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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.

Some analytical formulas are presented for torsional analysis of homogeneous hollow tubes. The cross section is supposed to consist of straight and circular segments. Thicknesses of segments of the cross section can be different. The problem is formulated in terms of Prandtl's stress function. The derived approximate formulas are so simple that computations can be carried out by a simple calculator. Several examples are presented to validate the formulation. The accuracy of formulas is verified by accurate finite element method solutions. It is seen that the error of the formulation is small and the formulas can be used for analysis of thin to moderately thick-walled hollow tubes.

This paper is concerned with the equivalent extensional and flexural rigidities of a single layer graphene sheet by treating it as a plane lattice structure made of tightly packed carbon atoms into an array of honeycomb-shaped cells. Each carbon atom is modeled as a node with concentrated atomic mass and prescribed six degrees of freedom. The covalent bond between adjacent carbon atoms provides axial, bending, and torsional stiffness. Using the Poisson's ratio of 0.16 and thickness of 3.4 Å , the equivalent Young's moduli are found to be approximately 0.112 TPa for bending and in the range of 1.03-1.04 TPa for in-plane modes. Subsequently, the graphene structure is simulated by a classical plate with prescribed geometric and mechanical properties. The in-plane and out-of-plane free vibration analyses of the rectangular plate provide the natural frequencies and associated mode shapes. Results are compared with eigen analyses of the lattice structure model for different sizes of graphene. Examples are considered to show close agreement in the results from these two methods. Mode shapes reveal that the lattice structure model shows symmetry about the horizontal and vertical axes and also about the diagonals.