Geometrically Nonlinear Static Analysis of an Embedded Multiwalled Carbon Nanotube and the van der Waals Interaction

Wu, Chih-Ping; Hong, Zong-Li; Wang, Yung-Ming

doi:10.1061/(ASCE)NM.2153-5477.0000134

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Technical Papers

Jun 22, 2017

Geometrically Nonlinear Static Analysis of an Embedded Multiwalled Carbon Nanotube and the van der Waals Interaction

Authors: Chih-Ping Wu [email protected], Zong-Li Hong [email protected], and Yung-Ming Wang [email protected]Author Affiliations

Publication: Journal of Nanomechanics and Micromechanics

Volume 7, Issue 4

https://doi.org/10.1061/(ASCE)NM.2153-5477.0000134

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Abstract

On the basis of Reissner’s mixed variational theorem (RMVT), rather than the principle of virtual displacement (PVD), the authors presented a nonlocal Timoshenko beam theory (TBT) for the geometrically nonlinear static analysis of multiwalled carbon nanotubes (MWCNT) embedded in an elastic medium. The embedded MWCNT was subjected to mechanical loads on its outer-most surface, with combinations of free, simply supported, and clamped edge conditions. The van der Waals interaction between any pair of walls constituting the MWCNT was considered, and the interaction between the MWCNT and its surrounding medium was simulated using the Pasternak-type foundation model. In the formulation, the governing equations of a typical wall and the associated boundary conditions were derived, in which von Kármán geometrical nonlinearity was considered. Eringen’s nonlocal elasticity theory was used to account for the small-length scale effect. The deformations induced in the embedded MWCNT were obtained using the differential quadrature method and a direct iteration approach. In the numerical examples, solutions of the RMVT-based nonlocal TBT converged rapidly, and the convergent solutions of its linear counterpart closely agreed with the analytical and numerical solutions of the PVD-based nonlocal beam theories available in the literature.

Introduction

From the time carbon nanotubes (CNT) were first discovered by Iijima (1991), the development of nanoscience and nanotechnologies has proceeded at a fast rate, as has the application of CNT to various advanced industries because of their superior mechanical, electrical, thermal, and chemical properties. To assess the performance of CNT and extend their lifetimes, a number of studies carried out the structural analyses of nonembedded and embedded single- and multiwalled (SW and MW) CNT. Basically, there are four kinds of simulation methods used to conduct such analyses: (1) atomistic modeling, (2) hybrid atomistic-continuum mechanics, (3) experimental methods, and (4) continuum mechanics, with the first three approaches being especially time-consuming and expensive. However, classical continuum mechanics-based theories are not able to capture the small-length scale effect and, therefore, some modified theories have been developed, such as the strain gradient elasticity, coupled-stress elasticity, surface elasticity, and Eringen’s nonlocal elasticity theories (ENET) (Eringen 1972, 1983, 2002), with ENET being the most popular and widely used in various nonlocal continuum mechanics theories. Some comprehensive literature surveys with regard to the vibration characteristics of CNT and the various simulation methods for the analysis of these also have been carried out (Arash and Wang 2012; Brischetto 2014; Eltaher et al. 2016; Hu et al. 2012; Gibson et al. 2007; Wang et al. 2016).

Some theoretical and experimental investigations (Fu et al. 2006; Walgraef 2007) have examined the behaviors of CNT under large external loads, in which their deformations are nonlinear, rather than linear. However, there remain relatively few articles carrying out the large deflection and large amplitude vibration analyses of CNT, as compared with those examining their linear counterparts. Among the various ENET-based nonlocal beam theories used for the analysis of CNT, the nonlocal Timoshenko beam theory (TBT) commonly is used in the open literature and has been shown to produce satisfactory results as seen in applications, such as the nonlocal classical beam theory (CBT), TBT, Reddy’s beam theory (RBT), Levinson’s beam theory (LBT), generalized shear deformation beam theory (GSDBT), and sinusoidal-shear deformation beam theory (SSDBT) (Aydogdu 2008, 2009; Murmu and Pradhan 2009a, b; Reddy 2007; Reddy and Pang 2008; Thai 2012; Thai and Vo 2012; Wang et al. 2008). The literature survey in the current work thus will focus on articles related to the geometrically nonlinear analysis of SW/MW CNT that are nonembedded and embedded in an elastic medium, and different boundary conditions.

On the basis of the principle of virtual displacement (PVD), Yang et al. (2010) developed a nonlocal TBT combined with the von Kármán geometrical nonlinearity (VKGN) for the nonlinear free vibration analysis of SWCNT, with combinations of simply-supported and clamped edges, in which the effects of transverse shear deformation and rotary inertia were considered. Within the framework of the nonlocal CBT and VKGN, Ghayesh (2014) investigated the nonlinear dynamic responses of SWCNT with various boundary conditions and subjected to the primary and superharmonic excitations. A parametric study with regard to the effects of the length-to-diameter ratio, forcing amplitude and excitation frequency on the dynamic responses of SWCNT also was undertaken. Ke et al. (2009) extended the work of Yang et al. (2010) to the large amplitude vibration analysis of double-walled (DW) CNT embedded in an elastic medium. In their formulation, the nonlocal TBT-based motion equations were derived using Hamilton’s principle; the interactions between the DWCNT and its surrounding medium were considered, and the van der Waals (vdW) interaction between adjacent walls were simulated using the Winkler-type foundation model with different spring constants. The differential quadrature (DQ) method (Bert and Malik 1997; Du et al. 1994; Wu and Lee 2001) in conjunction with a direct iterative method then was used to solve the resulting nonlinear equations. This issue then was reexamined by Ansari et al. (2012), in which a two-parameter Pasternak foundation model was used to simulate the interaction between the DWCNT and its surrounding medium. Moreover, an alternative approach, the strain gradient theory, also has been used to study various structural analyses of nanoscale beams, such as linear bending analysis (Lim and Wang 2007) and wave propagation analysis (Lim et al. 2015), in which the local higher-order strain gradients were considered in a global sense.

Because almost all of the two-dimensional (2D) nonlocal beam theories used for the analysis of SW/MW CNT were derived on the basis of the PVD rather than Reissner’s mixed variational theorem (RMVT) (Reissner 1984, 1986), Wu and Lai (2015a, b) and Wu and Liou (2016) thus developed an RMVT-based nonlocal TBT for the linear bending, free vibration, and buckling analyses of SWCNT, embedded and non embedded in an elastic medium, and with different boundary conditions, in which the DQ and differential reproducing kernel methods (Wang et al. 2010) were used. In the PVD-based beam theories, the generalized displacement components (i.e., the normal displacement and rotation) were regarded as the primary variables; however, except for the generalized displacement components, the generalized resultant forces (i.e., shear force and bending moment) also were regarded as the primary variables in the RMVT-based beam theories. In Wu and Lai (2015a, b), the highest differential orders of the Euler-Lagrange equations in the RMVT-based strong and weak formulations were shown to be one-half lower than those in the PVD-based ones. The performance of the RMVT-based nonlocal TBT was shown to be superior to that of the PVD-based one, in that the convergence rate of the RMVT-based nonlocal TBT was faster than that of the PVD-based nonlocal one. Moreover, the predictions of the generalized force resultants obtained using the RMVT-based nonlocal TBT were more accurate than those obtained using the PVD-based nonlocal one. Finally, the RMVT-based nonlocal TBT was less time-consuming than the PVD-based one. Because of these benefits of the RMVT-based nonlocal TBT, and the fact that there are relatively few articles regarding the nonlinear analysis of MWCNT as compared with those related to the linear analysis of these, the authors of this paper extended the earlier work regarding the linear static analysis of embedded SWCNT to the geometrically nonlinear static analysis of embedded MWCNT with combinations of free, simply-supported, and clamped edge conditions, in which the effect of vdW interaction was considered. In the formulation, the Euler-Lagrange equations of a typical wall of the MWCNT and the associated boundary conditions were derived by using the stationary principle of Reissner’s energy functional, in which the ENET and VKGN were used. The vdW interaction forces between any pair of walls constituting the MWCNT and the Pasternak-type foundation model used to simulate the interaction between the MWCNT and its surrounding medium were taken into account. A parametric study related to the influence of some crucial effects on the nonlinear static behavior of the embedded MWCNT was conducted, such as the small-length scale effect, vdW interaction, aspect ratios, different boundary conditions, and the stiffness of the foundation.

Geometrically Nonlinear RMVT-Based Local TBT

Kinematic and Kinetic Assumptions

In this work, the authors considered a moderately thick MWCNT with a length

L

and the thickness of the

k

th-wall of the MWCNT

h_{k}

(

k = 1 - N_{l}

), resting on a two-parameter Pasternak foundation model, and under the mechanical loads on the outer-most surface, in which

N_{l}

denotes the total number of walls constituting the CNT. The vdW interaction forces between any pair of walls were taken into account. A set of Cartesian coordinates (

x

,

y

,

z

) was located at the center of the MWCNT. The configuration, coordinates, and kinematics of an SWCNT are given in Fig. 1, which also can be used for a typical wall of the MWCNT by replacing the applied external load and recovery forces exerted from the foundation with the corresponding vdW interaction forces exerted from other walls. The external and internal loading system for a triple-walled (TW) CNT is shown in Fig. 2.

Fig. 1. Configuration, coordinates, and kinematics of an SWCNT [reprinted from *Composite Structures*, 122, Chih-Ping Wu and Wei-Wen Lai, “Reissner’s mixed variational theorem-based nonlocal Timoshenko beam theory for a single-walled carbon nanotube embedded in an elastic medium and with various boundary conditions,” pp. 390–404, Copyright (2015), with permission from Elsevier]

Fig. 2. External and internal force systems applied to a TWCNT

In the TBT, the shear deformation effect was considered to be a constant through the thickness coordinate of each wall, and the related displacement field of the

k

th-wall of the MWCNT is given as follows:

u_{1}^{k} (x, z) = u_{k} (x) - z ϕ_{k} (x)

(1)

u_{2}^{k} (x, z) = 0

(2)

u_{3}^{k} (x, z) = w_{k} (x)

(3)

where

u_{i}^{k} (x, z)

(

i = 1 - 3

) = displacement components of the

k

th-wall of the MWCNT in the

x

,

y

, and

z

directions, respectively;

u_{k} (x)

and

w_{k} (x)

= midplane displacement components of the

k

th-wall in the

x

and

z

directions; and

ϕ_{k} (x)

= total rotation of the

k

th-wall in the

x - z

plane. The deformation of a section in the

x - z

plane is shown in Fig. 1.

The strain-displacement relations of the

k

th-wall of the MWCNT accounting for the VKGN are given by

ϵ_{x}^{k} = u_{k, x} - z ϕ_{k, x} + (1 / 2) {(w_{k, x})}^{2}

(4)

γ_{x z}^{k} = - ϕ_{k} + w_{k, x}

(5)

ϵ_{y}^{k} = ϵ_{z}^{k} = γ_{y z}^{k} = γ_{x y}^{k} = 0

(6)

where

ϵ_{x}^{k}

,

ϵ_{y}^{k}

,

ϵ_{z}^{k}

,

γ_{x z}^{k}

,

γ_{y z}^{k}

, and

γ_{x y}^{k}

= strain components of the

k

th-wall of the MWCNT; and

f_{, x} = \partial f / \partial x

, in which

f = u_{k}

,

w_{k}

, and

ϕ_{k}

.

The nonzero stress components of the

k

th-wall of the MWCNT in the local TBT are given by

σ_{x}^{k} = E_{k} [u_{k, x} - z ϕ_{k, x} + (1 / 2) {(w_{k, x})}^{2}]

(7)

τ_{x z}^{k} = k_{c k} G_{k} (- ϕ_{k} + w_{k, x})

(8)

where

E_{k}

and

G_{k}

= Young’s and the shear moduli of the

k

th-wall, respectively;

k_{c k}

= shear stress correction factor of the

k

th-wall, which accounts for the nonuniform shear stress distribution through the thickness; and the value of

k_{c k}

will depend on the cross section and Poisson’s ratio, and is reviewed by Cowper (1966).

The axial force (

N_{k}

), moment (

M_{k}

), and shear force (

Q_{k}

) resultants of the

k

th-wall of the MWCNT are defined by

N_{k} = \int_{A_{k}} σ_{x}^{k} d A_{k}

(9)

M_{k} = - \int_{A_{k}} σ_{x}^{k} z d A_{k}

(10)

Q_{k} = - \int_{A_{k}} τ_{x z}^{k} d A_{k}

(11)

where the sign conventions of

N_{k}

,

M_{k}

, and

Q_{k}

can refer to Wu and Lai (2015a).

Using Eqs. (9)–(11), the authors can express the stress components (

σ_{x}^{k}

and

τ_{x z}^{k}

) in terms of the axial force, bending moment, and shear force of the

k

th-wall of the MWCNT as follows:

σ_{x}^{k} = \frac{N_{k}}{A_{k}} - z \frac{M_{k}}{I_{k}}

(12)

τ_{x z}^{k} = - \frac{Q_{k}}{A_{k}}

(13)

where

A_{k}

and

I_{k}

= cross sectional area and the moment inertia of the

k

th-wall of the MWCNT.

vdW Interaction

In this work, a formula of the vdW interaction force derived by He et al. (2005a, b), was used. In He et al., for an MWCNT, the vdW interaction force

(q_{i j})

of the

i

th-wall exerted from the

j

th-wall of a MWCNT is expressed as

q_{i j} = {\bar{C}}_{i j} Δ w_{i j}

, in which

Δ w_{i j}

is the relative displacement between the

i

th- and

j

th-walls, and

{\bar{C}}_{i j}

is the vdW interaction coefficient. According to He et al.,

{\bar{C}}_{i j}

is given in the following form:

{\bar{C}}_{i j} = - (\frac{1,001 π ϵ σ^{12}}{3 a^{4}} E_{i j}^{13} - \frac{1,120 π ϵ σ^{6}}{9 a^{4}} E_{i j}^{7}) R_{j}

(14)

where vdW interaction force is defined as the repulsive force when the value of

C_{i j}

is positive, whereas it is the attractive force when the value of

C_{i j}

is negative;

R_{j}

= midsurface radius of the

j

th-wall of the MWCNT;

a

= C–C bond length; and

a = 1.42 Å

,

ϵ = 2.968 meV = 4.7488 \times 10^{- 22} J

,

σ = 0.3407 nm

, and

E_{i j}^{m} = {(R_{i} + R_{j})}^{- m} \int_{0}^{π / 2} {(1 - K_{i j} \cos^{2} θ)}^{- m / 2} d θ

(15)

K_{i j} = 4 R_{i} R_{j} {(R_{i} + R_{j})}^{- 2}

(16)

Because the vdW interaction coefficient

{\bar{C}}_{i j}

with a unit of

N / {(nm)}^{3}

is derived for a cylindrical shell model, it cannot be directly applied to the current nonlocal TBT. The equivalent vdW interaction coefficient

C_{i j}

with a unit

N / {(nm)}^{2}

for a beam model, therefore, is derived as follows:

4 \int_{0}^{π / 2} {\bar{C}}_{i j} Δ w_{i j} R_{i} \sin^{2} θ d θ = (π R_{i} {\bar{C}}_{i j}) Δ w_{i j} = C_{i j} Δ w_{i j}

(17)

where

C_{i j}

is obtained as

C_{i j} = (π R_{i}) {\bar{C}}_{i j}

; and the corresponding schematic diagram is given in Fig. 3.

Fig. 3. Distribution of the vdW interaction force of the $i$ th-wall exerted from the $j$ th-wall of an MWCNT and its equivalent vdW interaction force for a beam model

Stationary Principle of Reissner’s Energy Functional

The stationary principle of Reissner’s energy functional was used to derive the governing equations of the geometrically nonlinear RMVT-based local TBT, for which the generalized displacement variables (

u_{k}

,

w_{k}

, and

ϕ_{k}

) and the generalized force resultant ones (

N_{k}

,

M_{k}

, and

Q_{k}

) were selected as the primary variables subject to variation, and the corresponding energy functional (

Π_{R}

) for the geometrically nonlinear RMVT-based local TBT is written in the form of

Π_{R} = \sum_{k = 1}^{N_{l}} Π_{R}^{k}

(18)

where

Π_{R}^{k}

= Reissner strain energy functional of the

k

th-layer of the MWCNT in the RMVT-based nonlocal TBT, and is given as follows:

Π_{R}^{k} = \int_{0}^{L} \int_{A_{k}} [σ_{x}^{k} ϵ_{x}^{k} + τ_{x z}^{k} γ_{x z}^{k} - {(σ_{x}^{k})}^{2} / (2 E_{k}) - {(τ_{x z}^{k})}^{2} / (2 k_{c k} G_{k})] d A_{k} d x + δ_{k N_{l}} \int_{0}^{L} (1 / 2) [k_{w} {(w_{k})}^{2} + k_{G} {(w_{k, x})}^{2}] d x + \int_{0}^{L} (q_{v d W}^{k} + δ_{k N_{l}} q) w_{k} d x + {\bar{N}}_{k 0} u_{k 0} - {\bar{N}}_{k L} u_{k L} + {\bar{M}}_{k 0} ϕ_{k 0} - {\bar{M}}_{k L} ϕ_{k L} - {\bar{V}}_{k 0} w_{k 0} + {\bar{V}}_{k L} w_{k L} = \int_{0}^{L} {N_{k} u_{k, x} + (N_{k} / 2) {(w_{k, x})}^{2} - [{(N_{k})}^{2} / (2 A_{k} E_{k})] + M_{k} ϕ_{k, x} - [{(M_{k})}^{2} / (2 E_{k} I_{k})] + Q_{k} (ϕ_{k} - w_{k, x}) - [{(Q_{k})}^{2} / (2 k_{c k} G_{k} A_{k})]} d x + δ_{k N_{l}} \int_{0}^{L} (1 / 2) [k_{w} {(w_{k})}^{2} + k_{G} {(w_{k, x})}^{2}] d x + \int_{0}^{L} (q_{v d W}^{k} + δ_{k N_{l}} q) w_{k} d x + {\bar{N}}_{k 0} u_{k 0} - {\bar{N}}_{k L} u_{k L} + {\bar{M}}_{k 0} ϕ_{k 0} - {\bar{M}}_{k L} ϕ_{k L} - {\bar{V}}_{k 0} w_{k 0} + {\bar{V}}_{k L} w_{k L}

(19)

where

k_{w}

and

k_{G}

= Winkler stiffness and shear modulus of the surrounding elastic medium, respectively; and

{\bar{N}}_{k 0}

,

{\bar{N}}_{k L}

,

{\bar{M}}_{k 0}

,

{\bar{M}}_{k L}

,

{\bar{V}}_{k 0}

, and

{\bar{V}}_{k L}

= applied axial forces, moments, and shear forces of the

k

th-wall of the MWCNT at the edges, and are shown in Fig. 1;

δ_{k l}

= Kronecker delta symbol, in which

δ_{k l} = 1

when

k = l

, whereas

δ_{k l} = 0

when

k \neq l

; and

q

and

q_{v d W}^{k}

= external load applied on the outer-most wall (

k = N_{l}

) and the vdW interaction force of the

k

th-wall exerted from other walls, respectively, the positive direction of which is defined to be downward. The term

q_{v d W}^{k} δ w_{k}

is given as follows:

q_{v d W}^{k} δ w_{k} = [\sum_{\binom{l = 1}{(l \neq k)}}^{N_{l}} (C_{k l} w_{l}) - (\sum_{\binom{l = 1}{(l \neq k)}}^{N_{l}} C_{k l}) w_{k}] δ w_{k}

(20)

where

C_{k l}

= vdW interaction coefficient.

Performing the first-order variation of the previously noted RMVT-based energy functional, and then using integration by parts, the authors obtain

δ \sum_{k = 1}^{N_{l}} Π_{R}^{k} = \sum_{k = 1}^{N_{l}} {\int_{0}^{L} {[u_{k, x} + (1 / 2) {(w_{k, x})}^{2} - N_{k} / (A_{k} E_{k})] δ N_{k} + (- N_{k, x}) δ u_{k} + [φ_{k, x} - (M_{k} / E_{k} I_{k})] δ M_{k} + (- M_{k, x} + Q_{k}) δ φ_{k} + [(φ_{k} - w_{k, x}) - Q_{k} / (k_{c k} G_{k} A_{k})] δ Q_{k} + [- {(N_{k} w_{k, x})}_{, x} + Q_{k, x} + q_{v d W}^{k} + δ_{k N_{l}} (q + k_{w} w^{k} - k_{G} w_{, x x}^{k})] δ w_{k}} d x + [(Q_{k 0} - N_{k 0} w_{k 0, x} - δ_{k N_{l}} k_{G} w_{k 0, x} - {\bar{V}}_{k 0}) δ w_{k 0} - (Q_{k L} - N_{k L} w_{k L, x} - δ_{k N_{l}} k_{G} w_{k L, x} - {\bar{V}}_{k L}) δ w_{k L} - (M_{k 0} - {\bar{M}}_{k 0}) δ φ_{k 0} + (M_{k L} - {\bar{M}}_{k L}) δ φ_{k L} - (N_{k 0} - {\bar{N}}_{k 0}) δ u_{k 0} + (N_{k L} - {\bar{N}}_{k L}) δ u_{k L}]}

(21)

Imposing the stationary principle of the previously noted energy functional (i.e.,

δ \sum_{k = 1}^{N_{l}} Π_{R}^{k} = 0

) leads to the governing equations of the geometrically nonlinear RMVT-based local TBT and the possible boundary conditions as follows:

Governing equations in the interior domain of the

k

th-wall of the MWCNT (

0 < x < L

)

δ u_{k} : - N_{k, x} = 0

(22)

δ w_{k} : Q_{k, x} - N_{k, x} w_{k, x} - N_{k} w_{k, x x} = - q_{v d W}^{k} - δ_{k N_{l}} q - δ_{k N_{l}} (k_{w} w_{k} - k_{G} w_{k, x x})

(23)

δ ϕ_{k} : - M_{k, x} + Q_{k} = 0

(24)

δ N_{k} : - N_{k} + A_{k} E_{k} ⌊ u_{k, x} + (1 / 2) {(w_{k, x})}^{2} ⌋ = 0

(25)

δ M_{k} : - M_{k} + E_{k} I_{k} ϕ_{k, x} = 0

(26)

δ Q_{k} : - Q_{k} + k_{c k} G_{k} A_{k} (ϕ_{k} - w_{k, x}) = 0

(27)

where

k = 1, 2, \dots, N_{l}

.

Possible boundary conditions at the edges of the MWCNT

At x = 0, either u_{k 0} = {\bar{u}}_{k 0} or N_{k 0} = {\bar{N}}_{k 0}

(28a)

either w_{k 0} = {\bar{w}}_{k 0} or Q_{k 0} - N_{k 0} (w_{k 0, x}) - δ_{k N_{l}} k_{G} (w_{k 0, x}) = {\bar{V}}_{k 0}

(28b)

either ϕ_{k 0} = {\bar{ϕ}}_{k 0} or M_{k 0} = {\bar{M}}_{k 0}

(28c)

At x = L, either u_{k L} = {\bar{u}}_{k L} or N_{k L} = {\bar{N}}_{k L}

(29a)

either w_{k L} = {\bar{w}}_{k L} or Q_{k L} - N_{k L} (w_{k L, x}) - δ_{k N_{l}} k_{G} (w_{k L, x}) = {\bar{V}}_{k L}

(29b)

either ϕ_{k L} = {\bar{ϕ}}_{k L} or M_{k L} = {\bar{M}}_{k L}

(29c)

Geometrically Nonlinear RMVT-Based Nonlocal TBT

Eringen’s Nonlocal Constitutive Relations

In the local elasticity theory, the stress components induced at a particular material point of the loaded elastic body depend only on the strain components induced at that point, whereas in the nonlocal elasticity theory, these will depend on the strain components induced at all the material points of the continuum because of the small-length scale effect. According to Eringen (1972, 1983), the nonlocal constitutive behavior of an elastic body can be written as follows:

(1 - μ \nabla^{2}) σ_{i j} = c_{i j k l} ϵ_{k l}

(30)

where

μ

= nonlocal parameter;

μ = {(e_{0} a_{0})}^{2}

, in which

a_{0}

is the internal characteristic length and

e_{0}

is a constant used to adjust the nonlocal continuum model by matching its results with some of the reliable results obtained by experiments or other models;

e_{0} a_{0} \leq 2.0 nm

for an SWCNT (Wang and Wang 2007);

c_{i j k l}

= stiffness coefficient of the elastic beam-like solid; and

σ_{i j}

and

ϵ_{k l}

= stress and strain components, respectively. The classical continuum mechanics should be recovered when the nonlocal parameter

(μ)

approaches zero.

Strong Formulation

Using Eq. (30), the constitutive equations of the

k

th-wall of an MWCNT for the nonlocal Timoshenko beam theory can be written as follows:

σ_{x}^{k} - μ σ_{x, x x}^{k} = E_{k} ϵ_{x}^{k}

(31)

τ_{x z}^{k} - μ τ_{x z, x x}^{k} = k_{c k} G_{k} γ_{x z}^{k}

(32)

Using Eqs. (31) and (32) in conjunction with Eqs. (9)–(11), the axial force, bending moment, and shear force resultants of the

k

th-wall of the MWCNT in terms of the generalized displacement components can be expressed as

N_{k} - μ N_{k, x x} = A_{k} E_{k} ⌊ u_{k, x} + (1 / 2) {(w_{k, x})}^{2} ⌋

(33)

M_{k} - μ M_{k, x x} = E_{k} I_{k} ϕ_{k, x}

(34)

Q_{k} - μ Q_{k, x x} = k_{c k} G_{k} A_{k} (ϕ_{k} - w_{k, x})

(35)

Eqs. (33)–(35) combined with Eqs. (22)–(24) constitute the set of governing equations of the geometrically nonlinear RMVT-based nonlocal TBT, and Eqs. (28a)–(28c) and (29a)–(29c) are the possible boundary conditions at the edges. These equations constitute the strong formulation of the geometrically nonlinear nonlocal TBT, and are a well-posed boundary value problem for the geometrically nonlinear bending analysis of an embedded MWCNT with assorted boundary conditions.

Weak Formulation

In contrast to the previously noted strong formulation, a weak formulation for the finite element (FE) analysis of the current issue can be obtained using the Galerkin approach, and is given as

\sum_{k = 1}^{N_{l}} {{\int}_{0}^{L} ((- N_{k, x}) δ u_{k} + [Q_{k, x} - N_{k, x} w_{k, x} - N_{k} w_{k, x x} + q_{v d W}^{k} + δ_{k N_{l}} q + δ_{k N_{l}} (k_{w} w_{k} - k_{G} w_{k, x x}) δ w_{k}] + (- M_{k, x} + Q_{k}) δ ϕ_{k} + {- (N_{k} / A_{k} E_{k}) + μ (N_{k, x x} / A_{k} E_{k}) + [u_{k, x} + (1 / 2) {(w_{k, x})}^{2}]} δ N_{k} + [- (M_{k} / E_{k} I_{k}) + μ (M_{k, x x} / E_{k} I_{k}) + ϕ_{k, x}] δ M_{k} + [- (Q_{k} / k_{c k} G_{k} A_{k}) + μ (Q_{k, x x} / k_{c k} G_{k} A_{k}) + (ϕ_{k} - w_{k, x})] δ Q_{k}) d x + [(Q_{k 0} - N_{k 0} w_{k 0, x} - δ_{k N_{l}} k_{G} w_{k 0, x} - {\bar{V}}_{k 0}) δ w_{k 0} - (Q_{k L} - N_{k L} w_{k L, x} - δ_{k N_{l}} k_{G} w_{k L, x} - {\bar{V}}_{k L}) δ w_{k L} - (M_{k 0} - {\bar{M}}_{k 0}) δ ϕ_{k 0} + (M_{k L} - {\bar{M}}_{k L}) δ ϕ_{k L} - (N_{k 0} - {\bar{N}}_{k 0}) δ u_{k 0} + (N_{k L} - {\bar{N}}_{k L}) δ u_{k L}]} = 0

(36)

Substituting the FE interpolation of each primary variable in Eq. (36), performing the integration by parts, and then integration through the element domain, the authors finally obtained the corresponding equilibrium equations for the FE method following a standard process. The relevant detailed derivation regarding the Galerkin approach can refer to Eltaher et al. (2013, 2014), and is omitted in this paper. In this work, the authors adopted a strong formulation-based numerical technique, namely the DQ method, to analyze the current issue. Because the DQ method commonly is used for the strong formulation-based numerical analysis, the detailed process can refer to Wu and Lai (2015a), and it is not repeated in this paper.

Applications

In this section, the DQ method was used to construct the weighting coefficients for the first- and higher-order derivatives of the primary variables in the spatial (

x

) domain, such that

{\frac{\partial F^{r} (x)}{\partial x^{r}} |}_{x = x_{i}} = \sum_{j = 1}^{n_{p}} A_{i j}^{(r)} F_{j}

(37)

where

r \geq 1

; and

F_{j}

=

{(u_{k})}_{j}

,

{(w_{k})}_{j}

,

{(ϕ_{k})}_{j}

,

{(N_{k})}_{j}

,

{(M_{k})}_{j}

, and

{(Q_{k})}_{j}

which are the nodal values of in- and out-of-plane deflections, rotation, axial force, bending moment, and shear force components of the

k

th-wall of the MWCNT at

x = x_{j}

.

Applying the DQ formulae [i.e., Eq. (37)] to the strong formulation of the RMVT-based nonlocal TBT for the nonlinear bending problem of an embedded MWCNT, which consists of the governing Eqs. (22)–(24) and (33)–(35) associated with the appropriate boundary conditions given in Eqs. (28a)–(28c) and (29a)–(29c), the authors obtained a set of simultaneous nonlinear algebraic equations. In these, four different boundary conditions were considered: (1) the simple-simple (SS), (2) clamped-simple (CS), (3) clamped-clamped (CC), and (4) clamped-free (CF) supports.

Applying the DQ formulae to the governing equations at the sampling nodes in the interior domain, the following is obtained:

δ {(u_{k})}_{i} : - \sum_{j = 1}^{n_{p}} A_{i j}^{(1)} {(N_{k})}_{j} = 0

(38)

δ {(w_{k})}_{i} : \sum_{j = 1}^{n_{p}} A_{i j}^{(1)} {(Q_{k})}_{j} - [\sum_{j = 1}^{n_{p}} A_{i j}^{(1)} {(N_{k})}_{j}] [\sum_{j = 1}^{n_{p}} A_{i j}^{(1)} {(w_{k})}_{j}] - {(N_{k})}_{i} [\sum_{j = 1}^{n_{p}} A_{i j}^{(2)} {(w_{k})}_{j}] = - {(q_{v d W}^{k})}_{i} - δ_{k N_{l}} q (x_{i}) - δ_{k N_{l}} [k_{w} {(w_{k})}_{i} - k_{G} \sum_{j = 1}^{n_{p}} A_{i j}^{(2)} {(w_{k})}_{j}]

(39)

δ {(ϕ_{k})}_{i} : - \sum_{j = 1}^{n_{p}} A_{i j}^{(1)} {(M_{k})}_{j} + {(Q_{k})}_{i} = 0

(40)

δ {(N_{k})}_{i} : {(N_{k})}_{i} - μ \sum_{j = 1}^{n_{p}} A_{i j}^{(2)} {(N_{k})}_{j} - A_{k} E_{k} \sum_{j = 1}^{n_{p}} A_{i j}^{(1)} {(u_{k})}_{j} - (A_{k} E_{k} / 2) [\sum_{j = 1}^{n_{p}} A_{i j}^{(1)} {(w_{k})}_{j}] [\sum_{j = 1}^{n_{p}} A_{i j}^{(1)} {(w_{k})}_{j}] = 0

(41)

δ {(M_{k})}_{i} : {(M_{k})}_{i} - μ \sum_{j = 1}^{n_{p}} A_{i j}^{(2)} {(M_{k})}_{j} - E_{k} I_{k} \sum_{j = 1}^{n_{p}} A_{i j}^{(1)} {(ϕ_{k})}_{j} = 0

(42)

δ {(Q_{k})}_{i} : {(Q_{k})}_{i} - μ \sum_{j = 1}^{n_{p}} A_{i j}^{(2)} {(Q_{k})}_{j} - k_{c k} G_{k} A_{k} {(ϕ_{k})}_{i} + k_{c k} G_{k} A_{k} \sum_{j = 1}^{n_{p}} A_{i j}^{(1)} {(w_{k})}_{j} = 0

(43)

where

i = 2, 3, \dots, (n_{p} - 1)

.

For a typical

k

th-wall, Eqs. (38)–(43) represent

6 (n_{p} - 2)

equations in terms of

6 n_{p}

unknowns, whereas there are only three boundary conditions at each boundary point, and, thus, the authors needed to supplement three more governing equations at each of these. Four different cases of boundary conditions were considered and given as follows:

1.

SS supports

{(u_{k})}_{1} = {(w_{k})}_{1} = {(M_{k})}_{1} = 0 and {(u_{k})}_{n_{p}} = {(w_{k})}_{n_{p}} = {(M_{k})}_{n_{p}} = 0

(44)

2.

CS supports

{(u_{k})}_{1} = {(w_{k})}_{1} = {(ϕ_{k})}_{1} = 0 and {(u_{k})}_{n_{p}} = {(w_{k})}_{n_{p}} = {(M_{k})}_{n_{p}} = 0

(45)

3.

CC supports

{(u_{k})}_{1} = {(w_{k})}_{1} = {(ϕ_{k})}_{1} = 0 and {(u_{k})}_{n_{p}} = {(w_{k})}_{n_{p}} = {(ϕ_{k})}_{n_{p}} = 0

(46)

4.

CF supports

{(u_{k})}_{1} = {(w_{k})}_{1} = {(ϕ_{k})}_{1} = 0 and ({N_{k})}_{n_{p}} = {(M_{k})}_{n_{p}} = [({Q_{k})}_{n_{p}} - k_{G} \sum_{j = 2}^{n_{p}} A_{n_{p} j}^{(1)} {(w_{k})}_{j}] = 0

(47)

Finally, the sets of resulting equations considering the previously noted boundary conditions of Cases 1–4 are

6 (n_{p} - 1) N_{l}

simultaneous algebraic equations in terms of

6 (n_{p} - 1) N_{l}

unknowns, when

k = 1 - N_{l}

, which represents a standard nonlinear boundary value problem. Because a direct iterative method was used to solve these nonlinear equations, they are rewritten in matrix form as follows:

{[\begin{matrix} K_{11}^{(m)} & K_{12}^{(m)} \\ K_{21}^{(m)} & K_{22}^{(m)} \end{matrix}] + [\begin{matrix} G_{11}^{(m - 1)} & 0 \\ G_{21}^{(m - 1)} & 0 \end{matrix}]} [\begin{matrix} X_{1}^{(m)} \\ X_{2}^{(m)} \end{matrix}] = (\begin{matrix} P \\ 0 \end{matrix})

(48)

where superscript

m = m

th-iteration; coefficients

G_{11}^{(m - 1)}

and

G_{21}^{(m - 1)}

are in terms of the determined variables

X_{1}^{(m - 1)}

and

X_{2}^{(m - 1)}

;

G_{11}^{(0)} = G_{21}^{(0)} = 0

, which means the linear solutions will be used as the initial guess of the unknowns to obtain the nonlinear ones, i.e., the convergent solutions obtained from the direct iterative method with an allowable error

\in_{a}

less than

10^{- 5}

, in which the allowable error

\in_{a}

is defined as

\in_{a} = \sqrt{[X^{(m)} - X^{(m - 1)}] • [X^{(m)} - X^{(m - 1)}]} / \sqrt{[X^{(m)}] • [X^{(m)}]}

, in which the symbol,

•

, denotes the inner product operation for two vectors; and

X^{(m)} = {[\begin{matrix} X_{1}^{(m)} & X_{2}^{(m)} \end{matrix}]}^{T}

.

Once Eq. (48) is solved, the nonlinear solutions of the nodal generalized displacement and force resultant components for each layer can be obtained.

Illustrative Examples

Pressure-Loaded, Nonembedded Nanobeams

Reddy and Pang (2008) and Reddy (2007) investigated the linear static behavior of a simply-supported rectangular nanobeam subject to a uniformly-distributed load [

\bar{q} (x) = q_{0}

] by using the PVD-based nonlocal RBT and TBT, respectively, in which the area cross section of the nanobeam (

A

) and its associated area moment of inertia (

I

) were

A = b h

and

I

=

b h^{3} / 12

. This issue also was studied by Thai (2012) and Thai and Vo (2012), using Thai’s nonlocal RBT and the nonlocal SBT, respectively. These solutions in the linear bending case available in the literature thus are used as the benchmark solutions to validate the accuracy and convergence rate of the present RMVT-based nonlocal TBT, and a comparison between the linear and nonlinear solutions also was carried out.

For comparison purposes, the dimensionless deflection and loading intensity are defined as

\bar{w} = w / h

and

\bar{q} = (q_{0} L^{4}) / (E h^{5})

. The material properties of the nanobeam are the Young’s modulus

(E) = 1 TPa

and Poisson’s ratio

(ν) = 0.3

, and the shear correction factor

(k_{c}) = 5 / 6

.

Table 1 shows results of the convergence study for the DQ solutions of the RMVT-based nonlocal TBT for the dimensionless midspan deflection (

{\bar{w}}_{c}

) of a simply-supported, nanobeam subjected to a uniformly-distributed load [

q (x) = q_{0}

], in which

b = h = 1 nm

,

L = 10 nm

, and the nonlocal parameter

(μ) = 0

, 1, and 2

{(n m)}^{2}

. In the implementation of the DQ method, the total number of sampling nodes (

n_{p}

) was taken to be 9, 11, 13, and 21, and the node distribution was selected as the roots of an

n_{p}

-order Chebyshev polynomial (Wu and Lai 2015a).

Table 1. Results of the Convergence Study for the DQ Solutions of RMVT-Based Nonlocal TBT for the Dimensionless Midspan Deflection (

| {\bar{w}}_{c} |

) of a Simply-Supported Nanobeam without Foundation Models, and under a Uniformly Distributed Load

$μ {(nm)}^{2}$	Theories	$n_{p}$	$\bar{q} = 2$ $\bar{q} = 4$ $\bar{q} = 6$ $\bar{q} = 8$
$μ {(nm)}^{2}$	Theories	$n_{p}$	Linear	Nonlinear	Linear	Nonlinear	Linear	Nonlinear	Linear	Nonlinear
0	Present	9	0.3203	0.2633	0.6406	0.4156	0.9609	0.5196	1.2812	0.6000
		11	0.3203	0.2633	0.6406	0.4156	0.9609	0.5196	1.2812	0.6000
		13	0.3203	0.2633	0.6406	0.4156	0.9609	0.5196	1.2812	0.6000
		21	0.3203	0.2633	0.6406	0.4156	0.9609	0.5196	1.2812	0.6000
	CBT	—	0.3125	N/A	0.6250	N/A	0.9375	N/A	1.2500	N/A
	TBT	—	0.3203	N/A	0.6406	N/A	0.9609	N/A	1.2812	N/A
	TBT	—	0.3203	N/A	0.6406	N/A	0.9609	N/A	1.2812	N/A
	RBT	—	0.3203	N/A	0.6406	N/A	0.9609	N/A	1.2812	N/A
	SSDBT	—	0.3203	N/A	0.6406	N/A	0.9608	N/A	1.2811	N/A
1	Present	9	0.3503	0.2768	0.7006	0.4279	1.0509	0.5302	1.4012	0.6091
		11	0.3503	0.2769	0.7006	0.4280	1.0509	0.5303	1.4012	0.6093
		13	0.3503	0.2769	0.7006	0.4280	1.0509	0.5303	1.4012	0.6093
		21	0.3503	0.2769	0.7006	0.4280	1.0509	0.5303	1.4012	0.6093
	CBT	—	0.3425	N/A	0.6850	N/A	1.0275	N/A	1.3700	N/A
	TBT	—	0.3503	N/A	0.7006	N/A	1.0509	N/A	1.4012	N/A
	TBT	—	0.3509	N/A	0.7019	N/A	1.0528	N/A	1.4037	N/A
	RBT	—	0.3509	N/A	0.7019	N/A	1.0528	N/A	1.4037	N/A
	SSDBT	—	0.3509	N/A	0.7018	N/A	1.0527	N/A	1.4036	N/A
2	Present	9	0.3803	0.2888	0.7606	0.4385	1.1409	0.5392	1.5212	0.6169
		11	0.3803	0.2889	0.7606	0.4386	1.1409	0.5394	1.5212	0.6171
		13	0.3803	0.2889	0.7606	0.4386	1.1409	0.5394	1.5212	0.6171
		21	0.3803	0.2889	0.7606	0.4386	1.1409	0.5394	1.5212	0.6171
	CBT	—	0.3725	N/A	0.7450	N/A	1.1175	N/A	1.4900	N/A
	TBT	—	0.3803	N/A	0.7606	N/A	1.1409	N/A	1.5212	N/A
	TBT	—	0.3816	N/A	0.7631	N/A	1.1145	N/A	1.5262	N/A
	RBT	—	0.3816	N/A	0.7631	N/A	1.1145	N/A	1.5262	N/A
	SSDBT	—	0.3815	N/A	0.7631	N/A	1.1446	N/A	1.5261	N/A

Table 1 shows that in the linear bending cases, the present DQ solutions converged rapidly, and the convergent solutions were in excellent agreement with the analytical solutions available in the literature using assorted PVD-based nonlocal theories, such as PVD-based nonlocal CBT (Reddy 2007), TBT (Reddy and Pang 2008), Reddy’s and Thai’s RBTs (Reddy 2007; Thai 2012), and SBT (Thai and Vo 2012), in which the convergent solutions were obtained when

n_{p} = 11

. The solutions of the dimensionless deflection obtained using CBT were lower than those obtained using other nonlocal theories, which means CBT always underestimates the deflections because the effect of transverse shear deformations are discarded in the nonlocal CBT. The deviations of the solutions of the midspan deflection obtained using the nonlocal CBT and RBT were approximately 2.5%. In the linear bending cases, the variation of the dimensionless midspan deflection with the dimensionless loading intensity (

\bar{q}

) was linear, although it appeared to be a higher-order polynomial function with a decreasing slope in the nonlinear bending cases. The deviation between the linear and nonlinear solutions became greater when the loading intensity was large. The ratios of nonlinear-to-linear solutions of dimensionless midspan deflection were 0.82, 0.65, 0.54, and 0.47 when the dimensionless loading intensity was

\bar{q} = 2

, 4, 6, and 8, respectively. The geometrically nonlinear effect stiffened the nanobeam and decreased the deflection of the nanobeam. In addition, the dimensionless midspan deflection increased when the value of the nonlocal parameter became greater, which means the small-length scale effect will soften the nanobeam.

Pressure-Loaded, Nonembedded SWCNT

In this section, the authors investigated the static behavior of zigzag (8, 0), (9, 0), and (11, 0) SWCNT with combinations of free, simply-supported, and clamped edges, and subjected to a uniformly distributed load, in which the geometric parameters and material properties of these SWCNT are given by Yang et al. (2010) as follows:

For the zigzag (8, 0) SWCNT, L = 4.8659 nm, 2 R = 0.626 nm, h = 0.34 nm, E = 1.1556 TPa, and υ = 0.19

(49a)

For the zigzag (9, 0) SWCNT, L = 4.8749 nm, 2 R = 0.705 nm, h = 0.34 nm, E = 1.1572 TPa, and υ = 0.19

(49b)

For the zigzag (11, 0) SWCNT, L = 4.8857 nm, 2 R = 0.861 nm, h = 0.34 nm, E = 1.1621 TPa, and υ = 0.19

(49c)

The dimensionless deflection (

\bar{w}

) is defined as

\bar{w} = w / D = w / (2 R)

, in which

R

and

D

are the midsurface radius and diameter of the SWCNT considered. The area of the cross section (

A

) and area moment of inertia (

I

) are

A = 2 π R h

, and

I = (π / 64) [{(2 R + h)}^{4} - {(2 R - h)}^{4}]

, respectively, and the shear correction factor

k_{c}

(Cowper 1966) is taken as

k_{c} = \frac{6 (1 + ν) {(1 + c_{r})}^{2}}{(7 + 6 ν) {(1 + c_{r}^{2})}^{2} + (20 + 12 ν) c_{r}^{2}}

(50)

in which

c_{r} = (2 R - h) / (2 R + h)

.

Table 2 shows the DQ solutions of the RMVT-based nonlocal TBT for the dimensionless midspan deflections of zigzag (8, 0), (9, 0), and (11, 0) SWCNT with SS, SC, CC, and CF edge conditions, in which the total number of sampling nodes is taken to be

n_{p} = 21

and the nonlocal parameters are

μ = 0

, 1, and

2 {nm}^{2}

. Table 2 shows that on the basis of the same load intensity, the midspan deflections obtained using the nonlinear formulation were always less than those obtained using the linear formulation for the cases of SS, CS, and CC edges, whereas they were nearly identical to each other for the CF case. This means the effect of geometrical nonlinearity stiffened these SWCNT for the cases of SS, CS, and CC edges, whereas it was very minor for the enhancement of the gross stiffness of these SWCNT for the CF case. The deviations between the nonlinear and linear deflections tended to increase when the load intensity became greater. The results also showed that the midspan deflections of these SWCNT for different boundary conditions were

CF > SS > CS > CC

, in which > means larger. The midspan deflections of these SWCNT increased when the value of the nonlocal parameter became greater in the cases of SS and CS, whereas those for the SWCNT with CC edges did not have much variation in the range of

μ = 0 - 2 {nm}^{2}

, which means the small-length scale effect on the deflections of these SWCNT with different boundary conditions was

SS > CS > CC

. However, contrary to the former three cases, the midspan deflections of these SWCNT decreased when the value of the nonlocal parameter became greater in the cases of CF edges. Moreover, on the basis of the same values of nonlocal parameters and load intensity, and the same boundary conditions, the midspan deflections of these SWCNTs were sorted as zigzag (8, 0) > zigzag (9, 0) > zigzag (11, 0), which also meant that the midspan deflections decreased when the radius of the SWCNT became larger.

Table 2. DQ Solutions of RMVT-Based Nonlocal TBT for the Dimensionless Midspan Deflections (

| {\bar{w}}_{c} |

) of Zigzag (8, 0), (9, 0), and (11, 0) SWCNTs with Various Boundary Conditions, and under a Uniformly Distributed Load

Zigzag (m, n)	BCs	$μ {(nm)}^{2}$	$q_{0} = 0.5$ ( $N / m$ )		$q_{0} = 1.0$ ( $N / m$ )		$q_{0} = 2.0$ ( $N / m$ )		$q_{0} = 3.0$ ( $N / m$ )
Zigzag (m, n)	BCs	$μ {(nm)}^{2}$	Linear	Nonlinear	Linear	Nonlinear	Linear	Nonlinear	Linear	Nonlinear
(8, 0)	SS	0	0.1284	0.1251	0.2569	0.2348	0.5137	0.4025	0.7706	0.5242
		1	0.1767	0.1656	0.3533	0.2923	0.7066	0.4631	1.0600	0.5800
		2	0.2249	0.1997	0.4498	0.3329	0.8996	0.5005	1.3493	0.6135
	CS	0	0.0588	0.0586	0.1176	0.1162	0.2352	0.2249	0.3528	0.3225
		1	0.0717	0.0711	0.1435	0.1388	0.2869	0.2574	0.4304	0.3540
		2	0.0847	0.0832	0.1693	0.1589	0.3387	0.2809	0.5080	0.3732
	CC	0	0.0333	0.0333	0.0666	0.0664	0.1331	0.1320	0.1997	0.1962
		1	0.0333	0.0332	0.0666	0.0662	0.1331	0.1307	0.1997	0.1921
		2	0.0333	0.0332	0.0666	0.0661	0.1331	0.1295	0.1997	0.1884
	CF	0	0.4329	0.4329	0.8658	0.8658	1.7315	1.7315	2.5973	2.5973
		1	0.3846	0.3846	0.7693	0.7693	1.5386	1.5386	2.3079	2.3079
		2	0.3364	0.3364	0.6728	0.6728	1.3457	1.3457	2.0185	2.0185
(9, 0)	SS	0	0.0859	0.0848	0.1718	0.1638	0.3437	0.2962	0.5155	0.3992
		1	0.1175	0.1137	0.2350	0.2106	0.4700	0.3540	0.7049	0.4563
		2	0.1491	0.1399	0.2981	0.2473	0.5962	0.3924	0.8944	0.4919
	CS	0	0.0404	0.0404	0.0809	0.0804	0.1617	0.1579	0.2426	0.2306
		1	0.0490	0.0488	0.0981	0.0964	0.1962	0.1846	0.2943	0.2614
		2	0.0576	0.0571	0.1153	0.1115	0.2306	0.2064	0.3459	0.2835
	CC	0	0.0234	0.0234	0.0468	0.0467	0.0936	0.0932	0.1404	0.1390
		1	0.0234	0.0234	0.0468	0.0467	0.0936	0.0927	0.1404	0.1374
		2	0.0234	0.0234	0.0468	0.0466	0.0936	0.0922	0.1404	0.1359
	CF	0	0.2890	0.2890	0.5780	0.5780	1.1561	1.1561	1.7341	1.7341
		1	0.2574	0.2574	0.5149	0.5149	1.0298	1.0298	1.5447	1.5447
		2	0.2259	0.2259	0.4517	0.4517	0.9035	0.9035	1.3552	1.3552
(11, 0)	SS	0	0.0431	0.0430	0.0863	0.0850	0.1725	0.1636	0.2588	0.2330
		1	0.0583	0.0577	0.1165	0.1124	0.2331	0.2074	0.3496	0.2837
		2	0.0734	0.0720	0.1468	0.1373	0.2936	0.2415	0.4404	0.3195
	CS	0	0.0215	0.0215	0.0431	0.0430	0.0862	0.0855	0.1292	0.1269
		1	0.0258	0.0258	0.0516	0.0514	0.1033	0.1012	0.1549	0.1483
		2	0.0301	0.0300	0.0602	0.0596	0.1204	0.1158	0.1806	0.1668
	CC	0	0.0130	0.0130	0.0261	0.0260	0.0521	0.0520	0.0782	0.0779
		1	0.0130	0.0130	0.0261	0.0260	0.0521	0.0519	0.0782	0.0776
		2	0.0130	0.0130	0.0261	0.0260	0.0521	0.0518	0.0782	0.0773
	CF	0	0.1445	0.1445	0.2889	0.2889	0.5778	0.5778	0.8668	0.8668
		1	0.1293	0.1293	0.2587	0.2587	0.5173	0.5173	0.7760	0.7760
		2	0.1142	0.1142	0.2284	0.2284	0.4568	0.4568	0.6851	0.6851

Pressure-Loaded TWCNT Embedded in an Elastic Medium

In this section, the authors studied the nonlinear static behavior of a TWCNT with fully simply-supported and clamped edges, and under a uniform load, in which the vdW interaction forces of any pair of walls constituting the TWCNT were considered and the Pasternak-type foundation model was used. As noted previously, a formula derived by He et al. (2005a, b) for the vdW interaction forces between any pair of walls was modified and adopted in this work, which is given in Eqs. (14) and (17). The geometric parameters and material properties of the TWCNT were

R_{1} ∶ R_{2} ∶ R_{3} = (0.68 nm) (1 ∶ 1.5 ∶ 2)

,

L / D_{3} = 5

,

E = 1 TPa

, and

υ = 0.19

, and the effective thickness for each wall was

h = 0.34 nm

.

R_{i}

and

D_{i}

were the midsurface radius and diameter of the

i

th-wall;

D_{i} = 2 R_{i}

and

i = 1 - 3

counted from the inner-most wall to the outer-most wall. The dimensionless load intensity (

\bar{q}

), deflection (

\bar{w}

), axial-spring constant (

K_{w}

), and shear modulus of the medium (

K_{G}

) are defined as

\bar{q} = q_{0} L^{4} / E {(R_{3})}^{5}

,

\bar{w} = w / D_{3}

,

K_{w} = k_{w} L^{4} / E I_{3},

and

K_{G} = k_{G} L^{2} / E I_{3}

, in which

I_{3}

denoted the area moment of inertia of the third wall, which was the outer-most wall of the TWCNT.

Fig. 4 shows the DQ solutions for the deflection functions of the third wall of the TWCNT along the length direction with different boundary conditions and nonlocal parameters by using the linear and nonlinear formulations of the RMVT-based nonlocal TBT, in which

K_{w} = K_{G} = 0

,

μ = 0

and

2 {nm}^{2}

, and

\bar{q} = 200

. Fig. 4 shows that the maximum deflection occurred at the midspan in the cases of SS and CC. The effects of geometrical nonlinearity and small-length scale on the deflection functions of the TWCNT with different boundary conditions were

SS > CC

, in which > means more significant. In addition, the geometrical nonlinearity effect on the deflection functions of the TWCNT was more significant than the small-length scale effect on these.

Fig. 4. DQ solutions for the deflection functions of the third wall of an isolated TWCNT along the length direction for different boundary conditions: (a) SS; (b) CC supports

Fig. 5 shows the variations of the dimensionless midspan deflection of the third wall of the TWCNT with increasing load intensity and different nonlocal parameters for different boundary conditions, in which

K_{w} = K_{G} = 0

,

μ = 0

and

2 {nm}^{2}

, and

\bar{q} = 0 - 200

. Again, it shows that in the linear cases, the variation of the dimensionless midspan deflection with the dimensionless loading intensity (

\bar{q}

) was linear, whereas it appeared to be a higher-order polynomial function with a decreasing slope in the nonlinear cases. The small-length scale effect on the midspan deflections of the TWCNT in the linear cases was more significant than that in the nonlinear cases.

Fig. 5. Variations of the dimensionless midspan deflection of the third wall of an isolated TWCNT with increasing load intensity for different boundary conditions: (a) SS; (b) CC supports

Figs. 6 and 7 show the variations of the dimensionless midspan deflection of the third wall of the TWCNT with increasing load intensity and different values of

K_{w}

and

K_{G}

for different boundary conditions, in which

μ = 1 {nm}^{2}

,

\bar{q} = 0 - 200

,

K_{G} = 0

, and

K_{w} = 25

, 50, and 100 in Fig. 6;

K_{w} = 50

and

K_{G} = 0

, 5, and 10 in Fig. 7. Figs. 6 and 7 show that the midspan deflections of the embedded TWCNT tended to decrease when the values of

K_{w}

and

K_{G}

became greater, which means the interaction effect between the TWCNT and its surrounding medium increased the gross stiffness of the TWCNT.

Fig. 6. Variations of the dimensionless midspan deflection of the third wall of a TWCNT resting on the Winkler-type foundation model with increasing load intensity for different boundary conditions: (a) SS; (b) CC supports

Fig. 7. Variations of the dimensionless midspan deflection of the third wall of a TWCNT resting on the Pasternak-type foundation model with increasing load intensity for different boundary conditions: (a) SS; (b) CC supports

Concluding Remarks

In this article, the authors developed the RMVT-based nonlocal TBT for the nonlinear static analysis of MWCNTs with combinations of free, simply-supported, and clamped edge conditions embedded in an elastic medium. The strong formulations of the RMVT-based nonlinear nonlocal TBT and its associated possible conditions were derived using the stationary principle of Reissner’s energy functional, in which the vdW interaction forces between any pair of walls constituting the MWCNT were considered and the interaction between the MWCNT and its surrounding medium was simulated using the Pasternak-type foundation model. The DQ method was used to discretize the generalized displacement and force resultant components, and a direct iteration method was used to solve the resulting system equations. The results showed that in the linear cases, the variation of the dimensionless midspan deflection with the dimensionless loading intensity was linear, whereas it appeared to be a higher-order polynomial function with a decreasing slope in the nonlinear cases. The geometrical nonlinear effect on the deflection functions of MWCNTs was more significant than the small-scale effect on these. The geometrical nonlinear effect on the static behavior of MWCNTs with different boundary conditions was significant for the SS, CS, and CC cases, rather than the CF case, and were

SS > CS > CC

, in which > means more significant.

Acknowledgments

This work was supported by the Ministry of Science and Technology of the Republic of China through Grant MOST 103-2221-E-006-064-MY3.

References

Ansari, R., Gholami, R., and Darabi, M. A. (2012). “Nonlinear free vibration of embedded double-walled carbon nanotubes with layerwise boundary conditions.” Acta Mech., 223(12), 2523–2536.

Crossref

Google Scholar

Arash, B., and Wang, Q. (2012). “A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes.” Comput. Mater. Sci., 51(1), 303–313.

Crossref

Google Scholar

Aydogdu, M. (2008). “Vibration of multi-walled carbon nanotubes by generalized shear deformation theory.” Int. J. Mech. Sci., 50(4) 837–844.

Crossref

Google Scholar

Aydogdu, M. (2009). “A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration.” Physica E, 41(9), 1651–1655.

Crossref

Google Scholar

Bert, C. W., and Malik, M. (1997). “Differential quadrature: A powerful new technique for analysis of composite structures.” Compos. Struct., 39(3–4), 179–189.

Crossref

Google Scholar

Brischetto, S. (2014). “A continuum elastic three-dimensional model for natural frequencies of single-walled carbon nanotubes.” Compos. Part B. Eng., 61, 222–228.

Crossref

Google Scholar

Cowper, G. (1966). “The shear coefficient in Timoshenko’s beam theory.” J. Appl. Mech., 33(2), 335–340.

Crossref

Google Scholar

Du, H., Lim, M., and Lin, R. (1994). “Application of generalized differential quadrature method to structural problems.” Int. J. Numer. Methods Eng., 37(11), 1881–1896.

Crossref

Google Scholar

Eltaher, M. A., Alshorbagy, A. E., and Mahmoud, F. F. (2013). “Vibration analysis of Euler-Bernoulli nanobeams by using finite element method.” Appl. Math. Modell., 37(7), 4787–4797.

Crossref

Google Scholar

Eltaher, M. A., Khairy, A., Sadoun, A. M., and Omar, F. A. (2014). “Static and buckling analysis of functionally graded Timoshenko nanobeams.” Appl. Math. Comput., 229, 283–295.

Crossref

Google Scholar

Eltaher, M. A., Khater, M. E., and Emam, S. A. (2016). “A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams.” Appl. Math. Modell., 40(5–6), 4109–4128.

Crossref

Google Scholar

Eringen, A. C. (1972). “Nonlocal polar elastic continua.” Int. J. Eng. Sci., 10(1), 1–16.

Crossref

Google Scholar

Eringen, A. C. (1983). “On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves.” J. Appl. Phys., 54(9), 4703–4710.

Crossref

Google Scholar

Eringen, A. C. (2002). Nonlocal continuum field theories, Springer, New York.

Google Scholar

Fu, Y. M., Hong, J. W., and Wang, X. Q. (2006). “Analysis of nonlinear vibration for embedded carbon nanotubes.” J. Sound Vibr., 296(4–5), 746–756.

Crossref

Google Scholar

Ghayesh, M. H. (2014). “Nonlinear size-dependent behaviour of single-walled carbon nanotubes.” Appl. Phys. A, 117(3), 1393–1399.

Crossref

Google Scholar

Gibson, R. F., Ayorinde, E. O., and Wen, Y. F. (2007). “Vibrations of carbon nanotubes and their composites: A review.” Compos. Sci. Technol., 67(1), 1–28.

Crossref

Google Scholar

He, X. Q., Kitipornchai, S., and Liew, K. M. (2005a). “Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction.” J. Mech. Phys. Solids, 53(2), 303–326.

Crossref

Google Scholar

He, X. Q., Kitipornchai, S., Wang, C. M., and Liew, K. M. (2005b) “Modeling of van der Waals force for infinitesimal deformation of multi-walled carbon nanotubes treated as cylindrical shells.” Int. J. Solids Struct., 42(23), 6032–6047.

Crossref

Google Scholar

Hu, Y. G., Liew, K. M., and Wang, Q. (2012). “Modeling of vibration of carbon nanotuibes.” Procedia Eng., 31, 343–347.

Crossref

Google Scholar

Iijima, S. (1991). “Synthesis of carbon nanotubes.” Nature, 354(6348), 56–58.

Crossref

Google Scholar

Ke, L. L., Xiang, Y., Yang, J., and Kitipornchai, S. (2009). “Nonlinear free vibration of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory.” Comput. Mater. Sci., 47(2), 409–417.

Crossref

Google Scholar

Lim, C. W., and Wang, C. M. (2007). “Exact variational nonlocal stress modeling with asymptotic higher-order strain gradients for nanobeams.” J. Appl. Phys., 101(5), 054312.

Crossref

Google Scholar

Lim, C. W., Zhang, G., and Reddy, J. N. (2015). “A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation.” J. Mech. Phys. Solids, 78, 298–313.

Crossref

Google Scholar

Murmu, T., and Pradhan, S. C. (2009a). “Small-scale effect on the vibration of nonuniform nanocantilever based on nonlocal elasticity theory.” Physica E, 41(8), 1451–1456.

Crossref

Google Scholar

Murmu, T., and Pradhan, S. C. (2009b). “Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory.” Comput. Mater. Sci., 46(4), 854–859.

Crossref

Google Scholar

Reddy, J. N. (2007). “Nonlocal theories for bending, buckling and vibration of beams.” Int. J. Eng. Sci., 45(2–8), 288–307.

Crossref

Google Scholar

Reddy, J. N., and Pang, S. D. (2008). “Nonlocal continuum theories of beams for the analysis of carbon nanotubes.” J. Appl. Phys., 103(2), 023511.

Crossref

Google Scholar

Reissner, E. (1984). “On a certain mixed variational theorem and a proposed application.” Int. J. Numer. Methods Eng., 20(7), 1366–1368.

Crossref

Google Scholar

Reissner, E. (1986). “On a mixed variational theorem and on shear deformable plate theory.” Int. J. Numer. Methods Eng., 23(2), 193–198.

Crossref

Google Scholar

Thai, H. T. (2012). “A nonlocal beam theory for bending, buckling, and vibration of nanobeams.” Int. J. Eng. Sci., 52, 56–64.

Crossref

Google Scholar

Thai, H. T., and Vo, T. P. (2012). “A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams.” Int. J. Eng. Sci., 54, 58–66.

Crossref

Google Scholar

Walgraef, D. (2007). “On the mechanics of deformation instabilities in carbon nanotubes.” Eur. Phys. J., 146(1), 443–457.

Google Scholar

Wang, C. M., Kitipornchai, S., Lim, C. W., and Eisenberger, M. (2008). “Beam bending solutions based on nonlocal Timoshenko beam theory.” J. Eng. Mech., 475–481.

Crossref

Google Scholar

Wang, K. F., Wang, B. L., and Kitamura, T. (2016). “A review on the application of modified continuum models in modeling and simulation of nanostructures.” Acta Mech. Sinica, 32(1), 83–100.

Crossref

Google Scholar

Wang, Q., and Wang, C. M. (2007). “The constitutive relation and small scale parameter of nonlocal continuum mechanics formodelling carbon nanotubes.” Nanotechnology, 18(7), 075702.

Crossref

Google Scholar

Wang, Y. M., Chen, S. M., and Wu, C. P. (2010). “A meshless collocation method based on the differential reproducing kernel interpolation.” Comput. Mech., 45(6), 585–606.

Crossref

Google Scholar

Wu, C. P., and Lai, W. W. (2015a). “Free vibration of an embedded single-walled carbon nanotube with various boundary conditions using the RMVT-based nonlocal Timoshenko beam theory and DQ method.” Physica E, 68, 8–21.

Crossref

Google Scholar

Wu, C. P., and Lai, W. W. (2015b). “Reissner’s mixed variational theorem-based nonlocal Timoshenko beam theory for a single-walled carbon nanotube embedded in an elastic medium and with various boundary conditions.” Compos. Struct., 122, 390–404.

Crossref

Google Scholar

Wu, C. P., and Lee, C. Y. (2001). “Differential quadrature solution for the free vibration analysis of laminated conical shells with variable stiffness.” Int. J. Mech. Sci., 43(8), 1853–1869.

Crossref

Google Scholar

Wu, C. P., and Liou, J. Y. (2016). “RMVT-based nonlocal Timoshenko beam theory for the stability analysis of embedded single-walled carbon nanotube with various boundary conditions.” Int. J. Struct. Stab. Dyn., 16(10), 1550068.

Crossref

Google Scholar

Yang, J., Ke, L. L., and Kitipornchai, S. (2010). “Nonlinear free vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory.” Physica E, 42(5), 1727–1735.

Crossref

Google Scholar

Information & Authors

Information

Published In

Journal of Nanomechanics and Micromechanics

Volume 7 • Issue 4 • December 2017

Copyright

History

Received: Oct 17, 2016

Accepted: Apr 21, 2017

Published online: Jun 22, 2017

Discussion open until: Nov 22, 2017

Published in print: Dec 1, 2017

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Chih-Ping Wu [email protected]

Professor, Dept. of Civil Engineering, National Cheng Kung Univ., Tainan 70101, Taiwan, ROC (corresponding author). E-mail: [email protected]

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Zong-Li Hong [email protected]

Graduate Student, Dept. of Civil Engineering, National Cheng Kung Univ., Tainan 70101, Taiwan, ROC. E-mail: [email protected]

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Yung-Ming Wang [email protected]

Associate Professor, Dept. of Civil Engineering, National Cheng Kung Univ., Tainan 70101, Taiwan, ROC. E-mail: [email protected]

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Abstract

Introduction

Geometrically Nonlinear RMVT-Based Local TBT

Kinematic and Kinetic Assumptions

vdW Interaction

Stationary Principle of Reissner’s Energy Functional

Geometrically Nonlinear RMVT-Based Nonlocal TBT

Eringen’s Nonlocal Constitutive Relations

Strong Formulation

Weak Formulation

Applications

Illustrative Examples

Pressure-Loaded, Nonembedded Nanobeams

Pressure-Loaded, Nonembedded SWCNT

Pressure-Loaded TWCNT Embedded in an Elastic Medium

Concluding Remarks

Acknowledgments

References

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