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

#### **2.1. Nonlocal piezoelasticity**

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

$$\sigma\_{ij}^{nl}(\mathbf{x}) = \int\_{\upsilon} \alpha(\left|\mathbf{x} - \mathbf{x}\right|, \tau) \sigma\_{ij}^{l} dV(\mathbf{x}'), \qquad \forall \mathbf{x} \in V \tag{1}$$

$$D\_k^{ul} = \int\_{\upsilon} \alpha(\left| \mathbf{x} - \mathbf{x}' \right| \tau) D\_k^l dV(\mathbf{x}'), \qquad \forall \mathbf{x} \in V \tag{2}$$

where *σij nl* and *σij <sup>l</sup>* are, respectively, the nonlocal stress tensor and local stress tensor; *Dk nl* and *Dk l* 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 [20]:

$$(\mathbf{1} - \mu \nabla^2) \sigma\_{\boldsymbol{\psi}}^{\boldsymbol{n}l} = \sigma\_{\boldsymbol{\psi}}^l \tag{3}$$

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 written as [22]:

$$(\mathbf{1} - \mu \nabla^2) \mathbf{D}\_k^{nl} = \mathbf{D}\_k^l. \tag{4}$$

#### **2.2. Mindlin plate theory**

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

$$\begin{aligned} \mu\_{\boldsymbol{x}}(\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}, t) &= \boldsymbol{z}\boldsymbol{\nu}\_{\boldsymbol{x}}(\boldsymbol{x}, \boldsymbol{y}, t), \\ \mu\_{\boldsymbol{y}}(\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}, t) &= \boldsymbol{z}\boldsymbol{\nu}\_{\boldsymbol{y}}(\boldsymbol{x}, \boldsymbol{y}, t), \\ \mu\_{\boldsymbol{z}}(\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}, t) &= \boldsymbol{w}(\boldsymbol{x}, \boldsymbol{y}, t), \end{aligned} \tag{5}$$

where *ψx*(*x*, *y*) and *ψy*(*x*, *y*) are the rotations of the normal to the mid-plane about x- and ydirections, respectively.

The von Kármán strains associated with the above displacement field can be expressed in the following form:

$$\mathcal{L}\_{xx} = \mathbf{z} \frac{\partial \boldsymbol{\nu}\_x}{\partial \mathbf{x}}, \boldsymbol{\varepsilon}\_{yy} = \mathbf{z} \frac{\partial \boldsymbol{\nu}\_y}{\partial \mathbf{y}}, \boldsymbol{\gamma}\_{yz} = \frac{\partial \mathbf{z}}{\partial y} + \boldsymbol{\nu}\_{y'} \boldsymbol{\gamma}\_{xz} = \frac{\partial \mathbf{z}}{\partial \mathbf{x}} + \boldsymbol{\nu}\_{x'} \boldsymbol{\gamma}\_{xy} = \mathbf{z} (\frac{\partial \boldsymbol{\nu}\_x}{\partial y} + \frac{\partial \boldsymbol{\nu}\_y}{\partial \mathbf{x}}), \tag{6}$$

where (*εxx*, *εyy*) are the normal strain components and (*γyz*, *γxz*, *γxy*) are the shear strain components.
