A new nonlocal bending model for Euler–Bernoulli nanobeams

This paper is concerned with the bending problem of nanobeams starting from a nonlocal thermodynamic approach. A new coupled nonlocal model, depending on two nonlocal parameters, is obtained by using a suitable definition of the free energy. Unlike previous approaches which directly substitute the expression of the nonlocal stress into the classical equilibrium equations, the proposed approach provides a methodology to recover nonlocal models starting from the free energy function. The coupled model can then be specialized to obtain a nanobeam formulation based on the Eringen nonlocal elasticity theory and on the gradient elastic model. The variational formulations are consistently provided and the differential equations with the related boundary conditions are thus derived. Nanocantilevers are solved in a closed-form and numerical results are presented to investigate the influence of the nonlocal parameters.     

Effect of small size on dispersion characteristics of wave in carbon nanotubes

Effect of small size on dispersion characteristics of waves in multi-walled carbon nanotubes is investigated using an elastic shell model. Dynamic governing equations of the carbon nanotube are formulated on the basis of nonlocal elastic theory. The relationship between wavenumber and frequency of wave propagation is obtained from the solution of the eigenvalue equations. The numerical results show that the dispersion characteristics of wave in the multi-walled carbon nanotube are affected by the small size. Effect of small size is not obvious for the smaller wavenumber, and it will arise and increase gradually with the increase of the wavenumber. Effect of the small size will decrease as the inner radius of carbon nanotubes increases. In addition, the explicit expressions of the cut-off frequencies are derived. The results show that the cut-off frequencies cannot be influenced by the small size of carbon nanotubes.     

温度场作用下输流悬臂碳纳米管的颤振失稳分析

以非局部弹性理论为基础,采用欧拉-伯努利梁模型,考虑碳纳米管的小尺度效应,应用哈密顿原理获得了温度场作用下的输流悬臂单层碳纳米管（SWCNT）的振动控制方程以及边界条件,依靠微分变换法（DTM法）对此高阶偏微分方程进行求解,通过数值计算研究了温度场中悬臂单层输流碳纳米管的振动与颤振失稳问题。结果表明：管内流体流速、温度场中温度变化情况与小尺度参数都会对系统振动频率以及颤振失稳临界流速产生影响。其中,小尺度效应将会降低悬臂输流系统的稳定性,使系统更为柔软;而高温场与低温场对系统动态失稳的影响不同,低温场中随温度变化值的增加,系统的稳定性提高;高温场这一作用效果恰好与之相反。     

On the plastic buckling of curved carbon nanotubes

This research, for the first time, predicts theoretically static stability response of a curved carbon nanotube(CCNT) under an elastoplastic behavior with several boundary conditions. The CCNT is exposed to axial compressive loads. The equilibrium equations are extracted regarding the Euler–Bernoulli displacement field by means of the principle of minimizing total potential energy.The elastoplastic stress-strain is concerned with Ramberg–Osgood law on the basis of deformation and flow theories of plasticity. To seize the nano-mechanical behavior of the CCNT, the nonlocal strain gradient elasticity theory is taken into account. The obtained differential equations are solved using the Rayleigh–Ritz method based on a new admissible shape function which is able to analyze stability problems. To authorize the solution, some comparisons are illustrated which show a very good agreement with the published works. Conclusively, the best findings confirm that a plastic analysis is crucial in predicting the mechanical strength of CCNTs.     

Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes

This paper investigates the transverse and torsional wave in single- and double-walled carbon nanotubes (SWCNTs and DWCNTs), focusing on the effect of carbon nanotube microstructure on wave dispersion. The SWCNTs and DWCNTs are modeled as nonlocal single and double elastic cylindrical shells. Molecular dynamics (MD) simulations indicate that the wave dispersion predicted by the nonlocal elastic cylindrical shell theory shows good agreement with that of the MD simulations in a wide frequency range up to the terahertz region. The nonlocal elastic shell theory provides a better prediction of the dispersion relationships than the classical shell theory when the wavenumber is large enough for the carbon nanotube microstructure to have a significant influence on the wave dispersion. The nonlocal shell models are required when the wavelengths are approximately less than 2.36×10⁻⁹ and 0.95×10⁻⁹ m for transverse wave in armchair (15,15) SWCNT and torsional wave in armchair (10,10) SWCNT, respectively. Moreover, an MD-based estimation of the scale coefficient e₀ for the nonlocal elastic cylindrical shell model is suggested. Due to the small-scale effects of SWCNTs and the interlayer van der Waals interaction of DWCNTs, the phase difference of the transverse wave in the inner and outer tube can be observed in MD simulations in wave propagation at high frequency. However, the van der Waals interaction has little effect on the phase difference of transverse wave.     

A nonlocal strain gradient shell model incorporating surface effects for vibration analysis of functionally graded cylindrical nanoshells

In this paper, a novel size-dependent functionally graded(FG) cylindrical shell model is developed based on the nonlocal strain gradient theory in conjunction with the Gurtin-Murdoch surface elasticity theory. The new model containing a nonlocal parameter, a material length scale parameter, and several surface elastic constants can capture three typical types of size effects simultaneously, which are the nonlocal stress effect, the strain gradient effect, and the surface energy effects. With the help of Hamilton's principle and first-order shear deformation theory, the non-classical governing equations and related boundary conditions are derived. By using the proposed model, the free vibration problem of FG cylindrical nanoshells with material properties varying continuously through the thickness according to a power-law distribution is analytically solved, and the closed-form solutions for natural frequencies under various boundary conditions are obtained. After verifying the reliability of the proposed model and analytical method by comparing the degenerated results with those available in the literature, the influences of nonlocal parameter, material length scale parameter, power-law index, radius-to-thickness ratio, length-to-radius ratio, and surface effects on the vibration characteristic of functionally graded cylindrical nanoshells are examined in detail.     

VIBRATION OF FLUID-FILLED MULTI-WALLED CARBON NANOTUBES SEEN VIA NONLOCAL ELASTICITY THEORY

Vibration characteristics of fluid-filled multi-walled carbon nanotubes axe studied by using nonlocal elastic Fliigge shell model. Vibration governing equations of an N-layer carbon nanotube are formulated by considering the scale effect. In the numerical simulations, the effects of different theories, small-scale, variation of wavenumber, the innermost radius and length of double- walled and triple-walled carbon nanotubes are considered. Vibrational frequencies decrease with an increase of scale coefficient, the innermost radius, length of nanotube and effects of wall number are negligible. The results show that the cut-off frequencies can be influenced by the wall number of nanotubes.     

An analytical symplectic approach to the vibration analysis of orthotropic graphene sheets

A nonlocal continuum orthotropic plate model is proposed to study the vibration behavior of single-layer graphene sheets (SLGSs) using an analytical symplectic approach.A Hamiltonian system is established by introduc-ing a total unknown vector consisting of the displacement amplitude,rotation angle,shear force,and bending moment. The high-order governing differential equation of the vibra-tion of SLGSs is transformed into a set of ordinary differential equations in symplectic space.Exact solutions for free vibra-tion are obtianed by the method of separation of variables without any trial shape functions and can be expanded in series of symplectic eigenfunctions. Analytical frequency equations are derived for all six possible boundary con-ditions. Vibration modes are expressed in terms of the symplectic eigenfunctions.In the numerical examples,com-parison is presented to verify the accuracy of the proposed method. Comprehensive numerical examples for graphene sheets with Levy-type boundary conditions are given.A para-metric study of the natural frequency is also included.     

Nonlinear vibration of embedded single-walled carbon nanotube with geometrical imperfection under harmonic load based on nonlocal Timoshenko beam theory

Based on the nonlocal continuum theory, the nonlinear vibration of an embedded single-walled carbon nanotube (SWCNT) subjected to a harmonic load is investigated. In the present study, the SWCNT is assumed to be a curved beam, which is unlike previous similar work. Firstly, the governing equations of motion are derived by the Hamilton principle, meanwhile, the Galerkin approach is carried out to convert the nonlinear integral-differential equation into a second-order nonlinear ordinary differential equation. Then, the precise integration method based on the local linearzation is appropriately designed for solving the above dynamic equations. Besides, the numerical example is presented, the effects of the nonlocal parameters, the elastic medium constants, the waviness ratios, and the material lengths on the dynamic response are analyzed. The results show that the above mentioned effects have influences on the dynamic behavior of the SWCNT.     

Torsional static and vibration analysis of functionally graded nanotube with bi-Helmholtz kernel based stress-driven nonlocal integral model

A torsional static and free vibration analysis of the functionally graded nanotube(FGNT)composed of two materials varying continuously according to the power-law along the radial direction is performed using the bi-Helmholtz kernel based stress-driven nonlocal integral model.The differential governing equation and boundary conditions are deduced on the basis of Hamilton’s principle,and the constitutive relationship is expressed as an integral equation with the bi-Helmholtz kernel.Several nominal variables are introduced to simplify the differential governing equation,integral constitutive equation,and boundary conditions.Rather than transforming the constitutive equation from integral to differential forms,the Laplace transformation is used directly to solve the integro-differential equations.The explicit expression for nominal torsional rotation and torque contains four unknown constants,which can be determined with the help of two boundary conditions and two extra constraints from the integral constitutive relation.A few benchmarked examples are solved to illustrate the nonlocal influence on the static torsion of a clamped-clamped(CC)FGNT under torsional constraints and a clamped-free(CF)FGNT under concentrated and uniformly distributed torques as well as the torsional free vibration of an FGNT under different boundary conditions.     

Postbuckling prediction of double-walled carbon nanotubes under axial compression

An elastic double-shell model is presented for the buckling and postbuckling of a double-walled carbon nanotube subjected to axial compression. The analysis is based on a continuum mechanics model in which each tube of a double-walled carbon nanotube is described as an individual elastic shell and the interlayer friction is negligible between the inner and outer tubes. The governing equations are based on the Karman–Donnell-type nonlinear differential equations. The van der Waals interaction between the inner and outer nanotubes and the nonlinear prebuckling deformations of the shell are both taken into account. A boundary layer theory of shell buckling is extended to the case of double-walled carbon nanotubes under axial compression. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. Numerical results reveal that the single-walled carbon nanotube and the double-walled carbon nanotube both have an unstable postbuckling behavior.     

磁场对不同温度场中输流悬臂碳纳米管动态特性的影响

本文在采用经典欧拉-伯努利梁模型的基础上,引入考虑小尺度效应的非局部弹性理论,着重研究不同温度场中输流悬臂单层碳纳米管系统（SWCNT）在外加纵向磁场作用下的颤振失稳问题。基于哈密顿原理获得了该流固耦合系统的振动控制方程及相应的边界条件,应用微分变换法（DTM法）求解此高阶偏微分方程,通过数值计算研究了不同温度场中施加纵向磁场对系统动力学特性的影响。结果表明：施加纵向磁场在不同温度场中都将增强输流悬臂碳纳米管的动态稳定性。然而,这种增强程度却与温度场的变化量有关,在不同温度变化量下,磁场对系统稳定性的增强程度有一个峰值,这意味着,实际应用中,为了提高这类流固耦合系统的动态稳定性,一味提高纵向磁场强度并不可取。     

Vibration characteristics of piezoelectric functionally graded carbon nanotube-reinforced composite doubly-curved shells

This paper presents an analytical solution for the free vibration behavior of functionally graded carbon nanotube-reinforced composite(FG-CNTRC) doubly curved shallow shells with integrated piezoelectric layers. Here, the linear distribution of electric potential across the thickness of the piezoelectric layer and five different types of carbon nanotube(CNT) distributions through the thickness direction are considered. Based on the four-variable shear deformation refined shell theory, governing equations are obtained by applying Hamilton's principle. Navier's solution for the shell panels with the simply supported boundary condition at all four edges is derived. Several numerical examples validate the accuracy of the presented solution. New parametric studies regarding the effects of different material properties, shell geometric parameters, and electrical boundary conditions on the free vibration responses of the hybrid panels are investigated and discussed in detail.     

多场作用下输流单层碳纳米管的动力学特性

以非局部弹性理论为基础,采用欧拉-伯努利梁模型,考虑管型区域内滑移边界条件以及碳纳米管的小尺度效应,应用哈密顿原理获得了温度场与轴向磁场共同作用下的输流单层固支碳纳米管（SWCNT）的振动控制方程以及边界条件,依靠微分变换法（DTM法）对此高阶偏微分方程进行求解,通过数值计算研究了多场中单层固支输流碳纳米管的振动与失稳问题。结果表明：温度场、轴向磁场强度、Knudsen数及小尺度参数都会对系统振动频率以及失稳临界流速产生影响。     

Free vibration of size-dependent magneto-electro-elastic nanoplates based on the nonlocal theory

In this paper, the free vibration of magneto- electro-elastic （MEE） nanoplates is investigated based on the nonlocal theory and Kirchhoff plate theory. The MEE nanoplate is assumed as all edges simply supported rectan gular plate subjected to the biaxial force, external electric potential, external magnetic potential, and temperature rise. By using the Hamilton＇s principle, the governing equations and boundary conditions are derived and then solved analytically to obtain the natural frequencies of MEE nanoplates. A parametric study is presented to examine the effect of the nonlocal parameter, thermo-magneto-electro-mechanical loadings and aspect ratio on the vibration characteristics of MEE nanoplates. It is found that the natural frequency is quite sensitive to the mechanical loading, electric loading and magnetic loading, while it is insensitive to the thermal loading.     

Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity

The axial vibration of single walled carbon nanotube embedded in an elastic medium is studied using nonlocal elasticity theory. The nonlocal constitutive equations of Eringen are used in the formulations. The effect of various parameters like stiffness of elastic medium, boundary conditions and nonlocal parameters on the axial vibration of nanorods is discussed. It is obtained that, the axial vibration frequencies of the embedded nanorods are highly over estimated by the classical continuum rod model which ignores the effect of small length scale.     

Application of nonlocal beam models for carbon nanotubes

Investigations of wave and vibration properties of single- or multi-walled carbon nanotubes based on nonlocal beam models have been reported recently. However, there are numerous inconsistencies in the handling of the governing equations, applied forces, and boundary conditions based on some of the reported nonlocal beam models. In this paper, the consistent equations of motion for the nonlocal Euler and Timoshenko beam models are provided, and some issues on the nonlocal beam theories are discussed. The models are then applied to the studies of wave properties of single- and double-walled nanotubes. The wave and vibration properties of the nanotubes based on the presented nonlocal beam equations are studied, and scale effects are discussed.     

Axial vibration analysis of nanocones based on nonlocal elasticity theory

Carbon nanocones have quite fascinating electronic and structural properties,whose axial vibration is seldom investigated in previous studies.In this paper,based ona nonlocal elasticity theory,a nonuniform rod model is applied to investigate the small-scale effect and the nonuniformeffect on axial vibration of nanocones.Using the modifiedWentzel-Brillouin-Kramers(WBK) method,an asymptoticsolution is obtained for the axial vibration of general nonuniform nanorods.Then,using similar procedure,the axial vibration of nanocones is analyzed for nonuniform parameters,mode number and nonlocal parameters.Explicit expressionsare derived for mode frequencies of clamped-clamped andclamped-free boundary conditions.It is found that axial vibration frequencies are highly overestimated by the classicalrod model because of ignorance of the effect of small lengthscale.     

Nonlinear free vibration of nanotube with small scale effects embedded in viscous matrix

In this paper, the nonlinear free vibration of the nanotube with damping effects is studied. Based on the nonlocal elastic theory and Hamilton principle, the governing equation of the nonlinear free vibration for the nanotube is obtained. The Galerkin method is employed to reduce the nonlinear equation with the integral and partial differential characteristics into a nonlinear ordinary differential equation. Then the relation is solved by the multiple scale method and the approximate analytical solution is derived. The nonlinear vibration behaviors are discussed with the effects of damping, elastic matrix stiffness, small scales and initial displacements. From the results, it can be observed that the nonlinear vibration can be reduced by the matrix damping. The elastic matrix stiffness has significant influences on the nonlinear vibration properties. The nonlinear behaviors can be changed by the small scale effects, especially for the structure with large initial displacement.