**3. Modeling of the problem**

**2. Review of nonlocal piezoelasticity and Mindlin plate theory**

 ts( ) ( , ) ( ), = - ¢ ¢ " Î ò *nl <sup>l</sup>*

 t( , ) ( ), ¢ ¢ " Î ò *nl <sup>l</sup>*

> <sup>2</sup> (1 ) , -Ñ = ms

<sup>2</sup> (1 ) . -Ñ = m

Based on the Mindlin plate theory, the displacement field can be expressed as [15-17]:

( , , , ) ( , , ), ( , , , ) ( , , ), ( , , , ) ( , , ),

= = =

*u xyzt z xyt u xyzt z xyt u xyzt wxyt*

*x x y y*

*z*

y

y

 s*nl l*

where the parameter *μ* =(*e*0*a*)2 denotes the small-scale effect on the response of structures in nanosize, and ∇2 is the Laplacian operator in the above equation. Similarly, Eq. (2) can be

for the piezoelectric material can be given as follows [22]:

 a

= a

s

Based on the theory of nonlocal piezoelasticity, the stress tensor and the electric displacement at a reference point depend not only on the strain components and electric-field components at same position but also on all other points of the body. The nonlocal constitutive behavior

*ij ij <sup>v</sup> x x x dV x x V* (1)

*k k <sup>v</sup> D x x D dV x x V* (2)

*ij ij* (3)

*nl l D D k k* (4)

*nl* and

(5)

*<sup>l</sup>* are, respectively, the nonlocal stress tensor and local stress tensor; *Dk*

 are the components of the nonlocal and local electric displacement; *α*(| *x* − *x* ′|, *τ*) is the nonlocal modulus; | *<sup>x</sup>* <sup>−</sup> *<sup>x</sup>* ′| is the Euclidean distance; and *<sup>τ</sup>* <sup>=</sup>*e*0*<sup>a</sup>* / *<sup>l</sup>* is defined such that *l* is the external characteristic length, *e*0 denotes constant appropriate to each material, and *a* is the internal characteristic length of the material. Consequently, *e*0*a* is a constant parameter which is obtained with molecular dynamics, experimental results, experimental studies, and molec‐ ular structure mechanics. The constitutive equation of the nonlocal elasticity can be written as

**2.1. Nonlocal piezoelasticity**

42 Graphene - New Trends and Developments

where *σij nl* and *σij*

*Dk l*

[20]:

written as [22]:

**2.2. Mindlin plate theory**

Consider a coupled SLGS–PVDF nanoplate system as depicted in Fig. 1, in which geometrical parameters of length *L* , width *b*, and thickness *h* are indicated. The SLGS and PVDF nanoplate are coupled by an elastic medium which is simulated by Pasternak foundation. As is wellknown, this foundation model is capable of both transverse shear loads (*kg*) and normal loads (*kw*). The PVDF nanoplate is subjected to external electric voltage *ϕ* in thickness direction which is used for buckling smart control of SLGS.

**Figure 1.** SLGS coupled by a Pasternak foundation with PVDF nanoplate subjected to applied external electric poten‐ tial in thickness direction.
