1. Introduction
Carbon nanotubes have been researched heavily for a variety of unique uses. Today, it is used in structural stiffness, creating vantablack, electronic devices, boosting solar energy, etc. Due to its excellent mechanical properties, it is used for reinforcing composites and many researchers have been paying attention to understand the behavior of CNTreinforced composites. Ahmed et al. (2013) dealt with the static and dynamic analysis of composite laminated plate (Graphite/Epoxy composite plate). The static analysis is related to the maximum deflection while the dynamic analysis is related to the natural frequency of the plate. Sahu and Dutta (2002) performed the dynamic stability of curved panels with cutouts. Mallikarjuna (1988) studied the dynamics of laminated composite plates with a higherorder theory and finite element discretization. Analysis of laminated CNTreinforced plates, in which the CNT distribution is UD, FGV, FGO, and FGX, using the elementfree kpRitz method is presented by Lei et al. (2016).
Most of the studies performed are based on twophase composites. As CNT is still expensive, it is not practical to use it in large quantities. Therefore, a third phase, carbon fiber or eglass fiber, can be added to the mix for the production cost efficiency of the composite. In the present work, the multiscale formulation is applied, which includes the HalpinTsai model as well as micromechanical approaches (Lee, 2018), for determining the effective material properties of the threephase composite (CNTFPC). The study focuses on dynamic responses of CNTFPC shell for different CNT weight ratios, layup angles, curvatures of the shell and central cutout sizes and the interactions of the responses are studied.
2. Multiscale formulation
In the multiscale formulation, initially, the matrix and CNT are combined forming the twophase composite called carbonnanotube reinforced composite (CNTRC). The threephase composite (CNTFPC) is formed by combining CNTRC with carbonfiber or eglass fiber (Han et al., 2007;Zuo et al., 2013). For the elastic properties of CNTFPC, the HalpinTsai equation and micromechanical approaches (Lee, 2018) are used. The Young’s modulus of CNTRC using HalpinTsai equation can be calculated by:
where,
in which, E^{cnr} , E^{re} and ${E}_{11}^{cnt}$ represent Young’s modulus of CNTRC, resin matrix and CNT, and l^{cnt}, d^{cnt} and t^{cnt} denote the length, diameter and thickness of CNT respectively. The longitudinal tensile modulus of CNTFPC is calculated using the following formula:
Using the semiempirical relation from the Halpin Tsai model, E_{22}, G_{12}, ν_{12}, can be calculated as:
where, Φ, Φ_{cnr} and Φ_{f} correspond to E_{22}, G_{12} and ν_{12} denoting the modulus of CNTFPC, E^{cnr} , G^{cnr} and ν^{cnr} denoting the modulus of CNTRC and E^{f}, G^{f} and ν^{f} denoting the modulus of fiber respectively. For transverse loading, the reinforcing efficiency factor (χ) can be seen in Eq. (4) and Eq. (5) which depends on the cross section of the fiber and geometry of packing. Generally, the value of χ is between 1 and 2. In this study, as the fibers are circular and arranged in square array, the value of χ is taken as 2. Using Eq. (4) and Eq. (5), E_{22} can be expressed as:
If the volume fraction of the fiber (V^{f}) is greater than 0.5, the estimated value of G_{12} is lower than the actual value. In this case, we use the equation which is derived experimentally by Hewitt and Malherbe (1970) for χ :
3. Numerical Simulation
In finite element analysis, when the model is damaged due to loading, the stiffness of the damaged model is mostly represented using a reduced stiffness matrix. The reduced stiffness matrix is a product of the sum of all the stiffness matrices of the elements and the stiffness reduction factor κ^{(n)} (Au et al., 2003;Mares and Surace, 1996):
where, ${\tilde{C}}^{\left(n\right)}$ and C^{(n)} denote the stiffnesses in damaged and undamaged state. The element stiffness matrix in damaged state represented using local coordinates is as follows:
where, B denotes the displacement differentiation matrix of element e. General equation of motion of a system subjected to dynamic loading is given by:
where, $\ddot{u}$, $\dot{u}$ and u denote the acceleration, velocity and displacement, m, c and k denote the mass, viscous damping and stiffness of the system and p(t) is the dynamic load. In our case, the damping can be removed. Newmark further extended the equation of motion in which the total time is divided into equal time intervals Δt (Bathe, 1996). The time function f(t) at time t=nΔt is denoted by:
The displacement solution for time t=(n+1)Δt of the damaged structure is given by:
where, $\widehat{f}$ represents the load at time t=(n+1)Δt, $\widehat{K}$ is triangularized system stiffness matrix and λ_{i}(i=0, 2, 3) are the time integration constants at time t. The solution of velocity and acceleration can also be found using Newmark’s Method.
4. Verification
The model is a square plate consisting of three layers (0˚/90˚/0˚) with all the sides clamped subjected to a dynamic load of q_{o}=10^{5} N/m^{2}. The properties of the laminated plate: a=b=0.25m, h=5cm, ρ=800 kg/m^{3}, E_{2}= 21×10^{9} N/m^{2}, E_{1}/E_{2}=25, ν=0.25 and G_{12}=G_{13}=G_{23}= 0.5E_{2}. The modeling was done in the Abaqus program. The graph, shown in Fig. 1, is a comparison with Ref. Reddy, (1983), Mallikarjuna, (1988) and Zhang and Xiao, (2017) which shows sufficient accuracy of the procedure followed in the Abaqus program.
5. Numerical examples
In the parametric studies, the dynamic characteristics are studied between different SWCNT weight ratios, curvatures, layup angles, and central cutout sizes. The geometric properties of the model are: a=b=1m, thickness=10mm. The boundary condition is shown in Fig. 2. The model varies from flat to different radii: 0.8m, 0.55m, 0.4m and 0.32m. The properties of the individual materials used in the composites are listed in Table 1 and the resulting material properties of CNTFPC for different SWCNT weight ratios using the multiscale formulation are listed in Table 2. Poisson’s ratio is 0.233. The loading point and the displacement measurement point are shown in Fig. 3 for CNT weight ratio, curvature, and layup angle cases, while for the central cutout case, it is shown in Fig. 8 and the loads applied are 100,000N and 1.0N respectively.
In Fig. 4, we considered a twolayered (0˚/90˚) composite shell with radius 0.8m having SWCNT weight ratios increasing till 8% from none. It is seen that the deflection decreases with the increase in the SWCNT weight ratio. This shows that there is an increase in stiffness with the increase in the SWCNT weight ratio. The decrease in the deflection is significantly higher when the weight ratio is increased till 2% and less significant as the weight ratio increases further. For this reason, as CNT is expensive, the addition of CNT more than 2% is clearly not beneficial.
Fig. 5 shows the comparison of twolayered (0˚/90˚) composite shell of the SWCNT weight ratio of 1% with different curvature radii. It is evident that the deflection decreases with the increase in curvature. This is due to the increment of membrane force with the increase of curvature. This concludes that there is increase in stiffness with more curvature resulting in less deflection.
Besides, we also studied the influence of different layup angles on the shell with the same thickness. Fig. 6 is the plot of a cylindrical shell of radius 0.8m and SWCNT weight ratio 0% with different layup angles: [0˚/90˚], [0˚/90˚/90˚/0˚], [0˚/90˚/0˚/90˚], [45˚/45˚], [45˚ /45˚/45˚/45˚] and [45˚/45˚/45˚/45˚]. Comparing six different layup angles, [0˚/90˚/90˚/0˚] shows the least deflection concluding that the layup angle is the stiffest among the six layup angles.
The dynamic responses of the composite plate with 8% SWCNT weight ratio for different central cutout sizes were also examined. The mass of the shell is reduced by 1%, 4%, 16% and 36% as shown in Fig. 7. The comparison graph, in Fig. 9, shows that the deflection of 36% cutout is maximum and of 16% is also significant. However, the deflections for the lower sized cutout are almost the same as the composite plate with no cutout. As the deflection is related to the stiffness as well as the mass of the shell which changes with the cutout size, there are less significant changes till 4% cutout and dramatic changes with bigger cutout sizes. We can see that for the cutout size equal to or higher than 16%, the dynamic response changes drastically which means that it is not desirable to use such sizes with the parameters used in the study.
6. Conclusion
The results of the study can be summarized as follows:

(1) The CNT reinforcement in the laminated composite shell structure increases the stiffness.

(2) As the curvature increases, the membrane force increases resulting in lower deflection.

(3) Using less than 2% CNT would be beneficial as no dramatic changes occur in deflection when more CNT is added.

(4) 0˚/90˚/90˚/0˚ layup angle is the stiffest among the six different layup angles used in the study.

(5) In this case of our study, cutout size higher than or equal to 16% is not desirable.
In this study, we focused on the linear dynamic loading of the composite shell but various other stresses also influence the structures from the practical viewpoint. The structures are agitated by nonlinear dynamic loads such as earthquake loads and wind loads. Besides, due to the temperature difference, thermal stresses also occur on the structures. For this reason, nonlinear dynamic analysis and heat transfer analysis should be considered in future work.