Size-Dependent Transverse Free Vibration of Bimodular Microbeams Based on the Modified Couple Stress Theory

The size-dependent transverse free vibration of microbeams made of bimodular elastic materials was investigated using the different moduli theory and the modified couple stress theory. It is known that many materials have different tensile and compression properties. However, this is not considered during microbeam modeling in most cases, and this omission could cause structural security problems in applications. Therefore, analyzing bimodular microbeams is important. First, the strain–displacement relationships were corrected using a simplified mechanical model based on the tension and compression subareas in the cross-section. Then, it was verified that the strain energy formula is consistent when bending both upward and downward, although the neutral layer position in a bimodular microbeam changes due to its asymmetric behavior during vibration, indicating that the effective bending stiffness of a bimodular microbeam is constant. Based on this invariant effective flexural stiffness, the governing equation and natural boundary conditions were derived by employing Hamilton’s principle. An analytical solution was obtained by the separation of variables method. A comparative study was conducted, with a particular focus on the influences of the bimodular property and the size-dependency of the microbeam on its free vibration. The results show that, compared with an unimodular microbeam made of isotropic bimodular materials, the actual bimodularity leads to an over-predicted effective stiffness when the bimodularity ratio, r , is less than 1, and underestimations when r > 1. For the bimodularity ratio and the material length scale parameter, isotropic bimodular microbeams exhibit a typical “hard-spring” behavior; for the height, microbeams display an opposite “soft-spring” behavior. This work can provide a reference for predicting the vibration properties of bimodular structures and for the structural design of microbeams made of bimodular materials.