*2.5. Finite Element Modelling of the Composite Plate*

In the present study, the finite element method is used for conducting free vibration analysis of the composite plates with graphene inclusion. For performing the vibration analysis, an appropriate mesh must be developed in the problem domain. Considering that the composite plate has a side-to-thickness ratio higher than 10, shell elements are an effective option for meshing the model. Here, 4-node, quadrilateral, stress/displacement shell elements with reduced integration and a large-strain formulation are used [37].

According to the ASTM D7264 standard, *w* × *l* × *t*, with *w*, *l* > 10 *t* rectangular composite plates are modelled, where *w* is the width, *l* is the length, and *t* is the thickness of the plate. The plates consist of 8 unidirectional layers. The thickness and the stacking sequence of all plates under consideration are 2 mm and [0/+45/−45/90]*s*, respectively. The model has been tested for its convergence in terms of mesh density and a mesh with a size of 0.5 mm for each element is finally chosen.

The hybrid composite plates of these characteristics, as well as different carbon nanostructure inclusion types and volume fractions, have been successfully validated [34] for bending loading conditions. Since the method is validated, it can be expanded for the solution of the free vibration problem. The elemental matrices and displacements are written in the global coordinate system by applying the appropriate transformation matrices. After generating the global stiffness matrix (*K*) and the global mass matrix (*M*), assembled from the elemental matrices using conventional finite element procedures and considering the undamped free vibrations of the composite plate, the equation of motion can be assembled as ..

$$\mathbf{M}d + \mathbf{K}d = 0\tag{17}$$

where *d* is the assembled displacement vector. By applying the boundary conditions on the composite plate, the eigenvalue problem can be solved using common finite element procedures, which reveal the natural frequencies and corresponding mode shapes of vibration.

#### **3. Results and Discussion**

In this work, polyester resin, graphene, and carbon fiber were selected as matrix material, nano inclusion and reinforcement, respectively. Physical properties of the components constituting the hybrid composite material are presented in Table 2. These properties are the basis for the estimation of the lamina physical properties presented in the next section.


**Table 2.** Physical properties of components of the composite material.
