Effects of van der Waals interactions on the nonlinear vibration of multi-layered graphene sheets

Abstract

This paper is concerned with the forced nonlinear vibration of multi-layered graphene sheets modelled at the atomic level by the lattice structure approach. In this, the covalent bond between two carbon atoms is assumed as a structural member with prescribed physical and material properties. An atom is treated as a nodal point with its own mass and six degrees of freedom. The highly nonlinear van der Waals interaction between adjacent graphene layers is fully incorporated in the model by placing it in the force vector. This adjustment significantly reduces the computational hardships due to nonlinearity and increases the efficiency of the method. Newmark's direct integration method is modified to address the nonlinearity in the load vector and used for the solution of the matrix equation governing the motion of the multi-layered graphene sheet. Double-layered square graphene with simply supported and clamped boundary conditions is analysed to examine the out-of-plane and in-plane vibrational characteristics. Also, in order to illustrate the applicability of the numerical method, analyses are carried out with the first- and second-order Taylor series approximations of the van der Waals interactions, influence of which is found to be quite significant in the bending modes of vibration, but it essentially does not have a role in the in-plane modes. The numerical method developed herein is quite appropriate with reference to the structural formation at the atomic scale and also more efficient than previous computational approaches by others.