Three Dimensional Fibre-Reinforce Anisotropic Half Space with Lagging Behavior in the Presence of Heat Source and Gravity

Abstract

A three dimensional fibre-reinforce thermoelastic model is proposed here for an anisotropic medium with lagging behaviour in the presence of heat source and gravity. In the context of memory dependent derivative, the basic governing equations and heat conduction equation are adopted here. The analytical solutions are obtained by forming the vector–matrix differential equation in the transform domain. The graphical representation and numerical analysis of diffrerent physical variables with respect to different space variables have been done to verify the initial and boundary conditions.

Appendix

$A_{11}=\lambda +2\alpha +4\mu _{L}-2\mu _{T}+\beta$, $A_{12}=\lambda +\alpha$, $A_{13}=\mu _{T}$, $A_{14}=\alpha +2\mu _{T}$, $A_{22}=\lambda +2\mu _{T}$

$$\begin{aligned} \beta _{1}= & {} \frac{A_{11}}{\lambda +2\mu }, \beta _{2}=-\frac{(\lambda +\alpha )i \psi _{2}}{\lambda +2\mu }, \beta _{3}=-\frac{(\lambda +\alpha )i \psi _{3}}{\lambda +2\mu },\\ \beta _{4}= & {} -\frac{\beta _{11}}{\gamma }(1+\frac{\epsilon _{10} G)}{\omega }, \beta _{5}=\frac{\lambda +\alpha }{\lambda +2\mu }, \beta _{6}=-\frac{A_{22}i \psi _{2}}{\lambda +2\mu },\\ \beta _{7}= & {} -\frac{\lambda i \psi _{3}}{\lambda +2\mu }, \beta _{8}=-\frac{\beta _{22}}{\gamma }(1+\frac{\epsilon _{10} G)}{\omega }, \beta _{9}=\frac{\lambda +\alpha }{\lambda +2\mu },\\ \beta _{10}= & {} -\frac{\lambda i \psi _{2}}{\lambda +2\mu }, \beta _{11}=-\frac{A_{22} i \psi _{3}}{\lambda +2\mu }, \beta _{12}=-\frac{\beta _{33}}{\gamma }(1+\frac{\epsilon _{10} G)}{\omega },\\ \beta _{13}= & {} -\frac{i \psi _{3}\mu _{T}}{\lambda +2\mu }, \beta _{14}=-\frac{i \psi _{2}\mu _{T}}{\lambda +2\mu }, \beta _{15}=-\frac{i \psi _{3}\mu _{T}}{\lambda +2\mu },\\ \beta _{16}= & {} \frac{\mu _{T}}{\lambda +2\mu }, \beta _{17}=-\frac{i \psi _{2}\mu _{T}}{\lambda +2\mu }, \beta _{18}=\frac{\mu _{T}}{\lambda +2\mu } \end{aligned}$$

$$\begin{aligned} \epsilon _{1}= & {} A_{11}+\mu _{0} {H_{0}}^{2}, \epsilon _{2}=\mu _{T}+\frac{P}{2}, \epsilon _{3}=A_{12}+\mu _{0} {H_{0}}^{2}-\frac{P}{2}+\mu _{T}, \epsilon _{4}=\frac{\rho g}{2 c_{0} \eta },\\ \epsilon _{5}= & {} -\beta _{11}\left( \frac{\lambda +2\mu }{\gamma }\right) \\ \epsilon _{6}= & {} c_{0}^2\left( \rho +\epsilon _{0}\mu _{0}^{2}H_{0}^{2}\right) , \epsilon _{7}=-\frac{2 \rho \varOmega _{2}}{c_0}, \epsilon _{8}=\frac{2 \rho \varOmega _{1}}{c_0}, \epsilon _{9}=-\rho \left( \omega _1^2+\varOmega _{2}^2\right) , \epsilon _{10}=\frac{\gamma }{c_{0}^2 \eta }\\ \epsilon _{11}= & {} A_{13}+\frac{P}{2}, \epsilon _{12}=A_{22}+\mu _{0} {H_{0}}^{2}, \epsilon _{13}=A_{13}+\frac{P}{2}, \epsilon _{14}=A_{12}+\mu _{0} {H_{0}}^{2}-\frac{P}{2}+\mu _{T},\\ \epsilon _{15}= & {} A_{14}+\mu _{0} {H_{0}}^{2}-\frac{P}{2}\\ \epsilon _{16}= & {} \frac{\rho g}{2 c_{0} \eta }, \epsilon _{17}=-\beta _{22} \left( \frac{\lambda +2\mu }{\gamma }\right) , \epsilon _{18}=\left( \rho +\epsilon _{0}\mu _{0}^{2} H_{0}^{2}\right) c_{0}, \epsilon _{19}=2 c_0 \rho , \epsilon _{20}=-\rho \varOmega _{2}^2,\\ \epsilon _{21}= & {} \rho \varOmega _{1}\varOmega _{2} \epsilon _{22}=A_{13}+\frac{P}{2}, \epsilon _{23}=A_{22}, \epsilon _{24}=A_{12}-\frac{P}{2}, \epsilon _{25}=A_{14}-\frac{P}{2},\\ \epsilon _{26}= & {} -\frac{\rho g }{2 c_0 \eta }, \epsilon _{27}=-\beta _{33}\left( \frac{\lambda +2\mu }{\gamma }\right) , \epsilon _{28}=\rho c_0^2, \epsilon _{29}=\rho \omega _1 \omega _2, \epsilon _{30}=-\rho \varOmega _1r,\\ \epsilon _{31}= & {} - 2 \rho c_0 \varOmega _1, \epsilon _{33}=\epsilon =\frac{\gamma ^{2}\theta _{0}}{\kappa \eta (\lambda +2\mu )}, \epsilon _{34}=- 1, \epsilon _{35}=\frac{\kappa \tau _{T} c_{0}^{2} \eta }{\kappa ^{*}}, \epsilon _{36}=\frac{\tau _{v}^{*}}{\kappa ^{*}}\\ \epsilon _{37}= & {} \frac{\tau _{\kappa }}{\kappa ^{*}}, \epsilon _{38}=\left( 1+\frac{\tau _{\theta } G}{ \omega c_0^{2} \eta } \right) (1+\epsilon _{35} s^{2}+\epsilon _{36} s), \epsilon _{39}=\epsilon _{37} s\left( 1+\frac{\tau _{q} G}{\omega c_{0}^{2}\eta }+\frac{\tau _{q}^{2} G s}{2\omega c_{0}^{4}\eta ^{2}}\right) \\ \epsilon _{40}= & {} \epsilon _{33} \epsilon _{37}s\left( 1+\frac{\tau _{q} G}{\omega c_{0}^{2}\eta }+\frac{\tau _{q}^{2} G s}{2\omega c_{0}^{4}\eta ^{2}}\right) , \epsilon _{41}=\epsilon _{34} \epsilon _{37}\left( 1+\frac{\tau _{q} G}{\omega c_{0}^{2}\eta }+\frac{\tau _{q}^{2} G s}{2\omega c_{0}^{4}\eta ^{2}}\right) \\ \epsilon _{51}= & {} s+\frac{\tau _{q}s G}{c_{0}^{2}\eta \omega }\!+\!\frac{\tau _{q}^{2}s^{2} G}{2c_{0}^{4}\eta ^{2}\omega }, \epsilon _{52}\!=\!-\left( s\left( 1+\frac{\tau _{T}G}{c_{0}^{2}\eta \omega }\right) \!+\!\frac{\kappa ^{*}}{\kappa c_{0}^{2}\eta }\left( 1+\frac{\tau _{v}^{*}G}{c_{0}^{2}\eta \omega }\right) \right) , \epsilon _{100}=\frac{\epsilon _{52}}{\epsilon _{51}} \end{aligned}$$

$$\begin{aligned} C_{61}= & {} s\frac{\epsilon _{19}}{\epsilon _{11}}, C_{62}=\frac{1}{\epsilon _{11}}(\epsilon _{12}\psi _{2}^2+\epsilon _{13}\psi _{3}^2+i \epsilon _{16}\psi _{3}+\epsilon _{18} s^{2}+\epsilon _{20}),\\ C_{63}= & {} \frac{1}{\epsilon _{11}}(-\epsilon _{15}\psi _{2}\psi _{3}+\epsilon _{21}+i\epsilon _{16}\psi _{2})\\ C_{64}= & {} \frac{\epsilon _{17} i \psi _{2}}{\epsilon _{11}}\left( 1+ \frac{\epsilon _{10} G}{\omega }\right) C_{65}=\frac{\epsilon _{14} i \psi _{2}}{\epsilon _{11}}),\\ C_{66}= & {} C_{67}=C_{68}=0\\ C_{71}= & {} s\frac{\epsilon _{31}}{\epsilon _{22}}, C_{72}=\frac{1}{\epsilon _{22}}(\psi _{2}\psi _{3}\epsilon _{25}+i\psi _{2}\epsilon _{26}+\epsilon _{29}),\\ C_{73}= & {} \frac{1}{\epsilon _{22}}(\psi _{2}^{2}\epsilon _{22}+i\psi _{3}^{2}\epsilon _{22}+\epsilon _{30}+s^2+\epsilon _{28})\\ C_{74}= & {} i \psi _{3} \frac{\epsilon _{27}}{\epsilon _{22}}\left( 1+\frac{\epsilon _{10}G}{\omega }\right) , C_{75}=\frac{1}{\epsilon _{22}}(i \psi _{3} \epsilon _{24}-\epsilon _{26}), C_{76}=C_{77}=C_{78}=0\\ C_{81}= & {} 0, C_{82}=-i \psi _{2} \frac{\epsilon _{40}}{\epsilon _{38}}, C_{83}=-i \psi _{3} \frac{\epsilon _{40}}{\epsilon _{38}}, C_{84}=\psi _{2}^{2}+ \psi _{3}^{2}+\frac{\epsilon _{39}}{\epsilon _{38}}, C_{85}=\frac{\epsilon _{40}}{\epsilon _{38}},\\ C_{86}= & {} C_{87}=C_{88}=0, C_{89}=\frac{\epsilon _{41}}{\epsilon _{38}} Q_{1} \end{aligned}$$

$$\begin{aligned} f_{11}= & {} C_{51}+\lambda C_{55}-\lambda ^{2},~~f_{12}=C_{52}+\lambda C_{56},~~f_{13}=C_{53}+\lambda C_{57},~~f_{14}=C_{54}+\lambda C_{58},\\ f_{21}= & {} C_{61}+\lambda C_{65},~~f_{22}=C_{62}+\lambda C_{66}-\lambda ^{2},~~f_{23}=C_{63}+\lambda C_{67},~~f_{24}=C_{64}+\lambda C_{68},\\ f_{31}= & {} C_{51}+\lambda C_{75},~~f_{32}=C_{72}+\lambda C_{76},~~f_{33}=C_{73}+\lambda C_{77}-\lambda ^{2},~~f_{34}=C_{74}+\lambda C_{78},\\ f_{41}= & {} C_{81}+\lambda C_{85},~~f_{42}=C_{82}+\lambda C_{86},~~f_{43}=C_{83}+\lambda C_{87},~~f_{44}=C_{84}+\lambda C_{88}-\lambda ^{2},\\ \end{aligned}$$

$$\begin{aligned}M_{21}=\ \left[ \begin{array}{cccc} C_{51} &{} \quad C_{52} &{} \quad C_{53} &{} \quad C_{54} \\ C_{61} &{} \quad C_{62} &{} \quad C_{63} &{} \quad C_{64} \\ C_{71} &{} \quad C_{72} &{} \quad C_{73} &{} \quad C_{74} \\ C_{81} &{} \quad C_{82} &{} \quad C_{83} &{} \quad C_{84} \end{array} \right] M_{22}=\ \left[ \begin{array}{cccc} C_{55} &{} \quad C_{56} &{} \quad C_{57} &{} \quad C_{58} \\ C_{65} &{} \quad C_{66} &{} \quad C_{67} &{} \quad C_{68} \\ C_{75} &{} \quad C_{76} &{} \quad C_{77} &{} \quad C_{78} \\ C_{85} &{} \quad C_{86} &{} \quad C_{87} &{} \quad C_{88} \end{array} \right] \end{aligned}$$

$R_{1j}=(\beta _{1}\lambda _{1}x_{1j}+\beta _{2}x_{2j}+\beta _{3}x_{3j}+\beta _{4}x_{4j})e^{\lambda _{j}x_{1}}$, $R_{2j}=(\beta _{5}\lambda _{1}x_{1j}+\beta _{6}x_{2j}+\beta _{7}x_{3j}+\beta _{8}x_{4j})e^{\lambda _{j}x_{1}}$, $R_{3j}=(\beta _{9}\lambda _{1}x_{1j}+\beta _{10}x_{2j}+\beta _{11}x_{3j}+\beta _{12}x_{4j})e^{\lambda _{j}x_{1}}$, $R_{4j}=(\beta _{13} x_{2j}+\beta _{14}x_{3j})e^{\lambda _{j}x_{1}}$,

$$\begin{aligned} \begin{array}{lll} R_{5j}=(\beta _{15} x_{1j}+\beta _{16}\lambda _{j} x_{3j})e^{\lambda _{j} x_{1}},&{}&{} R_{6j}=(\beta _{17} x_{1j}+\beta _{18}\lambda _{j} x_{2j})e^{\lambda _{j} x_{1}},\\ R_{7j}=\epsilon _{100}(\lambda {j}-i(\psi _{2}+\psi _{3}))x_{4j}e^{\lambda _{j}x_{1}},&{}&{} R_{8j}=x_{4j} e^{\lambda _{j}x_{1}}\\ \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{lll} N_{1j}=\frac{1}{\lambda _{j}}(\beta _{2}x_{1j}+\beta _{3}x_{3j}+\beta _{4}x_{4j}),&{}&{} N_{2j}=\frac{1}{\lambda _{j}}(\beta _{6}x_{2j}+\beta _{7}x_{3j}+\beta _{8}x_{4j}),\\ N_{3j}=\frac{1}{\lambda _{j}}(\beta _{10}x_{2j}+\beta _{11}x_{3j}+\beta _{12}x_{4j}),&{}&{} N_{4j}=\frac{1}{\lambda _{j}}(\beta _{13}x_{2j}+\beta _{14}x_{3j}),\\ N_{5j}=\frac{1}{\lambda _{j}}(\beta _{15}x_{1j}),&{}&{} N_{6j}=\frac{1}{\lambda _{j}}(\beta _{17}x_{1j}),\\ N_{7j}=-\frac{\epsilon _{100}i(\psi _{2}+\psi _{3})}{\lambda _{j}}x_{4j},&{}&{} N_{8j}=\frac{1}{\lambda _{j}}x_{4j}\\ \end{array} \end{aligned}$$

$$\begin{aligned} d_1= & {} \sum _{j=1}^{8}Q_j N_{1j},d_6=\sum _{j=1}^{8}Q_j N_{6j} \\ d_5= & {} \sum _{j=1}^{8}Q_j N_{5j},d_8=\sum _{j=1}^{8}Q_j N_{8j}+d_{10} \end{aligned}$$

$L_{1j}=\beta _{1}\lambda _{1}x_{1j}+\beta _{2}x_{2j}+\beta _{3}x_{3j}+\beta _{4}x_{4j}$, $L_{5j}=\beta _{15} x_{1j}+\beta _{16}\lambda _{j} x_{3j}$, $L_{6j}=\beta _{17} x_{1j}+\beta _{18}\lambda _{j} x_{2j}$, $L_{8j}=x_{4j}$

$$\begin{aligned} D= & {} \left| \begin{array}{cccc} L_{11} &{} \quad L_{12} &{} \quad L_{13} &{} \quad L_{14} \\ L_{61} &{} \quad L_{62} &{} \quad L_{63} &{} \quad L_{64} \\ L_{51} &{} \quad L_{52} &{} \quad L_{53} &{} \quad L_{54} \\ L_{81} &{} \quad L_{82} &{} \quad L_{83} &{} \quad L_{84} \end{array} \right| D_1=\left| \begin{array}{cccc} d_{1} &{} \quad L_{12} &{} \quad L_{13} &{} \quad L_{14} \\ d_{6} &{} \quad L_{62} &{} \quad L_{63} &{} \quad L_{64} \\ d_{5} &{} \quad L_{52} &{} \quad L_{53} &{} \quad L_{54} \\ d_{8} &{} \quad L_{82} &{} \quad L_{83} &{} \quad L_{84} \end{array} \right| D_2=\left| \begin{array}{cccc} L_{11} &{} \quad d_{1} &{} \quad L_{13} &{} \quad L_{14} \\ L_{61} &{} \quad d_{6} &{} \quad L_{63} &{} \quad L_{64} \\ L_{51} &{} \quad d_{5} &{} \quad L_{53} &{} \quad L_{54} \\ L_{81} &{} \quad d_{8} &{} \quad L_{83} &{} \quad L_{84} \end{array} \right| \\ D_3= & {} \left| \begin{array}{cccc} L_{11} &{} \quad L_{12} &{} \quad d_{1} &{} \quad L_{14} \\ L_{61} &{} \quad L_{62} &{} \quad d_{6} &{} \quad L_{64} \\ L_{51} &{} \quad L_{52} &{} \quad d_{5} &{} \quad L_{54} \\ L_{81} &{} \quad L_{82} &{} \quad d_{8} &{} \quad L_{84} \end{array} \right| D_4=\left| \begin{array}{cccc} L_{11} &{} \quad L_{12} &{} \quad L_{13} &{} \quad d_{1} \\ L_{61} &{} \quad L_{62} &{} \quad L_{63} &{} \quad d_{6} \\ L_{51} &{} \quad L_{52} &{} \quad L_{53} &{} \quad d_{5} \\ L_{81} &{} \quad L_{82} &{} \quad L_{83} &{} \quad d_{8} \end{array} \right| \end{aligned}$$