Nonlinear resonance of FG multilayer beam-type nanocomposites: Effects of graphene nanoplatelet-reinforcement and geometric imperfection

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Contents lists available at ScienceDirect

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www.elsevier.com/locate/aescte

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Nonlinear resonance of FG multilayer beam-type nanocomposites: Effects of graphene nanoplatelet-reinforcement and geometric imperfection

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a

School of Mechanical and Aerospace Engineering, Nanyang Technological University (NTU), Singapore 639798, Singapore b Institute of Solid Mechanics, Beihang University (BUAA), Beijing 100191, China

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a r t i c l e

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a b s t r a c t

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Article history: Received 3 October 2019 Received in revised form 3 December 2019 Accepted 11 January 2020 Available online xxxx

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Hu Liu a , Han Wu b , Zheng Lyu a,∗

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Keywords: Nonlinear vibration Functionally graded nanocomposite Graphene nanoplatelet Geometric imperfection Nanobeam

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In this paper, the nonlinear dynamic response of a FG multilayer beam-type nanocomposite reinforced with graphene nanoplatelet (GNP) by considering the initial geometric imperfection is investigated on the basis of nonlocal strain gradient Euler-Bernoulli beam theory. Four patterns of GNP distribution incorporating the uniform distribution (UD) and O-, X-, and A- FG pattern distributions are taken into account and the effective elastic properties of the beam-type nanocomposite are evaluated in the framework of Halpin-Tsai scheme. The first-order vibrational mode is employed to represent the initial geometric imperfection of the nonlinear FG beam-type nanocomposite. Correspondingly, the nonlinear amplitude-frequency response of the imperfect FG multilayer beam-type nanostructures subjected to the excitation resonance is analyzed with the aid of multiple scale method. Firstly, the present model is validated with a comparison of two previous works. Then, a comprehensive investigation is conducted to evaluate the effects of GNP distributed pattern, weight fraction of GNPs, geometric imperfection amplitude, boundary condition, excitation amplitude, nonlocal and strain gradient size scale parameters on the nonlinear frequency-response of FG multilayer beam-type nanostructures. The current work is beneficial for the application of GNP as reinforcement to enhance mechanical performances of nanostructures. © 2020 Elsevier Masson SAS. All rights reserved.

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1. Introduction

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With the rapid development of micro/nano-electro-mechanical systems (MEMS/NEMS) and biotechnology, nanocomposites, as new type of materials incorporating multi-phase solids, have attracted a great deal of interest in academic research and engineering. More and more scientists pay their attention to the application of nanocomposites in the design of micro/nano-scale devices, such as micro/nano-robots for drug delivery, nano-sensors, nano-actuators and nano-transducers [1,2]. Generally, the carbon-based nanofillers such as carbon nanotube (CNT) and graphene nanoplatelet (GNP) are used to improve their mechanical and thermal performance because they can provide high stiffness and strength without greatly increasing the total mass of structures [3–5]. A great challenge in using CNT is to disperse the CNT nanofillers uniformly in matrix. Fortunately, GNP provides much more effective choice due to their prominently high specific surface area to improve the load transfer between reinforcements and matrix [6–8]. It is reported that the effective Young’s modulus of epoxy nanocomposite can enhance 31% by reinforcing with 0.1% weight fraction GNP, while only an increment of 3% can be achieved by employing the CNT nanofillers [6]. As an ideal reinforcement, GNP has aroused significant research interests, which were mainly focused on static and dynamic behaviors of GNP-reinforced composites. Particularly, the reinforced composites with functionally graded (FG) material properties have aroused increasing concern [9]. By varying the material properties across one or more directions, the FG composites can achieve exceptional mechanical and physical properties [10,11]. Over the last decade, extensive investigations of GNP-reinforced FG composites have been carried out. For instance, Wang and his colleagues [12] focused on nonlinear bending of axially FG microbeam reinforced by graphene nanoplates.

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Corresponding author. E-mail addresses: [email protected], [email protected] (Z. Lyu).

https://doi.org/10.1016/j.ast.2020.105702 1270-9638/© 2020 Elsevier Masson SAS. All rights reserved.

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Fig. 1. Schematic view of a FG multilayer GNP-reinforced beam-type nanocomposite subjected to external harmonic force.

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Yang et al. [13] presented a theoretical modeling on the buckling behavior of a GNP-reinforced FG composite beam by using the first-order shear deformation beam theory. Thai et al. [14] studied the free vibration, buckling and bending performances of a GNP-reinforced FG composite plate based on the refined plate theory. Similar analysis on the free vibration of FG composite plates reinforced by GNPs with considering of the porosity distribution is investigated by Gao et al. [15]. Other studies on mechanical behaviors of GNP-reinforced FG composites were also reported in literature [16–21]. However, a lot of studies concentrated on the macro-scale composite structures, and the small-scale effect of nanocomposites is often neglected, which motivates this research. As proved by various nano-scale experiments, the size-dependent effect plays an important role on the mechanical behavior of nanocomposites. Due to the difficulty in controlling the nano-scaled experiments, and the great time-consuming of the atom dynamics simulation, the non-classical continuum mechanics gains much more popularity in revealing the size-scale behaviors of nanocomposites. The nonlocal elasticity and strain gradient elasticity are the most adopted theories, in which the former one suggests that the stress at an arbitrary point depends not only on the strain of this specific point but also on the strain of all other areas in the body, and it can present the stiffness-softening size-scale effect [21–23]. The latter one indicates that the stress in a reference point is related to the strain and its high-order gradient terms, which can provide the stiffness-hardening size-scale effect [24]. By combining the nonlocal and strain gradient terms into a uniform theory, the nonlocal strain gradient theory is developed to capture the size-dependent behaviors of nanocomposites [25]. Over the last decade, numerous researchers have focused on employing this advanced theory to detect the bending, buckling, and vibration characteristics of FG nanocomposites [26–29]. Particularly, some investigations on the basis of the nonlocal strain gradient theory have been dedicated by Sahmani et al. [30–32] to reveal the mechanical behaviors of GNP-reinforced FG nanocomposites. From the literatures reviewed, one should be noted that the issues mentioned above are all limited to the geometrically perfect nanocomposites, there are few works focused on the mechanical behaviors of GNP-reinforced FG nanocomposites with initial geometric imperfection. It is well-known that the machine manufacturing errors caused by the surface form error and local indentation are unavoidable [33–36]. For example, it is very hard to keep graphene nanoplatelet plane in a matrix due to the small stiffness in the thickness direction. These induced small initial geometric curvatures may lead to some peculiar mechanical properties, which should not be ignored. In the literature available, some attempts also have been carried to study the mechanical behaviors of imperfect nanostructures. Ghayesh is the pioneer in studying the initial geometric imperfection effect on the mechanical response of size-dependent micro-/nano structures [37–40]. In his work, it was found that the mechanical behaviors of nanostructures are greatly affected by the geometric imperfection. Besides, Gholami and Ansari [41] investigated the nonlinear resonant behavior of a shear deformable nanobeams involving geometric imperfection. For reinforced composite structures, Wu et al. [42] discussed the influence of geometric imperfection on the dynamic characteristics of FG composite beams reinforced by CNTs. Similar problem for CNTs- reinforced composite plates is reported by Thang et al. [43]. However, their studies are only confined to macro-scale reinforced composite structures. As a result, there is a strong encouragement to gain a deep understanding of the size-dependent behaviors of GNP-reinforced FG nanocomposites in the presence of geometric imperfection. To the best of our knowledge, investigations on nonlinear dynamics of FG multilayer beam-type nanostructures reinforced by GNPs with initial geometric imperfection are limited although few studies conduct the influence of GNPs on the mechanical behavior of nanostructure. This paper aims to investigate nonlinear resonance performances of GNPs-reinforced multilayer beam-type nanostructures by means of the nonlocal strain gradient Euler-Bernoulli beam theory. With the help of Hamilton principle, the equilibrium equation of the imperfect multilayer beam-type nanostructures is obtained, which is numerically solved on the basis of the multiple scale method. Several different parameters including the GNP distributed pattern and weight fraction, boundary condition, excitation amplitude, geometric imperfection amplitude, as well as size scale parameters on the amplitude-frequency response of the imperfect FG multilayer beam-type nanostructures are investigated in detail. 2. Theoretical formulations 2.1. Effective material properties

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As presented in Fig. 1, a FG multilayer beam-type nanostructure with length L and width b, reinforced by GNPs is taken into account. The nanostructure is made of a total number of N L layers with equivalent thickness h = h/ N L , in which h indicates the total thickness. In this configuration, the GNPs are distributed randomly and evenly as in matrix for each layer, and the weight fraction of GNPs in each layer is different, including one uniform distribution (UD pattern) and three FG distribution types as seen in Fig. 2. For FG-O pattern nanobeam, the middle has the maximum weight fraction of GNPs, while the up and bottom sides have the minimum one. On the contrary, the GNPs in FG-X pattern nanobeam is distributed in the opposite way. For FG-A pattern nanobeam, the weight fraction changes gradually from the up to the bottom side of the multilayer nanobeam. Such configurations are typical filler reinforced composites. Thus, the effective elastic properties of FG multilayer beam-type nanocomposites can be calculated using Halpin-Tsai model. On the basis of the Halpin-Tsai model, the elastic modulus in the k-th layer of the nanobeam with randomly distributed GNPs can be approximately determined as [44]

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E (k) =

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(k)

3 1 + ξL ηL V G N P 8 1 − η L V (k) GN P

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Fig. 2. FG material distribution patterns considered in the present study.

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(k)

+

5 1 + ξT ηT V G N P

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8 1 − η T V (k) GN P

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EM

(1)

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in which E M indicates Young’s modulus of the matrix phase, and the parameters EGN P EM EGN P EM

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ηL =

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−1 + ξL

,

EGN P EM EGN P EM

ηT =

−1 + ξT

ξL =

,

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2L G N P hG N P

,

ξT =

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ηL , ηT , ξL and ξT satisfy

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2b G N P

(2)

hG N P

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(k)

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where E GNP is Young’s modulus of GNPs, and L GNP , bGNP , hGNP are, respectively, the length, width and thickness of GNPs. Besides, V G N P in

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Eq. (1) represents the volume fraction of the GNPs in the k-th layer, which can be represented by using the weight fraction of GNPs g G P N , i.e.,

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(k)

(k)

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V GN P =

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ρ

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(k)



(k)

(k)

(k)

= ρG N P V G N P + ρ M 1 − V G N P

(3)

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(k)

gG N P

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(4)

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36 37 38

(5)

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In the present study, the nonlocal strain gradient Euler-Bernoulli beam theory is adopted to describe the size-dependent behaviors of FG multilayer beam-type nanostructures. The components of displacement in a Euler-Bernoulli beam has the following form

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∂ w (x, t ) u 1 (x, z, t ) = u (x, t ) − z , ∂x

u 2 (x, z, t ) = 0,

u 3 (x, z, t ) = w (x, t ) + w 0 (x)

(6)

in which u 1 , u 2 and u 3 in order refer to the displacements in an arbitrary point of the beam-type nanostructures across the axial (x) and transverse (z) directions. u and w are, respectively, the displacements in the middle surface of the nanobeam along x− and z− directions. In addition, w 0 stands for an initial displacement along the transverse direction used to represent the geometric imperfection of the beam-type nanostructures. According to the hypothesis of Euler-Bernoulli beam theory, the nonzero strain in the nanobeam with considering both the von Kármán hypotheses and the initial imperfection is given as

εxx =

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∂u 1 + ∂x 2



∂w ∂x

2 −z

∂ w ∂ w dw 0 + ∂ x dx ∂ x2

as

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U=

 1 2 V

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2

(7)

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In the framework of the nonlocal strain gradient theory, the strain energy of the FG multilayer beam-type nanostructures is expressed

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2.2. Nonlocal strain gradient Euler-Bernoulli beam theory

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in which g G∗ N P refers to the weight fraction of GNPs.

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⎧ ∗ g G N P , UDPattern ⎪ ⎪ ⎪

 ⎪  ⎪

⎪ N L +1 N L +1

⎨ 4g ∗ − k − / (2 + N L ) , FG − OPattern

GN P 2 2

=

  ⎪



N L +1 1 ∗ ⎪ ⎪ ⎪ 4g G N P 2 + k − 2 / (2 + N L ) , FG − XPattern ⎪ ⎪ ⎩ 2kg G∗ N P / (1 + N L ) , FG − APattern

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(k)

g G P N + (ρG N P /ρ M ) 1 − g G P N



In addition, the weight fraction of GNPs for different distribution patterns can be denoted as

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where ρ M and ρGNP correspond to the density of matrix and GNPs, respectively. Using the rule of mixtures, the effective density in the k-th layer reinforced by GNPs can be defined as

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gG P N

(k)

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σi j εi j + σi∗jm εi j,m dV

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(8)

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in which σi j and denoted as

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σi∗jm = l2 C i jkl

V

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α0 ( x − x , e0a)εkl dV ,

σi j = C i jkl

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σi∗jm are the stress tensors corresponding to the nonlocal and high-order nonlocal terms, respectively, which can be

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α1 ( x − x , e1a)εkl ,m dV

3

(9)

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6

where e 0 a and e 1 a are the size-dependent parameters to reflect the nonlocal stress effect, l represents a material length scale parameter involving the strain gradient effect [45]. For a specified material, these small-scale parameters e 0 a, e 1 a and l can be determined by matching the dispersion wave with atomic lattice dynamics model. C ijkl is the

elastic coefficients

of the beam-type nanostructure; εkl and εkl,m are, respectively, the components of strain and its gradient; α0 ( x − x , e0a) and α1 ( x − x , e1 a) are the kernel functions satisfying Eringen’s nonlocal relations, which can be obtained for the beam-type nanostructure by neglecting the size-dependent characteristic in the width and thickness directions, i.e.,

2

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11 12

(11)

∇ stands for

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23









∂2 ∂2 ∂ 2 (k) 2 1 − (ea) t = 1 − l + xx ∂ x2 ∂ x2 ∂ z2



2

E

εxx

(13)

δU =







⎡

∗ t xx δ εxx + σxxz δ εxx,z dV + ⎣

V



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=



⎤L

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0

0

2 ∂u ∗ ∂w ∂w ∗ ∂ w dw 0 ∗ dw 0 ∂ w ∗ ∂ w + N xx δ + N xx δ + N xx δ + N xx δ − M xx δ 2 ∂x ∂x ∂x ∂ x dx dx ∂ x ∂x

∗

2

(14)



NL  ⎜ ( N xx , M xx ) = b ⎝ k =1

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L

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0

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⎛

⎛

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⎟ {1, z} t xx dz⎠ ,

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(k)

NL  ⎜ ∗∗ ∗∗ , M xx )=b ( N xx ⎝ k =1

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⎞

z(k) z(k+1)

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⎞

52

(k) ⎟ {1, z} σxx dz⎠ ,

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z(k)

⎞

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z(k+1)

NL  ⎜ (k)∗ ⎟ ∗ N zz = b σxxz dz⎠ ⎝ k =1

z(k)

∗ ∗ , M xx )=b ( N xx

NL  k =1

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dx

where A indicates the section area of the beam-type nanostructures and the stress resultants are given as follows, z(k+1)

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∂u ∂ w ∂w ∂w ∂ w dw 0 dw 0 ∂ w ∗ ∂ w + N xx δ + N xx δ + N xx δ − M xx δ 2 − N zz δ 2 ∂x ∂x ∂x ∂ x dx dx ∂ x ∂x ∂x

⎛

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2

N xx δ

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∗ σxx δ εxx d A ⎦

A

L 



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∗ ∗ σxx δ εxx + σxxx δ εxx,x + σxxz δ εxx,z dV

V

=

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28

(k)

22

25

(k) (k)

where t xx is the total stress tensor for the nonlocal strain gradient beam-type nanostructures in the k-th layer. The virtual strain energy δ U of the FG multilayer beam-type nanostructures with considering the thickness effect can be achieved by [46–48]

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For the beam-type multilayer nanostructures, the general constitutive relation in the framework of nonlocal strain gradient theory can be represented as

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21

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(12)

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ε

t i j = σi j − ∇ σi jm

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2 (k) (k) xxm = l E xx,m

∗

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(k)∗

(10)

According to the nonlocal strain gradient theory, the total stress tensor t i j is defined as

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42

(k)

Note that e 0 a = e 1 a = ea is used in the above equation, and E (k) denotes the elasticity modulus in the k-th layer. Here, 2 the Laplacian operator and ∇ 2 = ∂∂x2 . Moreover, εxx and ε xx,x are, respectively, the axial strain and its gradient.

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2

(1 − (ea) ∇ )σ

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(k)

(1 − (ea)2 ∇ 2 )σxx = E (k) εxx

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V

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1

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⎛ ⎜ ⎝

z(k+1)

⎞

(k)∗ ⎟ {1, z} σxx dz⎠

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(15)

z(k)

where z(k) and z(k +1) are separately the coordinate position of the upper and bottom faces of the k-th layer, and b is the width of the beam-type nanostructures.

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By using Eq. (12), the resultants of moment M xx and the axial force N xx can be separately calculated as

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zτxx d A = A

A

N xx =

τxx d A = A

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25

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( A xx , B xx , D xx ) = b

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δK =

5

(16)

2 = −l A xx ∂∂ xw2 .

ρ

NL 









2



2

11



2

12

(17)



15

(18)

∂ ux ∂ ux δ ∂t ∂t

17 19 20

⎟

(19)

23



∂ uz ∂ uz δ ∂t ∂t



L  dV ≈

m0



∂w ∂w δ ∂t ∂t





+ m0

∂u ∂u δ ∂t ∂t

(20)

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⎞

32

z(k+1)

NL  ⎜ ⎟ m0 = b ρ (k) dz⎠ ⎝

(21)

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z(k)

36

The virtual work done due to the external harmonic force is obtained as

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L

38

F cos( t )δ wdx

(22)

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0

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where denotes the excitation frequency of the harmonic force. The Hamilton principle can be utilized for deriving the nonlinear governing equations, which reads

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t2

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δ(U − K − W )dt = 0

(23)

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t1

Substituting Eqs. (14), (20) and (21) into Eq. (23), and collecting the coefficients of δ u and δ w yields the following equilibrium equations 2

∂ N xx ∂ u − m0 2 = 0 ∂x ∂t   ∗ ∂ ∂ 2 N zz ∂ 2 M xx ∂ ∂w dw 0 ∂2 w + + δw : + N N + F cos ( t ) − m =0 xx xx 0 ∂x ∂x ∂x dx ∂ x2 ∂ x2 ∂t2 δu :

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(24)

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(25)

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2.3. Governing equations

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dx

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24 26



in which the mass inertia m0 can be given as

⎛

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z(k)

+

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1, z, z2 E (k) dz⎠



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0

δW =

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10

⎞

z(k+1)

⎜ ⎝



k =1

6

9

2

A

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The first variation of kinetic energy δ K for the FG multilayer beam-type nanostructures accounting both the longitudinal and transverse displacements is calculated as

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2

k =1

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3

8

⎛

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2

in which the stiffness terms A xx , B xx and D xx are defined as

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∂x

1

,

∂ w dw 0 ∂u 1 ∂ w ∂ N xx ∂ ∂ ∂ w = 1 − l2 2 A xx + + − 1 − l2 2 B xx 2 ∂x 2 ∂x ∂ x dx ∂ x2 ∂x ∂x ∂x      2 ∂ w dw 0 ∂u 1 ∂ w ∂ 2 M xx ∂2 ∂2 ∂2 w M xx − (ea)2 = 1 − l2 2 B xx + + − 1 − l2 2 D xx 2 2 ∂x 2 ∂x ∂ x dx ∂x ∂x ∂x ∂x

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32

xx

∗ ∂σ ∂ N xx ∗∗ σxx − xx d A = N xx − ∂x ∂x



2

N xx − (ea)2

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31

∂x ∗

∗∗ d A = M xx −

∂ M∗

Besides, with the help of Eqs. (7) and (13), the above equations can be re-expressed as

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xx

∗ In view of Eqs. (7), (11) and (15), one has N zz

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A

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∂σ ∗

σxx −

z



8 9



M xx =

5 6

5

For the axis dynamic problem, the axial inertial ∂ 2 u /∂ t 2 in Eq. (24) is neglected, indicating the axial force N xx is a constant. Hence, N xx in Eq. (17) can be re-expressed as



2

∂ N xx = N c = 1 − l ∂ x2 2



A xx

∂u 1 + ∂x 2



∂w ∂x

2





2

∂ w dw 0 ∂ + − 1 − l2 2 ∂ x dx ∂x



(26)

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∗ and N ∗∗ can be stated as With the help of Eq. (9), the relationship between the stress resultants N xx xx

∂ N ∗∗ ∗ N xx = l2 xx ∂x

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∂ w B xx ∂ x2

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(27)

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From Eqs. (16), (26) and (27), one can arrive the following relations



1



2

 ∂ u 1 ∂ w 2 ∂ w dw 0 ∂2 w N xx = A xx + + − B xx 2 ∂x 2 ∂x ∂ x dx ∂x  2  2 2 3 ∂ w ∂ w ∂ w dw 0 ∂ w d2 w 0 ∗ 2 ∂ u 2∂ w − N xx = A xxl + + + B l xx ∂ x ∂ x2 ∂ x dx2 ∂ x2 ∂ x2 dx ∂ x3 ∗∗

(28)

10 11 12 13 14 15

(29)

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!



"





2

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2

10 11 12

(30)

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15 16

Integrating both sides of Eq. (30) over the length of the beam-type nanostructures, as well as recalling that N xx and the above boundary conditions, yields

A xx

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L   1

L

2

∂w ∂x

2 −

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49

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= N xx

x= L

=0



2

∂ w B xx w 0 dx − L ∂ x2

L

64

(32)

25 26

M xx =

B xx A xx

 Nc +



B 2xx A xx

− D xx

2

∂ 1−l ∂ x2 2





2

2

2

2

2

N∗

∂ zz ∂ w ∂ w d w0 ∂ w + (ea)2 m0 2 − F cos( t ) − N c 2 − N c − ∂ x2 ∂t ∂x dx2 ∂ x2

 (33)

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By combining the above equation with Eq. (25), the governing equations in terms of the transverse deflection w can be finally derived,

34 35



   2 4 ∂2 ∂2 w B 2xx ∂4 w ∂ w 2 ∂ 1 − (ea) m + D − 1 − l + l2 A xx 4 − 0 xx 2 2 2 4 A xx ∂x ∂t ∂x ∂x ∂x ⎫ ⎧   L    

L 2 ⎬ ⎨ A 1 ∂ w 2 ∂2 w 2 2 2 ∂ w d w B ∂ w ∂ 0 xx xx = F cos( t ) 1 − (ea)2 2 + − w dx − dx 0 ⎩ L 2 ∂x L ∂x ∂ x2 dx2 ∂ x2 ∂ x2 ⎭

36

2

0

28

31

37

(34)

38 39 40 41

0

42

To obtain general results, the following dimensionless quantities are introduced

& x¯ =

x L

¯ = , w &

¯ = /

w r

¯0= , w

E M Im

ρM Am L 4

w0 r

Im

, r=

Am





, A¯ xx , B¯ xx , D¯ xx = ' h /2 

43

& E M Im

h

ρM Am L 4

, z¯ = , t¯ = t 

44

z

A xx

,

B xx r

,

D xx

E M Am E M I m E M I m

, μ=



, F¯ =

ea L

l

L

m0

L

r

ρM Am

¯0= , η= , β= , m

45

,

46

(35)

F L4

47 48

r E M Im

49 50



where we have ( A m , I m ) = b −h/2 1, z2 dz. Thus, the dimensionless form of the governing equation is calculated as

51 52



   2 4 ¯ ¯ ¯ ∂ w ∂2 ∂2 w B¯ 2xx ∂4 w 2 ∂ ¯ ¯ 1−μ m + D − 1 − η + η2 β 2 A¯ xx 4 − 0 xx 2 2 4 2 ¯ xx ∂ x¯ ∂ x¯ ∂ x¯ ∂ x¯ ∂ t¯ A ⎧ ⎫  1    

1 2 2 ⎨ ⎬ 2 2 ¯ 2 ¯ 2 ¯ ¯ ¯ ∂ ∂ w d w 1 w w ∂ w ∂ ∂ 0 ¯ xx ¯ t¯) ¯ xx ¯ ¯ ¯ = F¯ cos( 1 − μ2 2 + A − w d x − B d x 0 ⎩ 2 ∂ x¯ ∂ x¯ ∂ x¯ 2 dx¯ 2 ∂ x¯ 2 ∂ x¯ 2 ⎭

53 54

2

0

55

(36)

56 57 58 59

0

60

2.4. Boundary conditions and geometric imperfection

61 62

In the present study, the FG multilayer beam-type nanostructures with two types of the end are taken into account, i.e., Hinged (H) end:

63 64 65

65 66

21

24

∂ w dx ∂ x2

62 63

20

23

2

60 61

19

27

50 51

x=0

This equation indicates that both the transverse deflection w and geometric imperfection w 0 can alter the axial force N xx , which is also called the mid-plane stretching phenomenon. The bending moment M xx is obtained by substituting Eq. (25) into Eq. (18), i.e.,

47 48

(1)

0

44 45



(31)

22

0

33 34



(1)

N xx = N c =

17 18

u |x=0 = u |x= L = 0

31 32

13 14

The axial displacement boundary conditions for the FG multilayer beam-type nanostructures with immovable ends are

26 27

7 9





∂ w dw 0 ∂2 w ∂u 1 ∂ w 1 − l ∇ A xx + + − 1 − l2 ∇ 2 B xx 2 ∂x 2 ∂x ∂ x dx ∂x    2 ( 1 ) ∂ w dw 0 ∂u 1 ∂ w ∂ 2 w ∂ N xx = A xx + + = Nc − B xx 2 − ∂x 2 ∂x ∂ x dx ∂x ∂x 2

23 24

6 8

Combining Eq. (29) with Eq. (26) leads to

16 17

4 5

8 9

3

¯ = 0, w

M xx = 0

(37)

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1

Clamped (C) end:

2 3 4 5 6

7

1 2

¯ ∂w =0 ∂ x¯

¯ = 0, w

(38)

4

¯ (¯x, t¯) is assumed as Here, the Galerkin method is adopted to discretize the differential governing equation, the transverse deflection w the following series expansion

9

N 

¯ (¯x, t¯) = w

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

32 33 34

dt¯2

40

43 44 45

Fˆ = F¯

47 48

χ1 = ⎣

1

n (¯x)1 (¯x)dx¯ − μ2

0

⎡

χ2 = ⎣

59 60 61 62 63 64 65 66

n2 (¯x)dx¯ − μ2

1

0

1

21

(C − C)

22

(42)

0

⎤

23 24 25 26 27 28 29 31 32 34 35

¯ 0, n (¯x)n (¯x)dx¯ ⎦ m

⎤

n (¯x)(14) (¯x)dx¯ ⎦ A 0

⎧ ⎨ ⎩

¯ xx A 0 A

1

36 37

n (¯x)1 (¯x)dx¯ + B¯ xx

0

n (¯x)n (¯x)dx¯ − μ

¯ xx A 0 A 2

1

n (¯x)dx¯

0

⎡ ⎣

¯ xx A 2

2

1



1

n (¯x)1 (¯x)dx¯ − μ

38

⎭

40

39 41

,

42 43 44 45

0

2

⎤

n (¯x)1 (¯x)dx¯ ⎦

0

(4 )

0

1

0

46 47 48 49

,

50

(  )2 n (¯x) dx¯

51 52

0

⎡ 1 ⎤ 1

1

(  )2 ⎣ n (¯x)n (¯x)dx¯ − μ2 n (¯x)n(4) (¯x)dx¯ ⎦ n (¯x) dx¯ 0

⎫ ⎬

⎫ ⎤⎧

1

1 ⎬ ⎨ (4 ) n (¯x)n (¯x)dx¯ ⎦ A¯ xx A 0 n (¯x)1 (¯x)dx¯ + B¯ xx n (¯x)dx¯ ⎭ ⎩

0

0

χ3 = −

0

1



57 58

19

30

0

0

−

18 20

⎡ ⎤ 1 

1

1 B¯ 2xx ( 4 ) ( 6 ) (4 ) 2 2 2 ¯ ¯ ⎣ n (¯x)n (¯x)dx¯ − η + D xx − n (¯x)n (¯x)dx¯ ⎦ + η β A xx n (¯x)n (¯x)dx¯ ¯ xx A

54 56

1

0

53 55

1

0

50 52

χ0 = ⎣

n (¯x)dx¯ ,

49 51

(40)

3

⎡

1 ⎡

46

14

33

41 42

(C − H)

  ¯ t¯ + χ1 w (t¯) + χ2 w (t¯) + χ3 w (t¯) = Fˆ cos

in which

38

13

17

Inserting Eqs. (39) and (41) into Eq. (34), and multiplying the resultant equation with the eigenfunction n (¯x), then integrating within the range [0, 1], leads to

χ0

12

15

(41)

2

11

16

¯ 0 = A 0 1 (¯x) w

37 39

(H − H)

¯ 0 , a trigonometric function similar to the first-order vibrational mode of the perfect FG To describe the initial geometric imperfection w multilayer beam-type nanostructures with a given value of deflection A 0 is presented,

d2 w (t¯)

9 10

⎧ sin (γn x¯ ) , in which γn = nπ , ⎪ ⎪ ⎪ ⎪ ⎪ sin(λ )+sinh(λ ) ⎪ sin (γn x¯ ) − sinh (γn x¯ ) − cos(λn )+cosh(λn ) × ⎪ ⎪ n n ⎨ in which tan(γn ) = tanh(γn ), n (¯x) = [cos (γn x¯ ) − cosh (γn x¯ )] , ⎪ ⎪ ⎪ ⎪ sin (γ x¯ ) − sinh (γ x¯ ) − sin(γn )−sinh(γn ) × ⎪ n n ⎪ cos(γn )−cosh(γn ) ⎪ ⎪ ⎩ in which cos(γn ) cosh(γn ) = −1, [cos (γn x¯ ) − cosh (γn x¯ )] ,

35 36

(39)

in which w (t¯) denotes the time-dependent generalized coordinates; n (¯x) refers to the eigenfunction, which can be defined as the form of trigonometric functions and should satisfy the boundary conditions. In the present paper, three different boundary conditions are taken into account, i.e.,

30 31

6 8

w (t¯)n (¯x)

n =1

10 11

5 7

7 8

3

0

53 54

(43)

56

0

57 58

2.5. Solution procedure for nonlinear dynamic response

59

In this section, the multiple scale method is employed to analyze the amplitude-frequency response of imperfect FG multilayer beamtype nanostructures under excitation resonance. In this case, it is assumed that the excitation frequency is close to the natural frequency ω0 , and the closeness of the resonance is defined by

¯ = ω0 + ε λ 2

in which λ and

ε are, respectively, the detuning parameter and perturbation parameter.

55

(44)

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8

1

By introducing the following parameters,

2 3 4 5 6

w (t¯)

p (t¯) =

ε

d2 p (t¯)

8

dt¯2

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

Fˆ

ε 3 χ0

ω0 =

,

2

χ1 , χ0

γ1 =

χ2 , χ0

γ2 =

χ3 χ0

(45)

7

(46)

where ω0 stands for the linear natural frequency of the FG multilayer beam-type nanostructure. On the basis of the multiple scale method, the response p can be denoted as a function of scaled times T 0 , T 1 , T 2 and perturbation parameter ε as follows, 3

p ( T 0 , T 1 , T 2 ) = p 0 ( T 0 , T 1 , T 2 ) + ε p 1 ( T 0 , T 1 , T 2 ) + ε p 2 ( T 0 , T 1 , T 2 ) + O (ε )

(47)

in which T 0 = t¯, T 1 = εt¯, T 2 = ε 2 t¯. Accordingly, the time derivatives can be expanded to T i as

⎧ ⎨ ⎩

d dt¯ d dt¯2

(48)

34 35 36 37 38 39 40

2 0 p0 2 0 p1

ε : +ω =0 1 ε : +ω = −2D 0 D 1 p 0 − γ1 p 20 2 2 2 ε : D 0 p 2 + ω0 p 2 = f cos(ω0 T 0 + λ T 2 ) − 2D 0 D 1 p 1 − 2D 0 D 2 p 0 − D 21 p 0 − 2γ1 p 0 p 1 − γ2 p 30

(49)

43

46 47

50 51 52

55 56 57 58

61 62 63 64 65 66

20 21 22 23 24

(51)

25 26 27

31

29 30 32

D1 H = 0

(53)

γ1 p1 = 2 ω0



H2 3

¯ exp (2iω0 T 0 ) − H H

35



36

+ cc

(54)



2iω0 D 2 H + 3γ2 −

10γ12 3ω02



in which

2

2

f exp(iλ T 2 ) = 0

(55)

42 43 45 46

α exp(iϑ)

⎧ dα ⎨ dT = 2

(56)

dϑ dT 2

f sin(β) 2ω0

=

1 24ω03

⎧ dα ⎨ dT = 2 dβ dT 2



9γ2 ω02 − 10γ12



= λα −

λα −

1 24ω03



48 49

α3 −

(57)

f cos(β) 2ω0

51 52 53 54 55

f sin(β) 2ω0

56

1 24ω03



 2

9γ2 ω02 − 10γ1

By using the steady-state conditions



47

50

where β = λ T 2 − ϑ , and these equations can be finally simplified as

⎩α

40

44

α and ϑ are both real numbers in terms of T 2 . Inserting Eq. (56) into (55) and separating real and imaginary parts, one has

⎩α

39 41

1

¯ − H2 H

To solve the above equation, the function H is assumed to satisfy the following polar form

H=

37 38

where cc stands for the complex conjugate term. Similarly, introducing Eqs. (52) and (54) into Eq. (51) and eliminating secular terms leads to

1

33 34

which indicates that H is independent of T 1 . Then, the solution of Eq. (50) can be obtained

59 60

18

above equation into Eq. (50) and neglecting the secular terms, one has

53 54

14

28

48 49

13

¯ ( T 1 , T 2 ) exp(−iω0 T 0 ) p 0 = H ( T 1 , T 2 ) exp(iω0 T 0 ) + H (52) √ ¯ stands for its complex conjugate. Substituting the where i = −1, H represents an unknown function with respect to T 1 and T 2 , and H

44 45

12

(50)

The general solution of Eq. (49) is given as

41 42

11

19

32 33

10

17

= D 20 + 2ε D 0 D 1 + ε 2 (2D 0 D 2 + D 21 ) + O (ε 3 )

D 20 p 0 D 20 p 1

9

15

where D i = ∂/∂ T i , (i = 0, 1, 2). Substituting Eqs. (47) and (48) into Eq. (46), and separating the coefficients at each power of ε , results in 0

8

16

= D 0 + ε D 1 + ε 2 D 2 + O (ε 3 )

2

4 6

¯ t¯) + ω02 p (t¯) + εγ1 p 2 (t¯) + ε 2 γ2 p 3 (t¯) = ε 2 f cos(

2

3 5

The nonlinear differential equation (42) can be re-written as

7 9

f =

,

1

*

9γ2 ω02 − 10γ12



dβ dT 2

α3 +

= 0 and

dα dT 2



2

2

α3

f cos(β) 2ω0

=

f 2ω0

(58)

57 58 59

= 0, yields

60 61 62

(59)

The above equation gives the amplitude–frequency response of FG multilayer beam-type nanostructure subjected to excitation resonance, illustrating that the response amplitude α is a function of the detuning parameter λ and excitation amplitude f .

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Table 1 FG material properties used in comparison studies [49].

2 3 4 5

2

Material

Young modulus (GPa)

Mass density (kg/m3 )

3

Silicon Aluminum

E 1 = 210 E 2 = 70

ρ1 = 2370 ρ2 = 2700

4 5

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Fig. 3. Comparison between the present model and Nazemnezhad’s model [49] for nonlinear frequency ratio of perfect FG nonlocal nanobeams.

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39

40

40

Fig. 4. Comparison between the present model and Tang’s model [50] for nonlinear frequency-response of perfect FG nanobeams.

41

41 42 43 44 45 46 47 48 49 50

53 54 55 56 57 58 59 60 61 62 63 64 65 66

43

Before providing the numerical results, the reliability of the present model is confirmed by comparing with two previous models. In the first comparison model, the nonlinear frequency ratio (nonlinear frequency/ linear frequency) of a FG nonlocal nanobeam without considering the effect of geometric imperfection is examined, which gives a comparison with the work presented by Nazemnezhad and Hosseini-Hashemi [49]. Here, a H-H FG perfect nonlocal nanobeam with thickness h = 3 nm, length L = 20h, and width b = h is considered, and the material properties including Young modulus E and mass density ρ are varied along the thickness direction with the following criterion,



51 52

42

3. Comparison studies

E ( z) = ( E 1 − E 2 )



ρ (z) = (ρ1 − ρ2 )

2z + h 2h 2z + h 2h

m

44 45 46 47 48 49 50 51

+ E2

(60)

+ ρ2

(61)

52 53

m

54

where m denotes the FG index which governs the material distribution. The detailed material properties of the FG nanobeam are listed in Table 1. The maximum amplitude α = 1, geometric imperfection amplitude A 0 = 0, nonlocal size scale parameter μ = 0.1 are adopted, and the strain gradient size scale and thickness effects are both neglected in this comparison. The nonlinear frequency ratio as a function of the FG index for different nonlocal size scale parameters are compared as shown in Fig. 3. A good agreement you can see demonstrates the high accuracy of the present model. Fig. 4 gives the nonlinear frequency-response of a perfect H-H FG nanobeam under different FG indexes. In this comparison, the material properties also satisfy the index distribution presented by Eqs. (60) and (61), and the material properties are also derived from Table 1, the other parameters are set as f = 0.5, A 0 = 0 and μ = η = 0.1. Here, the thickness effect is not taken into account. The comparison results presented in this figure indicates that the nonlinear frequency-response given by our model is coincidence well with those predicted by Tang et al. [50], which validates the proposed model.

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Fig. 5. Nonlinear frequency-response of FG multilayer beam-type nanostructures for different GNP distribution types: (a) perfect, (b) imperfect.

20

4. Results and discussion

23 24 25 26 27 28 29 30

In this section, numerical studies are carried out to investigate the influence of GNP distribution type, weight fraction of GNPs, geometric imperfection amplitude, boundary condition, excitation amplitude, nonlocal and strain gradient size scale parameters on the nonlinear frequency-response of FG multilayer beam-type nanostructures. The nanostructures are composed of epoxy as polymer matrix and GNPs as reinforcements, whose material properties are derived from Refs. [9,51], i.e., the Young modulus and mass density of the epoxy matrix are E M = 3G P a, and ρ M = 1200kg /m3 , respectively, while the Young modulus and mass density of the GNPs-based reinforcement satisfy E GNP = 1.01T P a and ρGNP = 1062.5kg /m3 , respectively. Here, the geometrical parameters of the beam-type nanostructure are taken as h = 3nm, b = h, and L = 20h. In the following numerical analyses, the thickness effect is not taken into account unless otherwise specified. Besides, the total number of the multilayer beam-type nanostructure is assumed as N L = 6, and the GNPs-based reinforcement with length lGNP = 2.5nm, thickness hGNP = 0.3nm, width bGNP = 1.5nm is adopted. 4.1. Effect of GNPs

35 36 37 38 39 40 41 42 43 44 45 46 47 48

Fig. 5 presents the effect of GNP distribution type on the nonlinear frequency-response both for the perfect and imperfect FG multilayer beam-type nanostructures. Here, the H-H boundary condition is taken into account, and the weight fraction of GNPs g G∗ N P = 0.1, the excitation amplitude f = 0.5, and the non-dimensional size scale parameters μ = η = 0.1 are adopted. A 0 = 0 and 0.5 are, respectively, employed for the perfect and imperfect beam-type nanostructures. One can find that the FG multilayer beam-type nanostructure always bends to the right, i.e., exhibits a hardening effect. The amplitude of the forced vibration system is largest for the FG-A type GNP distribution, and followed by FG-X, UD, and FG-O type GNP distributions both for the perfect and imperfect nanobeams, indicating FG-O type distribution has the largest hardening effects, while that of FG-A type is the lowest. The derivation between FG-A type and the other three types of GNP distribution enlarges by considering the effect of geometric imperfection. In comparison with the perfect nanobeam, the hardening effect decreases for the imperfect counterpart for all types of GNP distribution, indicating that the geometric imperfection should not be neglected for the beam-type nanostructures. The influence of weight fraction of GNPs on the frequency-response of the nonlinear FG multilayer beam-type nanostructure is plotted in Fig. 6. In this calculation, an H-H beam-type nanostructure following FG-O type GNP distribution is taken into consideration. The parameters f = 0.5 and μ = η = 0.1 are used. It is demonstrated that the hardening behavior is intensified as the weight fraction of GNPs increases. The effect of weight fraction of GNPs is much more significant for the perfect beam-type nanostructure. The imperfect beam-type nanostructure displays lower hardening behavior in comparison with the perfect beam. 4.2. Effect of geometric imperfection

53 54 55 56 57 58 59

The nonlinear frequency-response curves of FG multilayer beam-type nanostructures under different geometric imperfection amplitudes are shown in Fig. 7 for various GNP distribution types. The parameters weight fraction of GNPs, excitation amplitude, and the non-dimensional size scale parameters are, respectively, specified as g G∗ N P = 0.1, f = 0.5, and μ = η = 0.1. The beam-type nanostructure without thickness effect and with H-H boundary condition is considered here. It is clear that the frequency-response curves bend severely by decreasing the value of geometric imperfection amplitudes, indicating the hardening behavior reduces as the geometric imperfection amplitude increases. The geometric imperfection shows the most influences on the beam-type nanostructures reinforced by GNPs with FG-A type, while the effect caused by the geometric imperfection is the lowest for the multilayer nanobeam with FG-X type GNP distribution. 4.3. Effect of size scale parameters

64 65 66

27 28 29 30 32 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 50 52 53 54 55 56 57 58 59 61 62

62 63

26

60

60 61

25

51

51 52

24

49

49 50

23

33

33 34

22

31

31 32

20 21

21 22

18 19

19

The influence of size scale parameters on the nonlinear frequency-response of FG multilayer beam-type nanostructures is investigated in this subsection. Here, the C-H beam-type nanostructure reinforced by GNPs with FG-O type distribution is taken into account. Fig. 8 shows the influence caused by the nonlocal size scale. The weight fractions of GNP g G∗ N P = 0.1, excitation amplitude f = 0.5, and strain gradient size scale parameter η = 0 are used. It is observed that the nonlocal size scale parameter reduces the amplitude of the forced vibration,

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Fig. 6. Nonlinear frequency-response of FG multilayer beam-type nanostructures for different weight fractions of GNPs g G∗ N P : (a) perfect, (b) imperfect.

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54 55

Fig. 7. Nonlinear frequency-response of FG multilayer beam-type nanostructures for different geometric imperfection amplitudes A 0 : (a) UD, (b) FG-O, (c) FG-X, (d) FG-A.

58 59 60 61 62 63 64 65 66

55 56

56 57

54

which indicates the nonlocal size scale parameter can strengthen the hardening effect of the FG multilayer beam-type nanostructure. It is also seen that the nonlinear frequency-response is sensitive to the geometric imperfection, i.e., the hardening behavior of the nonlinear forced vibration system decreases for the imperfect FG multilayer beam-type nanostructures. The strain gradient size scale parameter on the nonlinear frequency-response behavior of FG multilayer beam-type nanostructure is depicted in Fig. 9. In the calculation, we define the weight fractions of GNP, the excitation amplitude, and nonlocal size scale parameter as g G∗ N P = 0.1, f = 0.5, and μ = 0, respectively. It is shown that the hardening behavior of the FG multilayer beam-type nanostructure enhances with the decrease of the strain gradient size scale parameter both for the perfect and imperfect beam-type nanostructures. Besides, the strain gradient size scale shows much more influence on the perfect beam-type nanostructure, indicating the hardening behavior of FG multilayer beam-type nanostructure induced by the strain gradient size scale parameter reduces as the geometric imperfection is taken into account.

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Fig. 8. Nonlinear frequency-response of FG multilayer beam-type nanostructures for different nonlocal size scale parameters: (a) perfect, (b) imperfect.

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Fig. 9. Nonlinear frequency-response of FG multilayer beam-type nanostructures for different strain gradient size scale parameters: (a) perfect, (b) imperfect.

4.4. Effect of excitation amplitude

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The effect of external excitation amplitude on the nonlinear frequency-response curve of FG multilayer beam-type nanostructure is illustrated in Fig. 10. Here, a multilayer beam-type nanostructure with C-C boundary condition and reinforced by GNPs with FG-A distribution is studied. The parameters g G∗ N P = 0.1, μ = η = 0.1 are used for this numerical calculation. It is shown that the higher excitation amplitude leads to larger nonlinear amplitude and wider entire region of the resonant response of the FG multilayer beam-type nanostructure. In addition, the external excitation amplitude has no influence on the hardening behavior of FG multilayer beam-type nanostructure. At the same conditions, the geometric imperfection changes the nonlinear frequency-response of FG multilayer beam-type nanostructure. It suggests the coupled effect of excitation amplitude and geometric imperfection on nano-structural dynamic behavior is of significance for the research and design of FG multilayer beam-type nanostructures. 4.5. Effect of boundary condition

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In Fig. 11, the nonlinear frequency-response of FG multilayer beam-type nanostructures is compared for three different boundary conditions. Here, the GNPs distribution type is considered as FG-X, and the calculation parameters are carried out as g G∗ N P = 0.1, f = 0.5, A 0 = 0.5, and μ = η = 0.2. As depicted in this figure, the hardening effects for the H-H and C-C boundary conditions are the lowest and highest, respectively. Besides, we can also find the thickness effect can reduce the hardening effects for all boundary conditions. The derivation of the nonlinear frequency-response curve between the C-C and C-H beam-type nanostructures becomes reduced by accounting the effect of the thickness effect. 4.6. Effect of thickness

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The thickness effect on the nonlinear frequency-response of FG multilayer beam-type nanostructures is presented in Fig. 12 for two different external excitation amplitudes. In these analyses, the FG multilayer beam-type nanostructure with C-C boundary condition is considered. The distribution type of the reinforced GNPs is treated as FG-A, and the parameters g G∗ N P = 0.1, A 0 = 0.5, as well as μ = η = 0.1 are adopted. The external excitation amplitudes are set as f = 0.5 and 1.5 in Figs. 12(a) and (b), respectively. It is demonstrated that the external excitation amplitude has great influence on the region of the resonant response, while its influence on the nonlinear frequency-response behavior is independent on the value of the excitation amplitude. Especially, the nonlinear frequency-response curve

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Fig. 10. Nonlinear frequency-response of FG multilayer beam-type nanostructures for different excitation amplitudes f : (a) perfect, (b) imperfect.

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Fig. 11. Nonlinear frequency-response of FG multilayer beam-type nanostructures for different boundary conditions: (a) without thickness effect, (b) with thickness effect.

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Fig. 12. Influence of thickness on the nonlinear frequency-response of FG multilayer beam-type nanostructures: (a) f = 0.5, (b) f = 1.5.

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bends to the left more than the thickness effect is taken into account, illustrating the thickness effect can greatly reduce the hardening effects. 5. Conclusion remarks

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This paper is devoted to studying the nonlinear resonance behavior of FG multilayer beam-type nanostructures reinforced by GNPs with consideration of the initial geometric imperfection. The fundamental equation of the beam-type nanostructure is established based on Euler-Bernoulli beam theory involving the von Kármán geometrical nonlinear term and nonlocal strain gradient size scale effects. One uniform distribution (UD) and three FG types of GNP distribution incorporating O- pattern, X- pattern and A- pattern are evaluated. The

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amplitude-frequency response of imperfect FG multilayer beam-type nanostructures is investigated by using the multiple scale method, and influences caused by several different parameters are discussed in detail. The notable observations are presented as follows: 1. The weight fraction and distribution type of GNPs both show significant influences on the nonlinear frequency-response of FG multilayer beam-type nanostructures, i.e., the FG-O type distribution exhibits the largest hardening behavior, while that of FG-A type is the lowest. The weight fraction of GNPs can enhance the hardening behavior, while the thickness effect plays an opposite role on the nonlinear frequency-response of FG multilayer beam-type nanostructures. 2. The geometric imperfection can reduce the hardening behavior of FG multilayer beam-type nanostructure. The geometric imperfection plays the most important influence on FG-A type GNP distributed beam-type nanostructures, while its influence on the FG-X type GNP distribution is the lowest. Additionally, higher excitation amplitude yields larger nonlinear amplitude and wider entire region of the resonant response. 3. The nonlocal size scale can strengthen the hardening effect of the FG multilayer beam-type nanostructure, while an opposite change trend can be observed for the strain gradient size scale effect. The hardening behavior of the FG multilayer beam-type nanostructure with C-C boundary condition is the highest, and followed by C-H and C-C boundary conditions, respectively. Our preliminary studies suggest that the effect of graphene nanoplatelet-reinforcement geometric imperfection on dynamic behaviors of FG multilayer beam-type composite is of significance. Further work is needed to focus on comparison study of different types of geometric imperfections. Declaration of competing interest

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The authors declare that there is no conflict of interest in the manuscript titled “Nonlinear resonance of FG multilayer beam-type nanocomposites: Effects of graphene nanoplatelet-reinforcement and geometric imperfection”. References

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