The free vibration response of double-walled carbon nanotubes (DWCNTs) is investigated. The DWCNTs are modelled as two beams, interacting between them through the van der Waals forces, and the nonlocal Euler-Bernoulli beam theory is used. The governing equations of motion are derived using a variational approach and the free frequencies of vibrations are obtained employing two different approaches. In the first method, the two double-walled carbon nanotubes are discretized by means of the so-called “cell discretization method” (CDM) in which each nanotube is reduced to a set of rigid bars linked together by elastic cells. The resulting discrete system takes into account nonlocal effects, constraint elasticities, and the van der Waals forces. The second proposed approach, belonging to the semianalytical methods, is an optimized version of the classical Rayleigh quotient, as proposed originally by Schmidt. The resulting conditions are solved numerically. Numerical examples end the paper, in which the two approaches give lower-upper bounds to the true values, and some comparisons with existing results are offered. Comparisons of the present numerical results with those from the open literature show an excellent agreement.
1. Introduction
Carbon nanotubes (CNTs) constitute a prominent example of nanomaterials and nanostructures which have stimulated extensive research activities in science and engineering field. It is well known that CNTs are hollow cylindrical tubules composed of concentric graphitic shells with diameters on the scale of nanometers and, based on the number of walls, are designed as single-walled, double-walled and multiwalled nanotubes. In particular, the double-walled nanotubes can be viewed as two concentrically nested seamless grapheme cylinders bonded together by van der Waals forces.
Their discovery, since the publication of Iijima’s paper [1] in 1991, has attracted much attention by many researches and several studies have shown that the carbon nanotubes possess extraordinary mechanical, physical, and electrical properties. As has been pointed out in the literature, extensive studies have been conducted on studying the mechanical properties of single-walled carbon nanotubes (SWNTs) or multiwalled carbon nanotubes (MWNTs), and several investigations have been performed by employing computational and experimental methods [2, 3]. In this context, an excellent review article on the mechanical properties of nanotubes was published by Ruoff et al. [4], in which experimental measurements as well as theoretical predictions can be found.
In the earlier studies, the investigations on carbon nanotubes have mainly focused on numerous experiments [5] although these texts, at nanoscale, are very cumbersome. In addition, two different theoretical approaches have been developed: the atomistic and the continuum mechanics. Among the methods of atomistic simulations, the classical molecular dynamic (MD) simulations is the most common method in investigating the behaviour of CNTs [6, 7]. Since this approach is restricted to small-scale systems, the continuum modelling is considered to be a more appropriate method of investigating the structural behaviour of nanotubes. In the literature, there exist a lot of studies on analysing the bending, buckling, and postbuckling problems of nanotubes using Euler-Bernoulli [8, 9] and Timoshenko [10] beam models.
In recent years, due to the remarkable importance of nanostructures for many engineering and medical devices, research interest has grown on evaluating the vibrational properties of carbon nanotubes. The literature concerning the vibrational properties in CNTs is very rich and it is devoted to the dynamic problem of single and multiwalled carbon nanotubes. In this topic, the state of the art can be found in a review work by Gibson et al. [11]. In this paper, the authors report a coherent yet concise review of as many of these publications as possible and the main themes treated are the modelling and simulation of vibrating nanotubes. Extensive studies have been conducted to investigate the vibration behaviour by means of molecular dynamic simulations: for example, Ansari et al. in [12] have analyzed the vibrations characteristics of SWCNTs and DWCNTs under various layerwise boundary conditions at different lengths. The analysis performed and the results obtained show that the natural frequency of carbon nanotubes is strongly dependent on their boundary conditions especially when tubes are shorter in length. Moreover, several researchers implemented the elastic models of beams to study the dynamic problems, such as vibration and wave propagation, of carbon nanotubes [13, 14]. Xu et al. in [15, 16], for example, studied the free vibration of DWCNTs which consist of two coaxial single-walled CNT with interacting each other by the interlayer van der Waals forces. Therefore, the inner and outer tubes are modeled as two individual elastic beams and by using the Euler-Bernoulli beam model the exact solutions for natural frequencies, at different boundary conditions, have been derived. Also Elishakoff and Pentaras [17] deal with the evaluation of fundamental natural frequencies of DWCNTs under various boundary conditions and the expressions for the natural frequencies have been derived by applying the Bubnov-Galerkin and Petrov-Galerkin methods. In the paper by Natsuki et al. [18], instead, a theoretical approach to vibration characteristic analysis of CNTs with simply supported boundary condition has been presented. Applying the Euler-Bernoulli beam theory, the authors obtained the resonant frequencies of DWCNTs with different vibrational modes and they showed that the resonant frequencies decrease with increasing length of nanotubes.
Although the classical continuum methods are efficient in performing mechanical analysis of CNTs, their applicability to identify the small-scale effects on carbon nanotubes mechanical behaviours is questionable. The importance of size effect has been pointed out in a number of studies where the size dependence of the properties of nanotubes has been investigated. For example, Sun and Zhang, in [19], discussed the scarce applicability of continuous models to nanotechnology and proposed a semicontinuum model in studying nano-materials. The authors demonstrated that the values of the Young’s modulus and Poisson’s ratios depend on the number of atom layers in the thickness direction. These results show that the nanostructures and nanomaterials cannot be homogenized into a continuum. At this point, the nonlocal elastic continuum models are more pertinent in predicting the structural behaviour of nanotubes because of being capable of taking in the small-scale effects. It is well known that the nonlocal elasticity theory assumes that the stress state, at a given reference point, is considered to be a function of the strain field at all points of the body. The origins of the nonlocal theory of elasticity go to pioneering works, published in early 80s, by Eringen [20]. In [21] Reddy reports a complete development of the classical and shear deformation beam theories using the nonlocal constitutive differential equations and derived the solutions for bending, buckling, and natural frequencies problems of simply supported beams.
In recent years, many researchers have applied the nonlocal elasticity concept for the bending, buckling, and vibration analysis of nanostructures. Peddieson et al. [22] have used nonlocal Euler-Bernoulli model for static analysis of nanobeams and particular attention is paid to cantilever beams which are often used as actuators in small-scale systems. Further applications of the nonlocal elasticity theory have been employed in studying the buckling problem [23, 24] and vibration problems, by applying Euler-Bernoulli beam and shell theories and Timoshenko beam theory, in CNTs [23–32].
It is worth mentioning that, in the literature, most of the attention has been focused on deriving the variational formulation of equations and boundary conditions for single- and double-walled nanotubes undergoing vibrations with nonlocal elastic continuum methods. Reddy and Pang [33] reformulated the equation of motion of the Euler-Bernoulli and Timoshenko beam theories, using the nonlocal differential constitutive relations of Eringen. Following this approach, the equations of motion are used to evaluate the static bending, vibration, and buckling responses of beams with various boundary conditions. Adali [34] proposed a continuum model for studying the mechanical behaviour of multiwalled carbon nanotubes under compressive loads; the nonlocal theory of Euler-Bernoulli beams has been employed and the results are extended to multiwalled nanotubes subjected to transverse vibrations.
In this paper the free vibration frequencies of coaxial DWCNTs are detected, using two different approaches. The first method has already been used by the authors [35] and by Raithel and Franciosi [36] for different structural problems, and it has been properly modified for the title problem. The nanotube is reduced to a set of rigid bars, linked together by elastic cells, where masses and stiffnesses are supposed to be concentrated. The resulting discrete system is simple enough to allow to take into account nonlocal effects, constraint elasticities, and van der Walls forces, and a classical eigenvalue problem is reached, which can be easily handled by Mathematica [37].
The second proposed approach belongs to the semianalytical methods, and more precisely it can be considered an optimized version of the classical Rayleigh quotient, as proposed originally by Schmidt and then employed for various eigenvalue problems [38–40]. Basically, the trial function in the Rayleigh quotient is allowed to depend on parameters, and the resulting quotient is properly optimized. The resulting conditions can be solved numerically.
Numerical examples end the paper, in which the two approaches give lower-upper bounds to the true values, and some comparisons with existing values are offered.
2. Theoretical Approach
The beam structure, under consideration in Figure 1, is a concentric system of two nanotubes of cylindrical shape of length L, with Young’s modulus E and mass density ρ. For each nanotube the cross-sectional area Aj and the moment of inertia Ij are defined, where the index j=1,2 refers to the order of the nanotubes with the inner tube indicated by j=1 and the outer tube by j=2. For the DWCNTs, the main point in the analysis is the consideration of van der Waals (vdW) forces between the inner and outer tubes: the interaction pressure at any point between any two adjacent tubes depends on the difference of their deflections at that point. To take in to account the vdW forces, one defines the interaction coefficient c12 between the inner and outer nanotubes, which can be estimated approximately as [14]
(1)c12=320(2R1)erg/cm20.16a2,witha=1.42×10-9m,
where R1 is the inner radius of the wall and a is the carbon-carbon bond length.
Geometry of double-walled carbon nanotubes (DWCNTs).
As already said, the small-scale effect is taken into account by using the nonlocal theory for Euler-Bernoulli beams, so that the parameter η=e0a is introduced, where e0 is a constant which has to be experimentally determined for each material. In turn, a is an internal characteristic length, as already defined.
In order to analyze the dynamic behaviour of the system under consideration, the governing equations of motion, by considering the vdW forces and the small-scale effect, have been derived using a variational approach:
(2)T=12∑j=12∫0L[ρAj(∂vj∂t)2]dz,(3)E1=∑j=12∫0L[12EIj(∂2vj∂z2)2-η2ρAj(∂2vj∂t2)(∂2vj∂z2)]dz+12kjRLvj′2(z=0)+12kjTLvj2(z=0)+12kjRRvj′2(z=L)+12kjTRvj2(z=L)(∂2vj∂z2)2,(4)E2=12∫0L[c12(v2-v1)2+η2c12(∂v2∂z-∂v1∂z)2]dz,
where kjRL and kjTL, with j=1,2, as already defined above, are rotational and translational stiffness, respectively, at z=0, while, analogously, kjRR and kjTR are rotational and translational stiffness at z=L, respectively. In the above equations, the abscissa z represents the spatial coordinate while t is the time; in (2), (3), and (4) T denotes the kinetic energy, E1 is the strain energy and E2 is the potential energy due to vdW forces between the two nanotubes.
3. Discretization of DWCNTs by means of CDM Method
In this section the so-called “cell discretization method” (CDM), employed to analyze the dynamic behaviour of structure under consideration, is discussed. As already said, the two nanotubes are reduced to a set of t rigid bars with the same length l, linked together by n=t+1 elastic cells (see Figure 2). The moment of inertia Ij and the cross-sectional area Aj with j=1,2 will be evaluated at the cells abscissae, obtaining the concentrated stiffness k1i=EI1i/l, k2i=EI2i/l and the concentrated masses m1i=ρA1il, m2i=ρA2il for the inner and outer tubes, respectively. Both these quantities can be organized into the so-called unassembled stiffness diagonal matrix kj and the unassembled mass diagonal matrix mj, with dimension (n×n), j=1,2, for each of two nanotubes.
Structural system CDM.
In this way, the structure is reduced to a classical holonomic system, with 2n degrees of freedom, in particular, n vertical displacements v1,i, for inner tube, and n vertical displacements v2,i, for outer tube, at the cells abscissae will be conveniently assumed as Lagrangian coordinates and will be organized into the 2n-dimensional vector v. Moreover, for the inner and outer nanotubes the n-1 rotations of the rigid bars can be calculated as a function of the Lagrangian coordinates as follows:
(5)ϕ1,i=v1,i+1-v1,il,ϕ2,i=v2,i+1-v2,il
or, in matrix form: ϕ1=Vv1 and ϕ2=Vv2 where V is a rectangular transfer matrix with n-1 rows and n columns.
The relative rotations between the two faces of the elastic cells are given by
(6)ψj,1=ϕj,1,ψj,i=ϕj,i-ϕj,i-1,ψj,n=-ϕj,n-1
or in matrix form: ψ1=Δϕ1 for inner rigid bar and ψ2=Δϕ2 for outer rigid bar, where Δ is another rectangular transfer matrix with n rows and n-1 columns.
The strain energies, Lje with j=1,2, (the first terms of (3)), are given by
(7)L1e=12∑i=1nk1,iiψ1,i2,L2e=12∑i=1nk2,iiψ2,i2,
and they are concentrated at the cells of the inner and outer tubes, respectively.
The strain energies should be expressed as functions of the Lagrangian coordinates as follows:
(8)L1e=12ψ1Tk1ψ1=12ϕ1TΔTk1Δϕ1=12v1T(VTΔTk1ΔV)v1,L2e=12ψ2Tk2ψ2=12ϕ2TΔTk2Δϕ2=12v2T(VTΔTk2ΔV)v2
so that, the total strain energy can be expressed as
(9)Le=12vT(K100K2)v
with K1=(VTΔTk1ΔV) and K2=(VTΔTk2ΔV). The global assembled stiffness matrix assumes the following form:
(10)K=(K100K2).
The last term of (3), as function of the Lagrangian coordinates, assumes the following form:
(11)P1=12∑i=1nη2ρA1,iiv¨1,iψ1,i,P2=12∑i=1nη2ρA2,iiv¨2,iψ2,i
or:
(12)P1=12η2v¨1T(m1ΔV)v1,P2=12η2v¨2T(m2ΔV)v2.
By assembling the terms of the (12), one gets
(13)Pe=12v¨T(m1nl00m2nl)v,
where m1nl=(η2m1ΔV) and m2nl=(η2m2ΔV). The assembled mass matrix assumes the following form
(14)Mnl=(m1nl00m2nl).
The kinetic energy, (2), can be expressed as the sum of the following form terms:
(15)T1=12v˙1Tm1v˙1,T2=12v˙2Tm2v˙2
or, in assembled form
(16)T=12v˙T(m100m2)v˙.
The global assembled mass matrix is given by
(17)M=(m100m2).
The strain energy due to the vdW forces, (4), can be expressed as
(18)E2=12v1TC1v1+12v2TC1v2-v1TC1v2+12ϕ1TC2ϕ1+12ϕ2TC2ϕ2-ϕ1TC2ϕ2=12v1TC1v1+12v2TC1v2-v1TC1v2+12v1T(VTC2V)v1+12v2T(VTC2V)v2-v1T(VTC2V)v2,
where C2a=(VTC2V), so that (18) becomes
(19)E2=12v1TCtv1+12v2TCtv2-v1TCtv2
with Ct equal to
(20)Ct=C1+C2a.
The terms of matrix C1 are given by
(21)C1,ii=2l3c12,i=3,n-3C1,i+1,i=C1,ii+1=l3c12,i=2,n-2C1,11=l6c12,C1,22=l3c12,C1,12=C1,21=l12c12
and the terms of the matrix C2 assume, instead, the following form:
(22)C2,ii=η2c12l,i=1,n-1.C1 and C2a are two matrices with n rows and n columns and have half-bandwidths equal to 2 and they can be organized into a matrix with 2n rows and 2n columns; so that the matrix C takes the following form:
(23)C=(Ct-Ct-CtCt).
Finally, the strain energy terms of the flexible constraints at the ends are given by
(24)LjTL=12kjTLvj12,LjTR=12kjTRvjn2
with j=1,2, so that the assembled stiffness matrix K must be modified as follows:
(25)K[1,1]=K[1,1]+k1TL,K[n,n]=K[n,n]+k1TR,K[n+1,n+1]=K[n+1,n+1]+k2TL,K[2n,2n]=K[2n,2n]+k2TR.
The rotational stiffness of the constraints of each nanotube can be taken into account by summing up the corresponding flexibilities with the flexibilities of the rigid bars; for example, for the end constraints one gets
(26)k1[1,1]=k1[1,1]k1RLk1RL+k1[1,1],k1[n,n]=k1[n,n]k1RRk1RR+k1[n,n],k2[1,1]=k2[1,1]k2RLk2RL+k2[1,1],k2[n,n]=k2[n,n]k2RRk2RR+k2[n,n].
These terms will be organized in two matrices k1 and k2 furnished in (9).
Finally, the equation of motion can be written as
(27)Mtv¨+Ktv=0,
where Kt is the global assembled stiffness matrix
(28)Kt=K+C
and Mt the global assembled mass matrix
(29)Mt=-Mnl+M.
4. Numerical Comparisons
In order to show the potentialities of the proposed approach (CDM), several numerical examples have been performed, using a general code developed in Mathematica [37], and the obtained results are compared with those of available works in the literature and listed in bibliography. In the present study, the vibration analysis was carried out for DWCNTs with the same and different boundary conditions between inner and outer nanotubes, respectively. Some numerical comparisons have been performed with reference to Adali’s paper [34] in which the fundamental frequencies of clamped-free double-walled nanotubes have been computed by Rayleigh-Ritz (R-R) method.
As first numerical example, the free frequencies of vibration of simply supported-simply supported DWCNTs have been calculated using as approximation function ϕ(z)=sin(πz/L) and with the vertical displacement of the inner and outer nanotubes given by
(30)v1(z)=aϕ(z),v2(z)=bϕ(z)
which satisfy the boundary conditions at the simply supported ends. According to the to the Rayleigh-Ritz method, the following equation is obtained, corresponding to (45) of the paper [34]:
(31)(EI1ρA1L4ξ2ξ+c12ρA1-ω12)(EI2ρA1L4ξ2ξ+c12ρA1-A2A1ω12)-(c12ρA1)=0,
where
(32)ξ0=L-1∫0Lϕ(z)2dz,ξ1=L∫0L(dϕ(z)dz)2dz,ξ2=L3∫0L(d2ϕ(z)dz2)2dz,ξ=ξ0+η02ξ1,η0=ηL
with η=e0a. Substituting (32) into (31), the two fundamental frequencies values are deduced. As first case, setting the interaction coefficient c12 equal to zero, one obtains the first two frequencies values, λi=ωi2(ρA1L4/EI1), for inner and outer nanotubes, respectively. In the second one, putting c12=0.0694 TPa (31) gives the first fundamental frequency for DWCNTs.
The simply supported-simply supported nanotubes, under consideration, have the following mechanical and geometric properties: E=1.2 TPa, ρ=2.3 g/cm^{3}, L=100 nm, R1=0.35 nm, R2=0.69 nm, and t=0.34 nm, where R1 and R2 are the average radii of the inner and outer nanotubes and t is the thickness of the two nanotubes. In Table 1 the numerical comparisons with the results of the proposed method (CDM) and those obtained applying Rayleigh-Ritz method are reported. As it can be seen, the fundamental frequencies show that there is an excellent agreement among the results obtained by the two different numerical procedures.
Numerical comparison among R-R and CDM of simply supported DWCNTs: in columns 3 and 5 the first two dimensionless fundamental frequencies λ1 and λ2, with c12=0, are reported, while in column 4, the first frequency λ, with c12=0.0694 TPa, is listed.
η0
Method
λ1
λ
λ2
0
R-R
9.870
15.759
18.025
CDM
9.870
15.759
18.025
0.1
R-R
9.416
15.035
17.197
CDM
9.416
15.035
17.197
0.2
R-R
8.357
13.344
15.262
CDM
8.357
13.344
15.262
0.3
R-R
7.182
11.468
13.117
CDM
7.182
11.468
13.117
0.4
R-R
6.146
9.813
11.224
CDM
6.146
9.813
11.224
0.5
R-R
5.300
8.463
9.680
CDM
5.300
8.463
9.680
This system has been already solved by Reddy and Pang [33], in the absence of van der Waals interactions, using an exact approach. On the other hand, the van der Waals forces have been taken into account in [34], where an approximate Rayleigh-Ritz method has been adopted, using a single-term trial function.
In order to check the correctness of the numerical calculations of CDM, a numerical comparison with the results given by [34] is proposed applying an optimized version of the classical Rayleigh quotient, as proposed originally by Schmidt and then employed for various eigenvalue problems in [40]. In the Rayleigh-Schmidt (R-S) method, the components of displacement and rotation as suitable analytical approximation functions are assumed, in which one or more unknown parameters are presented. For example, the two-beam displacement approximation functions can be expressed as
(33)v1(z)=ϕ1(z)+a2ϕ2(z)+a3ϕ3(z),v2(z)=ϕ1(z)+b2ϕ2(z)+b3ϕ3(z).
The kinetic and strain energies and the elastic energy associated to the vdW forces can be expressed as
(34)T=12∫0L[ρA1ω2v12(z)+ρA2ω2v22(z)]dz,E1=12∫0L[EI1v2′′2(z)+EI2v2′′2(z)-η2ρA1ω2v1v1′′(z)-η2ρA2ω2v2v2′′(z)]dz,E2=12∫0L[c12(v2(z)-v1(z))2+η2c12(v2′(z)-v1′(z))2]dz.
After same algebra, one can write
(35)ω2=(∫0L[EI1v1′′2(z)+EI2v2′′2(z)]dz+∫0L[c12((v2′(z)-v1′(z))2(v2(z)-v1(z))2+η2(v2′(z)-v1′(z))2)]dz∫0L[EI1v′′12(z)+EI2v′′22(z)])×(∫0L[ρA1v12(z)+ρA2v22(z)]dz+η2∫0L[ρA1v1v1′′(z)+ρA2v2v2′′(z)]dz)-1.
Substituting in appropriate way (33) into (34), one gets
(36)T=12ω2∫0L[ρA1(ϕ1(z)+a2ϕ2(z)+a3ϕ3(z))2+ρA2(ϕ1(z)+b2ϕ2(z)+b3ϕ3(z))2]dz,E1=12∫0L[EI1(ϕ1′′(z)+a2ϕ2′′(z)+a3ϕ3′′(z))2+EI2(ϕ1′′(z)+b2ϕ2′′(z)+b3ϕ3′′(z))2-η2A1ρ(ϕ1(z)+a2ϕ2(z)+a3ϕ3(z))×(ϕ1′′(z)+a2ϕ2′′(z)+a3ϕ3′′(z))-η2A2ρ(ϕ1(z)+b2ϕ2(z)+b3ϕ3(z))×(ϕ1′′(z)+b2ϕ2′′(z)+b3ϕ3′′(z))(ϕ1′′(z)+a2ϕ2′′(z)+a3ϕ3′′(z))2]dz,E2=12∫0L[+a3ϕ3(z)))2c12((ϕ1(z)+b2ϕ2(z)+b3ϕ3(z))-(ϕ1(z)+a2ϕ2(z)+a3ϕ3(z)))2]dz+12∫0L[+a3ϕ3′(z)))2η2c12((ϕ1′(z)+b2ϕ2′(z)+b3ϕ3′(z))-(ϕ1′(z)+a2ϕ2′(z)+a3ϕ3′(z)))2]dz.
Therefore, ω2 depends on the unknown parameters a2, a3, b2, b3 which, in turn, can be obtained by minimizing (35). In fact, the properties of the Rayleigh quotient allow us to obtain the minimizing parameters by putting equal to zero the first derivatives of ω2.
Let us consider a clamped-clamped double-walled nanotubes having the same geometric and mechanical properties of the Example 1. The analysis is carried out applying the Rayleigh-Ritz and Rayleigh-Schmidt methods and assuming the following approximation functions:
(37)ϕ1(z)=1-cos(2πzL),ϕ2(z)=1-cos(4πzL),ϕ3(z)=1-cos(6πzL).
In Table 3, a numerical comparison with the results given by CDM and those obtained by R-R and R-S methods is considered. As shown, the CDM results are nearer to the Rayleigh-Schmidt values than to the Rayleigh-Ritz results.
In Table 4, the fundamental frequencies of a clamped-supported double-walled nanotube are considered, for the cases c12=0 and c12=0.0694 TPa. In the first case the nondimensional frequencies λ1 and λ2 have been calculated and in the second one the value of the first frequency λ has been determined.
Applying Rayleigh-Ritz and Rayleigh-Schmidt methods, the approximation functions assume the following form:
(38)ϕ1(z)=sin(πzL)sin(πz2L),ϕ2(z)=sin(2πzL)sin(3πz2L),ϕ3(z)=sin(3πzL)sin(5πz2L).
As one can see, also this numerical example confirms that the CDM results are nearer to the Rayleigh-Schmidt values than to the Rayleigh-Ritz results.
A further numerical example, of the first free frequency of vibration and for various scaling effect parameter (η0=0,0.5,0.7), is illustrated in Table 5. The different values of η0 have been chosen so that possible comparisons with other references can be deduced. The results are presented for a single-walled nanotube with various boundary conditions at two ends which are of a variety of combinations, namely, simply supported-simply supported (SS-SS), clamped-simply supported (CL-SS), clamped-clamped (CL-CL), and clamped-free (CL-FR). The results given by CDM have been obtained neglecting the vdW forces. The numerical comparison has been done between the values of Tables 1–4, of the papers [24, 30] and the results given by [23]. As one can note, there is an excellent agreement between the obtained results for SS-SS, CL-SS, and CL-CL cases. In the CL-Fr single-walled nanotube case and for scaling effect parameter η0=0.5, the calculations provide values higher than those obtained for η0=0 and this is impossible. The exact values are given by the proposed method (CDM), for scaling effect parameter η0=0.5–0.7, and reported in the last column of Table 5.
In Tables 6 and 7, the free frequencies values for clamped-sliding end (CL-SL) and sliding end-simply supported (SL-SS) double-walled nanotubes, having the same mechanical and geometric properties of the previous examples, are reported and obtained by the CDM.
In the Table 6, the clamped-sliding end (CL-SL) double-walled nanotube case is treated. The two first nondimensional frequencies λi have been obtained for kjTL=kjRL=kjRR=∞ and kjTR=0 with j=1,2. The case of sliding end-simply supported (SL-SS) for kjRL=kjTR=∞ and kjTL=kjRR=0 is reported in Table 7.
All previous numerical examples show that the nondimensional frequencies decrease as the small-scale parameter η0 increases as observed in similar studies on the free vibrations of SWCNTs [23] and DWCNTs (Tables 1, 2, 3, and 4) using the nonlocal theory.
Numerical comparison among [33, 34] and CDM of clamped-free DWCNTs: in columns 3 and 5 the first two dimensionless fundamental frequencies λ1 and λ2, with c12=0, are reported, while in column 4 the first frequency λ, with c12=0.0694 TPa, is listed.
η0
Method
λ1
λ
λ2
0
[34]
3.664
5.850
6.692
[33]
3.516
—
6.422
CDM
3.516
5.614
6.421
0.1
[34]
3.568
5.697
6.517
[33]
3.531
—
6.449
CDM
3.531
5.638
6.448
0.2
[34]
3.320
5.302
6.064
[33]
3.579
—
6.537
CDM
3.570
5.702
6.520
0.3
[34]
3.002
4.793
5.483
[33]
3.669
—
6.700
CDM
3.615
5.772
6.602
Numerical comparison among R-R, R-S, and CDM of clamped-clamped DWCNTs: in columns 3 and 5, the first two dimensionless fundamental frequencies λ1 and λ2, with c12=0, are reported, while in column 4 the first frequency λ, with c12=0.0694 TPa, is listed.
η0
Method
λ1
λ
λ2
0
R-R
22.793
36.394
41.628
R-S
22.410
35.783
—
CDM
22.373
35.722
40.859
0.1
R-R
21.427
34.212
39.132
R-S
21.137
33.750
—
CDM
21.109
33.704
38.551
0.2
R-R
18.449
29.458
33.694
R-S
18.303
29.225
—
CDM
18.289
29.202
33.402
0.3
R-R
15.422
24.625
28.166
R-S
15.359
24.525
—
CDM
15.353
24.515
27.001
0.4
R-R
12.934
20.652
23.622
R-S
12.907
20.609
—
CDM
12.905
20.605
21.139
0.5
R-R
11.005
17.571
20.098
R-S
10.992
17.552
—
CDM
10.991
17.550
17.273
Numerical comparison among R-R, R-S, and CDM of clamped-simply supported DWCNTs: in columns 3 and 5, the first two dimensionless fundamental frequencies λ1 and λ2, with c12=0, are reported, while in column 4, the first frequency λ, with c12=0.0694 TPa, is listed.
η0
Method
λ1
λ
λ2
0
R-R
15.799
25.227
28.855
R-S
15.723
25.105
—
CDM
15.418
24.618
28.158
0.1
R-R
14.901
23.801
27.224
R-S
14.846
23.705
—
CDM
14.600
23.311
26.662
0.2
R-R
12.928
20.642
23.611
R-S
12.893
20.587
—
CDM
12.746
20.351
23.278
0.3
R-R
10.876
17.366
19.863
R-S
10.857
17.336
—
CDM
10.777
17.208
19.682
0.4
R-R
9.162
14.628
16.732
R-S
9.151
14.612
—
CDM
9.106
14.540
16.630
0.5
R-R
7.818
12.483
13.228
R-S
7.881
12.473
—
CDM
7.784
12.428
14.216
Numerical comparison between the results obtained with CDM and [23, 24, 30] of a SWCNT and for different values of parameter η0.
η0
Method
SS-SS
Cl-SS
Cl-Cl
Cl-FR
0
[23]
9.8696
15.4182
22.3733
3.5160
[30]
9.8697
15.4182
22.3733
3.5160
[24]
9.8696
15.4182
22.3733
3.5160
CDM
9.8696
15.4180
22.3728
3.5160
0.5
[23]
5.3003
7.7837
10.9914
4.0882
[30]
5.3001
7.7835
10.9912
4.0881
CDM
5.3002
7.7837
10.9913
3.5874
0.7
[23]
4.0854
5.9362
8.3483
—
[30]
4.0852
5.9362
8.3483
—
CDM
4.0854
5.9362
8.3482
—
Numerical results for clamped-sliding end DWCNTs: in columns 1 and 3, the first two dimensionless fundamental frequencies λ1 and λ2, with c12=0, are reported, while in column 2, the first frequency λ, with c12=0.0694 TPa, is listed.
η0
λ1
λ
λ2
0
5.593
8.931
10.215
0.1
5.509
8.797
10.062
0.2
5.278
8.428
9.640
0.3
4.949
7.902
9.038
0.4
4.575
7.304
8.355
0.5
4.197
6.702
7.666
Numerical results for sliding end-simply supported DWCNTs: in columns 1 and 3, the first two dimensionless fundamental frequencies λ1 and λ2, with c12=0, are reported, while in column 2, the first frequency λ, with c12=0.0694 TPa, is listed.
η0
λ1
λ
λ2
0
2.467
3.940
4.506
0.1
2.438
3.892
4.452
0.2
2.354
3.759
4.299
0.3
2.232
3.564
4.077
0.4
2.090
3.336
3.816
0.5
1.941
3.099
3.545
In Table 8, a numerical comparison is illustrated between the results given by CDM and Rayleigh-Ritz methods and the results given by [27, 28, 32]. The structural system under consideration is a double-walled nanotube having the following mechanical and geometric properties: E=1 TPa, ρ=2.3 g/cm^{3}, L=14 nm, R1=0.35 nm, R2=0.7 nm, t=0.35 nm, and the vdW interaction coefficient is equal to c12=0.0694×1012 TPa.
Numerical comparison between the results obtained with CDM and [27, 28, 32] of a DWCNT and for different values of parameter η0.
Boundary condition
Ω1
Ω2
Ω3
Simply supported
η0=0
[32]
3.099
—
—
[27]
3.14
6.27
9.35
[28]
3.141
6.265
9.276
C.D.M.
3.141
6.265
8.275
R-R
3.141
—
—
η0=0.1
[32]
3.026
—
—
[27]
3.07
5.78
8.01
[28]
3.068
5.770
7.976
C.D.M.
3.068
5.780
8.036
R-R
3.068
—
—
Clamped
η0=0
[32]
4.482
—
—
[27]
4.73
7.82
10.82
[28]
4.726
7.796
10.654
C.D.M.
4.726
7.796
10.653
R-S
4.732
—
—
η0=0.1
[32]
4.359
—
—
[27]
4.59
7.12
9.19
[28]
4.590
7.105
9.123
C.D.M.
4.593
7.137
9.251
R-S
4.596
—
—
Propped
η0=0
[32]
3.802
—
—
[27]
3.93
7.05
10.09
[28]
3.925
7.035
9.981
C.D.M.
3.925
7.035
9.981
R-S
3.965
—
—
η0=0.1
[32]
3.701
—
—
[27]
3.82
6.45
8.60
[28]
3.819
6.444
8.557
C.D.M.
3.820
6.463
8.643
R-S
3.853
—
—
Cantilever
η0=0
[32]
1.88
4.69
7.82
[27]
1.875
4.690
7.797
C.D.M.
1.875
4.690
7.797
η0=0.1
[27]
1.88
4.55
7.13
[28]
1.879
4.544
7.111
C.D.M.
1.879
4.547
7.143
The numerical calculations have been performed for simply supported-simply supported, clamped-clamped, clamped-simply supported, and clamped-free double-walled nanotubes and setting the small-scale parameter equal to η0=0,0.1.
With reference to the paper [27], the free frequencies of double-walled carbon nanotubes (DWCNTs), Ωiωi2(ρATL4/EIT)4, with IT=I1+I2 and AT=A1+A2, are calculated. The structural system is modeled following the nonlocal Euler beams theory and using the Galerkin approach. In [32], the effects of small-scale parameters on the vibrations of DWCNTs, embedded in elastic medium and based on nonlocal Timoshenko theory, are examined in detail. Finally, in [28] the fundamental free frequencies of DWCNTs have been found by means of two analytical approaches in which solving the coupled governing equations of the motion are solved.
The numerical comparisons listed in Table 8 show that there is an excellent agreement between the CDM values and those obtained in [28].
In the following numerical examples the cases of double-walled carbon nanotubes with different boundary conditions between the inner and outer tubes is considered and the obtained results are compared with the values reported in [15]. Moreover, a numerical comparison is given for the case of clamped-clamped nanotubes between the CDM and the Rayleigh-Schmidt values. The nanotubes under consideration have the following mechanical and geometric properties: E=1 TPa, ρ=2.3 g/cm^{3}, L=14 nm, R1=0.35 nm, R2=0.7 nm, and t=0.34 nm, where R1 and R2 are the average radii of the inner and outer nanotubes and t is the thickness of the two nanotubes. The vdW interaction coefficient is equal to c12=71.11 GPa and the nonlocal effects are neglected. In Table 9, the first six free frequencies ωi for different boundary conditions are reported. As one can see, the obtained results with CDM method are lower than those by [15].
First free frequencies of vibrations of DWCNTs with different boundary conditions between inner and outer tubes, (for η=0).
Boundary condition
Method
ω1
ω2
ω3
ω4
ω5
ω6
ω7
Fr-Fr inner
[15]
1.040
2.84
5.140
7.890
8.130
8.380
9.350
Cl-Cl outer
CDM
1.021
2.786
5.760
6.270
6.669
7.915
8.083
Fr-Fr inner
[15]
0.170
1.040
2.890
5.290
6.550
7.890
8.170
Cl-Fr outer
CDM
0.163
1.024
2.833
5.210
6.491
7.860
8.041
SS-SS inner
[15]
1.050
2.840
5.180
7.290
7.890
8.240
9.080
Cl-Cl outer
CDM
1.0219
2.789
5.104
7.212
7.914
8.243
9.010
Cl-Cl inner
[15]
1.080
2.940
5.490
7.900
8.130
8.240
—
Cl-Cl outer
CDM
1.058
2.881
5.394
7.925
8.00
8.255
9.400
R-S
1.061
—
—
—
—
—
—
Recently, Hemmatnezhad and Ansari [31] have furnished a finite element formulation for the free vibration analysis of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory. In their analysis a numerical comparison is offered with the results given by [15] and reported in Table 4 in [31] for the case of DWCNTs so constrained: the simply supported-simply supported inner tube and clamped-clamped outer tube. In Table 10 the previous numerical comparison is proposed introducing the results given by CDM and differential quadrature method (DQM), developed in [36, 41].
Numerical comparison among [15, 31] CDM and DQM and the first three free frequencies of vibrations of DWCNTs are reported (for η=0).
L
Mode
[31]
[15]
CDM
DQM
14
ω1
0.9097
1.05
1.0298
1.0299
ω2
2.3601
2.84
2.7895
2.7896
ω3
4.1481
5.18
5.1043
5.1045
18
ω1
0.5751
0.64
0.6276
0.6276
ω2
1.4648
1.75
1.7202
1.7203
ω3
2.6153
3.36
3.3036
3.3039
24
ω1
0.3340
0.36
0.3550
0.3549
ω2
0.8390
1.00
0.9770
0.9767
ω3
1.5162
1.94
1.9046
1.9046
28
ω1
0.2482
0.27
0.2614
0.2612
ω2
0.6203
0.73
0.7200
7.1905
ω3
1.1261
1.43
1.4074
1.4066
As can be observed the first three free frequencies ωi values given by [15] overestimate the frequencies and the obtained results employing the CDM and DQM methods are lower than those by [15] while they greater than those by [31].
In Table 11, the nonlocal effects influence on the first three free frequencies of vibration are investigated, putting the small-scale parameter equal to η0=0.1,0.2.
Numerical comparison among [31] and CDM and the first three frequencies of vibrations of DWCNTs are reported (for η0=0.1 and η0=0.2).
L
Mode
η0=0.1
η0=0.2
[31]
CDM
[31]
CDM
14
ω1
0.8678
0.997
0.7103
0.866
ω2
1.7807
2.381
1.1764
1.726
ω3
2.7046
4.046
1.7232
2.583
18
ω1
0.5471
0.604
0.4328
0.524
ω2
1.0966
1.459
0.7206
1.045
ω3
1.6772
2.453
1.0936
1.563
24
ω1
0.3171
0.340
0.2446
0.295
ω2
0.6247
0.822
0.4089
0.588
ω3
0.9606
1.382
0.6379
0.879
28
ω1
0.2355
0.250
0.1800
0.2168
ω2
0.4610
0.604
0.3013
0.432
ω3
0.7103
1.016
0.4749
0.647
The numerical calculations relative to a DWCNTs, having various lengths (14 nm, 18 nm, 24 nm, and 28 nm), are performed and the results compared with the CDM method and FEM approach given by [31]. As shown, the results given by CDM method are greater than those by [31].
In the last Tables 12 and 13 the nondimensional free frequencies values are reported for different boundary conditions among inner and outer nanotubes, respectively, and for different values of the small-scale parameter η0. The numerical calculations have been performed using the geometrical and physical data given by [34]. In particular the first nondimensional frequency value, λ1=ω12(ρA1L4/EI1), is reported. As shown, the fundamental frequencies decrease with increasing values of the small-scale parameter η0.
Fundamental frequency of vibration λ1 of DWCNTs with different boundary conditions between inner and outer tubes and for η0[0.1–0.5].
η0
SS-SS innerCl-SS outer
SS-SS innerCl-Cl outer
SS-SS innerCl-Fr outer
SS-SS innerCl-Sl outer
0
24.567
35.558
24.564
35.408
0.1
23.304
33.683
23.303
33.684
0.2
20.346
29.187
20.346
29.188
0.3
17.204
24.504
17.204
24.504
0.4
14.537
20.597
14.537
20.597
0.5
12.426
17.543
12.426
17.543
Fundamental frequency of vibration λ1 of DWCNTs with different boundary conditions between inner and outer tubes and for η0[0.1–0.5].
η0
Cl-Fr innerCl-SS outer
Cl-Fr innerCl-Cl outer
Cl-Fr innerSS-Sl outer
Cl-Fr innerCl-Sl outer
0
24.618
35.577
8.231
8.916
0.1
23.311
33.694
8.676
8.796
0.2
20.351
29.195
8.322
8.427
0.3
17.208
24.510
7.809
7.901
0.4
14.540
20.601
7.225
7.304
0.5
12.428
17.547
6.634
6.702
5. Conclusions
Coaxial DWCNTs are modeled as two beams interacting between then through van der Waals forces, and nonlocal Euler-Bernoulli beam theory is employed in order to calculate the free vibration frequencies of the system. Two different numerical approaches are used in order to perform numerical comparisons. In the first method, the system, under consideration, has been modeled as a set of rigid bars linked together by elastic cells, where masses and stiffnesses are supposed to be concentrated. The resulting finite degree of freedom has allowed taking into account nonlocal effects, constraint elasticities, and van der Walls forces. The second proposed approach belongs to the semianalytical methods, and more precisely it can be considered an optimized version of the classical Rayleigh quotient, as proposed originally by Schmidt and then employed for various eigenvalue problems.
Several numerical examples have been treated in detail, comparing numerical and approximate results from the literature, and the proposed approaches have furnished excellent results.
More particularly, emphasis has been given to the influence of the small-scale parameter, of the length of the nanotubes and of the various boundary conditions on the free vibration frequency behaviour.
In the author’s opinion, the first method will be particularly useful—because of its intrinsic simplicity—in the future analysis of mass nanosensor and of nanotube in the presence of soil and follower forces.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
IijimaS.Helical microtubules of graphitic carbonKrishnanA.DujardinE.EbbesenT. W.YianilosP. N.TreacyM. M. J.Young's modulus of single-walled nanotubesDemczykB. G.WangY. M.CumingsJ.HetmanM.HanW.ZettlA.RitchieR. O.Direct mechanical measurement of the tensile strength and elastic modulus of multiwalled carbon nanotubesRuoffR. S.QianD.LiuW. K.Mechanical properties of carbon nanotubes: theoretical predictions and experimental measurementsTreacyM. M. J.EbbesenT. W.GibsonJ. M.Exceptionally high Young's modulus observed for individual carbon nanotubesTersoffJ.RuoffR. S.Structural properties of a carbon-nanotube crystalNardelliM. B.YakobsonB. I.BernholcJ.Brittle and ductile behavior in carbon nanotubesRuC. Q.Column buckling of multiwalled carbon nanotubes with interlayer radial displacementsWangQ.HuT.ChenQ. J.Bending instability characteristics of double walled nanotubesZhangY. Y.WangC. M.TanV. B. C.Buckling of multiwalled carbon nanotubes using Timoshenko beam theoryGibsonR. F.AyorindeE. O.WenY.-F.Vibrations of carbon nanotubes and their composites: a reviewAnsariR.AjoriS.ArashB.Vibrations of single- and double-walled carbon nanotubes with layerwise boundary conditions: a molecular dynamics studyYoonJ.RuC. J.MioduchowskiA.Non-coaxial resonance of an isolated multiwall carbon nanotubesYoonJ.RuC. Q.MioduchowskiA.Vibration of an embedded multiwall carbon nanotubeXuK.-Y.AifantisE. C.YanY.-H.Vibrations of double-walled carbon nanotubes with different boundary conditions between inner and outer tubesXuK. Y.GuoX. N.RuC. Q.Vibration of a double-walled carbon nanotube aroused by nonlinear intertube van der Waals forcesElishakoffI.PentarasD.Fundamental natural frequencies of double-walled carbon nanotubesNatsukiT.NiQ.-Q.EndoM.Analysis of the vibration characteristics of double-walled carbon nanotubesSunC. T.ZhangH.Size-dependent elastic moduli of platelike nanomaterialsEringenA. C.On differential equations of nonlocal elasticity and solutions of screw dislocation and surface wavesReddyJ. N.Nonlocal theories for bending, buckling and vibration of beamsPeddiesonJ.BuchananG. R.McNittR. P.Application of nonlocal continuum models to nanotechnologyGhannadpourS. A. M.MohammadiB.FazilatiJ.Bending buckling and vibration problems of nonlocal Euler beams using Ritz methodPradhaS. C.PhadikarJ. K.Bending, buckling and vibration analyses of nonhomogeneous nanotubes using GDQ and nonlocal elasticity theoryWangQ.VaradanV. K.Vibration of carbon nanotubes studied using nonlocal continuum mechanicsAnsariR.SahmaniS.Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam modelsShakouriA.LinR. M.NgT. Y.Free flexural vibration studies of double-walled carbon nanotubes with different boundary conditions and modeled as nonlocal Euler beams via the Galerkin methodEhteshamiH.HajabasiM. A.Analytical approaches for vibration analysis of multi-walled carbon nanotubes modeled as multiple nonlocal Euler beamsAnsariR.RouhiH.Analytical treatment of the free vibration of single-walled carbon nanotubes based on the nonlocal Flugge shell theoryWangC. M.ZhangY. Y.HeX. Q.Vibration of nonlocal Timoshenko beamsHemmatnezhadM.AnsariR.Finite element formulation for the free vibration analysis of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theoryKeL. L.XiangY.YangJ.KitipornchaiS.Nonlinear free vibration of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theoryReddyJ. N.PangS. D.Nonlocal continuum theories of beams for the analysis of carbon nanotubesAdaliS.Variational principles for transversely vibrating multiwalled carbon nanotubes based on nonlocal euler-bernoulli beam modelDe RosaM. A.LippielloM.Natural vibration frequencies of tapered beamsRaithelA.FranciosiC.Dynamic analysis of arches using Lagrangian approachWolframS.BertC. W.Application of a version of the Rayleigh technique to problems of bars, beams, columns, membranes, and platesLauraP. A. A.Optimization of variational methodsDe RosaM. A.FranciosiC.The optimized Rayleigh method and mathematica in vibrations and buckling problemsDe RosaM. A.LippielloM.Non-classical boundary conditions and DQM for double-beams