## Abstract

The traveling and standing flexural waves in the microbeam are studied based on the fraction-order nonlocal strain gradient elasticity in the present paper. First, the Hamilton’s variational principle is used to derive the governing equations and the boundary conditions with consideration of both the nonlocal effects and the strain gradient effects. The fraction-order derivative instead of the integer-order derivative is introduced to make the constitutive model more flexible while the integer-order constitutive model can be recovered as a special case. Then, the Euler–Bernoulli beam and the Timoshenko beam are both considered, and the corresponding formulations are derived. Two problems are investigated: (1) the dispersion of traveling flexural waves and the attenuation of the standing waves in the infinite beam and (2) the natural frequency of finite beam. The numerical examples are provided, and the effects of the nonlocal and the strain gradient effects are discussed. The influences of the fraction-order parameters on the wave motion and vibration behavior are mainly studied. It is found that the strain gradient effects and the nonlocal effect have opposite influences on the wave motion and vibration behavior. The fraction order also has evident influence on the wave motion and vibration behavior and thus can refine the prediction of the model.

## 1 Introduction

Driven by the rapid development of nanotechnology and nanoscience, nanoscale structures such as nanobeam, nanoplate, and nanoshell have gotten wide applications in nanoelectromechanical systems (NEMS) and therefore have also attracted great attention of many researcher [1,2]. The mechanical behaviors of these nanostructures are of great importance for better designing the small-scaled devices and systems. It is known that size effects should be taken into account for an accurate prediction of the static and dynamic responses of nanostructures when using continuum mechanics approaches. Since classical continuum theories cannot capture the size effects, several non-classical continuum theories have been proposed to assess the significant size effects on a small scale, such as the nonlocal elastic theory [3,4] and the strain gradient elastic theory [5,6].

The nonlocal elasticity theory proposed by Eringen [3,4] is the most widely used non-classical elastic theory. Different from the classical elasticity which assumes that the stress at a reference point only depends on the strain at the same point, the nonlocal elasticity assumes that the stress at a point is a function of strains at all points in an elastic continuum. Moreover, the nonlocal residual (also called the localization residual) needs to be incorporated into the governing equation and the boundary conditions so that the conservation law can be transformed from its integral form to a differential equation. Huang [7] addressed the nonlocal residual problem in detail and proposed a new concept of nonlocal stress instead of the Cauchy stress in the classic elasticity. In the nonlocal stress, the long-term interaction among the microstructure is incorporated. On the basis of nonlocal elasticity, the size-dependent mechanical behaviors of homogeneous or inhomogeneous nanobeams have been investigated by many researchers [8–11]. Their researches include the bending, buckling, vibration, and wave propagation behavior of nanobeams. It is concluded that the nonlocal effect has a significant influence on the static and dynamic characteristics. Nevertheless, the nonlocal elasticity captures only the stiffness softening effect; the stiffness enhancement effect reported in many experimental and theoretical investigations [12–14] cannot be included.

Strain gradient theory initiated by Mindlin [5,6] is another commonly used non-classical continuum theory that can capture the stiffness enhancement effect compared with the nonlocal elasticity. In the strain gradient theory, the total stress accounts for additional strain gradient terms to consider microstructural deformation mechanism at a small scale. For a homogeneous isotropic material, there are 16 additional independent higher-order material constants besides the two classical ones in Mindlin’s strain gradient theory. However, too many material constants make the application of the theory difficult because these higher-order material constants are usually difficult to be obtained from the experimental measurement. Due to the drawback, several studies have been performed to reduce the additional material constants. Fleck and Hutchinson [15,16] reformulated Mindlin’s simplified strain gradient theory, and the deformation gradient tensor is decomposed into two independent parts, the stretch gradient tensor and the rotation gradient tensor. The number of material length scale parameters is reduced to five. Lam et al. [13] proposed the modified strain gradient theory which incorporates only three additional material length scale parameters. Gourgiotis and Georgiadis [17] proposed the dipolar strain gradient elastic theory which incorporates only one additional material length scale parameters. Li and Wei [18–21] used the dipolar strain gradient elastic theory to study the microstructure effects of wave motion. Recently, Yang et al. [22] developed the modified couple stress theory which contains also only one additional material length scale parameter. Based on strain gradient theory and modified couple stress theory, various microstructure-dependent beam models have been developed to study the mechanical response of small-scaled beams [23–29]. These works indicate that the microstructure effect causes stiffness enhancement of the structures when higher-order strain mechanism is considered. From the aforementioned discussion, it is obvious that the nonlocal elasticity theory and the strain gradient theory are two entirely different non-classical continuum theories and describe two different size-dependent effects for the response of materials at a small scale.

The fraction-order calculus is the natural extension of the integer-order calculus. The integer-order calculus can be regarded as a special case of the fraction-order calculus. Due to the inherent nonlocality of the fraction-order derivative, fractional calculus has been applied to various fields to revise existing models in recent years. Zhang et al. [30] proposed a fractional order three-element model to accurately describe the viscoelastic dynamic properties of soil during vibratory compaction. Carpinteri et al. [31] made tentative investigation on using the fractional derivatives to model nonlocal elasticity. Rahimi et al. [32–34] proposed a generalized nonlocal stress–strain gradient theory by using the fractional conformable derivative. Compared with the integer-order differential model, the fraction-order differential model is generally more flexible but less reported in the investigation of wave propagation and vibration problems.

In this paper, the fraction-order nonlocal strain gradient elasticity which combines the merit of the nonlocal elasticity and the strain gradient elasticity is used to study the wave propagation problem and the vibration problem of the microbeam. Compared with the integer-order nonlocal gradient elasticity, the fraction-order nonlocal strain gradient elasticity is more flexible and the integer-order nonlocal gradient elasticity can be recovered as a special case. Based on the fraction-order nonlocal strain gradient model, the propagating and standing flexural waves in infinite Euler–Bernoulli beam (EBB) and Timoshenko beam are both investigated. Moreover, the standing wave in finite beam and the corresponding natural frequency are also studied. The influences of the nonlocal effects and the strain gradient effects on the wave motion and vibration behavior are discussed. The influences of the fraction order on wave motion and vibration behavior are mainly concerned.

## 2 Statement of Problem

### 2.1 Nonlocal Elasticity Theory.

*t*

_{ij}(

*r*) is a nonlocal stress tensor,

*σ*

_{ij}(

*r*′) is a local stress tensor,

*C*

_{ijkl}is the elastic modulus tensor of the classical elasticity,

*ɛ*

_{ij}(

*r*′) is a strain tensor at point

*r*′,

*ρ*is the material density,

*f*

_{i}is body force component per unit volume,

*α*

_{0}(|

*r*−

*r*′|,

*e*

_{0}

*a*) is an additional attenuation kernel function introduced to describe the nonlocal effect, and

*e*

_{0}is the nonlocal material constant and

*a*is an internal characteristic length.

*l*

_{1}=

*ea*, the constitutive equation in the nonlocal elastic theory can be obtained as follows:

### 2.2 Strain Gradient Theory.

*η*

_{ijk}is strain gradients, i.e.,

*η*

_{ijk}=

*ɛ*

_{ij,k}, $Dijklmn=l22\delta klCijmn$ is higher-order elastic tensor related to the strain gradient, where

*l*

_{2}is a material characteristic length parameter. From Eq. (6), we obtain

*a*)

*b*)

*σ*

_{ij}is the conventional stress tensor, which is a work conjugate of

*ɛ*

_{ij}, and

*τ*

_{ijk}is the higher-order stress tensor, which is a work conjugate of

*η*

_{ijk}.

*a*)

*b*)

*ξ*

_{ij}is the equivalent stress tensor under strain gradient theory.

*ρ*is the material mass density,

### 2.3 Nonlocal Strain Gradients.

*t*

_{ij}is the classical nonlocal stress, and $\tau \xafijm$ is the high-order nonlocal stress. The equivalent stress can be expressed as

*e*

_{0}=

*e*

_{1}=

*e*and that the nonlocal attenuation functions

*α*

_{0}(|

*r*−

*r*′|,

*e*

_{0}

*a*) and

*α*

_{1}(|

*r*−

*r*′|,

*e*

_{1}

*a*) are same. Applying the linear differential operators $\u03c2=1\u2212(ea)2\u22072=1\u2212l12\u22072$ on both sides of Eq. (12) yields

The constitutive relationship is more general one and reduces to the strain gradient elasticity when *l*_{1} = 0 and the nonlocal elasticity when *l*_{2} = 0 and the classic elasticity when *l*_{1} = *l*_{2} = 0.

*a*)

*b*)

*z*) is the Gamma function, i.e., $\Gamma (z)=\u222b0+\u221ee\u2212ttz\u22121dt(z\u2208C,Re(z)>0)$. In order to reflect the nonlocal effects in the stress–strain relation, the symmetric Caputo fractional differential is introduced

The validity of such an extension of integer-order constitutive relation to the fraction-order constitutive relation had been addressed in detail in the literature [36].

*x*are defined as

*a*)

*b*)

*f*(

*x*) =

*e*

^{ikx}, then

### 2.4 Derivation of Governing Equations.

*U*while the kinetic energy by

*T*, and the work done by external loads by

*W*. Hamilton’s principle states that

*D*is the normal gradient operator defined as $D=n\u22c5\u2207$. $n$ is unit outward vector normal to the surface

*S*of the body occupied volume

*V*.

*a*)

*b*)

### 2.5 Beams with Nonlocal Strain Gradients

EBBs with nonlocal strain gradients

*x*-axis is along the axis of the beam, the

*z*-axis is along the height of the beam, and the

*y*-axis is along the width of the beam. Assume that the displacement at any point on the cross section of the beam can be expressed as

*φ*= ∂

*w*/∂

*x*is the rotation angle of the cross section and

*w*is the flexural deflection.

*a*)

*b*)

*a*)

*b*)

*c*)

*E*is the elastic modulus tensor of the classical elasticity.

*I*is the rotational inertia. The external work under the homogeneous external load

*q*can be expressed as

Due to the arbitrariness of *δw*, we obtain the governing differential equation in terms of moment

*a*)

*b*)

*c*)

*α*and

*β*are material-dependent constants. Accordingly, the governing equations can be modified as

Timoshenko beam with nonlocal strain gradients

*a*)

*b*)

*φ*is the rotation angle of cross section and

*γ*is the shear angle of the beam. The non-zero strain gradients are

*a*,

*b*,

*c*)

*b*)

*c*)

*a*)

*b*)

*c*)

*d*)

*e*)

*τ*is the nominal shear stress on the cross section, which is evenly distributed throughout the cross section, $\tau ^$ is the true shear stress on the cross section, which is generally unevenly distributed across the cross section,

*κ*is the shear correction factor, and

*κτ*is the average shear stress in the cross section.

*a*)

*b*)

*E*is the tensile modulus while

*G*is the shear modulus.

*δw*and

*δφ*, it yields the governing differential equations

*a*)

*b*)

*a*)

*b*)

*c*)

*d*)

*a*)

*b*)

*a*)

*b*)

*a*)

*b*)

## 3 Dispersive Relation of Flexural Waves

### 3.1 Dispersive Relation of Flexural Waves for Euler–Bernoulli Beam.

*k*is the wave number, and

*ω*is the circular frequency. Inserting Eq. (56) into the governing equation of Euler beam, i.e., Eq. (40), and using Eq. (19), yields

The dispersion relation based on the nonlocal strain gradient with the spatial fractional derivative, i.e., Eq. (58), could reduce to that corresponding with either the strain gradient model by taking *l*_{1} = 0 and $\beta =1$ or the nonlocal stress model by taking *l*_{2} = 0 and $\alpha =1$. It could also reduce to the integer-order nonlocal strain model when *α* = *β* = 1. Hence, this new nonlocal strain gradient model with the fractional derivative is more generalized model.

### 3.2 Dispersive Relation of Flexural Waves for Timoshenko Beam.

*a*)

*b*)

*a*)

*b*)

*a*

_{3}=

*Gκ*/

*E*, other symbols are the same as in Eq. (58).

## 4 Standing Flexural Waves in a Finite Beam

### 4.1 Standing Flexural Waves in an Euler–Bernoulli Beam.

*ω*is the frequency of vibration. The vibration mode

*W*(

*x*) should be determined by boundary conditions. On the other hand, the vibration mode

*W*(

*x*) is the result of interference of two wave trains that travel along opposite directions from the view of wave propagation. The incident wave and the reflection waves at two ends of the Euler beam constitute the wave trains with opposite propagation directions. Therefore, the vibration mode

*W*(

*x*) can also be expressed as follows:

*a*)

*b*)

*c*)

*a*)

*b*)

*c*)

*a*)

*b*)

*c*)

*d*)

*e*)

*f*)

The condition existing non-zero solution requires that the coefficient matrix |*A*|_{6×6} is zero. Solving the coefficient determinant and using the dispersion relation *k*_{i} = *k*_{i}(*ω*), we can obtain the natural frequency spectrum of the Euler beam in the nonlocal strain gradient elasticity. After the natural frequency is determined, the amplitude ratio, i.e., $[1C2/C1C3/C1C4/C1C5/C1C6/C1]$, can be determined from Eq. (67), which can be inserted into Eq. (63) to obtain the vibration mode.

### 4.2 Standing Flexural Waves in a Timoshenko Beam.

*a*)

*b*)

*W*(

*x*) and Φ(

*x*) can also be expressed as

*a*)

*b*)

*D*

_{i}=

*χ*

_{i}

*C*

_{i}, $\chi i=\u2212iki(1\u2212\rho (1+l1\alpha cos\alpha \pi 2ki\alpha )\omega 2ki2G\kappa (1+l2\beta cos\beta \pi 2ki\beta ))$.

*a*)

*b*)

*c*)

*d*)

*a*)

*b*)

*c*)

*d*)

*a*)

*b*)

*c*)

*d*)

*e*)

*f*)

*g*)

*h*)

The condition that Eq. (73) has a non-zero solution requires the coefficient matrix, i.e., |*A*|_{8×8}, equal to zero. Solving the coefficient determinant and using the dispersion relation *k*_{i} = *k*_{i}(*ω*), we can obtain the natural frequency spectrum of the Timoshenko beam in the nonlocal strain gradient elasticity. After the natural frequency is determined, the amplitude ratio corresponding with the frequency spectrum, i.e.,$[1C2/C1C3/C1C4/C1C5/C1C6/C1$$C7/C1C8/C1]$, can be determined from Eq. (73), which can be inserted into Eq. (69) to obtain the vibration mode corresponding with the frequency spectrum.

## 5 Numerical Results and Discussions

*κ*takes different values for the different cross sections and, therefore, reflects the effects of cross-sectional shape. From the dispersion equations of the flexural waves, it is noted that the inertial moment

*I*and the cross-section area

*A*are not the determining geometric factors of the dispersion feature. It is the ratio of the inertial moment

*I*to the cross-sectional area

*A*, i.e., the radius of gyration $r=I/A$, which determines the dispersion feature. The microbeam is assumed to be homogeneous and isotropic. The equivalent material constants are assumed: Young's modulus

*E*= 0.72 TPa, Poisson's ratio

*v*= 0.254, mass density $\rho =2.3g/cm3$, and section correction factor $\kappa =10/9$. These parameters are all come from the study by Lim [41].The dimensionless nonlocal parameters and the strain gradient parameter are $l\xaf1=l1/r$ and $l\xaf2=l2/r$, respectively. After the material constants are given, the dispersion equation of the flexural waves can be expressed formally

For the standing wave problem, the slenderness ratio, i.e., *κ* = *L*/*r*, is a key factor. Moreover, the dimensionless frequency $\omega 0=\omega \omega L$$(\omega L=E\rho L2)$ is used instead.

Figure 1 shows the influences of the nonlocal parameter $l\xaf1$ on the dispersion relations of flexural waves in EBB. Different from the classic EBB, there is an extra standing wave generated by the strain gradient effects. The attenuation coefficient (imaginary wave number) satisfies $k\xaf=i((1l\xaf22+1))1/2$ when the frequency approaches zero for this extra standing wave. This implies that the attenuation coefficient becomes infinite and thus the standing wave disappears when strain gradient effects are not considered, i.e.,$l\xaf2=0$. Obviously, this extra standing wave is caused by the strain gradient effects. It is observed that the strain gradient effects on two standing waves are opposite. Figure 2 shows the influences of the strain gradient parameter $l\xaf2$ on the dispersion relations of flexural waves. By comparing Fig. 1 with Fig. 2, it is noted that the strain gradient effects and the nonlocal effects on the dispersion of flexural waves are opposite. The strain gradient effects increase the propagation speed (or decrease the wavenumber) while the nonlocal effects decrease the propagation speed (or increase the wavenumber). In particular, the attenuation coefficient (imaginary wave number) of the standing wave caused by the strain gradient effects is sensitive to the strain gradient parameter $l\xaf2$ while not sensitive to the frequency.

Figures 3 and 4 show the influences of nonlocal fractional order *α* and the strain gradient fractional order *β* on the dispersion of flexural waves in EBB, respectively. It is observed that the fraction-order nonlocal strain gradient model can provide a more refined description of the dispersion feature by elaborate adjustment of the nonlocal fractional order *α* and the strain gradient fractional order *β*. In general, the influences of the nonlocal fractional order *α* and the strain gradient fractional order *β* on the dispersion curves are opposite. Moreover, both fractional order *α* and *β* have opposite influences on the two standing waves. Compared with the integer-order nonlocal strain gradient model, the fraction-order nonlocal strain gradient model is a more flexible model to describe the frequency-dependent dispersion feature without the introduction of too many material parameters.

Figures 5 and 6 show the influence of the nonlocal parameter $l\xaf1$ and the strain gradient parameter $l\xaf2$ on the dispersion of flexural waves in TB. Different from EBB, there are two propagating flexural waves. Furthermore, the second propagating flexural wave has the cut-off frequency. Let the wavenumber is zero, we can get the cut-off frequency, i.e., $\omega 2=GA\beta /\rho I$. It is noted that the cut-off frequency is explicitly dependent on the shear modulus *G*. In other words, the cut-off frequency exists because we consider the shear deformation in the TB model. Moreover, two standing waves are both caused by the strain gradient effects. The imaginary wavenumber can be estimated by $k\xaf2=\u221212\u22121l\xaf22\xb112$ for TB when the frequency approaches zero. Obviously, the imaginary wavenumber tends to be infinite when the strain gradient parameter approaches zero. Two imaginary wavenumber appear when the strain gradient parameter is finite. It is also observed that the influences of the nonlocal parameter $l\xaf1$ and the strain gradient parameter $l\xaf2$ on the dispersion of flexural waves are opposite in TB as in EBB. However, their influences on two standing waves have the same trend but different frequency sensitivities. The imaginary wavenumber is sensitive to the nonlocal parameter $l\xaf1$ at high frequency while sensitive to the strain gradient parameter $l\xaf2$ at low frequency.

Figures 7 and 8 show the influences of nonlocal fractional order *α* and the strain gradient fractional order *β* on the dispersion of flexural waves in TB, respectively. As in EBB, the nonlocal fractional order *α* and the strain gradient fractional order *β* have opposite influences on the dispersion of flexural waves but have the same influences on the attenuation of two standing waves. The cut-off frequency is not affected by the nonlocal fractional order *α* and the strain gradient fractional order *β* at all.

Apart from the wave motion behavior, we also investigate the vibration mechanical behavior of EBB and TB by the application of the present fraction-order nonlocal strain gradient model. It is noted that the influences on EBB and TB are similar. Therefore, only the influences on the TB are reported here. Figure 9 shows the influences of the nonlocal parameters $l\xaf1$ and the strain gradient parameter $l\xaf2$ on the first natural frequency of TB. Evidently, the nonlocal effects decrease the first natural frequency while the strain gradient effects increase the first natural frequency. These are the results of the softening effects of nonlocal elasticity and the stiffening effects of strain gradient elasticity. Figure 10 shows the influences of the nonlocal fractional order *α* and strain gradient fraction order *β* on the first natural frequency. It is observed that the nonlocal fractional order *α* and strain gradient fraction order *β* have opposite influences on the natural frequency just like the influences of the nonlocal parameter $l\xaf1$ and the strain gradient parameter $l\xaf2$. The accurate characterization of the dynamic behavior of an actual beam can be acquired by appropriate adjustment of the nonlocal parameter $l\xaf1$ and the strain gradient parameter $l\xaf2$ as well as the nonlocal fractional order *α* and the strain gradient fraction order *β*.

## 6 Conclusion

The nonlocal elasticity and the strain gradient elasticity are often used to describe the size effects of material or structure at microscale. However, the nonlocal elasticity and the strain gradient elasticity are two fundamentally distinct size-dependent theories. In order to reflect the size effects of material more accurately, the strain gradient elasticity and the nonlocal elasticity should be incorporated and to get a more advanced elastic theory, i.e., nonlocal strain gradient elasticity. In this paper, the integer-order nonlocal strain gradient model is further modified as the fraction-order nonlocal strain gradient model by the introduction of the fraction-order derivatives. The dispersion of the flexural traveling wave and the attenuation of the flexural standing waves in a microbeam are both studied based upon the fraction-order nonlocal strain gradient model. Two models of beam, i.e., EBB and TB, are both considered. Apart from the wave motion problem, the vibration problem is also investigated. The main conclusions can be summarized as follows:

The nonlocal effect and the strain gradient effect have opposite influences on the dispersion of the flexural traveling waves and the attenuation of the flexural standing waves. Similarly, the first natural frequency increases by consideration of the strain gradient effects while decreases by consideration of nonlocal effects.

The strain gradient effect creates an extra standing wave for EBB while two standing waves for TB. The nonlocal effects do not create extra wave mode but affect the dispersion behavior of propagation flexural wave and the attenuation of the standing wave.

Different from EBB, there are two propagating flexural waves in the TB. The second propagating flexural wave has a cut-off frequency. The cut-off frequency is caused by the shear deformation and is not related to both the stain gradient effects and the nonlocal effects.

The introduction of fractional order derivative makes the nonlocal strain gradient model a more flexible and robust model. The nonlocal fractional order

*α*and the strain gradient fraction order*β*have different influences on both wave motion and vibration behavior. The actual dynamic behavior can be accurately characterized by the appropriate adjustment of them which can be considered as the material-dependent parameters.

## Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12072022, 11872105, and 11911530176) and the Fundamental Research Funds for the Central Universities (FRF-BR-18-008B and FRF-TW-2018-005).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.