**3.1. Constitutive relations for PVDF nanoplate**

In a piezoelectric material, application of an electric field will cause a strain proportional to the mechanical field strength, and vice versa. The constitutive equation for stresses *σ* and strains *ε* matrix on the mechanical side, as well as flux density D and field strength E matrix on the electrostatic side, may be arbitrarily combined as follows [23]:

$$
\begin{bmatrix}
\sigma\_{xx} \\
\sigma\_{yy} \\
\sigma\_{yz} \\
\sigma\_{zx} \\
\sigma\_{zy}
\end{bmatrix} = \begin{bmatrix}
\mathcal{C}\_{11}^{PVDF} & \mathcal{C}\_{12}^{PVDF} & 0 & 0 & 0 \\
\mathcal{C}\_{21}^{PVDF} & \mathcal{C}\_{22}^{PVDF} & 0 & 0 & 0 \\
0 & 0 & \mathcal{C}\_{44}^{PVDF} & 0 & 0 \\
0 & 0 & 0 & \mathcal{C}\_{55}^{PVDF} & 0 \\
0 & 0 & 0 & 0 & \mathcal{C}\_{66}^{PVDF}
\end{bmatrix} \begin{bmatrix}
\varepsilon\_{xx} \\
\varepsilon\_{yy} \\
\mathcal{V}\_{yz} \\
\mathcal{V}\_{zx} \\
\mathcal{V}\_{xy}
\end{bmatrix} - \begin{bmatrix}
0 & 0 & \varepsilon\_{31} \\
0 & 0 & \varepsilon\_{32} \\
0 & \varepsilon\_{24} & 0 \\
\varepsilon\_{15} & 0 & 0 \\
0 & 0 & 0
\end{bmatrix} \begin{bmatrix}
E\_{xx} \\
E\_{yy} \\
E\_{zz}
\end{bmatrix},
\tag{7}
$$

$$
\begin{Bmatrix} D\_{xx} \\ D\_{yy} \\ D\_{zz} \end{Bmatrix} = \begin{bmatrix} 0 & 0 & 0 & e\_{15} & 0 \\ 0 & 0 & e\_{24} & 0 & 0 \\ e\_{31} & e\_{32} & 0 & 0 & 0 \end{bmatrix} \begin{Bmatrix} \mathcal{E}\_{xx} \\ \mathcal{E}\_{yy} \\ \mathcal{Y}\_{yz} \\ \mathcal{Y}\_{zx} \\ \mathcal{Y}\_{xy} \end{Bmatrix} + \begin{bmatrix} \in & & & 0 & 0 \\ 0 & \in\_{22} & & 0 \\ 0 & 0 & \in\_{33} \end{bmatrix} \begin{Bmatrix} E\_{xx} \\ E\_{yy} \\ E\_{zz} \end{Bmatrix} \tag{8}
$$

where *Cij PVDF* , *eij* , and ∈*ij* denote elastic, piezoelectric, and dielectric coefficients, respectively. Also, electric field *Ei* (*i* = *x*, *y*, *z*) in terms of electric potential (*ϕ*) is given as follows [23]:

$$E = -\nabla \Phi.\tag{9}$$

The electric potential distribution in the thickness direction of the PVDF nanoplate in the form proposed by references [21] and [24] as the combination of a half-cosine and linear variation which satisfies the Maxwell equation is adopted as follows:

$$\Phi(\mathbf{x}, y, z, t) = -\cos(\frac{\pi z}{h})\phi(\mathbf{x}, y, t) + \frac{2zV\_0}{h}e^{i\Omega t} \tag{10}$$

where *ϕ*(*x*, *y*, *t*) is the time and spatial distribution of the electric potential caused by bending which must satisfy the electric boundary conditions, *V*0 is external electric voltage, and *Ω* is the natural frequency of the SLBNS which is zero for buckling analysis.

The strain energy of the PVDF nanoplate can be expressed as:

$$\mathrm{LI} = \frac{1}{2} \int\_{\Omega\_b} \int\_{-h/2}^{h/2} \left( \sigma\_{xx} \varepsilon\_{xx} + \sigma\_{yy} \varepsilon\_{yy} + \sigma\_{xy} \gamma\_{xy} + \sigma\_{xz} \gamma\_{xz} + \sigma\_{yz} \gamma\_{yz} - \mathbf{D}\_{xx} \mathbf{E}\_{xx} - \mathbf{D}\_{yy} \mathbf{E}\_{yy} - \mathbf{D}\_{zz} \mathbf{E}\_{zz} \right) \tag{11}$$

Combining Eqs. (6)–(11) yields:

#### Modeling and Control of a Smart Single-Layer Graphene Sheet http://dx.doi.org/10.5772/61277 45

$$\begin{split} \mathcal{U} &= \frac{1}{2} \int\_{\Omega\_{0}} \left( M\_{xx} \frac{\partial \boldsymbol{\nu}\_{x}}{\partial \boldsymbol{\varepsilon}} + M\_{yy} \frac{\partial \boldsymbol{\nu}\_{y}}{\partial \boldsymbol{\varepsilon}} + M\_{zy} \left( \frac{\partial \boldsymbol{\nu}\_{x}}{\partial \boldsymbol{\varrho}} + \frac{\partial \boldsymbol{\nu}\_{y}}{\partial \boldsymbol{\varepsilon}} \right) + Q\_{zz} \left( \frac{\partial \boldsymbol{w}\_{0}}{\partial \boldsymbol{\varepsilon}} + \boldsymbol{\nu}\_{x} \right) + Q\_{yz} \left( \frac{\partial \boldsymbol{w}\_{0}}{\partial \boldsymbol{\varrho}} + \boldsymbol{\nu}\_{y} \right) \right) d\boldsymbol{x} d\boldsymbol{y} \\ &- \frac{1}{2} \int\_{\Omega\_{0}} \int\_{-h/2}^{h/2} D\_{xx} \left( \cos(\frac{\pi z}{h}) \times \frac{\partial \boldsymbol{\rho}}{\partial \boldsymbol{\varepsilon}} \right) + D\_{yy} \left( \cos(\frac{\pi z}{h}) \times \frac{\partial \boldsymbol{\rho}}{\partial \boldsymbol{\varepsilon}} \right) - D\_{zz} \left( \frac{\pi}{h} \sin(\frac{\pi z}{h}) \phi + \frac{2V\_{0}}{h} \right) d\boldsymbol{z} d\boldsymbol{x} dy, \end{split} \tag{12}$$

where the stress resultant-displacement relations can be written as:

$$\left\langle (\mathbf{N}\_{xx'}\mathbf{N}\_{yy'}\mathbf{N}\_{xy})\iota(\mathbf{M}\_{xx'}\mathbf{M}\_{yy'}\mathbf{M}\_{xy})\right\rangle = \int\_{-h/2}^{h/2} \left\langle \sigma\_{xx'}\sigma\_{yy'}\tau\_{xy} \right\rangle \langle \mathbf{1}, \mathbf{z} \rangle d\mathbf{z},\tag{13}$$

$$\mathbb{E}\left\{\mathbf{Q}\_{xx'}\mathbf{Q}\_{yy}\right\} = \mathbf{K} \int\_{-h/2}^{h/2} \left\{\boldsymbol{\tau}\_{xx'}\boldsymbol{\tau}\_{yz}\right\} d\boldsymbol{z}\_{\boldsymbol{\tau}} \tag{14}$$

in which *K* is shear correction coefficient.

strains *ε* matrix on the mechanical side, as well as flux density D and field strength E matrix

11 12 31 21 22 32

00 00 0 0 000 0 0 0 000 0000

*yy yy xx*

*C C e C C e E*

*yz yz yy PVDF zx xz z*

ì ü é ù ì ü é ù ï ï ê ú ï ï ê ú ï ï ï ï ê ú ï ï ï ï <sup>=</sup> - ê ú í ý í ý ê ú ê ú ï ï ï ï ê ú ï ï ï ï ê ú ï ï ï ï ë û î þ ë û î þ

44 24 15 55

000 0 0 000 0 0

 e

 e

 g

*C e E*

 g

*C E e*

,

(7)

(8)

ì ü ï ï í ý ï ï î þ*<sup>z</sup>*

 g

, and ∈*ij* denote elastic, piezoelectric, and dielectric coefficients, respectively.

*E* = -ÑF. (9)

*xyzt xyt e h h* (10)

66

*C*

15 11 24 22 31 32 33

e

ì ü

*xx*

e

g

g

g

Also, electric field *Ei* (*i* = *x*, *y*, *z*) in terms of electric potential (*ϕ*) is given as follows [23]:

î þ

The electric potential distribution in the thickness direction of the PVDF nanoplate in the form proposed by references [21] and [24] as the combination of a half-cosine and linear variation

where *ϕ*(*x*, *y*, *t*) is the time and spatial distribution of the electric potential caused by bending which must satisfy the electric boundary conditions, *V*0 is external electric voltage, and *Ω* is

( )

*xx xx yy yy xy xy xz xz yz yz xx xx yy yy zz zz <sup>h</sup> <sup>U</sup> DE DE DE* (11)

 sg

<sup>0</sup> <sup>2</sup> ( , , , ) cos( ) ( , , ) p f<sup>W</sup> F =- + *i t <sup>z</sup> zV*

> sg

W - = ++++- - - ò ò *<sup>h</sup>*

the natural frequency of the SLBNS which is zero for buckling analysis.

 sg

The strain energy of the PVDF nanoplate can be expressed as:

*xy*

0 0 00 0 0 , 0 00 0 0

000 0 0 0

*yy xx xx yy yz yy zz xz zz*

ï ï ì ü <sup>é</sup> ùé ù ï ï <sup>Î</sup> ì ü ï ï <sup>ê</sup> úê ú ï ï ï ï í ý <sup>=</sup> <sup>ê</sup> úê ú í ý + Î í ý ï ï <sup>ê</sup> ï ï ï ï <sup>Î</sup> î þ <sup>ë</sup> ûë û ï ï î þ ï ï

*D E e D e E D E e e*

*PVDF*

on the electrostatic side, may be arbitrarily combined as follows [23]:

*xx xx*

*xy xy*

which satisfies the Maxwell equation is adopted as follows:

*PVDF*

*PVDF PVDF*

*PVDF PVDF*

s

44 Graphene - New Trends and Developments

s

s

s

s

where *Cij*

*PVDF* , *eij*

0

1 2 / 2 / 2

Combining Eqs. (6)–(11) yields:

se

 se The external work due to surrounding elastic medium can be written as:

$$\mathcal{W} = \frac{1}{2} \int\_0^L q \, w\_2 d\mathbf{x}\_{\prime} \tag{15}$$

where *q* is related to Pasternak foundation. Finally, using Hamilton's principles lead to the following governing equations:

$$\frac{\partial \mathbf{M}\_{\text{xx}}^{\text{PVDF}}}{\partial \mathbf{x}} + \frac{\partial \mathbf{M}\_{\text{xy}}^{\text{PVDF}}}{\partial \mathbf{y}} - \mathbf{Q}\_{\text{xx}}^{\text{PVDF}} = \mathbf{0},\tag{16}$$

$$\frac{\partial \mathbf{M}\_{xy}^{\text{PVDF}}}{\partial \mathbf{x}} + \frac{\partial \mathbf{M}\_{yy}^{\text{PVDF}}}{\partial y} - \mathbf{Q}\_{yy}^{\text{PVDF}} = \mathbf{0},\tag{17}$$

$$\frac{\partial \mathbf{Q}\_{xx}^{\text{FLEF}}}{\partial \mathbf{x}} + \frac{\partial \mathbf{Q}\_{yy}^{\text{FLEF}}}{\partial y} - \left(1 - \mu \nabla^2 \right) \left[k\_w \left(w\_1 - w\_2\right) - k\_x \nabla^2 \left(w\_1 - w\_2\right) + (N\_{zw} + N\_{zw}) \frac{\partial^2 w}{\partial x^2} + (N\_{ym}) \frac{\partial^2 w}{\partial y^2} = 0 \right], \tag{18}$$

$$\int\_{-h/2}^{h/2} \left[ \cos(\frac{\pi z}{h}) \frac{\partial D\_{xx}}{\partial \mathbf{x}} + \cos(\frac{\pi z}{h}) \frac{\partial D\_{yy}}{\partial \mathbf{x}} + \frac{\pi}{h} \sin(\frac{\pi z}{h}) D\_{zz} \right] dz = 0,\tag{19}$$

in which *kw* and *kg* are Winkler and shear coefficients of Pasternak medium, respectively. Also, mechanical force are zero (i.e. *Nxm* = *N ym* =0) and electrical force is *Nxe* =2*e*11*V*0. Substituting Eqs. (7) and (8) into Eqs. (13) and (14), the stress resultant-displacement relations and electric displacement for PVDF nanoplate can be obtained as follows:

$$(1 - \mu \nabla^2) M\_{\text{xx}}^{\text{PVDF}} = \frac{24 h e\_{31} \phi + \pi h^3 C\_{11}^{\text{PVDF}} \frac{\partial \mathcal{W}\_{\text{xx}}}{\partial \mathbf{x}} + \pi h^3 C\_{12}^{\text{PVDF}} \frac{\partial \mathcal{W}\_{\text{yy}}}{\partial \mathbf{y}}}{12 \pi} \,, \tag{20}$$

$$(1 - \mu \nabla^2) M\_{yy}^{\text{PVDF}} = \frac{24 h e\_{32} \phi + \pi h^3 \mathcal{C}\_{11}^{\text{PVDF}} \frac{\partial \mathcal{V} \boldsymbol{\nu}\_{yy}}{\partial \boldsymbol{y}} + \pi h^3 \mathcal{C}\_{12}^{\text{PVDF}} \frac{\partial \boldsymbol{\nu}\_{xx}}{\partial \mathbf{x}}}{12 \pi} \tag{21}$$

$$(1 - \mu \nabla^2) M\_{xy}^{\text{PVDF}} = \frac{\hbar^3 \mathcal{C}\_{\theta\theta}^{\text{PVDF}}}{12} (\frac{\partial \mathcal{W}\_{xx}}{\partial y} + \frac{\partial \mathcal{W}\_{yy}}{\partial x}),\tag{22}$$

$$\pi (1 - \mu \nabla^2) Q\_{\text{xx}}^{\text{PVDF}} = \frac{\text{Kh}\left(-2\varepsilon\_{15} \frac{\partial \phi}{\partial \mathbf{x}} + \mathbf{C}\_{\text{gg}}^{\text{PVDF}} \pi (\mathbf{y}\_{\text{xx}} + \frac{\partial w}{\partial y})\right)}{\pi},\tag{23}$$

$$(1 - \mu \nabla^2) Q\_{yy}^{PVDF} = \frac{Kh \left( -2\varepsilon\_{24} \frac{\partial \phi}{\partial \mathbf{x}} + C\_{44}^{PVDF} \pi \langle \boldsymbol{\nu}\_{yy} + \frac{\partial \boldsymbol{w}}{\partial \mathbf{y}} \rangle \right)}{\pi} \tag{24}$$

$$\int\_{-h/2}^{h/2} \left[ (1 - \mu \nabla^2) D\_{\text{xx}} \right] \cos(\frac{\pi z}{h}) \, dz = \left( \int\_{-h/2}^{h/2} e\_{15} \cos(\frac{\pi z}{h}) dz \right) \left( \frac{\partial w}{\partial \mathbf{x}} + \nu\_{\text{xx}} \right) + \left( \int\_{-h/2}^{h/2} e\_{11} \cos(\frac{\pi z}{h})^2 dz \right) \frac{\partial \phi}{\partial \mathbf{x}} \, \tag{25}$$

$$\int\_{-h/2}^{h/2} \left[ (1 - \mu \nabla^2) D\_{yy} \right] \cos(\frac{\pi z}{h}) \, dz = \left( \int\_{-h/2}^{h/2} e\_{24} \cos(\frac{\pi z}{h}) dz \right) \left( \frac{\partial w}{\partial y} + \mathcal{W}\_{yy} \right) + \left( \int\_{-h/2}^{h/2} e\_{22} \cos(\frac{\pi z}{h})^2 dz \right) \frac{\partial \phi}{\partial \mathbf{x}} \, \tag{26}$$

$$\begin{split} \left[\int\_{-h/2}^{h/2} (1-\mu \nabla^2) D\_{zz} \right] \frac{\pi}{h} \text{sinc}(\frac{\pi z}{h}) \, dz &= \left(\int\_{-h/2}^{h/2} e\_{31} \frac{\pi}{h} \text{sinc}(\frac{\pi z}{h}) z \, dz\right) \frac{\partial \nu\_{xx}}{\partial \mathbf{x}} + \left(\int\_{-h/2}^{h/2} e\_{32} \frac{\pi}{h} \text{sinc}(\frac{\pi z}{h}) z \, dz\right) \frac{\partial \nu\_{yy}}{\partial \mathbf{y}} \\ &- \left(\int\_{-h/2}^{h/2} e\_{33} \left(\frac{\pi}{h} \text{sinc}(\frac{\pi z}{h})\right)^2 dz \right) \phi. \end{split} \tag{27}$$

#### **3.2. Constitutive relations for SLGS**

The SLGS is subjected to uniform compressive edge loading along *x* and *y* axis. In order to obtain the governing equations of SLGS, the procedure outlined above for PVDF nanoplate can be expressed repeatedly by ignoring piezoelectric coefficient in Eq. (7) and electric displacement (Eq. (8)).

$$\frac{\partial \mathbf{M}\_{\text{xx}}^{\text{GS}}}{\partial \mathbf{x}} + \frac{\partial \mathbf{M}\_{\text{xy}}^{\text{GS}}}{\partial \mathbf{y}} - \mathbf{Q}\_{\text{xx}}^{\text{GS}} = \mathbf{0},\tag{28}$$

$$\frac{\partial \mathbf{M}\_{xy}^{\rm GS}}{\partial \mathbf{x}} + \frac{\partial \mathbf{M}\_{yy}^{\rm GS}}{\partial y} - \mathbf{Q}\_{yy}^{\rm GS} = \mathbf{0},\tag{29}$$

$$\frac{\partial \mathbf{Q}\_{xx}^{\rm CS}}{\partial \mathbf{x}} + \frac{\partial \mathbf{Q}\_{yy}^{\rm CS}}{\partial y} - \left(1 - \mu \nabla^2 \right) \left[ k\_w \left( \mathbf{w}\_2 - \mathbf{w}\_1 \right) - k\_g \nabla^2 \left( \mathbf{w}\_2 - \mathbf{w}\_1 \right) + \left( \mathbf{N}\_{zw} \right) \frac{\partial^2 w}{\partial \mathbf{x}^2} + \left( \mathbf{N}\_{ym} \right) \frac{\partial^2 w}{\partial y^2} = \mathbf{0} \right]. \tag{30}$$

It is noted that the superscript PVDF nanoplate in section 3.1 can be changed to *GS* in this section.
