1. INTRODUCTION
Combination of two or more materials with different properties resulting in a material with unique characteristics from the individual materials is a composite material. Basically, composite material is the combination of fiber/filament reinforcement and matrix, for examples, concrete, reinforced cement concrete, glass fiber reinforced plastic, carbon fiber reinforced polymer.
Carbon nanotubes is rolled chicken wirelike structure of carbon atoms. The length is about 0.25 μm and diameter about 12nm. CNTs have the highest tensile strength and tensile modulus among the materials discovered till date (Lee, 2018). There are two types of CNT: Singlewalled CNT (SWCNT) and Multiwalled CNT (MWCNT). SWCNT is a special type of carbon material made up of a single sheet of graphene rolled up and has a shape like a cylinder (Du et al., 2007;Kim et al., 2009). Cost of SWCNT is between 100700 USD per gram. It is used in this study. MWCNT, on the other hand, is made up of singlewalled carbon nanotubes arranged in concentric layers. It is stiffer and cheaper than SWCNT.
Uniform dispersion of CNTs in the matrix is one of difficulties that is faced while working with composites involving CNT. The CNTs get entangled during the production phase and the van der Waals force between the CNTs attract each other resulting in aggregation. There should also be effective stress transfer from a matrix to a reinforcement. Achieving suitable CNT matrix interfacial bonding for this transfer is another critical challenge (Lee, 2018). These challenges mentioned above hold back the performance of the resulting composites and the expected properties are not achieved.
Dynamic characteristics of CNT reinforced laminated shell structures depend on the CNT weight ratios and the shape of the shell. Thus, this study focuses on the interaction between CNT weight ratios, curvatures and central cutout sizes in free vibrations of SWCNT reinforced laminated composite cylindrical shells.
2. MULTISCALE FORMULATION
In this study, we combined epoxy resin and CNT first forming carbon nanotube reinforced composites (CNTRC). Later, CNTRC is combined with Eglass fibers forming carbon nanotubes/fiber/polymer composites (CNTFPC) (Han and Elliott, 2007;Zuo et al., 2013). The elastic properties of CNTRC and CNTFPC are calculated using the modified HalpinTsai equation and micromechanical approaches (Lee, 2018). From HalpinTsai equation, the Young's modulus of CNTRC:
where,
where, E^{cep}, E^{cp} and ${E}_{11}^{cn}$ are the Young’s moduli of CNTRC, epoxy resin and CNT, and l^{cn}, d^{cn} and t^{cn} are the length, diameter and thickness of the CNT respectively.
The longitudinal Young’s modulus of CNTFPC can be calculated by
E_{22}, G_{12}, ν_{12}, using HalpinTsai model, can be determined as
where, Φ, Φ_{cep}, and Φ_{f} denote material modulus of composites, corresponding matrix modulus, and corresponding fiber modulus, respectively. χ in Eq.(4) and Eq.(5) is called the reinforcing factor and it depends on the fiber geometry and packing geometry. The value of χ lies between 1.0 and 2.0. For circular fibers of square array, as in this study, the value of χ is 2. From Eq.(4) and Eq.(5), for E_{22}, the equations can be written as:
In the case of shear modulus (G_{12}), if the volume fraction of fiber (V^{f}) in the CNTFPC is more than 0.5, the Halpin Tsai equation gives the result lower than the actual. So, in this case, the equation from Hewitt and Malherbe (1970) is used to calculate χ :
This equation is derived from the experimental results.
Table 1 shows properties and definition of the materials used in this study.
3. NUMERICAL EXAMPLES
The study has been carried out for the fundamental frequencies of vibration. The modeling of different cases of the composite shell structures for the frequencies was done in the ABAQUS program. As shown in Table 2, natural frequencies are computed from the program and the results are compared with those of previous studies. The table shows sufficient accuracy of the procedure followed in the ABAQUS program.
The effect of different shapes, layup angles, sizes of cutout, CNT weight ratios were considered for the study. Square shell structure with sides 1.0m and thickness 10mm of different shapes with radius of 0.8m, 0.55m, 0.4m, 0.32m including flat are shown in Fig. 1. Fig. 2 shows shells with cutout reducing the mass by 1%, 4%, 16% and 36%. Layup angles of [0/90], [0/90/90/0], [0/90/0/90], [45/45], [45/45/45/45] and [45/45/ 45/45] are used for the numerical analysis.
Table 3 shows the material properties of CNTFPC for different CNT weight ratios calculated using the multiscale analysis discussed earlier. Fig. 3 shows the variation of fundamental frequency due to different curvatures. The flat model has the least frequency and as the curvature increases, the frequency also increases with more curvature. The figure also shows weight ratio. For the radius of 0.32m, the percentage that increas with it. This happens because the membrane force of frequency with the increase in CNT increase in frequencies from 0% through 8% CNT weight ratio is 6.38%, 2.95%, 1.82% 1.28%, 0.95%, 0.81%, 0.69% and 0.57% respectively. The increase in frequency is high at first but decreases with the increase in CNT weight ratio. As CNT is very expensive, adding more CNT is clearly not beneficial. Addition of less than 2% CNT would be better considering the costing. Effect of different layup angles on the frequency in the same thickness of composite is shown in Fig. 4. Comparing six different layup angles, [0/90/90/0] gives the highest frequency.
Cutouts are structural requirements in every structure. It may be provided to reduce weight, access the interior or to lay the lines for fuel or electricity. There are changes in the frequencies with the introduction of cutout as shown in Table 4. From the fundamental formula for the natural frequency, both the stiffness and mass change simultaneously when cutout is introduced to the structure which may result in increase or decrease of the frequency (Sahu and Datta, 2002;Rao and Krishnan, 1999). In this case, the frequency decreases with the increase in the size of the cutout. For 4.0% reduction cutout in the structure, the frequency with no CNT is 9.5924Hz which is less than the one without cutout. Simply, with the addition of 1% CNT, we get 10.195 Hz which means the frequency loss due to cutout is recovered (Lee and Park, 2009). Similarly, for 16% cutout, 2.0% CNT can be added to get 9.7978Hz. This shows that CNT reinforcement can be used to recover the frequency loss from cutout of laminated composite plates.
Fig. 5 and 6 are the mode shapes with and without SWCNT addition. There is no significant change in the mode shapes. But there are changes with the change in cutout size.
4. SUMMARY AND CONCLUSIONS
The result of the study can be summarized as follows:

1) CNT reinforcement in laminated composite shell structure increases the fundamental frequency.

2) As the curvature increases, the membrane force increases with it resulting in higher frequency.

3) CNT is expensive and use of less than 2% CNT would be better. [0/90/90/0] layup angle gives the highest frequency among the six different layup cases.

4) CNT reinforcement can be used to recover the frequency loss from cutout of laminated composite plates or shells.

5) There is negligible change in the mode shape due to the addition of CNT but significant change is seen when the cutout size changes.