Free vibrations of elastic beams by modified nonlocal strain gradient theory

International Journal of Engineering Science 133 (2018) 99–108

Contents lists available at ScienceDirect

International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci

Free vibrations of elastic beams by modified nonlocal strain gradient theory A. Apuzzo a, R. Barretta b,∗, S.A. Faghidian c, R. Luciano a, F. Marotti de Sciarra b a

Department of Civil and Mechanical Engineering, University of Cassino and Southern Lazio, via G. Di Biasio 43, Cassino, FR 03043, Italy Department of Structures for Engineering and Architecture, University of Naples Federico II, via Claudio 21, Naples 80125, Italy c Department of Mechanical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran b

a r t i c l e

i n f o

Article history: Received 12 August 2018 Revised 30 August 2018 Accepted 3 September 2018

Keywords: Bernoulli–Euler nano-beam Axial vibrations Flexural vibrations Nonlocal strain gradient elasticity Constitutive boundary conditions NEMS

a b s t r a c t Size-dependent axial and flexural free vibrations of Bernoulli–Euler nano-beams are investigated by the modified nonlocal strain gradient elasticity model presented in (Barretta & Marotti de Sciarra, 2018). The corresponding elastodynamic problem, with the natural constitutive boundary conditions, is solved by an effective analytical methodology. Axial and flexural fundamental frequencies are determined for cantilever and fully-clamped nanobeams. Effects of nonlocal and gradient scale parameters on fundamental frequencies are examined and compared with those obtained by Eringen’s local/nonlocal mixture model. New benchmarks are found for vibrations of beams. The adopted nonlocal strain gradient model, with the appropriate constitutive boundary conditions, is capable of capturing both softening and stiffening dynamical responses. Accordingly, it provides an advantageous approach for design and optimization of a wide range of nano-scaled beam-like components of Nano-Electro-Mechanical-Systems (NEMS). © 2018 Elsevier Ltd. All rights reserved.

1. Introduction The recent challenges in nano-engineering have increased the necessity to rigorously examine the mechanical behavior of nano-devices. Beam-like elements are fundamental structural parts of modern nano-electro-mechanical systems (NEMS) which exhibit size effects. Therefore, a profound understanding of analytical and numerical models is required for appropriate assessment and design of NEMS (Kumar, Singh, Hui, Feo, & Fraternali, 2018; Ramsden, 2016; Zhang & Hoshino, 2014). The analysis of nano-structures has become a topic of major concern in the current literature and an extensive variety of constitutive models is exploited in the community of Engineering Science to assess size phenomena. Instances are the following. Eringen’s nonlocal differential equation is employed to examine pull-in instabilities of nano-beams (Ouakad & Sedighi, 2016; Sedighi & Sheikhanzadeh, 2017; Sedighi, Keivani, & Abadyan, 2015; Sedighi, Daneshmand, & Abadyan, 2015, 2016), vibration of nano-rods (Numanog˘ lu, Akgöz, & Civalek, 2018), nonlinear functionally graded nano-plates (Srividhya, Raghu, Rajagopal, & Reddy, 2018) and wrinkling hierarchy in graphene (Zhao, Guo, & Lu, 2018). An enhanced Eringen’s differential law was proˇ posed by Barretta, Feo, Luciano, and Marotti de Sciarra (2016) and applied to torsion (Apuzzo, Barretta, Cana d¯ija et al., 2017) and flexure (Barretta, Feo, Luciano, Marotti de Sciarra, and Penna, 2016; Demir & Civalek, 2017). Eringen’s local/nonlocal twophase integral methodologies are exploited to study static (Acierno, Barretta, Luciano, Marotti de Sciarra, & Russo, 2017) and dynamic axial behaviors of nano-rods (Zhu & Li, 2017a), static flexure of slender (Fernández-Sáez, Zaera, Loya, & Reddy, 2016; ∗

Corresponding author. E-mail addresses: [email protected] (A. Apuzzo), [email protected] (R. Barretta), [email protected] (R. Luciano), [email protected] (F. Marotti de Sciarra).

https://doi.org/10.1016/j.ijengsci.2018.09.002 0020-7225/© 2018 Elsevier Ltd. All rights reserved.

100

A. Apuzzo et al. / International Journal of Engineering Science 133 (2018) 99–108

Koutsoumaris, Eptaimeros, & Tsamasphyros, 2017; Wang, Zhu, & Dai, 2016) and stubby nano-beams (Wang, Huang, Zhu, & Lou, 2018), as well as free vibrations (Fernández-Sáez & Zaera, 2017; Khanik, 2018) and buckling phenomena (Zhu, Wang, & ˇ ´ Cana d¯ija, Dai, 2017). Also, strain gradient models are employed for Bernoulli–Euler (Akgöz & Civalek, 2015; Barretta, Brcˇ ic, Luciano, & Marotti de Sciarra, 2017), Timoshenko (Marotti de Sciarra & Barretta, 2014) and shear deformable nano-beams (Akgöz & Civalek, 2014a), stability analysis of carbon nanotubes (Akgöz & Civalek, 2017a), dynamic torsion of nano-beams (Sedighi, Koochi, Keivani, & Abadyan, 2017), vibrations of three-dimensionally graded nano-beams (Hadi, ZamaniNejad, & Hosseini, 2018) and flexoelectric curved micro-beams (Qi, Huang, Fu, Zhou, & Jiang, 2018). Couple stress theories are utilized for buckling analysis (Akgöz & Civalek, 2014b; Jiao & Alavi, 2018; Taati, 2018), dynamics of elastic and viscoelastic nanobeams (Akgöz & Civalek, 2017b,2018; Attia & Abdel Rahman, 2018; Ghayesh, 2018; Ghayesh, & Farokhi, 2018) and Lamb wave dispersion in nano-plates (Ghodrati, Yaghootian, Ghanbar Zadeh, & Sedighi, 2018). Also, nonlocal strain gradient theory is used to study extension (Zhu & Li, 2017b), flexure (Li & Hu, 2016; Li, Tang, & Hu, 2018), vibration (Li, Li, & Hu, 2016; Lu, Guo, & Zhao, 2017), buckling of nano-beams (Li & Hu, 2015), dynamics of nano-tubes (Ghayesh & Farajpour, 2018; She, Ren, Yuan, & Xiao, et al., 2018) and nano-plates (Lu, Guo, & Zhao, 2018). Nonlinear analyses of Reissner nonlocal strain gradient beams are carried out in (Faghidian, 2018a) by using the variational formulation in (Faghidian, 2018b). Further overviews can be found in (Aifantis, 2016; Rafii-Tabar, Ghavanloo, & Fazelzadeh, 2016; Thai, Vo, Nguyen, & Kim, 2017). However, two basic pure nonlocal elasticity models are available in literature, depending on whether strain-driven or stress-driven formulations are adopted. The strain-driven nonlocal elasticity theory was introduced by Eringen (1983), wherein the stress field is the output of the integral convolution between elastic strain field and a smoothing kernel. Eringen’s integral model (EIM) can be replaced for unbounded continua with a differential law which is commonly named Eringen differential model (EDM). EIM cannot be exploited to assess size effects in structural problems in bounded domains, due to the confliction between constitutive boundary conditions and equilibrium (Romano, Barretta, Diaco, & Marotti de Sciarra, 2017; Romano, Barretta, & Diaco, 2017; Romano, Luciano, Barretta, & Diaco, 2018). Such an ill-posedness may be partly removed by the mixture Eringen integral model (MEIM) (Eringen, 1972, 1987), as thoroughly discussed in (Romano, Barretta, & Diaco, 2017; Romano, Luciano, Barretta, & Diaco, 2018). The stress-driven nonlocal integral model (SDM) was conceived by Romano and Barretta (2017a), where source and output fields of Eringen’s strain-driven integral law are swapped. The nonlocal elastic strain field is therefore defined as output of the integral convolution between stress field and a smoothing kernel. Unlike EIM, SDM in bounded domains leads to well-posed nano-structural problems of engineering interest (Romano & Barretta, 2017b). The stress-driven nonlocal integral approach has been effectively adopted in a variety of elastostatic, elastodynamic and thermoelastic problems of ˇ d¯ija, Feo, et al., 2018; Barretta, nano-mechanics (Apuzzo, Barretta, Luciano, Marotti de Sciarra, & Penna, 2017; Barretta, Cana ˇ d¯ija, Luciano, & Marotti de Sciarra, 2018; Barretta, Diaco, et al., 2018; Barretta, Fazelzadeh, Feo, Ghavanloo, & Luciano, Cana 2018; Barretta, Faghidian, & Luciano 2018; Barretta, Faghidian, Luciano, Medaglia, & Penna, 2018a; Barretta, Luciano, Marotti de Sciarra, & Ruta, 2018; Mahmoudpour, Hosseini-Hashemi, & Faghidian, 2018). The stress-driven nonlocal integral law, convexly combined with the local elastic law, leads to a mixture stress-driven integral model (MSDM) which provides also a viable approach to capture size effects in bounded nano-structures (Barretta, Fabbrocino, Luciano, & Marotti de Sciarra, 2018; Barretta, Faghidian, Luciano, Medaglia, & Penna, 2018b). Also, Eringen’s differential model (EDM) and strain gradient elasticity theory (SGT) were first unified by Aifantis (2003, 2011) to conceive the nonlocal strain gradient (NSG) elasticity theory. Lim, Zhang, and Reddy (2015) then combined the Eringen integral model (EIM) with the NSG to formulate a higherorder nonlocal theory. To include higher-order strain gradient effects within Eringen’s integral model (EIM), a second-order integro-differential nonlocal theory was recently contributed within a thermodynamic framework by Faghidian (2018c). According to nonlocal strain gradient theory, the stress field is the sum of two integral convolutions. The former one is performed between strain field and a smoothing kernel depending on a nonlocal parameter. The latter one, involving also a gradient parameter, is the gradient of the convolution between strain gradient field and a smoothing kernel. Such an integral constitutive law, if formulated on unbounded domains and equipped with Helmholtz’s bi-exponential kernels, may be correctly replaced with a differential relation, due to the tacit fulfillment of constitutive boundary conditions of vanishing at infinity. However, the nonlocal strain gradient problems of technical interest involve bounded domains, and hence, suitable constitutive boundary conditions are to be prescribed to close the constitutive law. This key issue has been solved in (Barretta & Marotti de Sciarra, 2018). It has been also pointed out that the differential law associated with the nonlocal strain gradient integral model of elasticity, equipped with appropriate constitutive boundary conditions, leads to well-posed nano-engineering problems. In the present study, axial and flexural free vibrations of Bernoulli–Euler nano-beams are formulated by modified nonlocal strain gradient integral theory presented in (Barretta & Marotti de Sciarra, 2018), equipped with the appropriate constitutive boundary conditions. Fundamental frequencies are analytically determined for cantilever and fully-clamped beams and compared with those obtained by Eringen’s local/nonlocal mixture model. The modified nonlocal strain gradient dynamical model, is demonstrated to be capable of exhibiting both softening and stiffening structural behaviors, and therefore, provides a convenient approach for assessment and design of a wide range of nano-devices exploited in NEMS applications. New benchmarks are also detected for axial and flexural free vibrations of nano-beams.

A. Apuzzo et al. / International Journal of Engineering Science 133 (2018) 99–108

101

2. Bernoulli–Euler elastic beams: modified nonlocal strain gradient theory An elastically homogeneous beam of length L, cross-sectional domain  of area A and material density ρ is considered here. Beam configuration is described by Cartesian coordinates, where it is assumed that the longitudinal axis passing through the line of cross-sectional centroids is denoted with x. The z axis is also assumed to be downward parallel to the flexure direction, the pair x –z identifies with the plane of flexure. The components of the displacement field in accordance to the Bernoulli–Euler beam kinematics write as

u1 = u(x, t ) − z∂x w(x, t ),

u2 = 0,

u3 = w(x, t )

(1)

where u and w, respectively, are the axial and transverse displacements of the cross-section centroid at time t. The only non-zero strain along the beam axis is then expressed by

εxx = ε (x, t ) − zχ (x, t ) = ∂x u(x, t ) − z∂xx w(x, t )

(2)

with ɛ and χ being the extension and curvature of the beam centroidal axis. Let axial force N and bending moment M be the dual fields of the extension ɛ and flexural curvature χ , respectively. The differential and boundary conditions of dynamic equilibrium of a Bernoulli–Euler beam in absence of applied external loading may be determined as (Reddy, 2017)

∂x N = Aρ ∂tt u ∂xx M = −Aρ ∂tt w + Iρ ∂xxtt w Nδ u|x=0,L = 0 M (∂x δ w )|x=0,L = (∂x M )δ w|x=0,L = 0 



(3)

with Aρ =  ρ (z )dA and Iρ =  cross-sectional mass and rotatory inertia. In the nonlocal strain gradient theory of elasticity, axial force N is expressed in terms of elastic extension ɛ and of its derivative along the beam axis ∂ x ɛ by (Lim et al., 2015)



N=

L 0

ρ (z )z2 dA

ϕ (x − x¯, c )(AE ε )(x¯, t )dx¯ − 2 ∂x





L

0

ϕ (x − x¯, c )(AE ∂x¯ ε )(x¯, t )dx¯

(4)

where AE =  E (z )dA is the local elastic axial stiffness defined as the average of the field of the Euler-Young modulus E over beam cross-section. The characteristic length ≥ 0 is also introduced to render dimensionally homogeneous the convolutions in Eq. (4). Following Lim et al. (2015), the following Helmholtz bi-exponential smoothing kernel is employed to describe the integral convolutions above



ϕ (x, c ) =

1 |x| exp − 2c c



(5)

where c > 0 represents the characteristic length of Eringen’s nonlocal elasticity depicting long-range interactions. The biexponential kernel fulfills positivity, parity, normalization and impulsivity on the real axis (Romano, et al., 2018). Following the mathematical formulation of Prop. 3.1 by Barretta and Marotti de Sciarra (2018), it may be shown that the nonlocal strain gradient constitutive law Eq. (4) equipped with the bi-exponential kernel Eq. (3) is equivalent to the differential constitutive relation

(AE ε )(x, t ) − 2 ∂xx (AE ε )(x, t ) = N (x, t ) − c2 ∂xx N (x, t )

(6)

with the following two constitutive boundary conditions (CBC) at the beam ends x = 0 and x = L

∂x N (0, t ) − 1c N (0, t ) =

2 c2

∂x N (L, t ) +

2 c2

1 N c

(L, t ) =

∂x ( AE ε ) ( 0, t )

(7)

∂x (AE ε )(L, t )

Similarly, the bending moment M of nonlocal strain gradient theory is described in terms of elastic flexural curvature χ and of its derivative ∂ x χ by (Lim et al., 2015)



M=

L 0

ϕ (x − x¯, c )(IE χ )(x¯, t )dx¯ − 2 ∂x



0

L

ϕ (x − x¯, c )(IE ∂x¯ χ )(x¯, t )dx¯

(8)

 where IE =  E (z )z2 dA designates the local elastic bending stiffness defined as the second moment of the field of EulerYoung elastic moduli E on the beam cross-section. As illustrated by Barretta and Marotti de Sciarra (2018), the nonlocal strain gradient constitutive law Eq. (8) equipped with the bi-exponential kernel Eq. (3) is equivalent to the differential constitutive relation

(IE χ )(x, t ) − 2 ∂xx (IE χ )(x, t ) = M (x, t ) − c2 ∂xx M (x, t )

(9)

with the two constitutive boundary conditions (CBC) at the beam ends x = 0andx = L

∂x M (0, t ) − 1c M (0, t ) =

2 c2

∂x M (L, t ) +

2 c2

1 M c

(L, t ) =

∂x (IE χ )(0, t )

∂x (IE χ )(L, t )

(10)

102

A. Apuzzo et al. / International Journal of Engineering Science 133 (2018) 99–108

It is worth underlining that, as proved by Barretta and Marotti de Sciarra (2018) and contrary to the claims of literature, nonlocal strain gradient integral laws for bounded nano-beams can be replaced with differential constitutive laws, provided that the appropriate constitutive boundary conditions are verified. 3. Free vibration analysis The size-dependent nonlocal strain gradient laws presented in Section 2 are adopted here to examine axial and flexural free vibrations of Bernoulli–Euler nano-beams. 3.1. Axial free vibrations The axial force N can be conveniently determined utilizing the differential condition of equilibrium Eq. (3) the kinematic compatibility of elastic extension ɛ Eq. (2) as





N = c2 ∂x Aρ ∂tt u + (AE ∂x u ) − 2 ∂xx (AE ∂x u )

1

along with

(11)

The differential equation of equilibrium governing axial vibrations writes thus as





c2 ∂xx Aρ ∂tt u + ∂x (AE ∂x u ) − 2 ∂xxx (AE ∂x u ) = Aρ ∂tt u

(12)

under the classical boundary conditions Eq. (3) 3 and constitutive boundary conditions Eq. (7). Natural frequencies and mode shapes are determined by separating spatial and time variables

u(x, t ) = U (x ) exp (i t )

(13)

√ with i = −1 and being the natural frequency of axial vibrations. The governing equation on the axial base function U(x) is determined by substituting Eq. (13) into Eq. (12)



1 − 2 ∂xx



   ∂x (AE ∂xU (x )) + 1 − c2 ∂xx Aρ 2U (x ) = 0

(14)

Such an equation Eq. (14) in the framework of nonlocal theory of elasticity is the same as the result introduced by Li et al. (2016), but subjected to fundamentally different higher-order constitutive boundary conditions of Eq. (7). For a uniform beam, the analytical solution of the preceding Eq. (14) can be expressed by

U (x ) = U1 sin α1 x + U2 cos α1 x + U3 sinh α2 x + U4 cosh α2 x

(15)

with Uk (k = 1..4 ) unknown constants needed to be determined by suitable boundary conditions, along with 

(ρ 2 c2 −E )+ (ρ 2 c2 −E )

α = 2 1

α = 2 2

(

)

− ρ 2 c2 −E +



2

+4E ρ 2 2

2E 2

(ρ 2 c2 −E )

2

(16) +4E ρ 2 2

2E 2

3.2. Flexural free vibrations Employing the differential condition of equilibrium Eq. (3)2 as well as the kinematic compatibility involving the flexural curvature χ Eq. (2), the bending moment M writes as





M = c2 −Aρ ∂tt w + Iρ ∂xxtt w + (IE ∂xx w ) − 2 ∂xx (IE ∂xx w )

(17)

Accordingly, the differential equation of equilibrium takes the form





c2 ∂xx −Aρ ∂tt w + Iρ ∂xxtt w + ∂xx (IE ∂xx w ) − 2 ∂xxxx (IE ∂xx w ) = −Aρ ∂tt w + Iρ ∂xxtt w

(18)

subjected to the classical boundary conditions Eq. (3) 4 and constitutive boundary conditions Eq. (10). Similarly, natural frequencies and mode shapes of flexural vibrations are evaluated by employing again the classical separation of spatial and time variables as

w(x, t ) = W (x ) exp (iωt )

(19)

√ with i = −1 and ω being the natural frequency of flexural vibrations. The equation governing the flexural base function W(x) is subsequently obtained by substituting Eq. (19) into Eq. (18)



1 − 2 ∂xx



   ∂xx (IE ∂xxW (x )) = 1 − c2 ∂xx Aρ ω2W (x ) − Iρ ω2 ∂xxW (x )

(20)

Eq. (20), governing flexural vibrations of Bernoulli–Euler nonlocal gradient nano-beams, coincides with the one introduced by Li et al. (2016), though equipped with fundamentally different higher-order constitutive boundary conditions of Eq. (10). For a uniform beam, the analytical solution of Eq. (20) can be expressed by (Barretta, et al., 2018a)

W (x ) =

6

Wk exp (ηk x )

(21)

k=1

where Wk (k = 1..6 ) are unknown constants desired to be determined by appropriate boundary conditions, along with ηk that are the roots of the characteristic equation



E I 2 η 6 − E I − ρ I ω 2 c 2



  η 4 − ω 2 ρ Ac 2 + ρ I ω 2 η 2 + ω 2 ρ A = 0

(22)

A. Apuzzo et al. / International Journal of Engineering Science 133 (2018) 99–108

103

Fig. 1. Effects of nonlocal and gradient parameters on the fundamental axial frequency of cantilever nano-beams.

Fig. 2. Effects of nonlocal and gradient parameters on the fundamental axial frequency of fully-clamped nano-beams.

4. Numerical results and discussion Axial and flexural fundamental frequencies of Bernoulli–Euler cantilever and fully-clamped nano-beams according to the nonlocal strain gradient theory, equipped with the appropriate constitutive boundary conditions, are numerically evaluated. The results are then compared with those obtained by Eringen’s local/nonlocal mixture model. A homogeneous fourth-order algebraic system in terms of the unknown constants Uk (k = 1..4 ) is first formulated for the axial problem, by adopting the solution form of the axial spatial base function Eq. (15) and imposing the classical boundary conditions Eq. (3)3 along with the constitutive boundary conditions Eq. (7). Similarly, as a result of enforcing the classical boundary conditions Eq. (3)4 together with the constitutive boundary conditions Eq. (10) in the solution form of the flexural spatial base function Eq. (21), a homogeneous sixth-order algebraic system in terms of the flexural unknown constants Wk (k = 1..6 ) is obtained. It is well-established that to get a non-trivial solution, the determinant of the coefficients of the homogeneous algebraic system has to vanish. The solution procedure leads to a highly nonlinear characteristic equation for each set of selected boundary conditions, in terms of fundamental frequencies, that can be numerically solved. The subsequent non-dimensional terms: slenderness ratio r¯, nonlocal and gradient parameters λc and λ , axial and flexural ¯ and ω fundamental frequencies

¯ are respectively introduced in all the illustrative results

r¯ = Lr , where r =



λc = Lc ,

λ = L ,

¯2=

L2 Aρ

2 π 2 AE ,

ω¯ 2 =

L4 Aρ IE

ω2

(23)

Iρ /Aρ denotes the radius of gyration of the cross-section. The variations of the normalized fundamental axial

¯ /

¯ LOC associated with the nonlocal strain gradient model, equipped with the proposed constitutive boundary frequencies

conditions and Eringen’s local/nonlocal mixture model in terms of the small-scale parameters are illustrated in Figs. 1 and 2 for cantilever and fully-clamped nano-beams. While the parameters λc and λ of NSG are respectively ranging in the set of {0+ , 0.2, 0.4, 0.6, 0.8, 1} and {0.1, 0.3, 0.5, 0.7, 1}, the mixture parameter m of the mixture Eringen integral model (MEIM) is assumed to vary in the set {0.1, 0.3, 0.5, 0.7}. The presented axial fundamental frequencies are also normalized employing the corresponding fundamental axial frequencies of the local beam model LOC . The non-dimensional axial frequencies of Bernoulli–Euler nano-beams in the framework of MEIM are also re-developed here following the analytical approach of

104

A. Apuzzo et al. / International Journal of Engineering Science 133 (2018) 99–108 Table 1 Normalized fundamental axial frequencies of cantilever nano-beams. ¯

¯ LOC

NSG

λc +

0 0.2 0.4 0.6 0.8 1

MEIM

λ = 0.1

λ = 0 . 3

λ = 0 . 5

λ = 0 . 7

λ = 1

m = 0.1

m = 0.3

m = 0.5

m = 0.7

1.00881 0.825677 0.68401 0.57862 0.499123 0.437734

1.04746 0.880371 0.752015 0.652907 0.575002 0.512614

1.07204 0.91814 0.802117 0.711743 0.639306 0.579944

1.08408 0.93752 0.829042 0.744967 0.677421 0.621704

1.09229 0.951042 0.848321 0.769457 0.706371 0.654384

0.999316 0.858328 0.743509 0.663064 0.607725 0.568678

0.999547 0.901055 0.82128 0.768441 0.732934 0.707709

0.999707 0.933679 0.880701 0.84622 0.823051 0.806547

0.999836 0.962028 0.931904 0.912477 0.899459 0.890218

Table 2 Normalized fundamental axial frequencies of fully-clamped nano-beams. ¯

¯ LOC

NSG

MEIM

λc

λ = 0.1

λ = 0 . 3

λ = 0 . 5

λ = 0 . 7

λ = 1

m = 0.1

m = 0.3

m = 0.5

m = 0.7

0+ 0.2 0.4 0.6 0.8 1

1.04609 0.702992 0.508701 0.394737 0.321349 0.27053

1.37139 0.92573 0.670681 0.520155 0.42311 0.355938

1.85837 1.2552 0.909472 0.705278 0.573627 0.482513

2.41097 1.62873 1.18015 0.915153 0.7443 0.626059

3.29032 2.22298 1.61076 1.24905 1.01584 0.854447

0.99863 0.73285 0.572463 0.484617 0.433774 0.402341

0.999093 0.80616 0.69233 0.636693 0.607212 0.590063

0.999412 0.867462 0.792766 0.758517 0.741011 0.731038

0.999672 0.923203 0.881586 0.863266 0.854086 0.848912

Fig. 3. Effects of nonlocal and gradient parameters on the fundamental flexural frequency of cantilever nano-beams.

Fernández-Sáez and Zaera (2017) while employing the solution technique of Barretta et al. (2018a). The numerical values of ¯ /

¯ LOC assessed by NSG and MEIM for cantilever and fully-clamped nano-beams normalized fundamental axial frequencies

are collected in Tables 1 and 2, respectively. The effects of nonlocal and gradient parameters on the normalized fundamental flexural frequency ω ¯ /ω ¯ LOC associated with the nonlocal strain gradient (NSG) model, equipped with the proposed constitutive boundary conditions and the mixture Eringen integral model (MEIM) are exhibited in Figs. 3 and 4 for cantilever and fully-clamped nano-beams. While the scale parameters λc ,λ of NSG and the mixture one m of MEIM is assumed ranging in the same set as the axial free vibrations, the nano-beam slenderness ratio is given by r¯ = 1/20. Once more, the illustrated fundamental flexural frequencies are normalized utilizing their corresponding local counterparts of Bernoulli–Euler beam model ωLOC . Additionally, following the analytical approach of Fernández-Sáez and Zaera (2017) while utilizing the solution technique of Barretta et al. (2018a), the non-dimensional flexural frequencies of Bernoulli–Euler nano-beams of the MEIM are re-developed here. The numerical values of normalized flexural frequencies ω ¯ /ω ¯ LOC evaluated by NSG and MEIM for cantilever and fully-clamped nano-beams are reported in Tables 3 and 4. It is deduced from the illustrations that both the axial and flexural frequencies associated with the mixture Eringen nonlocal integral model decrease either by increasing the characteristic parameter λc or by decreasing the mixture parameter m. This softening phenomenon of the Eringen nonlocal term has been formerly reported in the literature for elastostatic analysis of nano-beams subjected to torsion (Barretta et al., 2018b). Also, the fundamental axial and flexural frequencies of local beam theory can be recovered for vanishing non-dimensional characteristic parameter

A. Apuzzo et al. / International Journal of Engineering Science 133 (2018) 99–108

105

Fig. 4. Effects of nonlocal and gradient parameters on the fundamental flexural frequency of fully-clamped nano-beams.

Table 3 Normalized fundamental flexural frequencies of cantilever nano-beams. ω¯ ω¯ LOC

NSG

MEIM

λc

λ = 0.1

λ = 0 . 3

λ = 0 . 5

λ = 0 . 7

λ = 1

m = 0.1

m = 0.3

m = 0.5

m = 0.7

0+ 0.2 0.4 0.6 0.8 1

1.01778 0.742578 0.57857 0.472341 0.398493 0.344364

1.10709 0.842905 0.678222 0.566807 0.486605 0.426181

1.17522 0.919966 0.759563 0.648583 0.566748 0.503698

1.21277 0.964591 0.809559 0.701637 0.621223 0.558531

1.24021 0.998419 0.849128 0.745402 0.667894 0.607142

0.998535 0.788125 0.66311 0.586481 0.53713 0.503695

0.999096 0.855192 0.770458 0.721086 0.690134 0.669137

0.999415 0.904427 0.849209 0.817509 0.797545 0.783872

0.999674 0.945972 0.915037 0.89734 0.886152 0.87847

Table 4 Normalized fundamental flexural frequencies of fully-clamped nano-beams. ω¯ ω¯ LOC

NSG

MEIM

λc

λ = 0.1

λ = 0 . 3

λ = 0 . 5

λ = 0 . 7

λ = 1

m = 0.1

m = 0.3

m = 0.5

m = 0.7

0+ 0.2 0.4 0.6 0.8 1

1.18839 0.628169 0.393436 0.28134 0.217621 0.176963

2.15561 1.15636 0.725658 0.519051 0.401504 0.326484

3.33222 1.79059 1.12389 0.803918 0.621857 0.505661

4.55943 2.45124 1.53864 1.1006 0.851348 0.69227

6.4318 3.45877 2.17114 1.55303 1.20132 0.976846

0.997265 0.600143 0.450478 0.390592 0.362465 0.34744

0.998189 0.715079 0.617838 0.584221 0.569701 0.562296

0.998826 0.808152 0.747441 0.727722 0.71942 0.715237

0.999345 0.890238 0.857485 0.84721 0.842938 0.840798

λc → 0+ or neglecting the contribution of the nonlocal phase m → 1. Due to the ill-posedness of Eringen integral model, the mixture parameter cannot vanish in the MEIM (Romano, Barretta, Diaco, & Marotti de Sciarra, 2017; Romano, Barretta, & Diaco, 2017; Romano, Luciano, Barretta, & Diaco, 2018). As noticeably inferred from Figs. 1–4, both axial and flexural fundamental frequencies consistent with the nonlocal strain gradient theory, equipped with the constitutive boundary conditions, exhibit a stiffening behavior in terms of the gradientparameter λ and a softening behavior in terms of the nonlocal parameter λc for both cantilever and fully-clamped nano-beams. While both the fundamental axial and flexural frequencies of MEIM are strictly lower than the counterpart local natural frequencies due to the softening effect of the Eringen nonlocal theory, the nonlocal strain gradient theory of elasticity can lead to softening and stiffening structural responses via suitably driving the characteristic parameters λc and λ . Therefore, the proposed nonlocal strain gradient theory provides an efficient approach to model and assess significantly a wide class of nanomechanical problems. The nonlocal strain gradient constitutive law coincides with the Eringen integral convolution as the gradient parameter λ vanishes. In contrast to Eringen’s pure nonlocal integral model, the nonlocal strain gradient law results in well-posed problems also for bounded structures.

106

A. Apuzzo et al. / International Journal of Engineering Science 133 (2018) 99–108

5. Conclusion The size-dependent dynamic behavior of Bernoulli–Euler nano-beams has been investigated by modified nonlocal strain gradient elasticity theory (Barretta & Marotti de Sciarra, 2018). The considered integral laws, providing nonlocal gradient axial forces and bending moments in terms of axial strains, flexural curvatures and small-scale parameters, have been conveniently transformed into equivalent differential problems, equipped with natural constitutive boundary conditions. Axial and flexural free vibrations have been then studied. Exact fundamental frequencies for cantilever and fully-clamped nanobeams have been determined as functions of nonlocal and gradient small-scale parameters. The proposed model, able to describe both softening and stiffening structural responses, can therefore be advantageously employed to characterize sizedependent dynamical behaviors of a wide class of nano-beams used as sensors and actuators in modern NEMS applications. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ijengsci.2018.09. 002. References Acierno, S., Barretta, R., Luciano, R., Marotti de Sciarra, F., & Russo, P. (2017). Experimental evaluations and modeling of the tensile behavior of polypropylene/single-walled carbon nanotubes fibers. Composite Structures, 174, 12–18. Aifantis, E. C. (2003). Update on a class of gradient theories. Mechanics of Materials, 35(3–6), 259–280. Aifantis, E. C. (2011). On the gradient approach–Relation to Eringen’s nonlocal theory. International Journal of Engineering Science, 49(12), 1367–1377. Aifantis, E. C. (2016). Internal length gradient (ILG) material mechanics across scales and disciplines. Advances in Applied Mechanics, 49, 1–110. Akgöz, B., & Civalek, Ö. (2014a). Mechanical analysis of isolated microtubules based on a higher-order shear deformation beam theory. Composite Structures, 118, 9–18. Akgöz, B., & Civalek, Ö. (2014b). Thermo-mechanical buckling behavior of functionally graded microbeams embedded in elastic medium. International Journal of Engineering Science, 85, 90–104. Akgöz, B., & Civalek, Ö. (2015). Bending analysis of FG microbeams resting on Winkler elastic foundation via strain gradient elasticity. Composite Structures, 134, 294–301. Akgöz, B., & Civalek, Ö. (2017a). A size-dependent beam model for stability of axially loaded carbon nanotubes surrounded by Pasternak elastic foundation. Composite Structures, 176, 1028–1038. Akgöz, B., & Civalek, Ö. (2017b). Effects of thermal and shear deformation on vibration response of functionally graded thick composite microbeams. Composites Part B, 129, 77–87. Akgöz, B., & Civalek, Ö. (2018). Vibrational characteristics of embedded microbeams lying on a two-parameter elastic foundation in thermal environment. Composites Part B, 150, 68–77. ˇ Apuzzo, A., Barretta, R., Cana d¯ija, M., Feo, L., Luciano, R., & Marotti de Sciarra, F. (2017). A closed-form model for torsion of nanobeams with an enhanced nonlocal formulation. Composites Part B, 108, 315–324. Apuzzo, A., Barretta, R., Luciano, R., Marotti de Sciarra, F., & Penna, R. (2017). Free vibrations of Bernoulli–Euler nano-beams by the stress-driven nonlocal integral model. Composites Part B, 123, 105–111. Attia, M. A., & Abdel Rahman, A. A. (2018). On vibrations of functionally graded viscoelastic nanobeams with surface effects. International Journal of Engineering Science, 127, 1–32. Barretta, R., & Marotti de Sciarra, F. (2018). Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams. International Journal of Engineering Science, 130, 187–198. Barretta, R., Feo, L., Luciano, R., & Marotti de Sciarra, F. (2016). Application of an enhanced version of the Eringen differential model to nanotechnology. Composites Part B, 96, 274–280. Barretta, R., Feo, L., Luciano, R., Marotti de Sciarra, F., & Penna, R. (2016). Functionally graded Timoshenko nanobeams: A novel nonlocal gradient formulation. Composites Part B, 100, 208–219. ˇ ´ M., Cana Barretta, R., Brcˇ ic, d¯ija, M., Luciano, R., & Marotti de Sciarra, F. (2017). Application of gradient elasticity to armchair carbon nanotubes: Size effects and constitutive parameters assessment. European Journal of Mechanics - A/Solids, 65, 1–13. ˇ Barretta, R., Cana d¯ija, M., Feo, L., Luciano, R., Marotti de Sciarra, F., & Penna, R. (2018). Exact solutions of inflected functionally graded nano-beams in integral elasticity. Composites Part B, 142, 273–286. ˇ Barretta, R., Cana d¯ija, M., Luciano, R., & Marotti de Sciarra, F. (2018). Stress-driven modeling of nonlocal thermoelastic behavior of nanobeams. International Journal of Engineering Science, 126, 53–67. Barretta, R., Diaco, M., Feo, L., Luciano, R., Marotti de Sciarra, F., & Penna, R. (2018). Stress-driven integral elastic theory for torsion of nano-beams. Mechanics Research Communications, 87, 35–41. Barretta, R., Luciano, R., Marotti de Sciarra, F., & Ruta, G. (2018). Stress-driven nonlocal integral model for Timoshenko elastic nano-beams. European Journal of Mechanics - A/Solids, 72 275-28. Barretta, R., Faghidian, S. A., & Luciano, R. (2018). Longitudinal vibrations of nanorods by stressdriven integral elasticity. Mechanics of Advanced Materials and Structures article in press. Barretta, R., Fazelzadeh, S. A., Feo, L., Ghavanloo, E., & Luciano, R. (2018). Nonlocal inflected nano-beams: A stress-driven approach of bi-Helmholtz type. Composite Structures, 200, 239–245. Barretta, R., Faghidian, S. A., Luciano, R., Medaglia, C. M., & Penna, R. (2018a). Free vibrations of FG elastic Timoshenko nano-beams by strain gradient and stress driven nonlocal models. Composites Part B, 154, 20–32. Barretta, R., Fabbrocino, F., Luciano, R., & Marotti de Sciarra, F. (2018). Closed-form solutions in stress-driven two-phase integral elasticity for bending of functionally graded nano-beams. Physica E, 97, 13–30. Barretta, R., Faghidian, S. A., Luciano, R., Medaglia, C. M., & Penna, R. (2018b). Stress-driven two-phase integral elasticity for torsion of nano-beams. Composites Part B, 145, 62–69. Demir, Ç., & Civalek, Ö. (2017). On the analysis of microbeams. International Journal of Engineering Science, 121, 14–33. Eringen, A. C. (1972). Linear theory of nonlocal elasticity and dispersion of plane waves. International Journal of Engineering Science, 10, 425–435. Eringen, A. C. (1983). On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54, 4703–4710. Eringen, A. C. (1987). Theory of nonlocal elasticity and some applications. Res Mechanica, 21(4), 313–342. Faghidian, S. A. (2018a). On non-linear flexure of beams based on non-local elasticity theory. International Journal of Engineering Science, 124, 49–63. Faghidian, S. A. (2018b). Reissner stationary variational principle for nonlocal strain gradient theory of elasticity. European Journal of Mechanics - A/Solids, 70, 115–126.

A. Apuzzo et al. / International Journal of Engineering Science 133 (2018) 99–108

107

Faghidian, S. A. (2018c). Integro-differential nonlocal theory of elasticity. International Journal of Engineering Science, 129, 96–110. Fernández-Sáez, J., Zaera, R., Loya, J. A., & Reddy, J. N. (2016). Bending of Euler–Bernoulli beams using Eringen’s integral formulation: A paradox resolved. International Journal of Engineering Science, 99, 107–116. Fernández-Sáez, J., & Zaera, R. (2017). Vibrations of Bernoulli–Euler beams using the two-phase nonlocal elasticity theory. International Journal of Engineering Science, 119, 232–248. Ghayesh, M. H. (2018). Dynamics of functionally graded viscoelastic microbeams. International Journal of Engineering Science, 124, 115–131. Ghayesh, M. H., & Farajpour, A. (2018). Nonlinear mechanics of nanoscale tubes via nonlocal strain gradient theory. International Journal of Engineering Science, 129, 84–95. Ghayesh, M. H., & Farokhi, H. (2018). On the viscoelastic dynamics of fluid-conveying microtubes. International Journal of Engineering Science, 127, 186–200. Ghodrati, B., Yaghootian, A., Ghanbar Zadeh, A., & Sedighi, H. M. (2018). Lamb wave extraction of dispersion curves in micro/nano-plates using couple stress theories. Waves in Random and Complex Media, 28(1), 15–34. Hadi, A., ZamaniNejad, M., & Hosseini, M. (2018). Vibrations of three-dimensionally graded nanobeams. International Journal of Engineering Science, 128, 12–23. Jiao, P., & Alavi, A. H. (2018). Buckling analysis of graphene-reinforced mechanical metamaterial beams with periodic webbing patterns. International Journal of Engineering Science, 131, 1–18. Khanik, H. B. (2018). On vibrations of nanobeam systems. International Journal of Engineering Science, 124, 85–103. Koutsoumaris, C. Chr., Eptaimeros, K. G., & Tsamasphyros, G. J. (2017). A different approach to Eringen’s nonlocal integral stress model with applications for beams. International Journal of Solids and Structures, 112, 222–238. Kumar, R., Singh, R., Hui, D., Feo, L., & Fraternali, F. (2018). Graphene as biomedical sensing element: State of art review and potential engineering applications. Composites Part B, 134, 193–206. Li, L., & Hu, Y. (2015). Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory. International Journal of Engineering Science, 97, 84–94. Li, L., & Hu, Y. (2016). Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material. International Journal of Engineering Science, 107, 77–97. Li, L., Li, X., & Hu, Y. (2016). Free vibration analysis of nonlocal strain gradient beams made of functionally graded material. International Journal of Engineering Science, 102, 77–92. Li, L., Tang, H., & Hu, Y. (2018). The effect of thickness on the mechanics of nanobeams. International Journal of Engineering Science, 123, 81–91. Lim, C. W., Zhang, G., & Reddy, J. N. (2015). A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids, 78, 298–313. Lu, L., Guo, X., & Zhao, J. (2017). Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory. International Journal of Engineering Science, 116, 12–24. Lu, L., Guo, X., & Zhao, J. (2018). On the mechanics of Kirchhoff and Mindlin plates incorporating surface energy. International Journal of Engineering Science, 124, 24–40. Mahmoudpour, E., Hosseini-Hashemi, S. H., & Faghidian, S. A. (2018). Non-linear vibration analysis of FG nano-beams resting on elastic foundation in thermal environment using stress-driven nonlocal integral model. Applied Mathematical Modelling, 57, 302–315. Marotti de Sciarra, F., & Barretta, R. (2014). A gradient model for Timoshenko nanobeams. Physica E, 62, 1–9. Numanog˘ lu, H. M., Akgöz, B., & Civalek, Ö. (2018). On dynamic analysis of nanorods. International Journal of Engineering Science, 130, 33–50. Ouakad, H. M., & Sedighi, H. M. (2016). Rippling effect on the structural response of electrostatically actuated single-walled carbon nanotube based NEMS actuators. International Journal of Non-Linear Mechanics, 87, 97–108. Qi, L., Huang, S., Fu, G., Zhou, S., & Jiang, X. (2018). On the mechanics of curved flexoelectric microbeams. International Journal of Engineering Science, 124, 1–15. Rafii-Tabar, H., Ghavanloo, E., & Fazelzadeh, S. A. (2016). Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures. Physics Reports, 638, 1–97. Ramsden, J. J. (2016). Nanotechnology, an introduction (2nd ed.). New York: Elsevier. Reddy, J. N. (2017). Energy principles and variational methods in applied mechanics (3rd ed.). New Jersey: Wiley. Romano, G., & Barretta, R. (2017a). Nonlocal elasticity in nanobeams: The stress-driven integral model. International Journal of Engineering Science, 115, 14–27. Romano, G., & Barretta, R. (2017b). Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams. Composites Part B, 114, 184–188. Romano, G., Barretta, R., Diaco, M., & Marotti de Sciarra, F. (2017). Constitutive boundary conditions and paradoxes in nonlocal elastic nano-beams. International Journal of Mechanical Sciences, 121, 151–156. Romano, G., Barretta, R., & Diaco, M. (2017). On nonlocal integral models for elastic nanobeams. International Journal of Mechanical Sciences, 131, 490–499. Romano, G., Luciano, R., Barretta, R., & Diaco, M. (2018). Nonlocal integral elasticity in nanostructures, mixtures, boundary effects and limit behaviours. Continuum Mechanics and Thermodynamics, 30(3), 641–655. Sedighi, H. M., & Sheikhanzadeh, A. (2017). Static and dynamic pull-in instability of nano-beams resting on elastic foundation based on the nonlocal elasticity theory. Chinese Journal of Mechanical Engineering, 30, 385–397. Sedighi, H. M., Keivani, M., & Abadyan, M. (2015). Modified continuum model for stability analysis of asymmetric FGM double-sided NEMS: Corrections due to finite conductivity, surface energy and nonlocal effect. Composites Part B, 83, 117–133. Sedighi, H. M., Daneshmand, F., & Abadyan, M. (2015). Modified model for instability analysis of symmetric FGM double-sided nano-bridge: Corrections due to surface layer, finite conductivity and size effect. Composite Structures, 132, 545–557. Sedighi, H. M., Daneshmand, F., & Abadyan, M. (2016). Modeling the effects of material properties on the pull-in instability of nonlocal functionally graded nano-actuators. ZAMM - Journal of Applied Mathematics and Mechanics, 96, 385–400. Sedighi, H. M., Koochi, A., Keivani, M., & Abadyan, M. (2017). Microstructure-dependent dynamic behavior of torsional nano-varactor. Measurement, 111, 114–121. She, G.-L., Ren, Y.-R., Yuan, F.-G., & Xiao, W.-S. (2018). On vibrations of porous nanotubes. International Journal of Engineering Science, 125, 23–35. Srividhya, S., Raghu, P., Rajagopal, A., & Reddy, J. N. (2018). Nonlocal nonlinear analysis of functionally graded plates using third-order shear deformation theory. International Journal of Engineering Science, 125, 1–22. Thai, H.-T., Vo, T. P., Nguyen, T.-K., & Kim, S.-E. (2017). A review of continuum mechanics models for size-dependent analysis of beams and plates. Composite Structures, 177, 196–219. Taati, E. (2018). On buckling and post-buckling behavior of functionally graded micro-beams in thermal environment. International Journal of Engineering Science, 128, 63–78. Wang, Y. B., Zhu, X. W., & Dai, H. H. (2016). Exact solutions for the static bending of Euler–Bernoulli beams using Eringen’s two phase local/nonlocal model. AIP Advances, 6(8), 085114. Wang, Y., Huang, K., Zhu, X., & Lou, Z. (2018). Exact solutions for the bending of Timoshenko beams using Eringen’s two-phase nonlocal model. Mathematics and Mechanics of Solids article in press. Zhang, J. X. J., & Hoshino, K. (2014). Molecular sensors and nanodevices. New York: Elsevier. Zhao, J., Guo, X., & Lu, L. (2018). Small size effect on the wrinkling hierarchy in constrained monolayer graphene. International Journal of Engineering Science, 131, 19–25. Zhu, X., & Li, L. (2017a). Longitudinal and torsional vibrations of size-dependent rods via nonlocal integral elasticity. International Journal of Mechanical Sciences, 133, 639–650.

108

A. Apuzzo et al. / International Journal of Engineering Science 133 (2018) 99–108

Zhu, X., & Li, L. (2017b). Closed form solution for a nonlocal strain gradient rod in tension. International Journal of Engineering Science, 119, 16–28. Zhu, X., Wang, Y., & Dai, H.-H. (2017). Buckling analysis of Euler–Bernoulli beams using Eringen’s two-phase nonlocal model. International Journal of Engineering Science, 116, 130–140.