Size-dependent thermoelastic vibrations of Timoshenko nanobeams by taking into account dual-phase-lagging effect

Abstract

In this article, size-dependent modeling and analysis of thermoelastic coupling effect on the oscillations of Timoshenko nanobeams are carried out. Small-scale effect on the nanostructure and heat conduction is taken into account with the aid of nonlocal strain gradient theory (NSGT) together with dual-phase-lag (DPL) heat conduction model. In order to illustrate the influence of nonclassical scale parameters on the coefficients of governing equations, the normalized forms of size-dependent equations of motion and heat conduction are established by definition of some dimensionless parameters. These coupled differential equations are then solved in the Laplace domain to attain the analytical thermoelastic responses of a simply supported Timoshenko nanobeam subjected to a dynamic load. Through several numerical examples, a detailed parametric study is performed to illuminate the decisive role of nonlocal, strain gradient and phase lag parameters in thermoelastic behavior of Timoshenko nanobeams. Furthermore, comparing the results corresponding to various relative magnitudes of nonlocal and strain gradient length scale parameters confirms the potential of NSGT for covering both hardening and softening characteristic of nanoscale structures.

Appendices

Appendix A

$$ \begin{aligned} A_{1} & = \frac{1}{12}\left( \frac{h}{L} \right)^{2} \left( \frac{l}{L} \right)^{2} ,\quad A_{2} = K_{s} \left( \frac{G}{E} \right)\left( \frac{l}{L} \right)^{2} + \frac{1}{12}\left( \frac{h}{L} \right)^{2} ,\\ A_{3} &= K_{s} \left( \frac{G}{E} \right),\quad A_{4} = K_{s} \left( \frac{G}{E} \right)\left( \frac{l}{L} \right)^{2} , \\ A_{5} & = \frac{1}{12}\left( \frac{h}{L} \right)^{2} ,\quad A_{6} = \frac{1}{12}\left( \frac{h}{L} \right)^{2} \left( \frac{ea}{L} \right)^{2} ,\quad A_{7} = \left( \frac{ea}{L} \right)^{2} ,\quad A_{8} = \left( {\frac{{\tau_{1} }}{{\tau_{2} }}} \right)\left( \frac{h}{L} \right)^{2} , \\ A_{9} & = \pi^{2} \left( {\frac{{\tau_{1} }}{{\tau_{2} }}} \right),\quad A_{10} = \left( {\frac{{\tau_{q} }}{{\tau_{1} }}} \right)\left( \frac{h}{L} \right)^{2} ,\quad A_{11} = \left( \frac{h}{L} \right)^{2} + \pi^{2} \left( {\frac{{\tau_{T} }}{{\tau_{2} }}} \right) ,\\ A_{12} & = \left( {\frac{{\tau_{T} }}{{\tau_{2} }}} \right)\left( \frac{h}{L} \right)^{2} , \\ A_{13} & = \frac{{\Delta_{E} }}{1 - 2\nu }\left( {\frac{{\tau_{q} }}{{\tau_{1} }}} \right)\left( \frac{h}{L} \right)^{2} ,\quad A_{14} = \frac{{\Delta_{E} }}{1 - 2\nu }\left( \frac{h}{L} \right)^{2} \\ \end{aligned} $$

(A1)

Appendix B

$$ \begin{aligned} B_{1} & = C_{1} + C_{2} s^{2} ,\quad B_{2} = C_{3} ,\quad B_{3} = C_{4} ,\quad B_{4} = C_{5} + C_{6} s^{2} , \\ B_{5} & = C_{7} /\left( {s + \Omega \tau_{1} } \right),\quad B_{6} = C_{8} s + C_{9} s^{2} ,\quad B_{7} = C_{10} + C_{11} s + C_{12} s^{2} \\ \end{aligned} $$

(B1)

with

$$ \begin{aligned} C_{1} & = A_{1} \left( {n\pi } \right)^{4} + A_{2} \left( {n\pi } \right)^{2} + A_{3} ,\quad C_{2} = A_{6} \left( {n\pi } \right)^{2} + A_{5} ,\quad C_{3} = A_{4} \left( {n\pi } \right)^{3} + A_{3} \left( {n\pi } \right), \\ C_{4} & = A_{1} \left( {n\pi } \right)^{3} + A_{5} \left( {n\pi } \right),\quad C_{5} = A_{4} \left( {n\pi } \right)^{4} + A_{3} \left( {n\pi } \right)^{2} ,\quad C_{6} = A_{7} \left( {n\pi } \right)^{2} + 1, \\ C_{7} & = A_{5} , C_{8} = A_{13} \left( {n\pi } \right), \quad C_{9} = A_{14} \left( {n\pi } \right), \\ C_{10} & = A_{8} \left( {n\pi } \right)^{2} + A_{9} ,\quad C_{11} = A_{12} \left( {n\pi } \right)^{2} + A_{11} ,\quad C_{12} = A_{10} \\ \end{aligned} $$

(B2)

Appendix C

$$ a_{0} = C_{3} C_{7} C_{10} ,\quad a_{1} = C_{3} C_{7} C_{11} ,\quad a_{2} = C_{3} C_{7} C_{12} $$

(C1)

$$ \begin{aligned} b_{0} & = C_{1} C_{7} C_{10} ,\quad b_{1} = C_{1} C_{7} C_{11} + C_{4} C_{7} C_{8} ,\quad b_{2} = C_{1} C_{7} C_{12} + C_{2} C_{7} C_{10} + C_{4} C_{7} C_{9} , \\ b_{3} & = C_{2} C_{7} C_{11} ,\quad b_{4} = C_{2} C_{7} C_{12} \\ \end{aligned} $$

(C2)

$$ c_{1} = C_{3} C_{7} C_{8} ,\quad c_{2} = C_{3} C_{7} C_{9} $$

(C3)

$$ \begin{aligned} d_{0} & = C_{1} C_{5} C_{10} - C_{3}^{2} C_{10} ,\quad d_{1} = C_{1} C_{5} C_{11} + C_{4} C_{5} C_{8} - C_{3}^{2} C_{11} , \\ d_{2} & = C_{1} C_{5} C_{12} + C_{1} C_{6} C_{10} + C_{2} C_{5} C_{10} + C_{4} C_{5} C_{9} - C_{3}^{2} C_{12} , \\ d_{3} & = C_{1} C_{6} C_{11} + C_{2} C_{5} C_{11} + C_{4} C_{6} C_{8} , \\ d_{4} & = C_{1} C_{6} C_{12} + C_{2} C_{5} C_{12} + C_{2} C_{6} C_{10} + C_{4} C_{6} C_{9} , \\ d_{5} & = C_{2} C_{6} C_{11} ,\quad d_{6} = C_{2} C_{6} C_{12} \\ \end{aligned} $$

(C4)