Nonlinear low-velocity impact response of FG-GRC laminated plates resting on visco-elastic foundations

Composites Part B 144 (2018) 184–194

Contents lists available at ScienceDirect

Composites Part B journal homepage: www.elsevier.com/locate/compositesb

Nonlinear low-velocity impact response of FG-GRC laminated plates resting on visco-elastic foundations

T

Yin Fana,c, Y. Xiangc,d, Hui-Shen Shena,b,∗, D. Huie a

School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China c School of Computing, Engineering and Mathematics, Western Sydney University, Locked Bag 1797, Penrith, NSW 2751, Australia d Centre for Infrastructure Engineering, Western Sydney University, Locked Bag 1797, Penrith, NSW 2751, Australia e Department of Mechanical Engineering, University of New Orleans, New Orleans, LA 70148, USA b

A R T I C L E I N F O

A B S T R A C T

Keywords: Nano-structures Plates Laminates Analytical modeling Functionally graded materials

The nonlinear transient response of functionally graded graphene reinforced composite (FG-GRC) laminated plates resting on visco-Pasternak foundations in thermal environments under impact load is investigated in this paper. Each layer of a laminated plate is assumed to have the same thickness, but the volume fraction of graphene is assumed to be functionally graded in a piece-wise pattern along the plate thickness direction. The stiffness of FG-GRC is then obtained by an extended Halpin-Tsai model, where the graphene efficiency parameters are introduced and determined by molecular dynamics (MD) simulations. The impactor is assumed to be a metal sphere and the contact process between the impactor and the laminated plate is described by a modified Hertz model. The effects of the visco-Pasternak foundation and the temperature variation as well as the initial load are taken into consideration. In the framework of von Kármán type of kinematic nonlinearity, the motion equations of an FG-GRC laminated plate are established based on a higher-order shear deformation theory and solved by a two-step perturbation technique. Finally, the motion equations of the impactor and the FG-GRC laminated plate can be simultaneously solved by the Runge-Kutta approach. The numerical results illustrate the effects of functionally graded graphene distribution, foundation stiffness, temperature variation, initial in-plane load and different impactor velocities on the contact force and the deflection of the FG-GRC laminated plate.

1. Introduction Graphene is the two-dimensional one-atom-thick form of sp2 carbon atom allotropes that possesses extraordinary material properties such as super high strength and stiffness and superior electrical conductivity [1]. Graphene is the ideal nano-filler reinforcement agent for polymer [2] or metal [3] matrix to create high performing nanocomposites which have huge potentials in a wide range of engineering applications. The outstanding physicochemical properties of graphene [4–8], which traditional carbon fibers are unable to achieve, make it possible for graphene reinforced composites (GRCs) to replace the fiber reinforced composites in the future [9–12]. However, due to the weak physical interactions between graphene and polymer matrix, the transfer efficiency in GRCs is relatively low [13–15]. It is also reported that the continuous increase of graphene content in GRCs may also degrade the mechanical performance of GRCs [16]. In order to better utilize the low volume fraction of graphene in GRCs, the concept of functionally graded materials (FGMs) can be

∗

employed. The concept of FGMs was first presented and applied by Shen [17] for carbon nanotube (CNT) reinforced composite plates where the volume fraction of carbon nanotubes (CNTs) is functionally graded with a linear distribution along the plate thickness direction. Recently, the concept of FGM has also been applied for the nanocomposite structures containing graphene reinforcements [18–46]. Within the framework of three-dimensional elasticity theory, Yang and his co-authors [18–20] studied the bending behaviors of rectangular, circular, annular and elliptical functionally graded graphene platelet reinforced composite (FG-GPLRC) plates. In their studies, graphene platelet (GPL) was selected as the filler and whose weight fraction varies continuously and smoothly along the plate thickness direction. In the research work of Yang and his co-authors [21–29], the FG-GPLRC beams or plates were divided into N layers with each layer having the same isotropic material properties but different weight fraction of GPL. A modified Halpin-Tsai model [47] was employed to estimate the equivalent isotropic material properties of GPLRCs and based on this model, a systematical research on the mechanical behaviors, including

Corresponding author. School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China. E-mail address: [email protected] (H.-S. Shen).

https://doi.org/10.1016/j.compositesb.2018.02.016 Received 8 January 2018; Received in revised form 9 February 2018; Accepted 17 February 2018 Available online 22 February 2018 1359-8368/ © 2018 Elsevier Ltd. All rights reserved.

Composites Part B 144 (2018) 184–194

Y. Fan et al.

vibration [21,22,25], bending [23–25], dynamic instability [26] and buckling and postbuckling [23,27–29] of GRLRC beams and plates was carried out. In the same time, Shen and his co-authors [30–42] conducted numerous research works of FG-GRC structures. In their study, a GRC structure is made of laminated GRC layers which are reinforced by aligned single layer graphene sheets. The laminated layers can have different volume fraction of graphene reinforcement to form a piecewise functionally graded (FG) pattern. The anisotropic and temperature dependent material properties of GRC layers are obtained by the extended Halpin-Tsai model with the graphene efficiency parameters that can be determined by matching the results from the Halpin-Tsai model and the molecular dynamics (MD) simulations [48]. Utilizing this model, Shen and his co-authors investigated the nonlinear vibration of FG-GRC laminated beams [30], plates [31], cylindrical shells and panels [32,33], the nonlinear bending of FG-GRC laminated plates and panels [34,35], the nonlinear bending and thermal postbuckling of FGGRC laminated beams [36], and the postbuckling and thermal postbuckling of FG-GRC laminated plates [37–39], cylindrical shells and panels [40–42]. On the basis of above research achievements, other researchers have also carried out their studies related to FG-GRC structures [43–46]. For example, following the Yang's model, Gholami and Ansari [43] studied the large deflection of FG-GPLRC sinusoidal shear deformation plates subjected to uniform and sinusoidal transverse load. And the nonlinear instability of FG-GPLRC nanoshells under axial compressive load was examined by Sahmani and Aghdam [44]. Following the Shen's model, Mirzaei and Kiani [45] analyzed the thermal buckling response of FG-GRC laminated plates by using a non-uniform rational B-spline based isogeometric finite element method. Moreover, an investigation on thermal postbuckling of FG-GRC laminated beams was carried out by Kiani and Mirzaei [46]. However, to the authors' best knowledge, there has been no reported research on the responses of FG-GRC laminated plates subject to low-velocity impact. In the present work, the study on the low-velocity impact response of FG-GRC laminated plates is carried out. An extended Halpin-Tsai model is employed to estimate the material properties of GRCs, which is assumed to be dependent on temperature. In the framework of von Kármán displacement-strain relationships, a set of dynamic equations of GRC laminated plates can be established based on the Reddy's higher order shear deformation theory. In the set of equations, the plate-foundation interaction and the thermal effects are both taken into account. The motion equation of FG-GRC laminated plates can be derived by the Shen's two-step perturbation technique and then solved by the Runge-Kutta method simultaneously with the motion equation of the impactor. The numerical illustrations highlight the lowvelocity impact responses of FG-GRC plates resting on visco-Pasternak foundation under different sets of environmental conditions.

E22 = η2 G12 = η3

G 1 + 2(bG / hG ) γ22 VG G 1 − γ22 VG

Em (1b)

1 Gm G 1 − γ12 VG

(1c)

where E and G are the Young's modulus and shear modulus, respectively, while aG, bG and hG are the length, width and effective thickness of the graphene sheet. The subscripts G and m represent, respectively, graphene and matrix. V is volume fraction and Vm + VG = 1. In Eqs. G G G , γ22 and γ12 are, respectively, as follows (1a)–(1c), the expressions of γ11 G γ11 =

G E11 /E m − 1 G m E11/ E + 2aG / hG

(2a)

G γ22 =

G E22 /E m − 1 G m E22/ E + 2bG / hG

(2b)

G γ 12 =

G G12 /Gm − 1 G G12 /Gm

(2c)

The values of the graphene efficiency parameters will be determined by matching the MD simulation results [48] of elastic moduli of GRCs. According to the Schapery method [49], in the longitudinal and transverse directions, the thermal expansion coefficients α11 and α22 of GRCs can be, respectively, written as

α11 =

G G VG E11 α11 + Vm E mα m G + Vm E m VG E11

(3a)

G G α22 = (1 + ν12 ) VG α 22 + (1 + ν m) Vm α m − ν12 α11

(3b)

in which ν12 is the Poisson's ratio. For GRCs, the expressions of Poisson's ratio and density are G ν12 = VG ν12 + Vm ν m

(4a)

ρ = VG ρG + Vm ρm

(4b)

where ρG and ρm are the densities of the graphene and matrix, respectively. 2.2. Modified Hertz model In a case of quasi-static approximation, the Hertz contact law is acceptable in the analysis of low-velocity impact [50]. The contact process between an impactor and a plate is divided into two phases. In the loading phase, the total contact force Fc (t) is assumed to follow the Hertz contact law:

Fc (t ) = K c [δ (t )]r

(5)

where the local contact indentation δ(t) is defined by 2. Theoretical models

δ (t ) = W i (t ) − W (t )

2.1. Extended Halpin-Tsai model

in which W i (t ) denotes the displacement of the impactor and W (t ) represents the deflection of the plate at the impact location. According to the Hertz contact law, the power r = 1.5 is considered for the contact between two homogeneous isotropic solids. However, it has been reported that r = 1.5 is also available for laminated composite targets [51]. In Eq. (5), Kc is the contact stiffness and is defined by

It is assumed that a GRC layer is made of an isotropic polymer matrix reinforced by graphene sheet fillers. The graphene sheets are assumed to be aligned in the laminated GRC layer. The experimental results from Kuilla et al. [11] showed that the Young's moduli of GRC with aligned graphene sheets follow the Halpin-Tsai [47] theoretical prediction. However, it is also pointed out in Kuilla et al. [11] that the Halpin-Tsai model cannot be directly employed to compute the effective material properties of GRC when the volume fraction of graphene is at a high level. Hence, it is necessary to introduce efficiency parameters η1, η2 and η3 in the extended Halpin-Tsai model

E11 = η1

G 1 + 2(aG / hG ) γ11 VG G 1 − γ11 VG

Kc =

(6)

4 ∗ i E R 3

(7) ∗

i

where R is the radius of the impactor and the expression of E is written as −1

1 − ν iν i 1⎞ E ∗ = ⎜⎛ + ⎟ i E E z⎠ ⎝

Em

(8)

where E and ν are the Young's modulus and Poisson's ratio of the impactor, respectively, and Ez is the transverse Young's modulus at the top i

(1a) 185

i

Composites Part B 144 (2018) 184–194

Y. Fan et al.

surface of the plate. For traditional fiber reinforced composite, Ez is normally estimated as the value of E22. For functionally graded carbon nanotube reinforced composite, Fan and Wang [52,53] presented a novel definition of Ez, which is

Ez (t ) =

1 δ (t )

∫t

tu + δ (t )

u

L͠ 11 (W ) − L͠ 12 (Ψx ) − L͠ 13 (Ψy ) + L͠ 14 (F ) − L͠ 16 (M T )

(13a)

1 L͠ 21 (F ) + L͠ 22 (Ψx ) + L͠ 23 (Ψy ) − L͠ 24 (W ) = − L͠ (W + 2W ∗, W ) 2

(13b)

⎜

⎟

⎜

E22 (Z ) dZ

(9)

in which Z = tu is the top surface of the plate. However, benefit from the MD results of Lin et al. [48], we can employ the E33 of the GRCs as the transverse Young's modulus directly. In the unloading phase, the contact force Fc can be defined as

δ (t ) − δ0 ⎤ Fc (t ) = Fmax ⎡ ⎢ δ ⎦ ⎣ max − δ0 ⎥

= I9 (10)

= I9

(13c)

∂2Ψy ∂3W + I10 2 ∂t ∂Y ∂t 2

(13d)

in which the nonlinear operator L͠ ( ) is defined by

∂2 ∂2 ∂2 ∂2 ∂2 ∂2 L͠ ( ) = −2 + 2 2 ∂X ∂Y ∂X ∂Y ∂X ∂Y ∂Y 2 ∂X 2

3. Dynamic equations

(14)

and the definition of linear operators L͠ ij ( ) can be found in Shen [58]. The generalized inertia Ii (i = 1, 2, 3, 4, 5 and 7) is defined by

As shown in Fig. 1, it is assumed that a rectangular GRC laminated plate, whose four edges are all simply supported, with length a, width b and thickness h rests on a visco-Pasternak foundation in a uniform temperature field T. The impactor is supposed to be a steel sphere with radius Ri and initial impact velocity V0 . The impactor impacts the GRC laminated plate at the central point.

N

(I1, I2, I3, I4 , I5, I7) =

hk

∑ ∫

ρk (1, Z , Z 2, Z 3, Z 4, Z 6) dZ (15)

k = 1 hk − 1

and then we can obtain the expressions of I8, I9 and I10

3.1. Motion equations of FG-GRC laminated plate The foundation and the plate are assumed to be bonded perfectly, which means the foundation and the plate are not separated when displacement occurs. The coupling effect of visco-elastic foundation and plate can be expressed by an interaction force

I8 =

I2 I2 I1

I9 =

4 3h2

I10 =

− I3 −

4 I, 3h2 5

(I − ) ,

I2 I2 I1

5

I2 I4 I1

− I3

(16)

where

I2 = I2 − I5 = I5 −

(11)

I3 = I3 −

where the symbol ∇2 is the Laplace operator and is defined by

∂2 ∂2 + ∂X 2 ∂Y 2

∂3W ∂ 2Ψ + I10 2x 2 ∂X ∂t ∂t

L͠ 41 (W ) + L͠ 42 (Ψx ) + L͠ 43 (Ψy ) + L͠ 44 (F ) − L45 (N T ) − L46 (S T )

where Fmax and δmax are the maximum contact force and indentation. The local indentation δ0 equals to zero when δmax remains below a critical indentation during the loading phase [54]. It shows that the power s = 2.5 provides a good fit to the experimental data [55].

dW p (X , Y ) = K1 W − K2 ∇2 W + Cd dt

⎟

L͠ 31 (W ) + L͠ 32 (Ψx ) + L͠ 33 (Ψy ) + L͠ 34 (F ) − L͠ 35 (N T ) − L͠ 36 (S T )

s

∇2 =

∂3Ψy ⎞ ∂ 2W ∂3Ψx = L͠ (W + W ∗, F ) + L͠ 17 ⎛ 2 ⎞ + I8 ⎛⎜ +β ⎟ 2 ∂Y ∂t 2 ⎠ ⎝ ∂t ⎠ ⎝ ∂X ∂t ∂W ⎞ + qi − ⎛K1 W − K2 ∇2 W + Cd ∂t ⎠ ⎝

4 I, 3h2 4 4 I, 3h2 7 8 16 I + 4 I7 3h2 5 9h

(17)

NT ,

MT and

PT are,

In Eqs.(13a)–(13d), respectively, the forces, moments and higher order moments caused by the temperature change ΔT and are defined by

(12)

In Eq. (11), p is the force per unit area, K1 and K2 are the Winkler stiffness and the shearing layer stiffness of the foundation, respectively, Cd is the damping coefficient for the foundation, and t is time. The initial deflection W ∗ (X , Y ) herein is assumed to be caused only by the temperature change ΔT (ΔT = T – T0, where T0 is the reference temperature), while the additional deflection W (X , Y ) is caused by transverse impact load qi . Based on a higher order shear deformation plate theory [56], a set of general von Kármán-type equations [57] for a laminated plate consisting of N layers can be derived and expressed by

T T T ⎡ N x Mx Px ⎤ ⎢NT MT PT⎥ = y y ⎥ ⎢ y T T T⎥ ⎢ N xy M xy P xy ⎦ ⎣

N

hk

∑∫ i = 1 hk − 1

⎡ Ax ⎤ ⎢ Ay ⎥ (1, Z , Z 2) ΔTdZ ⎢A ⎥ ⎣ xy ⎦ k

(18a)

T

T ⎡ S x ⎤ ⎡ M xT ⎤ ⎡ Px ⎤ 4 ⎢ T⎥ ⎢ T⎥ ⎢ T⎥ ⎢ S y ⎥ = ⎢ M y ⎥ − 3h2 ⎢ P y ⎥ T⎥ ⎢ T ⎥ ⎢M T ⎥ ⎢ P xy S ⎣ ⎦ ⎣ xy ⎦ ⎣ xy ⎦

(18b)

where

⎡Q11 Q12 Q16 ⎤ ⎡ c 2 s2 ⎤ α ⎡ Ax ⎤ ⎡ 11 ⎤ ⎢ Ay ⎥ = −⎢Q12 Q22 Q26 ⎥ ⎢ s 2 c 2 ⎥ ⎢ α22 ⎥ ⎢ ⎥ ⎢ ⎢A ⎥ ⎥ − 2 cs 2 cs xy ⎦k ⎣ ⎦k ⎣ ⎦k ⎣Q16 Q26 Q66 ⎦k ⎣

(19)

in which the subscript k represents the kth layer. We assume that four simply supported edges of the GRC laminated plate are all immovable or movable in the plane. In the condition of inplane movable edges, it is supposed that an initially uniaxial load is acting in the X direction. Hence, the boundary conditions can be written as X = 0, a;

Fig. 1. Geometry and coordinate system of the GRC laminated plate.

186

Composites Part B 144 (2018) 184–194

Y. Fan et al.

W = Ψy = 0

(20a)

4. Solutions for dynamic equations

Mx = Px = 0

(20b)

U = 0 (immovable edges)

(20c)

Before carrying out the solution process, it is convenient to firstly define the following dimensionless quantities for the problem of lowvelocity impact response of the FG-GRC laminated plate. Introduce dimensionless coefficients

∫0

b

Nx dY + σx bh = 0 (movable edges)

(20d)

X

Y

x = πa,

Y = 0, b;

a

y = πb,

β = b,

(20e)

(Ψx , Ψy ) =

My = Py = 0

(20f)

λq =

V = 0 (immovable edges)

(20g)

(γT 3, γT 4, γT 6, γT 7) =

(20h)

(K2, k2) = K2 ⎛ 2 ∗ , ⎝ π D11

∫0

a

Eqs. (20c) and (20g) denote the case of in-plane immovable edges. They can also be expressed in detailed, respectively, by b

a

2

∫ ∫ ⎧⎨A11∗ ∂∂YF2 0

0

∗ + A12

⎩

∗ + ⎛B12 ⎝

∂ 2F 4 ∗ ∂Ψx ∗ ⎞ + ⎡ ⎛B11 − 2 E11 ⎢⎝ ∂X 2 3h ⎠ ∂X ⎣

2 2 4 ∗ ∂Ψy ⎤ 4 ⎛ ∗ ∂2W 1 ∂W ⎞ ∗ ∂ W ⎞ ⎞ − 2 E12 − 2 E11 + E12 − ⎛ 2 2 ⎥ ∂ Y h X Y X ∂ ∂ ∂ 3h 3 2 ⎠ ⎝ ⎠ ⎝ ⎠ ⎦ ⎜

⎟

⎜

b

∫∫ 0

0

⎜

γ170 = −

I1 E0 a2 ∗ , π 2ρ0 D11

E0 ρ0

,

σx b2h , ∗ D ∗ ]1/2 4π 2 [D11 22

H ∗ = H − EA−1 E

(Mx , Px ) =

∗

γ171 =

4E0 (I5 I1 − I4 I2) ∗ , 3ρ0 h2I1 D11

V0 =

(γ80, γ90, γ10) = (I8, I9, I10) ρ

aV0 ∗ D ∗ A∗ A∗ π D11 22 11 22

hk

∫ hk − 1

∗

L31 (W ) + L32 (Ψx ) − L33 (Ψy ) + γ14 L34 (F ) − L36 (S T ) = γ90

(Qij )k (1,

∂3W ∂ 2Ψ + γ10 2x ∂x ∂t 2 ∂t (28c)

¨ ∂W + γ10 Ψ¨y ∂y (28d)

Accordingly, the dimensionless nonlinear operator L ( ) can be written as

hk

k = 1 hk − 1

(28a)

(28b)

L41 (W ) − L42 (Ψx ) + L43 (Ψy ) + γ14 L44 (F ) − L46 (S T ) = γ90 β (23a)

∑ ∫

(27)

⎟

∗

(Qij )k (1, Z , Z 2, Z 3, Z 4, Z 6) dZ (i,

j = 1, 2, 6) (Aij , Dij , Fij ) =

m

⎡ Ax ⎤ (1, Z , Z 3) ΔTdZ ⎢ Ay ⎥ ⎣ ⎦

k = 1 hk − 1

Z 4 ) dZ

E0 ∗ , 0 D11

ρ0 E0

hk

Z 2,

b4 ⎞ , E0 h3 ⎠

⎟

L21 (F ) + γ24 L22 (Ψx ) + γ24 L23 (Ψy ) − γ24 L24 (W ) 1 = − γ24 β 2L (W + 2W ∗, W ) 2

Generally, A , D and H are symmetric matrices while B , E and F may not always be symmetric. Here, stiffness coefficients Aij, Bij, Dij, etc. of the plate are defined by

N

a4

(K1, k1) = K1 ⎛ 4 ∗ , ⎝ π D11

⎛Mx , 4 Px ⎞ a2 3h2 ⎠ ⎝ , ∗ [D ∗ D ∗ A∗ A∗ ]1/4 π 2 D11 11 22 11 11

⎜

(22)

∗

∑ ∫

,

T T a2 (Ax , Ay ) , ∗ D ∗ ]1/2 π 2 [D11 22

∂3Ψy ⎞ ∂ 2W ∂3Ψx = γ14 β 2L (W + W ∗, F ) + L17 ⎛ 2 ⎞ + γ80 ⎜⎛ +β ⎟ 2 ∂ y ∂t 2 ⎠ ∂ t ∂ x ∂ t ⎝ ⎠ ⎝ ∂W ⎞ − ⎛K1 W − K2 ∇2 W + Cd + λq ∂t ⎠ ⎝

A∗ = A−1 , B ∗ = −A−1 B, D∗ = D − BA−1 B, E ∗ = −A−1 E ,

N

a3 ∗ π 3D11

),

1/2

L11 (W ) − L12 (Ψx ) − L13 (Ψy ) + γ14 L14 (F ) − L16 (M T ) (21b)

The definition of the reduced matrices A∗, B∗, D∗, E∗, F∗ and H∗ are

(Aij , Bij , Dij , Eij, Fij , Hij ) =

4 T 4 T F , F 3h2 x 3h2 y

A∗

γ24 = ⎡ A11 ∗ ⎤ ⎣ 22 ⎦

,

It is worth noting that L15(NT) = L25(NT) = L35(NT) = L45(NT) = 0 when there is a uniform temperature field. Then, Eqs.(13a)–(13d) can be simplified and expressed in dimensionless form as follows

⎟

∗ ∗ N xT + A22 N yT ) ⎫ dYdX = 0 − (A12 ⎬ ⎭

∗

(γT 1, γT 2) =

E0 ρ0

b2 ⎞ , E0 h3 ⎠

N T T T ⎡ Ax Dx Fx ⎤ ⎢ A T DT FT ⎥ = − ∑ y y⎦ k=1 ⎣ y

2

∗

T y

πt a

,

where ρ0 and E0 are, respectively, the values of ρ and E at the reference temperature (T0 = 300 K). AxT , DxT , FxT , etc., are defined by

2 4 ∗ ∂Ψy ⎤ 4 ⎛ ∗ ∂2W 1 ∂W ⎞ ∗ ∗ ∂ W ⎞ ⎞ + ⎛B22 − 2 E22 − 2 E21 + E22 − ⎛ 2 h X Y ∂ ∂ 3h 3 2 ⎝ ∂Y ⎠ ⎝ ⎠ ∂Y ⎥ ⎝ ⎠ ⎦

F ∗ = F − EA−1 B,

T x

=

F ∗ D ∗ ]1/2 [D11 22

(26)

(21a)

⎟

(D , D ,

F=

m

2 2 ⎧A ∗ ∂ F + A ∗ ∂ F + ⎡ ⎛B ∗ − 4 E ∗ ⎞ ∂Ψx 22 12 2 ⎢ ⎝ 21 3h2 21⎠ ∂X ⎨ ∂X ∂Y 2 ⎣ ⎩ ⎜

22

,

⎟

∗ ∗ N xT + A12 N yT ) ⎫ dXdY = 0 − (A11 ⎬ ⎭ a

a2 ∗ π 2hD11

γ5 = − A12 ∗ ,

⎜

Cd = Cd

λx =

A∗

qi a4 , ∗ [D ∗ D ∗ A∗ A∗ ]1/4 π 4D11 11 22 11 22

a2

Nx dY = 0 (movable edges)

∗ D ∗ A∗ A∗ ]1/4 [D11 22 11 22 D ∗ 1/2 γ14 = ⎡ D22 , t ∗ ⎤ ⎣ 11 ⎦

(Ψx , Ψy ) a , π [D ∗ D ∗ A∗ A∗ ]1/4 11 22 11 22

W = Ψx = 0

W

W=

(i, j = 4,5) (23b)

L( ) =

3.2. Motion equations of impactor

∂2 ∂2 ∂2 ∂2 ∂2 ∂2 −2 + 2 2 2 2 ∂x ∂y ∂x ∂y ∂x ∂y ∂y ∂x

(29)

The longitudinal vibration of impactor is neglected in the current impact analysis. Hence, the motion equation of the impactor is expressed as

and the dimensionless linear operators Lij ( ) are all defined in Shen [58]. The boundary conditions Eqs. (20a)–(20h) can also be expressed in non-dimensional form as x = 0, π;

¨ i (t ) + Fc (t ) = 0 miW

W = Ψy = 0

(30a)

Mx = Px = 0

(30b)

δx = 0 (immovable edges )

(30c)

(24)

and the corresponding initial conditions for impactor are defined as

W i (0) = 0,

i

W˙ (0) = V0

(25) 187

Composites Part B 144 (2018) 184–194

Y. Fan et al.

1 π

∫0

π

β2

∂ 2F dy + 4λ x β 2 = 0 (movable edges ) ∂y 2

(3) (2) (1) Ψx (x , y, t ) = ε [C11 (t ) + C¨11 (t )]cos mx sin ny + ε 2C20 (t )sin 2 mx

(30d)

(3) (3) + ε 3 [D31 (t )cos 3 mx sin ny + D13 (t )cos mx sin 3 ny]

y = 0, π;

+ O (ε 4 )

(33c)

W = Ψx = 0

(30e)

My = Py = 0

(30f)

(3) (3) + ε 3 [D31 (t )sin 3 mx cos ny + D13 (t )sin mx cos 3 ny]

δ y = 0 (immovable edges )

(30g)

+ O (ε 4 )

1 π

∫0

π

β2

(3) (2) (1) Ψy (x , y, t ) = ε [D11 (t ) + D¨ 11 (t )]sin mx cos ny + ε 2D02 (τ )sin 2 ny

(1)

∂ 2F dx = 0 (movable edges ) ∂x 2

π

π

∫∫

(30h)

(1) + 2(g20 cos 2 mx + g02 cos 2 ny ) Φ (εA11 (t ))

0

0

2

3

(1) + g3 (εA11 (t )) sin mx sin ny

2

∂ 2W ∂ 2W ∂ 2W ⎞ ∂W ⎞ 1 − γ24 ⎛γ611 2 + γ244 β 2 2 + 2γ516 β − γ24 ⎛ ∂ x ∂ y ∂ x ∂ y ∂x ⎠ 2 ⎝ ⎝ ⎠ ⎟

2 +(γ24 γT 1 − γ5 γT 2) ΔT ] dxdy

δy =

1 4π 2β 2γ24

π

π

2

∫ ∫ ⎡⎢ ∂∂xF2 0

0

− γ5 β 2

⎣

(31a)

(1) εA11 = Wm − Θ3 Wm3 + ⋯

⎟

2

1 ∂ 2W ∂ 2W ∂ 2W ⎞ ∂W ⎞ − γ24 ⎛γ240 2 + γ622 β 2 2 + 2γ526 − γ24 β 2 ⎛ x y x y 2 ∂ ∂ ∂ ∂ ⎝ ⎠ ⎝ ∂y ⎠ ⎜

⎟

⎜

g0

⎟

+ (γT 2 − γ5 γT 1) ΔT ⎤ ⎥ dydx ⎦

Ψx (x , y, τ , ε ) =

∑ ε jψxj (x , y, τ )

∑ ε jf j (x , y, τ )

(32d)

j=0

λq (x , y, τ , ε ) =

Numerical results are presented in this section for low-velocity impact response of FG-GRC laminated plates on a visco-Pasternak foundation. First of all, the key to apply extended Halpin-Tsai model is to determine the effective material properties of the GRCs. We select Poly (methyl methacrylate) (PMMA) as the matrix material and its mechanical properties are Em = (3.52–0.0034T) GPa, ρm = 1150 kg/m3, νm = 0.34 and αm = 45(1 + 0.0005ΔT) × 10−6 K−1. The zigzag (referred to as 0-ply) graphene sheets with effective thickness hG = 0.188 nm and ρG = 4118 kg/m3 are selected as reinforcements, whose temperature-dependent material properties are listed in Table 1 [48]. The efficiency parameters corresponding to different temperatures and volume fractions and the transverse Young's modulus E33 are both shown in Table 2. In addition, it is assumed that out-plane shear moduli G13 = G23 = 0.5G12. It is assumed that there are totally ten plies in a square laminated plate and each layer has the same thickness of 0.2 mm. The plate has a ratio of width to thickness 10. We vary the volume fraction of graphene for each layer to achieve the functionally graded variation in the plate thickness direction. For example, the graphene volume fractions of the total ten layers along the thickness

(32c)

j=1

F (x , y, τ , ε ) =

5. Results and discussion

(32b)

∑ ε jψyj (x , y, τ )

∑ ε jλj (x , y, τ ) (32e)

j=1

where ε is a small perturbation parameter without any physical meaning and τ = εt is introduced to simplify perturbation procedure for solving the nonlinear dynamic response problem. Substituting Eqs. (32a)–(32e) into Eqs. (28a)–(28d), we obtain a set of perturbation equations for the different orders of ε. After solving the equations order by order, we finally obtain the asymptotic solutions up to the third order of ε: (1) (3) W (x , y, t ) = ε [A11 (t )sin mx sin ny] + ε 3 [A11 (t )sin mx sin my (3) (3) + A31 (t )sin 3 mx sin ny + A13 (t )sin mx sin 3 ny] + O (ε 4 )

Table 1 Temperature-dependent material properties for graphene nanosheet (aG = 14.76 nm, G bG = 14.77 nm, thickness hG = 0.188 nm, ν12 = 0.177, ρG = 4118 kg/m3) [48].

(33a) (0) 2 (0) 2 (1) (t ) + F (x , y, t ) = −B00 y /2−b00 x /2 + ε [B11

(3) B¨11 (t )]sin

mx sin ny

(2) 2 (2) 2 (2) y /2−b00 x /2 + B02 (t )cos 2 ny + ε 2 [−B00 (2) (3) (t )cos 2 mx ] + ε 3 [B31 (t )sin 3 mx sin ny + B20 (3) (t )sin mx sin 3 ny] + O (ε 4 ) + B13

(36)

A forth order Runge-Kutta numerical method is utilized to solve Eqs. (35) and (36). All symbols used in Eqs. (34)–(36) will be described in Appendix.

(32a)

j=1

Ψy (x , y, τ , ε ) =

¨ i = gi (W i − Wm)3/2 W

∑ ε jwj (x , y, τ ) j=1

3 d 2 (Wm) d (Wm) + gc + g1 (Wm) + g2 (Wm)2 + g3 (Wm)3 = gq (W i − Wm) 2 dt 2 dt (35)

After the process of non-dimensionalization, Eq. (24) can be rewritten as

(31b)

A two-step perturbation technique developed by Shen [58] is used to solve Eqs. (28a)–(28d). For the immovable boundary condition, the dimensionless initial deflection W ∗is caused by thermal bending moments and can be obtained by solving static equations of the plate [58] in thermal environment. Expanding the additional solutions, we have

W (x , y, τ , ε ) =

(34)

Substituting Eq. (34) into Eq. (33e) and applying the Galerkin procedure, we have

∂Ψy ⎞ ∂ 2F ∂Ψ + γ24 ⎛γ220 x + γ522 β x ∂y ⎠ ∂y 2 ∂ ⎝ ⎜

(33e)

Note that τ has been replaced back by t in the above solutions. (1) In Eq. (33a), (εA11 ) is taken as the second perturbation parameter relating to the dimensionless deflection W. From Eq. (33a), taking (x, y) = (π/2 m, π/2n) yields

⎟

⎜

2

(1) + 3g3 Φ (εA11 (t )) sin mx sin ny

2 2 ⎡γ 2 β 2 ∂ F − γ ∂ F + γ ⎛γ ∂Ψx + γ β ∂Ψy ⎞ 5 24 511 233 ⎢ 24 ∂y 2 ∂y ⎠ ∂ ∂x x2 ⎝ ⎣ ⎜

(1)

(1) λq (x , y, t ) = ε [g1 A11 (t ) + g0 εA¨11 (t ) + gc εA˙ 11 (t )]sin mx sin ny

In Eqs. (30c) and (30f), the detailed expressions of δx and δ y can be, respectively, written as

1 δx = 4π 2β 2γ24

(33d)

(33b) 188

T (K)

G E11 (GPa)

G E22 (GPa)

G G12 (GPa)

G α11 (× 10−6/K)

G α22 (× 10−6/K)

300 400 500

1812 1769 1748

1870 1763 1735

683 691 700

−0.90 −0.35 −0.08

−0.95 −0.40 −0.08

Composites Part B 144 (2018) 184–194

Y. Fan et al.

Table 2 Temperature-dependent efficiency parameters and transverse Young's modulus E33 for graphene/PMMA nanocomposites. T (K)

VG

η1

η2

η3

E33 (GPa)a

300

0.03 0.05 0.07 0.09 0.11 0.03 0.05 0.07 0.09 0.11 0.03 0.05 0.07 0.09 0.11

2.929 3.068 3.013 2.647 2.311 2.977 3.128 3.060 2.701 2.405 3.388 3.544 3.462 3.058 2.736

2.855 2.962 2.966 2.609 2.260 2.896 3.023 3.027 2.603 2.337 3.382 3.414 3.339 2.936 2.665

11.842 15.944 23.575 32.816 33.125 13.928 15.229 22.588 28.869 29.527 16.712 16.018 23.428 29.754 30.773

3.401 3.663 4.380 5.182 5.817 2.634 2.783 3.407 4.271 4.802 1.872 1.966 3.160 3.226 3.321

400

500

a

The values of E33 come from Lin et al. [48].

12

Contact force (kN)

Present FEM [59] 9

6

3

0

0

1

2

3

4 Fig. 4. Effect of FG distribution on low-velocity impact response of GRC laminated plate: (a) Contact force; (b) Central deflection.

Time (ms) Fig. 2. Comparisons of contact force for the composite laminated plate under low-velocity impact load.

direction are [0.11/0.09/0.07/0.05/0.03]s for FG-X, [0.03/0.05/0.07/ 0.09/0.11]s for FG-O, [(0.11)2/(0.09)2/(0.07)2/(0.05)2/(0.03)2] for FGV and [(0.03)2/(0.05)2/(0.07)2/(0.09)2/(0.11)2] for FG-Λ. For UD-GRC plate, the volume fraction of graphene for each layer is assumed to be fixed and equal to 0.07. Note that in all following numerical examples, only the cross-ply stacking sequence [0/90]5T is used unless otherwise stated. 5.1. Validation To validate the effectiveness and accuracy of present method on the low-velocity impact analysis, two examples for a composite laminated plate and an FG-CNTRC plate are re-calculated in this section. The first example is the comparison of contact force history during the process of low-velocity impact on a rectangular composite laminated plate. The plate is simply supported and the geometry of the plate is 127 × 76.2 × 4.65 mm3. The material properties of the composite are E11 = 129 GPa, E22 = 7.5 GPa, G12 = 3.5 GPa, ν12 = 0.33 and ρ = 1540 kg/m3. The stack sequence of the laminated plate is [45/90/45/0]3s. The specification of the impactor is Ei = 207 GPa, νi = 0.3, Mi = 6.18 kg and Ri = 6.35 mm. The initial impact velocity of the impactor is 1.76 m/s. Note that the definition of contact stiffness used in this example is the same as that in Vaziri et al. [59]. As shown in Fig. 2, the present results of contact force are slightly above the existing FEM

Fig. 3. Comparisons of contact force for the FG-CNTRC plate under low-velocity impact load.

189

Composites Part B 144 (2018) 184–194

Y. Fan et al.

Fig. 5. Effect of temperature change on low-velocity impact response of FG-X GRC laminated plate: (a) Contact force; (b) Central deflection.

Fig. 6. Effect of initial impact velocity on low-velocity impact response of the FG-X GRC laminated plate: (a) Contact force; (b) Central deflection.

results [59]. Fig. 3 shows another example for validation on the analysis of lowvelocity impact. In this example, a square plate is constitutive of CNTRC and three distributions, FG-X, FG-Λ and UD, of CNTs are considered. The plate has a thickness of 20 mm. The ratio of width to thickness is 10 and the detailed material properties of CNT and matrix can be found in Ebrahimi and Habibi [60]. The impactor is made of steel, whose modulus, Poisson's ratio, mass density and radius are 207 GPa, 0.3, 7960 kg/m3 and 6.35 mm, respectively. The plate is impacted at the central point by the impactor with an initial impact velocity 3 m/s. It is worth noting that the impactor used in the following parametric examples is the same as the one in this example, except that the diameter of the impactor is assumed to be equal to the thickness of the plate studied in the following parametric examples. It is observed in Fig. 3 that there is a good agreement between results obtained by the present method and the FEM method in Ebrahimi and Habibi [60].

for FG-O or FG- Λ plates is the lowest. Because the plates with FG-X or FG-V distribution have the largest transverse modulus at the contact surface while the transverse modulus of FG-O or FG-Λ plates at the contact surface is the lowest, we can conclude that the peak contact force mainly depends on the transverse modulus at the contact surface of the plate. As shown in Fig. 4(b), the FG-X GRC laminated plate experiences the lowest central deflection although it is subjected to the highest peak contact force. Due to its better performance against deflection, only the FG-X plate is taken into consideration in the following numerical examples. Figs. 5(a) and (b) illustrate the contact force between the impactor and the plate and the central deflection of the plate under different ambient temperatures 300 K, 400 K and 500 K, respectively. As the transverse modulus of the plate is decreased when temperature rises, the peak contact force is reduced as shown in Fig. 5(a). However, it is found in Fig. 5(b) that the central deflection of the FG-X GRC laminated plate is increased as the temperature increases. Fig. 6 presents the low-velocity impact behaviours of the FG-X GRC laminated plate under different initial impact velocities. We choose three velocities 3 m/s, 5 m/s and 7 m/s in this study. As expected, both contact force and central deflection are increased as the initial impact velocity increases. Fig. 7 illustrates the effect of visco-elastic foundation on the low-

5.2. Parametric study Fig. 4 depicts the comparison of low-velocity impact response of GRC laminated plates with four FG distributions (i.e. FG-V, FG-Λ, FG-X and FG-O). Besides, the UD-GRC laminated plate is also taken into consideration as a comparator. It can be seen from Fig. 4(a) that the peak contact force for FG-X or FG-V plates is the highest while the one 190

Composites Part B 144 (2018) 184–194

Y. Fan et al.

Fig. 7. Effect of visco-elastic foundation on low-velocity impact response of the FG-X GRC laminated plate: (a) Contact force; (b) Central deflection.

Fig. 8. Effect of initial uniaxial loads on low-velocity impact response of the FG-X GRC laminated plate: (a) Contact force; (b) Central deflection.

velocity impact response of the FG-X GRC laminated plate. The foundation stiffness are (k1, k2) = (1000, 100) for the Pasternak foundation, (k1, k2) = (1000, 100) with Cd = 1 or 2 for the visco-Pasternak foundation and (k1, k2) = (0, 0) with Cd = 0 for the plate without foundation. As shown in Fig. 7(a), the presence of the foundation has an effect on the contact force history. However, as observed in Fig. 7(b) the central deflection of the FG-X GRC laminated plate is decreased remarkably by increasing the stiffnesses or damping coefficients of the foundation. Unlike in the aforementioned numerical examples, an FG-X GRC laminated plate, whose four edges are all assumed to be in-plane movable, with [0/90/0/90/0]s lay-up is used in Fig. 8. In addition, we apply the initially unidirectional in-plane compressive and tensile loads on two opposite edges of the plate. Three in-plane loads Px/Pcr = −0.5, 0 and 0.5 are taken into account, where Pcr is the critical buckling load for the same plate. As shown in Fig. 8(a), the presence of the in-plane load has almost no effect on the peak contact force. However, the compressive load extends the time history of contact process while the time history is shortened when the tensile load is applied. It also can be found in Fig. 8(b) that the central deflection of the plate increases when an in-plane tensile load is applied while the central deflection is

decreased when an in-plane compressive load is applied.

6. Conclusions An analysis on the low-velocity impact response of FG-GRC laminated plates resting on a visco-elastic foundation in thermal environment has been presented. An extended Halpin-Tsai model is used to estimate the material properties of GRCs and a modified Hertz model is employed to analyze the contact process. The governing equations of the plate are based on a higher order plate theory with von Kármán geometric nonlinearity. The parametric studies on the effect of temperature, foundation stiffness, FG distribution, initial in-plane load and impact velocity have been carried out. The numerical results reveal that the effects of temperature change, initial impact velocity, foundation stiffness and initial in-plane load are all have a significant impact on the central deflection of FG-GRC laminated plates. It is also observed that the initial in-plane load and the foundation stiffness have a negligible effect on the contact force. Among the four FG distributions, FG-X plate performs the best against deflection in the process of impact. 191

Composites Part B 144 (2018) 184–194

Y. Fan et al.

Acknowledgments

Foundation of China (Grant 51779138), and the Australian Research Council (Grant DP140104156). The authors are very grateful for these financial supports.

This study was supported by the National Natural Science Appendix In Eqs. (34)–(36), (0) β 2B00 = γT 1 ΔT ,

m2 + γ5 n2β2

1

(2) β 2B00 = − 8 γ24 1

(2) b00 = − 8 γ24

(3) A11 =

(3) A13 = (3) A31 =

(0) b00 = γT 2 ΔT , (1) 2 (A11 ) ,

2 − γ2 γ24 5

2 n2β2 γ5 m2 + γ24 2 − γ2 γ24 5

(1) 2 (A11 ),

(2) 2 (2) 2 (2) (2) γ14 β 2 (B00 m + b00 n ) − 2γ14 m2n2β 2 (B20 + B02 ) ∗ (1) (A11 + A11 ), Q11 (2) 2γ14 m2n2β2B02

Q13 (2) 2γ14 m2n2β2B20

Q31

(1) ∗ (A11 + A11 ), (1) ∗ (A11 + A11 ),

g

(1) (1) B11 = γ24 g05 A11 , 06

(2) B20 =

(2) B02 =

γ24 n2β2 32m2γ6

γ24 n2β2

(1) 2 (A11 ) +

16m2γ6

∗ (1) A11 A11

γ24 m2 γ m2 ∗ (1) (1) 2 (A11 ) + 242 2 A11 A11 , 2 2 32n β γ7 16n β γ7 g

(3) (3) , B11 = γ24 g05 A11 06

∗

g (1) (3) B¨11 = −γ24 g05 A¨11 , 06

g

(3) (3) B13 = γ24 g135 A13 , 136

g

(3) (3) B31 = γ24 g315 A31 , 316

(

g g

g04

(1) C11 = m γ14 γ24 g02 g05 − (2) C20 =−

g00

00 06

8γ14 γ220 m3

( = m( γ γ

g g

( = m(

g

g04 g00

00 06

g132 g135 14 24 g g 130 136

∗ g04

g00

g310

g02 g05 ∗ g00 g06

00 06

8γ14 γ233 n3β3 γ41 + 4γ432 n2β2

( = nβ(γ γ

g g

( = nβ(

g

g03 g00

00 06

g131 g135 14 24 g g 130 136

g03 g00

−

∗ g03

g00

(1) 11

)A

(1) 11 ,

)A , )A , (3) 11

g133 g130

g

g313

310 316

g310

(3) D31 = nβ γ14 γ24 g311 g315 − (1) D¨ 11

(3) 31

(2) B02 ,

(3) D11 = nβ γ14 γ24 g01 g05 − (3) D13

)A , )A¨ ,

g314

g g

(3) 13

g130

g

(1) D11 = nβ γ14 γ24 g01 g05 − (2) D02 =−

(3) 11

310 316

− γ14 γ24

(

)A , )A ,

g134

−

(3) C31 = m γ14 γ24 g312 g315 − (3) C¨11

(1) 11 ,

(2) B20 ,

γ31 + 4γ320 m2

(3) C11 = m γ14 γ24 g02 g05 − (3) C13

)A

∗

(3) 13

)A

(3) 31 ,

)

g g (1) − γ14 γ24 g01 g05 A¨11 , 00 06

Θ3 = α313 + α331,

192

Composites Part B 144 (2018) 184–194

Y. Fan et al.

g0 = −γ170 + γ171 (m2 + n2β 2) + γ80 ⎡γ14 γ24 ⎢ ⎣ ∗ − g08 − γ14 γ24

∗ g07 g05

g06

(m2g02 + n2β 2g01) g05 g00 g06

−

m2g04 + n2β 2g03 ⎤ ⎥ g00 ⎦

,

g2 = g20 + g02,

g3 =

2 m4 + 2m2n2β 2γ5 + n4β 4γ24 n2β 2 1 m4 ⎞, γ14 γ24 ⎛⎜ + +2 ⎟ 2 2 16 γ γ γ − γ 7 6 24 5 ⎠ ⎝

gc = Cd, gq =

∗ ∗ ∗ ∗ 1/8 4a2K c [D11 D22 A11 A22 ] m n sin ⎛ π ⎞ sin ⎛ π ⎞, ∗ π 4D11 ⎝ 2 ⎠ ⎝2 ⎠

gi = −

∗ ∗ ∗ ∗ 1/8 a2K c ρ0 [D11 D22 A11 A22 ] π 2ME0

For different boundary conditions

g1 = g08 + γ14 γ24

g05 g07 g06

+ K1 + K2 (m2 + n2β 2) − γ14 (m2γT 1 + n2β 2γT 2) ΔT (immovable edges )

g g P g1 = ⎡g08 + γ14 γ24 05 07 + K1 + K2 (m2 + n2β 2) ⎤ ⎡1 − x ⎤ (movable edges ) ⎢ ⎥⎢ g P cr ⎥ ⎦ 06 ⎣ ⎦⎣ In the above equations (with others are defined as in Shen [58]) 1

( γ γ mnβ(

g20 = 2 γ14 γ24 m2n2β 2 g02 =

α313 =

1 2 14 24

γ8 γ6

2 2 2 γ9 γ7

n4β 4

γ14 γ24 , 16Q13 γ6

g

) ),

+ 2 g05 , 06

g05

+ 2g

α331 =

06

γ14 γ24 n4β 4 16Q31 γ7

References [1] Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SV, et al. Electric filed effect in atomically thin carbon films. Science 2004;306:666–9. [2] Suh J, Bae D. Mechanical properties of polytetrafluoroethylene composites reinforced with graphene nanoplatelets by solid-state processing. Compos Part B Eng 2016;95:317–23. [3] Choi BK, Yoon GH, Lee S. Molecular dynamics studies of CNT-reinforced aluminum composites under uniaxial tensile loading. Compos Part B Eng 2016;91:119–25. [4] Geim AK, Novoselov KS. The rise of graphene. Nat Mater 2007;6:183–91. [5] Bunch JS, Verbridge SS, Alden JS, van der Zande AM, Parpia JM, Craighead HG, et al. Impermeable atomic membranes from graphene sheets. Nano Lett 2008;8:2458–62. [6] Lee C, Wei X, Kysar JW, Hone J. Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science 2008;321:385–8. [7] Bolotin KI, Sikes KJ, Jiang Z, Klima M, Fudenberg G, Hone J, et al. Ultrahigh electron mobility in suspended graphene. Solid State Commum 2008;146:351–5. [8] Reddy CD, Rajendran S, Liew KM. Equilibrium configuration and continuum elastic properties of finite sized graphene. Nanotechnology 2006;17:864–70. [9] Rafiee MA, Rafiee J, Wang Z, Song H, Yu Z-Z, Koratkar N. Enhanced mechanical properties of nanocomposites at low graphene content. ACS Nano 2009;3:3884–90. [10] Liang J-Z, Du Q, Tsui GCP, Tang C-Y. Tensile properties of graphene nano-platelets reinforced polypropylene composites. Compos Part B Eng 2016;95:166–71. [11] Kuilla T, Bhadra S, Yao D, Kim NH, Bose S, Lee JH. Recent advances in graphene based polymer composites. Prog Polym Sci 2010;35:1350–75. [12] Wang F, Zhang Y, Zhang BB, Hong RY, Kumar MR, Xie CR. Enhanced electrical conductivity and mechanical properties of ABS/EPDM composites filled with graphene. Compos Part B Eng 2015;83:66–74. [13] Layek RK, Nandi AK. A review on synthesis and properties of polymer functionalized graphene. Polymer 2013;54:5087–103. [14] Hsiao MC, Liao SH, Yen MY, Liu PI, Pu NW, Wang CA, et al. Preparation of covalently functionalized graphene using residual oxygen-containing functional groups. ACS Appl Mater Interfaces 2010;2:3092–9. [15] Yuan B, Bao C, Song L, Hong N, Liew KM, Hu Y. Preparation of functionalized graphene oxide/polypropylene nanocomposite with significantly improved thermal stability and studies on the crystallization behavior and mechanical properties. Chem Eng J 2014;237:411–20. [16] Milani MA, Gonzalez D, Quijada R, Basso NRS, Cerrada ML, Azambuja DS, et al.

[17] [18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

193

Polypropylene/graphene nanosheet nanocomposites by in situ polymerization: synthesis, characterization and fundamental properties. Compos Sci Technol 2013;84:1–7. Shen H-S. Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments. Compos Struct 2009;91:9–19. Yang B, Kitipornchai S, Yang Y-F, Yang J. 3D thermos-mechanical bending solution of functionally graded graphene reinforced circular and annular plates. Appl Math Model 2017;49:69–86. Yang B, Yang J, Kitipornchai S. Thermoelastic analysis of functionally graded graphene reinforced rectangular plates based on 3D elasticity. Meccanica 2017;52:2275–92. Yang B, Mei J, Chen D, Yu F, Yang J. 3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates. Compos Struct 2018;184:1040–8. Feng C, Kitipornchai S, Yang J. Nonlinear free vibration of functionally graded polymer composite beams reinforced with graphene nanoplatelets (GPLs). Eng Struct 2017;140:110–9. Song M, Kitipornchai S, Yang J. Free and forced vibrations of functionally graded polymer composite plates reinforced with graphene nanoplatelets. Compos Struct 2017;159:579–88. Song M, Yang J, Kitipornchai S. Bending and buckling analyses of functionally graded polymer composite plates reinforced with graphene nanoplatelets. Compos Part B Eng 2018;134:106–13. Feng C, Kitipornchai S, Yang J. Nonlinear bending of polymer nanocomposite beams reinforced with non-uniformly distributed graphene platelets (GPLs). Compos Part B Eng 2017;110:132–40. Zhao Z, Feng C, Wang Y, Yang J. Bending and vibration analysis of functionally graded trapezoidal nanocomposite plates reinforced with graphene nanoplatelets (GPLs). Compos Struct 2017;180:799–808. Wu H, Yang J, Kitipornchai S. Dynamic instability of functionally graded multilayer graphene nanocomposite beams in thermal environment. Compos Struct 2017;162:244–54. Yang J, Wu H, Kitipornchai S. Buckling and postbuckling of functionally graded multilayer graphene platelet-reinforced composite beams. Compos Struct 2017;161:111–8. Song M, Yang J, Kitipornchai S, Zhu W. Buckling and postbuckling of biaxially compressed functionally graded multilayer graphene nanoplatelet-reinforced polymer composite plates. Int J Mech Sci 2017;131–132:345–55. Wu H, Kitipornchai S, Yang J. Thermal buckling and postbuckling of functionally

Composites Part B 144 (2018) 184–194

Y. Fan et al.

rectangular plates. Compos Struct 2017;180:760–71. [44] Sahmani S, Aghdam MM. Nonlinear instability of axially loaded functionally graded multilayer graphene platelet-reinforced nanoshells based on nonlocal strain gradient elasticity theory. Int J Mech Sci 2017;131–132:95–106. [45] Mirzaei M, Kiani Y. Isogeometric thermal buckling analysis of temperature dependent FG graphene reinforced laminated plates using NURBS formulation. Compos Struct 2017;180:606–16. [46] Kiani Y, Mirzaei M. Enhancement of non-linear thermal stability of temperature dependent laminated beams with graphene reinforcements. Compos Struct 2018;186:114–22. [47] Halpin JC, Kardos JL. The Halpin-Tsai equations: a review. Polym Eng Sci 1976;16:344–52. [48] Lin F, Xiang Y, Shen H-S. Temperature dependent mechanical properties of graphene reinforced polymer nanocomposites - a molecular dynamics simulation. Compos Part B Eng 2017;111:261–9. [49] Schapery RA. Thermal expansion coefficients of composite materials based on energy principles. J Compos Mater 1968;2:380–404. [50] Abrate S. Impact engineering of composite structures. Springer Press; 2011. [51] Kistler LS, Waas AM. Experiment and analysis on the response of curved laminated composite panels subjected to low velocity impact. Int J Impact Eng 1998;21:711–36. [52] Fan Y, Wang H. Nolinear low-velocity impact analysis of matrix cracked hybrid laminated plates containing CNTRC layers resting on visco-Pasternak foundation. Compos Part B Eng 2017;117:9–19. [53] Fan Y, Wang H. Nolinear low-velocity impact on damped and matrix-cracked hybrid laminated beams containing carbon nanotube reinforced composite layers. Nonlinear Dyn 2017;89:1863–76. [54] Setoodeh AR, Malekzadeh P, Nikbin K. Low velocity impact analysis of laminated composite plates using a 3D elasticity based layerwise FEM. Mater Des 2009;30:3795–801. [55] Sun CT, Chen JK. On the impact of initially stressed composite laminates. J Comput Math 1985;19:490–504. [56] Reddy JN. A refined nonlinear theory of plates with transverse shear deformation. Int J Solids Struct 1984;20:881–96. [57] Shen H-S. Kármán-type equations for a higher-order shear deformation plate theory and its use in the thermal postbuckling analysis. Appl Math Mech 1997;18:1137–52. [58] Shen H-S. A two-step perturbation method in nonlinear analysis of beams, plates and shells. Singapore: John Wiley & Sons Inc; 2013. [59] Vaziri R, Quan X, Olson MD. Impact analysis of laminated composite plates and shells by super finite elements. Int J Impact Eng 1996;18:765–82. [60] Ebrahimi F, Habibi S. Low-velocity impact response of laminated FG-CNT reinforced composite plates in thermal environment. Adv Nano Res 2017;5:69–97.

graded graphene nanocomposite plates. Mater Des 2017;132:430–41. [30] Shen H-S, Lin F, Xiang Y. Nonlinear vibration of functionally graded graphenereinforced composite laminated beams resting on elastic foundations in thermal environments. Nonlinear Dyn 2017;90:899–914. [31] Shen H-S, Xiang Y, Lin F. Nonlinear vibration of functionally graded graphenereinforced composite laminated plates in thermal environments. Comput Meth Appl Mech Eng 2017;319:175–93. [32] Shen H-S, Xiang Y, Fan Y. Nonlinear vibration of functionally graded graphenereinforced composite laminated cylindrical shells in thermal environments. Compos Struct 2017;182:447–56. [33] Shen H-S, Xiang Y, Fan Y, Hui D. Nonlinear vibration of functionally graded graphene-reinforced composite laminated cylindrical panels resting on elastic foundations in thermal environments. Compos Part B Eng 2018;136:177–86. [34] Shen H-S, Xiang Y, Lin F. Nonlinear bending of functionally graded graphene-reinforced composite laminated plates resting on elastic foundations in thermal environments. Compos Struct 2017;170:80–90. [35] Shen H-S, Xiang Y, Fan Y, Hui D. Nonlinear bending analysis of FG-GRC laminated cylindrical panels on elastic foundations in thermal environments. Compos Part B Eng 2018;141:148–57. [36] Shen H-S, Lin F, Xiang Y. Nonlinear bending and thermal postbuckling of functionally graded graphene-reinforced composite laminated beams resting on elastic foundations. Eng Struct 2017;140:89–97. [37] Shen H-S, Xiang Y, Lin F, Hui D. Buckling and postbuckling of functionally graded graphene-reinforced composite laminated plates in thermal environments. Compos Part B Eng 2017;119:67–78. [38] Shen H-S, Xiang Y, Lin F. Thermal buckling and postbuckling of functionally graded graphene-reinforced composite laminated plates resting on elastic foundations. Thin-Walled Struct 2017;118:229–37. [39] Yu Y, Shen H-S, Wang H, Hui D. Postbuckling of sandwich plates with graphenereinforced composite face sheets in thermal environments. Compos Part B Eng 2018;135:72–83. [40] Shen H-S, Xiang Y. Postbuckling behavior of functionally graded graphene-reinforced composite laminated cylindrical shells under axial compression in thermal environments. Comput Meth Appl Mech Eng 2018;330:64–82. [41] Shen H-S, Xiang Y. Postbuckling of functionally graded graphene-reinforced composite laminated cylindrical shells subjected to external pressure in thermal environments. Thin-Walled Struct 2018;124:151–60. [42] Shen H-S, Xiang Y, Fan Y. Postbuckling of functionally graded graphene-reinforced composite laminated cylindrical panels under axial compression in thermal environments. Int J Mech Sci 2018;135:398–409. [43] Gholami R, Ansari R. Large deflection geometrically nonlinear analysis of functionally graded multilayer graphene platelet-reinforced polymer composite

194