Torsional buckling analysis of MWCNTs considering quantum effects of fine scaling based on DFT and molecular mechanics method
Graphical abstract
Introduction
In the past two decades, carbon nanotubes (CNTs) [1] have evoked a great deal of interest due to their superior physical and chemical properties over other materials known to man. In terms of mechanical properties, CNTs have proved to be amongst the lightest, stiffest and strongest materials yet measured with high elastic modulus of greater than 1 TPa comparable to that of diamond and strengths many times higher than the strongest steel at a fraction of the weight. The buckling and bending analysis of CNTs have been extensively studied by means of experimental methods and molecular dynamics (MD) simulations [[2], [3], [4], [5], [6]]. Nevertheless, conducting controlled experiments at the nanometer dimension is expensive and has some limitations. In addition, MD simulations are very time consuming and the computational time increases as the number of atoms considered for simulation increases. The other way for modeling structures at the scale of nanometers is on the basis of continuum mechanics. Many researchers applied classical and modified continuum models in the analysis of nanostructures due to the computational efficiency as well as reasonable accuracy [[7], [8], [9], [10], [11]]. Kitipornchai et al. [12] offered a continuum model for the vibrations of multi-layered graphene sheets with simply-supported boundary conditions. Wang et al. [8] developed nonlocal elastic beam and shell models to study the elastic buckling of CNTs. Using continuum mechanics, Yao and Han [11] studied the axial buckling of MWCNTs under temperature field. Ansari et al. [13] investigated the vibrations of embedded multi-layered graphene sheets with different boundary conditions based upon nonlocal continuum mechanics. Although the continuum models have generated valuable results for comprehension the behavior of carbon nanotubes, they have at least two inadequacies in their analyses. First problem is the uncertainty that exist in defining nanotube wall thickness in the literature. Numerous researchers used the value of spacing of graphite (0.34 nm) for the wall thickness of CNTs in their analyses. However, Yakobson et al. [4] proposed the effective thickness of carbon nanotubes equal to 0.066 nm through comparison of the molecular dynamics simulation results with those of continuum model. Some other values of wall thickness were also offered for the continuum models [14,15]. Another defect of continuum models is concerned with the discrete nature of CNTs. In other words, continuum mechanics has not the potential to incorporate the effect of chirality into account. An alternative way for analyzing nanostructures is based upon the molecular mechanics, which is computationally efficient and can capture the chirality and size independence of CNTs. A review in the literature reveals that molecular mechanics models have been widely used to study the behavior and properties of CNTs. Based upon a molecular mechanics model, Chang and Gao [16] derived the closed-form expressions for elastic modulus and Poisson’s ratio of single-walled carbon naotubes (SWCNTs). Chang et al. [17] studied the elastic axial buckling of armchair and zigzag SWCNTs using molecular mechanics approach. They indicated that zigzag tubes are more stable than armchair tubes with the same diameter. Meo and Rossi used a finite element approach, based on the molecular mechanics theory on the use of non-linear and torsional springs for SWCNTs [18]. The torsional buckling of CNTs is of great technical importance to many research workers [[19], [20], [21], [22], [23], [24]]. Of these stu.dies, some have been conducted on the basis of the continuum models [[19], [20], [21], [22]] and some others on the basis of the atomistic models [23,24].
Dynamic instability of viscoelastic porous functionally graded (FG) nanobeam embedded on visco-Pasternak medium subjected to an axially oscillating loading as well as magnetic field was studied by Jalaeia and Civalek [25]. Their results revealed that with increasing power-law index and structural damping, the pulsation frequency decreases and so, instability region shifts to left side while as the magnetic field magnifies, the dynamic instability moves to right side. Also, it is represented that the porosity effect on the dynamic stability of FG nanobeam depends significantly on the values of power-low index and magnetic field. Free vibration analysis of nano-sized annular sector plate has been analyzed by Gurses et al. [26] using the nonlocal continuum theory. It was seen that the size effects are significant in the vibration analysis of nano-scaled annular sector plates and need to be included in the mechanical model. The longitudinal free vibration problem of a micro-scaled bar has been formulated by Akgöz and Civalek [27] using the strain gradient elasticity theory. It was observed that size effect is more significant when the ratio of the microbar diameter to the additional length scale parameter is small. It was also observed that the difference between natural frequencies predicted by non-classical and classical models becomes more prominent for both lower values of slenderness ratio of the microbar and for higher modes. Free vibration behavior of carbon nanotube-reinforced composite (CNTRC) microbeams has been investigated by Civalek et al. [28] and it was observed that the largest frequencies occur in X-Beam while O-Beam has the lowest ones. It is also found that the size effect is more prominent when the thickness of the beam is close to the length scale parameter and this effect nearly disappears as the thickness of the beam increases.
The fine scale is referred to the very small dimensions of the structure, in which each specific structure begins to expand to form its larger dimensions. The formation of the structures in very small scale, which are referred to as fine scale, have different properties and mechanical behavior from their expanded state, and with increasing the size and dimensions, these changes decrease and according to the type of material and its size and dimensions, the process of these changes is different. By increasing the length and dimensions of the structure, properties and mechanical behavior of the material, it approaches its stable state. Since structures with very small lengths have many applications in nanoscale, to increase the accuracy of calculations and avoid errors, the properties and mechanical behavior of the structure related to the specified size of the structure should be used. The quantum effects of finite scaling in very small sizes are so important that in some cases changing the size of the structure causes larger changes in the properties and mechanical behavior of one material from another. In other words, in some cases, instead of using one material, with higher or lower resistance, the same material with different dimensional sizes can be used. Since the effect of fine scaling on the buckling strain has never been investigated before for the MWCNTs, In this paper, quantum and molecular mechanics are used to study the quantum effects of fine scaling on the buckling strength of multi-walled carbon nanotubes under torsional loadings, as well as the effects of changes in length, diameter, chirality, wall number and length to diameter ratio of the structure. Density functional theory (DFT) along with the generalized gradient approximation (GGA) function is implemented to obtain the relevant elastic constants of the nanotubes. In the quantum mechanics method, the longitudinal range of the effect of quantum variables of fine scaling on the structural properties was determined. To study this range, specific lengths with different chiralities are considered and the coefficients needed for the molecular mechanics method have been obtained using the combination of quantum mechanics and molecular mechanics methods. Then the critical strain of multi-walled nanotubes with different lengths and chiralities has been obtained using molecular mechanics.
Section snippets
Potential energy
Based on the molecular mechanics modeling approach, the total potential energy, , can be expressed as the sum of the several energies due to the valence of bonded interactions or bonded and non-bonded interactionsin which , , , and are the energies relevant to the bond stretching, bond angle variation, bond inversion, and bond torsion, respectively. Moreover, and represent the energies of van der Waals and electrostatic interactions, respectively [[29]
Torsional buckling of chiral single-walled carbon nanotubes
The chirality of the structure of a single-walled carbon nanotube is often signified with a pair of integers (n,m). The following geometrical parameters can be considered for) n,mchiral ( carbon nanotubes before deformation aswhere
Results and discussion
In this section, according to what is shown in Fig. 3, Fig. 4, and Table 3, the structural properties and buckling behavior of four multi-walled nanotubes with specific diameters and lengths are compared. In this comparison, four models of multi-wall nanotubes have been considered, two nanotubes with armchair structure, and the other two nanotubes with zigzag structure. First, the characteristics of the nanosheets from which the nanotubes are emitted will be described. Armchair nanotubes are
Validation
Table 12 shows the data summery for graphene and Table 13 shows the ones obtained for the CNTs. Our results show a very good agreement with the ones presented in these table and the difference of our results with these results is due to the quantum effects of fine-scaling. In these tables, B shows the bulk modulus; σc, buckling stress; εc, buckling strain; σTS, ultimate tensile strength; σf, failure stress; εf, failure strain; E, Young’s modulus; G, shear modulus; ν, Poisson’s ratio; J, polar
Conclusion
In this paper, the quantum effects of fine scaling on the buckling strength of multi-walled carbon nanotubes with different chiralities have been studied through density functional theory and molecular mechanics method. First, using quantum mechanics, the effective range of quantum effects of fine scaling of about 20 Å was identified and then for nanosheets with specific widths within this range and zigzag and armchair chiralities the mechanical properties have been obtained by quantum
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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