# Free Vibrations and Flutter Analysis of Composite Plates Reinforced with Carbon Nanotubes
## 2 Materials and Methods  
### Micromechanical Modeling  
In nanosystem modeling, molecular dynamics simulations are the dominant methodology [29; 30; 31]. However, in the analysis of mechanical properties, the global response of the nanocomposite is expected, and, thus, micromechanical modeling appears to be an appropriate approach [32]. In micromechanical modeling, the effective (average) macroscopic behavior is based on the micromechanical structure, material properties, and volume fraction of constituents. For this study, reinforcement was assumed to be in the form of single-walled carbon nanotubes uniformly distributed in the epoxy matrix. Figure 1 schematically depicts the concept of a nanocomposite plate with uniformly distributed carbon nanotubes and is sufficient for this paper. However, for a better understanding of this topic, the reader can study the physical microscopic images of nanotubes, such as scanning electron microscopy (SEM) in 2D and X-ray computed tomography (XCT) in 3D. Examples of such images can be found in [33; 34; 35]. The uniform dispersion and alignment of carbon nanotubes in a polymer matrix are essential to the production of nanocomposites with improved mechanical properties [25; 26; 27; 28]. As shown by the experimental results of [PERSON] et al. [11], nanocomposites with highly aligned CNTs can be expected to exhibit the properties of typical long-fiber composites. The CNT/nanocomposite with unidirectional CNTs is an orthotropic body having nine independent material constants in the constitutive equations. For aligned and uniformly distributed CNTs in the matrix, a transversely isotropic body is assumed, which reduces the number of independent constants. The effective thickness of the CNTs is a crucial issue in estimating the material properties of CNTs. The influence of CNT thickness on the material properties is discussed, for example, in papers [36; 37; 38]. According to the molecular structural mechanics approach, a CNT is defined as a space frame. The covalent C-C bonds are represented as connecting beams, and the carbon atoms are connecting nodes. Tensile strength, bending stiffness, and torsional stiffness are calculated using the energy equivalence between local potential energies in computational chemistry and strain energies in structural mechanics. For example, based on the force field for benzene and assuming a C-C bond length of 0.142 nm, the cross-sectional diameter and Young's modulus of the beam element are 0.147 nm and  
Figure 1: The concept of the micromechanical modeling of the equivalent nanocomposite with aligned carbon nanotubes. 5.49 TPa, respectively [39]. However, many force field constants are used in the literature to analyze CNTs, resulting in different macroscopic material properties of the nanostructure. As a result, the scatter in the wall thickness of the CNTs is presented in the range of 0.066-0.34 nm [40]. In general, the mechanical properties of CNTs tend to increase with decreasing thickness. CNTs with a small diameter of less than 1 nm and a thickness of 0.34 nm are assumed in this paper. The material data of the nanocomposite's constituents are presented in Table 1. The current study assumes common material data for CNTs [36] and epoxy [12]. The effective material properties of CNT/nanocomposites are calculated based on [PERSON]'s model. [PERSON] [41] proposed an exact elasticity solution for a periodic system of fibers embedded in an isotropic matrix. [PERSON]'s elasticity approach incorporates the Hashin-Rosen and [PERSON] models for a transversely isotropic body.