Dual-phase-lag model on micropolar thermoelastic rotating medium under the effect of thermal load due to laser pulse

Abstract

The present investigation deals with the deformation in micropolar thermoelastic medium with dual-phase-lag theory due to rotation subjected to the thermal laser pulse. The normal mode analysis technique is used to solve the problem. The material is heated by a non-Gaussian laser beam with pulse duration of 0.02 ps. The closed-form expressions of the normal stress, the tangential stress, the couple stress, and the temperature distribution are obtained. The variation of considered variables is depicted graphically to show the effect of rotation and time. Some particular cases of interest are deduced from the present investigation.

Appendices

Appendix 1

$$L_{1} = \frac{ - 1}{{\gamma^{ * 8} - A\gamma^{ * 6} + B\gamma^{ * 4} - C\gamma^{ * 2} + F}},N_{1} = a_{2} (a_{6} \gamma^{ * } - a_{7} ),N_{2} = a_{2} (a_{8} \gamma^{ * 2} - a_{9} ),$$

$$N_{3} = N_{2} (\gamma^{ * } - a_{1} ) + a_{2} N_{1} ,N_{4} = \frac{{N_{1} (a_{3} \gamma^{ * } - a_{4} ) - a_{2} N_{2} }}{{a_{5} }},N_{5} = N_{2} \left( {\frac{ - \,2x}{{r^{2} }}} \right) + \gamma^{ * 2} N_{1} ,$$

$$N_{6} = N_{1} \left( {\frac{ - \,2x}{{r^{2} }}} \right) - \gamma^{ * } N_{2} ,N_{7} = \left( {\frac{ - \,2x}{{r^{2} }}} \right)N_{5} - \frac{{\lambda \gamma^{ * } }}{{\rho C_{0}^{2} }}N_{6} - N_{3} ,H_{1n} = \frac{{s_{1} k_{n}^{4} - s_{2} k_{n}^{2} - s_{3} }}{{a_{2} a_{6} k_{n}^{2} - a_{2} a_{7} }},$$

$$N_{8} = \frac{\lambda }{{\rho {\kern 1pt} {\kern 1pt} C_{0}^{2} }}\left( {\frac{ - \,2x}{{r^{2} }}N_{5} - \gamma^{ * } N_{6} } \right) - N_{3} ,N_{9} = \frac{\lambda }{{\rho C_{0}^{2} }}\left( {\frac{ - \,2x}{{r^{2} }}} \right)N_{5} - \gamma^{ * } N_{6} - N_{3} ,H_{4n} = iaH_{1n} - k_{n} ,$$

$$N_{10} = \frac{\mu + k}{{\rho C_{0}^{2} }}\left( {\frac{ - 2x}{{r^{2} }}} \right)N_{6} - \frac{\mu }{{\rho C_{0}^{2} }}\gamma^{ * } N_{5} + \frac{k}{{\rho C_{0}^{2} }}N_{4} ,f^{ * } (x,t) = \left[ {t + \tau_{q} \left( {1 - \frac{t}{{t_{0} }}} \right)} \right]\exp \left( { - \frac{{x^{2} }}{{r^{2} }} - \frac{t}{{t_{0} }}} \right),$$

$$N_{11} = \frac{\mu }{{\rho C_{0}^{2} }}\left( {\frac{ - 2x}{{r^{2} }}} \right)N_{6} - \frac{\mu + k}{{\rho C_{0}^{2} }}\gamma^{ * } N_{5} - \frac{k}{{\rho C_{0}^{2} }}N_{4} ,N_{12} = \frac{{\eta_{0}^{2} \gamma }}{{\rho C_{0}^{4} }}\left( {\frac{ - 2x}{{r^{2} }}} \right)N_{4} ,N_{13} = \frac{{\eta_{0}^{2} \gamma }}{{\rho C_{0}^{4} }}\gamma^{ * } N_{4} ,$$

$$H_{2n} = k_{n}^{2} H_{1n} - a_{1} + a_{2} ,H_{3n} = \frac{{(a_{3} k_{n}^{2} - a_{4} - a_{2} H_{1n} )}}{{a_{5} }},H_{6n} = iaH_{4n} - \frac{\lambda }{{\rho C_{0}^{2} }}k_{n} H_{5n} - H_{2n} ,$$

$$H_{5n} = - k_{n} H_{1n} + ia,H_{7n} = \frac{\lambda }{{\rho C_{0}^{2} }}\left( {ia{\kern 1pt} H_{4n} - k_{n} H_{5n} } \right) - H_{2n} ,H_{11n} = \frac{{\gamma \eta_{0}^{2} }}{{\rho C_{0}^{4} }}(iaH_{3n} ),$$

$$H_{8n} = - k_{n} H_{5n} + \frac{\lambda ia}{{\rho C_{0}^{2} }}H_{4n} - H_{2n} ,H_{9n} = \frac{1}{{\rho C_{0}^{2} }}\left[ {ia\left( {\mu + k} \right)H_{5n} - k_{n} \mu H_{4n} + kH_{3n} } \right],$$

$$H_{10n} = \frac{1}{{\rho C_{0}^{2} }}\left[ {\left( {\mu + k} \right)k_{n} H_{4n} + ia\mu H_{4n} - kH_{3n} } \right],H_{12n} = \frac{{\gamma \eta_{0}^{2} }}{{\rho C_{0}^{4} }}( - k_{n} H_{3n} ),n = 1,2,3,4.$$

Appendix 2

$$L_{2} = \frac{ - 1}{{\gamma^{ * 4} - A_{1} \gamma^{ * 2} + A_{2} }},N_{1}^{ * } = \gamma^{ * } - r_{1} ,N_{2}^{ * } = \left( {\frac{ - 2x}{{r^{2} }}} \right)^{2} + \frac{\lambda }{{\rho C_{0}^{2} }}\gamma^{ * 2} - N_{1}^{ * } ,$$

$$N_{4}^{ * } = \frac{\lambda }{{\rho C_{0}^{2} }}\left( {\frac{ - 2x}{{r^{2} }}} \right)^{2} + \gamma^{ * 2} ,N_{3} = N_{2} (\gamma^{ * } - a_{1} ) + a_{2} N_{1} ,N_{4} = \frac{{N_{1} (a_{3} \gamma^{ * } - a_{4} ) - a_{2} N_{2} }}{{a_{5} }},$$

$$N_{5} = N_{2} \left( {\frac{ - 2x}{{r^{2} }}} \right) + \gamma^{ * 2} N_{1} ,N_{6} = N_{1} \left( {\frac{ - 2x}{{r^{2} }}} \right) - \gamma^{ * } N_{2} ,N_{7} = \left( {\frac{ - 2x}{{r^{2} }}} \right)N_{5} - \frac{{\lambda \gamma^{ * } }}{{\rho C_{0}^{2} }}N_{6} - N_{3} ,$$

$$N_{3}^{ * } = \frac{\lambda }{{\rho C_{0}^{2} }}\left[ {\left( {\frac{ - 2x}{{r^{2} }}} \right)^{2} + \gamma^{ * 2} } \right],N_{5}^{ * } = \frac{k + 2\mu }{{\rho C_{0}^{2} }}\left( {\frac{2x}{{r^{2} }}\gamma^{ * } } \right),N_{6}^{ * } = \frac{1}{{\rho C_{0}^{2} }}\left[ {\left( {\frac{ - 2x}{{r^{2} }}} \right)^{2} (k + \mu ) + \mu \gamma^{ * 2} } \right],$$

$$H_{1\hbar }^{ * } = \left( {\alpha_{\hbar }^{2} - r_{1} } \right),H_{2\ell }^{ * } = \frac{{a_{3} \alpha_{\ell }^{2} - r_{2} }}{{a_{5} }},H_{3\hbar }^{ * } = \frac{{\lambda \alpha_{\hbar }^{2} }}{{\rho C_{0}^{2} }} - a^{2} - H_{1\hbar }^{ * } ,H_{4\ell }^{ * } = ia\left( {\alpha_{\ell } - \frac{{\lambda \alpha_{\ell } }}{{\rho C_{0}^{2} }}} \right),$$

$$H_{5\hbar }^{ * } = \frac{\lambda }{{\rho C_{0}^{2} }}\left( {\alpha_{\hbar }^{2} - a^{2} } \right) - H_{1\hbar }^{ * } ,H_{6\hbar }^{ * } = \alpha_{\hbar }^{2} - \frac{{\lambda a^{2} }}{{\rho C_{0}^{2} }} - H_{1\hbar }^{ * } ,H_{7\ell }^{ * } = ia\left( {\frac{{\lambda \alpha_{\ell } }}{{\rho C_{0}^{2} }} - \alpha_{\ell } } \right),$$

$$H_{8\hbar }^{ * } = \frac{k + 2\mu }{{\rho C_{0}^{2} }}( - \;ia\alpha_{\hbar } ),H_{9\ell }^{ * } = \frac{1}{{\rho C_{0}^{2} }}\left[ {kH_{2\ell }^{ * } - a^{2} (k + \mu )} \right] + \alpha_{\ell }^{ * } ,$$

$$H_{10\hbar }^{ * } = \frac{1}{{\rho C_{0}^{2} }}\left[ {\mu \alpha_{\hbar }^{ * 2} - a^{2} (k + \mu )} \right],H_{11\ell }^{ * } = \frac{1}{{\rho C_{0}^{2} }}\left[ { - ia\mu \alpha_{\ell }^{ * } - ia\alpha_{\ell }^{ * } (k + \mu ) - kH_{2\ell }^{ * } } \right],$$

$$H_{12\ell }^{ * } = \frac{{\gamma \eta_{0}^{2} }}{{\rho C_{0}^{4} }}iaH_{2\ell }^{ * } ,H_{13\ell }^{ * } = \frac{{\gamma \eta_{0}^{2} }}{{\rho C_{0}^{4} }}( - \;\alpha_{\ell } )H_{2\ell }^{ * } ,\hbar = 1,2,\ell = 3,4.$$