Nonlocal Elastic Shell Research Articles

Nonlinear modal analysis of multi-walled nanotube oscillations using nonlocal anisotropic elastic shell model

Nonlinear modal analysis of multi-walled nanotube oscillations using nonlocal anisotropic elastic shell model

Nonlocal anisotropic elastic shell model for vibrations of double-walled carbon nanotubes under nonlinear van der Waals interaction forces

In this paper, a novel nonlocal anisotropic elastic shell model is developed to investigate the nonlinear vibrations of double-walled carbon nanotubes (DWCNTs) in the framework of Sanders–Koiter shell theory. Van der Waals interaction forces between the two concentric single-walled carbon nanotubes (SWCNTs) composing a DWCNT are modelled via Lennard-Jones potential and He’s formulation. In the linear vibration analysis, the displacement field of each SWCNT is expanded by means of a double mixed series in terms of Chebyshev orthogonal polynomials along the longitudinal direction and harmonic functions along the circumferential direction, and Rayleigh–Ritz method is considered to get approximate natural frequencies and modal shapes. In the nonlinear vibration analysis, the three displacements of each SWCNT are re-expanded by means of the approximate eigenfunctions derived in the linear analysis, and an energy approach based on Lagrange equations is adopted to obtain a set of nonlinear ordinary differential equations of motion, which is then solved numerically. Molecular dynamics simulations are performed in order to calibrate the proper value of nonlocal parameter to be inserted in the constitutive equations of the proposed elastic continuum model. A simplified linear distribution of van der Waals interaction forces is initially adopted to analyse the nonlinear vibrations of DWCNTs, obtaining a hardening nonlinear behaviour. By considering a more realistic nonlinear distribution of van der Waals interaction forces, a stronger hardening nonlinear behaviour is found.

Propagation of waves with nonlocal effects for vibration response of armchair double-walled CNTs

In this paper, vibration characteristics of double-walled carbon nanotubes (CNTs) are studied based upon nonlocal elastic shell theory. The significance of small scale is being perceived by developing nonlocal Love shell model. The wave propagation approach has been utilized to frame the governing equations as eigen value system. The influence of nonlocal parameter subjected to diverse end supports has been overtly analyzed. An appropriate selection of material properties and nonlocal parameter has been considered. The influence of changing mechanical parameter Poisson's ratio has been investigated in detail. It is found that the frequencies decrease as nonlocal parameter increases and for the certain values of nonlocal parameter against range of Poisson ratio rise slowly with length double-walled CNTs. The dominance of boundary conditions via nonlocal parameter is shown graphically. The results generated furnish the evidence regarding applicability of nonlocal shell model and also verified by earlier published literature.

Prediction of small-scale vibration of zigzag DWCNTs: Numerical approach

In this article, free vibration attributes of zigzag double-walled carbon nanotubes (CNTs) based on nonlocal elastic shell model have been investigated. The impact of small scale is being perceived by establishing Flugge shell model. The wave propagation is engaged to frame the ruling equations as eigen value system. The influence of nonlocal parameter subjected to different end supports has been overtly examined. A suitable choice of material properties and nonlocal parameter been focused to analyze the vibration characteristics. The new set of inner and outer tubes radii investigated in detail against aspect ratio and length. The dominance of boundary conditions via nonlocal parameter is shown graphically. The results generated furnish the evidence regarding applicability of nonlocal shell model and also verified by earlier published literature.

Non-local orthotropic elastic shell model for vibration analysis of protein microtubules

Vibrational analysis in microtubules is examined based on the nonlocal theory of elasticity. The complete analytical formulas for wave velocity are obtained and the results reveal that the small scale effects can reduce the frequency, especially for large longitudinal wave-vector and large circumferential wave number. It is seen that the small scale effects are more significant for smaller wave length. The methods and results may also support the design and application of nano devices such as micro sound generator etc. The effects of small scale parameters can increase vibrational frequencies of the protein microtubules and cannot be overlooked in the analysis of vibrating phenomena. The results for different modes with nonlocal effect are checked.

Nonlocal vibration of DWCNTs based on Flügge shell model using wave propagation approach

In this article, free vibration attributes of double-walled carbon nanotubes based on nonlocal elastic shell model have been investigated. For this purpose, a nonlocal Flugge shell model is established to observe the small scale effect. The wave propagation is employed to frame the governing equations as eigenvalue system. The influence of nonlocal parameter subjected to different end supports has been overtly examined. A suitable choice of material properties and nonlocal parameter been focused to analyze the vibration characteristics. The new set of inner and outer tubes radii investigated in detail against aspect ratio and length. The dominance of boundary conditions via nonlocal parameter is shown graphically. The results generated furnish the evidence regarding applicability of nonlocal shell model and also verified by earlier published literature.

Non-local effect on the vibration analysis of double walled carbon nanotubes based on Donnell shell theory

Abstract This piece of work describes a free vibration analysis of double walled carbon nanotubes (DWCNTs) based on non-local elastic shell model. The classical Donnell's shell theory is engrossed to account the effects of small scale by merging Eringen's non-local elasticity equations into it. The wave propagation approach is implemented to analyse the influence of various edge conditions and also van der Waals interaction forces between inner and outer tubes given due consideration. An eigen value problem is formed to investigate the frequency spectra of DWCNTs in relevance of aspect ratio and distinct values of non-local parameter. The results in current study show the ability to perform free frequency characteristics of DWCNTs by developed non-local elastic shell model with appropriate choice of material properties has been certified by experimental comparisons with molecular dynamic data and also verified by earlier computations.

Nonlinear vibrations characteristics of single-walled carbon nanotubes by nonlocal elastic shell model

The model of single-walled carbon nanotubes nonlinear vibrations, which is based on shell theory, is derived. The system of partial differential equations with respect to three displacements projections is obtained using the Sanders–Koiter shell theory and nonlocal elasticity. The system of nonlinear ordinary differential equations, which describes the nanotubes vibrations, is obtained by the Galerkin method. The harmonic balanced method is used to calculate free nonlinear vibrations. The bifurcation behavior of CNT vibrations owing to the Naimark–Sacker bifurcation is analyzed. Almost periodic vibrations of carbon nanotubes are investigated numerically. The nanotube periodic and almost periodic vibrations have three displacements projections u,v,w, which are commensurable. This is essential difference of the nanotube vibrations from macroshells.

Rayleigh-Ritz Vibrational Analysis of Multiwalled Carbon Nanotubes Based on the Nonlocal Flügge Shell Theory

A nonlocal elastic shell model considering the small scale effects is developed to study the free vibrations of multiwalled carbon nanotubes subject to different types of boundary conditions. Based on the nonlocal elasticity and the Flügge shell theory, the governing equations are derived which include the interaction of van der Waals forces between adjacent and nonadjacent layers. To analytically solve the problem, the Rayleigh-Ritz method is employed. In the present analysis, different combinations of layerwise boundary conditions are taken into account. Some new intertube resonant frequencies and the associated noncoaxial vibrational modes are identified owing to incorporating circumferential modes into the shell model.

Nonlocal Flügge Shell Model for Vibrations of Double-Walled Carbon Nanotubes With Different Boundary Conditions

In this paper, the vibrational behavior of double-walled carbon nanotubes (DWCNTs) is studied by a nonlocal elastic shell model. The nonlocal continuum model accounting for the small scale effects encompasses its classical continuum counterpart as a particular case. Based upon the constitutive equations of nonlocal elasticity, the displacement field equations coupled by van der Waals forces are derived. The set of governing equations of motion are then numerically solved by a novel method emerged from incorporating the radial point interpolation approximation within the framework of the generalized differential quadrature method. The present analysis provides the possibility of considering different combinations of layerwise boundary conditions. The influences of small scale factor, layerwise boundary conditions and geometrical parameters on the mechanical behavior of DWCNTs are fully investigated. Explicit expressions for the nonlocal frequencies of DWCNTs with all edges simply supported are also analytically obtained by a nonlocal elastic beam model. Some new intertube resonant frequencies and the corresponding noncoaxial vibrational modes are identified due to incorporating circumferential modes into the shell model. A shift in noncoaxial mode numbers, not predictable by the beam model, is also observed when the radius of DWCNTs is varied. The results generated also provide valuable information concerning the applicability of the beam model and new noncoaxial modes affecting the physical properties of nested nanotubes.

Thermal Effect on Dynamic Stability of Single-Walled Carbon Nanotubes in Low and High Temperatures Based on Nonlocal Shell Theory

This paper studies the thermal effect on dynamic stability of single-walled carbon nanotubes (SWCNTs) embedded in polymer matrix in low and high temperatures under impact loads. To this end, we present a nonlocal elastic shell model including thermal and small-size effects. Further, numerical calculations are presented for SWCNTs. Results show that the increasing in temperature field has the effect of increasing the dynamic buckling loads at low or room temperature. However, this effect leads to decrease the dynamic buckling loads at high temperature. In addition, it is observed that nonlocal dynamic buckling loads increase by an increase in the Neperian frequency.

Scale Coefficient, Length, Mode and Radius Effect on Critical Axial Compressed Buckling Load of Carbon Nanotubes via Nonlocal Continuum Model

Some exact concise analytic solutions of critical axial compressed buckling load for carbon nanotubes are derived via nonlocal beam. Scale coefficient, length, mode and radius effect on nonlocal critical axial compressed buckling load of CNTs is established and can be analyzed in terms of the general solutions. Radius effect on nonlocal critical axial compressed buckling load is only found through nonlocal elastic shell model but not derived via nonlocal elastic beam model. Numerical calculations of CNTs show that local critical axial compressed buckling load through local elastic theory is overestimated. Scale coefficient, length, mode and radius effect should be taken into account in predicting more accurate results for mechanical behaviors of CNTs via continuum model.

Free vibration analysis of single- and double-walled carbon nanotubes based on nonlocal elastic shell models

In the present study, the free vibration characteristics of single- and double-walled carbon nanotubes (SWCNTs and DWCNTs) are investigated on the basis of a nonlocal elastic shell model. Eringen’s nonlocal elasticity equations are applied to the classical Donnell shell theory to incorporate the size-effects into the vibration analysis of carbon nanotubes (CNTs). An exact solution is developed for the governing equations of the nonlocal elastic shell model with the inclusion of size effects. Molecular dynamics (MD) simulations are performed to obtain fundamental frequencies of SWCNTs and DWCNTs with different values of aspect ratio and types of chirality. To derive the appropriate values of a nonlocal parameter for vibrations of SWCNTs and DWCNTs, the results of the continuum model are matched with those of MD simulations. This study shows that the small scale effects in the nonlocal model make nanotubes more flexible.

Investigation on Size Effect of Cantilever Carbon Nanotube

Nonlocal elastic shell model based on the semi-moment theory is developed and applied to investigate the small scale effect on the bending problem of the cantilever carbon nanotube (CNT) with a vertical concentrated load applied at its tip. The small-scale effect is taken into account and is incorporated in the formulation. Analytical expressions of the stress are derived for the nonlocal elastic bending problem. It is obvious to observe significant small-scale effects on the stress resultants. The smaller the radius is, the more obvious the scale effect appears. The numerical results show that the scale effect cannot be ignored for CNTs of small radius.

Prediction of nonlocal scale parameter for carbon nanotubes

Based on the theory of nonlocal elasticity, a nonlocal shell model accounting for the small scale effect is developed for the bending characteristics of CNTs subjected to the concentrated load. With this nonlocal shell model, explicit expressions are derived for the bending solutions. To extract the proper values of nonlocal scale parameter, we have made molecular dynamics (MD) simulations for various radii and lengths of armchair and zigzag CNTs, the results of which are matched with those of nonlocal continuum model. It is found that the present nonlocal elastic shell model with its appropriate values of nonlocal scale parameter has the capability to predict the bending behavior of CNTs, which is comparable with the results of MD simulations. Moreover, exact closed form solutions for the nonlocal scale parameter for zigzag and armchair CNTs are obtained. The results show that nonlocal scale parameter is independent of the length of CNTs, and dependent on the radius of CNTs.

Torsional instability of carbon nano-peapods based on the nonlocal elastic shell theory

In this paper a shell formulation is proposed for analyzing the torsional instability of carbon nano-peapods (CNPs), i.e., the hybrid structures composed of C60 fullerenes encapsulated inside carbon nanotubes (CNTs), based on the nonlocal elasticity theory. The nonlocal elasticity theory, as a well-known non-classical continuum theory, is capable to capture small scale effects which appear due to the discontinuities in nano-structures. Based on the derived formulation, the critical torsional moments for a pristine (10,10) CNT and C60@(10,10) CNP are investigated as case studies. The results for the (10,10) CNT are compared with those of the available molecular dynamics simulations in the literature, and accordingly the appropriate value of the small scale coefficient appearing in the constitutive equations of the nonlocal theory is estimated for CNTs. Then, the critical torsional moment for the C60@(10,10) CNP is predicted. It is observed that the presence of the encapsulated C60 fullerenes inside the (10,10) CNT causes an increase in the torsional instability resistance of the CNT more than 100%.

Analytical Treatment of the Free Vibration of Single-Walled Carbon Nanotubes Based on the Nonlocal Flugge Shell Theory

In the current work, the vibration characteristics of single-walled carbon nanotubes (SWCNTs) under different boundary conditions are investigated. A nonlocal elastic shell model is utilized, which accounts for the small scale effects and encompasses its classical continuum counterpart as a particular case. The variational form of the Flugge type equations is constructed to which the analytical Rayleigh–Ritz method is applied. Comprehensive results are attained for the resonant frequencies of vibrating SWCNTs. The significance of the small size effects on the resonant frequencies of SWCNTs is shown to be dependent on the geometric parameters of nanotubes. The effectiveness of the present analytical solution is assessed by the molecular dynamics simulations as a benchmark of good accuracy. It is found that, in contrast to the chirality, the boundary conditions have a significant effect on the appropriate values of nonlocal parameter.

Nonlocal anisotropic elastic shell model for vibrations of single-walled carbon nanotubes with arbitrary chirality

A nonlocal anisotropic elastic shell model is developed to study the effect of small scale on shell-like vibration of single-walled carbon nanotubes (SWCNTs) with arbitrary chirality. Anisotropic elastic shell model is reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion are derived and analytical solution for the vibration of anisotropic SWCNTs is presented by using the Flügge shell theory and complex method. The suggested model is justified by a good agreement between the results given by the present model and available data in literature. Furthermore, the model is used to elucidate the effect of small scale on the vibration of zigzag, armchair and chiral SWCNTs. Our results show that small scale is essential for vibration of SWCNTs when the axial wave-length is not extremely long. Moreover, the results show that local model substantially overestimates vibrational frequencies of almost all modes.

Thermal effect on axial buckling behavior of multi-walled carbon nanotubes based on nonlocal shell model

In the current study, a nonlocal elastic shell model is developed to investigate the axially compressed buckling response of multi-walled carbon nanotubes (MWCNTs) considering thermal environment effect. To this end, Eringen's nonlocal elasticity equations are incorporated into the classical Donnell shell theory to represent the MWCNTs as elastic multi-walled shell models coupled with the van der Waals interaction forces between the adjacent layers. Exact solution is presented to obtain the critical axial buckling loads of the nanotubes in thermal environment corresponding to different values of nonlocal elasticity parameter, axial and circumferential wavenumbers, temperature change and aspect ratio of the nanotube. It is found that the effect of small-scale is more prominent for MWCNTs having smaller diameters and a fewer number of walls. It is further found that the low and high temperature environments have contrary influences on the axial buckling of MWCNTs.

Thermal Buckling of Multiwalled Carbon Nanotubes Using a Semi-Analytical Finite Element Approach

In this article, a semi-analytical finite element approach based on the nonlocal elastic shell model is proposed to study the thermal buckling of multiwalled carbon nanotubes (MWCNTs). By taking nonlocality into consideration, the small scale effect is incorporated into the model presented herein. Based upon the Eringen nonlocal elasticity, the displacement field equations coupled by the van der Waals interaction force are derived. Comprehensive results for the thermal buckling of multiwalled carbon nanotubes are given. The influences of the small scale parameter and boundary conditions on the thermal instability of MWCNTs are examined. It is indicated that the possibility of the radial buckling mode of deformation due to radially elevated temperature is significantly higher than that of axial buckling mode of deformation due to uniformly reduced temperature when carbon nanotubes experience thermal loads.