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 [811]. 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 [1214] 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 [1821] 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 [2329]. 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. [3234] 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.

In classical elasticity, the state of a point in an object is only related to the infinitesimal neighborhood of the point. In the nonlocal elasticity, any substance has a local structure, and these internal structures interact with each other through the long-range forces between them. Therefore, the state of a point in a substance is affected by other points of matter, but the degree of influence of different points is different. The nonlocal elastic theory considers the long-range force effect between the microstructures and assumes the stress at each point is not only related to the strain at that point but also related to the strain at all points in its neighborhood. Therefore, the nonlocal elasticity modifies the classical elastic equation as following:
tij,j(r)+fi(r)=ρu¨i(r)
(1)
tij(r)=Vα0(|rr|,e0a)σij(r)dV
(2)
σij(r)=Cijklεkl(r)
(3)
εij(r)=12[ui,j(r)+uj,i(r)]
(4)
where tij(r) is a nonlocal stress tensor, σij(r′) is a local stress tensor, Cijkl is the elastic modulus tensor of the classical elasticity, ɛij(r′) is a strain tensor at point r′, ρ is the material density, fi is body force component per unit volume, α0(|rr′|, e0a) is an additional attenuation kernel function introduced to describe the nonlocal effect, and e0 is the nonlocal material constant and a is an internal characteristic length.
Eringen [4] introduces a Green’s function of a linear differential operator with constant coefficients ς=1(ea)22 (where 2 is the Laplacian) such that ςα(|rr|,ea)=δ(rr). Then, applying the differential operator to Eq. (2), and assuming l1 = ea, the constitutive equation in the nonlocal elastic theory can be obtained as follows:
(1l122)tij=Cijklεkl
(5)

2.2 Strain Gradient Theory.

The classical elasticity assumes that the stress at one point is only related to the strain at that point. However, the strain gradient elasticity considers that the stress at one point is related not only to the strain at that point but also to the strain gradient at that point. Therefore, the strain energy density is not only related to strain but also related to strain gradient, i.e.
U(εij,ηijk)=12Cijklεijεkl+12Dijklmnηijkηlmn
(6)
where ηijk is strain gradients, i.e., ηijk = ɛij,k, Dijklmn=l22δklCijmn is higher-order elastic tensor related to the strain gradient, where l2 is a material characteristic length parameter. From Eq. (6), we obtain
σij=Uεij=Cijklεkl
(7a)
τijk=Uηijk=Dijklmnηlmn
(7b)
where σij is the conventional stress tensor, which is a work conjugate of ɛij, and τijk is the higher-order stress tensor, which is a work conjugate of ηijk.
Then, according to the Hamiltonian variational principle, the governing equation can be expressed as
ξij,j+fi=ρu¨i
(8a)
ξij=σijτijk,k=Cijkl(1l222)εkl
(8b)
where ξij is the equivalent stress tensor under strain gradient theory. ρ is the material mass density,

2.3 Nonlocal Strain Gradients.

The nonlocal elastic theory and the strain gradient elastic theory are two different non-classical elastic theories that describe two different size-dependent effects. The nonlocal strain gradient elasticity combines the merits of the two non-classic elasticities. The stress field of the nonlocal strain gradient theory considers not only the nonlocal long-range effect of the material but also the influence of the high-order stress caused by the strain gradient. The corresponding internal energy density potential, the stress, and the high-order stress are
U=12CijklεijVα0(|rr|,e0a)εkl(r)dV+12l22Cijklεij,mVα1(|rr|,e1a)εkl,m(r)dV
(9)
tij(r)=Uεij=Vα0(|rr|,e0a)Cijklεkl(r)dV
(10)
τ¯ijm=Uεij,m=l22Vα1(|rr|,e1a)Cijklεkl,m(r)dV
(11)
where tij is the classical nonlocal stress, and τ¯ijm is the high-order nonlocal stress. The equivalent stress can be expressed as
ζij=tijτ¯ijm,m
(12)
Let us assume e0 = e1 = e and that the nonlocal attenuation functions α0(|rr′|, e0a) and α1(|rr′|, e1a) are same. Applying the linear differential operators ς=1(ea)22=1l122 on both sides of Eq. (12) yields
(1l122)ζij=Cijkl(1l222)εkl
(13)

The constitutive relationship is more general one and reduces to the strain gradient elasticity when l1 = 0 and the nonlocal elasticity when l2 = 0 and the classic elasticity when l1 = l2 = 0.

Compared with the integer-order derivative, the fractional order derivative can bridge the gap between two adjacent integer-order derivatives. Furthermore, the fractional derivative will degrade to the integer-order derivative when the fraction order tends to an integer. This shows that Eq. (13) can be modified by replacing the integer differential with the fractional differential, i.e.,
(1l1αDα)ζij=Cijkl(1l2βDβ)εkl
(14)
Here, the left and right Caputo fractional differentials are used, which are defined as, respectively, [35]
CDxαw(x)=1Γ(na)xw(n)(ξ)(xξ)αn+1dξ(n=[α]+1,n1<αn,ξ<x)
(15a)
xCDαw(x)=(1)n1Γ(na)xw(n)(ξ)(ξx)αn+1dξ(n=[α]+1,n1<αn,ξ>x)
(15b)
where Γ(z) is the Gamma function, i.e., Γ(z)=0+ettz1dt(zC,Re(z)>0). In order to reflect the nonlocal effects in the stress–strain relation, the symmetric Caputo fractional differential is introduced
Dxa=12(CDxa+xCDa)
(16)

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].

In order to solve the fractional order differential equation, the Fourier transform method is often used. The Fourier transform and the inverse Fourier transform about x are defined as
[f(x)](s1)=f^(s1)=+f(x)eis1xdx
(17a)
1[f^(s1)]=f(x)=12π+f(s1)eis1xds1
(17b)
It is known that [f(n)(x)](s1)=(is1)nf^(s1) for any integer-order derivative. For the fraction-order derivative, it is also valid [37,38], i.e., [Dxαf(x)]=(is1)αf^(s1); [xDαf(x)]=(is1)αf^(s1). Performing the Fourier transform on Eq. (16) leads to
[Dxαf(x)](s1)==12[(is1)α+(is1)α]f^(s1)==cos(πα2)s1αf^(s1)
(18)
In particular, when f(x) = eikx, then
Dxα(eikx)=x1[x[Dxα(eikx)](s1)]=x1[cosαπ2s1α2πδ(s1k)]=cosαπ2kαeikx
(19)

2.4 Derivation of Governing Equations.

Here, Hamilton’s principle is applied to derive the governing equations. Let the nonlocal elastic energy be denoted by U while the kinetic energy by T, and the work done by external loads by W. Hamilton’s principle states that
δ0t(W+TU)dt=0
(20)
The external work can be expressed as
W=VfudV+S(t¯(0)u+t¯(1)Du)dS
(21)
where, f is the body force per unit volume. t¯(0) and t¯(1) are the classical traction and couple vector per unit surface area. D is the normal gradient operator defined as D=n. n is unit outward vector normal to the surface S of the body occupied volume V.
The kinetic energy is
T=Vρ2(u˙u˙)dV
(22)
The elastic energy based on the nonlocal strain gradient model can be expressed in a vector form as
U=V(t:u+τ¯u)dV
(23)
Further, substituting Eqs. (21)(23) into Eq. (20) yields
0=0tV[(tτ¯)+fρu¨]δudVdt+0tS(t(0)t¯(0))δudSdt+0tS(t(1)t¯(1))δ(Du)dSdt
(24)
where t(0)=n(tτ¯)+(nnτ¯)(n)+Ds(nτ¯); t(1)=nnτ¯; Ds=nn.
Due to the arbitrariness of δu and δ(Du), we obtain the governing differential equation
(tτ¯)+f=ρu¨
(25)
and boundary conditions
t(0)=t¯(0)oru=u¯
(26a)
t(1)=t¯(1)orDu=Du¯
(26b)
where t¯(0), t¯(1), u¯, and Du¯ are the specified function at the boundary.

2.5 Beams with Nonlocal Strain Gradients

  1. EBBs with nonlocal strain gradients

Consider an Euler beam, the cross-sectional area of the beam is A, the coordinate system is established such that the 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
u=zφ(x,t),v=0,w=w(x,t)
(27)
where φ = ∂w/∂x is the rotation angle of the cross section and w is the flexural deflection.
The corresponding non-zero strain and the strain gradient are
εxx=zφx=z2wx2
(28a)
ηxxx=εxx,x=z3wx3,ηxxz=εxx,z=2wx2
(28b)
The non-zero nonlocal stress and the high-order stress are
txx=Vα0(|rr|,l1)Eεxx(r)dV
(29a)
τ¯xxx=l22Vα0(|rr|,l1)Eεxx,x(r)dV
(29b)
τ¯xxz=l22Vα0(|rr|,l1)Eεxx,z(r)dV
(29c)
The constitutive relation of an Euler beam under the nonlocal strain gradient theory can be simplified as
(1l122)ζxx=E(1l222)εxx
(30)
where E is the elastic modulus tensor of the classical elasticity.
The elastic strain energy is
δU=V(txxδεxx+τ¯xxxδηxxx+τ¯xxzδηxxz)dV=0LMδwdx0LMhδwdx0LPδwdx,
(31)
where
M=AtxxzdA,P=AτxxzdA,Mh=AτxxxzdA
(32)
The kinetic energy is
δK=0LAρutδutdAdx+0LAρwtδwtdAdx=0LρIφtδφtdx+0LρAwtδwtdx,
(33)
where I is the rotational inertia. The external work under the homogeneous external load q can be expressed as
δW=0Lqδwdx
(34)
Hamilton’s principle, i.e., Eq. (14), yields
0=0t(Mhδw)|0Ldt+0t[(M+pMhx)δw]|0Ldt0t[x(M+PMhx)δw]|0Ldt+0t0L[2x2(M+PMhx)]δwdxdt+0L(ρAwtδw)|0tdx0L0tρA2wt2δwdtdx0t0Lqδwdxdt
(35)

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

2x2(M+PMhx)(ρA2wt2+q)=0
(36)
and boundary conditions at the ends of the beam
Mh=0w=0
(37a)
M+pMhx=0orw=0
(37b)
x(M+PMhx)=0orw=0
(37c)
Further, regardless of the external force’s work, the governing equations, i.e., Eq. (36), can also be expressed in terms of flexural deflection as
ρA[2wt2l124wt2x2]=EI(4wx4l226wx6)EAl224wx4
(38)
The stress–strain constitutive equation with the spatial fractional differential for the Euler beam is obtained by replacing the Laplacian operator in Eq. (30) with the symmetric Caputo fractional differential
(1l12αDx2α)ζxx=(1l22βDx2β)Eεxx
(39)
where α and β are material-dependent constants. Accordingly, the governing equations can be modified as
(1l12αDx2α)ρA2wt2=EI(1l22βDx2β)4wx4EAl22βDx2β2wx2
(40)
  1. Timoshenko beam with nonlocal strain gradients

The non-zero strain in the Timoshenko beam can be expressed as
εxx=zφx
(41a)
2εxz=wxφ=γ
(41b)
where φ is the rotation angle of cross section and γ is the shear angle of the beam. The non-zero strain gradients are
ηxxx=εxx,x=z2φx2
(42a, b, c)
ηxxz=εxx,z=φx
(42b)
2ηxzx=2εxz,x=2wx2φx
(42c)
The non-zero nonlocal stress and the higher-order stress are
txx=Vα0(|rr|,l1)EεxxdV
(43a)
τ¯xxx=l22Vα0(|rr|,l1)Eεxx,x(r)dV
(43b)
τ¯xxz=l22Vα0(|rr|,l1)Eεxx,z(r)dV
(43c)
txz=2Vα0(|rr|,l1)Gεxz(r)dV
(43d)
τ¯xzx=2l22Vα0(|rr|,l1)Gεxz,x(r)dV
(43e)
The shear force on the cross section of the beam can be expressed as
Q=Aτ^dA=Aκτ
(44)
where τ is the nominal shear stress on the cross section, which is evenly distributed throughout the cross section, τ^ 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.
The general constitutive relation in the nonlocal strain gradient elasticity can be simplified as
(1l122)ζxx=E(1l222)zφx
(45a)
(1l122)ζxz=Gκ(1l222)(wxφ)
(45b)
where E is the tensile modulus while G is the shear modulus.
The elastic deformation energy is
δU=V(txxδεxx+τ¯xxxδηxxx+τ¯xxzδηxxz)dV=+V(2txzδεxz+2τ¯xzxδηxzx)dV=0LMδφxdx0LMhδ2φx2dx0LPδφxdx+0LQδ(wxφ)dx+0LQhδ(2wx2φx)dx
(46)
where
M=AtxxzdA,Mh=Aτ¯xxxzdA,P=Aτ¯xxzdA,Q=AtxzdA,Qh=Aτ¯xzxdA
(47)
The kinetic energy is
δK=0LAρutδutdAdx+0LAρwtδwtdAdx=0LρIφtδφtdx+0LρAwtδwtdx
(48)
In the case of the lateral uniform external load, the external work can be expressed as
δW=0Lqδwdx
(49)
According to the Hamilton’s principle, we obtain
0=0t(Mhδφx)|0Ldt+0t[(M+PMhx)δφ]|0Ldt0t[Qhδ(wxφ)]|0Ldt0t0L(Mx+Px2Mhx2Q+Qhx)δφdxdt0t[(QQhx)δw]|0Ldt0t0L(2Qhx2Qx)δwdxdt+0L(m2φtδφ)|0tdx+0L(m0wtδw)|0tdx0L0tm22φt2δφdtdx0L0tm02wt2δwdtdx+0t0Lqδwdxdt
(50)
Due to the arbitrariness of δw and δφ, it yields the governing differential equations
2Qhx2Qx+ρA2wt2q=0
(51a)
Mx+Px2Mhx2Q+Qhx+ρI2φt2=0
(51b)
and boundary conditions at the ends of the beam
Mh=0orφx=0
(52a)
M+PMhx=0orφ=0
(52b)
QQhx=0orw=0
(52c)
Qh=0orwxφ=0
(52d)
Using Eq. (47), without considering the work done by external forces, the governing equations can also be expressed in terms of flexural deflection and rotation angle as
ρA(2wt2l124wt2x2)=GAκ(2wx2φxl224wx4+l223φx3)
(53a)
ρI(2φt2l124φt2x2)=GAκ(wxφl223wx3+l222φx2)+EI(2φx2l224φx4)+EAl222φx2
(53b)
The stress–strain constitutive equations with the spatial fractional order derivative are obtained by replacing the Laplacian operator in Eq. (45) with the symmetric Caputo fractional differential
(1l12αDx2α)ζxx=(1l22βDx2β)Eεxx
(54a)
(1l12αDx2α)ζxz=(1l22βDx2β)Gκγxz
(54b)
Accordingly, the governing equations are modified as
ρA(1l12αDx2α)2wt2=GAκ(2wx2φxl22βDx2β2wx2+l22βDx2βφx)
(55a)
ρI(1l12αDx2α)2φt2=GAκ(wxφl22βDx2βwx+l22βDx2βφ)+EI(2φx2l22βDx2β2φx2)+EAl22βDx2βφ
(55b)

3 Dispersive Relation of Flexural Waves

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

The flexural wave solution can be expressed as
w(x,t)=w^ei(kxωt)
(56)
where w^ is the wave amplitude, 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
(1cos(απ)l12αk2α)ρAω2=EI(1cos(βπ)l22βk2β)k4EAcos(βπ)l22βk2β
(57)
After treatment of non-dimension, Eq. (57) becomes
a2k¯4+2βk¯4+a2k¯2+2βa1ω¯2k¯2α+ω¯2=0
(58)
where k¯=kr, ω¯=ω/ωe, ωe=EAρI, l¯1=l1/r, l¯2=l2/r, r=IA, a1=cos(απ)l¯12α, a2=cos(βπ)l¯22β.

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 l1 = 0 and β=1 or the nonlocal stress model by taking l2 = 0 and α=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.

The flexural wave solution for a Timoshenko beam can be expressed as
w(x,t)=w^ei(kxωt)
(59a)
φ(x,t)=φ^ei(kxωt)
(59b)
where w^ and φ^ are the wave amplitude. Inserting Eq. (59) into the governing equation, i.e., Eq. (55), and using Eq. (19), we can get
(ρω2ρω2l12αcos(απ)k2αGκk2+Gκl22βcos(βπ)k2+2β)w^=(GκikGκil22βcos(βπ)k1+2β)φ^
(60a)
(GAκik+GAκil22βcos(βπ)k1+2β)w^=(ρIω2ρIω2l12αcos(απ)k2αGAκ+GAκl22βcos(βπ)k2βEIk2+EIl22βcos(βπ)k2+2β+EAl22βcos(βπ)k2β)φ^
(60b)
Equation (60) can be rewritten in a matrix form Ax=0(x={w^,φ^}T), with elements of the coefficient matrix A(2×2) as
a11=ρω2ρω2l12αcos(απ)k2αGκk2+Gκl22βcos(βπ)k2+2β,a12=GκikGκil22βcos(βπ)k1+2β,a21=GAκik+GAκil22βcos(βπ)k1+2β,a22=ρIω2ρIω2l12αcos(απ)k2αGAκ+GAκl22βcos(βπ)k2βEIk2+EIl22βcos(βπ)k2+2β+EAl22βcos(βπ)k2β
The condition existing non-zero solution of Eq. (60) requires that the coefficient determinant is zero, which leads to a dimensionless dispersion relation
a3a22k¯4(1+β)2a3a2k¯2(2+β)ω¯2(1+a3)a1a2k¯2(1+α+β)+a3a22k¯2(1+2α)+a3k¯4+ω¯2(1+a3)a1k¯2+2α+(ω¯2a3+a3ω¯2)a2k¯2+2β+ω¯2a12k¯4αω¯2(1+a3)a1a2k¯2(α+β)ω¯2(1+a3)k¯2ω¯2(2ω¯2a3)a1k¯2α+ω¯2(1+a3)a2k¯2β+ω¯4ω¯2a3=0
(61)
where a3 = /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.

Considering the free vibration problem of the Euler beam (thus, the lateral load is zero) [39,40]. The vibration solution of Eq. (40) can be expressed as
w(x,t)=W(x)eiωt
(62)
where 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:
W(x)=C1eik1x+C2eik2x+C3eik3x+C4eik4x+C5eik5x+C6eik6x
(63)
According to Eq. (37), the boundary conditions at two ends of cantilever beam are
(w)x=0=0
(64a)
(w)x=0=0
(64b)
(w)x=0=0
(64c)
(Mh)x=L=0,
(65a)
(M+PMhx)x=L=0
(65b)
[x(M+PMhx)]x=L=0
(65c)
Substituting Eq. (63) into Eqs. (64) and (65) yields
C1+C2+C3+C4+C5+C6=0
(66a)
ik1C1+ik2C2+ik3C3+ik4C4+ik5C5+ik6C6=0
(66b)
k12C1+k22C2+k32C3+k42C4+k52C5+k62C6=0
(66c)
Γ1C1eik1L+Γ2C2eik2L+Γ3C3eik3L+Γ4C4eik4L+Γ5C5eik5L+Γ6C6eik6L=0
(66d)
Π1C1eik1L+Π2C2eik2L+Π3C3eik3L+Π4C4eik4L+Π5C5eik5L+Π6C6eik6L=0
(66e)
Σ1C1eik1L+Σ2C2eik2L+Σ3C3eik3L+Σ4C4eik4L+Σ5C5eik5L+Σ6C6eik6L=0
(66f)
where Γi=EIl2βcosβπ2ikiβ+1, Πi=EIki2+El2βcosβπ2kiβ(A+Iki2), and Σi=EIiki3+El2βcosβπ2ikiβ+1(A+Iki2). Equation (66) can be rewritten in the matrix form
[A]6×6[C1C2C3C4C5C6]T=0
(67)

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 ki = ki(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.

The vibration solution of Eq. (55) can be expressed as follows:
w(x,t)=W(x)eiωt
(68a)
φ(x,t)=Φ(x)eiωt
(68b)
Similar to the treatment for the Euler beam, the vibration mode W(x) and Φ(x) can also be expressed as
W(x)=C1eik1x+C2eik2x+C3eik3x+C4eik4x+C5eik5x+C6eik6x+C7eik7x+C8eik8x(69a)
(69a)
Φ(x)=D1eik1x+D2eik2x+D3eik3x+D4eik4x+D5eik5x+D6eik6x+D7eik7x+D8eik8x
(69b)
where Di = χiCi, χi=iki(1ρ(1+l1αcosαπ2kiα)ω2ki2Gκ(1+l2βcosβπ2kiβ)).
According to Eq. (52), the boundary conditions at two ends of cantilever beam can be expressed as
(φ)x=0=0
(70a)
(φx)x=0=0
(70b)
(w)x=0=0
(70c)
(wx)x=0=0
(70d)
(Mh)x=L=0
(71a)
(M+PMhx)x=L=0
(71b)
(QQhx)x=L=0
(71c)
(Qh)x=L=0
(71d)
Substituting Eq. (69) into Eqs. (70) and (71) yields
D1+D2+D3+D4+D5+D6+D7+D8=0
(72a)
ik1D1+ik2D2+ik3D3+ik4D4+ik5D5+ik6D6+ik7D7+ik8D8=0
(72b)
C1+C2+C3+C4+C5+C6+C7+C8=0
(72c)
ik1C1+ik2C2+ik3C3+ik4C4+ik5C5+ik6C6+ik7C7+ik8C8=0
(72d)
Γ1D1eik1L+Γ2D2eik2L+Γ3D3eik3L+Γ4D4eik4L+Γ5D5eik5L+Γ6D6eik6L+Γ7D7eik7L+Γ8D8eik8L=0
(72e)
Π1D1eik1L+Π2D2eik2L+Π3D3eik3L+Π4D4eik4L+Π5D5eik5L+Π6D6eik6L+Π7D7eik7L+Π8D8eik8L=0
(72f)
Σ1C1eik1L+Σ2C2eik2L+Σ3C3eik3L+Σ4C4eik4L+Σ5C5eik5L+Σ6C6eik6L+Σ7C7eik7L+Σ8C8eik8L=0
(72g)
Υ1C1eik1L+Υ2C2eik2L+Υ3C3eik3L+Υ4C4eik4L+Υ5C5eik5L+Υ6C6eik6L+Υ7C7eik7L+Υ8C8eik8L=0
(72h)
where Γi=EIl2βcosβπ2kiβ, Πi=(EI+EAl22)ikiEIl2βcosβπ2ikiβ+1, Σi=GAκ(ikiχi)(1l2βcosβπ2kiβ), and Υi=GAκl2βcosβπ2kiβ(ikiχi).
Equation (72) can be rewritten in the matrix form
[A]8×8[C1C2C3C4C5C6C7C8]T=0
(73)

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 ki = ki(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/C1C7/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

In this section, some numerical examples are presented for both EBB and TB. The cross section of microbeam is not limited to circular or rectangular. Although the circular or rectangular cross section is the one which is met the most often. In fact, the shear correction factor 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 ρ=2.3g/cm3, and section correction factor κ=10/9. These parameters are all come from the study by Lim [41].The dimensionless nonlocal parameters and the strain gradient parameter are l¯1=l1/r and l¯2=l2/r, respectively. After the material constants are given, the dispersion equation of the flexural waves can be expressed formally
f(E,v,ρ,k,r,ω,l1,l2)=0
Performing the dimensional analysis leads to the non-dimensional form of dispersion equation
f(1,v,1,kr,1,ωωe,l1r,l2r)=0

For the standing wave problem, the slenderness ratio, i.e., κ = L/r, is a key factor. Moreover, the dimensionless frequency ω0=ωωL(ωL=EρL2) is used instead.

Figure 1 shows the influences of the nonlocal parameter l¯1 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¯=i((1l¯22+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¯2=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¯2 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¯2 while not sensitive to the frequency.

Fig. 1
Influence of the nonlocal parameter l¯1 on the dispersion of flexural waves in EBB: (a) traveling wave, (b) standing wave caused by strain gradient effects, and (c) Standing wave
Fig. 1
Influence of the nonlocal parameter l¯1 on the dispersion of flexural waves in EBB: (a) traveling wave, (b) standing wave caused by strain gradient effects, and (c) Standing wave
Close modal
Fig. 2
Influences of the strain gradient parameter l¯2 on the dispersion of flexural waves in EBB: (a) traveling wave, (b) standing wave caused by strain gradient effects, and (c) standing wave
Fig. 2
Influences of the strain gradient parameter l¯2 on the dispersion of flexural waves in EBB: (a) traveling wave, (b) standing wave caused by strain gradient effects, and (c) standing wave
Close modal

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.

Fig. 3
Influences of nonlocal fractional order α on the dispersion of flexural waves in EBB: (a) traveling wave, (b) standing wave caused by strain gradient effects, and (c) standing wave
Fig. 3
Influences of nonlocal fractional order α on the dispersion of flexural waves in EBB: (a) traveling wave, (b) standing wave caused by strain gradient effects, and (c) standing wave
Close modal
Fig. 4
Influences of strain gradient fractional order β on the dispersion of flexural waves in EBB: (a) traveling wave, (b) standing wave caused by strain gradient effects, and (c) standing wave
Fig. 4
Influences of strain gradient fractional order β on the dispersion of flexural waves in EBB: (a) traveling wave, (b) standing wave caused by strain gradient effects, and (c) standing wave
Close modal

Figures 5 and 6 show the influence of the nonlocal parameter l¯1 and the strain gradient parameter l¯2 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., ω2=GAβ/ρI. 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¯2=121l¯22±12 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¯1 and the strain gradient parameter l¯2 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¯1 at high frequency while sensitive to the strain gradient parameter l¯2 at low frequency.

Fig. 5
Influence of the nonlocal parameter l¯1 on the dispersion of flexural waves in TB: (a) traveling wave, (b) traveling wave with cut-off frequency, (c) first standing wave caused by strain gradient effects, and (d) second standing wave caused by strain gradient effects
Fig. 5
Influence of the nonlocal parameter l¯1 on the dispersion of flexural waves in TB: (a) traveling wave, (b) traveling wave with cut-off frequency, (c) first standing wave caused by strain gradient effects, and (d) second standing wave caused by strain gradient effects
Close modal
Fig. 6
Influence of the strain gradient parameter l¯2 on the dispersion of flexural waves in TB: (a) traveling wave, (b) traveling wave with cut-off frequency, (c) first standing wave caused by strain gradient effects, and (d) second standing wave caused by strain gradient effects
Fig. 6
Influence of the strain gradient parameter l¯2 on the dispersion of flexural waves in TB: (a) traveling wave, (b) traveling wave with cut-off frequency, (c) first standing wave caused by strain gradient effects, and (d) second standing wave caused by strain gradient effects
Close modal

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.

Fig. 7
Influences of nonlocal fractional order α on the dispersion of flexural waves in TB: (a) traveling wave, (b) traveling wave with cut-off frequency, (c) first standing wave caused by strain gradient effects, and (d) second standing wave caused by strain gradient effects
Fig. 7
Influences of nonlocal fractional order α on the dispersion of flexural waves in TB: (a) traveling wave, (b) traveling wave with cut-off frequency, (c) first standing wave caused by strain gradient effects, and (d) second standing wave caused by strain gradient effects
Close modal
Fig. 8
Influences of strain gradient fractional order β on the dispersion of flexural waves in TB: (a) traveling wave, (b) traveling wave with cut-off frequency, (c) first standing wave caused by strain gradient effects, and (d) second standing wave caused by strain gradient effects
Fig. 8
Influences of strain gradient fractional order β on the dispersion of flexural waves in TB: (a) traveling wave, (b) traveling wave with cut-off frequency, (c) first standing wave caused by strain gradient effects, and (d) second standing wave caused by strain gradient effects
Close modal

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¯1 and the strain gradient parameter l¯2 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¯1 and the strain gradient parameter l¯2. The accurate characterization of the dynamic behavior of an actual beam can be acquired by appropriate adjustment of the nonlocal parameter l¯1 and the strain gradient parameter l¯2 as well as the nonlocal fractional order α and the strain gradient fraction order β.

Fig. 9
Influences of nonlocal parameters l¯1 and strain gradient parameters l¯2 on the first natural frequency of TB: (a) nonlocal parameters and (b) strain gradient parameter
Fig. 9
Influences of nonlocal parameters l¯1 and strain gradient parameters l¯2 on the first natural frequency of TB: (a) nonlocal parameters and (b) strain gradient parameter
Close modal
Fig. 10
Influences of the nonlocal fractional order α and strain gradient fraction order β on the first natural frequency of TB: (a) nonlocal fractional order and (b) strain gradient fraction order
Fig. 10
Influences of the nonlocal fractional order α and strain gradient fraction order β on the first natural frequency of TB: (a) nonlocal fractional order and (b) strain gradient fraction order
Close modal

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:

  1. 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.

  2. 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.

  3. 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.

  4. 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.

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