Curvature effects on axially compressed buckling of a small-diameter double-walled carbon nanotube

The curvature effects of interlayer van der Waals (vdW) forces on axially compressed buckling of a double-walled carbon nanotube (DWNT) of diameter down to 0.7 nm are studied. Unlike most existing models which assume that the interlayer vdW pressure at a point between the inner and outer tubes depends merely on the change of the interlayer spacing at that point, the present model considers the dependence of the interlayer vdW pressure on the change of the curvatures of the inner and outer tubes at that point. A simple expression is derived for the curvature-dependence of the interlayer vdW pressure in which the curvature coefficient is determined. Based on this model, an explicit formula is obtained for the axial buckling strain. It is shown that neglecting the curvature effect alone leads to an under-estimate of the critical buckling strain with a relative error up to −7%, while taking the average radius of two tubes as the representative radius and the curvature effect leads to an over-estimate of the critical buckling strain with a relative error up to 20% when the inner radius downs to 0.35 nm. Therefore, the curvature effects play a significant role in axially compressed buckling problems only for DWNTs of very small radii. In addition, our results show that the effect of the vdW interaction pressure prior to buckling of DWNTs under pure axial stress is small enough and can be negligible whether the vdW interaction curvature effects are neglected or not.     

Modeling of van der Waals force for infinitesimal deformation of multi-walled carbon nanotubes treated as cylindrical shells

This paper models van der Waals (vdW) force for axially compressed multi-walled carbon nanotubes (CNTs), whereby each tube is treated as a cylindrical shell continuum. Explicit formulas are derived for predicting the critical axial strain of a triple-walled CNT using a more refined vdW model. The analysis of a cylindrical shell continuum model of multi-walled CNTs using this refined vdW force model is carried out to study the influence of the effect of vdW interaction between different layers of a CNT and the size effect of a CNT on the vdW interaction. It is shown herein that the greatest contribution to the vdW interaction comes from the adjacent layers and the contribution from a remote layer may be neglected. The vdW interaction is found to be strongly dependent on the radius of the tube, especially when the radius is small enough (<7 nm). When the radius is large enough (>40 nm), the vdW interaction coefficient c_ij can be taken as a constant value (i.e. independent of radius). However, these constant values are different for the vdW interaction between two different layers of a multi-walled CNT. The effect of the vdW interaction on the critical axial strain of a triple-walled CNT for the cases of before and after buckling is also examined for various innermost radii.     

Effects of layer number and initial pressure on continuum-based buckling analysis of multi-walled carbon nanotubes accounting for van der Waals interaction

The structural instability of multi-walled carbon nanotubes (MWCNTs) has captured extensive attention due to the unique characteristic of extremely thin hollow cylinder structure. The previous studies usually focus on the buckling behavior without considering the effects of the wall number and initial pressure. In this paper, the axial buckling behavior of MWCNTs with the length-to-outermost radius ratio less than 20 is investigated within the framework of the Donnell shell theory. The governing equations for the infinitesimal buckling of MWCNTs are established, accounting for the van der Waals (vdW) interaction between layers. The effects of the wall number, initial pressure prior to buckling, and aspect ratio on the critical buckling mode, buckling load, and buckling strain are discussed, respectively. Specially, the four-walled and twenty-walled CNTs are studied in detail, indicating the fact that the buckling instability may occur in other layers besides the outermost layer. The obtained results extend the buckling analysis of the continuum-based model, and provide theoretical support for the application of CNTs.     

Small scale effect on flow-induced instability of double-walled carbon nanotubes

Small scale effect on flow-induced instability of double-walled carbon nanotubes (DWCNTs) is investigated using an elastic shell model based on Donnell’s shell theory. The dynamic governing equations of DWCNTs are formulated on the basis of nonlocal elasticity theory, in addition, the van der Waals (vdW) interaction between the inner and outer walls is taken into account in the nonlocal shell modeling. The instability of DWCNTs that is induced by a pressure-driven steady flow is investigated. The numerical computations indicate that as the flow velocity increases, DWCNTs have a destabilizing way to get through multi-bifurcations of the first and second bifurcations in turn. It is concluded that the natural frequency of DWCNTs and the critical flow velocity of the flow-induced instability are strictly related to the ratio of the length to the outer radius of DWCNTs, the pressure of the fluid and the small scale effects. Furthermore, it is interesting to observe that as the small scale effects are considered, the natural frequencies and the critical flow velocities of DWCNTs decrease as compared to the results with the classical (local) continuum mechanics, therefore, the small scale effects play an important role on performing the instability analysis in the fluid-conveying DWCNTs.     

Dynamic stability of electrostatic torsional actuators with van der Waals effect

The influence of van der Waals (vdW) force on the stability of electrostatic torsional nano-electro-mechanical systems (NEMS) actuators is analyzed in the paper. The dependence of the critical tilting angle and voltage is investigated on the sizes of structure with the consideration of vdW effects. The pull-in phenomenon without the electrostatic torque is studied, and a critical pull-in gap is derived. A dimensionless equation of motion is presented, and the qualitative analysis of it shows that the equilibrium points of the corresponding autonomous system include center points, stable focus points, and unstable saddle points. The Hopf bifurcation points and fork bifurcation points also exist in the system. The phase portraits connecting these equilibrium points exhibit periodic orbits, heteroclinic orbits, as well as homoclinic orbits.     

Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction

Explicit formulas are derived for the van der Waals (vdW) interaction between any two layers of a multi-walled carbon nanotube (CNT). Based on the derived formulas, an efficient algorithm is established for the buckling analysis of multi-walled CNTs, in which individual tubes are modeled as a continuum cylindrical shell. The explicit expressions are also derived for the buckling of double-walled CNTs. In previous studies by Ru (J. Appl. Phys. 87 (2000b) 7227) and Wang et al. (Int. J. Solids Struct. 40 (2003) 3893), only the vdW interaction between adjacent two layers was considered and the vdW interaction between the other two layers was neglected. The vdW interaction coefficient was treated as a constant that was not dependent on the radii of the tubes. However, the formulas derived herein reveal that the vdW interaction coefficients are dependent on the change of interlayer spacing and the radii of the tubes. With the increase of radii, the coefficients approach constants, and the constants between two adjacent layers are about 10% higher than those reported by Wang et al. (Int. J. Solids. Struct. 40 (2003) 3893). In addition, the numerical results show that the vdW interaction will lead to a higher critical buckling load in multi-walled CNTs. The effect of the tube radius on the critical buckling load of a multi-walled CNT is also examined.     

Nonlocal beam theory for nonlinear vibrations of embedded multiwalled carbon nanotubes in thermal environment

A nonlocal elastic beam model is developed to investigate the small scale effects on the large-amplitude vibration analysis of embedded multiwalled carbon nanotubes (MWCNTs) at an elevated temperature. The nested slender nanotubes are coupled with each other through the van der Waals (vdW) interlayer interaction. The curvature-dependent vdW force employed incorporates not only pairwise nearest-neighbor but also nonneighbor interactions between nested nanotubes. The incremental harmonic balance method is adopted to analytically solve the nonlinear equations that are governed by the vibrations of nested nanotubes. The influences of small scale parameter, geometrical parameters, temperature rise, and the elastic medium are fully examined.     

Dynamic bifurcations analysis of a micro rotating shaft considering non-classical theory and internal damping

In this study, the dynamic bifurcation of a viscoelastic micro rotating shaft is investigated. The non-classical theory (the modified couple stress theory) and the Kelvin Voigt model are used for modeling the viscoelastic micro shaft. The transverse equations of motion are derived using the variational approach. The reduced order model of the system is obtained by the Galerkin method. Using the Routh–Hurwitz criteria the stability regions of the system are extracted in which the effect of the length scale parameter is significant. Using the center manifold theory and the normal form method the double zero eigenvalue bifurcation is analyzed. The results show that the internal and external damping coefficients, the rotational speed and the material length scale parameter influence the critical speed, amplitude, and phase of a non-trivial solution, and radius of limit cycle (periodic solution). Also, it is seen that by increasing the dimensionless length scale parameter (material length scale per radius of the shaft) the radius of the limit cycle is decreased, whereas the critical rotational speed and the rate of the phase are increased. However, the radius of the limit cycle concerning the classical theory is higher than that of regarding the modified couple stress theory. Furthermore, with an increase of the external damping coefficient the radius of the limit cycle is linearly decreased; however, the critical speed of the system is increased. Additionally, by decreasing length scale parameter the results of the modified couple stress theory approach the classical theory ones.     

不同参数下环形通道内湍流旋流流动的模拟

应用一种合理考虑湍流一旋流相互作用及湍流脉动各向异性的新的代数ＲｅｙｎｏｌｄＳ应力模型,对环形通道内的湍流旋流流动进行了数值模拟．研究了旋流数、进口轴向速度和内外半径比等参数对环形通道内湍流旋流流动的影响,以及由此产生的流场变化对强化环形通道内传热的作用．     

考虑范德华力曲率效应的双壁碳纳米管外压屈曲

针对双壁碳纳米管外压屈曲问题,研究了层间范德华力的曲率效应对临界外压的影响。应用弹性双层圆柱壳模型,考虑层间范德华力不仅与层间距有关而且与挠度曲率的变化有关,导出了外压屈曲临界压力解析公式。计算得出在不同半径、不同长细比下,外压屈曲临界压力的数值结果,并与经典壳的结果和忽略范德华力曲率效应的结果做了比较。结果显示,对于小半径的双壁碳纳米管曲率效应对外压屈曲有效明显的影响。     

Stability and bifurcation analysis in tri-neuron model with time delay

A simple delayed neural network model with three neurons is considered. By constructing suitable Lyapunov functions, we obtain sufficient delay-dependent criteria to ensure global asymptotical stability of the equilibrium of a tri-neuron network with single time delay. Local stability of the model is investigated by analyzing the associated characteristic equation. It is found that Hopf bifurcation occurs when the time delay varies and passes a sequence of critical values. The stability and direction of bifurcating periodic solution are determined by applying the normal form theory and the center manifold theorem. If the associated characteristic equation of linearized system evaluated at a critical point involves a repeated pair of pure imaginary eigenvalues, then the double Hopf bifurcation is also found to occur in this model. Our main attention will be paid to the double Hopf bifurcation associated with resonance. Some Numerical examples are finally given for justifying the theoretical results.     

A glance on the effects of temperature on axisymmetric dynamic behavior of multiwall carbon nanotubes

In this paper the effects of temperature on theradial breathing modes(RBMs) and radial wave propagation in multiwall carbon nanotubes(MWCNTs) are investigated using a continuum model of multiple elastic isotropicshells.The van der Waals forces between tubes are simulatedas a nonlinear function of interlayer spacing of MWCNTs.The governing equations are solved using a finite elementmethod.A wide range of innermost radius-to-thickness ratioof MWCNTs is considered to enhance the investigation.Thepresented solution is verified by comparing the results withthose reported in the literature.The effects of temperature onthe van der Waals interaction coefficient between layers ofMWCNTs are examined.It is found that the variation of thevan der Waals interaction coefficient at high temperature issensible.Subsequently,variations of RBM frequencies andradial wave propagation in MWCNTs with temperatures upto 1 600 K are illustrated.It is shown that the thick MWCNTs are more sensible to temperature than the thin ones.     

Analytical modeling of static behavior of electrostatically actuated nano/micromirrors considering van der Waals forces

In this paper,the effect of van der Waals(vdW)force on the pull-in behavior of electrostatically actuatednano/micromirrors is investigated.First,the minimum potential energy principle is utilized to find the equation governing the static behavior of nano/micromirror under electrostatic and vdW forces.Then,the stability of static equilibrium points is analyzed using the energy method.It is foundthat when there exist two equilibrium points,the smaller oneis stable and the larger one is unstable.The effects of different design parameters on the mirror’s pull-in angle andpull-in voltage are studied and it is found that vdW forcecan considerably reduce the stability limit of the mirror.Atthe end,the nonlinear equilibrium equation is solved numerically and analytically using homotopy perturbation method(HPM).It is observed that a sixth order perturbation approximation can precisely model the mirror’s behavior.The results of this paper can be used for stable operation design andsafe fabrication of torsional nano/micro actuators.     

Post-critical behavior of Beck's column with a tip mass

This study examines how a tip mass with rotary inertia affects the stability of a follower-loaded cantilevered column. Using nonlinear modeling and perturbation analysis, expressions are set up for determining the stability of the straight column and the amplitude of post-critical flutter oscillations. Bifurcation diagrams are given, showing how the vibration amplitude changes with follower load and other parameters. These results agree closely with numerical simulation. It is found that sufficiently large values of tip mass rotary inertia can change the primary bifurcation from supercritical into subcritical. This can imply very large motions for follower loads just beyond critical, contrasting the finite amplitude motions accompanying supercritical bifurcations. Also, the straight column may be destabilized by a sufficiently strong disturbance at loads far below the value of critical load predicted by linear theory. A similar change in bifurcation is found to occur with increased external (as compared to internal) damping, and with a shortening in column length. These effects are not revealed by linear modeling and analysis, which may consequently fail to predict even qualitatively the real critical load for a column with tip mass.     

Multiple Duffing problem in a folding structure with hilltop bifurcation and possible imperfections

We review a multiple Duffing oscillation, based on static bifurcation theory. We find that it is useful to consider the structural instability of a folding truss with possible imperfections as a theoretical model for a Duffing problem with multiple potential wells. Theoretical bifurcation analysis revealed that the equilibrium path on this model has a “hilltop bifurcation.” In addition, we have considered the elastic (in-)stability of several folding models with imperfections. The present model is very sensitive near a critical point, leading to strong geometrical nonlinearity. We found that there are both global and local dynamic behaviours that are related to bifurcation and imperfect influences, which correspond to the structure of the multiple homo- and heteroclinic orbits. We suggest a theoretical model for hilltop bifurcation, based on the static bifurcation problem and perturbation theory, to assist in the identification of the structural mechanisms of the global and local dynamics of different paths. Such models are very useful for investigating the essential and invariant nonlinear phenomena of the extended Duffing oscillator model.     

The lowest stability and bifurcation in supercritical Taylor vortices

Here, we numerically investigate the lowest stability and bifurcation boundary of supercritical Taylor vortices in flows with different wavenumbers and for various radius ratios; the radius ratios range from those corresponding to axisymmetrical Taylor vortex flow (TVF) to those corresponding to wavy vortex flow (WVF). The variation in the wavenumber of a supercritical TVF is found to affect the stability of the flow, because the wavenumber of the Taylor vortices remains constant only when the flow is quasi-static. The variation in the wavenumber is examined and found to be significant when the radius ratio is less than 0.7842. The results for TVF are compared with those for the flow during the quasi-static transition from TVF to WVF.     

Oscillatory instability of the Couette flow between two unidirectionally rotating cylinders

In the problem of stability of the Couette flow between unidirectionally rotating cylinders, bifurcation points of codimensions 1, 2, and 3 are calculated. These points are interesting because in their small neighborhood the sequences of fluid flow bifurcations, including the development of chaotic regimes, can be studied analytically by completely correct methods. It is found that at large rotation velocities of both cylinders the pattern of the neutral curves corresponding to bifurcation points of codimension 1 changes and various intersections of two and three neutral curves (bifurcation points of codimensions 2 and 3) appear.     

Effects of supported angle on stability and dynamical bifurcations of cantilevered pipe conveying fluid

The effects of the supported angle on the stability and dynamical bifurcations of an inclined cantilevered pipe conveying fluid are investigated. First, a theoretical model of the pipe is developed through the force balance and stress-strain relationship. Second, the response surfaces, stability, and critical lines of the typical hanging system (H-S) and standing system (S-S) are discussed based on the modal analysis. Last, the bifurcation diagrams of the pipe are presented for different supported angles. It is shown that pipes will undergo a series of bifurcation processes and show rich dynamic phenomena such as buckling, Hopf bifurcation, period-doubling bifurcation, chaotic motion, and divergence motion.