1 Introduction

With high output power and simple manufacturability, micro/nano scale piezoelectric structures have a widespread application in micro-nano-electro-mechanical systems (NEMS/MEMS) and biotechnology devices or systems, including biosensors, actuators, transistors, probes, and resonators^[1-4]. Such devices or systems, thus, are referred to as micro electromechanical systems, and are promising in the field of renewable/alternative energy. The structural sizes of these systems range from hundreds of nanometers to micrometers typically. Because of their small size, the behaviors and properties of micro/nano scale structures are very different from those of macrostructures^[5-7]. To explain the physical phenomena and predict the properties of micro/nano scale structures, two characteristics are treated as main factors affecting the properties of microstructures. One is the nature and scale of internal microstructure unit. Based on this factor, some non-classical theories are developed to account for microstructures related effects, such as the strain gradient theory^[8-9], the couple stress theory^[10-11], the micromorphic theory^[12-13], and the nonlocal theory^[14-15]. These high-order theories contain some new material constants whose values are excessively difficult to be determined. In addition, few experiments have been done to test length scale parameters in the constitutive equations of high-order theories. The other factor that affects the properties of microstructures is the free surface of the structure. Micro/nano scale structures have larger surface to volume ratios, and the energy of surface atoms differs from that of the bulk atoms. Therefore, surface effects of microstructures dominate material's physical and chemical properties^[16-18]. There are also some surface models studying the effects of surface properties on the behaviors of micro/nano materials. Different from high-order theories, the new material parameters can be easily measured by the nano-indentation^[19] and the first-principle method^[20]. Therefore, this paper mainly focuses on the analysis of surface effects and surface theories.

The surface elasticity theory, which was proposed by Gurtin and Murdoch^[21-22], is exploited extensively in investigating surface effects of elastic materials based on the assumption that the displacement between the bulk material and the surface is continuous. For convenience and efficiency, the linear surface elasticity theory is usually used to study the bending and buckling behavior of nanowires and nanobeams^[23-26]. Among these studies, Lu et al.^[23] investigated the effect of surface stress on the resonance frequency of a cantilever sensor by incorporating strain-dependent surface stress terms into the equations of motion. Their study implies that the surface stress can affect the stress state of the bulk beam. To find the surface effects on the flexural deformation of nanobeams, Wang and Feng^[24] proposed a generalized Young-Laplace model with equivalent distributed terms in the governing equation of the beam. He and Lilley^[25] then incorporated the generalized Young-Laplace model into the Euler-Bernoulli beam theory to study the surface effect on bending resonance of nanowires with different boundary conditions. However, Park and Klein^[27] pointed out that, due to the presence of the surface stresses, the nanowires are initially out of equilibrium, and a compressive relaxation strain should be added at the free end for the fixed/free nanowires. They used the surface Cauchy-Born model to numerically calculate and analyze the surface effect on the resonant property of fixed/fixed and fixed/free nanowires. Song et al.^[28] demonstrated that the Young-Laplace model was not able to predict the bending behavior of nanowires with different boundary conditions correctly owing to the fact that the surface-induced initial stresses were neglected. By comparison with the existing experimental and computational results, it is verified that the surface-induced initial stresses could greatly affect the overall mechanical properties of nanowires. Likewise, other researchers^[29-32] took into account the surface-induced deformation or stresses (or strains) as well.

Even though the Young-Laplace model is in violation of Newton's third law and ignores the initial surface-induced stresses (or strains), it is still in use currently. Huang and Yu^[33] established the surface energy function and governing equations to investigate the effect of surface piezoelectricity on the behavior of a piezoelectric ring. Yan and Jiang^[34-35] studied the surface effects on the behaviors of piezoelectric nanobeams and nanowires. Wang and Feng^[36] and Samaei et al.^[37] extended their Young-Laplace model to piezoelectric nanowires in order to analyze the surface effect on the vibration and buckling of the nanowires. Dai et al.^[20] estimated the surface properties of dielectric materials by using a combination of a theoretical framework and atomistic calculations. Although Jiang et al.^[38] analyzed the influence of internal residual stress induced by the surface stresses on the vibration of piezoelectric nanobeams, they only studied the simply supported case, and other kinds of beams, however, have not yet been reported.

This paper aims to develop a continuum model for the piezoelectric Euler-Bernoulli nanobeam with surface effects considering the surface-induced fields in the bulk beam and the surface area change. The remainder of this paper is organized as follows. In Section 2, the initial strain and electric field of the bulk beam induced by the surface residual stress are determined by the energy minimization. In Section 3, the new governing equations for the doubly-clamped beam and the cantilever beam are deduced by incorporating the nonlinear Green-Lagrangian strain into the Euler-Bernoulli model with consideration of the surface area change. Section 4 provides the static bending and variational solutions of doubly-clamped and cantilever beams. Section 5 deals with the numerical results by comparing the proposed model with the Young-Laplace model. Conclusions are finally drawn in Section 6.

2 Equilibrium state of piezoelectric nanobeam with surface stress effects in absence of external loading

The piezoelectric nanobeam with surface effects is generally decomposed into the bulk part and the surface layer with the negligible thickness. The initial state of the surface is characterized by a uniform initial surface residual stress σ₀. Without any external loading, the whole piezoelectric nanobeam should be in an equilibrium state. In comparison, for the piezoelectric nanobeam with different boundary conditions, the equilibrium states have a big difference. For the doubly-clamped piezoelectric nanobeam, it is assumed to be fabricated through a top-down or etching process. Therefore, it is unable to deform the doubly-clamped nanobeam due to the constraints at both ends, i.e., no field occurs in the bulk of the nanobeam, as shown in Fig. 1(a). For the cantilever piezoelectric nanobeam, because of the free end, surface tension will induce an axial deformation. In its equilibrium state, there should be strain and electric field in the bulk of the beam, as shown in Fig. 1(b). It should be mentioned that the nanobeams in Fig. 1 are slender beams with the large aspect ratio and polarized in the z-direction. Therefore, only the electric field in the z-direction is considerable.

In order to determine the strain and the electric field induced by the initial surface residual stress in the cantilever beam, the potential energy should be obtained. In the bulk part of the piezoelectric nanobeam, the axial stress and the electric displacement can be expressed as

(1)

where c₁₁, e₃₁, and κ₃₃ are the elastic modulus, the piezoelectric constant, and the dielectric constant, respectively.

Through the Gaussian theorem in electro statics (i.e., =0) and without electric loads, the electric displacement is zero in the thickness direction of the beam^[39], which leads to

(2)

As the thickness of surface layer is negligible, the electric displacement on the surface is neglected. Therefore, only the surface stress is considered,

(3)

where c₁₁^s is the surface elastic modulus, and e₃₁^s is the surface piezoelectric constant. The continuity condition between the bulk and surface requires . Then, in the absence of external loading, the beam potential energy can be written as

(4)

where A is the initial cross-section, and C is the initial perimeter of the cross-section.

By virtue of minimization of Π, is determined as

(5)

By combining Eq. (1) and Eq. (5), it gives

(6)

For the doubly-clamped beam,

(7)

3 Governing equation of piezoelectric nanobeam with different boundary conditions

Consider a piezoelectric nanobeam with a rectangular cross section. The thickness, width, and length are denoted by h, b, and L, respectively. The Cartesian coordinate is shown in Fig. 2, where the x-axis is coincident with the neutral axial along the length of the beam, the y-axis is along the width of the beam, and the z-axis is parallel to the thickness of the beam. Based on the Euler-Bernoulli beam model, the displacement field and electric potential in the beam can be expressed as a function of (x, t),

(8)

where u_x, u_y, and u_z are x-, y-, z-components of the displacement vector u of the point (x, y, z) on a beam cross-section. u and w are, respectively, x- and z-components of the displacement vector of the point (x, 0, 0) in the centroidal axis.

The initial state for this section is the equilibrium state studied in Section 2. Therefore, the total strains in the beam consisting of two parts, the strain induced by σ₀ and the strain induced by external loads, can be given as

(9)

Because the boundary condition can affect the beam deflection, the strain expressions for beams with the different boundary conditions are different. For beams with the doubly-clamped boundary condition, no axial deformation is allowed, i.e., u(x, t)=0. And because of =0, Eq. (9) is simplified as

(10)

For the cantilever beam, the resultant axial force and axial strain generated by bending when z=0 should be vanished at the free end, that is,

(11)

Therefore, the total strain can be rewritten as

(12)

The constitutive equations for the bulk piezoelectric material can be written as

(13)

According to Gauss's law, the open circuit condition, and the expression of D_z, the electric field can be expressed as

(14)

Based on the classical piezoelectricity theory, for a linear case, the internal energy density of the bulk piezoelectric nanobeam can be expressed as

(15)

Similar to Eq. (3), the constitutive equation for the surface layer under deformation can be written as

(16)

For the nanobeam, because of the large ratio of surface to volume, the surface area change is considerable after deformation. The internal energy density of the surface layer defined in the deformed surface area is written as

(17)

The relation between the deformed surface area A' and the initial surface area A for the one-dimensional beam is^[30]

(18)

Because the energy is a scalar, the surface internal energy density defined in the initial area can be obtained by multiplying Eq. (17) and the ratio of the deformed surface area to the initial surface area, i.e.,

(19)

By adding the bulk and surface internal energy, the total internal energy for the piezoelectric nanobeam can be obtained by

(20)

Substituting Eqs. (15) and (19) into Eq. (20) and by means of the expression of strains (Eq. (10) or Eq. (12)) and the electric field (Eq. (14)), we can finally obtain the following equations:

for the doubly-clamped beam,

(21)

and for the cantilever beam,

(22)

where I is the bulk moment of inertia, and I_s is the surface moment of inertia.

The kinetic energy of the piezoelectric nanobeam is given by

(23)

where ρ is the mass density.

According to Hamilton's principle^[40], the actual motion minimizes the difference of the kinetic energy and total internal energy. Then, the governing equation of the beams with surface effects and surface-induced initial fields can be obtained by

(24)

For the doubly-clamped beam,

(25)

For the cantilever beam,

(26)

When the surface area change is neglected, the only difference between the doubly-clamped beam and the cantilever beam is P^*. Consistent with the governing equation form of the elastic cantilever beam in Refs. [22] and [30], we have

(27)

However, the Young-Laplace model in Refs. [34] and [35] did not take into account the boundary condition effect. Regardless of the doubly-clamped nanobeam or the cantilever beam, their governing equations are the same as Eq. (24) with

(28)

By comparing the parameters in Eq. (25) with those in Eq. (28), it is found that, when the surface area change is not considered, the governing equation derived by our present formulation for the doubly-clamped beam is the same as that derived by Yan and Jiang^[34].

4 Solution of doubly-clamped and cantilever piezoelectric nanobeams 4.1 Static bending

For the doubly-clamped beam, the governing equation of static bending problem can be expressed as

(29)

where q is the applied uniform load. Using the boundary conditions, i.e., w|_{x=0, x=L} = 0, w'|_{x=0, x=L} =0, the solution of deflection can be obtained as

(30)

with P^* > 0, and

(31)

with P^* < 0.

For the cantilever beam, the governing equation of static bending problem is written as

(32)

When only a concentrated force P is applied at the free end of the cantilever beam, the boundary conditions can be given as

(33)

with . Then, the general solution of deflection is obtained as

(34)

4.2 Free vibration

The natural vibration mode of the beam is set to the form of

(35)

where ω is the angular frequency of the nanobeam.

Then, Eq. (24) can be reduced to an ordinary differential equation with respect to the vertical displacement function W(x),

(36)

For the doubly-clamped beam, solving Eq. (36) yields the general solution of W(x),

(37)

where c_i (i=1, 2, 3, 4) are undetermined coefficients. β₁ and β₂ satisfy

(38)

Making use of the boundary conditions of the doubly-clamped beam,

(39)

the final characteristic equation of the angular resonant frequency ω can be derived as

(40)

While for the cantilever beam, on account of P^* =0, the solution of Eq. (31) can be written as

(41)

where C_i (i=1, 2, 3, 4) are undetermined coefficients, and β is defined by

(42)

The boundary conditions of the cantilever beam can be written as

(43)

Combining Eqs. (41) and (43), we can obtain the final governing equation of the angular resonant frequency ω,

(44)

5 Numerical results and discussion

Due to the lack of experimental results for piezoelectric cantilever nanobeams, by setting e₃₁ =e₃₁^s =0 in Eqs. (26) and (27), the present model for the cantilever beam degenerates into the elastic model to validate our proposed model. Then, the effective elastic stiffness E_eff of the cantilever ZnO nanowire predicted by the present model is compared with the results obtained by the existing continuum model^[28] and the experimental model^[17], where E_eff =(f^*)²E with f^* being the ratio of the resonant frequency calculated with the surface elasticity and surface-induced initial stresses to that calculated without the surface elasticity and surface-induced initial stresses. The material parameters of ZnO are the same as those used in Ref. [28], i.e., E=140 GPa, E^s=267 N/m, σ₀ =-0.91 N/m, and ρ=5 600 kg/m³. Figure 3 shows that the effective elastic stiffness predicted by the present model is consistent with that predicted by the experimental measurement^[17] and also the same as that predicted by the continuum model of Song et al.^[28] which also considered surface elasticity and surface-induced initial stresses.

For piezoelectric nanobeams, the Young-Laplace model proposed by Yan and Jiang^[34] is used as the benchmark. As only a few piezoelectric materials' surface constants are available in the literature, we choose PZT-5H for calculation, with the surface elastic constant c₁₁^s of 7.56 N/m and the surface piezoelectric constant e₃₁^s of -3× 10^-8 C/m. According to the approximation^{[16, 41]}, the residual surface stress σ₀ is set to the order of 0.1 N/m—1 N/m. The bulk material constants of PZT-5H^[34] are taken as c₁₁=126 GPa, e₃₁=-6.5 C/m², κ₃₃ =1.3 × 10^-8 C/(V·m), and the mass density ρ=7 500 kg/m³. The geometric size of the nanobeam is given as L=20h and b=2h.

In terms of the doubly-clamped beam, the difference of the governing equations between our proposed model and the Young-Laplace model of Yan and Jiang^[34] is (EI)^*. Because our proposed model uses the geometric nonlinear strain and considers the surface area change, there may be some differences in the deflection and vibrational frequency predicted by these two models. However, when σ₀=0, the governing equations of our proposed model and Young-Laplace model^[34] are the same. Figure 4 shows the normalized deflection w₁/w₀ of the doubly-clamped beam with σ₀ taking different values, where w₀ is the deflection of the doubly-clamped beam without the surface effect. The results in Fig. 4 reflect that the deflection predicted by our proposed model is very close to that predicted by the Young-Laplace model^[34]. Only when σ₀ =-1 and h < 50 nm, the deflection predicted by our proposed model is a little larger than that predicted by the Young-Laplace model^[34]. Compared with the beam deflection obtained by the case of σ₀=0, it is obvious that the residual surface stress σ₀ has great effects on the deflection of the doubly-clamped beam. Surface effects are mainly induced by the residual surface stress.

Figure 5 shows the normalized first-order resonant frequency ω₁/ω₀ of the doubly-clamped beam as a function of the beam thickness under the condition of the residual surface stress σ₀ taking different values, where . Because the surface area change is considered in our proposed model, the resonant frequency predicted by the present model is slightly larger than that predicted by the Young-Laplace model in Ref. [34] when σ₀ is positive. The result is opposite, when σ₀ is negative. Meanwhile, when σ₀=0, there is no initial surface stress for the doubly-clamped beam. The first-order resonant frequencies predicted by the present model and the Young-Laplace model in Ref. [34] are the same. By comparing the resonant frequencies of doubly-clamped beams with and without initial surface stress in Fig. 5, the residual surface stress σ₀ can also greatly affect the resonant frequency. The smaller the scale of the doubly-clamped beam is, the greater the effect of σ₀ on the resonant frequency is.

The major difference between the Young-Laplace models for the piezoelectric material and our proposed model is the governing equation for the cantilever beam. For our proposed model, the governing equation of the cantilever beam does not have the term of , i.e., P^* =0, because of the effect of the boundary condition or the surface-induced initial strain and electric field. In other words, under the influence of surface stresses, the free end of the cantilever beam leads to the existence of surface-induced fields in the bulk beam. Then, the surface-induced strain and electric field can counteract the effect of the vertical loading due to the surface stress, i.e., . This is consistent with the analysis in Ref. [22].

Figures 6 and 7 show variations of the normalized deflection and the first-order resonant frequency versus the beam thickness with σ₀ taking different values. The results show the normalized deflection and the first-order resonant frequency predicted by our proposed model are dramatically different from those predicted by the Young-Laplace model^[34]. For our proposed model, whether the residual surface stress is positive, negative or zero, the normalized deflection and the first-order resonant frequency only have slight changes with w/w₀ < 1 and ω₁/ω₀ > 1. It means that the surface residual stress has few effects on the deflection and resonant frequency of the cantilever beam, and surface effects have slight effects on the behaviors of static bending and free vibration of the cantilever beam. However, for the Young-Laplace model proposed by Yan and Jiang^[34], the results reflect the surface residual stress σ₀ has significant influences not only on the deflection of the cantilever beam but also on the first-order resonant frequency of the beam. Meanwhile, the direction of residual stress σ₀ determines the change of the normalized deflection and the first-order resonant frequency. When σ₀ =1, w/w₀ > 1 or ω₁/ω₀ < 1, the normalized deflection and frequency decrease with the decreasing beam size. When σ₀ =-1, w/w₀ < 1 or ω₁/ ω₀ > 1, the normalized deflection and frequency increase with the decreasing beam sizes. According to Yan and Jiang's model^[34], surface effects are mainly induced by the surface residual stress σ₀, and have significant effects on the behaviors of the cantilever beam, which is in conflict with the results of the proposed model.

For the elastic material, it was reported by Song et al.^[28] that the surface model without the surface-induced stresses dramatically overestimates the surface stress effects owing to the ignoring of the effects of initial stress. Likewise, the Young-Laplace model does not consider the induced residual strain σ₀ and the electric field. Such similar results can be observed in the piezoelectric materials, i.e., the surface stress effects on the piezoelectric cantilever nanobeam may be overestimated as well.

By synthesizing the results of the doubly-clamped beam and cantilever beam, it is concluded that the boundary conditions of the beam and surface-induced fields in the bulk beam are vital for the analysis of surface effects on the piezoelectric nanobeam behaviors.

6 Conclusions

The boundary condition effects and the surface-induced fields (strain and electric field) are essential for investigating the impact of surface stress on the vibration problems of piezoelectric nanobeams. Because of the boundary conditions, the initial states of the doubly-clamped beam and the cantilever beam are totally different. Based on the initial equilibrium state, by applying the nonlinear Green-Lagrangian strain to the Euler-Bernoulli beam theory and considering the surface area change, this paper develops a new formulation of piezoelectric nanobeams with surface stress effects. By comparison with the Young-Laplace models for the piezoelectric material, we obtain the following findings:

(Ⅰ) For the doubly-clamped beam, the form of governing equation derived by the proposed model is similar to that derived by the surface model without the induced bulk stress σ₀. The major difference between the deflections and resonant frequencies predicted by these two models is induced by the surface area change.

(Ⅱ) For the cantilever beam, due to the fact that the surface-induced initial strain and electric field are ignored in the existing piezoelectric surface model, there is a significant discrepancy between our model and the Young-Laplace model. Through changing the values of residual surface stress σ₀, the deflection and first-order resonant frequency predicted by our proposed model do not change a lot. On the contrary, the deflection and first-order resonant frequency predicted by the model without the induced bulk stress σ₀ are affected greatly. Similar to the comparison results of elastic materials^[28], it is concluded that the surface model without induced fields σ₀ will overestimate the surface stress effects.