Nanoscale vibration characterization of multi-layered graphene sheets embedded in an elastic medium

Detailed studies on the nanoscale vibration characteristics of multi-layered graphene sheets (MLGSs) that are embedded in an elastic medium are carried out using continuum-based modelling and Generalized Differential Quadrature (GDQ) method. Natural frequencies and their associated vibration modes of practical interest of single-layered and triple-layered graphene sheets, as well as general MLGSs that are embedded in an elastic medium are established. Numerical simulations are conducted to examine the effects of van der Waals (vdW) interactions, which are present as bonding forces between the layers, on nanoscale vibration natural frequencies and their mode shapes. The results show that for a general MLGSs embedded in an elastic medium, vibration modes can in general be classified into three families - lower classical synchronized modes which are independent of van der Waals forces and are somewhat sensitive to the surrounding elastic medium, middle van der Waals enhanced modes which are largely determined by the presence of van der Waals interactions and are hence less sensitive to the changes of the surrounding elastic medium, and higher mixed modes which are combinations of classical synchronized modes and van der Waals enhanced modes. Detailed characterizations of these modes from their derived mode shapes have been achieved for the typical case of an embedded triple-layered GSs, as well as general embedded MLGSs. Effects of Winkler modulus K_W, the shear layer modulus G_b, different boundary conditions, aspect ratio β and the number L of graphene layers on nanoscale vibration properties have been examined in detail. The results presented in this paper, for the first time, provide accurate and wholesome studies and characterizations on the interesting nanoscale vibration properties of multi-layered graphene sheets embedded in an elastic medium and the results obtained will certainly be useful to those who are concerned with the dynamics of embedded graphene sheets which are increasingly being deployed for various innovative engineering applications such as nano-electro-mechanical systems (NEMS).     

Surface effects on nonlinear vibration of embedded functionally graded nanoplates via higher order shear deformation plate theory

In this paper, free vibration behavior of functionally nanoplate resting on a Pasternak linear elastic foundation is investigated. The study is based on third-order shear deformation plate theory with small scale effects and von Karman nonlinearity, in conjunction with Gurtin–Murdoch surface continuum theory. It is assumed that functionally graded (FG) material distribution varies continuously in the thickness direction as a power law function and the effective material properties are calculated by the use of Mori–Tanaka homogenization scheme. The governing and boundary equations, derived using Hamilton's principle are solved through extending the generalized differential quadrature method. Finally, the effects of power-law distribution, nonlocal parameter, nondimensional thickness, aspect of the plate, and surface parameters on the natural frequencies of FG rectangular nanoplates for different boundary conditions are investigated.     

Thermo-mechanical vibration analysis of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets based on a higher-order shear deformation beam theory

This article proposes a higher-order shear deformation beam theory for free vibration analysis of functionally graded carbon nanotube-reinforced composite sandwich beams in a thermal environment. The temperature-dependent material properties of functionally graded carbon nanotube-reinforced composite beams are supposed to vary continuously in the thickness direction and are estimated through the rule of mixture. The governing equations and boundary conditions are derived by using Hamilton's principle, and the Navier solution procedure is used to achieve the natural frequencies of the sandwich beam in a thermal environment. A parametric study is led to carry out the effects of carbon nanotube volume fractions, slenderness ratio, and core-to-face sheet thickness ratio on free vibration behavior of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets. Numerical results are also presented in order to compare the behavior of sandwich beams including uniformly distributed carbon nanotube-reinforced composite face sheets to those including functionally graded carbon nanotube-reinforced composite face sheets.     

A nonlinear thick plate formulation based on the modified strain gradient theory

A geometric nonlinear first-order shear deformation theory-based formulation is presented to analyze microplates. The formulations derived herein are based on a modified strain gradient theory and the von Karman nonlinear strains. The modified strain gradient theory includes five material length scale parameters capable to capture the size effects in small scales. The governing equations of motion and the most general form of boundary conditions of an arbitrary-shaped plate are derived using the principle of virtual displacements. The analysis is general and can be reduced to the modified couple stress plate model or the classical plate model.     

Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory

A single-elastic beam model has been developed to analyze the thermal vibration of single-walled carbon nanotubes (SWCNT) based on thermal elasticity mechanics, and nonlocal elasticity theory. The nonlocal elasticity takes into account the effect of small size into the formulation. Further, the SWCNT is assumed to be embedded in an elastic medium. A Winkler-type elastic foundation is employed to model the interaction of the SWCNT and the surrounding elastic medium. Differential quadrature method is being utilized and numerical solutions for thermal-vibration response of SWCNT is obtained. Influence of nonlocal small scale effects, temperature change, Winkler constant and vibration modes of the CNT on the frequency are investigated. The present study shows that for low temperature changes, the difference between local frequency and nonlocal frequency is comparatively high. With embedded CNT, for soft elastic medium and larger scale coefficients (e₀a) the nonlocal frequencies are comparatively lower. The nonlocal model-frequencies are always found smaller than the local model-frequencies at all temperature changes considered.     

Buckling analysis of corrugated plates using a mesh-free Galerkin method based on the first-order shear deformation theory

This paper deals with elastic buckling analysis of stiffened and un-stiffened corrugated plates via a mesh-free Galerkin method based on the first-order shear deformation theory (FSDT). The corrugated plates are approximated by orthotropic plates of uniform thickness that have different elastic properties along the two perpendicular directions of the plates. The key to the approximation is that the equivalaent elastic properties of the orthotropic plates are derived by applying constant curvature conditions to the corrugated sheet. The stiffened corrugated plates are analyzed as stiffened orthotropic plates. The stiffeners are modelled as beams. The stiffness matrix of the stiffened corrugated plate is obtained by superimposing the strain energy of the equivalent orthotropic plate and the beams after implementing the displacement compatibility conditions between the plate and the beams. The mesh free characteristic of the proposed method guarantee that the stiffeners can be placed anywhere on the plate, and that remeshing is avoided when the stiffener positions change. A few selected examples are studied to demonstrate the accuracy and convergence of the proposed method. The results obtained for these examples, when possible, are compared with the ANSYS solutions or other available solutions in literature. Good agreement is evident for all cases. Some new results for both trapezoidally and sinusoidally corrugated plates are then reported.     

A nth-order meshless generalization of Reddy’s third-order shear deformation theory for the free vibration on laminated composite plates

In this paper, a nth-order shear deformation theory is proposed to analyze the free vibration of laminated composite plates. The present nth-order shear deformation theory satisfies the zero transverse shear stress boundary conditions on the top and bottom surface of the plate. Reddy’s third-order theory can be considered as a special case of present nth-order theory (n = 3). Natural frequencies of the laminated composite plates with various boundary conditions, side-to-thickness ratios, material properties are computed by present nth-order theory and a meshless radial point collocation method based on the thin plate spline radial basis function. The results are compared with available published results which demonstrate the accuracy and efficiency of present nth-order theory.     

Small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium

As a first endeavor, the small scale effect on the thermal buckling characteristic of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium is investigated. The surrounding elastic medium is modeled as the two-parameter elastic foundation. The formulation is derived using the classical plate theory (CPT) in conjunction with the nonlocal elasticity theory. The solution procedure is based on the transformation of the governing equations from physical domain to computational domain and then discretization of the spatial derivatives by employing the differential quadrature method (DQM) as an efficient and accurate numerical tool. The fast rate of convergence of the method is shown and the results are compared against existing results in literature. Then, the influence of small scale parameter in combination with the elastic medium parameters, geometrical shape and the boundary conditions on the thermal buckling load of the nanoplates is investigated.     

A cell‐based smoothed discrete shear gap method (CS‐FEM‐DSG3) based on the C0‐type higher‐order shear deformation theory for dynamic responses of Mindlin plates on viscoelastic foundations subjected to a moving sprung vehicle

A cell‐based smoothed discrete shear gap method (CS‐FEM‐DSG3) based on the first‐order shear deformation theory (FSDT) was recently proposed for static and dynamic analyses of Mindlin plates. In this paper, the CS‐FEM‐DSG3 is extended to the C⁰‐type higher‐order shear deformation plate theory (C⁰‐HSDT) and is incorporated with damping–spring systems for dynamic responses of Mindlin plates on viscoelastic foundations subjected to a moving sprung vehicle. At each time step of dynamic analysis, one four‐step procedure is performed including the following: (1) transformation of the weight of a four‐wheel vehicle into the sprung masses at wheels; (2) dynamic analysis of the sprung mass of wheels to determine the contact forces; (3) transformation of the contact forces into loads at nodes of plate elements; and (4) dynamic analysis of the plate elements on viscoelastic foundations. The accuracy and reliability of the proposed method are verified by comparing its numerical solutions with those of other available numerical results. Copyright © 2014 John Wiley & Sons, Ltd.