Analysis of Wave Motion in Micropolar Thermoelastic Medium Based on Moore–Gibson–Thompson Heat Equation Under Non-local and Hyperbolic Two-Temperature

Abstract

The aim of the present investigation is to examine the impacts of non-local, hyperbolic two-temperature (HTT) and impedance parameters on the propagation of plane waves in the context of the micropolar thermoelastic medium under Moore–Gibson–Thompson (MGT) heat equation. The problem is formulated for two dimensional and simplified with the aid of dimensionless quantities and potential functions. A reflection technique is used to solve the problem. The amplitude ratios of reflected waves namely longitudinal displacement wave (LD-wave), thermal wave (T-wave), coupled transverse wave (CD-I wave), and coupled micro-rotational wave (CD-II wave) are obtained against the angle of incidence by applying impedance boundary restrictions. The characteristics of non-local, HTT and impedance parameters on amplitude ratios have been depicted graphically. Some special cases are also obtained for the present study. Physical views presented in the article may be useful for the composition of new materials, geophysics, earthquake engineering, and other scientific disciplines.

Appendices

Appendix I

$$\begin{aligned} {{\text{a}}}_{1}&=\frac{\uplambda +\upmu }{\uprho {{\text{c}}}_{1}^{2}},{{\text{a}}}_{2}=\frac{\upmu +{\text{K}}}{\uprho {{\text{c}}}_{1}^{2}},{{\text{a}}}_{3}=\frac{{\text{K}}{\upgamma }_{1}{{\text{T}}}_{0}}{{\uprho }^{2}{{\text{c}}}_{1}^{4}},{\text{ a}}_{4}=\frac{{\upgamma }_{1}{{\text{T}}}_{0}}{\uprho {{\text{c}}}_{1}^{2}},{{\text{a}}}_{5}=\frac{\upgamma }{\uprho {{\text{c}}}_{1}^{2}\widehat{{\text{j}}}},{{\text{a}}}_{6}=\frac{{{\text{Kc}}}_{1}^{2}}{\widehat{{\text{j}}}{\upgamma }_{1}{\upomega }_{1}^{2}{{\text{T}}}_{0}}, \\ {{\text{a}}}_{7}&=\frac{2{\text{K}}}{\widehat{{\text{j}}}\uprho {\upomega }_{1}^{2}},{{\text{a}}}_{8}=\frac{{{\text{K}}}^{*}}{{\upomega }_{1}{{\text{K}}}_{1}^{*}},{{\text{a}}}_{9}=\frac{{\upgamma }_{1}{{\text{c}}}_{1}^{2}}{{\upomega }_{1}{{\text{K}}}_{1}^{*}},{{\text{a}}}_{10}=\frac{\upmu }{{\upgamma }_{1}{{\text{T}}}_{0}},{{\text{a}}}_{11}=\frac{{\text{K}}}{{\upgamma }_{1}{{\text{T}}}_{0}},{{\text{a}}}_{12}=\frac{{\text{K}}}{\uprho {{\text{c}}}_{1}^{2}}, \\ {{\text{a}}}_{13}&=\frac{\uplambda }{{\upgamma }_{1}{{\text{T}}}_{0}},{{\text{a}}}_{14}=\frac{\left(\uplambda +2\upmu +{\text{K}}\right)}{{\upgamma }_{1}{{\text{T}}}_{0}},{{\text{a}}}_{15}=\frac{\upgamma {\upomega }_{1}^{2}}{\uprho {{\text{c}}}_{1}^{4}} \\ \end{aligned} $$

Appendix II

$$ \begin{aligned} {{\text{A}}}_{01}&=\frac{\left({\upiota \upomega }+{{\text{a}}}_{8}\right)+{\upiota \upomega }{\uptau }_{{\text{e}}}\left(1-{{\upxi }_{1}}^{2}{\upomega }^{2}+{{\text{a}}}_{4}{{\text{a}}}_{9}+{\upomega }^{2}{\upvarsigma }\right)}{\upiota {{\upomega \uptau }}_{{\text{e}}}},\hfill\\{{\text{A}}}_{02}&=\frac{\left({\upiota \upomega }+{{\text{a}}}_{8}\right)(1-{{\upxi }_{1}}^{2}{\upomega }^{2})+\upiota {\uptau }_{{\text{e}}}{\upomega }^{3}{\upvarsigma }\left(1-{{\upxi }_{1}}^{2}{\upomega }^{2}+{{\text{a}}}_{4}{{\text{a}}}_{9}\right)}{\upiota {{\upomega \uptau }}_{{\text{e}}}}, \\ {{\text{A}}}_{03}&=\frac{{\upomega }^{2}\left({{\text{a}}}_{2}+{{\text{a}}}_{5}+{{\text{a}}}_{7}{{\upxi }_{1}}^{2}-{\upomega }^{2}\left({{\upxi }_{1}}^{2}+{{\upxi }_{2}}^{2}\right)\right)-{{\text{a}}}_{2}{{\text{a}}}_{7}}{{{\text{a}}}_{7}-{\upomega }^{2}},\hfill\\ {{\text{A}}}_{04}&=\frac{{\upomega }^{4}\left({{\text{a}}}_{5}{{\upxi }_{1}}^{2}-{\upomega }^{2}{{\upxi }_{1}}^{2}{{\upxi }_{2}}^{2}+{{\text{a}}}_{2}{{\upxi }_{2}}^{2}\right)-{{\upomega }^{2}({\text{a}}}_{2}{{\text{a}}}_{5}+{{\text{a}}}_{3}{{\text{a}}}_{6})}{{{\text{a}}}_{7}-{\upomega }^{2}},\\ {\uptau }_{{\text{e}}}&=\left({\uptau }_{0}-\frac{\upiota }{\upomega }\right).\\ \end{aligned} $$

Appendix III

$${{\text{d}}}_{{\text{i}}}=\frac{\left[{\upomega }^{2}\left(1+{{\upxi }_{1}}^{2}{{\upkappa }_{{\text{i}}}}^{2}\right)-{{\upkappa }_{{\text{i}}}}^{2}\right]}{{{\text{a}}}_{4}\left(1-{\upomega }^{2}{\upvarsigma }{{\upkappa }_{{\text{i}}}}^{2}\right)}, {{\text{f}}}_{{\text{i}}}=\frac{{{\text{a}}}_{2}{\upkappa }_{{\text{j}}}^{2}-{\upomega }^{2}\left(1+{{\upxi }_{1}}^{2}{{\upkappa }_{{\text{j}}}}^{2}\right)}{{{\text{a}}}_{3}},\left({\text{i}}={1,2}\right),\left({\text{j}}={3,4}\right).$$

Appendix IV

$${{\text{b}}}_{1{\text{p}}}=-\left[{{\text{a}}}_{13}{{\text{sin}}}^{2}{\uptheta }_{{\text{p}}}+{{\text{a}}}_{14}{{\text{cos}}}^{2}{\uptheta }_{{\text{p}}}\right]{\upkappa }_{{\text{p}}}^{2}-{{\text{L}}}_{1},{{\text{b}}}_{1{\text{q}}}=\left[{\upkappa }_{{\text{q}}}^{2}\left({{\text{a}}}_{13}-{{\text{a}}}_{14}\right){\cos }{\uptheta }_{{\text{q}}}+{\upkappa }_{{\text{q}}}{{\text{z}}}_{1}{\upiota \omega }\right]{\sin }{\uptheta }_{{\text{q}}},$$

$${{\text{b}}}_{2{\text{p}}}=-{\upkappa }_{{\text{p}}}^{2}{\sin }{\uptheta }_{{\text{p}}}{\cos }{\uptheta }_{{\text{p}}}\left({2{\text{a}}}_{10}+{{\text{a}}}_{11}\right)+{\upkappa }_{{\text{p}}}{{\text{z}}}_{2}{\upiota \omega sin }{\uptheta }_{{\text{p}}}, {{\text{b}}}_{2{\text{q}}}={\upkappa }_{{\text{q}}}^{2}\left[{({\text{a}}}_{10}+{{\text{a}}}_{11}){{\text{cos}}}^{2}{\uptheta }_{q}-{{\text{a}}}_{10}{{\text{sin}}}^{2}{\uptheta }_{{\text{q}}}\right]-{{\text{L}}}_{2},$$

$${{\text{b}}}_{3{\text{p}}}=0,{{\text{b}}}_{3{\text{q}}}={{\text{a}}}_{15}{{\text{f}}}_{{\text{p}}}\upiota {\upkappa }_{{\text{q}}}{\text{cos}}{\uptheta }_{{\text{p}}}+{{\upomega z}}_{3}{{\text{f}}}_{{\text{p}}},{{\text{b}}}_{4{\text{p}}}=(1-{\upvarsigma }{\upkappa }_{{\text{p}}}^{2}){{\text{d}}}_{{\text{p}}}\left[{\upiota {\upkappa }_{{\text{p}}}{\text{K}}}_{1 }^{*}{\text{cos}}{\uptheta }_{{\text{p}}}+{{\upomega z}}_{4}\right],$$

$${{\text{b}}}_{4{\text{q}}}=0,{{\text{L}}}_{1}=\left(1-{\upvarsigma }{\upkappa }_{{\text{p}}}^{2}\right){{\text{d}}}_{{\text{p}}}-{\upkappa }_{{\text{p}}}{{\text{z}}}_{1}{\upiota \omega cos}{\uptheta }_{{\text{p}}}, {\upvarsigma }=\frac{{\upbeta }^{*}}{{{\text{w}}}^{2}},{{\text{L}}}_{2}={{\text{a}}}_{12}{{\text{f}}}_{{\text{p}}}+{\upkappa }_{{\text{q}}}{{\text{z}}}_{2}{\upiota \omega cos}{\uptheta }_{q},$$

$$\left({\text{p}}={1,2}\right),\left({\text{q}}={3,4}\right),$$

where ${{\text{R}}}_{1}, {{\text{R}}}_{2},{{\text{R}}}_{3}$ and ${{\text{R}}}_{4}$ are the amplitude ratios of reflected LD-wave, reflected T-wave, CD-I wave and CD-II wave making an angle ${\uptheta }_{1}, {\uptheta }_{2}, {\uptheta }_{3},$ and ${\uptheta }_{4}$ as shown in Fig. 1 and are given by

$${{\text{R}}}_{1}=\frac{{{\text{A}}}_{1}}{{{\text{A}}}^{*}}, {\text{ R}}_{2}=\frac{{{\text{A}}}_{2}}{{{\text{A}}}^{*}}, {{\text{R}}}_{3}=\frac{{{\text{B}}}_{1}}{{{\text{A}}}^{*}}, {{\text{R}}}_{4}=\frac{{{\text{B}}}_{2}}{{{\text{A}}}^{*}},$$

For incident LD-wave, ${{\text{A}}}^{*}={{\text{A}}}_{01},$

$${{\text{Y}}}_{1}=-{{\text{b}}}_{11},{{\text{Y}}}_{2}={{\text{b}}}_{21},{{\text{Y}}}_{3}={{\text{b}}}_{31},{{\text{Y}}}_{4}={{\text{b}}}_{41}.$$

For incident T-wave, ${{\text{A}}}^{*}={{\text{A}}}_{02},$

$${{\text{Y}}}_{1}=-{{\text{b}}}_{12},{{\text{Y}}}_{2}={{\text{b}}}_{22},{{\text{Y}}}_{3}={{\text{b}}}_{32},{{\text{Y}}}_{4}={{\text{b}}}_{42}.$$

For incident CD I-wave, ${{\text{A}}}^{*}={{\text{B}}}_{01},$

$${{\text{Y}}}_{1}={{\text{b}}}_{13},{{\text{Y}}}_{2}={-{\text{b}}}_{23},{{\text{Y}}}_{3}={{\text{b}}}_{33},{{\text{Y}}}_{4}={{\text{b}}}_{43}.$$

For incident CD II-wave, ${{\text{A}}}^{*}={{\text{B}}}_{02},$

$${{\text{Y}}}_{1}={{\text{b}}}_{14},{{\text{Y}}}_{2}={-{\text{b}}}_{24},{{\text{Y}}}_{3}={{\text{b}}}_{34},{{\text{Y}}}_{4}={{\text{b}}}_{44}.$$