An exact solution for vibrations of postbuckled microscale beams based on the modified couple stress theory

The modified couple stress theory, as a theory capable of capturing size effects, is implemented to study the vibration characteristic of a postbuckled microbeam. To this end, a modified couple stress Euler–Bernoulli beam model containing geometric nonlinearity is considered. Within the framework of a variational formulation and based on Hamilton’s principle, the governing equation and corresponding boundary conditions are derived. By eliminating time-dependent terms, the governing equation of vibration is reduced to that of buckling problem for the microbeam subjected to an axial load. The critical buckling loads and their corresponding mode shapes are predicted through an exact solution for various boundary conditions. Afterwards, the vibration analysis of a simply-supported microbeam is investigated around the obtained postbuckling configuration. It is found that the stiffness of microbeam predicted by the modified couple stress model is higher than that predicted by the classical model. Additionally, it is demonstrated that the natural frequencies by considering all of the vibration modes except the first mode are independent of the buckling load. The influences of the dimensionless length-scale parameter, Poisson’s ratio, various boundary conditions and the number of buckled modes on the critical buckling loads and natural frequency are fully investigated.