Various gradient elasticity theories in predicting vibrational response of single-walled carbon nanotubes with arbitrary boundary conditions

Abstract

This article deals with the development and use of different gradient beam theories in order to predict the vibrational behavior of single-walled carbon nanotubes (SWCNTs). To address the problem of free vibration, the Euler–Bernoulli and Timoshenko beam theories in conjunction with the gradient elasticity theories including stress, strain and combined strain/inertia are implemented. The generalized differential quadrature method is employed to numerically solve the problem which can treat various boundary conditions. The results generated from the present gradient models are compared with those from molecular dynamics simulations as a benchmark of good accuracy and the proper values of small length scales used in the gradient models are proposed. This study shows prominent differences between various gradient models when the nanotube becomes very short (for aspect ratios of approximately lower than six). It is indicated that applying the strain gradient elasticity by incorporation of inertia gradients yields more reliable results especially for shorter length SWCNTs on account of two small-scale factors related to the inertia and strain gradients. Moreover, since with the reduction in the aspect ratio of nanotubes the effects of boundary conditions become dominant, a discussion is given to investigate the influence of end conditions on the vibrational characteristics of SWCNTs.

Keywords

Carbon nanotubes vibrations gradient beam model molecular dynamics simulation generalized differential quadrature method

1. Introduction

Of all of the nanostructured materials, carbon nanotubes (CNTs) (Iijima, 1991) have acquired paramount importance in the context of nanotechnology, as evidenced by the large number of research papers that have emerged in the technical literature. This is mainly due to their remarkable physical, electrical, and mechanical properties (Rafii-Tabar, 2004; Ho et al., 2004; Manchado et al., 2005; Sumfleth et al., 2010; Bellucci, 2008). Investigations on mechanical characteristics of CNTs have focused both on theoretical and experimental methods. Since performing controlled experiments at the nanometer scale is very complicated and prohibitively expensive, the theoretical modeling and analysis of nanostructured materials has become increasingly important.

The theoretical modeling of nanomaterials can be divided into two main categories: atomistic models and continuum models. In spite of the ascendency of the atomistic models compared with the continuum-based models, in engineering applications where the structures may be in a scale of several micrometers, the atomistic simulation will be computationally expensive which makes it difficult to analyze such structures. Consequently, continuum models have been one of the most applied models in this field due to their computational efficiency as well as satisfactory accuracy. Many researchers have successfully applied the continuum mechanics to describe the mechanical behavior of nanostructures (Pantano et al., 2004; Dequesnes et al., 2004; Gibson et al., 2007; Ouakad and Younis, 2010, 2011a,b; Ansari and Hemmatnezhad, 2011; Ansari et al., 2011a,b,c,d,e,f,g,h). However, the classical continuum models have a serious shortcoming; the size effects cannot be captured by these models. Hence, the extension of the classical continuum mechanics to accommodate the size dependence of nanostructures becomes a topic of major concern. So far, different modified continuum theories have been developed to solve boundary-value problems by the concentration on the size effects and elimination of singularities from crack tips and dislocation lines which can be categorized into two groups: integral models and gradient models. According to Eringen (1983, 2002), in the integral models, the nonlocal terms are incorporated into integral equations of elasticity. Eringen’s nonlocal elasticity has gained much popularity among the research workers owing to its efficiency as well as simplicity in analyzing the behavior of various nanostructures. There are many researches in which the behavior of nanostructured materials is studied via this format of continuum mechanics (Peddieson et al., 2003; Sudak, 2003; Wang, 2005; Li and Kardomateas, 2007; Hu et al., 2008; Shen and Zhang, 2010a,b; Ansari et al., 2010a,b, 2011a,b,c,d,e,f,g,h, Ansari and Rouhi, 2011) The gradient elasticity theories initiated by Aifantis (1984) can also be regarded as alternative approaches to the nonlocal elasticity.

More recently, attention has focused on the gradient elasticity theories to study the size effect on different problems of nano- and micro-structures (Askes and Aifantis, 2009; Wang, 2010; Aifantis, 2011; Akgoz and Civalek, 2011). However, to the best of the authors’ knowledge, a few papers published on the application of these theories to CNT problems. In addition, short CNTs are finding significant application in nanodevices (Yoon et al., 2011) and there are a number of research papers in which the behavior of CNTs in the aspect ratio range of $L / D < 6.0$ has been studied (Wang et al., 2006; Duan et al., 2007; Zhang et al., 2009; Shen and Zhang, 2010a,b; Sismek, 2011a,b; Wang et al., 2012). Since size effects become more prominent when the length of the nanotube is scaled down to deep submicrometer dimensions, it is necessary to give further investigations based on the gradient theories to study the mechanical behavior of short CNTs.

Motivated by this, in the present work the authors investigate the vibrational characteristics of SWCNTs on the basis of different gradient elasticity theories with the focus on lower values of aspect ratio. To this end, the Euler–Bernoulli beam (EBB) and Timoshenko beam (TB) theories are implemented into which the stress gradient (or nonlocal), strain gradient and combined strain/inertia gradient theories are incorporated. Since the boundary conditions considerably affect the mechanical behavior of nanostructured materials (Ansari et al., 2011a,b,c,d,e,f,g,h), developing powerful numerical solution methods capable of treating different boundary conditions is undoubtedly essential and can play an important role in the advancement of nanotechnology. Thus, to obtain the natural frequencies of SWCNTs with different end conditions, the generalized differential quadrature (GDQ) numerical method has been employed herein. The GDQ method has shown excellent accuracy, efficiency and great potential in solving complicated partial differential equations (Ansari et al., 2010a,b, 2011a,b,c,d,e,f,g,h). To validate the present analysis and in order to obtain the proper values of small length scales used in the gradient theories, molecular dynamics (MD) simulations are also conducted for an armchair SWCNT with different aspect ratios and end conditions. Comparative results between various gradient elasticity theories and also different beam assumptions are presented in predicting the free vibration response of SWCNTs.

2. Gradient elasticity theories

According to Eringen (1983, 2002), in the nonlocal elasticity theory, the stress tensor at a reference point x is considered to be a functional of the strain tensor at all the points of the body. The nonlocal constitutive equation in the stress gradient (nonlocal) theory can be given as

σ_{ij} - ℓ^{2} σ_{ij, mn} = C_{ijkl} ɛ_{kl}

(1)

where

σ_{ij}

and

ɛ_{ij}

are the stress and strain tensors, respectively, and

C_{ijkl}

denotes the fourth-order elasticity tensor; ℓ is an internal length scale obtainable by experiment or microscopic models. The strain gradient combined with inertia gradients counterpart of Equation (1) can be expressed as Askes and Aifantis (2009)

ρ ({\overset{··}{u}}_{i} - ℓ_{m}^{2} {\overset{··}{u}}_{i, mn}) = C_{ijkl} (u_{k, jl} + ℓ^{2} u_{k, jlmn})

(2)

in which ρ is the mass density and u_i is the displacement. It should be noted that when

ℓ_{m}^{=} 0

the theory reduces to the strain gradient theory. From Equations (1) and (2), one can find that the stress gradient and the combined strain/inertia gradient theories differ by their number of small length scales. According to Equation (1), the stress gradient format has only one small length scale (ℓ), whereas from Equation (2), the combined strain/inertia theory has two small length scales, ℓ and

ℓ_{m}

which are related to the strain and inertia gradients, respectively. The strain gradient length scale ℓ is related to the representative volume element (RVE) size in elastostatics and the inertia gradient length scale

ℓ_{m}

is related to the RVE size for elastodynamics (Askes and Aifantis, 2009)

3. Beam theories on the basis of gradient elasticity

Consider an SWCNT of length L, Young’s modulus E, Poisson’s ratio ν, shear modulus $G,$ mass density ρ, cross-sectional area A and cross-sectional moment of inertia I as depicted in Figure 1. According to different gradient elasticity theories, the stress, strain and inertia gradient terms are incorporated into the EBB and TB governing equations for the free vibration of SWCNTs as follows.

Figure 1.

Schematic of a single-walled carbon nanotube.

3.1. Stress gradient beam models

In the case of stress gradient (nonlocal) elasticity the axial and shear equivalents of Equation (1) are written as

σ - ℓ^{2} σ_{, xx} = E ɛ

(3a)

τ - ℓ^{2} τ_{, xx} = G γ

(3b)

where

τ = σ_{xy}

and ɛ, γ are the axial normal and shear strains, respectively. The classical EBB governing equation is given by

- EI \frac{\partial^{4} w}{\partial x^{4}} = ρ A \frac{\partial^{2} w}{\partial t^{2}} - ρ I \frac{\partial^{4} w}{\partial x^{2} \partial t^{2}}

(4)

in which w is the transverse displacement. Using Equation (3a) along with the EBB assumptions, the stress gradient format of Equation (4) can be given as

- EI \frac{\partial^{4} w}{\partial x^{4}} = ρ A \frac{\partial^{2} w}{\partial t^{2}} - (ℓ^{2} ρ A + ρ I) \frac{\partial^{4} w}{\partial x^{2} \partial t^{2}} + ℓ^{2} ρ I \frac{\partial^{6} w}{\partial x^{4} \partial t^{2}}

(5)

Also, the conventional TB governing equations read as

κ_{s} GA (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial ψ}{\partial x}) = ρ A \frac{\partial^{2} w}{\partial t^{2}}

(6a)

- κ_{s} GA (\frac{\partial w}{\partial x} + ψ) + EI \frac{\partial^{2} ψ}{\partial x^{2}} = ρ I \frac{\partial^{2} ψ}{\partial t^{2}}

(6b)

where ψ is the angular displacement of the beam and

κ_{s}

stands for the shear correction factor. With the use of Equations (3) and also the TB hypotheses, the stress gradient versions of Equations (6) are expressed in the forms

κ_{s} GA (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial ψ}{\partial x}) = ρ A \frac{\partial^{2} w}{\partial t^{2}} - ℓ^{2} ρ A \frac{\partial^{4} w}{\partial x^{2} \partial t^{2}}

(7a)

- κ_{s} GA (\frac{\partial w}{\partial x} + ψ) + EI \frac{\partial^{2} ψ}{\partial x^{2}} = ρ I \frac{\partial^{2} ψ}{\partial t^{2}} - ℓ^{2} ρ I \frac{\partial^{4} ψ}{\partial x^{2} \partial t^{2}}

(7b)

3.2. Strain gradient beam models

Hooke’s law for the stress and strain relations in the context of strain gradient elasticity is formulated as follows

σ = E (ɛ + ℓ^{2} ɛ_{, xx})

(8a)

τ = G (γ + ℓ^{2} γ_{, xx})

(8b)

Based on the Equation (8a), the governing equation of the strain gradient EBB model is given by

- EI (\frac{\partial^{4} w}{\partial x^{4}} + ℓ^{2} \frac{\partial^{6} w}{\partial x^{6}}) = ρ A \frac{\partial^{2} w}{\partial t^{2}} - ρ I \frac{\partial^{4} w}{\partial x^{2} \partial t^{2}}

(9)

Furthermore, for the case of TB model we have

κ_{s} GA (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial ψ}{\partial x}) + κ_{s} GA ℓ^{2} (\frac{\partial^{4} w}{\partial x^{4}} + \frac{\partial^{3} ψ}{\partial x^{3}}) = ρ A \frac{\partial^{2} w}{\partial t^{2}}

(10a)

\begin{matrix} - κ_{s} GA (\frac{\partial w}{\partial x} + ψ) - κ_{s} GA ℓ^{2} (\frac{\partial^{3} w}{\partial x^{3}} + \frac{\partial^{2} ψ}{\partial x^{2}}) \\ + EI (\frac{\partial^{2} ψ}{\partial x^{2}} - ℓ^{2} \frac{\partial^{4} ψ}{\partial x^{4}}) = ρ I \frac{\partial^{2} ψ}{\partial t^{2}} \end{matrix}

(10b)

3.3. Combined strain/inertia gradient beam models

The strain gradient elasticity by incorporating inertia gradients yields the following stress–strain relations

σ = E (ɛ + ℓ^{2} ɛ_{, xx}) + ρ ℓ_{m}^{2} \overset{··}{ɛ}

(11a)

τ = E (γ + ℓ^{2} γ_{, xx}) + ρ ℓ_{m}^{2} \overset{··}{γ}

(11b)

Thus, the governing equation of the EBB model with strain and inertia gradients is of the form

- EI (\frac{\partial^{4} w}{\partial x^{4}} + ℓ^{2} \frac{\partial^{6} w}{\partial x^{6}}) = ρ A \frac{\partial^{2} w}{\partial t^{2}} - ρ I (\frac{\partial^{4} w}{\partial x^{2} \partial t^{2}} - ℓ_{m}^{2} \frac{\partial^{6} w}{\partial x^{4} \partial t^{2}})

(12)

Moreover, for the TB model with the combined strain/inertia gradients one can obtain

\begin{matrix} κ_{s} GA (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial ψ}{\partial x}) + κ_{s} GA ℓ^{2} (\frac{\partial^{4} w}{\partial x^{4}} + \frac{\partial^{3} ψ}{\partial x^{3}}) \\ = ρ A \frac{\partial^{2} w}{\partial t^{2}} - κ_{s} ρ A ℓ_{m}^{2} (\frac{\partial^{4} w}{\partial x^{2} \partial t^{2}} + \frac{\partial^{3} ψ}{\partial x \partial t^{2}}) \end{matrix}

(13a)

\begin{matrix} - κ_{s} GA (\frac{\partial w}{\partial x} + ψ) - κ_{s} GA ℓ^{2} (\frac{\partial^{3} w}{\partial x^{3}} + \frac{\partial^{2} ψ}{\partial x^{2}}) \\ + EI (\frac{\partial^{2} ψ}{\partial x^{2}} + ℓ^{2} \frac{\partial^{4} ψ}{\partial x^{4}}) = ρ I \frac{\partial^{2} ψ}{\partial t^{2}} \\ + κ_{s} ρ A ℓ_{m}^{2} (\frac{\partial^{3} w}{\partial x \partial t^{2}} + \frac{\partial^{2} ψ}{\partial t^{2}}) - ρ I ℓ_{m}^{2} \frac{\partial^{4} ψ}{\partial x^{2} \partial t^{2}} \end{matrix}

(13b)

4. Generalized differential quadrature method

The fundamental idea of the differential quadrature method lies in the approximation of the partial derivative of a function with respect to a coordinate at a discrete point as a weighted linear sum of the function values at all discrete points along that coordinate direction. Let $\frac{\partial^{r} f}{\partial x^{r}}$ be the rth derivative of a function $f (x)$ which can be expressed as a linear sum of the function values

\frac{\partial^{r} f (x)}{\partial x^{r}} |_{x = x_{P}} = \sum_{Q = 1}^{n} A_{PQ}^{(r)} f (x_{P})

(14)

where n is the number of total discrete grid points used in the approximation process and

A_{PQ}^{(r)}

are weighting coefficients. The weighting coefficients of the first derivative are determined by

A_{PQ}^{(1)} = \frac{M (x_{P})}{(x_{P} - x_{Q}) M (x_{Q})} (P, Q = 1, 2, \dots, n; P \neq Q)

(15)

where

M (x_{P}) = Π_{Q = 1; Q \neq P}^{n} (x_{P} - x_{Q})

The weighting coefficients of higher-order derivatives can be obtained through the following recurrence relation

A_{PQ}^{(r)} = {\begin{matrix} r [A_{PQ}^{(r - 1)} A_{PQ}^{(1)} - \frac{A_{PQ}^{(r - 1)}}{x_{p} - x_{q}}], P \neq Q \\ - \sum_{Q = 1}^{n} A_{PQ}^{(r)}, P = Q (P, Q = 1, 2, \dots, n; 2 \leq r \leq n - 1) \end{matrix}} .

(16)

4.1. GDQ analog of the governing equations

The field variables ( $w$ and $ψ$ ) are assumed to have a periodic solution of the form

w (x, t) = W (x) e^{j ω t}

(17)

ψ (x, t) = Ψ (x) e^{j ω t}

(18)

where W and Ψ are nodal values of the field variables. By applying the GDQ method, the discrete counterparts of the governing differential equations corresponding to different gradient beam models at the rth given point can be expressed as follows.

For the EBB model:

classical theory

- EI \sum_{s = 1}^{n} A_{rs}^{(4)} W_{s} = - ρ A ω^{2} W_{r} + ρ I ω^{2} \sum_{s = 1}^{n} A_{rs}^{(2)} W_{s}

(19)

stress gradient theory

\begin{matrix} - EI \sum_{s = 1}^{n} A_{rs}^{(4)} W_{s} = - ρ A ω^{2} W_{r} \\ + (ℓ^{2} ρ A + ρ I) ω^{2} \sum_{s = 1}^{n} A_{rs}^{(2)} W_{s} \\ - ℓ^{2} ρ I ω^{2} \sum_{s = 1}^{n} A_{rs}^{(4)} W_{s} \end{matrix}

(20)

strain gradient theory

\begin{matrix} - EI (\sum_{s = 1}^{n} A_{rs}^{(4)} W_{s} + ℓ^{2} \sum_{s = 1}^{n} A_{rs}^{(6)} W_{s}) \\ = - ρ A ω^{2} W_{r} + ρ I ω^{2} \sum_{s = 1}^{n} A_{rs}^{(2)} W_{s} \end{matrix}

(21)

combined strain/inertia gradient theory

\begin{matrix} - EI (\sum_{s = 1}^{n} A_{rs}^{(4)} W_{s} + ℓ^{2} \sum_{s = 1}^{n} A_{rs}^{(6)} W_{s}) \\ = - ρ A ω^{2} W_{r} + ρ I ω^{2} (\sum_{s = 1}^{n} A_{rs}^{(2)} W_{s} - ℓ_{m}^{2} \sum_{s = 1}^{n} A_{rs}^{(4)} W_{s}) \end{matrix}

(22)

For the TB model:

classical theory

κ_{s} GA \sum_{s = 1}^{n} A_{rs}^{(2)} W_{s} + κ_{s} GA \sum_{s = 1}^{n} A_{rs}^{(1)} φ_{s} = - ρ A ω^{2} W_{r}

(23a)

- κ_{s} GA \sum_{s = 1}^{n} A_{rs}^{(1)} W_{s} + EI \sum_{s = 1}^{n} A_{rs}^{(2)} Ψ_{s} - κ_{s} GA Ψ_{r} = - ρ I ω^{2} Ψ_{r}

(23b)

stress gradient theory

\begin{matrix} κ_{s} GA (\sum_{s = 1}^{n} A_{rs}^{(2)} W_{s} + \sum_{s = 1}^{n} A_{rs}^{(1)} Ψ_{s}) \\ = - ρ A ω^{2} W_{r} + ℓ^{2} ρ A ω^{2} \sum_{s = 1}^{n} A_{rs}^{(2)} W_{s} \end{matrix}

(24a)

\begin{matrix} - κ_{s} GA (\sum_{s = 1}^{n} A_{rs}^{(1)} W_{s} + Ψ_{r}) + EI \sum_{s = 1}^{n} A_{rs}^{(2)} Ψ_{s} \\ = - ρ I ω^{2} Ψ_{r} + ℓ^{2} ρ I ω^{2} \sum_{s = 1}^{n} A_{rs}^{(2)} Ψ_{s} \end{matrix}

(24b)

strain gradient theory

\begin{matrix} κ_{s} GA (\sum_{s = 1}^{n} A_{rs}^{(2)} W_{s} + \sum_{s = 1}^{n} A_{rs}^{(1)} Ψ_{s}) \\ + κ_{s} GA ℓ^{2} (\sum_{s = 1}^{n} A_{rs}^{(4)} W_{s} + \sum_{s = 1}^{n} A_{rs}^{(3)} Ψ_{s}) \\ = - ρ A ω^{2} W_{r} \end{matrix}

(25a)

\begin{matrix} - κ_{s} GA (\sum_{s = 1}^{n} A_{rs}^{(1)} W_{s} + Ψ_{r}) \\ - κ_{s} GA ℓ^{2} (\sum_{s = 1}^{n} A_{rs}^{(3)} W_{s} + \sum_{s = 1}^{n} A_{rs}^{(2)} Ψ_{s}) \\ + EI \sum_{s = 1}^{n} A_{rs}^{(2)} Ψ_{s} + EI ℓ^{2} \sum_{s = 1}^{n} A_{rs}^{(4)} Ψ_{s} \\ = - ρ I ω^{2} Ψ_{r} \end{matrix}

(25b)

combined strain/inertia gradient theory

\begin{matrix} κ_{s} GA (\sum_{s = 1}^{n} A_{rs}^{(2)} W_{s} + \sum_{s = 1}^{n} A_{rs}^{(1)} Ψ_{s}) \\ + κ_{s} GA ℓ^{2} (\sum_{s = 1}^{n} A_{rs}^{(4)} W_{s} + \sum_{s = 1}^{n} A_{rs}^{(3)} Ψ_{s}) \\ = - ρ A ω^{2} W_{r} \\ + κ_{s} ρ A ℓ_{m}^{2} ω^{2} (\sum_{s = 1}^{n} A_{rs}^{(2)} W_{s} + \sum_{s = 1}^{n} A_{rs}^{(1)} Ψ_{s}) \end{matrix}

(26a)

\begin{matrix} - κ_{s} GA (\sum_{s = 1}^{n} A_{rs}^{(1)} W_{s} + Ψ_{r}) \\ - κ_{s} GA ℓ^{2} (\sum_{s = 1}^{n} A_{rs}^{(3)} W_{s} + \sum_{s = 1}^{n} A_{rs}^{(2)} Ψ_{s}) \\ + EI \sum_{s = 1}^{n} A_{rs}^{(2)} Ψ_{s} + EI ℓ^{2} \sum_{s = 1}^{n} A_{rs}^{(4)} Ψ_{s} \\ = - ρ I ω^{2} Ψ_{r} - κ_{s} ρ A ℓ_{m}^{2} ω^{2} (\sum_{s = 1}^{n} A_{rs}^{(1)} W_{s} + Ψ_{r}) \\ + ρ I ℓ_{m}^{2} ω^{2} \sum_{s = 1}^{n} A_{rs}^{(2)} Ψ_{s} \end{matrix}

(26b)

4.2. GDQ analog of the boundary conditions

Using the GDQ approximation, the discretized counterparts of different boundary conditions at the rth given point for each beam theory are as follows.

For EBB theory:

simply supported–simply supported

W_{r} = 0, \sum_{s = 1}^{n} A_{rs}^{(2)} W_{s} = 0 at edges x = 0, L

(27)

clamped–clamped

W_{r} = 0, \sum_{s = 1}^{n} A_{rs}^{(1)} W_{s} = 0 at edges x = 0, L

(28)

clamped–simply supported

W_{r} = 0, \sum_{s = 1}^{n} A_{rs}^{(1)} W_{s} = 0 at edges x = 0

(29a)

W_{r} = 0, \sum_{s = 1}^{n} A_{rs}^{(2)} W_{s} = 0 at edges x = L

(29b)

clamped–free

W_{r} = 0, \sum_{s = 1}^{n} A_{rs}^{(1)} W_{s} = 0 at edges x = 0

(30a)

\sum_{s = 1}^{n} A_{rs}^{(2)} W_{s} = 0, \sum_{s = 1}^{n} A_{rs}^{(3)} W_{s} = 0 at edge x = L

(30b)

For the TB theory:

simply supported–simply supported

W_{r} = 0, \sum_{s = 1}^{n} A_{rs}^{(1)} Ψ_{s} = 0 at edges x = 0, L

(31)

clamped–clamped

W_{r} = 0, Ψ_{r} = 0 at edges x = 0, L

(32)

clamped–simply supported

W_{r} = 0, Ψ_{r} = 0 at edges x = 0

(33a)

W_{r} = 0, \sum_{s = 1}^{n} A_{rs}^{(1)} Ψ_{s} = 0 at edge x = L

(33b)

clamped–free

W_{r} = 0, Ψ_{r} = 0 at edges x = 0

(34a)

\sum_{s = 1}^{n} A_{rs}^{(1)} Ψ_{s} = 0, Ψ_{r} + \sum_{s = 1}^{n} A_{rs}^{(1)} W_{s} = 0 at edge x = L

(34b)

Rearranging the quadrature analogs of governing equations and boundary conditions within the framework of a generalized eigenvalue problem yields the natural frequencies of SWCNT.

5. Numerical results and discussion

First, the convergence characteristics and accuracy of the GDQ-based solutions presented herein are carefully assessed. In the numerical evaluations, the effective thickness of SWCNTs is assumed to be equal to the spacing of graphite

(h = 0.34 nm)

. Also, Poisson’s ratio ν, mass density ρ and Young’s modulus E are assumed to be

0.3

2300 Kkg / m^{3}

and

1.1 TPa

, respectively. The fundamental frequencies of an SWCNT with three different boundary conditions based on the classical EBB and TB theories for up to

n = 16

grid sampling points are given in Table 1. This table shows quite clearly the converging trend of the present numerical solution with increasing number of grid points.

Table 1.

Convergence study of fundamental frequencies (THz) of an SWCNT with different boundary conditions corresponding to different beam models ( $L / D = 10, R = 0.5 nm$ )

Number of grid points	Boundary conditions
	Simply supported–simply supported		Clamped–clamped		Clamped–simply supported
	EBB	TB	EBB	TB	EBB	TB
4	0.3537	0.3441	0.8281	0.7680	0.5508	0.5388
6	0.3723	0.3509	0.8435	0.7808	0.5825	0.5382
8	0.3713	0.3688	0.8408	0.8080	0.5796	0.5680
10	0.3713	0.3680	0.8409	0.8056	0.5797	0.5658
12	0.3713	0.3680	0.8409	0.8056	0.5797	0.5659
14	0.3713	0.3680	0.8409	0.8056	0.5797	0.5659
16	0.3713	0.3680	0.8409	0.8056	0.5797	0.5659

EBB, Euler–Bernoulli beam; TB, Timoshenko beam.

Adjustment of the small length scales used in the gradient models plays an important role in order to precisely predict the response of nanotubes. Some recent research has proposed the values of small-scale factors by comparing the results of continuum models with those of MD simulations (Ansari and Rouhi, 2011; Duan et al., 2007). Following the aforementioned treatment, in the present study, the proper values of small length scales corresponding to an (8,8) armchair SWCNT with four commonly used boundary conditions are determined through matching the fundamental frequencies calculated by the gradient continuum beam models with those obtained from MD simulations. To this end, the adaptive intermolecular reactive empirical bond order (AIREBO) potential is adopted in the NanoHive simulator (Nanorex Inc., 2005) to perform the MD simulations presented in the current analysis. Furthermore, the nanotubes are allowed to vibrate freely for 10,000 timesteps of $0.5 fs$ .

Table 2 shows the calibrated ratios of

R / ℓ

(the ratio of nanotube radius over small length scale ℓ) and

ℓ_{m} / ℓ

for different beam theories and boundary conditions. The least-squares method is used to determine the best value for these ratios so that the sum of squares of errors between the results from MD simulations and the corresponding values from the gradient beam models are minimized for a relatively large range of nanotube aspect ratio. The results listed in Table 2 indicate that the significance of the small size effects on the vibrational behavior of SWCNTs is dependent on the boundary conditions.

Table 2.

Calibrated ratios of $R / ℓ$ and $ℓ_{m} / ℓ$ for different beam theories and boundary conditions

Boundary conditions	EBB		TB
Boundary conditions	$R / ℓ$	$ℓ_{m} / ℓ$	$R / ℓ$	$ℓ_{m} / ℓ$
Simply supported–simply supported	2.6	40	2.6	22.6
Clamped–clamped	2.3	20.15	2.3	33.2
Clamped–simply supported	2.1	23.8	2.1	47.75
Clamped–free	1.2	54	1.2	85

EBB, Euler–Bernoulli beam; TB, Timoshenko beam.

Tables 3 and 4 present the values of fundamental frequency of (8,8) SWCNTs with four commonly used boundary conditions obtained from MD simulations and also the EBB and TB models based on the classical, stress, strain and combined strain/inertia gradient elasticity theories. It is observed that for both EBB and TB models the results of gradient elasticity theories and MD simulations are in good agreement. These tables indicate that for all types of end conditions the frequency of SWCNTs decreases as the aspect ratio increases. One can find that in the range of aspect ratio considered here, the discrepancy between the results of different gradient theories is not significant. However, for shorter length SWCNTs application of the combined strain/inertia gradient theory leads to lower values of frequency which are in closer agreement with the MD results. This is mainly due to the presence of extra material length scale parameter

ℓ_{m}

in the constitutive equation of this theory.

Table 3.

Fundamental frequencies (THz) predicted by different Euler–Bernoulli beam models and by the MD simulations for (8,8) SWCNTs corresponding to various boundary conditions

$L / D$	MD simulation	Classical theory	Stress gradient theory	Strain gradient theory	Combined strain/ inertia gradient theory
Simply supported–simply supported
8.3	0.5299	0.5406	0.5400	0.5400	0.5298
10.1	0.3618	0.3662	0.3659	0.3659	0.3627
13.7	0.1931	0.1985	0.1984	0.1984	0.1979
17.3	0.1103	0.1244	0.1244	0.1244	0.1243
20.9	0.0724	0.0854	0.0854	0.0854	0.0853
24.5	0.0519	0.0621	0.0621	0.0621	0.0621
28.1	0.0425	0.0473	0.0473	0.0473	0.0473
31.6	0.0358	0.0372	0.0372	0.0372	0.0372
35.3	0.0287	0.0298	0.0298	0.0298	0.0298
39.1	0.0259	0.0244	0.0244	0.0244	0.0244
Clamped–clamped
8.3	1.1838	1.2235	1.2215	1.2215	1.1839
10.1	0.8077	0.8293	0.8284	0.8283	0.8163
13.7	0.4374	0.4498	0.4495	0.4495	0.4475
17.3	0.2602	0.2819	0.2818	0.2818	0.2813
20.9	0.1769	0.1935	0.1934	0.1934	0.1933
24.5	0.1313	0.1407	0.1407	0.1407	0.1406
28.1	0.1028	0.1072	0.1072	0.1072	0.1072
31.6	0.0816	0.0844	0.0844	0.0844	0.0844
35.3	0.0661	0.0677	0.0676	0.0676	0.0676
39.1	0.0576	0.0554	0.0554	0.0554	0.0554
Clamped–simply supported
8.3	0.8211	0.8436	0.8421	0.8421	0.8212
10.1	0.5602	0.5717	0.5710	0.5710	0.5643
13.7	0.3034	0.3100	0.3098	0.3098	0.3087
17.3	0.1767	0.1943	0.1942	0.1942	0.1940
20.9	0.1182	0.1333	0.1333	0.1333	0.1332
24.5	0.0875	0.0970	0.0969	0.0969	0.0969
28.1	0.0704	0.0739	0.0739	0.0739	0.0739
31.6	0.0559	0.0582	0.0582	0.0582	0.0581
35.3	0.0455	0.0466	0.0466	0.0466	0.0466
39.1	0.0398	0.0382	0.0382	0.0382	0.0382
Clamped–free
8.3	0.1899	0.1939	0.1939	0.1939	0.1899
10.1	0.1298	0.1310	0.1311	0.1311	0.1298
13.7	0.0683	0.0709	0.0709	0.0709	0.0707
17.3	0.0376	0.0444	0.0444	0.0444	0.0443
20.9	0.0244	0.0304	0.0304	0.0304	0.0304
24.5	0.0181	0.0221	0.0221	0.0221	0.0221
28.1	0.0138	0.0169	0.0169	0.0169	0.0169
31.6	0.0117	0.0133	0.0133	0.0133	0.0133
35.3	0.0102	0.0106	0.0106	0.0106	0.0106
39.1	0.0092	0.0087	0.0087	0.0087	0.0087

Table 4.

Fundamental frequencies (THz) predicted by different Timoshenko beam models and by the MD simulations for (8,8) SWCNTs corresponding to various boundary conditions

$L / D$	MD simulation	Classical theory	Stress gradient theory	Strain gradient theory	Combined strain/ inertia gradient theory
Simply supported–simply supported
8.3	0.5299	0.5337	0.5332	0.5332	0.5299
10.1	0.3618	0.3630	0.3628	0.3628	0.3617
13.7	0.1931	0.1976	0.1975	0.1975	0.1973
17.3	0.1103	0.1240	0.1240	0.1240	0.1240
20.9	0.0724	0.0852	0.0852	0.0852	0.0852
24.5	0.0519	0.0620	0.0620	0.0620	0.0620
28.1	0.0425	0.0472	0.0472	0.0472	0.0472
31.6	0.0358	0.0372	0.0372	0.0372	0.0372
35.3	0.0287	0.0298	0.0298	0.0298	0.0298
39.1	0.0259	0.0244	0.0244	0.0244	0.0244
Clamped–clamped
8.3	1.1838	1.3221	1.3189	1.3188	1.1837
10.1	0.8077	0.9101	0.9086	0.9086	0.8633
13.7	0.4374	0.5013	0.5009	0.5009	0.4933
17.3	0.2602	0.3165	0.3163	0.3163	0.3144
20.9	0.1769	0.2180	0.2179	0.2179	0.2173
24.5	0.1313	0.1589	0.1589	0.1589	0.1586
28.1	0.1028	0.1213	0.1212	0.1212	0.1211
31.6	0.0816	0.0955	0.0955	0.0955	0.0955
35.3	0.0661	0.0767	0.0766	0.0766	0.0766
39.1	0.0576	0.0628	0.0628	0.0628	0.0628
Clamped–simply supported
8.3	0.8211	0.9322	0.9297	0.9296	0.8212
10.1	0.5602	0.6373	0.6361	0.6361	0.5996
13.7	0.3034	0.3486	0.3482	0.3482	0.3421
17.3	0.1767	0.2193	0.2192	0.2192	0.2177
20.9	0.1182	0.1508	0.1508	0.1508	0.1503
24.5	0.0875	0.1098	0.1098	0.1098	0.1096
28.1	0.0704	0.0837	0.0837	0.0837	0.0836
31.6	0.0559	0.0660	0.0659	0.0659	0.0659
35.3	0.0455	0.0529	0.0529	0.0529	0.0529
39.1	0.0398	0.0433	0.0433	0.0433	0.0433
Clamped–free
8.3	0.1899	0.2185	0.2187	0.2187	0.2052
10.1	0.1298	0.1481	0.1482	0.1482	0.1438
13.7	0.0683	0.0803	0.0803	0.0803	0.0796
17.3	0.0376	0.0504	0.0504	0.0504	0.0502
20.9	0.0244	0.0346	0.0346	0.0346	0.0345
24.5	0.0181	0.0251	0.0251	0.0251	0.0251
28.1	0.0138	0.0191	0.0191	0.0191	0.0191
31.6	0.0117	0.0151	0.0151	0.0151	0.0151
35.3	0.0102	0.0121	0.0121	0.0121	0.0121
39.1	0.0092	0.0099	0.0099	0.0099	0.0099

In order to investigate the difference between various gradient models, the fundamental frequencies of fully clamped SWCNTs are shown in Figures 2 and 3 for lower values of aspect ratio ranging from 3 to 8 corresponding to the EBB and TB theories, respectively. From these figures, it is observed that the discrepancy between different gradient beam models becomes more pronounced for SWCNTs of very low aspect ratios. With increasing the ratio of length to diameter, the difference between various gradient models decreases so that the frequency envelopes tend to converge. According to Figures 2 and 3, for aspect ratios higher than approximately 6.5 and 7.25 corresponding to the EBB and TB theories, respectively, the classical, stress, strain and combined strain/inertia gradient models give almost identical frequencies. It is further observed that the results of combined strain/inertia gradient model are lower than those of other gradient models. This observation means that the strain/inertia gradient model can accommodate the results of other models.

Figure 2.

Variation of fundamental frequency with aspect ratio for (8,8) armchair SWCNTs based on the Euler–Bernoulli beam theory ( $ℓ = 0.6 nm, ℓ_{m} / ℓ = 3$ ).

Figure 3.

Variation of fundamental frequency with aspect ratio for (8,8) armchair SWCNTs based on the Timoshenko beam theory ( $ℓ = 0.6 nm, ℓ_{m} / ℓ = 3$ ).

The effect of boundary conditions on the vibration characteristics of SWCNT can be studied in Figure 4. In this figure the fundamental frequencies of combined strain/inertia gradient TB model are plotted against the aspect ratio corresponding to four commonly used boundary conditions. As can be inferred from this figure, the value of frequency is greatly affected by the type of boundary conditions especially for shorter length nanotubes. As the length increases, frequencies tend to decrease and the effect tube end conditions diminish so that the frequency envelopes tend to converge. According to the results shown in this figure, the values of frequency related to the SWCNT with clamped–free boundary conditions are smaller than those of fully simply supported SWCNT and the latter ones are also smaller than those of SWCNT with clamped–simply supported boundary conditions. Also, the highest values of frequency are associated with fully clamped SWCNT.

Figure 4.

Variation of fundamental frequency with aspect ratio for (8,8) SWCNTs with different boundary conditions based on the combined strain/inertia gradient Timoshenko beam model ( $ℓ = 0.6 nm, ℓ_{m} / ℓ = 3$ ).

Figure 5 provides a comparison between the EBB and TB theories in predicting the vibrational behavior of SWCNTs. The fundamental frequencies obtained from the combined strain/inertia gradient EBB and TB models for simply supported–simply supported end conditions are plotted in this figure for length of nanotube ranging from 4 to 16 nm. As would be expected, the difference between the results of these beam theories is relatively more prominent at low values of length. Figure 5 clarifies that for the length up to approximately 10 nm, the EBB theory tends to overestimate the frequencies of SWCNT. For instance, at $L = 4 nm$ the percentage of relative error between the results predicted by the EBB and TB theories is 13.3%. The reason is that unlike the EBB model, the TB model considers the effects of transverse shear deformation and rotary inertia. This makes the TB gradient model generally more preferable than the EBB counterpart.

Figure 5.

Comparison between the fundamental frequencies of fully simply supported (8,8) SWCNT obtained from the combined strain/inertia gradient Euler–Bernoulli beam (EBB) and Timoshenko beam (TB) models ( $ℓ = 0.6 nm, ℓ_{m} / ℓ = 3$ ).

6. Conclusion

In this work, based on the gradient elasticity, the free vibration of SWCNTs was studied. To this end, the EBB and TB models within the framework of the stress, strain and combined strain/inertia gradient theories were developed. The generalized differential quadrature method was employed to obtain the natural frequencies of SWCNTs with different boundary conditions. To validate the present methodology, MD simulations were also conducted for an armchair SWCNT with different aspect ratios and boundary conditions. Moreover, the appropriate values of small length scales were extracted through comparison between the results of continuum models and those from MD simulations. It was concluded that the calibrated values of small length scales are dependent on the boundary conditions. Numerical results showed that the discrepancy between different gradient theories is more pronounced for shorter nanotubes and also applying the combined strain/inertia gradient theory for shorter SWCNTs leads to very accurate results which are in excellent agreement with the MD ones due to two small length scales relevant to the inertia and strain gradients. The effect of boundary conditions on the free vibration of SWCNTs was studied in this work and it was observed that the difference between the results of SWCNTs with various end conditions is more prominent for lower values of aspect ratio. Furthermore, comparison between the results of EBB and TB theories revealed that the EBB model tend to overestimate the frequencies of short SWCNTs.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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