Analytical prediction of Young's modulus of carbon nanotubes using a variational method

Molecular mechanics and solid mechanics are linked to establish, a nanoscale analytical continuum theory for determination of stiffness and Young's modulus of carbon nanotubes. A space-frame structure consisted of representative unit cells has been introduced to describe the mechanical response of carbon nanotubes to the applied loading. According to this assumption a novel unit cell, given the name mechanical unit cell here is introduced to construct a graphene sheet or the wall of the carbon nanotubes. Incorporating the Morse potential function with the strain energy of the mechanical unit cells in a carbon nanotube is the key point of this study. The structural model of the carbon nanotube is solved to obtain its Young's modulus by using the principle of minimum total potential energy. It was found that the Young's modulus of the zigzag and armchair single-walled carbon nanotubes are 1.42 and 1.30 TPa, respectively. The results indicate sensitivity of the stiffness and Young's modulus of carbon nanotubes to chirality but show no dependence on its diameter. The presented analytical investigation provides a very simple approach to predict the Young's modulus of carbon nanotubes and the obtained results are in good agreement with the existing experimental and theoretical data.     

Dynamical behaviors of fluid-conveyed multi-walled carbon nanotubes

Based on an elastic beam model and potential flow theory, and by adopting N-mode Galerkin discretization technique, the dynamical stability behaviors of fluid-conveyed multi-walled carbon nanotubes (MWCNTs) are studied. The influences of the flow inside the innermost tube and the van der Waals (vdW) interaction between any two walls on the instabilities of the CNTs-fluid system are discussed on the numerical simulations in detail. Also, the effects of the innermost tube radius, the length and the layer quantity of MWCNTs on the critical velocities of the destabilized system are intensively analyzed and compared. The results show that the critical velocities increase sharply as the radius becomes larger, the layer quantity increases, and the length decreases. The bifurcations happen in turn of divergence and Hopf types.     

Noncoaxial vibration of fluid-filled multi-walled carbon nanotubes

This paper is concerned with the free vibration of the fluid-filled multi-walled carbon nanotubes (MWCNTs) with simply supported ends. Based on simplified Donnell’s cylindrical shell model and potential flow theory, the effect of internal fluid on the coupling vibration of the MWCNTs-fluid system is discussed in detail. The results show that the resonant frequencies are decreased due to the effect of the fluid, and the fluid has only a little influence on the associated amplitude ratio in MWCNTs corresponding to the natural resonant frequency (frequency of the innermost tube), while plays a significant role in the associated amplitude ratios corresponding to the intertube resonant frequency. For the natural resonant frequency, the vibration mode is coaxial. However, for the intertube resonant frequency, the system shows complex noncoaxial vibration, which plays a critical role in electronic and transport properties of carbon nanotubes (CNTs).     

Approximate solutions to the nonlinear vibrations of multiwalled carbon nanotubes using Adomian decomposition method

This paper applies the Adomian decomposition method (ADM) to the search for the approximate solutions to the problem of the nonlinear vibrations of multiwalled carbon nanotubes embedded in an elastic medium. A multiple-beam model is utilized in which the governing equations of each layer are coupled with those of its adjacent ones via the van der Waals inter layer forces. The amplitude–frequency curves for large-amplitude vibrations of single-walled, double-walled and triple-walled carbon nanotubes are obtained. The influence of changes in material constants of the surrounding elastic medium and the effect of changes in nanotube geometrical parameters on the vibration characteristics are studied by comparing the results with those from the open literature. This method needs less work in comparison with the traditional methods and decreases considerable volume of calculation, and it’s powerful mathematical tool for solving wide class of nonlinear differential equations. Special attention is given to prove the convergence of the method. Some examples are given to illustrate the determination approximate solutions of the proposed problem.     

Nonlinear separation approach to inverse variational inequalities

In this paper, we employ the image space analysis to investigate an inverse variational inequality (for short, IVI) with a cone constraint. By virtue of the nonlinear scalarization function commonly known as the Gerstewitz function, three nonlinear weak separation functions, two nonlinear regular weak separation functions and a nonlinear strong separation function are first introduced. Then, by these nonlinear separation functions, theorems of the weak and strong alternative and some optimality conditions for IVI with a cone constraint are derived without any convexity. In particular, a global saddle-point condition for a nonlinear function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Finally, two gap functions and an error bound for IVI with a cone constraint are obtained.     

Effects of a small length scale on vibrations of an embedded double-walled carbon nanotube

Extensive continuum analyses are carried out to estimate the influence of matrix stiffness, a small length scale, and intertubular radial displacements on free vibrations of an individual double-walled carbon nanotybe. The analyses are based on both local and classical Euler–Bernoulli and Timoshenko elasticity theories with concentricity and nonconcentricity assumptions. The effect of a small length scale is incorporated in the formulations. New intertubular resonant frequencies are calculated based on these theories. Detailed results are demonstrated for the resonant frequencies as functions of matrix stiffness and the small length scale. The results indicate that the internal radial displacement and the stiffness of the surrounding matrix can greatly affect the resonant frequencies, especially at higher frequencies, and thus the latter does not keep the otherwise concentric structure at ultrahigh frequencies. More over, at high frequencies and small aspect ratios, the effect of the small length scale be comes more significant.     

Analytical analysis of buckling and post-buckling of fluid conveying multi-walled carbon nanotubes

In this work, buckling and post-buckling analysis of fluid conveying multi-walled carbon nanotubes are investigated analytically. The nonlinear governing equations of motion and boundary conditions are derived based on Eringen nonlocal elasticity theory. The nanotube is modeled based on Euler–Bernoulli and Timoshenko beam theories. The Von Karman strain–displacement equation is used to model the structural nonlinearities. Furthermore, the Van der Waals interaction between adjacent layers is taken into account. An analytical approach is employed to determine the critical (buckling) fluid flow velocities and post-buckling deflection. The effects of the small-scale parameter, Van der Waals force, ends support, shear deformation and aspect ratio are carefully examined on the critical fluid velocities and post-buckling behavior.     

Nonlinear vibrations of a viscoelastic beam

Averaging [2, 3] is used to examine the free transverse oscillations of a viscoelastic beam supported at the ends, the rheological properties being described by a certain cubic integral relationship. The forced principal and fractional resonant oscillations are considered for n=1, these being excited by a transverse periodic force; the limiting oscillations are determined that correspond to the principal resonance.M. T. Urazbaev Institute of Building Mechanics and Seismic Stability, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, No. 5, pp. 939–943, September–October, 1973.     

Nonlinear vibrations of a slider-crank mechanism

A nonlinear vibration analysis of the slider-crank mechanism used in reciprocating machines such as internal-combustion engines and air compressors is formulated by the principles of variational mechanics. A dynamically equivalent system is employed in which the connecting rod is modelled as a pendulum oscillating about a reciprocating piston and an approximate solution is derived by an application of the Ritz averaging method. Numerical results for a typical example are presented and discussed.     

Nonlinear electroelastostatics: a variational framework

Different formulations of the constitutive laws and governing equations for nonlinear electroelastic solids are reviewed and two new variational principles for electroelastostatics are introduced. One is based on use of the electrostatic scalar potential and one on the vector potential, combined with the deformation function. In each case Lagrangian forms of the electric variables are used. Their connections with several formulations of nonlinear electroelasticity in the literature are established and some differences highlighted.     

Nonlinear electroelastostatics: a variational framework

Different formulations of the constitutive laws and governing equations for nonlinear electroelastic solids are reviewed and two new variational principles for electroelastostatics are introduced. One is based on use of the electrostatic scalar potential and one on the vector potential, combined with the deformation function. In each case Lagrangian forms of the electric variables are used. Their connections with several formulations of nonlinear electroelasticity in the literature are established and some differences highlighted.      

Rigorous van der Waals effect on vibration characteristics of multi-walled carbon nanotubes under a transverse magnetic field

An analytical method is presented to investigate rigorous van der Waals interaction effect on vibration characteristics of multi-walled carbon nanotubes embedded in matrix under a transverse magnetic field. Each of the concentric tubes of multiwall carbon nanotubes is considered as an individual elastic shell and coupled with any two walls through a rigorous van der Waals interaction being dependent on the change of interlayer spacing and the radii of tubes. Results show that the rigorous van der Waals interaction effect makes the lowest magneto-vibration frequency of multi-walled carbon nanotubes decrease and the highest magneto-vibration frequency increase. The effect of rigorous van der Waals interaction on magneto-elastic vibrations of multi-walled carbon nanotubes is dependent on the transverse magnetic strength and the matrix constrained stiffness.     

Properties of a composite prepared using a concentrate of carbon nanotubes in polyethylene

Results of an investigation into the properties of polyethylene (PE) with small, no more than 5 wt.%, additions of multiwall carbon nanotubes (CNTs) are reported. Specimens of the composite were prepared using a concentrate containing 31.6 wt.% of nanotubes in the polyethylene matrix. The concentrate was fabricated by a patent in situ polymerization method. Experimental data on the influence of CNT additions on the thermograms of differential scanning calorimetry, the crystallinity of the polyethylene matrix, and the indices of mechanical properties (yield stress, strength, elastic modulus, ultimate elongation, and long-term creep) of PE/CNT composite are obtained. A theoretical analysis of elastic properties of the PE/CNT composite was carried out by using the Mori–Tanaka theory of an equivalent medium. The calculation results are compared with experimental data.     

Rotating response on the vibrations of functionally graded zigzag and chiral single walled carbon nanotubes

This study deals with the vibration analysis of zigzag and chiral rotating functionally graded carbon nanotubes (FG-CNT) invoking Love's shell theory using wave propagation approach. The frequency equation is formed in the eigenvalue form. It has been shown that with the increase of angular speed, frequencies of forwarding curve decrease and backward curve increase. The phenomena of frequency versus length-and height-to-radius ratios are noted as decreasing and increasing, respectively, for rotating CNTs. The backward and forward frequency curves of clamped-free are lower throughout the computation than the clamped-clamped zigzag and chiral carbon nanotube depending upon the rotating speed. MATLAB software is used to calculate the rotating (backward and forward) frequencies of SWCNTs and the frequency peaks in the present results show excellent stability across a wide range of parameters. Using geometrical and material parameters, the vibration results are given in tabular and graphical form. It is thus desirable to produce more precise estimations of the vibrational frequencies of CNTs. The present results are compared with earlier literature using simply supported boundary conditions and show a good coincidence.     

Nonlinear vibration analysis of double-walled carbon nanotubes based on nonlocal elasticity theory

The nonlinear free vibration of double-walled carbon nanotubes based on the nonlocal elasticity theory is studied in this paper. The nonlinear equations of motion of the double-walled carbon nanotubes are derived by using Euler beam theory and Hamilton principle, with considering the von Kármán type geometric nonlinearity and the nonlinear van der Waals forces. The surrounding elastic medium is formulated as the Winkler model. The harmonic balance method and Davidon–Fletcher–Powell method are utilized for the analysis and simulation of the nonlinear vibration. The simulation results show that the nonlocal parameter, aspect ratio and surrounding elastic medium play more important roles in the nonlinear noncoaxial vibration than those in the coaxial vibration of the double-walled carbon nanotubes. The noncoaxial vibration amplitudes of only considering nonlinear van der Waals forces are larger than those of considering both geometric nonlinearity and nonlinear van der Waals forces.     

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.     

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.