On Finite Element Vibration Analysis of Carbon Nanotubes

In this chapter, a finite element formulation is proposed to study the natural frequencies of double-walled carbon nanotubes modeled as, both, local and nonlocal Euler-Bernoulli beams, coupled with van der Waals interaction forces. The formulation uses Galerkin-weighted residual approach and employs Hermite cubic polynomial function to derive the linear eigenvalue problem. Natural frequencies are found for clamped-free, clamped-clamped and simply supported-simply supported boundary conditions. The results are in good agreement with the formulations found in the literature. The effect of nonlocal factor on the natural frequencies of the system is found out by comparing local and nonlocal results. Additionally, the universality of the proposed model is proven by application to a double-elastic Euler-Bernoulli beam. This formulation paves way for Finite Element Method (FEM) analysis of multi-walled CNTs—either locally or nonlocally

Nonlinear Vibration Analysis of Thermo-Magneto-Mechanical Piezoelectric Nanobeam Embedded in Multi-Layer Elastic Media based on Nonlocal Elasticity Theory

The present article focuses on the investigations of electromechanical thermo-magnetic coupled effects on the nonlinear vibration of single-walled carbon nanobeam embedded in Winkler, Pasternak, quadratic and cubic nonlinear elastic media for simply supported and clamped boundary conditions are investigated. From the parametric studies, it is shown that the frequency of the nanobeam increases at low temperature but decreases at the high temperatures. The nonlocal parameter decreases the frequencies of the piezoelectric nanobeam. An increase in the quadratic nonlinear elastic medium stiffness causes a decrease in the first mode of the nanobeam with clamped-clamped supports and an increase in all modes of the simply supported nanobeam at both low and high temperature. When the magnetic force, cubic nonlinear elastic medium stiffness, and amplitude increase, there is an increase in all mode frequency of the nanobeam. A decrease in Winkler and Pasternak elastic media constants and increase in the nonlinear parameters of elastic medium results in an increase in the frequency ratio. The frequency ratio increases as the values of the dimensionless nonlocal, quadratic and cubic elastic medium stiffness parameters increase. However, the frequency ratio decreases as the values of the temperature change, magnetic force, Winkler and Pasternak layer stiffness parameters increase. An increase in the temperature change at high temperature reduces the frequency ratio but at low or room temperature, increase in temperature change, increases the frequency ratio of the structure nanotube. This work will greatly benefit in the design and applications of nanobeams in thermal and magnetic environments

Nonlinear Vibration Analysis of Thermo-Magneto-Mechanical Piezoelectric Nanobeam Embedded in Multi-Layer Elastic Media based on Nonlocal Elasticity Theory

The present article focuses on the investigations of electromechanical thermo-magnetic coupled effects on the nonlinear vibration of single-walled carbon nanobeam embedded in Winkler, Pasternak, quadratic and cubic nonlinear elastic media for simply supported and clamped boundary conditions are investigated. From the parametric studies, it is shown that the frequency of the nanobeam increases at low temperature but decreases at the high temperatures. The nonlocal parameter decreases the frequencies of the piezoelectric nanobeam. An increase in the quadratic nonlinear elastic medium stiffness causes a decrease in the first mode of the nanobeam with clamped-clamped supports and an increase in all modes of the simply supported nanobeam at both low and high temperature. When the magnetic force, cubic nonlinear elastic medium stiffness, and amplitude increase, there is an increase in all mode frequency of the nanobeam. A decrease in Winkler and Pasternak elastic media constants and increase in the nonlinear parameters of elastic medium results in an increase in the frequency ratio. The frequency ratio increases as the values of the dimensionless nonlocal, quadratic and cubic elastic medium stiffness parameters increase. However, the frequency ratio decreases as the values of the temperature change, magnetic force, Winkler and Pasternak layer stiffness parameters increase. An increase in the temperature change at high temperature reduces the frequency ratio but at low or room temperature, increase in temperature change, increases the frequency ratio of the structure nanotube. This work will greatly benefit in the design and applications of nanobeams in thermal and magnetic environments

Nonlinear Vibrations of Multiwalled Carbon Nanotubes under Various Boundary Conditions

The present work deals with applying the homotopy perturbation method to the problem of the nonlinear oscillations of multiwalled carbon nanotubes embedded in an elastic medium under various boundary conditions. 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 interlayer forces. The amplitude-frequency curves for large-amplitude vibrations of single-walled, double-walled, and triple-walled carbon nanotubes are obtained. The influences of some commonly used boundary conditions, changes in material constant of the surrounding elastic medium, and variations of the nanotubes geometrical parameters on the vibration characteristics of multiwalled carbon nanotubes are discussed. The comparison of the generated results with those from the open literature illustrates that the solutions obtained are of very high accuracy and clarifies the capability and the simplicity of the present method. It is worthwhile to say that the results generated are new and can be served as a benchmark for future works

Buckling analysis of a non-concentric double-walled carbon nanotube

On the basis of a theoretical study, this research incorporates an eccentricity into a system of compressed double-walled carbon nanotubes (DWCNTs). In order to formulate the stability equations, a kinematic displacement with reference to the classical beam hypothesis is utilized. Furthermore, the influence of nanoscale size is taken into account with regard to the nonlocal approach of strain gradient, and the van der Waals interaction for both inner and outer tubes is also considered based on the Lennard–Jones model. Galerkin decomposition is employed to numerically deal with the governing equations. It is evidently demonstrated that the geometrical eccentricity remarkably affects the stability threshold and its impact is to increase the static stability of DWCNTs

Post-critical buckling of truncated conical carbon nanotubes considering surface effects embedding in a nonlinear Winkler substrate using the Rayleigh-Ritz method

This research predicts theoretically post-critical axial buckling behavior of truncated conical carbon nanotubes (CCNTs) with several boundary conditions by assuming a nonlinear Winkler matrix. The post-buckling of CCNTs has been studied based on the Euler-Bernoulli beam model, Hamilton's principle, Lagrangian strains, and nonlocal strain gradient theory. Both stiffness-hardening and stiffness-softening properties of the nanostructure are considered by exerting the second stress-gradient and second strain-gradient in the stress and strain fields. Besides small-scale influences, the surface effect is also taken into consideration. The effect of the Winkler foundation is nonlinearly taken into account based on the Taylor expansion. A new admissible function is used in the Rayleigh-Ritz solution technique applicable for buckling and post-buckling of nanotubes and nanobeams. Numerical results and related discussions are compared and reported with those obtained by the literature. The significant results proved that the surface effect and the nonlinear term of the substrate affect the CCNT considerably

Damped forced vibration analysis of single-walled carbon nanotubes resting on viscoelastic foundation in thermal environment using nonlocal strain gradient theory

In this paper, the damped forced vibration of single-walled carbon nanotubes (SWCNTs) is analyzed using a new shear deformation beam theory. The SWCNTs are modeled as a flexible beam on the viscoelastic foundation embedded in the thermal environment and subjected to a transverse dynamic load. The equilibrium equations are formulated by the new shear deformation beam theory which is accompanied with higher-order nonlocal strain gradient theory where the influences of both stress nonlocality and strain gradient size-dependent effects are taken into account. In this new shear deformation beam theory, there is no need to use any shear correction factor and also the number of unknown variables is the only one that is similar to the Euler-Bernoulli beam hypothesis. The governing equations are solved by utilizing an analytical approach by which the maximum dynamic deflection has been obtained with simple boundary conditions. To validate the results of the new proposed beam theory, the results in terms of natural frequencies are compared with the results from an available well-known reference. The effects of nonlocal parameter, half-wave length, damper, temperature and material variations on the dynamic vibration of the nanotubes, are discussed in detail. Keywords: Forced vibration, Single walled carbon nanotube, A new refined beam theory, Higher-order nonlocal strain gradient theory, Dynamic deflectio

Low-frequency linear vibrations of single-walled carbon nanotubes: Analytical and numerical models

Low-frequency vibrations of single-walled carbon nanotubes with various boundary conditions are considered in the framework of the Sanders–Koiter thin shell theory. Two methods of analysis are proposed. The first approach is based on the Rayleigh–Ritz method, a double series expansion in terms of Chebyshev polynomials and harmonic functions is considered for the displacement fields; free and clamped edges are analysed. This approach is partially numerical. The second approach is based on the same thin shell theory, but the goal is to obtain an analytical solution useful for future developments in nonlinear fields; the Sanders–Koiter equations are strongly simplified neglecting in-plane circumferential normal strains and tangential shear strains. The model is fully validated by means of comparisons with experiments, molecular dynamics data and finite element analyses obtained from the literature. Several types of nanotubes are considered in detail by varying aspect ratio, chirality and boundary conditions. The analyses are carried out for a wide range of frequency spectrum. The strength and weakness of the proposed approaches are shown; in particular, the model shows great accuracy even though it requires minimal computational effort

Variational Principles for Buckling of Microtubules Modeled as Nonlocal Orthotropic Shells

A variational principle for microtubules subject to a buckling load is derived by semi-inverse method. The microtubule is modeled as an orthotropic shell with the constitutive equations based on nonlocal elastic theory and the effect of filament network taken into account as an elastic surrounding. Microtubules can carry large compressive forces by virtue of the mechanical coupling between the microtubules and the surrounding elastic filament network. The equations governing the buckling of the microtubule are given by a system of three partial differential equations. The problem studied in the present work involves the derivation of the variational formulation for microtubule buckling. The Rayleigh quotient for the buckling load as well as the natural and geometric boundary conditions of the problem is obtained from this variational formulation. It is observed that the boundary conditions are coupled as a result of nonlocal formulation. It is noted that the analytic solution of the buckling problem for microtubules is usually a difficult task. The variational formulation of the problem provides the basis for a number of approximate and numerical methods of solutions and furthermore variational principles can provide physical insight into the problem