*3.1. E*ff*ect of Graphene on Composite Lamina Physical Properties*

Given the physical properties of the components, the physical properties of graphene-added unidirectional carbon fiber-reinforced lamina are evaluated theoretically, based on the proposed method, and the outcome is provided in Table 3. Note that *E*1, *E*2, *G*12, *G*23, as well as ρ*<sup>c</sup>* increase as the volume fraction of graphene increases. Specifically, there is an 34.6%, 542%, 533%, and 686% increase in *E*1, *E*2, *G*12, and *G*<sup>23</sup> with the addition of 50 vol.% graphene, respectively. Simultaneously, there is only 13.6% increase in the mass density of the lamina due to the presence of graphene since it is denser than polyester resin.

#### *3.2. Vibrational Characteristics of the Composite Plate*

A numerical calculation of the natural frequencies and mode shapes of vibration can be carried out for a composite plate, according to the procedure described in the previous section for different boundary conditions. Here, we assume two types of supports, (a) the longitudinal and transversal edges of the rectangular plate are clamped, CCCC (clamped—clamped—clamped—clamped), and (b) the longitudinal and transversal edges of the plate are simply supported SSSS (simple—simple—simple—simple). Figure 2 depicts several basic bending mode shapes of vibration

modes; however, it affects their sequence.

of the rectangular plate. The general nomenclature for the mode shapes, here, is (*a*,*b*), where *a* and *b* correspond to the number of ridges or valleys existing in the mode at longitudinal and transverse direction of the plate. Note that Figure 2 depicts the transverse modes of the CCCC case; however, the general shape (*a*,*b*) exists in the SSSS case as well. The difference in the shape is mainly on the boundaries, where the rotation of the degrees of freedom are free and fixed in SSSS and CCCC boundary condition, respectively. The structure of the first three (basic) modes of vibration depends on the plate geometric and stiffness characteristics. In general, the presence of graphene into the composite seems not to influence the shape of the vibration modes; however, it affects their sequence. *Materials* **2020**, *13*, x FOR PEER REVIEW 7 of 15 *3.2. Vibrational Characteristics of the Composite Plate* 


**Table 3.** Physical properties of components of the composite material. A numerical calculation of the natural frequencies and mode shapes of vibration can be carried

**Figure 2.** Mode shapes of vibration of a composite plate with *w* = 25 mm, *l* = 64 mm, and CCCC boundary conditions. **Figure 2.** Mode shapes of vibration of a composite plate with *w* = 25 mm, *l* = 64 mm, and CCCC boundary conditions.

Figure 3 depicts the natural frequency, *f*0, of the composite plate without the presence of graphene considering various configurations. Specifically, the width remains constant *w* = 25 mm, while the

Figure 3 depicts the natural frequency, *f* <sup>0</sup>, of the composite plate without the presence of graphene considering various configurations. Specifically, the width remains constant *w* = 25 mm, while the length is *l* = 25, 30, 35, 45, 85, 145, 245 mm. Figure 3a illustrates the natural frequency variation concerning CCCC boundary conditions. Figure 3b describes the natural frequency variation regarding SSSS boundary conditions. The natural frequencies of the CCCC case are greater than the corresponding ones of SSSS, as expected. *Materials* **2020**, *13*, x FOR PEER REVIEW 8 of 15 SSSS boundary conditions. The natural frequencies of the CCCC case are greater than the corresponding ones of SSSS, as expected. *Materials* **2020**, *13*, x FOR PEER REVIEW 8 of 15 SSSS boundary conditions. The natural frequencies of the CCCC case are greater than the corresponding ones of SSSS, as expected.

**Figure 3.** Natural frequency of the composite plate without the presence of graphene considering various configurations for (**a**) CCCC, and (**b**) SSSS boundary conditions. **Figure 3.** Natural frequency of the composite plate without the presence of graphene considering various configurations for (**a**) CCCC, and (**b**) SSSS boundary conditions. **Figure 3.** Natural frequency of the composite plate without the presence of graphene considering various configurations for (**a**) CCCC, and (**b**) SSSS boundary conditions.

#### *3.3. Effect of Graphene on Natural Frequancies Assuming under CCCC Boundary Conditions 3.3. E*ff*ect of Graphene on Natural Frequancies Assuming under CCCC Boundary Conditions 3.3. Effect of Graphene on Natural Frequancies Assuming under CCCC Boundary Conditions*

In order to study the effects of graphene inclusion on vibration behavior of laminated composite plates under CCCC boundary conditions, various values of volume fraction of graphene are studied. Specifically, eleven different cases are considered, i.e., *Vgr* = 0, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, and 0.50. Figure 4 illustrates the effect of graphene volume fraction on (1,1) (Figure 4a), (1,2) (Figure 4b), (2,1) (Figure 4c), and (2,2) (Figure 4d) transverse vibrations. As observed, the higher the volume fraction of graphene, the higher the frequency of graphene. This observation is true for all the different transverse modes. Specifically, there is an up to 73.2%, 82.7%, 72.6%, and 82.4% increase in (1,1), (1,2), (2,1), and (2,2) mode shape, respectively, with the addition of 50 vol.% graphene. In order to study the effects of graphene inclusion on vibration behavior of laminated composite plates under CCCC boundary conditions, various values of volume fraction of graphene are studied. Specifically, eleven different cases are considered, i.e., *Vgr* = 0, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, and 0.50. Figure 4 illustrates the effect of graphene volume fraction on (1,1) (Figure 4a), (1,2) (Figure 4b), (2,1) (Figure 4c), and (2,2) (Figure 4d) transverse vibrations. As observed, the higher the volume fraction of graphene, the higher the frequency of graphene. This observation is true for all the different transverse modes. Specifically, there is an up to 73.2%, 82.7%, 72.6%, and 82.4% increase in (1,1), (1,2), (2,1), and (2,2) mode shape, respectively, with the addition of 50 vol.% graphene. In order to study the effects of graphene inclusion on vibration behavior of laminated composite plates under CCCC boundary conditions, various values of volume fraction of graphene are studied. Specifically, eleven different cases are considered, i.e., *Vgr* = 0, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, and 0.50. Figure 4 illustrates the effect of graphene volume fraction on (1,1) (Figure 4a), (1,2) (Figure 4b), (2,1) (Figure 4c), and (2,2) (Figure 4d) transverse vibrations. As observed, the higher the volume fraction of graphene, the higher the frequency of graphene. This observation is true for all the different transverse modes. Specifically, there is an up to 73.2%, 82.7%, 72.6%, and 82.4% increase in (1,1), (1,2), (2,1), and (2,2) mode shape, respectively, with the addition of 50 vol.% graphene.

**Figure 4.** *Cont*.

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*Materials* **2020**, *13*, x FOR PEER REVIEW 9 of 15

**Figure 4.** Normalized frequency versus volume fraction of graphene considering (**a**) (1,1), (**b**) (1,2), (**c**) (2,1), and (**d**) (2,2) vibration mode for the CCCC boundary condition. **Figure 4.** Normalized frequency versus volume fraction of graphene considering (**a**) (1,1), (**b**) (1,2), (**c**) (2,1), and (**d**) (2,2) vibration mode for the CCCC boundary condition. **Figure 4.** Normalized frequency versus volume fraction of graphene considering (**a**) (1,1), (**b**) (1,2), (**c**) (2,1), and (**d**) (2,2) vibration mode for the CCCC boundary condition.

Figure 5 shows the effect of graphene volume fraction on (1,3) (Figure 5a), (3,1) (Figure 5b), (2,3) (Figure 5c), (3,2) (Figure 5d), and (3,3) (Figure 5e) transverse vibrations. As described previously, the higher the volume fraction of graphene, the higher the natural frequency of graphene. This observation is true for all different transverse modes. Specifically, there is an up to 91.5%, 71.5%, 91.4%, 82% and 90.6% increase in (1,3), (3,1), (2,3), (3,2), and (3,3), respectively, with the addition of 50 vol.% graphene. It is noted that the graphene increases slightly more the normalized frequency of higher order modes than basic ones. Figure 5 shows the effect of graphene volume fraction on (1,3) (Figure 5a), (3,1) (Figure 5b), (2,3) (Figure 5c), (3,2) (Figure 5d), and (3,3) (Figure 5e) transverse vibrations. As described previously, the higher the volume fraction of graphene, the higher the natural frequency of graphene. This observation is true for all different transverse modes. Specifically, there is an up to 91.5%, 71.5%, 91.4%, 82% and 90.6% increase in (1,3), (3,1), (2,3), (3,2), and (3,3), respectively, with the addition of 50 vol.% graphene. It is noted that the graphene increases slightly more the normalized frequency of higher order modes than basic ones. Figure 5 shows the effect of graphene volume fraction on (1,3) (Figure 5a), (3,1) (Figure 5b), (2,3) (Figure 5c), (3,2) (Figure 5d), and (3,3) (Figure 5e) transverse vibrations. As described previously, the higher the volume fraction of graphene, the higher the natural frequency of graphene. This observation is true for all different transverse modes. Specifically, there is an up to 91.5%, 71.5%, 91.4%, 82% and 90.6% increase in (1,3), (3,1), (2,3), (3,2), and (3,3), respectively, with the addition of 50 vol.% graphene. It is noted that the graphene increases slightly more the normalized frequency of higher order modes than basic ones.

**Figure 5.** *Cont*.

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**Figure 5.** Normalized frequency versus volume fraction of graphene considering (**a**) (1,3), (**b**) (3,1), (**c**) (2,3), (**d**) (3,2), and (**e**) (3,3) vibration mode for CCCC boundary condition. **Figure 5.** Normalized frequency versus volume fraction of graphene considering (**a**) (1,3), (**b**) (3,1), (**c**) (2,3), (**d**) (3,2), and (**e**) (3,3) vibration mode for CCCC boundary condition.

#### *3.4. Effect of Graphene on Natural Frequancies Assuming SSSS Boundary Conditions 3.4. E*ff*ect of Graphene on Natural Frequancies Assuming SSSS Boundary Conditions*

In order to study the effects of graphene inclusion on vibration behavior of laminated composite plates under SSSS boundary conditions, various values of volume fraction of graphene are considered. In order to study the effects of graphene inclusion on vibration behavior of laminated composite plates under SSSS boundary conditions, various values of volume fraction of graphene are considered.

Once again, 11 different cases are considered, i.e., *Vgr* = 0, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, and 0.50. Figure 6 illustrates the effect of graphene volume fraction on (1,1) (Figure 6a), (1,2) (Figure 6b), (2,1) (Figure 6c), and (2,2) (Figure 6d) transverse vibrations. As observed, the higher the volume fraction of graphene, the higher the frequency of graphene for all different transverse modes. Specifically, there is an up to 64.9%, 60.9%, 63.4%, and 70.1% increase in (1,1), (1,2), (2,1), and (2,2) mode shape, respectively, with the addition of 50 vol.% graphene. Once again, 11 different cases are considered, i.e., *Vgr* = 0, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, and 0.50. Figure 6 illustrates the effect of graphene volume fraction on (1,1) (Figure 6a), (1,2) (Figure 6b), (2,1) (Figure 6c), and (2,2) (Figure 6d) transverse vibrations. As observed, the higher the volume fraction of graphene, the higher the frequency of graphene for all different transverse modes. Specifically, there is an up to 64.9%, 60.9%, 63.4%, and 70.1% increase in (1,1), (1,2), (2,1), and (2,2) mode shape, respectively, with the addition of 50 vol.% graphene.

Figure 7 shows the effect of graphene volume fraction on (1,3) (Figure 7a), (3,1) (Figure 7b), (2,3) (Figure 7c), (3,2) (Figure 7d), and (3,3) (Figure 7e) transverse vibrations. As previously, the higher the volume fraction of graphene, the higher the frequency of graphene. This observation is true for all the different transverse modes. Specifically, there is an 79.3%, 60.9%, 79.2%, 69.1% and 80% increase in (1,3), (3,1), (2,3), (3,2), and (3,3), respectively, with the addition of 50 vol.% graphene. As in SSSS case, the graphene seems to increase slightly more the normalized frequency of higher order modes than basic ones. Figure 7 shows the effect of graphene volume fraction on (1,3) (Figure 7a), (3,1) (Figure 7b), (2,3) (Figure 7c), (3,2) (Figure 7d), and (3,3) (Figure 7e) transverse vibrations. As previously, the higher the volume fraction of graphene, the higher the frequency of graphene. This observation is true for all the different transverse modes. Specifically, there is an 79.3%, 60.9%, 79.2%, 69.1% and 80% increase in (1,3), (3,1), (2,3), (3,2), and (3,3), respectively, with the addition of 50 vol.% graphene. As in SSSS case, the graphene seems to increase slightly more the normalized frequency of higher order modes than basic ones.

*Materials* **2020**, *13*, x FOR PEER REVIEW 11 of 15

*Materials* **2020**, *13*, x FOR PEER REVIEW 11 of 15

**Figure 6.** Normalized frequency versus volume fraction of graphene considering (**a**) (1,1), (**b**) (1,2), (**c**) (2,1), and (**d**) (2,2) vibration mode for SSSS boundary condition. **Figure 6.** Normalized frequency versus volume fraction of graphene considering (**a**) (1,1), (**b**) (1,2), (**c**) (2,1), and (**d**) (2,2) vibration mode for SSSS boundary condition. **Figure 6.** Normalized frequency versus volume fraction of graphene considering (**a**) (1,1), (**b**) (1,2), (**c**) (2,1), and (**d**) (2,2) vibration mode for SSSS boundary condition.

**Figure 7.** *Cont*.

*Materials* **2020**, *13*, x FOR PEER REVIEW 12 of 15

**Figure 7.** Normalized frequency versus volume fraction of graphene considering (**a**) (1,3), (**b**) (3,1), (**c**) (2,3), (**d**) (3,2), and (**e**) (3,3) vibration mode for the SSSS boundary condition. **Figure 7.** Normalized frequency versus volume fraction of graphene considering (**a**) (1,3), (**b**) (3,1), (**c**) (2,3), (**d**) (3,2), and (**e**) (3,3) vibration mode for the SSSS boundary condition.

In the previous cases (SSSS and CCCC), the behavior of the normalized frequency of various vibration mode versus the graphene volume fraction has been depicted concerning different L/w ratio. It is observed that the slope of this curve changes with different L/w ratio, and the trend is also different for the different vibration mode. The effect is more obvious in lower order than higher order modes. This phenomenon may be explained by the variation of the stiffness and mass (density) characteristics of the structure due to the presence of graphene that differs from each other (Table 3). Nevertheless, as is well known, mass and stiffness ratio directly affect the vibration frequency. In the previous cases (SSSS and CCCC), the behavior of the normalized frequency of various vibration mode versus the graphene volume fraction has been depicted concerning different L/w ratio. It is observed that the slope of this curve changes with different L/w ratio, and the trend is also different for the different vibration mode. The effect is more obvious in lower order than higher order modes. This phenomenon may be explained by the variation of the stiffness and mass (density) characteristics of the structure due to the presence of graphene that differs from each other (Table 3). Nevertheless, as is well known, mass and stiffness ratio directly affect the vibration frequency.

#### *3.5. Average Effect of Graphene on Natural Frequencies 3.5. Average E*ff*ect of Graphene on Natural Frequencies*

In order to estimate the average effect of graphene on the natural frequencies of the composite plate, the mean value of the normalized frequency is calculated for every graphene volume fraction considering all modes shape of vibration. Figure 8 depicts the average effect of graphene concerning both CCCC and SSSS boundary conditions. It is obvious that the effect of graphene is greater in CCCC than SSSS boundary conditions. For a rough estimation of the effect for *Vgr* greater than zero, a linear function can be used, i.e., In order to estimate the average effect of graphene on the natural frequencies of the composite plate, the mean value of the normalized frequency is calculated for every graphene volume fraction considering all modes shape of vibration. Figure 8 depicts the average effect of graphene concerning both CCCC and SSSS boundary conditions. It is obvious that the effect of graphene is greater in CCCC than SSSS boundary conditions. For a rough estimation of the effect for *Vgr* greater than zero, a linear function can be used, i.e.,

$$\frac{f}{f\_0} = \varepsilon V\_{\mathcal{S}^r} + d \tag{18}$$

݂ ݂ where *c* is the slope, and *d* is the intercept. In the CCCC case, *c* = 1.4164 and *d* = 1.0745 with R<sup>2</sup> = 0.9848. In the SSSS case, *c* = 1.1797 and *d* = 1.0533 with R<sup>2</sup> = 0.9885. where *c* is the slope, and *d* is the intercept. In the CCCC case, *c* = 1.4164 and *d* = 1.0745 with R2 = 0.9848. In the SSSS case, *c* = 1.1797 and *d* = 1.0533 with R2 = 0.9885.

*Materials* **2020**, *13*, x FOR PEER REVIEW 13 of 15

**Figure 8.** Average effect of graphene on natural frequency of the composite plate of graphene for CCCC, and SSSS boundary conditions. **Figure 8.** Average effect of graphene on natural frequency of the composite plate of graphene for CCCC, and SSSS boundary conditions.

#### **4. Conclusions 4. Conclusions**

A multi-scale procedure has been applied in the investigation of the vibration behavior of laminated composite plates reinforced by carbon fibers and graphene. According to analysis results, the following important general conclusions have been drawn: A multi-scale procedure has been applied in the investigation of the vibration behavior of laminated composite plates reinforced by carbon fibers and graphene. According to analysis results, the following important general conclusions have been drawn:


The dispersion of graphene particles in a polymer matrix is a big challenge, and there exists a practical restriction on the addition of graphene above a certain level (maximum of 10 wt.%), owing to the development of agglomerates [39]. Here, we considered a uniform distribution of the graphene in the polymer matrix concerning low- and high-volume fractions. Therefore, the present approach may be used as a design tool for low-volume fractions and be able to reveal the optimum vibrational performance for higher ones. The dispersion of graphene particles in a polymer matrix is a big challenge, and there exists a practical restriction on the addition of graphene above a certain level (maximum of 10 wt.%), owing to the development of agglomerates [39]. Here, we considered a uniform distribution of the graphene in the polymer matrix concerning low- and high-volume fractions. Therefore, the present approach may be used as a design tool for low-volume fractions and be able to reveal the optimum vibrational performance for higher ones.

Future work concerns the attempt of introduction of a correction in the present computational approach considering the non-uniform distribution of nanoparticles in the polymer matrix. Future work concerns the attempt of introduction of a correction in the present computational approach considering the non-uniform distribution of nanoparticles in the polymer matrix.

**Author Contributions:** Conceptualization, S.G.; methodology, S.G.; software, S.G., G.G. and S.M.; validation, S.G., G.G. and S.M.; investigation, S.G.; writing—original draft preparation, S.G.; writing—review and editing, G.G. and S.M.; visualization, S.G., G.G. and S.M.; supervision, S.G.; All authors have read and agreed to the published version of the manuscript. **Author Contributions:** Conceptualization, S.G.; methodology, S.G.; software, S.G., G.G. and S.M.; validation, S.G.,G.G. and S.M.; investigation, S.G.; writing—original draft preparation, S.G.; writing—review and editing, G.G. and S.M.; visualization, S.G., G.G. and S.M.; supervision, S.G.; All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding

**Funding:** This research received no external funding **Conflicts of Interest:** The authors declare no conflict of interest. **Conflicts of Interest:** The authors declare no conflict of interest.
