Coupled longitudinal-transverse-rotational free vibration of post-buckled functionally graded first-order shear deformable micro- and nano-beams based on the Mindlin's strain gradient theory

Presented herein is a comprehensive study on the size-dependent coupled longitudinal-transverse-rotational free vibration behavior of post-buckled functionally graded (FG) micro- and nano-beams based on the most general Mindlin's strain gradient theory. The current model enables us to incorporate size effects via introducing material length scale parameters and is developed in the framework of the first-order shear deformable beam model and the von Karman geometric nonlinearity. The FG micro- and nano-beams, whose volume fraction is expressed by using a power law function, are assumed to be made of a mixture of metals and ceramics. By using Hamilton's principle, the nonlinear governing equations and associated boundary conditions are derived for FG micro- and nano-beams in the postbuckling domain. Afterwards, the governing equations and boundary conditions are discretized using the generalized differential quadrature (GDQ) method in conjunction with a direct approach without linearization, before solving numerically by Newton's method. The effects of length scale parameter, length-to-thickness ratio, material gradient index and boundary conditions on the postbuckling path and frequency of FG micro- and nano-beams are carefully investigated. Finally, numerical results obtained from both the modified strain gradient theory (MSGT) and modified couple stress theory (MCST) are compared.