#### Abstract

A novel electric Gibbs function was proposed for the piezoelectric microbeams (PMBs) by employing a modified couple stress theory. Based on the new Gibbs function and the Euler-Bernoulli beam theory, the governing equations which incorporate the effects of couple stress, flexoelectricity, and piezoelectricity were derived for the mechanics of PMBs. The analysis of the effective bending rigidity shows the effects of size and flexoelectricity can greaten the stiffness of PMBs so that the natural frequency increases significantly compared with the Euler-Bernoulli beam, and then the mechanical and electrical properties of PMBs are enhanced compared to the classical beam. This study can guide the design of microscale piezoelectric/flexoelectric structures which may find potential applications in the microelectromechanical systems (MEMS).

#### 1. Introduction

Piezoelectricity is a well-studied electromechanical coupling effect in which the mechanical strain leads to electric polarization in piezoelectric crystals, or vice versa. Due to the excellent electromechanical characteristics at microscale [1], piezoelectric based microstructures have found a wide range of applications in microtechnology, like microtransducers, microgenerators, microresonantors, and so forth [2, 3]. Particularly, these above microstructures are quasi one-dimensional structures which can be efficiently characterized by simple Euler-Bernoulli beam theory. Flexoelectricity [4–7] is, however, the coupling between the mechanical strain gradient and the electric polarization, and it is a universal electromechanical mechanism in all insulators including piezoelectric materials [8–11]. Shen and Hu [12, 13] introduced an electric Gibbs function to consider both of the piezoelectricity and flexoelectricity and derived the governing equations for dielectric materials. Flexoelectricity has also been found to be useful when it is used for sensing purpose [14, 15]. Moreover, the flexoelectric structures are theoretically predicted to be more sensitive when scaled down to microdomains [16, 17], yet their work did not take the size effect at microscale into account.

At microscale, size effect may be significant and even dominate the behavior of structures. Several strain gradient theories were dedicated to investigating the size effect [18–21]. Among these work, Yang et al. [20] developed a modified couple stress theory with only one high-order material constant to account for the size effect on the microscale structures which makes the size effect more convenient to express. Afterwards, many efforts were devoted to studying the size effect on the mechanical behaviors of microbeams using the modified couple stress theory [22–25]. Among these, Park and Gao [22] developed a new model for the bending of a Bernoulli-Euler beam. Ma et al. [23] employed the Timoshenko beam model to study the size-dependence of static bending and vibration behaviors. Reddy [24] modified Euler-Bernoulli and Timoshenko models for functionally graded beams and studied the size-dependence of the deflection, vibration, and buckling behaviors for beam models. In these studies, the modified couple stress theory was used for elastic beams whereas the electromechanical coupling effects, for example, piezoelectric effect and flexoelectric effect, were not considered. Recently, the modified couple stress theory was utilized by Ansari et al. [25] for discussing the vibration characteristics of microbeams with piezoelectricity; however, flexoelectricity was not taken into account. Li et al. [26] reformulated a flexoelectric theory to study the bending behavior of PMBs with flexoelectricity. However, the vibration behaviors of PMBs with flexoelectricity have not been reported to date, which may be as important as the bending behavior in MEMS.

The goal of this paper is to study the natural frequency of PMBs with considering the size effect, piezoelectricity, and flexoelectricity. To achieve this goal, together with the Euler-Bernoulli beam theory, a new Gibbs function needs to be defined by employing a modified couple stress theory to derive the governing equations for the PMBs. The proposed study can be fundamental for the vibration behaviors of PMBs which may have potential application in the microelectromechanical systems (MEMS).

This paper is organized as follows. After introduction, Section 2 presents an electric Gibbs function based on the flexoelectricity theory and the modified couple stress theory. The dynamics equation and boundary conditions of piezoelectric beams were then derived by using the new Gibbs function and the Euler-Bernoulli beam model. Afterwards, the first-order natural frequencies under different boundary conditions of the current model and the traditional Euler-Bernoulli model were displayed. In particular, the natural frequency of simply supported PMBs subjected to a voltage across the thickness is solved as an example. Then the relationship between the frequency and the effective bending stiffness was discussed and employed to study the effects of couple stress, piezoelectricity, and flexoelectricity in Section 3. The new findings were summarized in Section 4.

#### 2. Dynamics Equation of Piezoelectric Microbeams with Flexoelectricity

An Euler-Bernoulli beam in Figure 1 is considered:

To account for the piezoelectric effect, flexoelectric effect, and nonlocal mechanical and electrical effects, an electric Gibbs energy density including the energy of elasticity, piezoelectricity, flexoelectricity, and couple stress was proposed bywhere is dielectric constant tensor, and are electric field vectors, is the nonlocal electrical coupling coefficient tensor, is elastic stiffness tensor, is strain tensor, is piezoelectric coefficient tensor, is a fourth-order tensor of the flexoelectric coefficient, and and are the gradients of strain and electric field. is deviator part of couple stress tensor and is the symmetric part of curvature tensor, which, using the modified couple stress theory [20, 22], are defined aswhere is Lamé’s constants, is the material length scale parameter, is the rotation vector, and is the gradient of displacement vector, respectively.

Thus the constitutive equations for piezoelectric Euler-Bernoulli beams can be derived aswhere , , and denote the higher-order stress tensor, electric displacement vector, and electric quadrupole, respectively. In the absence of free charges, Gauss’s law requires

Substituting (4) into (5) leads to

The electrical boundary conditions are given bywhere is the electric potential. Combining with (4)–(7) yieldswith . Neglecting the high-order nonlocal electric effect in (8) by assuming yields the following equations [27, 28]:

The elastic energy , kinetic energy , and external work are calculated by the following equations:

Hamilton’s principle reads

Then the governing equation for their transverse vibration can be derived by the vibrational principlewhere

For facilitating, make the substitution of , , , and , and then we have

If the flexoelectric effect is neglected, the effective bending rigidity can be reduced to

If the working temperature of PMBs is higher than Curie’s point, the piezoelectric effect will disappear and then the effective bending rigidity can be rewritten as

The geometric parameters are defined as with and being the cross section area and inertia moment of beams. The natural frequencies for PMBs with different boundary conditions, such as simply supported (S-S), cantilever (C-F), and clamped-clamped (C-C) types, can be calculated based on the following characteristic equations: wherewith representing the angular resonant frequency.

#### 3. Results and Discussion

In this section the formulae derived in Section 2 will be used to study the fundamental vibration of the PMBs. The emphasis is on the effects of size, piezoelectricity, and flexoelectricity. To this end, BaTiO_{3} was chosen as the material of the PMBs where the material constants are listed in Table 1. The cross section of BaTiO_{3} beam is assumed to be square; that is, .

Employing (13), (18), and (19) and the material constants in Table 1, we can easily get the first-order natural frequency of the PMBs . Moreover, the first natural frequency of traditional Euler-Bernoulli beam model ( for C-F type, for S-S type, and for C-C type) is employed for comparison. Here denotes the natural frequency of the present model; that is, both effects of piezoelectricity and flexoelectricity are taken into consideration. Correspondingly, represents the natural frequency when only piezoelectric effect is considered. The difference between the present model and the traditional model can be seen in Figure 2. Hereby PMBs have *μ*m and *μ*m.

From Figure 2 we can see that the differences in the frequencies between the present model and the traditional model increase significantly when the beam length decreases to less than 400 *μ*m and then turn to be nearly the same when the beam length increases to more than 800 *μ*m. In general, the trends in the frequencies of the present model resemble that of the traditional model because the frequency equation of and are very alike with small difference between the bending stiffness and .

##### 3.1. Effect of Piezoelectricity and Flexoelectricity

Throughout the following sections of this work we will calculate the first normalized natural frequencies and , and, for facilitating the discussion, the simply supported (S-S) boundary condition was chosen as a representative. The normalized frequencies are shown in Figure 3 for PMBs where length-to-thickness ratio is fixed at 20 and the length increases from 20 *μ*m to 1000 *μ*m. Accordingly the thickness grows from 1 *μ*m to 50 *μ*m and the material length scale parameter is defined by as others do in [25, 26] which increases from 0.5 *μ*m to 25 *μ*m. In addition, the voltage , 0, and 2 V are applied to reveal the effects of piezoelectricity and flexoelectricity.

**(a)**

**(b)**

In Figure 3(a), where only piezoelectric effect is considered (i.e., without flexoelectricity), = 1.461 is obtained at showing that the size effect increases by 46.1% for all PMBs of different length and length scale parameter . Equations (13) and (14) show the contribution of the size effect on the effective bending stiffness of the PMBs. At *μ*m, increases to 1.465 when V is applied whereas it deceases to 1.457 at V. This result shows that the external voltage leads to the increase (or decrease) of the frequency of PMBs in the presence of a negative (or positive) voltage. A 4 V external voltage can tune the frequency of PMBs by 0.55%, and the tuning of frequency can be greater under greater external voltage. Such a piezoelectric effect can be explained by the electromechanical coupling term in and the extra-axial force in (13) and (14). Specifically, such a piezoelectric effect decreases monotonically with rising length and shows significant length-dependence. As a result with = −2 V or 2 V approaches when the length is sufficiently large; for example, *μ*m. In this case the piezoelectric effect turns out to be negligible.

Different from Figure 3(a), both effects of piezoelectricity and flexoelectricity are considered in Figure 3(b) for the normalized frequency . Thus the difference between the results in Figures 3(a) and 3(b) [i.e., ] reveals the effect of flexoelectricity on the normalized fundamental frequency. First it is noted in Figure 3(b) that, at , becomes length-dependent, which is in contrast to the length-independent in Figure 3(a). At *μ*m flexoelectric effect slightly increases the normalized frequency by 0.73% relative to . The relative increment then decreases with rising and becomes negligible for long and thick PMBs. Qualitatively similar trend is observed for the flexoelectric effect at V, which raises the normalized frequency by 0.26% at *μ*m. The increase of the frequency decreases with rising and almost vanishes at *μ*m. On the other hand, when positive voltage 2 V is applied, it is noted in Figure 3(b) that first decreases with rising and reaches the lowest value at *μ*m. It then increases with increase of and reaches close to when the length is greater than 900 *μ*m. This behavior of at 2 V is qualitatively different from the corresponding -dependence of in Figure 3(a) and can be attributed to the competition between the reversed effects of piezoelectricity (with positive voltage) and the flexoelectricity on the normalized frequency. Here it is noted that considering flexoelectricity always leads to the increase of the frequency independent of the voltage applied. This behavior can be attributed to the fact that flexoelectricity influences the frequency via the electromechanical coupling term in found in (14), which always greatens the effective bending stiffness and also makes the natural frequency size-dependent. Though it can be seen from Figure 3(a) the positive voltage decreases the natural frequency with decreasing , when is shorter than 302 *μ*m flexoelectric effect dominates the electromechanical behaviors resulting in a significant rise in the natural frequency.

##### 3.2. Size Effect on the Electromechanical Coupling in Vibration

After investigating the effects of piezoelectricity and flexoelectricity, we would like to further reveal the influence of the size effect of piezoelectricity and flexoelectricity. Here we first calculated the -dependence of (without flexoelectricity) in Figure 4 for PMBs subjected to a voltage of −2 V, 0, and 2 V, respectively. In the calculations two groups of PMBs are considered, which have an aspect ratio and 40, respectively. In addition the scale material constant is assumed to be a constant and increases from 0 (classical theory) to 2 *μ*m. Figure 4 shows that in general increases with growing for all the PMBs under a given voltage. It is worth mentioning that at the results remain unchanged no matter what aspect ratio is considered. Specifically, the curves associated with different nonzero voltages are almost parallel with one another. These results indicate that the change in normalized frequency, that is, due to piezoelectric effect, is not very sensitive to the variation of the scale parameter (the subscript shows the voltage applied). On the other hand, as rises with rising , the relative change due to piezoelectric effect decreases significantly with increasing (the subscript shows the voltage considered). For example, at relative change due to V ( = −2 V) is 0.56% (−0.56%) for PMBs with = 40 h. However, at = 2 *μ*m such a relative change decreases to 0.47% at = −2 V and −0.48% at = 2 V. In addition it is also noticed that for a constant the piezoelectric effect is less pronounced for shorter PMBs with greater = 20.

Next we shall further exam the influence of the size effect on the effect of flexoelectricity by calculating the -dependence of in Figure 5. It is easy to understand that reflects the pure effect of flexoelectricity. Figure 5 shows that, at , decreases from around 0.98 × 10^{−3} to 0.90 × 10^{−3} when increases from 0 to 2.0 *μ*m. These results indicate that the flexoelectricity increases the frequency of PMBs by less than 0.1%, and such an effect is found to be even smaller for greater or larger size effect. In addition, at = −2 V (or 2 V) the effect of flexoelectricity on PMB increases (decreases) slightly but the influence of voltage becomes smaller for greater . Indeed it is seen from Figure 5 that, in terms of when , is 0.32% higher than but it declines to 0.27% when = 2 *μ*m. In other words, by increasing the effective bending stiffness by flexoelectricity increases; nevertheless the frequency difference of between existing external voltage and no external voltage diminishes.

#### 4. Conclusions

A novel electric Gibbs function was proposed to describe the size effect on the electromechanical coupling behaviors of PMBs by incorporating a modified couple stress theory into Euler-Bernoulli beam model. Based on the obtained model, the effects of piezoelectricity and flexoelectricity and the size effect were examined for the vibration behavior of PMBs. It is found that, at the microscale, the size effect can raise the effective stiffness of the structure and thus increases the natural frequency. With the assumed value of the material length scale parameter , the couple stress dominates the mechanical behavior of the PMBs and increases with rising . Piezoelectric effect is achieved, which can increase or decrease the frequency depending on the axial force induced by external voltage. In particular, the contribution of piezoelectric effect to the natural frequency was found to be decreasing with rising or geometric size of PMBs. In the meantime, the flexoelectricity can also raise the structural stiffness via an electromechanical coupling term, which is relatively small but detectable and always increases the frequency of PMBs. With the development of material science, the higher performance of flexoelectric materials might play an more important role in MEMS.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

The research reported herein was supported under NSFC (no. 11272138) and Specialized Research Fund for the Doctoral Program of Higher Education of China (no. 20123227130002).