Free vibration of embedded double-walled carbon nanotubes considering nonlinear interlayer van der Waals forces

In this article, nonlinear free vibration of embedded double-walled carbon nanotubes (DWCNTs) duo to the nonlinear interlayer van der Waals (vdW) force is studied based on the nonlocal Euler-Bernoulli beam theory. The interlayer vdW force is modeled as a nonlinear function of inner and outer tubes deflections considering the variation of the interlayer distance along the circumference of DWCNTs. The harmonic balance method is applied to analyze the relationship between the deflection amplitudes and the frequencies of in-phase and out-of-phase free vibrations for DWCNTs. Finally, the influences of the nonlocal parameter, surrounding elastic medium, nanotube length, end condition and vibrational mode on the nonlinear free vibration properties of DWCNTs are discussed in detail.     

Prediction of biaxial buckling behavior of single-layered graphene sheets based on nonlocal plate models and molecular dynamics simulations

The biaxial buckling behavior of single-layered graphene sheets (SLGSs) is studied in the present work. To consider the size-effects in the analysis, Eringen’s nonlocal elasticity equations are incorporated into the different types of plate theory namely as classical plate theory (CLPT), first-order shear deformation theory (FSDT), and higher-order shear deformation theory (HSDT). An exact solution is conducted to obtain the critical biaxial buckling loads of simply-supported square and rectangular SLGSs with various values of side-length and nonlocal parameter corresponding to each type of nonlocal plate model. Then, molecular dynamics (MD) simulations are performed for a series of armchair and zigzag SLGSs with different side-lengths, the results of which are matched with those obtained by the nonlocal plate models to extract the appropriate values of nonlocal parameter relevant to each type of nonlocal elastic plate model and chirality. It is found that the present nonlocal plate models with their proposed proper values of nonlocal parameter have an excellent capability to predict the biaxial buckling response of SLGSs.     

Nonlocal effect on axially compressed buckling of triple-walled carbon nanotubes under temperature field

This paper studies the small scale effect on the buckling behaviors of triple-walled carbon nanotubes (TWCNTs) with the initial axial stress under the temperature field. The TWCNTs are modeled as three elastic shells coupled together through vdW interaction between different layers. Buckling governing equations of CNTs are firstly formulated on the basis of nonlocal elastic theory and the small scale effect on CNTs buckling results with the change of temperature are then achieved. The results show that the critical buckling load is dependent on the temperature, scale parameter and wavenumber. Some conclusions are drawn that small scale effect will arise gradually with the increases of wavenumber, and the temperature can influence the ratio between the nonlocal buckling load and the corresponding local load. Furthermore, with or without effects of nonlocal considered, the same results is obtained that the axial buckling load increases as the value of temperature increases at low and room temperature condition, while at high temperature condition the axial buckling load decreases as the value of temperature increases.     

Sandwich plate model of multilayer graphene sheets for considering interlayer shear effect in vibration analysis via molecular dynamics simulations

Van der Waals (vdWs) bindings acting as interlayer shear force between graphene layers of multi-layer graphene sheets (MLGSs) are considered in vibration analysis in the present study. To idealize the structure of MLGS incorporating interlayer shear interactions, a sandwich model (SM) is represented which laminates the graphene layers. The layers stick together with vdWs bonds. The bonds are modeled as core layers between every two adjacent layers. Molecular dynamic (MD) simulation is carried out to validate the results obtained by SM. Afterward, the values for bending rigidity and layers thicknesses are obtained so as to match SM frequencies with MD results. It is observed the SM can predict the vibration behavior of MLGSs well for different values of aspect ratio. The present paper deals with a new method of incorporating shear effect and a novel investigation of integrating SM with MD.     

Nonlinear vibration and instability of embedded double-walled boron nitride nanotubes based on nonlocal cylindrical shell theory

This paper investigates the nonlinear vibration and instability of the embedded double-walled boron nitride nanotubes (DWBNNTs) conveying viscous fluid based on nonlocal piezoelasticity cylindrical shell theory. The elastic medium is simulated as Winkler–Pasternak foundation, and adjacent layers interactions are assumed to have been coupled by van der Walls (vdW) force evaluated based on the Lennard–Jones model. The nonlinear strain terms based on Donnell’s theory are taken into account. The Hamilton’s principle is employed to obtain coupled differential equations, containing displacement and electric potential terms. Differential quadrature method (DQM) is applied to estimate the nonlinear frequency and critical fluid velocity for clamped supported mechanical and free electric potential boundary conditions at both ends of the DWBNNTs. Results indicated that some parameters including nonlocal parameter, elastic medium’s modulus, aspect ratio and vdW force have significant influence on the vibration and instability of the DWBNNT while the fluid viscosity effect is negligible. In addition, the low aspect ratio should be taken into account for DWBNNT in optimum design of nano/micro devices.     

Elastic buckling and vibration analyses of orthotropic nanoplates using nonlocal continuum mechanics and spline finite strip method

This paper addresses the elastic buckling and vibration characteristics of isotropic and orthotropic nanoplates using finite strip method. In order to consider small scale effect, Eringen’s nonlocal continuum elasticity is employed. The governing nanoplate equations are derived using the principle of virtual work while B3-spline finite strip method is applied to the buckling and vibration analyses. The buckling load and vibration frequency of graphene sheets, which are subjected to biaxial compression and pure shear loading, are determined whilst the effects of different parameters such as sheet size, nonlocal parameter, aspect ratio and boundary conditions are investigated. The interaction curves of the critical biaxial compression loading as well as the interaction curves of the critical uniaxial compression and shear loading are also obtained. It is shown that small scale effect plays considerable role in the analysis of small sizes plates.     

Thermal-mechanical vibration and instability of a fluid-conveying single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory

Based on the theories of thermal elasticity mechanics and nonlocal elasticity, an elastic Bernoulli-Euler beam model is developed for thermal-mechanical vibration and buckling instability of a single-walled carbon nanotube (SWCNT) conveying fluid and resting on an elastic medium. The finite element method is adopted to obtain the numerical solutions to the model. The effects of temperature change, nonlocal parameter and elastic medium constant on the vibration frequency and buckling instability of SWCNT conveying fluid are investigated. It can be concluded that at low or room temperature, the fundamental natural frequency and critical flow velocity for the SWCNT increase as the temperature change increases, on the other hand, while at high temperature the fundamental natural frequency and critical flow velocity decrease as the temperature change increases. The fundamental natural frequency for the SWCNT decreases as the nonlocal parameter increases, both the fundamental natural frequency and critical flow velocity increase as elastic medium constant increases.     

Mechanical stability of functionally graded stiffened cylindrical shells

This paper is concerned with the elastic buckling of stiffened cylindrical shells by rings and stringers made of functionally graded materials subjected to axial compression loading. The shell properties are assumed to vary continuously through the thickness direction. Fundamental relations, the equilibrium and stability equations are derived using the Sander’s assumption. Resulting equations are employed to obtain the closed-form solution for the critical buckling loads. The results show that the inhomogeneity parameter and geometry of shell significantly affect the critical buckling loads. The analytical results are compared and validated using the finite element method.     

基于非局部应变梯度理论功能梯度纳米板的弯曲和屈曲研究

以纳米机器人等智能器件中的功能梯度纳米板结构为研究对象,基于非局部应变梯度理论,研究了其弯曲和屈曲问题.推导了一般情况下的功能梯度纳米板运动方程,弯曲和屈曲作为其特例可简化而成.分析了非局部尺度参数、材料特征尺度参数、梯度指数、纳米板尺寸等对弯曲挠度和临界屈曲载荷的影响.结果表明:不同高阶连续介质力学理论下的最大挠度都随梯度指数的增大而增大,正方形纳米板挠度较小,且板厚越大,弯曲挠度越小;最大挠度随非局部尺度参数的增大而增大,随材料特征尺度参数的增大而减小.临界屈曲载荷随梯度指数的增大而减小,随板厚、长宽比的增大而增大,随非局部尺度的增大而减小,随材料特征尺度的增大而增大.非局部应变梯度高阶弯曲和屈曲中存在结构软化与硬化机制,两个内特征参数之间具有耦合效应,当非局部尺度大于材料特征尺度时,非局部效应在功能梯度纳米板力学性能中占主导作用;当材料特征尺度大于非局部尺度时,应变梯度效应占主导作用.解析结果还证明了当非局部尺度等于材料特征尺度时,非局部应变梯度理论结果退化为经典结果.     

Nonlinear in-plane buckling of fixed shallow functionally graded graphene reinforced composite arches subjected to mechanical and thermal loading

The nonlinear in-plane buckling analysis for fixed shallow functionally graded (FG) graphene reinforced composite arches which are subjected to uniform radial load and temperature field is presented in this paper. The arch is composed of multiple graphene platelet reinforced composite (GPLRC) layers with gradient changes of concentration of graphene platelets (GPLs) in each layer. The principle of virtual work, combined with the effective materials properties estimated by the Halpin-Tsai micromechanics model for GPLRC layer, is used to derive the nonlinear buckling equilibrium equations of the FG-GPLRC arch, and then the analytical solutions for the limit point and bifurcation buckling loads are obtained. Comprehensive parametric studies are conducted to explore the effects of various distribution patterns and geometries of GPL, temperature field and arch geometry on the nonlinear equilibrium path and buckling behavior of the composite arch. The influence of temperature on the geometric parameters which are defined as switches between limit point buckling, bifurcation buckling and no buckling are also discussed. It is found that a higher temperature field can increase the buckling loads of the FG-GPLRC arch but reduce the value of the minimum geometric parameters that switching the buckling modes. The results also show that even a small amount of GPLs filler content can increase the buckling loads of the FG-GPLRC arch considerably, and distributing more GPLs near the surface layers is the best pattern to enhance the buckling performances of FG-GPLRC arches.     

阶梯柱屈曲的改进Fourier级数分析

该文对阶梯柱的弹性屈曲问题进行了研究.首先基于改进Fourier级数法采用局部坐标逐段建立阶梯柱的位移函数表达式,然后由带约束的势能变分原理得到含屈曲荷载的线性方程组,利用线性方程组有非零解的条件把问题转化为矩阵特征值问题得到临界载荷,最后讨论方法中的参数取值,并把结果与已有文献和有限元的结果比较,从而验证方法的精度.所提模型在阶梯柱的两端和变截面处引入横向弹簧和旋转弹簧,通过改变弹簧的刚度值模拟不同的边界.所提方法在工程设计中能比较精确地确定各种弹性边界条件下阶梯柱的临界载荷.     

A general nonlocal nonlinear model for buckling of nanobeams

This study presents a unified model for the nonlocal response of nanobeams in buckling and postbuckling states. The formulation is suitable for the classical Euler–Bernoulli, first-order Timoshenko, and higher-order shear deformation beam theories. The small-scale effect is modeled according to the nonlocal elasticity theory of Eringen. The equations of equilibrium are obtained using the principle of virtual work. The stress resultants are developed taking into account the nonlocal effect. Analytical solutions for the critical buckling load and the amplitude of the static nonlinear response in the postbuckling state are obtained. It is found out that as the nonlocal parameter increases, the critical buckling load reduces and the amplitude of buckling increases. Numerical results showing variation of the critical buckling load and the amplitude of buckling with the nonlocal parameter and the length-to-height ratio for simply supported and clamped–clamped nanobeams are presented.     

Calibration of Eringen's small length scale coefficient for buckling circular and annular plates via Hencky bar-net model

This paper is focused on the modeling of circular and annular graphene sheets via Hencky bar-net model (HBM¹) and calibrating the Eringen's small length scale coefficient e₀ in Eringen's nonlocal theory. The buckling solutions of circular and annular graphene sheets based on Eringen's nonlocal continuum plate theory are first obtained. On the other hand, HBM is developed to model the same structure from the discrete view. HBM is a grid system comprising rigid bars and arcs connected by frictionless hinges with elastic rotational and torsional springs. By regarding the length of straight segments in HBM equal to the characteristic length of Eringen's nonlocal model (ENM²) and matching their solutions, the Eringen's small length scale coefficient e₀ is calibrated. It is found that for circular graphene sheet, e₀ = 0.258 for clamped edge and e₀ = 0.300 for simply supported edge. For annular graphene sheet, e₀ is dependent on the inner to outer radius ratio χ and boundary conditions. The scale coefficient e₀ takes 0.307–0.367 for clamped edges while 0.219–0.290 for simply supported edges with χ varying from 0.2 to 0.8. Another finding is that the graphene sheet will buckle with a very small load when its dimension is large, regardless of models adopted. However for small dimensions, ENM and HBM predict lower buckling loads than the classical local model because the scale effect is more obvious.     

Surface effects on the buckling behaviors of piezoelectric cylindrical nanoshells using nonlocal continuum model

The focus of this paper is on the analytical buckling solutions of piezoelectric cylindrical nanoshells under the combined compressive loads and external voltages. To capture the small-scale characteristics of the nanostructures, the constitutive equations with the coupled nonlocal and surface effects are adopted within the framework of Reddy's higher-order shell theory. The governing equations involving the displacements and induced piezoelectric field are solved by employing the separation of variables. The derived accurate solutions conclude that bucking critical stresses should go down rapidly while the nonlocal effects reach a certain level. With the enhancing surface effects, the stability of piezoelectric cylindrical nanoshells can be improved significantly. Meanwhile, the induced electric field also plays an important role in elevating the buckling critical stresses. For the nanoshells with remarkable nonlocal effects, boundary conditions, shell length and radius have little influence on the buckling solutions. The detailed effects of the boundary conditions, geometric parameters, material properties and applied voltages are discussed.     

Unified theory of 2n+1 order size-dependent beams: Mathematical difficulty for functionally graded size-effect parameters solved

New insights on theoretical modeling of size-dependent functionally graded (FG) nanobeams are provided by establishing a unified theory of 2n+1 order shear deformable model with the aids of nonlocal strain gradient elasticity. The unified model covers Euler-type (n = 0), Reddy-type (n = 1), 5th (n = 2), 7th (n = 3) order beam and etc., and the limiting situation n → ∞ predicts nonlocal strain gradient Timoshenko model. The mathematical difficulty for FG nonlocal parameter is particularly emphasized, and an attempt is made for the first time to overcome the difficulty. Theoretically, the governing equations and boundary conditions of 2n+1 order nonlocal strain gradient beams, especially with FG nonlocal parameter and FG strain gradient parameter, are systematically formulated. The difficulty for FG nonlocal parameter is satisfactorily solved with by adopting the present 2n+1 order beam theory. Analytically, solutions to bending and buckling analyses within the unified model are obtained, from which the analytical solutions for Euler- and Timoshenko-type beam can be recovered. Numerically, bending deflection and buckling critical load for Euler beam, Reddy beam, 5th-11th order beam and Timoshenko beam are depicted, of which the benchmark solutions for the 5th to 11th order beam are given for the first time. Meanwhile, potential extensions of the unified model into fractional order is discussed, where benchmark solutions for n = 1.1, 0.88, 0.77, 0.4and0.2 are listed. The influences of FG nonlocal parameter, dimensionless height and Poisson's ratio (or the ratio E/G) on the bending deflection and buckling critical load are systematically studied. The present work mainly contributes to theoretical developments and greatly facilitates the mechanical analysis of beam-type structures.     

Interaction curves for vibration and buckling of thin-walled composite box beams under axial loads and end moments

Interaction curves for vibration and buckling of thin-walled composite box beams with arbitrary lay-ups under constant axial loads and equal end moments are presented. This model is based on the classical lamination theory, and accounts for all the structural coupling coming from material anisotropy. The governing differential equations are derived from the Hamilton’s principle. The resulting coupling is referred to as triply flexural–torsional coupled vibration and buckling. A displacement-based one-dimensional finite element model with seven degrees of freedoms per node is developed to solve the problem. Numerical results are obtained for thin-walled composite box beams to investigate the effects of axial force, bending moment, fiber orientation on the buckling loads, buckling moments, natural frequencies and corresponding vibration mode shapes as well as axial-moment–frequency interaction curves.     

Buckling analyses of three characteristic-lengths featured size-dependent gradient-beam with variational consistent higher order boundary conditions

In this work, a three characteristic-lengths featured size-dependent gradient-beam is constructed by adopting the modified nonlocal model, resulting in much more general constitutive equation with stress gradient up to four-order and strain gradient to two-order. The six-order differential governing equation for transverse displacement is formulated. All boundary conditions especially variational consistent higher order boundary conditions of the present model are derived with the aid of weighted residual approach. The closed-form solutions to critical buckling loads under different sets of boundary conditions are systematically formulated with higher order boundary conditions incorporated. The numerical results show that both nonlocal parameters have significant effect on the buckling behaviors. Meanwhile, if two nonlocal parameters are taken as same, the present results cannot always reduce to that from Eringen's nonlocal model. Due to its clear physical meaning, the present model is expected to be widely adopted in mechanical analyses of nano-structures.     

Vibration analysis of Euler–Bernoulli nanobeams by using finite element method

In the present study, an efficient finite element model for vibration analysis of a nonlocal Euler–Bernoulli beam has been reported. Nonlocal constitutive equation of Eringen is proposed. Equations of motion for a nonlocal Euler–Bernoulli are derived based on varitional statement. The finite element method is employed to discretize the model and obtain a numerical approximation of the motion equation. The model has been verified with the previously published works and found a good agreement with them. Vibration characteristics, such as fundamental frequencies, are illustrated in graphical and tabulated form. Numerical results are presented to figure out the effects of nonlocal parameter, slenderness ratios, rotator inertia, and boundary conditions on the dynamic characteristics of the beam. The above mention effects play very important role on the dynamic behavior of nanobeams.     

Computational Multi-Scale Stability Analysis of Periodic Electroactive Polymer Composites at Finite Strains

This work is dedicated to multi-scale stability analysis, especially macroscopic and microscopic stability analysis of periodic electroactive polymer (EAP) composites with embedded fibers. Computational homogenization is considered to determine the response of materials at macro-scale depending on the selected representative volume element (RVE) at micro-scale [4, 5]. The quasi-incompressibility condition is considered by implementing a four-field variational formulation on the RVE, see [7]. Based on the works [1–3, 6, 8] the macroscopic instabilities are determined by the loss of strong ellipticity of homogenized moduli. On the other hand, the bifurcation type microscopic instabilities are treated exploiting the Bloch-Floquet wave analysis in context of finite element discretization, which allows to detect the changed critical size of periodicity of the microstructure and critical macroscopic loading points. Finally, representative numerical examples are given which demonstrate the onset of instability surfaces, the stable macroscopic loading ranges, and further a periodic buckling mode at a microscopic instability point is presented. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)