Surface effects on the vibration and buckling of piezoelectric nanoplates

The enhanced piezoelectricity and unique coupling between piezoelectric and semiconducting properties of piezoelectric nanomaterials make them attractive for applications as sensors, resonators, generators and transistors in the nanoelectromechanical systems (NEMS) [1–6]. To better understand the underlying mechanisms and improve the performances of these advanced devices, some fundamental issues must be clearly addressed, for example, the vibration and buckling behaviors of piezoelectric nanostructures.

Different from their macroscpoic counterparts, nanomaterials exhibit size-dependent mechanical and physical properties due to their large surface area to volume ratio. For example, experimental investigations and atomistic simulations have demonstrated that the elastic constants or the piezoelectric coefficients of some piezoelectric nanomaterials increase dramatically with the decrease of the material size to the nanoscale [7–10]. In the literature, modified continuum mechanics theories have also been adopted as alternative and cost-effective tools to study the surface effects on the size-dependent properties of elastic nanostructures with various configurations [11–15] based on the surface elasticity model developed by Gurtin and Murdoch [16]. However, this surface elasticity model is not sufficient in predicting the size-dependent properties of piezoelectric nanomaterials since it neglects surface piezoelectricity effect, which is unique for piezoelectric materials. As pointed out by Tagantsev [17], the effect of surface piezoelectricity may become significant with the miniaturization of piezoelectric materials into a nanoscale size. Therefore, it is essential to incorporating the effect of the surface piezoelectricity to investigate the electroelastic properties of piezoelectric nanomaterials. As an extension of the surface elasticity model, a surface piezoelectricity model with the consideration of the surface piezoelectricity as well as the residual surface stress and surface elasticity, was first proposed by Huang and Yu [18] to study the electroelastic responses of a piezoelectric nanoring. Based on this model, the surface effects on the bending, vibration and buckling behaviors of piezoelectric nanobeams have been recently investigated in our previous work [19, 20]. It is found that the influence of the surface piezoelectricity on the static and dynamic behaviors of the piezoelectric nanobeams is significant. Li et al. [21] studied the surface effects on the wrinkling of a piezoelectric nanofilm on a compliant substrate. Their results showed that the wavelength and amplitude of the wrinkling were significantly affected by the surface parameters for the films with nanoscale thickness. However, the investigation of the surface effects on two-dimensional piezoelectric nanostructures is very limited. In order to enrich the studies on the plate-like piezoelectric nanostructures, the present work aims to develop a modified piezoelectric plate model based on the classical Kirchhoff plate theory and the surface piezoelectricity model to study the surface effects on the vibration and buckling behaviors of a piezoelectric nanoplate (PNP). Simulation results will show how the influence of the surface effects on the resonant frequency and the critical buckling potential changes with the PNP thickness and aspect ratio. In addition, the possibility of frequency tuning via applied electric potentials will also be investigated.

The problem considered is a rectangular PNP with thickness h, in-plane length a and width b, as illustrated in fig. 1. A Cartesian coordinate (x₁, x₂, x₃) is used to describe the plate with x₁ and x₂ axes and the origin at the mid-plane of the undeformed plate, and x₃-axis in the thickness direction. The PNP is poled along the x₃-direction and is subjected to an electric voltage V . Following Kirchhoff's hypotheses, the displacements at any point of the plate are expressed as

$$\begin{eqnarray} &&u_{\alpha}(x_1,x_2,x_3,t)=u_{\alpha}^0(x_1,x_2,t)-x_3 \displaystyle\frac{\partial u_3(x_1,x_2,x_3,t)}{\partial x_{\alpha}},\quad\\ &&u_3(x_1,x_2,x_3,t)=w(x_1,x_2,t), \end{eqnarray} \tag{ 1 }$$

where u⁰_α are the in-plane displacements of the mid-plane and w is the transvere displacement. Such mid-plane displacements describing the membrane deformations may be induced by the in-plane applied mechanical load, the applied electrical load due to the electromechanical coupling or the residual surface stress induced relaxation [22, 23]. The usual convention of summation over repeated indices is used here, e.g., Greek indices run from 1 to 2. Accordingly, the in-plane strains for the Kirchhoff plate can be written as

$$\begin{equation} \varepsilon_{\alpha\beta}=\frac{1}{2}\left(\displaystyle\frac{\partial u_{\alpha}^0}{\partial {x_\beta}}+\displaystyle\frac{\partial u_{\beta}^0}{\partial {x_\alpha}}\right)-x_3\frac{\partial^2 w}{\partial {x_\alpha} \partial {x_\beta}}. \end{equation} \tag{ 2 }$$

The electric field E is assumed to exist only along the x₃-direction and can be expressed in terms of the electric potential Φ as

$$\begin{equation} E_3=-\frac{\partial \Phi}{\partial x_3}. \end{equation} \tag{ 3 }$$

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Following the surface piezoelectricity model, the plate itself is composed of a bulk part and the upper and lower surface layers with negligible thickness. The constitutive equations for the surface layers of the PNP are different from the bulk, which are given as

$$\begin{eqnarray} &&\sigma_{\alpha \beta}^{\rm s}=\sigma_{\alpha \beta}^0+c_{\alpha \beta \gamma \delta}^{\rm s}\varepsilon_{\gamma \delta}-e_{3\alpha \beta}^{\rm s}E_{3},\\ &&D_{\gamma}^{\rm s}=D_{\gamma}^0+e_{\alpha \beta \gamma}^{\rm s}\varepsilon_{\alpha \beta}+\kappa_{\gamma \delta}^{\rm s}E_\delta, \end{eqnarray} \tag{ 4 }$$

where σ^s_αβ and D^s_γ are surface stresses and surface electric displacements; c^s_αβγδ, e^s_3αβ and κ^s_γδ are elastic, piezoelectric and dielectric constants for the surfaces; σ⁰_αβ and D⁰_γ are residual surface stresses and residual surface electric displacements. Following the assumption [24] that the stress component in the x₃-direction is negligible, the in-plane stresses σ_αβ and electric displacement D₃ for the bulk are expressed as

$$\begin{equation} \sigma_{\alpha \beta}=c_{\alpha \beta \gamma \delta}\varepsilon_{\gamma \delta}-e_{3 \alpha \beta}E_{3}, \quad D_3=e_{3 \alpha \beta}\varepsilon_{\alpha \beta}+\kappa_{33}E_{3}, \end{equation} \tag{ 5 }$$

with c_αβγδ, e_3αβ and κ₃₃ being the bulk elastic, piezoelectric and dielectric constants for the plane stress problem.

According to the generalized Young-Laplace equations, the existence of surface can be represented by the traction jumps T_i exerting on the bulk of the plate, which can be expressed in terms of the surface stresses as

$$\begin{equation} T_\alpha=\frac{\partial \sigma^{\rm s}_{\alpha \beta}}{\partial {x_\beta}}, \quad T_3^u=\sigma_{\alpha \beta}^{\rm s}\frac{\partial^2 w}{\partial x_{\alpha} \partial x_{\beta}}, \quad T_3^l=-\sigma_{\alpha \beta}^{\rm s}\frac{\partial^2 w}{\partial x_{\alpha} \partial x_{\beta}}.\nonumber \end{equation} \tag{ 6 }$$

The superscripts u and l denote the upper and lower surfaces of the plate, respectively. It should be noted that the electric displacement jump across the surfaces is zero.

In the absence of free electric charges, the electric displacement should satisfy the Gauss's law

$$\begin{equation} \frac{\partial D_3}{\partial x_3}=0. \end{equation} \tag{ 7 }$$

Applying the electric boundary conditions Φ(h/2) = V and Φ(−h/2) = 0, the electric potential and the electric field distribution can be determined.

For the transverse vibration of the PNP, the motion equation is derived as

$$\begin{equation} \frac{\partial^2 M^*_{\alpha \beta}}{\partial x_{\alpha} \partial x_{\beta}}-N^*_{\alpha \beta}\frac{\partial^2 w}{\partial x_{\alpha} x_{\beta}}+\rho h \ddot{w}-\displaystyle\frac{1}{12}\rho h^3 \frac{\partial^2 \ddot{w}}{\partial x_{\alpha}^2}=0, \end{equation} \tag{ 8 }$$

where ρ is the mass density, N*_αβ, M*_αβ are the generalized resultant forces and moments for the PNP with the consideration of surface effects, i.e.,

$$\begin{eqnarray} &&N_{\alpha \beta}^*=N_{\alpha \beta}+(\sigma_{\alpha \beta}^{\rm s})^u+(\sigma_{\alpha \beta}^{\rm s})^l,\\ &&M_{\alpha \beta}^*=M_{\alpha \beta}-\displaystyle\frac{h}{2}\left[\left(\sigma_{\alpha \beta}^{\rm s}\right)^u-\left(\sigma_{\alpha \beta}^{\rm s}\right)^l\right], \end{eqnarray} \tag{ 9 }$$

with N_αβ and M_αβ being the axial forces and bending moments in the bulk of the plate defined as

$$\begin{equation} N_{\alpha \beta}=\int_{-h/2}^{h/2}\sigma_{\alpha \beta} {\rm d} x_3, \quad M_{\alpha \beta}=-\int_{-h/2}^{h/2}\sigma_{\alpha \beta}x_3 {\rm d}x_3. \end{equation} \tag{ 10 }$$

It should be noted that the subscripts in the material constants as shown in eqs. (4) and (5) can be relabeled in the contracted notation due to the symmetry of stress and strain tensors following $11 \rightarrow 1; 22 \rightarrow 2; 33 \rightarrow 3; 23 \rightarrow 4; 13 \rightarrow 5; 12 \rightarrow 6$. In the following formulation and discussion, the material constants in the contracted notation, i.e., c₁₁,c₁₂,c₆₆,c^s₁₁,c^s₁₂,c^s₆₆,e₃₁ and e^s₃₁ will be used. After the manipulation of the equations above, the transverse motion equation of the PNP with surface effects can be derived in terms of w(x₁,x₂,t) as

$$\begin{eqnarray} &&D_{11}\displaystyle\left(\frac{\partial^4 w}{\partial x_1^4}+\displaystyle\frac{\partial^4 w}{\partial x_2^4}\right)+2\left(D_{12}+2D_{66}\right)\displaystyle\frac{\partial^4 w}{\partial x_1^2 x_2^2}+\rho h \ddot{w}\nonumber\\ &&-\displaystyle\frac{1}{12}\rho h^3\displaystyle\left(\frac{\partial^2 \ddot{w}}{\partial x_1^2}+\displaystyle\frac{\partial^2 \ddot{w}}{\partial x_2^2}\right)= N^*_{11}\displaystyle\frac{\partial^2 w}{\partial x_1^2}+2N^*_{12}\displaystyle\frac{\partial^2 w}{\partial x_1 \partial x_2}\nonumber\\ &&+N^*_{22}\displaystyle\frac{\partial^2 w}{\partial x_2^2}, \end{eqnarray} \tag{ 11 }$$

with

$$\begin{eqnarray} &&D_{11}=\left(c_{11}+\displaystyle\frac{e_{31}^2}{\kappa_{33}}\right)\displaystyle\frac{h^3}{12}+\left(c_{11}^{\rm s}+\displaystyle\frac{e_{31}^{\rm s}e_{31}}{\kappa_{33}}\right)\displaystyle\frac{h^2}{2},\\ &&D_{12}=\left(c_{12}+\displaystyle\frac{e_{31}^2}{\kappa_{33}}\right)\displaystyle\frac{h^3}{12}+\left(c_{12}^{\rm s}+\displaystyle\frac{e_{31}^{\rm s}e_{31}}{\kappa_{33}}\right)\displaystyle\frac{h^2}{2},\\ &&D_{66}=c_{66}\displaystyle\frac{h^3}{12}+c_{66}^{\rm s}\displaystyle\frac{h^2}{2},\\ &&N^*_{11}=2\left(\sigma_{11}^0+c_{11}^{\rm s}\displaystyle\frac{\partial u_1^0}{\partial x_1}+c_{12}^{\rm s}\frac{\partial u_2^0}{\partial x_2}+e_{31}^{\rm s}\displaystyle\frac{V}{h}\right)\\ &&~~~~~~~~+\left(c_{11}\displaystyle\frac{\partial u_1^0}{\partial x_1}+c_{12}\frac{\partial u_2^0}{\partial x_2}+e_{31}\displaystyle\frac{V}{h}\right)h,\\ &&N^*_{12}=2\left[\sigma_{12}^0+c_{66}^{\rm s}\left(\displaystyle\frac{\partial u_1^0}{\partial x_2}+\frac{\partial u_2^0}{\partial x_1}\right)\right]\\ &&~~~~~~~~+c_{66}\left(\displaystyle\frac{\partial u_1^0}{\partial x_2}+\frac{\partial u_2^0}{\partial x_1}\right)h,\\ &&N^*_{22}=2\left(\sigma_{22}^0+c_{12}^{\rm s}\displaystyle\frac{\partial u_1^0}{\partial x_1}+c_{11}^{\rm s}\frac{\partial u_2^0}{\partial x_2}+e_{31}^{\rm s}\displaystyle\frac{V}{h}\right)\\ &&~~~~~~~~+\left(c_{12}\displaystyle\frac{\partial u_1^0}{\partial x_1}+c_{11}\frac{\partial u_2^0}{\partial x_2}+e_{31}\displaystyle\frac{V}{h}\right)h. \end{eqnarray} \tag{ 12 }$$

Obviously, the in-plane constraints for the PNP must be prescribed first to solve eq. (11). For a clamped-clamped plate with four edges being fully restrained, it is reasonable to set the displacements u⁰_α(x₁,x₂) as zero, which was adopted by Zhao et al. [24] for a conventional piezoelectric plate. Moreover, the residual surface stress are assumed as σ⁰₁₁ = σ⁰₂₂ = σ⁰ and σ⁰₁₂ = 0. The harmonic solution of eq. (11) takes

$$\begin{equation} w(x_1,x_2,t)=W(x_1,x_2)e^{i\omega t}, \end{equation} \tag{ 13 }$$

where ω is the resonant frequency and W(x₁,x₂) represents the vibration mode. Substituting eq. (13) into eq. (11) results in

$$\begin{eqnarray} &&D_{11}\left(\displaystyle\frac{\partial^4 W}{\partial x_1^4}+\frac{\partial^4 W}{\partial x_2^4}\right)+2\left(D_{12}+2D_{66}\right)\displaystyle\frac{\partial^4 W}{\partial x_1^2 \partial x_2^2}\\ &&-\rho h \omega^2 W+\displaystyle\frac{1}{12}\rho h^3\omega^2\left(\frac{\partial^2 W}{\partial x_1^2}+\frac{\partial^2 W}{\partial x_2^2}\right)=\\ &&f\left(\displaystyle\frac{\partial^2 W}{\partial x_1^2}+\frac{\partial^2 W}{\partial x_2^2}\right), \end{eqnarray} \tag{ 14 }$$

with $f=2\left (\sigma ^0+e_{31}^{\mathrm { s}}V/h\right )+e_{31}V$ being the biaxial force induced by the applied electrical load and residual surface stress. Such a force may cause the buckling of PNPs.

In order to do the vibration and buckling analysis, the Ritz method [25] is adopted to get the approximate solutions. The weak form of the variational statement of eq. (14) is written as

$$\begin{eqnarray} && 0=\int_{0}^b\int_{0}^a\bigg \{D_{11}\left[\frac{\partial^2 W}{\partial x_1^2}\frac{\partial^2 \left(\delta W\right)}{\partial x_1^2}+\frac{\partial^2 W}{\partial x_2^2}\frac{\partial^2 \left(\delta W\right)}{\partial x_2^2}\right]\nonumber\\ &&+D_{12}\left[\frac{\partial^2 W}{\partial x_2^2}\frac{\partial^2 \left(\delta W\right)}{\partial x_1^2}+\frac{\partial^2 W}{\partial x_1^2}\frac{\partial^2 \left(\delta W\right)}{\partial x_2^2}\right]\nonumber\\ &&+4D_{66}\frac{\partial^2 W}{\partial x_1 \partial x_2}\frac{\partial^2 \left(\delta W\right)}{\partial x_1 \partial x_2}-\omega^2\rho h W \delta W\nonumber\\ &&-\frac{1}{12}\omega^2\rho h^3\left[\frac{\partial W}{\partial x_1} \frac{\partial \left(\delta W\right)}{\partial x_1}+\frac{\partial W}{\partial x_2}\frac{\partial \left(\delta W\right)}{\partial x_2}\right]\nonumber\\ &&+f\left[\frac{\partial W}{\partial x_1}\frac{\partial \left(\delta W\right)}{\partial x_1}+\frac{\partial W}{\partial x_2}\frac{\partial \left(\delta W\right)}{\partial x_2}\right]\bigg \}{\rm d}x_1{\rm d}x_2. \end{eqnarray} \tag{ 15 }$$

According to the Ritz method, the transverse deflection of the PNP can be approximated by

$$\begin{equation} W(x_1,x_2)\approx\sum_{i=1}^m\sum_{j=1}^nc_{ij}X_i(x_1)Y_j(x_2), \end{equation} \tag{ 16 }$$

where c_ij are the unknown constants, and X_i(x₁) and Y_j(x₂) are the coordinate functions satisfying the boundary conditions. For the clamped-clamped PNP, W should satisfy W = 0 and ∂W/∂x₁ = 0 at x₁ = 0 and x₁ = a; W = 0 and ∂W/∂x₂ = 0 at x₂ = 0 and x₂ = b. Accordingly, X_i(x₁) and Y_j(x₂) are chosen as [25]

$$\begin{eqnarray} &&X_i(x_1)=\left(\displaystyle\frac{x_1}{a}\right)^{i+1}-2\left(\displaystyle\frac{x_1}{a}\right)^{i+2}+\left(\displaystyle\frac{x_1}{a}\right)^{i+3},\\ &&Y_j(x_2)=\left(\displaystyle\frac{x_2}{b}\right)^{j+1}-2\left(\displaystyle\frac{x_2}{b}\right)^{j+2}+\left(\displaystyle\frac{x_2}{b}\right)^{j+3}. \end{eqnarray} \tag{ 17 }$$

Substituting eq. (16) into eq. (15), we have

$$\begin{equation} \left([R]-\omega^2[B]\right)\{c\}=0, \end{equation}\noindent \tag{ 18 }$$

with

$$\begin{eqnarray} &&R_{(ij)(kl)}=\nonumber\\ &&\int_{0}^b\int_{0}^a \bigg[D_{11}\displaystyle\left(\frac{\partial^2 X_i}{\partial x_1^2}Y_j \displaystyle\frac{\partial^2 X_k}{\partial x_1^2}Y_l+X_i \displaystyle\frac{\partial^2 Y_j}{\partial x_2^2}X_k \displaystyle\frac{\partial^2 Y_l}{\partial x_2^2}\right)\nonumber\\ &&+D_{12}\left(X_i \displaystyle\frac{\partial^2 Y_j}{\partial x_2^2}\displaystyle\frac{\partial^2 X_k}{\partial x_1^2}Y_l+\displaystyle\frac{\partial^2 X_i}{\partial x_1^2}Y_j X_k \displaystyle\frac{\partial^2 Y_l}{\partial x_2^2}\right)\nonumber\\ &&+4D_{66}\displaystyle\frac{\partial X_i}{\partial x_1}\displaystyle\frac{\partial Y_j}{\partial x_2}\frac{\partial X_k}{\partial x_1}\displaystyle\frac{\partial Y_l}{\partial x_2}\nonumber\\ &&+f\displaystyle\left(\frac{\partial X_i}{\partial x_1}Y_j \displaystyle\frac{\partial X_k}{\partial x_1}Y_l+X_i \displaystyle\frac{\partial Y_j}{\partial x_2}X_k \displaystyle\frac{\partial Y_l}{\partial x_2}\right)\bigg]{\rm d}x_1{\rm d}x_2, \end{eqnarray} \noindent \tag{ 19 }$$

and

$$\begin{eqnarray} &&\hskip-8pt B_{(ij)(kl)} = \int_{0}^b\int_{0}^a\bigg[\rho h X_i X_k Y_j Y_l\nonumber\\ &&\hskip-8pt +\frac{1}{12}\rho h^3\left(\frac{\partial X_i}{\partial x_1}\frac{\partial X_k}{\partial x_1}Y_j Y_l+X_i X_k \frac{\partial Y_j}{\partial x_2}\frac{\partial Y_l}{\partial x_2}\right)\bigg]{\rm d}x_1{\rm d}x_2.\quad\quad \end{eqnarray} \noindent \tag{ 20 }$$

Then, the resonant frequency ω of the PNP can be obtained by solving the characteristic equations of (18).

Due to the inherent electromechanical coupling of piezoelectric materials, the applied electrical load generates in-plane forces when the in-plane displacements of the plate are constrained, which may cause the mechanical buckling of the PNP once the resulting forces become compressive. Moreover, the surface stresses may also contribute to these in-plane forces, as shown in eq. (9). Therefore, it will be interesting to investigate the buckling behavior of the PNP with surface effects. The equation governing the buckling behaviors of PNPs can be determined by letting ω = 0 in eq. (14). Similarly, substituting eq. (16) into eq. (15) with ω = 0 results in

$$\begin{equation} \left([R']-f[B']\right)\{c\}=0, \end{equation} \noindent \tag{ 21 }$$

with

$$\begin{eqnarray} &&R'_{(ij)(kl)}=\nonumber\\ &&\int_{0}^b\int_{0}^a \bigg[D_{11}\displaystyle\left(\frac{\partial^2 X_i}{\partial x_1^2}Y_j \displaystyle\frac{\partial^2 X_k}{\partial x_1^2}Y_l+X_i \displaystyle\frac{\partial^2 Y_j}{\partial x_2^2}X_k \displaystyle\frac{\partial^2 Y_l}{\partial x_2^2}\right)\nonumber\\ &&+D_{12}\left(X_i \frac{\partial^2 Y_j}{\partial x_2^2} \displaystyle\frac{\partial^2 X_k}{\partial x_1^2}Y_l+\displaystyle\frac{\partial^2 X_i}{\partial x_1^2}Y_j X_k \frac{\partial^2 Y_l}{\partial x_2^2}\right)\nonumber\\ &&+4D_{66}\displaystyle\frac{\partial X_i}{\partial x_1}\displaystyle\frac{\partial Y_j}{\partial x_2}\frac{\partial X_k}{\partial x_1}\displaystyle\frac{\partial Y_l}{\partial x_2}\bigg]{\rm d}x_1{\rm d}x_2. \end{eqnarray} \noindent \tag{ 22 }$$

and

$$\begin{eqnarray} &&B'_{(ij)(kl)}=\nonumber\\ &&-\int_{0}^b\int_{0}^a\left(\frac{\partial X_i}{\partial x_1}Y_j \frac{\partial X_k}{\partial x_1}Y_l+X_i \frac{\partial Y_j}{\partial x_2}X_k \frac{\partial Y_l}{\partial x_2}\right) {\rm d}x_1{\rm d}x_2.\nonumber\\ \end{eqnarray} \tag{ 23 }$$

After determining f from the characteristic solutions of (21), the critical electric potential for buckling is calculated as

$$\begin{equation} V=\frac{f-2\sigma^0}{e_{31}+2e_{31}^{\rm s}/h}. \end{equation} \tag{ 24 }$$

To quantitatively show the surface effects on the vibration and buckling behaviors of the PNP, PZT-5H is chosen as an example material for case study. Its macroscopic material constants are taken as c₁₁ = 102 GPa, c₁₂ = 31 GPa, e₃₁ = −17.05 C/m² and κ₃₃ = 1.76 × 10⁻⁸ C/Vm. Since the surface material constants are not completely available due to the lack of atomic calculations and experiments, the values adopted in [18] are used in the current work, i.e., c^s₁₁ = 7.56 N/m, e^s₃₁ = −3.0 × 10⁻⁸ C/m. In addition, the other surface material constants are taken as σ⁰ = 1.0 N/m, c^s₁₂ = 3.3 N/m and c^s₆₆ = 2.13 N/m.

For a PNP (a = b = 20h) subjected to V = − 0.1 V, fig. 2 shows the normalized first mode resonant frequency vs. the plate thickness, where ω^V is the resonant frequency of the PNP without the surface effects. With the considered values of the surface material constants, it is found that the separate influence of the surface piezoelectricity is obvious the largest in comparison with the residual surface stress and the surface elasticity for the PNP with small thickness, for example, h < 30 nm. This observation indicates the importance of applying the surface piezoelectricity model to investigate the mechanical property of piezoelectric nanostructures. With the increase of the plate thickness h, the influence of the surface effects diminishes and this normalized resonant frequency tends to approach 1. As reported in [13], the residual surface stress may vary from a negative value to a positive one. The combined surface effects with setting σ⁰ = 0 N/m and σ⁰ = −1.0 N/m are also provided in this figure for comparison with σ⁰ = 1.0 N/m. The obvious discrepancy among these curves indicates that this surface piezoelectricity model is sensitive to the values of surface material constants. Figure 3 plots the normalized resonant frequency vs. the plate thickness for the PNP with different in-plane aspect ratio, b/a = 0.25, 0.5 and 1.0 for example. It is observed that the influence of the surface effects on the normalized resonant frequency increases as b/a increases for a given a/h. When the in-plane aspect ratio b/a is fixed, the influence of the surface effects also increases with the aspect ratio a/h. These results indicate that the surface effects are more prominent for PNPs with smaller thickness and larger in-plane area.

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Figure 4 depicts the variation of ω/ω⁰ vs. h with ω⁰ being calculated without considering the surface effects and the applied electric potential. It is seen that the influence of the surface effects is significantly affected by the applied electrical load, which is similar to that observed for a simply supported piezoelectric nanobeam [20]. The variation of the resonant frequency with the applied electrical load suggests possible frequency tuning of PNPs by the applied electric potentials. Such frequency tuning concept is expected to provide guidelines for the design and applications of PNPs as resonators. It is interesting to note that with the increase of the electric potential or the decrease of the plate thickness, the resonant frequency may drop down. This phenomenon indicates a possible mechanical buckling of the PNP caused by the combined electrical load and the surface effects, which is an important issue needs to be addressed in order to keep the mechanical integrity of the structures.

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Figure 5 shows the normalized critical electric potential V_cr/V⁰_cr for buckling (V⁰_cr is calculated without surface effects) vs. the plate thickness h. Similar to the trend observed in fig. 2, the surface effects on V_cr/V⁰_cr are more prominent for smaller h. Although the combined surface effects are not significant, the separate influence of the surface piezoelectricity and the residual surface stress is obvious when h is small, for example, h < 30 nm. It is the competition between these two surface effects that ends up with a smaller combined effect. Again this result suggests the importance of incorporating the surface piezoelectricity into the plate model to study the mechanical behavior of the PNP, otherwise may lead some misleading predictions. The influence of the surface effects on the critical electric potential of the PNP with different aspect ratios is displayed in fig. 6. It is found that the influence of the surface effects is significantly affected by the aspect ratio. When a/h = 20, the surface effects decrease the critical electric potential of the PNP for b/a = 0.25, 0.5 and 1.0. However, such a trend is changed for a PNP with larger surface area, a/h = 30 and b/a = 1.0 for example. This phenomenon could be explained by studying the individual surface effect as presented in fig. 7. Since the influence of the surface elasticity is much smaller than that of the residual surface stress and surface piezoelectricity, we just ignore the surface elasticity here. From fig. 7, it is found that the residual surface stress always increases the critical electric potential, and such effect is further enhanced with the increase of the in-plane aspect ratio b/a. However, the surface piezoelectricity always decreases the critical electric potential and its influence is independent of the aspect ratio b/a. Therefore, the residual surface stress becomes dominant with the increase of b/a and a/h, resulting the trend change for the combined surface effects on the critical electric potential observed in fig. 6.

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In summary, a modified plate model is developed based on the classical Kirchhoff plate theory and the surface piezoelectricity model to investigate the surface effects on the vibration and buckling of the PNPs. Simulation results show that the surface piezoelectricity has a significant effect on the resonant frequency and critical electric potential for buckling, indicating the importance of using the surface piezoelectricity model in predicting the mechanical behavior of the PNPs. The surface effects on the resonant frequency of the PNPs are found more prominent for the PNPs with smaller thickness and larger in-plane aspect ratio, while the influence of the surface piezoelectricity on the critical electric potential for buckling is independent of such aspect ratio. This study also suggests the possible frequency tuning of PNPs via applied electric potentials. The fundamental investigations on the mechanical behavior of PNPs carried out in this work might be helpful for the design and applications of PNP-based nanodevices.

This work was supported by Natural Sciences and Engineering Research Council of Canada (NSERC) and Academic Development Fund of UWO.