New criteria for nanoscale slender beams and thin plates: Low frequency domain of flexural wave

Abstract The classical Euler–Bernoulli beam model and Kirchhoff plate model are very useful on the macroscopic scale. In the context of Eringen’s nonlocal elasticity theory, this article aims to develop the criteria of the applicability of nanoscale Euler–Bernoulli beam and Kirchhoff plate models based on the Timoshenko beam model and the Mindlin plate model via the wave propagation theory. The corresponding governing differential equations for the nanoscale Timoshenko beam and Mindlin plate are derived by the Hamilton’s principle, and the dispersion equations of wave are then obtained. By applying Taylor expansion to the corresponding solutions of the dispersion equations, new criteria are developed, simultaneously taking into account the effects of nonlocal parameters and material properties. When the nonlocal parameter is set to zero, the present criteria may be readily degenerated to their macroscopic counterparts. According to the present criteria, this article systematically evaluates the existing studies in literature. Various works in literature did not consider the effect of the nonlocal parameter, and hence, failed to satisfy the application conditions of the Euler–Bernoulli beam and Kirchhoff plate models on the nanoscale. The work in this article is of scientific significance to various studies on nanostructures.


Introduction
With the rapid development of the internet and artificial intelligence technologies, nanoelectromechanical systems with different functions are widely applied in various fields such as microchips, flexible electronics, and wireless communications [1]. Nanobeams and nanoplates serve as the load-bearing structures in nanoelectromechanical systems. Typical applications include nanosensors, nanoactuators, atomic force microscopes, and nanoswitches [2]. As the characteristic scale of structural members decreases to the order of nanometer, the mechanical behaviors of nanoscale structures are quite distinct from their macroscopic counterparts [3][4][5].
Among the aforementioned nonclassical continuum theories, the differential-form constitutive model of NET was widely adopted in predicting the mechanical behavior of nanoscale structures, owing to its simplicity. For instance, Reddy [14] derived the equations of motion for nonlocal Euler-Bernoulli beam model and presented the analytical solutions of bending, vibration and buckling under simplysupported boundary condition. Thai [15] proposed a nonlocal shear deformation beam theory to predict the bending, buckling and vibration responses of nanotubes. Mohamed et al. [16] proposed a six-order compact finite difference method for the vibration analysis of nonlocal Euler-Bernoulli beams in elastic medium. Utilizing the nonlocal Euler-Bernoulli beam model, Şimşek [17] explored the static bending, buckling, free and forced vibrations of functionally graded nanotubes. Li et al. [18] investigated the effect of boundary constraints on the free vibration characteristics of Euler-Bernoulli and Timoshenko nanobeams. Recently, a unified nonlocal formulation for the bending, buckling and vibration analysis of nanobeams is developed by Rohit and Sayyad [19].
Making use of NET, scholars investigated the mechanical behaviors of nanoplates on bending, buckling, vibration, and so forth. For example, Pradhan and Phadikar [20] developed the nanoscale Kirchhoff and Mindlin plates theories. Liu et al. [21] investigated the transverse dynamic behavior of graphene nanosheets or other plate-like nanostructures with axial motion. Making use of the two-variable refined plate theory, Malekzadeh and Shojaee [22] investigated the free vibration of nanoplates. Utilizing the size-dependent Kirchhoff and Mindlin plate models, Lu et al. [23] revealed the coupling effects of nonlocal stress, strain gradient and surface energy on the dynamic response of nanoplates. A comprehensive literature review on the state-of-the-art of the mechanical behaviors of nanostructures is far beyond the scope of the present work presentation. The interested readers may refer to the recent review articles [24][25][26][27][28].
Through a thorough literature survey, the classical Euler-Bernoulli beam and Kirchhoff plate models are widely adopted to study the mechanical behaviors of thin nanostructures owing to their simplicity and elegance. The corresponding macroscopic criteria are taken for granted to make sense on the nanoscale. However, very few scholar is aware of the fact that the microscopic criteria of the classical models should take the nonlocal parameter e 0 a into consideration.
Within the framework of NET, this article aims to establish the criteria of the applicability of the nanoscale Euler-Bernoulli beam and Kirchhoff plate models based on the Timoshenko beam model and the Mindlin plate model via the wave propagation theory. The present new criteria simultaneously take into account the effects of nonlocal parameters and material properties and may be degenerated to the macroscopic counterparts. The phase diagrams k f e 0 a $ k f h (k f represents the wave number of flexural wave; h is the height of beams or plates) are also presented according to the present criteria.
This article is organized as follows. Section 2 is dedicated to the dispersion equation for nanoscale Timoshenko beams. In section 3, the corresponding criterion for nanoscale slender beams in the low frequency domain is presented. In section 4, the phase diagrams k f e 0 a $ k f h are drawn. The parameters adopted in literature are checked to identify whether they meet the present criteria or not. Some concluding remarks are drawn in section 5.

Dispersion equation of nanoscale Timoshenko beams
Consider a Timoshenko nanobeam of length l, width b and height h. The coordinate system is established as shown in Figure 1. The equations governing the flexual wave propagating in the Timoshenko beams turn out to be Àh=2 qz 2 dz: The symbol e 0 a is a nonlocal parameter, which is related to molecular properties such as molecular dimension and bonding strength. The symbol e 0 is a nonlocal material constant that may be approximated experimentally or by matching the dispersion curves of plane waves to those of atomic lattice dynamics. The symbol a is an intrinsic characteristic length of the material, such as lattice parameters, C-C bond lengths, and particle spacing [9].
Through a thorough literature survey, quite few studies on determinations of e 0 a have been conducted to date, with an exception of Ref. [29]. It is pointed out in Ref. [29] that the use of the nonlocal parameter in variety of the previously conducted studies was artificial where no definite values were set for the nonlocal parameter. In light of the definition of e 0 a in Ref. [9], the values of e 0 a are determined in Ref. [29] for various materials by matching the dispersion curves of plane waves to those of atomic lattice dynamics.
The symbol w represents the displacement component of the neutral plane of the beam along the z-axis. The symbol w x represents the angle of rotation of the transverse normal of the cross section with respect to the x-axis. The symbol j is the shear correction factor. The derivation of Eq. (1) is detained in Appendix A.
Combining Eqs. (1a) and (1b) and eliminating the parameter w x give the governing equation in terms of the deflection w as D @ 4 w @x 4 þ I 0 @ 2 w @t 2 À I 0 D jGh þ I 2 @ 4 w @x 2 @t 2 þ I 0 I 2 jGh @ 4 w @t 4 À I 0 e 0 a ð Þ 2 @ 4 w @x 2 @t 2 þ I 0 D jGh þ I 2 e 0 a ð Þ 2 @ 6 w @x 4 @t 2 À 2I 0 I 2 e 0 a ð Þ 2 jGh @ 6 w @x 2 @t 4 þ I 0 I 2 e 0 a ð Þ 4 jGh @ 8 w @x 4 @t 4 ¼ 0 On letting the nonlocal parameter e 0 a ¼ 0 in Eq. (2), one arrives at 3) is in agreement with the governing equation of the Timoshenko beam on the macroscale [30]. Thus, the correctness of the governing equation Eq. (2) is verified to some extent. On (2) with W denoting the wave amplitude, the dispersion equation may be obtained where x and k x stand for the circular frequency and wave number, respectively. Equation (4) may be further transformed into where k l , k t , and k f represent the wave numbers of longitudinal wave, transverse wave, and flexural wave, respectively [31] In fact, Eq. (5) provides the basis of proposing the criterion in the next section.

Criterion of nanoscale slender beams
In the low-frequency case k f ) k t > k l , and k f ! 10 3 k t 1 , combining the definitions of the wave number of flexural and transverse waves in Eq. (6) yields If the wave number of the flexural wave k f is taken as the reference scale, then a key dimensionless wave number can be introduced as k x ¼ k x =k f , from which Eq. (5) may be expressed as follows In the case of e 0 a ¼ 0, Eq. (9) is reduced to the dispersion equation for the Timoshenko beams on the macroscopic scale [32] k To some extent, the correctness of the dispersion equation Eq. (9) is verified. In fact, Eq. (9) is a quartic equation in terms of the variable for k x , which has four distinct roots. The root may be derived explicitly by symbolic computational softwares (e.g., MATLAB 2 ). Their expressions are extremely lengthy and complicated. However, it is seen that the solutions depend upon two parameters k f e 0 a and k f h in their expressions.
Under the conditions of low frequency and the inequalities holding true the parameters k f e 0 a and k f h may be regarded as two small parameters. Applying multivariate Taylor expansion to the aforementioned solutions, the leading order term of the dimensionless solutions of wave number k ð1Þ x and the first two terms of the dimensionless solutions k ð2Þ x may be obtained as k where D 1 is a correction of the term k ð2Þ x to the leading term k ð1Þ x , and its expression is In summary, the solutions to Eq. (9) may be put in the following unified form From a physical point of view, k x should be a positive real number. Therefore, only the solutions k ð1Þ x ¼ 1 and k ð2Þ x ¼ 1 þ D 1 make sense. In the following derivation, attentions are confined to these two solutions. It is found that k ð1Þ x ¼ 1 is identical to the solution pertinent to the slender beam in the macroscopic case (i.e., the solution of Eq. 10) [32].
This article aims to develop the criterion of the applicability of the nanoscale Euler-Bernoulli beam model. When the value of k ð2Þ x ¼ 1 þ D 1 is sufficiently accurate and the value of D 1 is far less than 1, the effects of the height h and the nonlocal parameter e 0 a of the nanoscale Timoshenko beam may be neglected. Under such a circumstance, the Timoshenko beam may be regarded as a slender beam. According to Ref. [32], the value of D 1 in this article is set to 10%, without losing 1 There is no rigorous mathematical definition of the inequality A)B. In practice, when A!10B holds, one can claim A) B. In this article, Eq. (7) is used as one condition of criterion. Owing to the fact, k f ! 10 3 k t > 10 2 k t > 10k t , the present definition of "A)B" would not affect the present results in section 4. rationality. Therefore, when the value of D 1 is less than 10%, the classical Euler-Bernoulli assumptions on deformation may be used, without losing too much accuracy.
Consequently, the following criterion of Euler-Bernoulli beam model for nanoscale slender beams at low frequency may be proposed as where the wave number and wavelength of flexural wave are related by k f ¼ 2p=k f : Equations (15) and (16) are one of the main results of this article, which are new to literature as well.
If the nonlocal parameter vanishes e 0 a ¼ 0, Eq. (15) may be degenerated to the criterion for slender beams on the macroscopic scale as In the case of ¼ 0:3 and j ¼ 0:85, Eq. (17) is readily reduced to This is identical to the classical result in literature [32]. Thus, the rationality of this criterion is verified to a certain extent.
Geometrically, Eq. (15) represents an elliptical region in the parametric space of k f e 0 a and k f h. Further, the parameters k f e 0 a and k f h should satisfy the prerequisites specified in Eqs. (7) and (11). Hence, Eqs. (7), (11), and (15) together constitute the criterion in terms of the geometrical and material parameters. If the criterion were satisfied, the Timoshenko beam model may be replaced by the Euler-Bernoulli beam model since the correction term D 1 is less than 10%, which is an acceptable derivation in engineering senses. Otherwise, a big error larger than 10% would take place. For a nanobeam with the parameters meeting Eq. (18) but failing to satisfy Eq. (15), it would lead to an inaccurate prediction to study the mechanical behavior of a beam using the Euler-Bernoulli beam model.
The derivation procedure of criterion of nanoscale slender beams may be readily extended to develop the criterion for nanoscale thin plates. Consider a Mindlin nanoplate of length l, width b, and height h. The Cartesian coordinate system is established as shown in Figure 2. To establish the criterion for thin nanoplate, we consider flexural wave propagating along the x-direction in the plate. Although the Mindlin plate is a two-dimensional structure, the problem under consideration is of one-dimensional structure, and turns out to be a plane-strain beams problem, as pointed out by Guo et al. [33].
Since the derivation procedure of criterion for nanoscale thin plates is quite similar to that of the slender beams, it is shown in Appendix B. The criterion turns out to be It is noted that Eq. (19) may be directly obtained by replacing with =ð1 À Þ in Eq. (15), which is consistent with the intrinsic relations between plane-stress and planestrain problems in elasticity [34].

Numerical results and discussion
In this section, the present new criteria would be used to check the appropriatenesses of the Euler-Bernoulli beam model and the Kirchhoff plate model, which were adopted in the existing studies to characterize nanobeams and nanoplates. To this end, numerous studies in literature are chosen. All the data in the figures and tables in this section originate directly from those references, and the frequencies x satisfy Eq. (7) or (B.21). The shear correction factor j tends to infinity since shear deformations were not considered in these two classical models. In the limit case j ! 1 in Eqs. (15) and (B.29), the criteria for both slender beams and thin plates become It should be pointed out that in Eq. (20) the expression for the flexural wave number k f for slender beams is different from that of thin plates. The expressions of k f for beams and plates are given in Eqs. (6) and (B.20), respectively. As the value of k f is related to the cross-sectional properties I 0 and D (or D 0 ) for both beams and plates, the formulae of I 0 and D (or D 0 ) for typical geometric sections in engineering are presented in Appendix C for readers' convenience.
Geometrically, Eq. (20) symbolizes an elliptical area with the semi-major axis ffiffiffiffiffiffiffiffiffi ffi 24=5 p and the semi-minor axis ffiffiffiffiffiffiffi ffi 2=5 p : Owing to the fact ffiffiffiffiffiffiffiffiffi ffi 24=5 p > 1, the parameter space of k f h and k f e 0 a turns out to be a truncated elliptical area, as shown in Figures 3 and 4.
Similarly, the criterion for the thin plates may be simplified as by inserting Eq. (B.20) and x 1 4 e 0 a h 2 þ 1 48

Euler-Bernoulli beam model
In this subsection, the parameters in typical references associated with the nanobeams, where the Euler-Bernoulli model was used, are substituted into Eq. (6) to calculate the values of k f , k f e 0 a, and k f h sequentially. In the phase diagram k f e 0 a $ k f h determined by Eq. (20), the locations (k f e 0 a, k f h) corresponding to the references are presented in Figure 3. It is seen that the parameters adopted in various references meet the criterion (17) on the macroscopic scale, but fail to satisfy the requirement of criterion (20) on the microscopic scale.
As an illustration, from the parameters in Refs. [44,49], the values of D 1 are computed and tabulated in Table 1. It is evident that the value of D 1 ! 0.1 and the corresponding parameters do not satisfy the microscopic criterion in Eq. (20) for slender beams. The values of D 1 corresponding to other references, which do not satisfy Eq. (20), are given in Supporting Information of this article.

Kirchhoff plate model
In this subsection, the values of k f , k f e 0 a, and k f h are calculated sequentially for the relevant references, by following the same procedure as the Euler-Bernoulli beams. The locations (k f e 0 a, k f h) corresponding to these references are shown in Figure 4, which is actually the phase diagram k f e 0 a $ k f h determined by Eq. (20). It is seen that the parameters adopted in some references, satisfy the macroscopic criterion (B.31) for thin plates, but fail to meet the microscopic criterion (20).
As an example, according to the parameters adopted by Refs. [23,65], the values of D 2 are evaluated, which are tabulated in Table 2. It is seen that the D 2 is not smaller than 0.1, and the corresponding parameters do not satisfy   Note: the following dimensionless parameters employed in these two references: l ¼ 10 nm, E ¼ 30 Â 10 6 , ¼ 0:3, q ¼ 1, and x ¼ xl 2 ffiffiffiffiffiffiffiffiffiffiffi qA=EI p : the thin plate criterion in Eq. (20). The values of D 2 corresponding to other references, which do not satisfy Eq. (20), are given in Supporting Information of this article. The phase diagrams k f e 0 a $ k f h are given according to the present new criteria. It is found that a large number of existing studies fail to satisfy the present criteria. In this case, fairly big errors would occur if the Euler-Bernoulli beam model or Kirchhoff plate model is employed when wave propagation or vibration problems are investigated, as shown in section 4. The present criteria provide a solid theoretical basis for an accurate prediction of the vibration and wave behaviors of beam or plate structures on the nanoscale, and hence, enjoy a significant scientific merit.

Concluding remarks
It should be noted that the present criteria are valid only in the low frequency domain. In the high frequency domain, the corresponding standards should take in different forms. This work is underway and would be reported in another journal paper.   Note: the following dimensionless parameters used in these two references: l ¼ 10 nm, E ¼ 30 Â 10 6 , ¼ 0:3, q ¼ 1, and where dT, dV, and dW denote the variations of the kinetic energy, strain energy, and work done by external forces, respectively. It is noted that the equations of motion for the plate would be also derived by Eq. (A.4). The kinetic energy variation of Timoshenko beams is Sorting the terms according to the order of z gives rise to From the definition of mass moment of inertia, one arrives at where the point ðÁÞ above the variable indicates the derivative with respect to time. From the relations among the axial force N, flexural moment M and shear force Q and the corresponding stress components (i.e., Eq. A.11), one comes to Since no external force acts on the beam, the work done by the external force is zero, i.e., W ¼ 0. By substituting Eqs. (A.7) and (A.10) into Eq. (A.4), integrating by parts and setting the coefficients of the variables du, dw, and dw x equal to zero, the following equilibrium equations are obtained as du : and the corresponding boundary conditions are The boundary conditions in Eq. (A.13) are consistent with the counterparts in Refs. [69][70][71].

Appendix B:
The detailed derivation of the criterion of thin plates is presented here.
In the context of NET, the relations between the nonclassical stress r with r 2 ¼ @ 2 =@x 2 þ @ 2 =@y 2 : According to the Mindlin plate model [72], the displacement field in the plate takes the form u x x, y, z, t ð Þ¼ u x, y, t ð Þþ zw x x, y, t ð Þ (B.2a) u y x, y, z, t ð Þ¼ v x, y, t ð Þþ zw y x, y, t ð Þ (B.2b) u z x, y, t ð Þ¼ w x, y, t ð Þ (B.2c) where v represents the displacement component of the neutral plane of the plate along the y-axis. The symbol w y represents the rotation angle of the transverse normal of the plate with respect to the y-axis. From the geometric relations, the nonzero strain components within the Mindlin plate may be obtained as e yy ¼ @u y @y ¼ @v @y þ z @w y @y (B.3b) c xy ¼ @u x @y þ @u y @x ¼ @u @y þ @v @x þ z @w x @y þ @w y @x ! (B.3c) The kinetic energy variation of Mindlin plates is Arranging the terms according to the order of z gives From the definition of mass moment of inertia, one has The variation of strain energy of the plates reads