NANO LETTERS
Variational Principles for Transversely Vibrating Multiwalled Carbon Nanotubes Based on Nonlocal Euler-Bernoulli Beam Model
2009 Vol. 9, No. 5 1737-1741
Sarp Adali* School of Mechanical Engineering, UniVersity of KwaZulu-Natal, Durban, South Africa Received September 6, 2008; Revised Manuscript Received December 24, 2008
ABSTRACT Variational principles are derived for multiwalled carbon nanotubes undergoing vibrations. Derivations are based on the continuum modeling with the Euler-Bernoulli beam representing the nanotubes and small scale effects taken into account via the nonlocal elastic theory. Hamilton’s principle for multiwalled nanotubes is given and Rayleigh’s quotient for the frequencies is derived for nanotubes undergoing free vibrations. Natural and geometric boundary conditions are derived which lead to a set of coupled boundary conditions due to nonlocal effects.
The objective of the present study is to derive the variational principles and the applicable boundary conditions involving the transverse vibrations of multiwalled carbon nanotubes based on the nonlocal theory of elasticity which accounts for the small scale effects. Variational principles for multiwalled nanotubes under buckling loads were obtained1 where a continuum model based on nonlocal theory of EulerBernoulli beams was employed. In the present study these results are extended to the case of multiwalled nanotubes undergoing transverse vibrations, and the Hamilton’s principle is derived. Understanding vibrational characteristics of carbon nanotubes is of practical importance in many present and potential applications of nano structures. Continuum models have been used extensively for this purpose with Euler-Bernoulli beam model being employed in several studies involving the vibrations of and wave propagation in single-walled and multiwalled nanotubes.2-10 Further studies on the subject include an experimental study of the vibrations of single-walled nanotubes11 and a review of the modeling techniques for carbon nanotubes undergoing vibrations and a discussion of related subjects.12 These studies used the classic continuum models that are local in the sense that the stress at a given point is only related to the strain at the same point, as such small scale effects were not taken into account, which can be substantial in nanosized components due to the atomic scale of the lattice structure. However the stress at a given point can be related to the strain at all points in the body using a nonlocal elastic theory that is * To whom correspondence should be addressed. E-mail:
[email protected]. 10.1021/nl8027087 CCC: $40.75 Published on Web 04/03/2009
2009 American Chemical Society
capable of taking small length effects into account. Nonlocal continuum mechanics have been developed in a number of studies13,14 intheearly1970s.NonlocaltheoryofEuler-Bernoulli beams has also been employed first for nanotube buckling15 and later in a number of studies involving the vibrations of carbon nanotubes16-22 and nanosized beams23-25 which showed that small scale effects contribute significantly to the mechanical behavior of nanostructures. These studies mostly involved simply supported boundary conditions and analytical expressions for the solutions. As such there is a need for the development of variational formulations that can be used in the implementation of approximate and numerical methods of solutions. Moreover variational principles lead to the derivation of correct natural and geometric boundary conditions that are fairly complicated due to coupling and small size effects. Variational principles for a system differential equations can be obtained in a number of ways. In the present study, a semi-inverse method developed by He26,27 will be employed. This method was applied to several problems of mathematical physics governed by a system of differential equations.28-32 Variational Principle. The multiwalled nanotube is defined as a concentric system of n nanotubes of cylindrical shape of length L. It lies on a Winkler foundation with foundation modulus k and is subject to an axial stress of σx, which can be tensile or compressive in which case the compressive stress is less than the buckling stress. The differential equations governing the transverse vibrations of double-walled nanotubes under axial loading based on
the nonlocal theory of elasticity19 can be generalized to multiwalled nanotubes and are given as D1(w1, w2) ) L1(w1) - c12(w2 - w1) + η2c12 l
(
∂2w2 ∂x2
∂2w1
-
∂x2
)
Di(wi-1, wi, wi+1) ) Li(wi) + c(i-1)i(wi - wi-1) -
2
η ci(i+1)
(
∂2wi+1 2
∂2wi
-
2
∂x
∂x
)
(
∂2wi ∂x2
-
∂2wi-1 ∂x2
)
1 2
∫∫ t2
L
t1
0
+ (2)
) 0 for i ) 2, 3, ..., n - 1
{( ) ( ) ( ) [ ( ) ( ) ]} ∂2wi
EIi
2
- FAi
2
∂x
η2Ai F
)0 (1)
ci(i+1)(wi+1 - wi) - η2c(i-1)i
Ui(wi) )
∂2wi ∂x ∂ t
(
η2c(n-1)n
2
L1(w1) +
δFj
∑ δw j)1
2
∂ wn
-
2
∂ wn-1 2
∂x
∂x
)
1
∂x4
2
+ FAi
∂ wi
Li(wi) +
(3)
1
)
Vn-1(wn-2, wn-1, wn) + Vn(wn-1, wn)
t2
L
0
∫∫ t2
L
t1
0
F1(w1, w2)dx dt
(4)
∑
∑
n
Ln(wn) +
∫ ∫ (w + η ( ∂x ) )dx dt + ∫ ∫ (- fw + F (w , w ))dx dt k 2
t2
t1
with Ui(wi) given by 1738
t2
t1
L
0
2 n
2
∂wn
(5)
(6)
n
n
n-1
n
n
) Ln(wn) +
n
∑ ∂w
∂Fj
j)n-1
n
(11)
n
-
∂ ∂Fj ∂x ∂wnx j)n-1 (12)
where i ) 2, 3,..., n-1 and the subscripts x and t denote differentiations with respect to x and t, and δFi/δwi is the variational derivative defined as
( ) ( ) ( ) ( ) ( )
δFi ∂Fi ∂ ∂Fi ∂ ∂Fi ∂2 ∂Fi ) + 2 + ... δwi ∂wi ∂x ∂wix ∂w ∂wit ∂x ∂wixx ∂ ∂Fi ∂2 ∂Fi + 2 + ... ∂t ∂wit ∂t ∂wit
(13)
Comparing eqs 10-12 with 1-3, we observe that the following equations have to be satisfied for Euler-Lagrange equations to represent the governing equations 1-3, viz. 2
∑ j)1
i+1
δFj
∑ δw
j)i-1
(
δFj ∂2w2 ∂2w1 - 2 ) -c12(w2 - w1) + η2c12 δw1 ∂x2 ∂x
)
(14)
) c(i-1)i(wi - wi-1) - ci(i+1)(wi+1 - wi) -
i
(
∂2wi ∂x2
-
∂2wi-1 ∂x2
)
(
+ η2ci(i+1)
∂2wi+1
(
∂x2
-
∂2wi ∂x2
)
(15)
)
δFj ∂2wn ∂2wn-1 ) c(n-1)n(wn - wn-1) - η2c(n-1)n δwn ∂x2 ∂x2 j)n-1 (16) n
∑
From eqs (14)-(16), it follows that (7)
2
L
0
δFj
∑ δw
η2c(i-1)i
Fi(wi-1, wi, wi+1)dx dt for i ) 2, 3, ..., n - 1
Vn(wn-1, wn) ) Un(wn) +
(10)
i+1
∂ ∂Fj ) f(x, t) ∂t ∂wnt j)n-1
where t1
∂ ∂Fj )0 ∂t ∂w1t
∂ ∂Fj )0 ∂t ∂wit j)i-1
V(wi, w2, ..., wn) ) V1(w1, w2) + V2(w1, w2, w3) + ... +
∫∫
-
1x
n
where the index i ) 1,2,...,n refers to the order of the nanotubes with the innermost nanotube indicated by i ) 1 and the outermost nanotube by i ) n with 0 e x e L and t1 e t e t2 with wi(x, t) denoting the transverse deflection of the ith nanotube. In eq 4, δin is the Kronecker’s delta with δin ) 0 for i * n and δnn ) 1, E is the Young’s modulus, Ii is the moment of inertia, and Ai is the cross-sectional area of the ith nanotube. The coefficient c(i-1)i is the interaction coefficient of van der Waals forces between the (i - 1)th and ith nanotubes with i ) 2,...,n. The constant η ) e0a is a material parameter defining the small scale effect in nonlocal elastic theory of beams where e0 is an experimentally determined constant and a is an internal characteristic length. In order to derive the variational principle, we first define a trial variational functional V(w1, w2,..., wn) given by
Vi(wi-1, wi, wi+1) ) Ui(wi) +
∂Fj
j)1
i+1
j)n-1
) (
V1(w1, w2) ) U1(w1) +
(9)
δFj ∂Fj ∂ ∂Fj ) Li(wi) + δw ∂w ∂x ∂wix i i j)i-1 j)i-1 j)i-1
-
∂x2 4 ∂ wi ∂2wn η2Ai F 2 2 + σx 4 + δink wn - η2 2 ∂x ∂ t ∂x ∂x
(
∂t2 4 ∂ wi
∂ wi
dx dt
i+1
2
+ A i σx
-
) ∑ ( ) ∑ ( ) ∑ ( ) ∑ ( ) ∑ ( ) 2
∂Fj
j)1
i+1
) f(x, t)
2
2
∂x2
j)1
nanotube and Li(wi) is the differential operator given by ∂ wi
∂2wi
+ σx
∑ ∂w - ∑ ∂x∂ ( ∂w 2
) L1(w1) +
where f(x,t) is the external force acting on the outermost
Li(wi) ) EIi
2
2
2
∂wi ∂x
- A i σx
where i ) 1, 2,..., n. In eqs 6-8, Fi(wi-1, wi, wi+1) denotes the unknown functions of wiand its derivatives and should be determined such that the differential eqs 1-3 correspond to the Euler-Lagrange equations of the variational functional (5) which are given by
l
Dn(wn-1, wn) ) Ln(wn) + c(n-1)n(wn - wn-1) -
4
2
∂wi ∂t
(8)
(
)
c12 c12 ∂w2 ∂w1 2 (w2 - w1)2 + η2 (17) 4 4 ∂x ∂x c(i-1)i ci(i+1) (wi - wi-1)2 + (wi+1 - wi)2 + Fi(wi-1, wi, wi+1) ) 4 4 η2c(i-1)i ∂wi ∂wi-1 2 η2ci(i+1) ∂wi+1 ∂wi 2 + 4 ∂x ∂x 4 ∂x ∂x for i ) 2, 3, ..., n - 1 (18) F1(w1, w2) )
(
)
(
)
Nano Lett., Vol. 9, No. 5, 2009
(
)
c(n-1)n η2c(n-1)n ∂wn ∂wn-1 2 (wn - wn-1)2 + 4 4 ∂x ∂x (19)
Fn(wn-1, wn) )
With Fi given by eqs 17-19, we observe that eqs 10-12 are equivalent to eqs 1-3, viz. 2
D1(w1, w2) ) L1(w1) +
δFj
∑ δw
)0
δFj )0 δw i j)i-1
(21)
δFj ) f(x, t) δwn j)n-1
(22)
∑
n
∑
Hamilton’s Principle. The Hamilton’ principle can be expressed as
∫
t2
t1
(23)
WE(t) )
∫
L
0
( ( )
∑∫
1 2 i)1
L
0
( ( ) -
Aiσx ∂wi 2 ∂x i)n
∑∫
1 2 i)1
PE1(t) )
L
0
2
2
∂wi ∂t
pAi
2
-
∂ wi ∂x ∂ t
+ η2AiF
( )
UFVi(Wi) )
Yi(Wi) )
i)n
∑∫
(
L
0
(
c(i-1)i(wi - wi-1) + η c(i-1)i 2
2
)
∂wn ∂x
d Wi
(
4
))
dx
1⁄2ωt
dx
η2Ai Fω2
d2Wi dx2
- σx
d4Wi dx4
d Wi dx2
) (
L
EIi
0
dx2
)
dWi dx
d2Wi
2
( )) dWi dx
2
-
( ) )} 2
d2Wi
+ η2
dx
dx2
(30)
2
dx - σxZi(Wi) - ω2Yi(Wi)
dx2
(31)
( )) ∫ (( )
dWi 2 dx, dx Ai L dWi Zi(Wi) ) 2 0 dx Wi2 +η2
n
ω2 ) min
k 2
i
dx2
2
+ η2
0
i)1
∫ (W L
0
( ) d2Wi
L
n
+η
( ))
2
d2Wi
2
dx
dx2
(32)
2
dx - σxZi(Wi) +
dWn dx
2
n
dx +
∑ ∫ F dx i)1
L
0
i
n
(33)
∑ Y (W ) i
i
i)1
where Fi (i ) 1, 2,..., n) are given by eqs 17-19 with wi(x,t) replaced by Wi(x), and Wi(x) ∈ C2(0,L). Boundary Conditions. Next we take the variations of the functional 5 using eqs 6-9 with respect to wi in order to derive the natural and geometric boundary conditions applicable to the nonlocal model of the vibrations of the multiwalled nanotubes. Let δwi denote the variation of wi which vanishes at t ) t1 and t ) t2, that is, δwi(x,t1) ) 0 and δwi(x,t2) ) 0. We observe that the first variations of V(w1, w2,..., wn-1, wn) with respect to wi, denoted by δwiV, can be obtained by integration by parts and are given by δw1V ) δw1V1 + δw1V2 )
∫∫ t2
t1
L
0
D1(w1w2)δw1 dx dt +
∫
t2
t1
i+1
δwiV )
∑δ
wiVj )
∫∫ t2
t1
t1
(29)
2
( ))
wi
t2
d2Wn
(( )
( )
(
∫
+
The variational principle for the case of free vibrations is the same as the one given by eqs 5-9 with the deflection Nano Lett., Vol. 9, No. 5, 2009
∫
∑ 21 ∫ EI
j)i-1
+ δink Wn - η2
1 2
(28)
2
- pAiω2Wi + Aiσx
dx
2
where ω is the vibration frequency. The equations governing the free vibrations are obtained by substituting eq 28 into eqs 1-3 with f(x,t) ) 0 and replacing the deflection wi(x,t) by Wi(x), partial differentiation ∂x by dx, and the operator Li(wi) by LFVi(Wi) given by LFVi(Wi) ) EIi
0
dx
In eqs 24-27, KE is the kinetic energy, WE is the work done by external forces, PE1 is the potential energy of deformation, and PE2 is the potential energy due to van der Waals forces between the nanotubes. Free Vibrations. In the present section, the variational principle in the case of freely vibrating multiwalled nanotubes is given. Let the harmonic motion of the ith nanotube be expressed as
4
L
2
∂wi ∂wi-1 ∂x ∂x
(
- pAiω2 W2i + η2
2
From eqs 5 and 31-32, the Rayleigh quotient is obtained as
( ))
(27)
wi(x, t) ) Wi(x)e(-1)
∫
(24)
(26) 1 PE2(t) ) 2 i)1
FAi 2
(25)
+ kw2n + kη2
∂x
dx
- f(x, t)wn(x, t) dx
2
2
{( )
2
The functions Fi(Wi-1, Wi, Wi+1) are of the same form as given by eqs 17-19 with partial derivatives replaced by ordinary derivatives with respect to x. Next, the Rayleigh quotient is obtained for the vibration frequency ω noting that
2
2
η Aiσx ∂ wi 2 ∂x2
∂2wi
( )) 2
2
(( ) EIi
0
d2Wi
EIi
Aiσx
where KE(t) )
∫
L
where
[(δKE(t) - (δWE(t) + δPE1(t) + δPE2(t)))]dt ) 0
i)n
1 2
(20)
i+1
Dn(wn-1, wn) ) Ln(wn) +
UFVi(Wi) )
1
j)1
Di(wi-1, wi, wi+1) ) Li(wi) +
wi(x,t) replaced by Wi(x), the double integrals replaced by the space integrals with respect to x, that is, ∫L0 Fi(Wi-1, Wi, Wi+1)dx, and Ui(wi) replaced by UFVi(Wi) given by
L
0
(34)
Di(wi-1, wi, wi+1)δwi dx dt +
∂ Ωi(0, L, t)dt
δwnV ) δwnVn-1 + δwnVn )
∂Ω1(0, L, t)dt
∫∫ t2
t1
L
0
for i ) 2, ..., n - 1
(35)
Dn(wn-1, wn)δwn dx +
∫
t2
t1
∂Ωn(0, L, t)dt
(36)
where δΩi(0,L,t) is the boundary term defined as 1739
∂Ωi(0, L, t) ) (EI1 - η A1σx) 2
3
2
η A1σx)
∂ w1 ∂x3
δw1
(
|
∂2w1 ∂x2
|
δw1
x)0
x)L
+ η A1F 2
x)0
Table 1. Fundamental Frequencies of Double-Walled Nanotubes Based on Nonlocal Theory with λ1 ) ω1(FA1L4/EI1)1/2
x)L
+ (-EI1 +
| ) | x)L
∂3w1
δw1 + x)0 ∂x∂t2 x)L
∂w1 ∂w2 - η2c12 (-A1σx + η c12) δw1 x)0 ∂x ∂x 2
∂Ωi(0, L, t) ) (EIi - η 3
η2Aiσx)
∂ wi 3
∂x
δwi
)
η (c(i-1)i + ci(i+1)) 2
|
2
x)L
x)0
3
∂x
+ (-EIi +
3
+ η2AiF
∂ wi ∂x ∂ t2
δwi
|
x)0
δwn
|
2
∂x
x)L
x)0
δwn
|
3
+ η2AnF
∂ wn ∂x∂t2
)) |
| ( ) |
+ (-Anσx + x)L
∂wn ∂wn-1 - η2c(n-1)n δwn η (c(n-1)n + k)) ∂x ∂x x)0 2
(39)
where i ) 2, 3,..., n-1 and δwi′ is the derivative of δwi with respect to x. Using the fundamental lemma of calculus of variations the boundary conditions at x ) 0,L are obtained as (EIi - η2Aiσx)
∂2wi
)0
2
∂wi )0 ∂x
or
∂x
for i ) 1, 2, ..., n (40)
(-EI1 - η2A1σx)
(-EIi + η2Aiσx)
∂3w1 3
∂x
∂3wi
+ η2A1F
2
+ (-A1σx + η2c12)
∂x ∂ t ∂w 2 )0 η2c12 ∂x
+ η2AiF
3
∂3w1
∂3wi 2
or
∂w1 ∂x
w1 ) 0
3
(-EIn + η2Anσx)
∂ wn
)
(42)
3
+ η2AnF
∂ wn
+ (-Anσx + η2(c(n-1)n + ∂x ∂ t2 ∂wn-1 ∂wn - η2c(n-1)n )0 or wn ) 0 k)) ∂x ∂x ∂x
3
(41)
+ (-Aiσx + η2(c(i-1)i +
∂x ∂x ∂ t ∂wi-1 ∂wi+1 ∂wi 2 - η c(i-1)i + c(i+1) ci(i+1))) )0 or ∂x ∂x ∂x wi ) 0 for i ) 2, ..., n - 1
(
(43)
It is observed that in the nonlocal elastic formulation of the problem (i.e., η > 0), the natural boundary conditions are coupled and these boundary conditions uncouple for η ) 0. Example. The fundamental frequency ω1 of a clampedfree double-walled nanotube with σx ) k ) 0 is computed by Rayleigh-Ritz method using a one-term trial function φ(x) ) 1 - cos(πx/2L) with the deflections of the inner and outer nanotubes given by W1(x) ) aφ(x) and W2(x) ) bφ(x), respectively, which satisfy the boundary conditions at the 1740
3.664 3.568 3.320 3.002 2.679 2.385
5.850 5.697 5.302 4.793 4.280 3.808
6.692 6.517 6.064 5.483 4.893 4.356
∑U
2
FVi(Wi) +
i)1
∑∫ i)1
L
0
Fi dx
(44)
and setting dV/da ) 0, dV/db ) 0, we obtain a system of two homogeneous equations the determinant of which is given by
(
x)L
x)0
0.0 0.1 0.2 0.3 0.4 0.5
2
+ (-EIn +
δwn
λ1 (outer tube)
V(W1, W2) )
x)L
x)L
x)0
λ1 (double-walled)
+ ((-Aiσx +
(
∂2wn
λ1 (inner tube)
clamped end (x ) 0). Substituting W1(x) and W2(x) into the variational principle
x)L
∂wi ∂wi-1 ∂wi+1 - η2 c(i-1)i + ci(i+1) δwi x)0 ∂x ∂x ∂x (38)
3
∂ wn
x)L
Aiσx) 2 δwi x)0 ∂x
∂Ωi(0, L, t) ) (EIn - η2Anσx) η2Anσx)
|
∂2wi
(37)
η0
)(
)
EI1 ξ2 c12 EI2 ξ2 c12 A2 2 + + - ω12 - ω1 4 ξ FA1 FA1L FA1L4 ξ FA1 A1 c12 2 )0 FA1
( )
(45)
where ξ0 ) L-1 ∫L0 φ2 dx, ξ1 ) L ∫L0 [(dφ)/(dx)]2dx, ξ2 ) L3 ∫L0 [(d2φ)/(dx2)]2dx, ξ ) ξ0 + η20ξ1, η0 ) η/L. Setting the determinant (45) equal to zero, the fundamental frequency is computed as the root of a quadratic equation. From eq 45, the frequencies ω11 and ω12 for the inner and outer nanotubes can be computed by setting the interaction coefficient c12 ) 0. This computation gives λ11 ) 3.664 and λ12 ) 6.692 where λ1i ) ω1i(FA1L4/EI1)1/2. These values represent an error of less than 4.3% using the one-term trial function φ(x) ) 1 - cos(πx/2L) as compared to the exact ) 3.515 and λexact ) 6.420. Numerical results values33 λexact 11 12 for the double-walled carbon nanotubes are given for E ) 1.2 TPa, c12 ) 0.0694 TPa, L ) 100 nm, R1 ) 0.35 nm, R2 ) 0.69 nm, t ) 0.34 nm where R1 and R2 are the average radii of the inner and outer nanotubes and t is the spacing between the nanotubes. The above values of the physical and geometric constants are taken from Sudak.15 The fundamental frequencies given in Table 1 show that the frequencies decrease as the small scale parameter η0 increases as observed in similar studies on the free vibrations of single and multiwalled nanotubes using the nonlocal theory16,18,21,23-25 with the local theory (η0 ) 0) giving the highest frequency. Moreover the frequencies of the singlewalled nanotubes are lower and upper bounds for those of the double-walled nanotubes. Concluding Remarks. The variational formulations for the free and forced vibrations of multiwalled carbon nanotubes were given using a continuum formulation based on nonlocal Euler-Bernoulli theory and a semi-inverse approach for the derivation of variational principles. Hamilton’s principle and Rayleigh quotient for free vibrations were obtained as well as the natural boundary conditions. A numerical example to compute the fundamental frequencies was given based on the Rayleigh-Ritz method. The variational principles presented here form the basis of several approximate and numerical methods of solution and facilitates the implementation of complicated boundary conditions. Nano Lett., Vol. 9, No. 5, 2009
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Nano Lett., Vol. 9, No. 5, 2009
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