In this thesis, we focus on evaluating the mechanical properties of single-walled carbon nanotube (SWCNT) based mainly on the stress-strain relationship. Therefore, we used the coupled molecular-structural approach to study the nonlinear variation of the stress-strain relationship for different configurations. Then, we come up with compact formulas in order to predict the nonlinear stress-strain relationship.
Application of CNTs
Characterization of CNTs
In this study, the focus is on predicting the elasticity of CNTs using existing theories. The elasticity of CNTs is paramount in establishing mathematical modeling of mechanical applications at the nanoscale.
Important parameters of CNTs
Geometry and Configuration of CNTs
Classification of CNTs
There are two models that describe the structures of multiwalled nanotubes. a) Russian doll model - sheets of graphite are arranged in concentric cylinders, e.g.
Inter-atomic potentials used in CNTs
Important additional pair energies include repulsive potentials, van der Waals energies, interactions involving polar and polarizability of molecules, interactions involving hydrogen bonding, and intermolecular strong energies which include covalent and coulombic interactions. Abell gave an expression for the chemical bond energy Eb over nearest neighbors as. For a range of interatomic distance ranging from 0.05 nm to 2.5 nm, the comparison of different potential models as evaluated in MATLAB is shown in Figure (3).
Literature Survey on analysis of mechanical properties of CNTs
Various approaches available for studies on CNTs
Molecular Mechanics Approach
CNT size effects on Elasticity
Based on the molecular mechanics approach, considering a rod-helix model, a closed-form solution is found to derive the analytical expression for the elasticity of carbon nanotubes, based on chirality. We utilized the work done by Tienchong et al, to verify the effects of chirality and elasticity on the frequency of CNTs. This gives insight into the essence of finding elastic properties, which are very important in applications in the field of sensor technology.
Using this expression, the chirality dependence and the magnitude of elasticity of CNTs were plotted. To calculate the modulus of elasticity, the thickness of CNTs is considered as 0.08 nm in this case. At higher diameters (more than 1 nm) of CNTs, the elasticity value is almost the same for both configurations.
When these elasticity values for different configurations are used to calculate the resonance frequency of the model given in the motivation section, it shows the importance of the prediction of elasticity at low diameters. Therefore, the prediction of the size, chirality dependence of the CNTs is more important in sensor applications in particular.
Stress-Strain Relationship for Armchair CNT
From Estretch, the stretch force can be calculated as. 13) Using the Eangle term, the moment function interms of Δθ. For the armchair configuration, bond lengths are a, b, b and bond angles are α, β, β as shown in fig 8(b). Considering, “t” as the thickness of CNT, the axial stress on the obtained unit cell is defined as,.
In this work, we tried to achieve a closed-form solution that can provide accurate results compared to the original work. Neglecting the term (Δα)4 in the above expression as the change in bond angle is much smaller compared to the bond stretch,. As can be seen from the graph, the stress-strain relationship is linear up to a strain of 0.08 and then nonlinearity begins.
It can also be noted that the behavior of the armchair configuration is almost the same for almost all diameters, and therefore the elastic properties are insensitive to the size of the armchair CNTs. However, there will be a slight variation in the elastic properties at higher loads and is slightly dependent on the diameter, especially in the non-linear zone. The present work is compared with the work of Xiao et al and the results are very satisfactory.
The error in the stress–strain relationship for both models increases with increasing chirality index of the armchair CNTs, but the percentage error, as shown in Figure 15 , is almost negligible.
Stress-Strain Relationship for zigzag CNT
Neglecting the (Δα)4 term in the above expression, Δα can be derived as,. 40) Using Δα in the stress and strain expression, we get For small changes in bond stresses, the stress and strain behavior of the unit cell is calculated using the above analytical expressions in MATLAB over different diameters of the zigzag configuration (fig 11). Unlike the armchair configuration, the behavior of zigzag CNTs depends on their diameters and this effect is more in the nonlinear region at higher strains.
Also, the linear relationship only exists up to a strain of 0.06, and then the nonlinear effects are more dominant. Finally, it is evident that zigzag CNTs are sensitive to their chirality index number. Compared with the results of the existing work of Xiao et al , the results are very similar and the effect of neglecting the fourth-order term of the bond angle change is zero.
Beam Element Modeling of CNT
For the classical structural mechanics approach, the strain energies for a beam structure are - Under pure axial force. Therefore, a direct relationship between the parameters of the molecular mechanics and structural mechanics can be established as. The final values for the beam element to be used in FEM procedure are evaluated as d=.
Modeling tools and approach for ANSYS analysis
ANSYS has been widely used for static and modal analysis of armchair, zigzag CNT and Graphene configurations. In ANSYS, two-node beam element 188 is used for CNT analysis, which has 6 degrees of freedom at each node. Care must be taken that the initial slope of the nonlinear stress data is equal to the initial value of the input elasticity.
From the modified Morse potential, the force versus bond stress effect is introduced into the beam element by calculating the stress-strain relationship for the beam dimensions. Using this force function, the stress developed in the beam element is calculated against the bond stresses. The dimensions required for calculating the stress and strain in the beam element are already available from the previous section.
This stress-strain relationship is used to define the nonlinearity of the beam element in the ANSYS package. The density of the beam element is needed for the modal analysis of the CNT/Graphene structures in ANSYS. Since a ray element contains two carbon atoms at its nodes, the total mass of the ray element is two-thirds of a carbon atom.
With the previously calculated beam dimensions, the effective density of the beam element is found to be 5,501x103 Kg m-3.
Scaling Factors and details of CNT/Graphene Models on ANSYS
Since the CNT/Graphene structure is a honeycomb structure, each carbon atom is divided between three beam elements. The effective mass contribution of a carbon atom to each radius is one third of its mass (the mass of a carbon atom is 19.9x10-27 Kg). Information on dimensions, number of nodes, number of elements and maximum applied displacement for different configurations of CNTs and Graphene are listed in the table below.
Comparison of the ANSYS simulation results
On the other hand, the zigzag configuration (fig.14) is relatively sensitive to the size of the CNT. The cross-sectional effects on the behavior of the zigzag CNTs are more at larger strain values. The linear region of the zigzag configuration can be observed up to the 0.06 strain limit and then it quickly becomes 0.08 strain value.
By comparing the stress-strain behavior of the two CNT configurations and the corresponding graphene configurations, the rolling effect of graphene in CNTs can be analyzed. However, there is no significant effect of rolling on the zigzag configuration of CNTs. The stress values for a given strain for the ANSYS model are smaller compared to the rod spiral model because the analysis is based on the rod spiral model on a single unit and the results are not sufficiently capable of capturing the behavior of the entire structure.
Because the ANSYS model uses a FEM background, the effects of neighboring atoms and bonds are nicely simulated. Figure 18 shows the onset of the decrease in instantaneous elasticity resulting from the stress–strain behavior of the armchair and zigzag configuration. Also, the initial modulus of elasticity of 0.8 TPa for CNTs is in agreement with most of the available literature.
The force vs displacement for each of the configurations of CNTs and graphene is plotted in Fig.19.
Continuum Models for Modal Analysis
The modal frequencies obtained from the ANSYS simulations compared with the theoretical natural frequencies calculated from the above formulas are listed in Tables 3 and 4. The results match satisfactorily with the continuum models at higher L/D ratios for all the configurations in both fixed fixed and cantilever end conditions.
Bending Modes and Fundamental Frequencies
The first fundamental frequency for buckling for different end conditions, over different L/D ratios for the armchair and zigzag configuration of CNTs are shown below. 24, for each configuration at different end conditions, the fundamental frequencies increase with decreasing L/D ratios.
Different modes of vibrations
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