**Appendix A**

2 2 2 2 22 22 55 44 11 2 2 2 2 2 2 4 4 2 42 2 2 2 42 4 4 2 4 22 2 22 4 55 12 44 13 22 22 14 2 2 p p p p p p p p p p m p p p p æ ö =- - + + + - ç ÷ è ø æ ö æ öæ ö ç ÷ - - + -- + ç ÷ç ÷ è ø è øè ø = - = æ ö =- + ç ÷ <sup>ç</sup> è ø *GS GS w g w w g g GS GS g KC m h KC n h m n <sup>L</sup> k k L b Lb k m m mn k n mn n k k L L Lb b Lb b KC m h <sup>L</sup> L KC n h <sup>L</sup> b m n L k L b* 2 2 4 4 4 42 2 2 2 42 4 4 2 4 22 2 22 4 55 21 2 23 2 23 11 66 22 2 2 55 2 3 23 12 66 44 31 2 32 ( 12 12 ( ) <sup>12</sup> p p p p p p m p p p p p p - - <sup>÷</sup> æ ö æ öæ ö ç ÷ + + ++ + ç ÷ç ÷ è ø è øè ø = - =- + + = - + = - = *w w w g g GS GS GS GS GS GS GS k k m m mn k n mn n K k L L Lb b Lb b KC m h <sup>L</sup> L Cm h Cn h <sup>L</sup> KC h L b m nh L CC Lb KC n h <sup>L</sup> b m nh <sup>L</sup>* 3 12 66 ( ) <sup>12</sup> + *GS GS C C Lb*

2 23 2 23 66 11 33 2 2 44 22 22 41 2 2 2 2 2 42 4 4 2 2 4 4 2 42 2 22 4 2 4 22 2 2 2 2 55 44 31 44 2 2 12 12 ( 2 p p p p p p p p p p m p p =- - æ ö =- - + - ç ÷ è ø æ ö æ öæ ö ç ÷ + ++ + + ç ÷ç ÷ è ø è øè ø = - -- *GS GS GS w g w w g g PVDF PVDF Cm h Cn h <sup>L</sup> KC h L b m n L kk L b k n mn n k m m mn k k b Lb b L L Lb KC m h KC n h <sup>e</sup> <sup>L</sup> L b* 2 2 22 22 0 2 2 2 44 22 4 4 2 42 2 42 2 2 2 42 4 4 31 0 31 0 4 2 4 22 2 2 2 22 4 55 45 44 46 2 15 47 2 2 2 2 2 p p p p p p p p p p p m p p p æ ö ++ + - ç ÷ è ø <sup>æ</sup> æ ö æ öö <sup>ç</sup> - - + + -- + ç ÷ ç ÷÷ <sup>ç</sup> <sup>÷</sup> <sup>è</sup> è ø è øø = - = - = + *w g w w g g PVDF PVDF V m m n k k L Lb e Vm k m m mn e Vm n k n mn n k k L L L Lb Lb b Lb b KC m h <sup>L</sup> L KC n h <sup>L</sup> b Khe m <sup>K</sup> <sup>L</sup> L* 2 24 2 55 54 p p = - *PVDF he n b KC m h <sup>L</sup> L*

smart control of the SLGS using elastically bonded PVDF nanoplate which is subjected to external voltage is the main contribution of the present paper. The elastic medium between SLGS and PVDF nanoplate is simulated by a Pasternak foundation. The governing equations are obtained based on nonlocal Mindlin plate theory so that the effects of small-scale, elastic medium coefficient, mode number, and graphene length are discussed. The results indicate that the imposed external voltage is an effective controlling parameter for buckling of the SLGS. It is found that the effect of external voltage becomes more prominent at higher nonlocal parameter and shear modulus. It is also observed that for a given length, the SLGS with negative external voltage will buckle first as compared to the SLGS with positive one. The results of this study are validated as far as possible by the buckling of SLGS in the absence of PVDF nanoplate, as presented by [16, 17, 28, and 29]. Finally, it is hoped that the results presented in this paper would be helpful for study and design of bonded systems based on

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*k m m mn k n mn n k k L L Lb b Lb b*

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smart control and electromechanical systems.

55

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54 Graphene - New Trends and Developments

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**Appendix A**

$$\begin{aligned} L\_{35} &= -\frac{C\_{10}^{\text{PDF}}m^2\pi^2h^3}{12L^2} - \frac{C\_{80}^{\text{PDF}}m^2\pi^2h^3}{12b^2} - KC\_{39}^{\text{PDF}}h\\ L\_{56} &= -\frac{m\pi^2h^3h}{12Lb^2} (-C\_{12}^{\text{PDF}} + C\_{69}^{\text{PDF}})\\ L\_{59} &= \frac{2h\omega\_{31}m + 2Kh\omega\_{33}m}{L}\\ L\_{54} &= -\frac{KC\_{49}^{\text{PDF}}m\pi h}{12Lb^2} \\ L\_{68} &= -\frac{m\pi^2h^3h}{12Lb^2}(C\_{12}^{\text{PDF}} + C\_{69}^{\text{PDF}})\\ L\_{68} &= -\frac{C\_{10}^{\text{PDF}}m^2\pi^2h}{12L^2} - \frac{C\_{11}^{\text{PDF}}m^2\pi^2h^3}{12b^2} - KC\_{44}^{\text{PDF}}h\\ L\_{78} &= \frac{2hc\_{29}^2 + 2Khc\_{29}}{b} \frac{2h\omega\_{31}}{b} \\ L\_{78} &= -\frac{4m^2h^2b^2c\_{27}\omega\_{44} + 4\pi^2h^2L^2c\_{2,4}\pi}{2L^2b^2h} \\ L\_{78} &= -\frac{4mh^2b^2c\_{12} + 4L^2bc\_{13}mh^2}{2L^2b^2h} \\ L\_{78} &= -\frac{4m^2h^2c\_{29} + 4L^2bc\_{23}mh^2}{2L^2b^2h} \\ L\_{79} &= -\frac{h^2m^2\pi^2 \left(c\_{10} + \frac{c\_{23}}{2}\right)^2 + h^2n^2\pi^2}{2L^2b^2h} \\ L\_{712$$
