Double-Walled Carbon Nanotubes
R. Ansari, Rajabiehfard and B. Arash
2. Governing Equations for CNTs
2.1. Shell Equations Based on Nonlocal Elasticity
The concept of non-locality is inherent in solid-state physics where the nonlocal attractions of atoms are prevalent. According to Eringen [54, 55], unlike the conventional local elasticity, in the nonlocal continuum theory, it is assumed that the stress at a point is a function of strains at all points in the continuum. To bring the non-locality into formulation, we employ the nonlocal constitutive equation given by Eringen [55]
(1)
where is the nonlocal parameter or characteristic length; t is the macroscopic stress tensor at a point. In the limit when the characteristic length goes to zero, nonlocal elasticity reduces to classical (local) elasticity. The stress tensor is related to strain by generalized Hooke‘s law as
(2)
here S is the fourth order elasticity tensor and ―:‖ denotes the double dot product. Hooke‘s law for the stress and strain relation is thus expressed by
(3)
Based on the FOSD theory, the three-dimensional displacement components , and in the , and directions, respectively, as shown in Fig. 1, are assumed to be
(4) where are the reference surface displacements and are the rotations of transverse normal about the -axis and -axis, respectively.
The middle surface strains and , the shear strains and and the middle surface curvatures and are given by
(5)
Figure 1. A double-walled CNT embedded in an elastic medium.
In the nonlocal elastic shell theory, the stress and moment resultants are defined based on the stress components in Eq. (3) and thus can be expressed as follows by referencing the kinematic relations in coupled Donnell theory
(6-1)
(6-2)
(6-3)
(6-4)
(6-5)
(6-6)
(6-7)
(6-8) In the multiple-shell model used herein, each tube of DWCNTs is described as an individual cylindrical shell of radius , length , thickness , as in Fig. 1. If and denote the longitudinal and circumferential coordinates, respectively, the governing equations on the basis of the Donnell shell theory are given as
(7) is the pressure exerted on the tube through the vdW interaction forces and/or the interaction between the tube and the surrounding elastic medium. To confine the effect of the surrounding medium on the outermost layers in the calculation of , the Kronecker delta is introduced as
(8) The vdW model employed captures the effects of the interlayer vdW interactions of all layers in a MWNT and accounts for the curvature dependence of the vdW interactions, which is proposed by [56]
(9) in which the vdW coefficients representing the pressure increment contributing to layer i from layer j are given by
(10) where is the C-C bond length, the depth of the potential, a parameter that is determined by the equilibrium distance and is the radius of jth layer and denotes the elliptic integral defined as
(11) here m is an integer and the coefficient is given by
(12)
The surrounding elastic medium interacts with the outermost layer of the DWCNT under consideration. To include this interaction, the Winkler foundation model is employed as
(13) where is the Winkler foundation modulus, which depends on the material properties of the elastic medium.
2.2. Field Equations
For the th tube of a DWCNT, by the use of Eqs. (5), the Eqs. (6) can be stated in terms of the five field variables
(14)
2.3. Solution Procedure
For DWCNT, the displacement field components are assumed to be functions of circumferential wave number, n, and the axial wave number, m. It is a simple task to indicate that for simply supported boundary conditions, the field equations admit solutions of the form
(15) After introducing Eq. (15) into Eqs (14), these equations can be written in the matrix form
(16)
where
where
,
where the operators are given in the Appendix, and
where
,
where the operators are given in the Appendix, and ,
3. Results and Discussion
The geometries and the mechanical properties of each layer of CNTs are
, , , .
Figure 2. Cross-sectional view of a double-walled CNT under the vdW interactions.
Figure 3. Variation of natural frequency with circumferential wave number for a double-walled CNT
with simply supported end conditions ( ).
Presented graphically in Fig. 3 is the natural frequency of a double-walled CNT versus circumferential mode number for several values of the small length scale ranging from
(corresponding to the classical/ local continuum model) to . One can observe from this Figure that the lowest natural frequency decreases as the small length scale increases. It physically means that the small-scale effects in the nonlocal model make nanotubes more flexible. It is further observed that the magnitude of decrease in natural frequencies corresponding to higher circumferential modes is considerably higher than those corresponding to lower ones.
Figure 4. Variation of natural frequencies of a double-walled CNT over a wide range of its aspect ratio ( ).
The variation of natural frequencies of a double-walled CNT over a wide range of its aspect ratio for various nonlocal parameters is plotted in Fig. 4. The values of nonlocal parameter are assumed to be varied from to . The profound effects of the small length scale on the natural frequencies of the CNT are seen from Fig. 4, especially for shorter CNTs and higher values of nonlocal parameter. As the ratio of length- to-innermost radius increases, natural frequencies tend to decrease, and the effects of small length scale diminish so that the frequency envelopes tend to converge. This observation means that the classical continuum model would give a reasonable prediction in the study of nanotubes of high aspect ratios for which the whole structure can be homogenized into a continuum.
Figure 5. Effect of the small length scale on the natural frequencies ratio for double-walled CNTs with various length-to-innermost radius ( ).
To further investigate the influence of the small length scale on the natural frequencies of nanotubes, the ratio of nonlocal frequency to local frequency will be discussed later on. The frequency ratios corresponding to various length-to-innermost radius ratios for a double- walled CNT are graphed in Fig. 5. It is observed that the effects of the small length scale are more prominent for shorter length CNTs as the name implies. In the prediction of natural frequencies of a CNT of via a classical continuum model, for instance, a relative error of 26.5% for the nonlocal parameter is introduced. This relative error reduces to about 9% when the aspect ratio of the CNT is increased by .
Figure 6. Effect of the surrounding medium on the natural frequencies of a double-walled
CNT ( ).
The local and nonlocal natural frequencies over a broad range of for a triple-walled CNT with and without being embedded in the surrounding medium are indicated in Figure 6.
As seen from this Figure, the effect of the surrounding elastic medium on natural frequencies of CNTs is negligible for . However, the frequency difference due to this effect becomes more pronounced when the ratio of length-to-innermost radius increases until it reaches a maximum value corresponding to infinitely long CNTs. It is also seen that the effect of small length scale on the natural frequency becomes smaller as the aspect ratio of nanotube increases, regardless of the medium effect.
Figure 7. Effect of the small length scale on the frequency ratios corresponding to various foundation moduli for a double-walled CNT with all edges simply-supported ( ).
Fig. 7 is presented to further examine the role of the surrounding medium in the small size dependence of the natural frequencies of a double-walled simply supported CNT. It is observed that the relative error deceases when the modulus of the elastic medium becomes larger. Physically speaking, in the study of vibration characteristics of CNTs without being embedded in the surrounding medium, more care must be taken over the dependence of the natural frequencies on the small length scale. For the given nonlocal parameter , the relative error in predicting the frequencies varies from about 17.5% to less than 1%, which correspond to and , respectively.
Conclusion
On the basis of the theory of nonlocal continuum mechanics, the free vibration characteristics of double-walled carbon nanotubes for a simply supported boundary condition were studied. The equations of motion of a double-walled carbon nanotube were derived
based on the coupled Donnell theory and the Eringen nonlocal elasticity. The following findings are summarized:
• The significance of the small size effects on the natural frequencies of double-walled carbon nanotubes is shown to be dependent on the geometric sizes of CNT and the elastic surrounding medium.
• The small-scale effects in the nonlocal continuum model make small-size CNTs more flexible. In other words, the classical continuum model tends to overestimate the natural frequencies of small size nanotubes, and one must recourse to the nonlocal version to reduce the relative error. As the small-scale parameter increases, the frequencies obtained for the nonlocal shell become smaller than those for its local counterpart.
• The natural frequencies corresponding to higher vibration modes are more sensitive to the small length scale.
• The existence of the elastic medium significantly enhances the values of natural frequencies particularly for lower ones.
• The small size effect becomes more pronounced when the modulus of the elastic medium becomes smaller.