# Analytical Solution for Static and Dynamic Analysis of Graphene-Based Hybrid Flexoelectric Nanostructures

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## Abstract

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## 1. Introduction

^{2}hybridized carbon atoms with remarkably high elastic modulus, tensile strength and high specific surface area (~3 times that of CNTs), that has encouraged huge interests among research and industrial communities. It has been reported in many experimental studies that graphene-based composites show substantial enhancement in mechanical properties as that of CNT-based composites [12]. In this work, they reported that the use of small quantity of graphene in the conventional matrices is found to alter and improve the properties of resulting nanocomposite significantly. Very few research studies [13,14,15] have been dedicated to the introduction of graphene or its derivatives as the modifiers/interphase to the conventional bulk composites in order to improve their multifunctional properties. The thickness of the graphene is in the order of nanometer. Hence, the concepts of validation of continuum mechanics at such nanoscales are still doubtful. However, classical homogenization strategies for graphene-reinforced composites are commonly implemented in the scientific literature with relatively reasonable outcomes. Therefore, many researchers [16,17,18] considered graphene layers as continuum plate in different types of analysis. Hence, in this paper, we are concentrating on graphene as nanofillers in a piezoelectric composite which was not conveyed in scientific literatures earlier.

## 2. Materials and Methods

#### 2.1. Assumptions of Kirchhoff’s Plate Theory

- Straight lines normal to the mid-surface (transverse normal) before deformation remain straight after deformation.
- The transverse normal are inextensible.
- The thickness of the plate does not change during a deformation.
- The transverse normal rotate in such a way that they remain normal to the middle surface after deformation.

#### 2.2. Closed-Form Solution for Static Analysis of GRPC Plates

#### 2.3. Closed-Form Solution for Modal Analysis Considering Free Vibration of GRPC Plates

## 3. Results

#### 3.1. Static Deflection of Hybrid Flexoelectric GRPC Plate

#### 3.2. Modal Analysis of Hybrid Flexoelectric GRPC Plate

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The detailed flowchart of electromechanical analysis of hybrid graphene-reinforced piezoelectric composite (GRPC) plate.

**Figure 3.**Effect of different flexoelectric coefficients on the center deflection of hybrid flexoelectric plate $(\mathrm{m}\text{}=\text{}1\text{};\text{}\mathrm{n}\text{}=\text{}1).$

**Figure 4.**Variation of center deflection of hybrid plate with respect to aspect ratio. (

**a**) m = 1; n = 1; (

**b**) m = 3; n = 3; (

**c**) m = 5; n = 5.

**Figure 5.**Effect of different graphene content on the center deflection of hybrid flexoelectric plate $(\mathrm{m}\text{}=\text{}3\text{};\text{}\mathrm{n}\text{}=\text{}3).$

**Figure 6.**Variation of maximum deflection against the hybrid flexoelectric plate thickness $(\mathrm{m}\text{}=\text{}3\text{};\text{}\mathrm{n}\text{}=\text{}3).$

**Figure 7.**Variation of normalized bending rigidity against the hybrid flexoelectric plate thickness.

**Figure 8.**Effect of different graphene content on the resonant frequency of hybrid flexoelectric plate with respect to plate thickness for mode $(1,1).$

**Figure 10.**Variation of resonant frequency against plate aspect ratio under mode $(1,1)$ and mode $(2,2)$, considering effect of flexoelectricity $(\mathrm{a}=\mathrm{b}=1200\text{}\mathrm{n}\mathrm{m})$.

**Figure 11.**Variation of resonant frequency against plate thickness under different modes, considering the effect of flexoelectricity $(\mathrm{a}=\mathrm{b}=45\text{}\mathrm{h})$.

**Table 1.**Effective properties of hybrid GRPC $({\mathrm{v}}_{\mathrm{g}}=0.2{\mathrm{v}}_{\mathrm{p}})$.

Material | ${\mathbf{C}}_{11}\text{}(\mathbf{G}\mathbf{P}\mathbf{a})$ | ${\mathbf{C}}_{12}\text{}(\mathbf{G}\mathbf{P}\mathbf{a})$ | ${\mathbf{C}}_{66}\text{}(\mathbf{G}\mathbf{P}\mathbf{a})$ | ${\mathbf{e}}_{31}\text{}(\mathbf{C}/{\mathbf{m}}^{2})$ | ${\mathbf{\chi}}_{33}\times {10}^{-9}\text{}(\mathbf{F}/\mathbf{m})$ |
---|---|---|---|---|---|

GRPC | 112.43 | 3.34 | 2.03 | −6.9337 | 3.264 |

**Table 2.**Three-dimensional representation of deflection of hybrid flexoelectric plate, with and without considering flexoelectricity for different values of m and n.

(m, n) | Without Flexoelectric Effect | With Flexoelectric Effect |
---|---|---|

$\mathrm{m}=1$; $\mathrm{n}=1$ | ||

$\mathrm{m}=3$; $\mathrm{n}=3$ |

**Table 3.**Deflection hybrid flexoelectric plate, with and without considering flexoelectricity for different loadings $(\mathrm{m}\text{}=1,3\text{};\text{}\mathrm{n}\text{}=1,3).$

Load | $\mathrm{m}=1;\mathrm{n}=1$ | $\mathrm{m}=3;\mathrm{n}=3$ |

Point | ||

In-line | ||

UDL | ||

UVL |

**Table 4.**Deflection hybrid flexoelectric plate, with and without considering flexoelectricity for different loadings $(\mathrm{m}\text{}=\text{}5,7\text{};\text{}\mathrm{n}\text{}=\text{}5,7).$

Load | $\mathrm{m}=5;\mathrm{n}=5$ | $\mathrm{m}=7;\mathrm{n}=7$ |

Point | ||

In-line | ||

UDL | ||

UVL |

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**MDPI and ACS Style**

Shingare, K.B.; Naskar, S. Analytical Solution for Static and Dynamic Analysis of Graphene-Based Hybrid Flexoelectric Nanostructures. *J. Compos. Sci.* **2021**, *5*, 74.
https://doi.org/10.3390/jcs5030074

**AMA Style**

Shingare KB, Naskar S. Analytical Solution for Static and Dynamic Analysis of Graphene-Based Hybrid Flexoelectric Nanostructures. *Journal of Composites Science*. 2021; 5(3):74.
https://doi.org/10.3390/jcs5030074

**Chicago/Turabian Style**

Shingare, Kishor Balasaheb, and Susmita Naskar. 2021. "Analytical Solution for Static and Dynamic Analysis of Graphene-Based Hybrid Flexoelectric Nanostructures" *Journal of Composites Science* 5, no. 3: 74.
https://doi.org/10.3390/jcs5030074