**4. Solution procedure**

(7) and (8) into Eqs. (13) and (14), the stress resultant-displacement relations and electric

+ +

*he h C h C*

+ +

*he h C h C*

(1 ) , <sup>12</sup>

(1 ) , <sup>12</sup>

¶ ¶ -Ñ =

3 <sup>2</sup> <sup>66</sup> (1 ) ( ), <sup>12</sup>

(1 ) ,

(1 ) ,

/ 2 / 2 / 2 <sup>2</sup> <sup>2</sup> <sup>15</sup> <sup>11</sup> / 2 / 2 / 2 (1 ) cos( ) cos( ) cos( ) ,

/ 2 / 2 / 2 <sup>2</sup> <sup>2</sup> <sup>24</sup> <sup>22</sup> / 2 / 2 / 2 (1 ) cos( ) cos( ) cos( ) ,

(1 ) sin( ) sin( ) sin( )

*h h*

 p

<sup>æ</sup> ö æ ¶ <sup>ö</sup> ¶ é ù - Ñ <sup>=</sup> <sup>ç</sup> ÷ ç <sup>+</sup> <sup>÷</sup> ë û <sup>è</sup> ø è ¶ ¶ <sup>ø</sup>

*h hh yy xx*

 pp

p

æ ö æ ö - Î ç ÷ ç ÷ è ø è ø

 p

æ öæ ö æ ö ¶ ¶ é ù - Ñ <sup>=</sup> ç ÷ç ÷ ç ÷ ++ Î ë û è øè ø ¶ ¶ è ø òòò

 p

<sup>æ</sup> öæ ö æ ¶ ¶ö é ù - Ñ <sup>=</sup> <sup>ç</sup> ÷ç ÷ ç ++ Î <sup>÷</sup> ë û <sup>è</sup> øè ø è ¶ ¶ <sup>ø</sup> òòò

¶ ¶ è ø -Ñ =

¶ ¶ è ø -Ñ =

*PVDF PVDF yy xx*

15 55

24 44

*<sup>z</sup> z w <sup>z</sup> <sup>D</sup> dz e dz dz*

*<sup>z</sup> z w <sup>z</sup> <sup>D</sup> dz e dz dz*

2

*<sup>z</sup> dz*

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

*zz z <sup>D</sup> dz e zdz <sup>e</sup> zdz*

sin( ) .

f

f


¶ ¶ -Ñ =

3 3 31 11 12

y

p

3 3 32 11 12

p

y

¶ ¶

2 ()

2 ()

æ ö ¶ ¶ ç ÷ -+ +

*PVDF*

p y

> y

> y

y

f

*hh hh x hh y*

æ ö ¶ ¶ ç ÷ -+ +

*PVDF*

p

p

*<sup>w</sup> Kh e C*

*<sup>w</sup> Kh e C*

p y

¶ ¶

*y x*

y p

*PVDF PVDF yy xx*

*x y <sup>M</sup>* (20)

p

*y x <sup>M</sup>* (21)

y

*xx*

*yy*

*<sup>h</sup> h x h x* (25)

*<sup>h</sup> h y h x* (26)

*x y <sup>Q</sup>* (24)

p

p

 p

p

 f

> f

y

(27)

*x y <sup>Q</sup>* (23)

*PVDF yy PVDF xx*

¶ ¶

¶ ¶

y

y

(22)

displacement for PVDF nanoplate can be obtained as follows:

24

24

*xy h C <sup>M</sup>*

f p

f p

2

2

m

*PVDF xx*

*PVDF yy*

m

2

2

p

p

pp

**3.2. Constitutive relations for SLGS**

m

m

m

m

*PVDF xx*

*PVDF yy*



/ 2 / 2 / 2 <sup>2</sup> 31 32 / 2 / 2 / 2

> / 2 33 / 2


*h h*

ò


*zz h hh*

ò òò

*h hh yy yy h hh*

*h hh xx xx h hh*

m

m

46 Graphene - New Trends and Developments

Steady-state solutions to the governing equations of the plate motion and the electric potential distribution which relate to the simply supported boundary conditions and zero electric potential along the edges of the surface electrodes can be assumed as [17,21]:

1 1 1 <sup>1</sup> <sup>1</sup> 1 2 2 2 <sup>2</sup> <sup>2</sup> 2 2 cos( )sin( ) sin( )cos( ) ( , ,) ( , ,) sin( )sin( ) ( , ,) ( , , ) cos( )sin( ) ( , ,) sin( )cos( ) ( , ,) ( , ,) p p y p p y y y p p p p y y y p p y f ì ü ï ï ï ï ï ï ï ï í ý = ï ï ï ï ï ï ï ï ï ï î þ *x y x y x x y y mx ny L b mx ny xyt L b xyt mx ny <sup>w</sup> L b w xyt mx ny xyt L b xyt mx ny w xyt L b xyt <sup>w</sup>* 2 . sin( )sin( ) sin( )sin( ) p p p p f ì ü ï ï í ý ï ï î þ *mx ny L b mx ny L b* (31)

As mentioned above, it is assumed that the SLGS plate is free from any transverse loadings. Uniform compressive edge loading along *x* and *y* axis are *Nxm* = − *P* and *N ym* = −*kP*, respectively. Substituting Eq. (31) into Eqs. (16)–(19) and (28)–(30) yields:

$$
\begin{bmatrix} L\_{11} & L\_{12} & L\_{13} & L\_{14} & 0 & 0 & 0 \\ L\_{21} & L\_{22} & L\_{23} & 0 & 0 & 0 & 0 \\ L\_{31} & L\_{32} & L\_{33} & 0 & 0 & 0 & 0 \\ L\_{41} & 0 & 0 & L\_{44} & L\_{45} & L\_{46} & L\_{47} \\ 0 & 0 & 0 & L\_{54} & L\_{55} & L\_{56} & L\_{57} \\ 0 & 0 & 0 & L\_{64} & L\_{65} & L\_{66} & L\_{67} \\ 0 & 0 & 0 & L\_{74} & L\_{75} & L\_{76} & L\_{77} \end{bmatrix} + P\_{55} \eta\_{mn} \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} \overline{w}\_{1} \\ \overline{w}\_{1}} \\ \overline{w}\_{1}} \\ \overline{w}\_{2} \\ \overline{w}\_{2} \\ 0 \\ 0 \\ \end{bmatrix} \tag{32}$$

where *ξmn* =1 <sup>+</sup> *<sup>μ</sup>*(( *<sup>m</sup><sup>π</sup> <sup>L</sup>* ) 2 <sup>+</sup> ( *<sup>n</sup><sup>π</sup> b* ) 2 ), *ηmn* =( *<sup>m</sup><sup>π</sup> <sup>L</sup>* ) 2 <sup>+</sup> *<sup>k</sup>*( *<sup>n</sup><sup>π</sup> b* ) 2 and *L ij* are defined in Appendix A. Finally, buckling load of the system (*P*) can be calculated by solving the above equation.
