Vibration Characteristics of Piezoelectric Microbeams Based on the Modified Couple Stress Theory

Abstract

The vibration behavior of piezoelectric microbeams is studied on the basis of the modified couple stress theory. The governing equations of motion and boundary conditions for the Euler-Bernoulli and Timoshenko beam models are derived using Hamilton’s principle. By the exact solution of the governing equations, an expression for natural frequencies of microbeams with simply supported boundary conditions is obtained. Numerical results for both beam models are presented and the effects of piezoelectricity and length scale parameter are illustrated. It is found that the influences of piezoelectricity and size effects are more prominent when the length of microbeams decreases. A comparison between two beam models also reveals that the Euler-Bernoulli beam model tends to overestimate the natural frequencies of microbeams as compared to its Timoshenko counterpart.

1. Introduction

Microbeams in sensors, actuators, and micro-/nanoelectromechanical systems are widely used. Examples can be found in vibration shock sensors [1], electrostatically excited microactuators [2, 3], resonant testing method, and atomic force microscope (AFM) [4, 5]. Some researchers experimentally revealed the size-dependent deformation and vibration behavior of microstructures [6–8]. These experiments showed that the consideration of size-dependent behavior is necessary in the analysis of microbeams. The size-dependent behavior of structures at micron/submicron scales cannot be predicted by classical continuum theories. Therefore, size-dependent continuum models were developed based on some nonclassical continuum theories such as strain gradient theories [9–11], nonlocal elasticity theories [12], and couple stress theories [13–15]. The classical couple stress elasticity theory was introduced by some researchers in 1960s. This theory contains two higher-order material length scale parameters in addition to two Lamé constants. Using this theory, the static and dynamic torsions of a circular cylindrical microbar were investigated by Zhou and Li [16]. The resonant frequencies of a microbeam were studied by Kang and Xi [17]. Using the couple stress theory, Asghari et al. [18] studied the size effects in Timoshenko beams.

Due to difficulties in determining the material length scale parameters, a modified couple stress theory with only one additional length scale parameter was introduced by Yang et al. [19]. Using the modified couple stress theory, the static mechanical properties of Euler-Bernoulli beam model were investigated by Park and Gao [20], and they explicated the bending test outcomes of an epoxy polymeric beam. A Timoshenko beam model was introduced by Ma et al. [21] to study the size-dependent static bending and free vibration properties. Based on the couple stress theory, Reddy [22] defined nonlinear size-dependent Euler-Bernoulli and Timoshenko functionally graded beam models and by using the Navier solution determined the natural frequency, buckling load, and deflection for beams with simply supported boundary conditions. The modified couple stress theory was also used by Kong et al. [23] to obtain the governing equation and boundary and initial conditions of an Euler-Bernoulli microbeam. Asghari et al. [24] presented a nonlinear Timoshenko beam model based on the modified couple stress theory and determined the nonlinear size-dependent static and vibration behavior. The size-dependent behavior of microbeams made of functionally graded material (FGM) using the modified couple stress theory was studied in [25]. Asghari et al. [26] presented a size-dependent formulation for FGM Timoshenko beams on the basis of the modified couple stress theory. Xia et al. [27] presented a nonlinear Euler-Bernoulli beam model based on the modified couple stress theory. The nonlinear size-dependent static bending and buckling and free vibration of beams were investigated in their work. Utilizing the modified couple stress Euler-Bernoulli beam theory, Wang et al. [28] analyzed the nonlinear free vibration behavior of microbeams. They concluded that the nonlinear vibration frequency obtained by their model is higher than that predicted by the classical continuum theory.

Piezoelectric components are widely used in various micro- and nanodevices. Recently, the analysis of nanoscale piezoelectric structures has attracted considerable attention because of their increasing applications. An analytical model was presented by Gheshlaghi and Hasheminejad [29] to predict surface effects on the free vibration of piezoelectric nanowires. They derived the governing equation of motion, the exact expressions for the natural frequencies, and the fundamental buckling voltage for simply supported piezoelectric nanowires. With the consideration of surface effects and surface piezoelectricity, Yan and Jiang [30] studied the electromechanical coupling behavior of piezoelectric nanowires using the Euler-Bernoulli beam theory. The effects of surface stresses on the vibration and buckling of piezoelectric nanowires were analyzed by Wang and Feng [31] by using the Euler-Bernoulli beam model. Surface effect on the buckling of piezoelectric nanofilms was investigated by Zhang et al. [32]. Yan and Jiang [33] studied electromechanical response of a curved piezoelectric nanobeam with the consideration of surface effects. K. Wang and B. Wang [34] analyzed the surface effects on the buckling of piezoelectric nanobeams. The nonlinear vibration of piezoelectric nanobeams was studied by Ke et al. [35] based on the nonlocal theory. Li et al. [36] investigated the postbuckling of piezoelectric nanobeams with surface effects. The effect of electrostatic force on piezoelectric nanobeams was studied by Liang and Shen [37]. Yan and Jiang [38] studied the vibration and buckling behavior of piezoelectric nanobeams with surface effects.

This paper presents the vibrational analysis of piezoelectric microbeams based on the modified couple stress theory. The governing equations of motion and associated boundary conditions for both Euler-Bernoulli and Timoshenko beam models are derived on the basis of Hamilton’s principle. For microbeams under simply supported boundary conditions, an exact expression of natural frequencies is presented. The obtained natural frequencies are normalized by using the natural frequency of classical Euler-Bernoulli beam model and then are used to illustrate the influences of piezoelectricity and size effects on the vibrational behavior of piezoelectric microbeams.

2. Modified Couple Stress Theory

In this theory, for infinitesimal deformation, the strain energy density of a linear elastic material is described by 1𝑢=2𝜎_𝑖𝑗𝜀_𝑖𝑗+𝑚_𝑖𝑗𝜒_𝑖𝑗𝑖,𝑗=1,2,3. (1)

For isotropic cases, the components of (1) are as follows: 𝜎_𝑖𝑗=𝜆𝑡𝑟(𝜀)𝛿_𝑖𝑗+2𝜇𝜀_𝑖𝑗,𝜀_𝑖𝑗=12(∇𝑢)_𝑖+(∇𝑢)^𝑇_𝑖,𝑚_𝑖𝑗=𝛽𝜒_𝑖𝑗=2𝑙²𝜇𝜒_𝑖𝑗,𝜒_𝑖𝑗=12(∇𝜃)_𝑖+(∇𝜃)^𝑇_𝑖, (2)(3)(4)(5) where 𝜎_𝑖𝑗, 𝜀_𝑖𝑗, 𝜒_𝑖𝑗, and 𝑚_𝑖𝑗 represent the components of the stress tensor, the strain tensor, the curvature tensor, and the deviatoric part of couple stress tensor, respectively. Also, 𝑢 and 𝜃 denote the displacement vector and the rotation vector, respectively. It should be noted that 𝜃=curl(𝑢)/2. In addition, 𝜆 and 𝜇 denote the Lamé constants, and 𝑙 is the material length scale parameter. It is recalled that 𝜆=𝐸𝜈/(1+𝜈)(1−2𝜈) and =𝐸/2(1+𝜈), where 𝐸 and 𝜈 are Young’s modulus and Poisson’s ratio, respectively. The parameter 𝛽 is a higher-order modulus and is considered as the rotational modulus that indicates the resistance of the material versus the gradient of its elements rotation. This parameter is relevant to the shear modulus 𝜇 and the length scale parameter 𝑙 as 𝛽=2𝑙²𝜇.

3. Derivation of Governing Equations and Boundary Conditions

For an Euler-Bernoulli beam, the displacement field is expressed as 𝑢_𝑥()=−𝑧𝜕𝑤𝑥,𝑡,𝑢𝜕𝑥_𝑦𝑢=0,_𝑧=𝑤(𝑥,𝑡), (6) in which 𝑢_𝑥, 𝑢_𝑦, and 𝑢_𝑧 denote the displacement along the axes 𝑥, 𝑦, and 𝑧, respectively, and 𝑤(𝑥,𝑡) is the lateral deflection of the beam.

Also, the displacement field of a Timoshenko beam is 𝑢_𝑥(),𝑢=𝑧𝜓𝑥,𝑡_𝑦𝑢=0,_𝑧=𝑤(𝑥,𝑡), (7) where 𝜓(𝑥,𝑡) denotes the rotation angle of the beam cross-section about the 𝑦-axis.

The nonzero component of the strain tensor for an Euler-Bernoulli beam is expressed as 𝜀_𝑥𝑥𝜕=−𝑧²𝑤𝜕𝑥² (8) and, for a Timoshenko beam, the nonzero components of the strain tensor are in the forms 𝜀_𝑥𝑥=𝑧𝜕𝜓,𝜀𝜕𝑥_𝑥𝑧=12𝜓(𝑥,𝑡)+𝜕𝑤.𝜕𝑥 (9)

From 𝜃=curl(𝑢)/2, the relations for the Euler-Bernoulli beam are written as 𝜃_𝑦=−𝜕𝑤𝜃𝜕𝑥^𝑥=𝜃_𝑧=0. (10)

Substituting (10) into (5), the expression for the only nonzero component of the symmetric part of the curvature tensor is obtained as follows: 𝜒_𝑥𝑦=𝜒_𝑦𝑥1=−2𝜕²𝑤𝜕𝑥². (11)

And, for the Timoshenko beam, the obtained relations are 𝜃_𝑦=12𝜓(𝑥,𝑡)−𝜕𝑤𝜕𝑥,𝜃_𝑥=𝜃_𝑧=0. (12)

In addition, substitution of (12) into (5) gives the expression for the only nonzero component of the symmetric part of the curvature tensor for this beam model as 𝜒_𝑥𝑦=𝜒_𝑦𝑥=14𝜕𝜓−𝜕𝜕𝑥²𝑤𝜕𝑥². (13)

For Euler-Bernoulli beam, substitution of (11) into (4) yields the nonzero component of the deviatoric part of the couple stress tensor in terms of the kinematic parameters as 𝑚_𝑥𝑦=−𝑙²𝜇𝜕²𝑤𝜕𝑥². (14)

For the Timoshenko beam, (13) can be substituted into (4) to get 𝑚_𝑥𝑦=2𝑙²𝜇14𝜕𝜓−1𝜕𝑥4𝜕²𝑤𝜕𝑥². (15)

The linear constitutive equations for a homogeneous orthotropic piezoelectric material are in the following forms: 𝜎_𝑥𝜎_𝑦𝜎_𝑧𝜏_𝑦𝑧𝜏_𝑧𝑥𝜏_𝑥𝑦=𝐸₁₁𝐸₁₂𝐸₁₃𝐸000₁₂𝐸₂₂𝐸₂₃𝐸000₁₃𝐸₂₃𝐸₃₃000000𝐸₄₄000000𝐸₅₅000000𝐸₆₆𝜀_𝑥𝑥𝜀_𝑦𝑦𝜀_𝑧𝑧𝜀_𝑦𝑧𝜀_𝑧𝑥𝜀_𝑥𝑦−0000𝑒₅00𝑒₁0𝑒₂0𝑒₃𝑒₄0𝜑0000^𝑥𝜑_𝑦𝜑_𝑧,𝐷_𝑥𝐷_𝑦𝐷_𝑧=0000𝑒₅0000𝑒₄𝑒00₁𝑒₂𝑒₃𝜀000^𝑥𝑥𝜀_𝑦𝑦𝜀_𝑧𝑧𝜀_𝑦𝑧𝜀_𝑧𝑥𝜀_𝑥𝑦+𝜂₁000𝜂₂000𝜂₃𝜑_𝑥𝜑_𝑦𝜑_𝑧, (16) in which 𝐸_𝑖𝑗, 𝑒_𝑖, and 𝜂_𝑖 represent the matrices of elastic, piezoelectric, and dielectric constants, respectively. In the piezoelectric materials, the electric-field components are determined by the electric potential Φ as 𝜑_𝑥=−𝜕Φ𝜕𝑥,𝜑_𝑧=−𝜕Φ𝜕𝑧,𝜑_𝑦=0. (17)

The poling direction of the piezoelectric medium is along the positive 𝑥-axis, where (𝑥,𝑧) is a rectangular Cartesian coordinate system as shown in Figure 1.

Figure 1 

A piezoelectric microbeam with rectangular cross-section and its coordinate system.

Numerical simulations in the literature reveal that the electric potential along the microbeam (𝑥-axis) except in the vicinity of two ends is almost constant which results in 𝜑_𝑥≤𝜑_𝑧 [40].

From (16), the electric displacements and stresses for a piezoelectric microbeam are obtained as follows: 𝐷_𝑥=𝑒₅𝜀_𝑥𝑧+𝜂₁₁𝜑_𝑥,𝐷_𝑧=𝑒₁𝜀_𝑥𝑥+𝜂₃₃𝜑_𝑧,𝜎_𝑥=𝐸₁₁𝜀_𝑥𝑥−𝑒₁𝜑_𝑧,𝜎_𝑥𝑧=𝜏_𝑥𝑧=𝐸₅₅𝜀_𝑥𝑧−𝑒₅𝜑_𝑥. (18)(19)

It is mentioned that 𝜑_𝑥≤𝜑_𝑧; 𝜂₁₁ and 𝜂₃₃ are on the same order, so the electric displacement 𝐷_𝑥 in comparison with 𝐷_𝑧 is insignificant.

In the absence of electric charges, electrostatic equilibrium condition is represented as follows [31]: 𝜕𝐷_𝑥+𝜕𝑥𝜕𝐷_𝑧𝜕𝑧=0. (20)

For Euler-Bernoulli beam model, by substituting (6), (8), (18), and (19) into (20) and with neglecting the electric displacement 𝐷_𝑥 in comparison with 𝐷_𝑧, an ordinary differential equation (ODE) is obtained. The expression for electric potential as the solution of the ODE is obtained as follows [41, 42]: 𝑒Φ=−¹𝜆₃₃𝑧²2𝜕²𝑤𝜕𝑥²+𝐶₁𝑧+𝐶₂. (21)

Assuming Φ(−ℎ/2)=0 and Φ(ℎ/2)=2𝑉 gives 𝑒Φ=−¹2𝜆₃₃𝑧²−ℎ²4𝜕²𝑤𝜕𝑥²+1+2𝑧ℎ𝑉. (22)

For the Timoshenko beam model, substitution of (7), (9), (18), and (19) into (20) yields an ODE, and, by solving it, the electric potential is represented as 𝑒Φ=⁵+2𝑒₁4𝜆₃₃𝜕𝜓+𝑒𝜕𝑥4𝜆⁵₃₃𝜕²𝑤𝜕𝑥²𝑧²+𝐶₁𝑧+𝐶₂. (23) Without loss of generality, assuming Φ(−ℎ/2)=0 and Φ(ℎ/2)=2𝑉 leads to 𝑒Φ=⁵+2𝑒₁4𝜆₃₃𝑧²−ℎ²4𝜕𝜓+𝑒𝜕𝑥4𝜆⁵₃₃×𝑧²−ℎ²4𝜕²𝑤𝜕𝑥²+1+2𝑧ℎ𝑉. (24)

For the Euler-Bernoulli beam model, from (8) and (18)–(22), the nonzero components of stress tensor can be written as follows: 𝜎_𝑥𝜕=−𝑧𝜆+2𝜇²𝑤𝜕𝑥²𝑒−𝑧²¹𝜆₃₃𝜕²𝑤𝜕𝑥²+2𝑒₁𝑉ℎ. (25)

The resultant axial force, which is induced by the applied electric potential, is in the form 𝑁_𝑥𝑥=𝑏^ℎ/2_−ℎ/2𝜎_𝑥𝑑𝑧=2𝑉𝑏𝑒₁. (26)

For the Timoshenko beam model, from (9), (18)–(20), (23), and (24), the nonzero components of stress tensor are obtained as 𝜎_𝑥=𝑧𝜆+2𝜇𝜕𝜓𝑒𝜕𝑥+𝑧¹𝑒₅2𝜆₃₃+𝑒²₁𝜆₃₃𝜕𝜓𝑒𝜕𝑥+𝑧¹𝑒₅2𝜆₃₃𝜕²𝑤𝜕𝑥²+2𝑒₁𝑉ℎ,𝜎_𝑥𝑧=𝜎_𝑧𝑥=𝜏_𝑥𝑧=2𝑘_𝑠𝜇12𝜓()+1𝑥,𝑡2𝜕𝑤,𝜕𝑥 (27) where 𝐾_𝑠 is the shear coefficient which for rectangular cross-section is 𝐾_𝑠=5(1+𝜐)/(6+5𝜐).

The resultant axial force is given by 𝑁_𝑥𝑥=𝑏^ℎ/2_−ℎ/2𝜎_𝑥𝑥𝑑𝑧=2𝑉𝑏𝑒₁. (28)

By using (8), (1)–(5), (11), and (14), the expressions for the elastic and kinetic energies of the Euler-Bernoulli beam are obtained in the forms 1𝑈=2^𝐿₀𝜎_𝑖𝑗𝜀_𝑖𝑗+𝑚_𝑖𝑗𝜒_𝑖𝑗=𝑑𝐴𝑑𝑥_𝐿0𝐴−𝑧𝜕𝜆+2𝜇²𝑤𝜕𝑥²𝑒−𝑧²¹𝜆₃₃𝜕²𝑤𝜕𝑥²+2𝑒₁𝑉ℎ×−12𝑧𝜕²𝑤𝜕𝑥²+12𝜇𝑙²𝜕²𝑤𝜕𝑥²1𝑑𝐴𝑑𝑥,𝑇=2^𝐿_0𝐴𝜌𝜕𝑢_𝑥𝜕𝑡²+𝜕𝑢_𝑧𝜕𝑡²=1𝑑𝐴𝑑𝑥2^𝐿₀𝜌𝑧²𝜕𝜕𝑡𝜕𝑤𝜕𝑥²+𝜕𝑤𝜕𝑡²𝑑𝐴𝑑𝑥. (29)

Also, external work is calculated as follows: 𝑊^ext=12^𝐿₀𝑁_𝑥𝑥𝜕𝑤𝜕𝑥²𝑑𝑥. (30)

The Hamilton principle, which is utilized to derive the governing equations of motion, is in the form ^𝑡2_𝑡1𝛿𝑇−𝛿𝑈+𝛿𝑊^ext𝑑𝑡=0. (31)

Using (31), the governing equations and corresponding boundary conditions are obtained as 𝑒𝜆+2𝜇𝐼+²¹𝐼𝜆₃₃+𝐴𝜇𝑙²𝜕⁴𝑤𝜕𝑥⁴−2𝑉𝑏𝑒₁𝜕²𝑤𝜕𝑥²𝜕=𝜌𝐴²𝑤𝜕𝑡²,𝜕𝑀𝜕𝑥^𝑥𝑥+𝑌_𝑥𝑦^𝑥=𝐿_𝑥=0=0or𝛿𝑊|^𝑥=𝐿_𝑥=0=0,𝑀_𝑥𝑥+𝑌_𝑥𝑦^𝑥=𝐿_𝑥=0=0or𝛿𝜕𝑤𝜕𝑥^𝑥=𝐿_𝑥=0=0, (32)(33)(34) where 𝑀_𝑥𝑥=𝑧𝜎_𝑥𝑥+𝑒𝑑𝐴=−𝐼𝜆+2𝜇²¹𝜆₃₃𝜕²𝑤𝜕𝑥²,𝑌_𝑥𝑦=𝑚_𝑥𝑦𝑑𝐴=−𝐴𝜇𝑙²𝜕²𝑤𝜕𝑥². (35)

For the Timoshenko beam model, by using (1)–(5), (9), (12), and (15), the expressions for the elastic and kinetic energies are obtained as follows: 1𝑈=2^𝐿₀𝜎_𝑖𝑗𝜀_𝑖𝑗+𝑚_𝑖𝑗𝜒_𝑖𝑗=𝑑𝐴𝑑𝑥_𝐿0𝐴2𝜇𝑘_𝑠12𝜓()+1𝑥,𝑡2𝜕𝑤𝜕𝑥²+12𝜆+2𝜇𝑧𝜕𝜓𝑒𝜕𝑥+𝑧¹𝑒₅2𝜆₃₃+𝑒²₁𝜆₃₃×𝜕𝜓𝑒𝜕𝑥+𝑧¹𝑒₅2𝜆₃₃𝜕²𝑤𝜕𝑥²+2𝑒₁𝑉ℎ×𝑧𝜕𝜓𝜕𝑥+2𝜇𝑙²×14𝜕𝜓−1𝜕𝑥4𝜕²𝑤𝜕𝑥²²1𝑑𝐴𝑑𝑥,𝑇=2^𝐿_0𝐴𝜌𝜕𝑢_𝑥𝜕𝑡²+𝜕𝑢_𝑧𝜕𝑡²=1𝑑𝐴𝑑𝑥2^𝐿₀𝜌𝑧²𝜕𝜓𝜕𝑡²+𝜕𝑤𝜕𝑡²𝑑𝐴𝑑𝑥. (36)

In this model, the external work is computed as 𝑊^ext=12^𝐿₀𝑁_𝑥𝑥𝜕𝑤𝜕𝑥²𝑑𝑥. (37)

Using (31), the governing equations and corresponding boundary conditions are obtained in the forms 𝑘_𝑠𝜕𝜇𝐴²𝑤𝜕𝑥²+𝜕𝜓𝜕𝑥−2𝑉𝑏𝑒₁𝜕²𝑤𝜕𝑥²+14𝜇𝐴𝑙²×𝜕³𝜓𝜕𝑥³−𝜕⁴𝑤𝜕𝑥⁴−𝑒₁𝑒₅𝐼4𝜆₃₃𝜕³𝜓𝜕𝑥³𝜕=𝜌𝐴²𝑤𝜕𝑡²,𝜆+2𝜇𝐼+2𝑒²₁+𝑒₁𝑒₅2𝜆₃₃1𝐼+4𝜇𝐴𝑙²𝜕²𝜓𝜕𝑥²+𝑒₁𝑒₅𝐼4𝜆₃₃𝜕³𝑤𝜕𝑥³−𝑘_𝑠𝜇𝐴𝜕𝑤()−1𝜕𝑥+𝜓𝑥,𝑡4𝜇𝐴𝑙²𝜕³𝑤𝜕𝑥³𝜕=𝜌𝐼²𝜓𝜕𝑡²,𝑀_𝑥𝑥+𝑄_𝑥+12𝑌_𝑥𝑦^𝑥=𝐿_𝑥=0=0or𝛿𝜓^𝑥=𝐿_𝑥=0=0=0,𝑄_𝑥+12𝑌_𝑥𝑦−𝐼𝜆+2𝜇+2𝑒²₁+𝑒₁𝑒₅2𝜆₃₃𝜕²𝜓𝜕𝑥^2𝑥=𝐿_𝑥=0|=0or𝛿𝑤^𝑥=𝐿_𝑥=0=0,12𝑌_𝑥𝑦+𝑒₁𝑒₅𝐼2𝜆₃₃𝜕𝜓𝜕𝑥^𝑥=𝐿_𝑥=0=0or𝛿𝜕𝑤𝜕𝑥^𝑥=𝐿_𝑥=0=0, (38)(39)(40)(41)(42) where 𝑀_𝑥𝑥=𝑧𝜎_𝑥𝑥𝑑𝐴=𝐼𝜆+2𝜇+2𝑒²₁+𝑒₁𝑒₅2𝜆₃₃×𝜕𝜓+𝑒𝜕𝑥¹𝑒₅𝐼2𝜆₃₃𝜕²𝑤𝜕𝑥²,𝑌_𝑥𝑦=𝑚_𝑥𝑦1𝑑𝐴=2𝐴𝜇𝑙²𝜕𝜓−𝜕𝜕𝑥²𝑤𝜕𝑥²,𝑄_𝑥=𝜎_𝑥𝑧𝑑𝐴=𝑘_𝑠𝜇𝐴𝜕𝑤().𝜕𝑥+𝜓𝑥,𝑡 (43)

4. Navier Solution of the Governing Equations

In order to solve the governing equations of motion, the Navier solution is utilized for both Euler-Bernoulli and Timoshenko beam models and the exact expressions of natural frequencies are derived. For the Euler-Bernoulli beam model, assume the periodic solution is as follows: 𝑤(𝑥,𝑡)=𝑊(𝑥)𝑒^𝑖𝜔𝑡, (44) where 𝜔 is the natural frequency of vibration. Substitution of (44) into (32) gives 𝑟₁𝑑⁴𝑊(𝑥)𝑑𝑥⁴+𝑟₂𝑑⁴𝑊(𝑥)𝑑𝑥⁴−𝑟₃(𝑊(𝑥))=0, (45) where 𝑟₁=𝑒𝜆+2𝜇𝐼+²¹𝐼𝜆₃₃+𝐴𝜇𝑙²,𝑟₂=−2𝑏𝑉𝑒₁,𝑟₃=𝜌𝐴𝜔². (46)

The general solution is in the form 𝑊(𝑥)=𝑎₁sin𝛼𝑥+𝑎₂cos𝛼𝑥+𝑎₃sin𝛽𝑥+𝑎₄cos𝛽𝑥, (47) where 𝛼²=12𝑟₁𝑟₂+𝑟²₂+4𝑟₁𝑟₃,𝛽²=12𝑟₁−𝑟₂+𝑟²₂+4𝑟₁𝑟₃, (48) and 𝑎₁ to 𝑎₄ are the integration constants which can be defined by imposing the problem boundary conditions. For a simply supported piezoelectric microbeam, the expression for 𝑛th natural frequencies of vibration is obtained as follows: ()×𝜔=𝑛𝜋−𝜌𝐴−𝜋²𝑒𝜆+2𝜇𝐼+²¹𝐼𝜆₃₃+𝐴𝜇𝑙²𝑛²+2𝑏𝑒₁𝑉𝐿^21/2×𝜌𝐴𝐿²⁻¹. (49)

For the Timoshenko beam model, assume the periodic solutions are as follows: 𝑤(𝑥,𝑡)=𝑐₁sin𝑛𝜋𝑥𝐿𝑒^𝑖𝜔𝑡,𝜓(𝑥,𝑡)=𝑐₂cos𝑛𝜋𝑥𝐿𝑒^𝑖𝜔𝑡. (50)(51)

Substituting (50) and (51) into (38) and (39) and then rearranging them in matrix form yield 𝐴₁₁𝐴₁₂𝐴₂₁𝐴₂₂𝑐₁𝑐₂=00, (52) where 𝐴₁₁=14𝜇𝐴𝑙²𝑛²𝜋²𝐿³−14𝑒₁𝑒₅𝐼𝑛³𝜋³𝜆₃₃𝐿³−𝑘_𝑠𝜇𝐴𝑛𝜋𝐿,𝐴₁₂=−𝑘_𝑠−𝜇𝐴𝜆+2𝜇𝐼+2𝑒²₁+𝑒₁𝑒₅2𝜆₃₃1𝐼+4𝜇𝐴𝑙²𝑛²𝜋²×𝐿²⁻¹+𝜌𝐼𝜔²,𝐴₂₁𝑘=−^𝑠𝜇𝐴𝑛²𝜋²𝐿²+𝜌𝐴𝜔²+2𝑉𝑏𝑒₁𝑛²𝜋²𝐿²−14𝜇𝐴𝑙²𝑛⁴𝜋⁴𝐿⁴,𝐴₂₂=14𝜇𝐴𝑙²𝑛³𝜋³𝐿³−1414𝑒₁𝑒₅𝐼𝑛³𝜋³𝜆₃₃𝐿³−𝑘_𝑠𝜇𝐴𝑛𝜋𝐿. (53)

For a nontrivial solution of (52), the determinant of the coefficient matrix must be equal to zero that yields a characteristic equation with the eigenvalues which are the 𝑛th natural frequencies of vibration for a simply supported Timoshenko beam. The expression for the natural frequency of this beam model is given in the appendix.

5. Results and Discussion

In this section, numerical results are presented in order to demonstrate the influences of piezoelectricity and size effects on the vibration behavior of simply supported piezoelectric microbeams. The material properties of the piezoelectric microbeam are given in Table 1 [39]. Also, microbeam’s height ℎ and its width 𝑏 are assumed to be equal (𝑏=ℎ).

Table 1 

Material properties of piezoelectric microbeam constituents [39].

The natural frequency is normalized with respect to the frequency of classical Euler-Bernoulli beam model in the form 𝜔₀=(𝑛𝜋/𝐿)²𝐸𝐼/𝜌𝐴. Also, the results are given for the first mode of vibration (𝑛=1).

Figure 2 shows the variation of the normalized natural frequency of the Euler-Bernoulli microbeam with the length of beam for different values of voltage (𝑉=0,±1volt) and different values of ℎ/𝑙 and with the assumption of 𝐿/ℎ=20. Similar graphs for the Timoshenko microbeam are displayed in Figure 3 by assuming that 𝐿/ℎ=10. It is clear that, by increasing the length, the natural frequency for the negative values of voltage increases and for the positive values decreases. As it can be seen from these figures, for a given value of length, by increasing the value of thickness-to-length scale parameter ratio ℎ/𝑙, the natural frequency decreases. In addition, it can be found from these figures that, for all values of ℎ/𝑙, the value of normalized natural frequency is higher than the one obtained from the classical theory (𝑙=0). For short microbeams, the effects of piezoelectricity and size effects on the normalized natural frequency are more prominent. The results of the normalized natural frequency for the negative and positive values of voltage are converged after a specific length of beam and the influences of piezoelectricity vanish.

(a)

(b)

(c)

(d)

Figure 2 

Variation of the normalized natural frequency of Euler-Bernoulli microbeam with length for different values of voltage (𝐿/ℎ=20): (a) ℎ/𝑙=2, (b) ℎ/𝑙=5, (c) ℎ/𝑙=10, and (d) 𝑙=0.

(a)

(b)

(c)

(d)

Figure 3 

Variation of the normalized natural frequency of Timoshenko microbeam with length for different values of voltage (𝐿/ℎ=10): (a) ℎ/𝑙=2, (b) ℎ/𝑙=5, (c) ℎ/𝑙=10, and (d) 𝑙=0.

Figure 4 demonstrates the variation of the normalized natural frequency of Euler-Bernoulli microbeam with voltage for different values of length-to-thickness ratio (𝐿/ℎ=20 and 40) and different values of ℎ/𝑙. Similar graphs for the Timoshenko microbeam are shown in Figure 5 for different values of length-to-thickness ratio (𝐿/ℎ=10, 20, and 40). As it can be observed from these figures, increasing the value of length-to-thickness ratio leads to the higher values of the normalized natural frequency and, also in the higher values of voltage, the normalized natural frequency becomes zero and buckling occurs. For the Timoshenko beam model, the influences of length-to-thickness ratio, voltage, and length scale parameter are similar to those in the Euler-Bernoulli beam model. In both beam models, the influences of piezoelectricity increase by the increment in length-to-thickness ratio.

(a)

(b)

(c)

(d)

Figure 4 

Variation of the normalized natural frequency of Euler-Bernoulli microbeam with voltage for different values of 𝐿/ℎ: (a) ℎ/𝑙=2, (b) ℎ/𝑙=5, (c) ℎ/𝑙=10, and (d) 𝑙=0.

(a)

(b)

(c)

(d)

Figure 5 

Variation of the normalized natural frequency of Timoshenko microbeam with voltage for different values of 𝐿/ℎ: (a) ℎ/𝑙=2, (b) ℎ/𝑙=5, (c) ℎ/𝑙=10, and (d) 𝑙=0.

Figure 6 presents the variation of the normalized natural frequency of Euler-Bernoulli and Timoshenko microbeams with the length for different values of 𝑙/ℎ and with the assumption of 𝐿/ℎ=40. In this figure, the frequency of Euler-Bernoulli beam model has been compared with the frequency of the Timoshenko beam model. As it can be seen from this figure, the Euler-Bernoulli beam model tends to overestimate the natural frequencies as compared to the Timoshenko beam model. In addition, the comparison between two graphs demonstrates that increasing the value of thickness-to-length scale parameter ratio causes the better agreement between two curves.

(a)

(b)

Figure 6 

Variation of the normalized natural frequency of Euler-Bernoulli and Timoshenko microbeams with length (𝐿/ℎ=40): (a) ℎ/𝑙=2 and (b) ℎ/𝑙=10.

6. Conclusion

In this paper, an exact solution was obtained for the vibration analysis of piezoelectric microbeams on the basis of the modified couple stress theory using both Euler-Bernoulli and Timoshenko beam models. First, the electric-field equations and the governing equations of motion were derived using Hamilton’s principle. Then, the Navier method was utilized for the simply supported piezoelectric microbeams to solve the governing equations and to achieve an exact expression for natural frequencies. By using the normalized natural frequencies, the effects of piezoelectricity and small scale for both beam models were illustrated. The numerical results showed that, for the slender microbeams with larger length-to-thickness ratio, the effect of piezoelectricity is more considerable and, by increasing the length of microbeam, piezoelectricity and size effects gradually vanish. By increasing the value of thickness-to-length scale parameter ratio, the normalized natural frequencies decrease for both beam models. Also, it was indicated that the Euler-Bernoulli beam model tends to overestimate the natural frequencies of microbeams in comparison with the Timoshenko beam model.

Appendix

Exact expression for the natural frequency of the Timoshenko beam is as follows: 1𝜔=−41𝜆₃₃𝜌𝐼𝐴𝐿²×√2𝜆₃₃×𝜆𝜌𝐼𝐴₃₃𝑘_𝑠𝜇𝐴²𝐿⁴+4𝜆₃₃𝐼𝐿²𝑘_𝑠𝜇𝐴𝑛²𝜋²−8𝜆₃₃𝐼𝐿²𝑏𝑉𝑒₁𝑛²𝜋²+4𝜆₃₃×𝜆+2𝜇𝐼+2𝑒²₁+𝑒₁𝑒₅2𝜆₃₃𝐼+14𝜇𝐴𝑙²𝑛²𝜋²𝐴𝐿²+𝜆₃₃𝐼𝜇𝐴𝑙²𝑛⁴𝜋⁴+16𝜆²₃₃𝑘²_𝑠𝜇²𝐴⁴𝐿⁸+16𝜆²₃₃×𝜆+2𝜇𝐼+2𝑒²₁+𝑒₁𝑒₅2𝜆₃₃𝐼+14𝜇𝐴𝑙²²×𝑛⁴𝜋⁴𝐴²𝐿⁴+𝜆²₃₃𝐼²𝜇²𝐴²𝑙⁴𝑛⁸𝜋⁸+32𝜆²₃₃𝑘²_𝑠𝜇²𝐴³𝐿⁶𝐼𝑛²𝜋²+64𝜆²₃₃𝑘_𝑠𝜇𝐴²𝐿⁶𝐼𝑏𝑉𝑒₁𝑛²𝜋²+32𝜆²₃₃𝑘_𝑠𝜇𝐴³𝐿⁶×𝜆+2𝜇𝐼+2𝑒²₁+𝑒₁𝑒₅2𝜆₃₃𝐼+14𝜇𝐴𝑙²𝑛²𝜋²𝜋²−40𝜆²₃₃𝑘_𝑠𝜇²𝐴³𝐿⁴𝐼𝑙²𝑛⁴𝜋⁴+16𝜆²₃₃𝐼²𝐿⁴𝑘²_𝑠𝜇²𝐴²𝑛⁴𝜋⁴−64𝜆²₃₃𝐼²𝐿⁴𝑘_𝑠𝜇𝐴𝑛⁴𝜋⁴𝑏𝑉𝑒₁−32𝜆²₃₃𝐼𝐿⁴𝑘_𝑠𝜇𝐴²𝑛⁴𝜋⁴×𝜆+2𝜇𝐼+2𝑒²₁+𝑒₁𝑒₅2𝜆₃₃𝐼+14𝜇𝐴𝑙²+8𝜆²₃₃𝐼²𝐿²𝑘_𝑠𝜇²𝐴²𝑛⁶𝜋⁶𝑙²+64𝜆²₃₃𝐼²𝐿⁴𝑏²𝑉²𝑒²₁𝑛⁴𝜋⁴+64𝜆²₃₃𝐼𝐿⁴𝑏𝑉𝑒₁𝑛⁴𝜋⁴×𝜆+2𝜇𝐼+2𝑒²₁+𝑒₁𝑒₅2𝜆₃₃𝐼+14𝜇𝐴𝑙²𝐴−16𝜆²₃₃𝐼²𝐿²𝑏𝑉𝑒₁𝑛⁶𝜋⁶𝜇𝐴𝑙²−8𝜆²₃₃𝜆+2𝜇𝐼+2𝑒²₁+𝑒₁𝑒₅2𝜆₃₃𝐼+14𝜇𝐴𝑙²𝑛⁶𝜋⁶𝐴²𝐿²𝐼𝜇𝑙²+4𝐼𝐴³𝐿²𝑛⁶𝜋⁶𝜇²𝑙⁴𝜆²₃₃−8𝐼²𝐴²𝐿²𝑛⁶𝜋⁶𝜇𝑙²𝜆₃₃𝑒₁𝑒₅+4𝐼³𝐴𝐿²𝑛⁶𝜋⁶𝑒²₁𝑒²₅+32𝐼²𝐴²𝐿⁴𝑛⁴𝜋⁴𝑒₁𝑒₅𝑘_𝑠𝜇𝜆₃₃^1/21/2. (A.1)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.