lateral vibrations of single-layered graphene sheets using doublet mechanics

this paper investigates the lateral vibration of single-layered graphene sheets based on a new theory called doublet mechanics with a length scale parameter. after a brief reviewing of doublet mechanics fundamentals, a sixth order partial differential equation that governs the lateral vibration of single-layered graphene sheets is derived. using doublet mechanics, the relation between natural frequency and length scale parameter is obtained in the lateral mode of vibration for single-layered graphene. it is shown that length scale parameter plays a significant role in the lateral vibration behavior of single-layered graphene sheets. such effect decreases the natural frequency compared to the predictions of the classical continuum mechanics models. however with increasing the length of the plate, the effect of scale parameter on the natural frequency decreases. for validating the results of this method, the results obtained herein are compared with the existing nonlocal and molecular dynamics results and good agreement with the latter is observed.