## Abstract

A size-dependent elasticity theory, founded on variationally consistent formulations, is developed to analyze the wave propagation in nanosized beams. The mixture unified gradient theory of elasticity, integrating the stress gradient theory, the strain gradient model, and the traditional elasticity theory, is invoked to realize the size effects at the ultra-small scale. Compatible with the kinematics of the Timoshenko–Ehrenfest beam, a stationary variational framework is established. The boundary-value problem of dynamic equilibrium along with the constitutive model is appropriately integrated into a single function. Various generalized elasticity theories of gradient type are restored as particular cases of the developed mixture unified gradient theory. The flexural wave propagation is formulated within the context of the introduced size-dependent elasticity theory and the propagation characteristics of flexural waves are analytically addressed. The phase velocity of propagating waves in carbon nanotubes (CNTs) is inversely reconstructed and compared with the numerical simulation results. A viable approach to inversely determine the characteristic length-scale parameters associated with the generalized continuum theory is proposed. A comprehensive numerical study is performed to demonstrate the wave dispersion features in a Timoshenko–Ehrenfest nanobeam. Based on the presented wave propagation response and ensuing numerical illustrations, the original benchmark for numerical analysis is detected.

## 1 Introduction

Nanostructured materials, such as carbon nanotubes (CNTs) and graphene, demonstrate significant yet peculiar physical properties at the ultra-small scale, and thus, found a wide range of implications in pioneering nanoengineering systems [15]. In view of the failure of the traditional continuum theory in the prediction of size-dependent phenomena at the nanoscale, appropriate generalized continuum theories, capable of realizing such nanoscopic features, should be implemented [6].

Various generalized continuum theories are proposed in the literature for accurate description of the behavior of continua with nanostructures; among abundant frameworks on the subject, the gradient theories of elasticity [79] and the nonlocal elasticity model [10] have drawn more attention. The gradient elasticity theories are founded on the main postulate that the mechanical response of nanocontinua depends on the traditional field variables along with the gradients of various orders up to a value characterizing the nonsimplicity grade of the material [11]. The strain gradient elasticity models of continuum mechanics can properly describe the stiffening response of nanostructures, and therefore, are the main focus of recent nanoscopic investigations [1217]. As a different line of thought, the effect of long-range interactions is taken into consideration in the framework of the nonlocal continuum mechanics where the behavior of a system at a reference point of the domain depends on the state of all the systems, i.e., the reference point along with distant points, through the nonlocal integral convolution. Due to the intrinsic simplicity of the nonlocal elasticity model, the nonlocal differential formulation attracted a particular interest in the literature and is widely utilized to capture the size effect in nanostructures [1824]. Nevertheless, serious doubts raised as the nonlocal differential model is applied to continua with finite boundaries [25].

Neither the gradient elasticity models nor the nonlocal elasticity theory is capable of covering the broad spectrum of physical features at the ultra-small scale; the hybrid continuum mechanics theories are, accordingly, introduced via the unification of dissimilar generalized theories. The instances are the nonlocal strain gradient theory [26,27], the nonlocal modified gradient theory [28,29], and the nonlocal surface elasticity [30,31]. The hybrid continuum theories constitute a promising current investigation as recent advances on the matter are addressed in Refs. [3238].

Advanced techniques founded on the wave propagation theory can be implemented as efficient nondestructive approaches to characterize the physical properties of nanocontinua, and therefore, stimulated a great deal of interest in the recent literature [3944]. The challenge of proposing a consistent variational framework to analyze the wave propagation in nanosized beams is met here. To account for the size effect, the mixture unified gradient theory of elasticity, integrating the stress gradient theory, the strain gradient model, and the traditional elasticity theory is invoked. The rest of the paper is structured as follows; to appropriately account for the shear deformation and rotary inertia effects, an elastic short stubby nanobeam consistent with the Timoshenko–Ehrenfest beam kinematics is considered in Sec. 2. A stationary variational framework, based on ad hoc functional space of kinetic test fields, is developed which consistently integrates the governing equations into a solitary functional. The corresponding boundary-value problem of dynamic equilibrium is determined and enriched with an appropriate form of nonstandard extra boundary conditions. The mixture unified gradient theory of elasticity is invoked for nanoscopic study of the flexural wave propagation in Sec. 3. As a range of gradient elasticity theories are consistently unified within the framework of the mixture unified gradient elasticity, the size-effect phenomena associated with the propagating wave in the continua with nanostructural features can be efficiently realized. The propagation characteristics of flexural waves are analytically addressed and a closed-form solution of the phase velocity is detected. Section 4 is devoted to perform a comprehensive numerical study to illustrate the wave dispersion features in a Timoshenko–Ehrenfest nanobeam wherein the nanoscopic effects corresponding to the length-scale parameters are thoroughly examined and discussed. Section 4 is, furthermore, enriched by making a thorough comparison between the wave propagation results determined via the proposed analytical approach and the counterpart results detected based on numerical simulation for CNTs. A viable methodology to calibrate the characteristic length-scale parameters associated with the generalized continuum theory is introduced. Section 5 summarizes the main outcomes and draws the concluding remarks.

## 2 Mixture Unified Gradient Elasticity Framework

The boundary-value problem associated with the dynamic equilibrium of nanocontinua with the finite domain is well-established to consist of differential conditions of equilibrium, traditional and nonstandard extra boundary conditions along with the constitutive laws. All of the aforementioned governing equations can be integrated into a single function within the stationary variational framework [45]. Another noteworthy advantage of the stationary variational principle lies in improving the accuracy and convergence rate of the approximate series solution, as evinced in Refs. [46,47], in comparison with the traditional series solution techniques [4850].

To develop the consistent stationary variational principle associated with the flexural wave propagation in nanosized beams, let us consider a prismatic elastic beam of length L which is referred to the orthogonal Cartesian coordinates where the x abscissa coincides with the longitudinal centroidal axis and the z ordinate is oriented downwards parallel to the beam height. The beam ends are restrained to prevent any rigid motion. The beam is also assumed to be subjected to a generalized transversal body force per unit length $q¯$ and an applied generalized distributed flexural couple $p¯$. The material is characterized by the material density ρ, the elastic modulus E, and the shear modulus G. In view of the fundamental assumption of the Timoshenko–Ehrenfest beam model, the beam cross-section is assumed to exhibit no warping after deformation, but not necessarily orthogonal to the deflected centroidal axis [51]. These kinematics assumptions can be achieved by taking the subsequent displacement field
$u1=−zψ(x,t),u2=0,u3=w(x,t)$
(1)
with ψ and w representing the cross-sectional rotation and the transverse displacement of the beam. The associated strain field, i.e., normal strain ɛ and the shear strain γ thus reads as
$ε=−z∂xψ(x,t)=−zκ(x,t),γ=−ψ(x,t)+∂xw(x,t)$
(2)
with κ = ∂xψ being the flexural curvature of the centroidal axis of the beam. The consistent form of the variational functional $ℑ$ within the framework of the mixture unified gradient theory of elasticity is proposed as
$ℑ=∫0L[−M0(x,t)κ(x,t)+Q0(x,t)γ(x,t)−M1(x,t)∂xκ(x,t)+Q1(x,t)∂xγ(x,t)−q¯(x)w(x,t)−p¯(x)ψ(x,t)−12EI(M0(x,t))2−ℓc22EI(∂xM0(x,t))2−12kGA(Q0(x,t))2−ℓc22kGA(∂xQ0(x,t))2−12EI(αℓc2+ℓs2)(M1(x,t))2−ℓc22EI(αℓc2+ℓs2)(∂xM1(x,t))2−12kGA(αℓc2+ℓs2)(Q1(x,t))2−ℓc22kGA(αℓc2+ℓs2)(∂xQ1(x,t))2]dx$
(3)
where A and I, respectively, designate the cross-sectional area and the second moment of area about the flexure axis. The flexural resultants M0 and M1 are, correspondingly, the dual mathematical fields of the flexural curvature κ and its derivative along the beam axis ∂xκ. Likewise, the shear resultants Q0 and Q1 are defined as dual mathematical fields of the shear strain γ and its derivative along the beam axis ∂xγ, respectively. The significance of the stress gradient theory and the strain gradient model is, correspondingly, addressed by the stress gradient characteristic length ℓc and the strain gradient length-scale parameter ℓs. The mixture parameter is, furthermore, represented by α ∈ [0, 1]. To recognize the nonuniform shear distribution across the beam thickness, the shear coefficient $k$ is implemented within the Timoshenko–Ehrenfest beam theory [51].
Consistent with the stationary variational principle, all the kinematic and kinetic field variables are selected as the primary variables subject to variation. Assuming the virtual kinetic field variables to have compact support on the domain, the first variation of the functional $ℑ$, subsequent to integration by parts, writes as
$δℑ=∫0L[−M0(x,t)δκ(x,t)−M1(x,t)∂xδκ(x,t)+Q0(x,t)δγ(x,t)+Q1(x,t)∂xδγ(x,t)−q¯(x)δw(x,t)−p¯(x)δψ(x,t)−δM0(x,t)(κ(x,t)+1EIM0(x,t)−ℓc2EI∂xxM0(x,t))−δM1(x,t)(∂xκ(x,t)+1EI(αℓc2+ℓs2)M1(x,t)−ℓc2EI(αℓc2+ℓs2)∂xxM1(x,t))+δQ0(x,t)(γ(x,t)−1kGAQ0(x,t)+ℓc2kGA∂xxQ0(x,t))+δQ1(x,t)(∂xγ(x,t)−1kGA(αℓc2+ℓs2)Q1(x,t)+ℓc2kGA(αℓc2+ℓs2)∂xxQ1(x,t))]dx−ℓc2EI∂xM0(x,t)δM0(x,t)|0L−ℓc2kGA∂xQ0(x,t)δQ0(x,t)|0L−ℓc2EI(αℓc2+ℓs2)∂xM1(x,t)δM1(x,t)|0L−ℓc2kGA(αℓc2+ℓs2)∂xQ1(x,t)δQ1(x,t)|0L$
(4)

As the virtual kinetic test field variables are selected in such a way to have compact support on the domain, the boundary terms in $δℑ$ vanish. Integrating by part while employing the kinematic compatibility conditions, consequently, yields

$δℑ=∫0L[(∂xM0(x,t)−∂xxM1(x,t)−Q0(x,t)+∂xQ1(x,t)−p¯(x))δψ(x,t)−(∂xQ0(x,t)−∂xxQ1(x,t)+q¯(x))δw(x,t)−δM0(x,t)(∂xψ(x,t)+1EIM0(x,t)−ℓc2EI∂xxM0(x,t))−δM1(x,t)(∂xxψ(x,t)+1EI(αℓc2+ℓs2)M1(x,t)−ℓc2EI(αℓc2+ℓs2)∂xxM1(x,t))+δQ0(x,t)((−ψ(x,t)+∂xw(x,t))−1kGAQ0(x,t)+ℓc2kGA∂xxQ0(x,t))+δQ1(x,t)((−∂xψ(x,t)+∂xxw(x,t))−1kGA(αℓc2+ℓs2)Q1(x,t)+ℓc2kGA(αℓc2+ℓs2)∂xxQ1(x,t))]dx+(−M0(x,t)+∂xM1(x,t))δψ|0L+(Q0(x,t)−∂xQ1(x,t))δw|0L−M1(x,t)∂xδψ|0L−Q1(x,t)δ(−ψ+∂xw)|0L$
(5)
Prescribing the stationarity of the functional $δℑ=0$ leads to the differential and boundary conditions of equilibrium for a mixture unified gradient Timoshenko–Ehrenfest beam as
$−∂x(M0(x,t)−∂xM1(x,t))+(Q0(x,t)−∂xQ1(x,t))+p(x)=ρI∂ttψ(x,t)∂x(Q0(x,t)−∂xQ1(x,t))+q(x)=ρA∂ttw(x,t)(M0(x,t)−∂xM1(x,t))δψ|0L=(Q0(x,t)−∂xQ1(x,t))δw|0L=0M1(x,t)|0L=Q1(x,t)|0L=0$
(6)
where the flexural curvature and shear strain fields are assumed to have arbitrary variations. The d’Alembert principle is also applied to establish the dynamic form of the differential conditions of equilibrium [45]. To further simplify the governing equations, the flexural moment M and the shear force Q are introduced as
$M(x,t)=M0(x,t)−∂xM1(x,t)Q(x,t)=Q0(x,t)−∂xQ1(x,t)$
(7)
The boundary-value problem of the dynamic equilibrium of a Timoshenko–Ehrenfest beam consistent with the mixture unified gradient theory is, accordingly, simplified as
$−∂xM(x,t)+Q(x,t)+p(x)=ρI∂ttψ(x,t)∂xQ(x,t)+q(x)=ρA∂ttw(x,t)M(x,t)δψ|0L=Q(x,t)δw|0L=0M1(x,t)|0L=Q1(x,t)|0L=0$
(8)
As a well-established privilege of employing the stationary variational principle, the constitutive laws of the resultant fields are cast as ordinary differential relations; the flexural resultants M0, M1 and the flexural moment M are, accordingly, expressed by
$M0(x,t)−ℓc2∂xxM0(x,t)=−EI∂xψ(x,t)M1(x,t)−ℓc2∂xxM1(x,t)=−EI(αℓc2+ℓs2)∂xxψ(x,t)M(x,t)−ℓc2∂xxM(x,t)=−EI(∂xψ(x,t)−(αℓc2+ℓs2)∂xxxψ(x,t))$
(9)
Likewise, the constitutive laws of the shear resultants Q0, Q1 and the shear force Q are determined as
$Q0(x,t)−ℓc2∂xxQ0(x,t)=kGA(−ψ(x,t)+∂xw(x,t))Q1(x,t)−ℓc2∂xxQ1(x,t)=kGA(αℓc2+ℓs2)(−∂xψ(x,t)+∂xxw(x,t))Q(x,t)−ℓc2∂xxQ(x,t)=kGA[(−ψ(x,t)+∂xw(x,t))−(αℓc2+ℓs2)(−∂xxψ(x,t)+∂xxxw(x,t))]$
(10)
while the constitutive relations of the flexural and shear resultants are suitably enriched with the gradient length-scale parameters, the effects of the traditional elasticity model are incorporated via the mixture parameter m, as well. The conceived stationary variational framework, thus, represents an apposite generalized continuum theory for nanoscopic analysis of the field quantities at the ultra-small scale.
Notably, the constitutive relations of the resultant fields are of higher order in comparison with the traditional elasticity ones; the determined boundary-value problem should be therefore supplied with a consistent form of nonstandard extra boundary conditions as Eq. (8)4. In order to properly describe the nanoscopic behavior of beam-type structures at the ultra-small scale, explicit formulae of the flexural resultant M1 and the shear resultant Q1 should be properly determined; otherwise, it will lead to an erroneous inference of the size-dependent response of nanostructures [21]. Subsequent to some straightforward mathematics, the flexural and the shear resultants can be determined as
$M1(x,t)=ℓc2(αℓc2+ℓs2)ℓc2(1−α)−ℓs2(ρA∂xttw(x,t)−∂xq(x)−ρI∂xxttψ(x,t)+∂xxp(x))−EI(αℓc2+ℓs2)(∂xxψ(x,t)−ℓc2(αℓc2+ℓs2)ℓc2(1−α)−ℓs2∂xxxxψ(x,t))Q1(x,t)=ℓc2(αℓc2+ℓs2)ℓc2(1−α)−ℓs2(ρA∂xttw(x,t)−∂xq(x))+kGA(αℓc2+ℓs2)(−∂xψ(x,t)+∂xxw(x,t))−kGAℓc2(αℓc2+ℓs2)2ℓc2(1−α)−ℓs2(−∂xxxψ(x,t)+∂xxxxw(x,t))$
(11)

A range of generalized continuum theories of the gradient type, capable of realizing the size-dependency of field variables at the nanoscale, can be retrieved as particular cases of the conceived mixture unified gradient theory of elasticity. The traditional constitutive model of the Timoshenko–Ehrenfest beam can be restored via either setting the mixture parameter to unity in the absence of the strain gradient length-scale parameter or vanishing the gradient characteristic lengths for a fixed mixture parameter.

In the absence of the mixture parameter, the constitutive relations of the flexural moment and the shear force consistent with the unified gradient elasticity theory can be restored as [28]
$M(x,t)−ℓc2∂xxM(x,t)=−EI(∂xψ(x,t)−ℓs2∂xxxψ(x,t))Q(x,t)−ℓc2∂xxQ(x,t)=kGA[(−ψ(x,t)+∂xw(x,t))−ℓs2(−∂xxψ(x,t)+∂xxxw(x,t))]$
(12)
The constitutive model of the moment and shear resultants corresponding to the stress gradient theory [8] is recovered as the strain gradient length-scale parameter tends to zero in Eq. (12) as
$M(x,t)−ℓc2∂xxM(x,t)=−EI∂xψ(x,t)Q(x,t)−ℓc2∂xxQ(x,t)=kGA(−ψ(x,t)+∂xw(x,t))$
(13)
Likewise, the constitutive law of the resultant fields consistent with the strain gradient theory [11] is obtained via vanishing the stress gradient characteristic length in Eq. (12) as
$M(x,t)=−EI(∂xψ(x,t)−ℓs2∂xxxψ(x,t))Q(x,t)=kGA[(−ψ(x,t)+∂xw(x,t))−ℓs2(−∂xxψ(x,t)+∂xxxw(x,t))]$
(14)
Lastly, the flexural moment and the shear force associated with the recently established mixture stress gradient theory [47] can be retrieved as the strain gradient length-scale parameter approaches zero in Eqs. (9) and (10) as
$M(x,t)−ℓc2∂xxM(x,t)=−EI(∂xψ(x,t)−αℓc2∂xxxψ(x,t))Q(x,t)−ℓc2∂xxQ(x,t)=kGA[(−ψ(x,t)+∂xw(x,t))−αℓc2(−∂xxψ(x,t)+∂xxxw(x,t))]$
(15)

As the softening and stiffening responses of nanosized structures can be effectively addressed within the established variationally consistent elasticity framework, the developed generalized continuum theory is fruitfully invoked for nanoscopic study of the propagation of flexural waves in nanostructures.

## 3 Wave Propagation Characteristics

The traditional theory of elasticity is insufficient in properly describing the dispersion phenomenon associated with the propagating waves with wavelength comparable to the characteristic lengths of the nanosized continua of interest. In contrast, the introduced mixture unified gradient theory of elasticity can be efficiently implemented for an accurate description of the dispersion behavior of waves in continua with nanoscopic features. Wave propagation analysis, additionally, can be fruitfully employed for the identification of the characteristic length-scale parameters associated with the generalized continuum theories [28,29,36,37].

The scattering of flexural waves is examined here within the context of the mixture unified gradient theory of elasticity with application to a Timoshenko–Ehrenfest nanobeam. To examine the phase velocity of propagating waves, the differential conditions of equilibrium are described in terms of the displacement field variables. To this end, the flexural moment and the shear force associated with the mixture unified gradient beam are first determined in view of the established constitutive relations (9)3 and (10)3 and the differential conditions of dynamic equilibrium Eq. (8)1,2 as
$M(x,t)=ℓc2ρA∂ttw(x,t)−ℓc2ρI∂xttψ(x,t)−EI(∂xψ(x,t)−(αℓc2+ℓs2)∂xxxψ(x,t))Q(x,t)=ℓc2ρA∂xttw(x,t)+kGA[(−ψ(x,t)+∂xw(x,t))−(αℓc2+ℓs2)(−∂xxψ(x,t)+∂xxxw(x,t))]$
(16)
where the transversal body force and the distributed flexural couple are allowed to vanish in the elastodynamic analysis. Substituting the detected results of the flexural and shear resultants into the differential conditions of dynamic equilibrium yields a set of governing equations for the mixture unified gradient Timoshenko–Ehrenfest beam
$ℓc2ρI∂xxttψ(x,t)+EI(∂xxxψ(x,t)−(αℓc2+ℓs2)∂xxxψ(x,t))−ρI∂ttψ(x,t)+kGA[(−ψ(x,t)+∂xw(x,t))−(αℓc2+ℓs2)(−∂xxψ(x,t)+∂xxxw(x,t))]=0ℓc2ρA∂xxttw(x,t)−ρA∂ttw(x,t)+kGA[(−∂xψ(x,t)+∂xxw(x,t))−(αℓc2+ℓs2)(−∂xxxψ(x,t)+∂xxxxw(x,t))]=0$
(17)
The propagation of flexural waves is examined for continua with unbounded domain, the condition of decay at infinity is therefore assumed; i.e., vanishing of the boundary conditions at infinity is tacitly met. For flexural waves propagating in nanosized beams, the analytical solution of the wave dispersion takes the form of
$ψ(x,t)=ψ¯exp(iκ(x−vt)),w(x,t)=w¯exp(iκ(x−vt))$
(18)
with $i=−1$, κ and v, respectively, representing the wave number and the phase velocity along with $ψ¯$ and $w¯$ being the coefficients of the wave amplitude. Imposing the wave dispersion solutions Eq. (18) to the governing equations Eq. (17) yields a homogeneous set of algebraic equations that requires to be singular to have a nontrivial solution. Vanishing of the determinant of coefficients of the homogeneous algebraic system, consequently, results in the characteristic equation of the flexural wave propagating in a mixture unified gradient Timoshenko–Ehrenfest beam as
$v4κ4(1+ℓc2κ2)2(ρAρI)−v2κ2(1+ℓc2κ2)(1+κ2(αℓc2+ℓs2))((EIρA)κ2+kGA(ρA+κ2ρI))+κ4(EIkGA)(1+κ2(αℓc2+ℓs2))2=0$
(19)

The determined characteristic equation of wave propagation encompasses two positive roots for each wave number; as the lower phase velocity corresponds to the acoustic mode, the higher phase velocity corresponds to the optical mode [52].

The sought phase velocity of propagating flexural waves in the Timoshenko–Ehrenfest beam consistent with the mixture unified gradient theory of elasticity can be explicitly expressed as
$v=1κ(1+κ2(αℓc2+ℓs2))2(1+ℓc2κ2)(ρAρI)(EIκ2ρA+kGA(ρA+κ2ρI)+(EIρA)2κ4+2EIkGAκ2ρA(ρA−κ2ρI)+(kGA)2(ρA+κ2ρI)2)1/2$
(20)

Notably, the wave propagation response of a Timoshenko–Ehrenfest nanobeam within the framework of the unified gradient theory is retrieved in the absence of the mixture parameter [52,53].

## 4 Numerical Illustrations and Discussion

The established formulations associated with the mixture unified gradient theory give rise to three length-scale parameter; namely the stress gradient characteristic length ℓc, the strain gradient length-scale parameter ℓs and the mixture parameter α. These intrinsic parameters associated with the conceived generalized continuum theory can be calibrated in comparison with the limited data detected by numerical simulation or experimental measurements. Efficacy of the established size-dependent elasticity theory in describing the propagation behavior of flexural waves is evinced via examining the phase velocity detected by numerical simulations for (10, 10) armchair CNTs [54]. The unknown parameters of a physical field can be detected by applying the inverse theory approach wherein the discrepancy between the reconstructed field variables and the limited data measurements is minimized [55,56]. The material properties of a (10, 10) armchair CNT are characterized by the material density ρ = 2237 kg/m3, the elastic modulus E = 470 GPa, and the shear modulus G = 196 GPa. The (10, 10) armchair CNT is also treated as a thin-shell beam with the cross-sectional area $A=2πrℏ$ and the second moment of area $I=πr3ℏ$ where r = 0.678 nm and $ℏ=0.0617nm$, respectively, represent the radius and thickness of the (10, 10) armchair CNT [57]. The shear coefficient $k=2(1+ν)/(4+3ν)$ is also applied for the thin-walled circular cross-section [51].

The phase velocity of propagating flexural waves in a (10, 10) armchair CNT is reconstructed by implementing a nonlinear least square optimization approach. The calibrated length-scale parameters are, accordingly, determined as ℓc = 0.83907 nm,  ℓs = 0.19565 nm and α = 0.00026. The numerical simulation data along with the reconstructed phase velocity of propagating waves in a mixture unified gradient Timoshenko–Ehrenfest beam are demonstrated in Fig. 1 which is additionally enriched by illustrating the space-frame structural model of the (10, 10) armchair CNT. As deducible from the comparison made, the mixture unified gradient theory can successfully capture the wave propagation characteristics on the specified range of wave numbers.

Fig. 1
Fig. 1
Close modal

The demonstrated results reveal that the qualitative aspects of the propagation behavior can be realized with admissible accuracy in the context of the mixture unified gradient elasticity theory yielding tolerable discrepancy between the inversely identified phase velocity and the numerical simulation data. A viable approach to inversely determine the characteristic length-scale parameters associated with the generalized continuum theories is, accordingly, proposed.

Nanoscopic effects of the length-scale parameters on the propagation response of flexural waves are also graphically demonstrated and discussed. For the sake of consistency of illustrations, the nondimensional form of the stress gradient characteristic parameter ζ, the strain gradient characteristic parameter η, the wave number $κ¯$, and the phase velocity $v¯$ are introduced as
$ζ=ℓcL,η=ℓsL,κ¯=κL,v¯=vLρAEI$
(21)

The phase velocity of flexural waves is, also, normalized with respect to the phase velocity of flexural waves consistent with the traditional Timoshenko–Ehrenfest beam $v¯0$. The 3D variation of the normalized phase velocity of the wave propagation in terms of the length-scale parameters along with the logarithmic scaling of the nondimensional wave number $κ¯$ is examined in Figs. 24. While nanoscopic effects of the stress gradient and the strain gradient characteristic parameters on the wave propagation response are, correspondingly, examined in Figs. 2 and 3, the size effect of the mixture parameter on the wave dispersion behavior is investigated in Fig. 4. The stress gradient characteristic parameter is considered to range in the interval [0, 1] in Fig. 2 as three values of the mixture parameter α = 0, 1/2, 1 are prescribed for a fixed strain gradient characteristic parameter η = 1/4. Likewise, the strain gradient characteristic parameter is ranging in the interval [0, 1] in Fig. 3 while three values of the mixture parameter α = 0, 1/2, 1 are applied for a prescribed value of the stress gradient characteristic parameter ζ = 1/4. As three distinct conditions of ζ = 1/3 > η = 1/4, ζ = η = 1/3, and ζ = 1/4 < η = 1/3 are chosen for the stress gradient and the strain gradient characteristic parameters in Fig. 4, the mixture parameter has also the ranging set [0, 1]. In all the ensuing numerical illustrations, the (logarithm of) nondimensional wave number [58] is ranging in the interval [10−1, 10+1].

Fig. 2
Fig. 2
Close modal
Fig. 3
Fig. 3
Close modal
Fig. 4
Fig. 4
Close modal

The propagating wave response is well-recognized to be insensitive to the nanoscopic physical properties for large wavelengths corresponding to low wave numbers. The phase velocity of flexural waves in the context of the mixture unified gradient theory, accordingly, remains unaffected by the characteristic parameters for lower wave numbers. The influence of the characteristic parameters on the wave propagation response is particularly enhanced at higher wave numbers. The wave propagation response of a mixture unified gradient Timoshenko–Ehrenfest beam is realized to be less affected by the characteristic parameters for nonvanishing values of the mixture parameter. The wave propagation response of flexural waves within the framework of the mixture unified gradient theory coincides with the phase velocity of the waves propagating in a traditional Timoshenko–Ehrenfest beam as the gradient characteristic parameters vanish, or alternatively, as the mixture parameter tends to unity in the absence of the strain gradient characteristic parameter.

## 5 Concluding Remarks

The stationary variational framework, in accordance with the mixture unified gradient theory of elasticity, is established to examine the propagation of flexural waves in nanobeams. Kinematics of the elastic short stubby beam is assumed to be consistent with the Timoshenko–Ehrenfest beam model suitably comprising the shear deformation and the rotary inertia effects. The mixture unified gradient theory consistently integrates the stress gradient theory, the strain gradient model, and the traditional elasticity theory, and thus, is capable of capturing the size-effect phenomenon associated with the propagating wave in a continuum with nanostructural features. The boundary-value problem of the dynamic equilibrium, the (traditional and nonstandard extra) boundary conditions as well as the constitutive relations are all incorporated into a single function. To properly describe the nanoscopic behavior of beam-type structures at the nanoscale, explicit formulae of the flexural and shear resultants, associated with the nonstandard extra boundary conditions, are determined. A variety of generalized continuum theories of the gradient type is demonstrated to be retrieved as particular cases of the developed mixture unified gradient theory of elasticity under ad hoc assumptions.

The propagation characteristics of flexural waves are analytically addressed and a closed-form solution of the phase velocity is determined. A comprehensive numerical analysis is performed to demonstrate the wave dispersion features in a mixture unified gradient Timoshenko–Ehrenfest nanobeam wherein the length-scale effects of the characteristic parameters are thoroughly discussed. The foreseeable nanoscopic features of the flexural waves propagating in a mixture unified gradient Timoshenko–Ehrenfest beam are confirmed, i.e., the softening response in terms of the stress gradient characteristic parameter is realized while the stiffening behavior of the strain gradient characteristic parameter and the mixture parameter is also captured. It is, furthermore, demonstrated that the nanoscopic effects on the phase velocity of wave propagation are merely significant when the wavelength is small enough to be commensurate with the gradient length-scale parameters. Effectiveness of the mixture unified gradient theory in describing the propagation characteristics of flexural waves is revealed through the consistent reconstruction of the phase velocity detected by numerical simulations for carbon nanotubes. The qualitative aspects of the wave propagation response in carbon nanotubes are evinced to be captured with an admissible accuracy. A practical approach to inversely determine the characteristic length-scale parameters associated with the generalized continuum theory is proposed.

It is hoped that the established variationally consistent elasticity framework and the ensuing numerical results introduce a viable methodology for the efficient dynamic examination of nanoscopic features associated with structural elements of ground-breaking nanoengineering systems.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

No data, models, or code were generated or used for this paper.

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