{17} ,\quad C = g_{20} /g_{17} ,\quad E = g_{21} /g_{17} ,\quad F = g_{22} /g_{17} ,\quad g_{1} = \varepsilon_{4} \omega \alpha^{2} - a_{1} \varepsilon_{1} \omega^{2} ,} \hfill \\ {g_{2} = - \varepsilon_{4} \omega A_{1} + a_{1} \varepsilon_{1} \omega^{2} b^{2} ,\quad g_{3} = a_{1} \left( {\varepsilon b^{2} + \omega^{2} } \right) + a^{\prime}_{0} \varepsilon_{4} \omega ,\quad g_{4} = A_{2} \varepsilon_{4} \omega ,\quad g_{5} = A_{3} + A_{5} - a_{3} a_{4} ,\quad g_{6} = A_{3} A_{5} - a_{3} a_{4} b^{2} ,} \hfill \\ {g_{7} = g_{2} - g_{1} g_{5} ,\quad g_{8} = - g_{2} g_{5} + g_{1} g_{6} + g_{4} A_{4} ,\quad g_{9} = g_{2} g_{6} - g_{4} A_{4} A_{5} ,\quad g_{10} = - g_{3} - a_{1} \varepsilon g_{5} ,\quad g_{11} = g_{3} g_{5} + a_{1} \varepsilon g_{6} ,} \hfill \\ {g_{12} = g_{3} g_{6} ,\quad g_{13} = a_{6} \left( {\varepsilon b^{2} + \omega^{2} } \right) + A_{6} \varepsilon ,\quad g_{14} = \varepsilon_{4} a_{9} \omega + A_{6} \left( {\varepsilon b^{2} + \omega^{2} } \right),\quad g_{15} = \varepsilon_{1} a_{6} \omega^{2} b^{2} + A_{6} \varepsilon_{1} \omega^{2} - a_{8} \varepsilon_{4} \omega ,} \hfill \\ {g_{16} = \varepsilon_{4} a_{8} \omega b^{2} - \varepsilon_{1} \omega^{2} A_{6} b^{2} ,\quad g_{17} = \varepsilon \left( {a_{6} g_{1} + a_{1} \varepsilon_{1} \omega^{2} a_{6} } \right),\quad g_{18} = - a_{6} \left( {\varepsilon g_{7} + \varepsilon_{1} \omega^{2} g_{10} } \right) + g_{13} g_{1} + a_{1} \varepsilon g_{15} ,} \hfill \\ {g_{19} = - a_{6} \left( {\varepsilon g_{8} + \varepsilon_{1} \omega^{2} g_{11} } \right) - g_{13} g_{7} + g_{14} g_{1} - a_{1} \varepsilon g_{16} - g_{10} g_{15} ,} \hfill \\ \quad {g_{20} = - a_{6} \varepsilon g_{9} + g_{13} g_{8} - g_{14} g_{7} + g_{10} g_{16} + g_{11} g_{15} + g_{12} a_{6} \varepsilon_{1} \omega^{2} ,\quad g_{21} = - g_{13} g_{9} + g_{14} g_{8} - g_{11} g_{16} + g_{12} g_{15} .} \hfill \\ \end{array}$$

(B3)