Buckling and free vibration of shallow curved micro/nano-beam based on strain gradient theory under thermal loading with temperature-dependent properties

Abstract

In this study, influences of a uniform thermomechanical loading in buckling and free vibration of a curved FG microbeam have been investigated, based on strain gradient theory (SGT) theory and Timoshenko beam model. Distribution of structural materials varies continuously in thickness direction due to power-law exponent. Unlike classical models, this novel model employs three length scale parameters which can capture the size effect. This work is based on SGT theory and Timoshenko beam model. Governing equation of motion and associated boundary condition have been developed based on Hamilton’s principle, which is the specified case of virtual work theorem. In continuance, final differential equations were solved by Navier’s solution method and the results have been presented. Moreover, influences of dimensionless length-to-thickness ratio (aspect ratio), dimensionless length scale parameter, power-law exponent, temperature difference and arc angle for various values of mode numbers on natural frequency and critical temperature by considering temperature-dependent material properties have been investigated. In order to validate accomplished study, some of the results were compared with those of previous works. It has been concluded that applying a thermomechanical loading on a FG microbeam causes the natural frequency to become more sensitive about variations of geometrical, physical and mechanical properties and characteristics.

Appendix

$$n = 2l_{0}^{2} ,\quad m = 2l_{1}^{2} ,\quad c = \frac{1}{4}l_{2}^{2}$$

(32)

$$\begin{aligned} \begin{array}{*{20}l} {J_{1} = \left( {\frac{27}{15}m} \right),} \hfill & {J_{2} = \left( {n + \frac{1}{5}m} \right),} \hfill & {J_{3} = \left( {n + \frac{3}{5}m} \right)} \hfill \\ {J_{4} = \left( {\frac{8}{15}m + c} \right), \, } \hfill & {J_{5} = \left( {\frac{16}{15}m + c} \right), \, } \hfill & {J_{6} = \left( {\frac{7}{6}m + c} \right)} \hfill \\ {J_{7} = \left( {n + \frac{13}{6}m} \right), \, } \hfill & { \, J_{8} = \left( {\frac{12}{5}m + c} \right),} \hfill & {J_{9} = \left( {n + \frac{13}{5}m} \right)} \hfill \\ {J_{10} = \left( {\frac{4}{15}m + c} \right), \, } \hfill & {J_{11} = \left( {n + \frac{4}{15}m} \right), \, } \hfill & {J_{12} = \left( {n + \frac{76}{15}m} \right)} \hfill \\ {J_{13} = \left( {n + m} \right),} \hfill & {J_{14} = \left( {\frac{29}{15}m + c} \right),} \hfill & { \, J_{15} = \left( {\frac{7}{15}m} \right)} \hfill \\ {J_{16} = \left( n \right),} \hfill & { \, J_{17} = \left( {\frac{22}{15}m} \right), \, } \hfill & {J_{18} = \left( {\frac{28}{15}m + c} \right)} \hfill \\ {J_{19} = \left( m \right),} \hfill & { \, J_{20} = \left( {n + \frac{8}{15}m} \right), \, } \hfill & { \, J_{21} = \left( {n + \frac{6}{5}m} \right)} \hfill \\ {J_{22} = \left( {\frac{37}{15}m + c} \right)} \hfill & {} \hfill & {} \hfill \\ \end{array} \hfill \\ \hfill \\ \end{aligned}$$

(33)

$$\begin{array}{*{20}l} {I_{1} = \left( {J_{3} + J_{14} } \right),} \hfill & {I_{2} = \left( {J_{1} + J_{2} + J_{5} } \right), \, } \hfill & {I_{3} = \left( {J_{2} + J_{18} } \right), \, } \hfill & {I_{4} = \left( {J_{4} + J_{7} } \right)} \hfill \\ {I_{5} = \left( {J_{4} + J_{9} } \right),} \hfill & {I_{6} = \left( {J_{4} + J_{12} } \right),} \hfill & {I_{7} = \left( {J_{3} + J_{8} } \right),} \hfill & {I_{8} = \left( {J_{3} + J_{15} } \right)} \hfill \\ {I_{9} = \left( {J_{5} + J_{13} } \right), \, } \hfill & {I_{10} = \left( {J_{4} + J_{16} } \right),} \hfill & { \, I_{11} = \left( {J_{14} + J_{21} } \right),} \hfill & {I_{12} = \left( {J_{10} + J_{12} } \right)} \hfill \\ {I_{13} = \left( {J_{5} + J_{16} } \right)} \hfill & {} \hfill & {} \hfill & {} \hfill \\ \end{array}$$

(34)