Effect of Inclined Load and Initial Stress on Plane Waves of Thermoelastic Rotating Medium via Three-Phase-Lag Model

Abstract

In this work, the effect of initial stress and oblique loading on plane waves in thermoelastic media will be investigated in the context of three-phase-lag theory. The entire elastic medium rotates at a uniform angular velocity. The problem is solved numerically by normal modal analysis. Plot and analyze the numerical results for temperature, displacement components, and stress. Graphical results show that the effects of phase delay, tilt angle, and initial stress parameters are evident. Variations in these quantities are plotted in the three-phase hysteresis model (3PHL) and the type III Green and Naghdi theory (G-N III) for insulated boundaries to show the effect of initial stress and angle of inclination in the medium.

Appendix

${{a}_{2}} = {{a}^{2}} - {{b}^{2}} - {{\Omega }^{2}},$ ${{a}_{3}} = 2ib\Omega ,$ ${{a}_{4}} = - {{\Omega }^{2}} - {{b}^{2}},$ ${{a}_{5}} = 2ib\Omega ,$ ${{b}_{1}} = \;{{b}_{7}}{{b}_{9}},$ ${{b}_{2}} = {{b}_{1}}{{a}^{2}},$ ${{b}_{3}} = K{\text{*}} - ib{{b}_{5}} - {{b}_{6}},$ ${{b}_{4}}\, = \,K{\text{*}}{{a}^{2}} - ib{{b}_{5}}{{a}^{2}} - {{a}^{2}}{{b}_{6}} - {{b}_{7}}{{b}_{8}}$, ${{b}_{5}} = K{{\eta }_{0}} + K{\text{*}}{{\tau }_{\nu }}$, ${{b}_{6}} = K{{\eta }_{0}}{{\tau }_{T}}{{b}^{2}}$, ${{b}_{7}} = 1 - ib{{\tau }_{q}} - \frac{1}{2}{{b}^{2}}\tau _{q}^{2}$, ${{b}_{8}} = \rho C_{0}^{2}{{C}_{E}}{{b}^{2}},$ b₉ = $\frac{{{{\gamma }_{1}}{{T}_{0}}{{b}^{2}}}}{\rho },$ ${{b}_{{10}}} = {{b}_{3}}{{a}_{1}},$ ${{b}_{{11}}} = {{b}_{3}}{{a}_{1}}{{a}^{2}} + {{a}_{4}}{{b}_{3}} + {{b}_{4}}{{a}_{1}} + {{a}_{2}}{{b}_{3}}{{a}_{1}} - {{b}_{1}}{{a}_{1}},$ b₁₂ = ${{b}_{4}}{{a}^{2}}{{a}_{1}}\, + \,{{b}_{4}}{{a}_{4}}\, + \,{{a}_{2}}{{b}_{3}}{{a}_{1}}{{a}^{2}}\, + \,{{a}_{2}}{{b}_{3}}{{a}_{4}}$ + b₄a₂a₁ – ${{a}_{1}}{{a}^{2}}{{b}_{1}}\, - \,{{b}_{1}}{{a}_{4}}\, - \,{{a}_{1}}{{b}_{2}}\, + \,{{a}_{3}}{{a}_{5}}{{b}_{3}},$ ${{b}_{{13}}} = {{b}_{4}}{{a}^{2}}{{a}_{1}}{{a}_{2}} + {{a}_{2}}{{b}_{4}}{{a}_{4}} - {{a}^{2}}{{a}_{1}}{{b}_{2}} - {{b}_{2}}{{a}_{4}} + {{b}_{4}}{{a}_{3}}{{a}_{5}},$ ${{L}_{0}} = \frac{{{{b}_{{11}}}}}{{{{b}_{{10}}}}},$ L₁ = $\frac{{{{b}_{{12}}}}}{{{{b}_{{10}}}}},$ L₂ = $\frac{{{{b}_{{13}}}}}{{{{b}_{{10}}}}},$ H_1n = $\frac{{{{b}_{2}} - {{b}_{1}}k_{n}^{2}}}{{{{b}_{3}}k_{n}^{2} - {{b}_{4}}}},$ H_2n = $- \frac{{{{a}_{5}}}}{{{{a}_{1}}(k_{n}^{2} - {{a}^{2}}) - {{a}_{4}}}},$ H_3n = $ia - {{k}_{n}}{{H}_{{2n}}},$ H_4n = $- ({{k}_{n}} + ia{{H}_{{2n}}}),$ H_5n = $\frac{\lambda }{{\rho {{C}_{0}}^{2}}}(k_{n}^{2} - {{a}^{2}}) - \frac{{2\mu }}{{\rho {{C}_{0}}^{2}}}({{a}^{2}} + ia{{k}_{n}}{{H}_{{2n}}})$ – H_1n, H_6n = $\frac{\lambda }{{\rho {{C}_{0}}^{2}}}(k_{n}^{2} - {{a}^{2}}) + \frac{{2\mu }}{{\rho {{C}_{0}}^{2}}}(k_{n}^{2} + ia{{k}_{n}}{{H}_{{2n}}})$ – H_1n, H_7n = $\frac{\mu }{{\rho C_{0}^{2}}}( - 2{{k}_{n}}ia + {{H}_{{2n}}}k_{n}^{2} + {{a}^{2}}{{H}_{{2n}}}) - \frac{p}{2}\frac{{{{\gamma }_{1}}{{T}_{0}}}}{{\rho C_{0}^{2}}}(k_{n}^{2} - {{a}^{2}}){{H}_{{2n}}}.$