Nonlinear free vibration of size-dependent microbeams with nonlinear elasticity under various boundary conditions

In this study, nonlinear couple stress-strain constitutive relationships in the modified couple stress theory (MCST) are derived on the basis of previous classical stress-strain constitutive relationships of nonlinear elasticity materials. Hamilton's principle is employed to obtain higher-order nonlinear governing equations within the framework of the updated MCST, von Kármán geometric nonlinearity and Bernoulli-Euler beam theory. These mathematical formulations are solved numerically by the differential quadrature method together with an iterative algorithm to determine the nonlinear dynamic features of microbeams with four groups of boundary conditions. A detailed parametric study is conducted to analyze the influences of nonlinear elasticity properties on the nonlinear free vibration characteristics of the microbeams. Results show that these microbeams exhibiting nonlinear couple constitutive relationships have lower frequencies than their approximately simplified linear couple constitutive relationships. In addition, the frequencies of microbeams with nonlinear elasticity properties decrease as the vibration amplitude increases.