1 Introduction

Micro/nanomechanical sensors or actuators, e.g., atomic force microscope cantilever tips, nanobeams, nanorods, and nanowires, have attracted much attention over the past decade. Due to the high sensitivity and increased stability, they have been widely used to measure the physical and mechanical parameters such as mass, hardness, Young's modulus, pressure, and adhesion energy in micro/nanoscales. When classical continuum beam theories, e.g., the Euler-Bernoulli beam theory, the Rayleigh beam theory, and the Timoshenko beam theory^[1-2], are employed for the analysis of these small scale structures, we find that the small scale effect in micro/nanostructures cannot be described due to its scale-free nature. An effective method is molecular dynamic (MD) simulation, which can discretize a micro/nanostructure composed of atoms and molecules. However, the computational cost is rather expensive, considering the long computing time and high requirements in computer performance. Therefore, the nonlocal continuum theory^[3] is developed on the basis of the classical continuum theory by assuming that the stress at a point is a function of the strains at all points of the body. In comparison, the classical continuum theory is referred to as the local theory because it states that the stress at a point only depends on the strain at this point. In view of the scale dependence of the nonlocal effect^[4], the nonlocal theory can better predict the static or dynamic behaviors of a micro/nanostructure. Wang and Hu^[5] modeled the flexural wave propagation in a carbon nanotube with the nonlocal theory and MD simulation, respectively, and showed that the nonlocal theory could predict the MD results better than the local theory when the phase velocity induced by the scale effect for a large wave number decreased. Murmu and Adhikari^[6] investigated the natural frequencies of a single-walled carbon nanotube with the tip mass, and demonstrated the validity by comparing the results obtained from the nonlocal theory and MD simulation. The nonlocal theory has been widely applied for the buckling^[7-10] and vibration analysis^[11-17] of micro/nanostructures. Xu^[12] considered the free transverse vibrations of nanoscaled to microscaled beams with the nonlocal elasticity theory, and found that the nonlocal effect was important for a nanoscale beam. Lei et al.^[15] conducted the frequency response analysis for the nonlocal viscoelastic damped nanobeams under different boundary conditions with transfer function methods.

As mentioned previously, the nonlocal effect should be considered when the length of the beams reaches the micro/nanoscale, at which the surface effects will become important because of the increasing ratio of the surface/interface area to the volume. Therefore, considerable research has been carried out on the investigation of the surface effects in the axial buckling^[18-19] or transverse vibration^[18-22] of a micro/nanobeam. Gurtin et al.^[23] evaluated the effect of the surface stress on the natural frequency of nanobeams. Lu et al.^[20] developed the model of Gurtin et al.^[23] by including the strain-dependent surface stress terms into the motion equation. Wang and Feng^[21] developed a sandwich-beam model to examine the effects of the surface elasticity and residual surface stress on the vibration of microbeams through the generalized Laplace-Young equation^[24]. Then, the sandwich-beam model was extended to analyze the surface effects of nanowires^[18] and nanotubes^[25].

It is noted that the nonlocal effect and surface effects are considered separately in the above references. However, these effects can all have significant effects on the natural frequencies when the beam size shrinks to microns or nanometers^[17-26]. Therefore, we will study the free vibration of a nonlocal Rayleigh beam^[14] with the consideration of the surface elasticity and residual surface stress, i.e., both the nonlocal and the surface effects are considered. Three boundary conditions, i.e., incorporating hinged-hinged, clamped-clamped, and clamped-hinged ends, are examined. For the hinged-hinged beam, an exact and explicit frequency equation is derived with the Rayleigh beam theory, which recovers to a classical Euler-Bernoulli beam by neglecting the effects of the nonlocal elasticity, the surface elasticity, the residual surface stress, and the rotatory inertia. However, the frequency equations obtained for the clamped-clamped and clamped-hinged beams are both transcendental equations, which can only be solved by numerical computations. In fact, a simple explicit frequency equation is desired in the engineering applications to directly characterize the dependence of the natural frequency on the above mentioned effects. Therefore, the Fredholm integral equation^[27-28] will be used in this paper to derive the approximate fundamental frequency equations for the clamped-clamped and clamped-hinged beams due to the importance of the fundament frequency in micro/nanobeams. To the best of the authors' knowledge, the explicit frequency equations have not been reported. Even though the effect of the shear deformation is neglected in this study, the proposed equations can be well applicable for the vibration analysis of various micro/nanomechanicalsensors or actuators^[6-27]. However, when the beam length is insufficiently long or the wave number is very large, the effect of the shear deformation must be considered. In our subsequent works, the explicit frequency equations, incorporating the shear deformation, will be derived.

2 Mathematical modeling

In this section, the governing equation of a nonlocal elastic micro/nanobeam with the surface effects will be derived. To consider the surface effects, the sandwich-beam model with two surface layers and one bulk layer presented by Wang and Feng^[21] is adopted, and the thickness of the surface layers is assumed to approach to zero. The effective bending rigidity of the sandwich beam is given by

(1)

where E, b, and h are Young's modulus, the width, and the thickness of the bulk layer, respectively. I=bh³/12 is the second moment of the area of the bulk layer. E^s is the tensile stiffness of one surface layer, and it is assumed that the two surface layers have identical E^s. The Laplace-Young equation^[24] is introduced to characterize the effect of the residual surface stress by expressing the stress jump < σ_ij⁺-σ_ij^-> across a surface as follows:

(2)

where σ_αβ^s and κ_αβ (α, β=1, 2) are the surface stresses and the surface curvature, respectively, and n_i (i=1, 2, 3) are the unit normal vectors of the surface. The Laplace-Young equation^[24] gives the transverse load on the micro/nanobeam as follows:

(3)

where τ^u and τ^b are the residual surface tensions of the upper and lower surfaces of the beam, respectively. w is the transverse displacement. In this study, x and y denote the coordinates along the beam length and thickness, which are measured from the left end and the mid-plane of the beam, respectively.

To consider the nonlocal elasticity effect, the Eringen normal stress-strain relation of the micro/nanobeam^[11-17] is given as follows:

(4)

where σ_xx and ε_xx are the normal stress and strain, respectively. E_eff is the effective Young's modulus induced by the bulk layer and two surface layers. l_c is an internal characteristic length of the nonlocal effect.

According to the Euler-Bernoulli beam theory, the strain-displacement relation of the beam can be written as follows:

(5)

Multiplying Eq. (4) by -ydA and integrating over the cross-section area A of the beam, we have

(6)

where M is the bending moment, and (EI)_eff is expressed by Eq. (1).

Based on the equilibrium of the force and the moment, we have

(7a)

(7b)

where Q is the shear force, and ρ is the density of the beam. It is noted that the last term of Eq. (7b) represents the effect of the rotatory inertia.

Substituting Eq. (7) into Eq. (6) leads to the following nonlocal bending moment and shear force:

(8a)

(8b)

Substituting Eq. (8b) into Eq. (7a), we can obtain the governing equation of the nonlocal elastic micro/nanobeam with the surface effects as follows:

(9)

We introduce the following dimensionless quantities:

(10)

where L is the beam length. Then, Eq. (9) can be nondimensionalized as follows:

(11)

where

(12)

Physically, A₁ and A₂ indicate the effects of the surface elasticity and the residual surface tension, respectively, A₃ indicates the slenderness ratio of the beam to the rotatory inertia effect, and A₄ is a dimensionless scale parameter characterizing the nonlocal effect. It is noted that, when A₃=A₄=0, the governing equation becomes the one derived by Wang and Feng^[21] for the local Euler-Bernoulli beam incorporating the surface effects. When A₁=A₂=A₃=0, the equation becomes the formulation for the nonlocal Euler-Bernoulli beam without the surface effects^[11]. When A₁=A₂=A₃=A₄=0, the governing equation of the classical Euler-Bernoulli beam is obtained.

To obtain the natural frequencies of free vibration of the beam, we take

(13)

where ω is the dimensionless circular frequency of the vibration.

Substituting Eq. (13) into Eq. (11), we have

(14)

where the primes denote the derivative with respect to ξ.

We set Y(ξ)=Ye^-ikξ, and substitute it into Eq. (14) to lead to the following characteristic equation:

(15)

The solutions of Eq. (15) can be given as follows:

(16a)

(16b)

Therefore,

(17)

The frequency is also called the cut-off frequency^[15-29], which means that the vibration frequency of the beam cannot be beyond it. Therefore, the term (1+A₁+A₂A₄-A₃A₄ω²) in Eq. (16) is always positive for all the possible vibration frequencies, which leads to k₁²>0 and k₂² < 0. Thus, the solution of Eq. (14) has the following form:

(18)

where the constants C_i (i=1, 2, 3, 4) are determined from the boundary conditions, , and .

3 Exact frequency equation

In this section, the exact natural frequency equations of the nonlocal elastic micro/nanobeam will be derived by considering three types of boundary conditions, i.e., hinged-hinged, clamped-clamped, and clamped-hinged ends. To better present these boundary conditions, the bending moment in Eq. (8a) is nondimensionalized and decoupled as follows:

(19)

where

(20)

For the hinged-hinged beam, the transverse displacement and bending moment at the two ends are zero. Therefore, the corresponding expressions of the boundary condition can be written as follows:

(21)

Substituting Eqs. (18) and (20) into Eq. (21), we obtain a series of algebraic equations with respect to C_i (i=1, 2, 3, 4) as follows:

(22a)

(22b)

(22c)

(22d)

The existence of a nontrivial solution in Eq. (22) needs the determinant of the coefficients matrix to be zero. Therefore, we have

(23)

According to Eq. (16), (β₁²+β₂²)²≠0. Hence, we have sin β₁=0 or sinh β₂=0. sinh β₂=0 yields β₂=0, and further leads to ω=0, which is the case of the static analysis. Therefore, sinh β₂≠0. The remaining term in Eq. (23) is sin β₁=0, which leads to β₁=nπ l (n=1, 2, 3, ...). Based on Eq. (16), β₁=nπ gives

(24)

The two roots of Eq. (24) are

(25a)

(25b)

It is noted that ω₁ is the cut-off frequency, and it is obtained as the wave number k approaches to infinity, which is unphysical and should be ignored. Hence, ω₂ is kept for the root of Eq. (24), and it can be rewritten as follows:

(26)

Equation (26) is the dimensionless exact and explicit natural frequency equation of the nonlocal elastic micro/nanobeam with the rotatory inertia and surface effects for the hinged-hinged boundary conditions. It is noted that, when A₃=A₄=0, Eq. (26) becomes

which is the frequency equation derived by Wang and Feng^[21] for the local Euler-Bernoulli beam with the consideration of the surface effects. When A₁=A₂=A₃=0, Eq. (26) becomes

which is the previously obtained frequency expression of the nonlocal Euler-Bernoulli beam neglecting the surface effects^[11]. When A₁=A₂=A₃=A₄=0, Eq. (26) recovers the classical equation of ω_n=n²π².

For the clamped-clamped beam, the boundary conditions are

(27)

In conjunction with Eqs. (18) and (27), the natural frequency of the beam can be obtained by

(28)

For the clamped-hinged beam, we have the four boundary conditions as follows:

(29)

and the corresponding natural frequency equation can be determined as follows:

(30)

4 Approximate fundamental frequency equation

In the preceding section, the exact frequency equations of the nonlocal clamped-clamped and clamped-hinged beams by incorporating the rotatory inertia and surface effects have been derived and presented in Eqs. (28) and (30). Unlike Eq. (26), for the hinged-hinged beam, Eqs. (28) and (30) are both transcendental equations, which means that the numerical calculation such as the Newton-Raphson method must be used to determine the exact natural frequency values. This is rather inconvenient for engineering applications. In practice, a simple explicit expression is required to reveal the dependence of the natural frequency on the nonlocal elasticity, the surface elasticity, the residual surface stress, and the rotatory inertia. Therefore, in this section, the Fredholm integral equation^[27-28] is adopted to derive the approximate fundamental frequency equation for the clamped-clamped and clamped-hinged beams.

Integrating Eq. (14) four times with respect to ξ from 0 to ξ, we have

(31a)

(31b)

(31c)

(31d)

where D_i (i=1, 2, 3, 4) are the coefficients determined from the boundary conditions, and the different boundary conditions produce the distinct sets of D_i. Substituting these four coefficients obtained for the specific boundary conditions into Eq. (31d) yields the required Fredholm integral equation.

We first consider a nonlocal elastic micro/nanobeam incorporating the rotatory inertia and surface effects with clamped-clamped ends. In conjunction with Eqs. (27) and (31), D_i can be calculated as follows:

(32a)

(32b)

(32c)

(32d)

where

(33)

Therefore, Eq. (31d) can be rewritten in the following form by substituting Eq. (32) into it:

(34)

where

(35a)

(35b)

This is the resulting Fredholm integral equation derived for the clamped-clamped beam, and the problem solving the fundamental frequency of the beam has been transformed to determine the eigenvalue problem of the Fredholm integral equation. In order to achieve this, an assumption expression of Y(ξ) satisfying the boundary conditions as shown in Eq. (27) needs to be proposed to characterize the first-order mode shape of the beam with the effects of the nonlocal elasticity, the surface elasticity, the residual surface stress, and the rotatory inertia. In this study, we adopt the first-order mode shape of the free vibration of the classical Euler-Bernoulli beam as the assumption expression. Then, we have

(36)

Substituting Eq. (36) into Eq. (34) and integrating both sides with respect to ξ from 0 to 1, we obtain the approximate dimensionless fundamental frequency of the clamped-clamped beam as follows:

(37)

In the following part, the Fredholm integral equation for a clamped-hinged beam with the consideration of the nonlocal elasticity, the surface elasticity, the residual surface stress, and the rotatory inertia effects is derived.

We insert the boundary conditions in Eq. (29) into Eq. (31) to have

(38a)

(38b)

(38c)

(38d)

Subsequently, the Fredholm integral equation of the clamped-hinged beam can be obtained by substituting Eq. (38) into Eq. (31d), i.e.,

(39)

where

(40a)

(40b)

It is noticeable that the generalized expression form of the integral equation is the same as Eq. (34) for the clamped-clamped beam apart from the distinction in actual formulas of the coefficients D_i as presented in Eqs. (32) and (38), respectively. Similarly, the first-order mode shape of the free vibration of the classical Euler-Bernoulli beam with clamped-hinged ends is used as the assumption expression of Y(ξ) to calculate the eigenvalue of the integral equation as follows:

(41)

Then, substituting Eq. (41) into Eq. (39) and integrating both sides with respect to ξ from 0 to 1, we obtain the approximate dimensionless fundamental frequency of the clamped-hinged beam as follows:

(42)

5 Results and discussion

At first, the accuracy of the approximate fundamental frequency equations presented in Eqs. (37) and (42) is verified by comparing with the exact values calculated from Eqs. (28) and (30). Then, the effects of the nonlocal elasticity, the surface elasticity, and the residual surface stress on the natural frequency of the beam are investigated in detail. For all the computations, A₁ and A₂ characterizing the surface effects both range from -0.2 to 0.2^[21], and A₄ indicating the nonlocal effect varies from 0.0 to 0.1^[29]. A₃ indicating the slenderness of the beam is fixed as 0.001 (h/L≈1/10), which corresponds to a slender beam. For a short beam, e.g., h/L=1/5, the shear deformation should be included to better exhibit the beam deformation. In this study, a Rayleigh beam model neglecting the effect of shear deformation is considered, and a small slender ratio with A₃=0.001 is adopted in the following analyses.

5.1 Validity of approximate fundamental frequencies

To verify the accuracy of the approximate dimensionless fundamental frequencies given in Eqs. (37) and (42), the exact frequency values are obtained by numerically solving Eqs. (28) and (30), and the comparative results are presented in Tables 1 and 2. It is observed that the approximate results are in excellent agreement with the exact ones, and the largest relative errors are 0.71% and 0.32% for the clamped-clamped beam and the clamped-hinged beam, respectively, by taking into account all possible combinations of A₁, A₂, and A₄. The evolution curves of the relative error versus A₁, A₂, and A₄ are plotted in Figs. 1 and 2 to further demonstrate the validity of the approximate fundamental frequency equations. It is clearly noted that the relative errors for these two cases both increase with the increase in A₄. This is because that the accuracy of the approximate fundamental frequencies depends significantly on the assumption expression of Y(ξ), which is represented by the first-order mode shape of the free vibration of the classical Euler-Bernoulli beam. As A₄ increases, the proposed mode shape gradually deviates from the actual one, resulting in a rising relative error. Even so, the approximate fundamental frequency equations still predict a reliable result with high accuracy for wide ranges of A₁, A₂, and A₄.

Another alternative in proposing the assumption expression of Y(ξ) is to expand it as power series^[27-30]. Taking the clamped-clamped beam as an example, the power series expansion is given as follows:

(43)

Apparently, the above expression meets the boundary conditions in Eq. (27). Because an explicit equation for the fundamental frequency needs to be determined, the first order in Eq. (43) is only required, which means N=1. Then, substituting it into Eq. (34) and integrating both sides with respect to ξ from 0 to 1, we obtain

(44)

By comparing the approximate values given by Eqs. (37)and (44) with the exact ones, it is noted from Fig. 3 that Eq. (37) provides a better prediction for the fundamental frequency. Therefore, regarding the first-order mode shape of the free vibration of the classical Euler-Bernoulli beam as an assumption expression of Y(ξ) is a better option.

6 Effects of A₁, A₂, and A₄ on the natural frequencies

In this subsection, the effects of A₁, A₂, and A₄ on the natural frequencies of the micro/nanobeam, where three types of boundary conditions are examined. As discussed in Section 1, considerable research has been carried out to separately investigate the effects of the nonlocal elasticity, the surface elasticity, and the residual surface stress^{[6-14, 20-21, 25, 31]}. Therefore, in this study, we mainly consider the combined effects of the above parameters. In order to reveal effectively these impacts, a frequency ratio is defined as the ratio of the dimensionless frequency for the nonlocal micro/nanobeams with the surface effects to the corresponding local beam neglecting the surface effects.

Figures 4 and 5 display the variation trends of the frequency ratio versus A₁, A₂, and A₄ for the hinged-hinged beam. It is noted that the natural frequency of the nonlocal micro/nanobeam decreases as the scale parameter A₄ increases (see Fig. 4), which is especially remarkable for higher-order vibration modes (see Fig. 5). Therefore, the nonlocal effect cannot be ignored for a micro/nanobeam. Besides, A₁ and A₂, indicating the surface effects, present a significant effect on the frequency ratio. Positive A₁ and A₂ will enlarge the natural frequency of a nonlocal elastic beam, and the natural frequency of the beam will be reduced by negative A₁ and A₂^[21]. Similar evolution trends are also observed for the clamped-clamped beam (see Fig. 6) and clamped-hinged beam (see Fig. 7). Therefore, when the beam size reduces to microns or nanometers, the effects of the nonlocal elasticity, the surface elasticity, and the residual surface stress should all be taken into account to obtain the accurate predictions of the natural frequencies. It is worth noting that the explicit frequency equations as shown in Eqs. (26), (37), and (42) have been proposed in this study to exhibit the relationship between the natural frequency and the above parameters (A₁, A₂, , A₃, and A₄), avoiding complicated numerical computations.

7 Conclusions

In this study, the governing equation of a nonlocal elastic micro/nanobeam with the consideration of the surface effects is deduced based on the Eringen nonlocal elasticity theory^[3] and the Rayleigh beam theory. Three types of boundary conditions, i.e., hinged-hinged, clamped-clamped, and clamped-hinged ends, are examined. For a hinged-hinged beam, an exact and explicit natural frequency equation is obtained. For the clamped-clamped and clamped-hinged beams, the Fredholm integral equation^[27-28] is used to determine the approximate fundamental frequency equations. The approximate equations provide accurate predictions in the fundamental frequency with the largest relative errors of 0.71% and 0.32%, respectively, by comparing with the exact results obtained from the numerical computations. In summary, the explicit frequency equations for three types of boundary conditions are proposed to characterize the relationship between the natural frequency and the effects of the nonlocal elasticity, the surface elasticity, the residual surface stress, and the rotatory inertia. The proposed equations can be easily extended to many other materials and structures, e.g., nanowires, nanorods, and nanotubes.

Acknowledgements The authors would like to thank School of Civil and Environmental Engineering at Nanyang Technological University, Singapore for kindly supporting this research topic.