Influence of in-plane pre-load on the vibration frequency of circular graphene sheet via nonlocal continuum theory

In this study, the free vibration behavior of circular graphene sheet under in-plane pre-load is studied. By using the nonlocal elasticity theory and Kirchhoff plate theory, the governing equation is derived for single-layered graphene sheets (SLGSs). The closed-form solution for frequency vibration of circular graphene sheets under in-plane pre-load has been obtained and nonlocal parameter appears into arguments of Bessel functions. The results are subsequently compared with valid result reported in the literature. The effects of the small scale, pre-load, mode number and boundary conditions on natural frequencies are investigated. The results are shown that at smaller radius of circular nanoplate, the effect of in-plane pre-loads is more importance.     

Nonlocal effect on the free vibration of short nanotubes embedded in an elastic medium

In this paper, the small size effect on the free vibration behavior of finite length nanotubes embedded in an elastic medium is investigated. The problem is formulated based on the three-dimensional (3D) nonlocal elasticity theory. Since the 3D nonlocal constitutive relations in a cylindrical coordinate system are used, in addition to displacement components, the stress tensor components are chosen as degrees of freedom. The surrounding elastic medium is modeled as the Winkler’s elastic foundation. The differential quadrature method as an efficient and accurate numerical tool in conjunction with the series solution is used to discretize the governing equations. Very fast rate of convergence of the method is demonstrated. The effects of the nonlocal parameter together with the other geometrical parameters and also the stiffness parameter of the elastic medium on the natural frequencies are studied.     

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.     

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 of circular/annular Mindlin nanoplates via nonlocal elasticity

The present study proposes an analytical solution for the axisymmetric/asymmetric buckling analysis of moderately thick circular/annular Mindlin nanoplates under uniform radial compressive in-plane load. In order to consider small-scale effects, nonlocal elasticity theory of Eringen is employed. To ensure the efficiency and stability of the present methodology, the results are compared with other ones presented in the literature. Further the exact closed-form solution is obtained using three potential functions. In addition, the effect of small scales on buckling loads for different parameters such as geometry of the nanoplate, boundary conditions, and axisymmetric/asymmetric mode numbers, is investigated. It is observed that the buckling mode shape for annular nanoplates, which corresponds to the lowest critical buckling load, may be axisymmetric or asymmetric depending on boundary conditions, inner to outer radius ratios, and thickness of the nanoplate. In other words, for stiffer boundary conditions and smaller inner to outer radius ratios, the mode shape corresponding to the lowest critical buckling load is an asymmetric mode. Also, the difference between axisymmetric and asymmetric buckling loads for higher mode numbers, greater thickness to outer radius ratios and smaller outer radii decreases by increasing the nonlocal parameter.     

Nonlocal elasticity effect on vibration of in-plane loaded double-walled carbon nano-tubes

Summary Vibration and buckling of in-plane loaded simply supported double-walled carbon nanotubes were investigated using the nonlocal Timoshenko-beam theory. The influence of in-plane loads on the natural frequencies was determined. The results show that while the natural frequencies decrease with increasing compressive in-plane loads, an increase in frequencies is observed for tension type of in-plane loads. Effects of in-plane loads are more pronounced for lower modes, and some mode changes are observed at critical in-plane loads. A comparison of nonlocal elasticity solutions with local elasticity solutions indicates that the nonlocal effects should be considered for higher modes of vibration of double-walled carbon nanotubes.     

Thermal buckling analysis of nanoplates based on nonlocal elasticity theory with four-unknown shear deformation theory resting on Winkler–Pasternak elastic foundation

The thermal buckling analysis of nanoplates is based on nonlocal elasticity theory with four-unknown shear deformation theory resting on Winkler–Pasternak elastic foundation. The nanoplate is assumed to be under three types of thermal loadings, namely uniform temperature rise, linear temperature rise, and nonlinear temperature rise through the thickness. The theory involves four unknown variables with small-scale effects, as against five in the case of other higher-order theories and first-order shear deformation theory. Closed-form solution for theory was also presented. Results are presented to discuss the influences of the nonlocal parameter, aspect ratio, side-to-thickness ratio, and elastic foundation parameters on the thermal buckling characteristics of analytical rectangular nanoplates.     

Stability analysis of an embedded single-walled carbon nanotube with small initial curvature based on nonlocal theory

Many experimental observations have shown that most nanostructures, such as carbon nanotubes, are often characterized by a certain degree of waviness along their axial direction. This geometrical imperfection has profound effects on the mechanical behavior of carbon nanotubes. In the present work, stability of a slightly curved carbon nanotube under lateral loading is investigated based on Eringen's nonlocal elasticity theory. Euler Bernoulli and Timoshenko beam theories are employed to obtain equilibrium equations. Winkler-Pasternak elastic foundation is used to approximate the effect of matrix. Effects of initial curvature, nonlocal parameter, beam length, and elastic foundation parameters on initiation of critical conditions are investigated.     

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.     

Forced vibration of sinusoidal FG nanobeams resting on hybrid Kerr foundation in hygro-thermal environments

Size-dependent forced vibration behavior of functionally graded (FG) nanobeams subjected to an in-plane hygro-thermal loading and lateral concentrated and uniform dynamic loads is investigated via a higher-order refined beam theory, which captures shear deformation influences needless of any shear correction factor. The nanobeam is in contact with a three-parameter Kerr foundation consisting of upper and lower spring layers as well as a shear layer. Hygro-thermo-elastic material properties of the nanobeam are described via power-law distribution considering exact position of the neutral axis. Through nonlocal elasticity theory of Eringen and Hamilton's principle, the governing equations of higher-order FG nanobeams on Kerr foundation under dynamic loading are derived. These equations are solved for simply-supported and clamped-clamped boundary conditions. A detailed parametric study is performed to show the importance of moisture concentration rise, temperature rise, material composition, nonlocality, Kerr foundation parameters, and boundary conditions on forced vibration characteristics and resonance frequencies of FG nanobeams. As a consequence, Kerr foundation parameters lead to a significant delay in the occurrence of resonance frequencies.     

Nonlocal free vibration of orthotropic non-prismatic skew nanoplates

The free vibration of orthotropic non-prismatic skew nanoplate based on the first-order shear deformation theory (FSDT) in conjunction with Eringen’s nonlocal elasticity theory is presented. As a simple, accurate and low computational effort numerical method, the differential quadrature method (DQM) is employed to solve the related differential equations. For this purpose, after deriving the equations of motion and the related boundary conditions, they are transformed from skewed physical domain to rectangular computational domain of DQM and accordingly discretized. After validating the formulation and method of solution, the effects of nonlocal parameter in combination with geometrical parameters and boundary conditions on the natural frequencies of the orthotropic skew nanoplates are investigated.     

Thermo-mechanical vibration of orthotropic cantilever and propped cantilever nanoplate using generalized differential quadrature method

In this article, the vibration frequency of an orthotropic nanoplate under the effect of temperature change is investigated. Using nonlocal elasticity theory, governing equations are derived. Based on the generalized differential quadrature method for cantilever and propped cantilever boundary conditions, the frequencies of orthotropic nanoplates are considered and the obtained results are compared with valid reported results in the literature. The effects of temperature variation, small scale, different boundary conditions, aspect ratio, and length on natural nondimensional frequencies are studied. The present analysis is applicable for the design of rotating and nonrotating nano-devices that make use of thermo-mechanical vibration characteristics of nanoplates.     

Nonlocal buckling and vibration analysis of thick rectangular nanoplates using finite strip method based on refined plate theory

The buckling and vibration of thick rectangular nanoplates is analyzed in this article. A graphene sheet is theoretically assumed and modeled as a nanoplate in this study. The two-variable refined plate theory (RPT) is applied to obtain the differential equations of the nanoplate. The theory accounts for parabolic variation of transverse shear stress through the thickness of the plate without using a shear correction factor. Besides, the analysis is based on the nonlocal theory of elasticity to take the small-scale effects into account. For the first time, the finite strip method (FSM) based on RPT is employed to study the vibration and buckling behavior of nanoplates and graphene sheets. Hamilton’s principle is employed to obtain the differential equations of the nanoplate. The stiffness, stability and mass matrices of the nanoplate are formed using the FSM. The displacement functions of the strips are evaluated using continuous harmonic function series which satisfy the boundary conditions in one direction and a piecewise interpolation polynomial in the other direction. A matrix eigenvalue problem is solved to find the free vibration frequency and buckling load of the nanoplates subjected to different types of in-plane loadings including the uniform and nonuniform uni-axial and biaxial compression. Comparison studies are presented to verify the validity and accuracy of the proposed nonlocal refined finite strip method. Furthermore, a number of examples are presented to investigate the effects of various parameters (e.g., boundary conditions, nonlocal parameter, aspect ratio, type of loading) on the results.     

考虑小尺度效应的微圆轴扭转振动

在微机电系统中,微纳米构件常常表现出尺度效应。基于非局部弹性理论,建立了微圆轴的扭转振动模型,并结合3种常见的边界条件,给出了具体的算例。结果表明:对比于经典连续力学,非局部弹性理论预言的圆轴扭转振动固有频率下降,并且微圆轴的外特征尺度即横截面半径越小,二者相差越大;振动频率的阶数越高,影响也越明显。随着截面半径的增加,振动频率下降并且非局部尺度效应逐渐消失。同时考察了扭转振动的模态函数和相对转角,发现前者与经典弹性理论结果一致。此外还讨论了材料内禀尺度的选取问题,以数值算例证明了内禀尺度与材料晶格常数非常接近,晶格常数可近似用作微纳米力学中材料的内禀尺度参数。     

Vibration analysis of orthotropic graphene sheets embedded in Pasternak elastic medium using nonlocal elasticity theory and differential quadrature method

In this paper, the small scale effect on the vibration analysis of orthotropic single layered graphene sheets embedded in elastic medium is studied. Elastic theory of the graphene sheets is reformulated using the nonlocal differential constitutive relations of Eringen. Both Winkler-type and Pasternak-type foundation models are employed to simulate the interaction between the graphene sheet and surrounding elastic medium. Using the principle of virtual work the governing differential equations are derived. Differential quadrature method is employed to solve the governing differential equations for various boundary conditions. Nonlocal theories are employed to bring out the small scale effect of the nonlocal parameter on the natural frequencies of the orthotropic graphene sheets embedded in elastic medium. Further, effects of (i) nonlocal parameter, (ii) size of the graphene sheets, (iii) stiffness of surrounding elastic medium and (iv) boundary conditions on non-dimensional vibration frequencies are investigated.     

An efficient numerical method for analyzing the thermal effects on the vibration of embedded single-walled carbon nanotubes based on the nonlocal shell model

Employing the variational differential quadrature (VDQ) method, the effects of initial thermal loading on the vibrational behavior of embedded single-walled carbon nanotubes (SWCNTs) based on the nonlocal shell model are studied. According to the first-order shear deformation theory and considering Eringen's nonlocal elasticity theory, the energy functionality of the system is presented and discretized using the VDQ method. The effects of thermal loading and elastic foundation are simultaneously taken into account. The use of the numerical discretization technique in the context of variational formulation reduces the order of differentiation in the governing equations and consequently improves the convergence rate. The accuracy of the present model is first checked by comparison with molecular dynamics simulation results and those of other methods. The effects of involved parameters are then investigated on the fundamental frequencies of thermally preloaded embedded SWCNTs. The results imply that the thermal loading has a significant effect on the vibration analysis of embedded SWCNTs.     

Influence of initial shear stress on the vibration behavior of single-layered graphene sheets embedded in an elastic medium based on Reddy's higher-order shear deformation plate theory

In this article, the small-scale effect on the vibration behavior of orthotropic single-layered graphene sheets is studied based on the nonlocal Reddy's plate theory embedded in elastic medium considering initial shear stress. Elastic theory of the graphene sheets is reformulated using the nonlocal differential constitutive relations of Eringen. To simulate the interaction between the graphene sheet and surrounding elastic medium we used both Winkler-type and Pasternak-type foundation models. The effects of initial shear stress and surrounding elastic medium and boundary conditions on the vibration analysis of orthotropic single-layered graphene sheets are studied considering five different boundary conditions. Numerical approach of the obtained equation is derived by differential quadrature method. Effects of shear stress, nonlocal parameter, size of the graphene sheets, stiffness of surrounding elastic medium, and boundary conditions on vibration frequency rate are investigated. The results reveal that as the stiffness of the surrounding elastic medium increases, the nonlocal effect decreases. Further, the nonlocal effect increases as the size of the graphene sheet is decreased. It is also found that the frequency ratios decrease with an increase in vibration modes.     

Longitudinal varying elastic foundation effects on vibration behavior of axially graded nanobeams via nonlocal strain gradient elasticity theory

In this research, vibration characteristics of axially functionally graded nanobeams resting on variable elastic foundation are investigated based on nonlocal strain gradient theory. This nonclassical nanobeam model contains a length scale parameter to explore the influence of strain gradients and also a nonlocal parameter to study the long-range interactions between the particles. The present model can degenerate into the classical models if the material length scale parameter and the nonlocal stress field parameter are both taken to be zero. Elastic foundation consists of two layers: a Winkler layer with variable stiffness and a Pasternak layer with constant stiffness. Linear, parabolic and sinusoidal variations of Winkler foundation in longitudinal direction are considered. Material properties are graded axially via a power-law distribution scheme. Hamilton's principle is employed to derive the governing equations that are solved applying a Galerkin-based solution for different boundary edges. Comparison study is also performed to verify the present formulation with those of previous papers. Results are presented to investigate the influences of the nonlocal and length scale parameters, various material compositions, elastic foundation parameters, type of foundation and various boundary conditions on the vibration frequencies of AFG nanobeams in detail.     

Buckling analysis of micro-/nano-scale plates based on two-variable refined plate theory incorporating nonlocal scale effects

This article presents the buckling analysis of isotropic nanoplates using the two variable refined plate theory and nonlocal small scale effects. The two variable refined plate theory takes account of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the plate, hence it is unnecessary to use shear correction factors. Nonlocal governing equations of motion for the nanoplate are derived from the principle of virtual displacements. The closed-form solution for buckling load of a simply supported rectangular nanoplate subjected to in-plane loading has been obtained by using the Navier’s method. Numerical results obtained by the present theory are compared with available exact solutions in the literature. The effect of nonlocal scaling parameter, mode numbers and aspect ratios of the nanoplates on buckling load are investigated and discussed in detail in the present work. It can be concluded that the present theory, which does not require shear correction factor, is not only simple but also comparable to the first-order and higher order shear deformable theory.