Nonlinear low-velocity impact analysis of matrix cracked hybrid laminated plates containing CNTRC layers resting on visco-Pasternak foundation

Accepted Manuscript Nonlinear low-velocity impact analysis of matrix cracked hybrid laminated plates containing CNTRC layers resting on visco-Pasternak foundation Yin Fan, Hai Wang PII:

S1359-8368(17)30091-4

DOI:

10.1016/j.compositesb.2017.02.010

Reference:

JCOMB 4893

To appear in:

Composites Part B

Received Date: 10 January 2017 Revised Date:

4 February 2017

Accepted Date: 8 February 2017

Please cite this article as: Fan Y, Wang H, Nonlinear low-velocity impact analysis of matrix cracked hybrid laminated plates containing CNTRC layers resting on visco-Pasternak foundation, Composites Part B (2017), doi: 10.1016/j.compositesb.2017.02.010. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Nonlinear low-velocity impact analysis of matrix cracked hybrid laminated plates containing CNTRC layers resting on

Yin Fan, Hai Wang*

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visco-Pasternak foundation

School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

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Abstract:

This paper investigates the low-velocity impact response of a shear deformable laminated

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plate which contains both carbon nanotube reinforced composite (CNTRC) layers and fiber reinforced composite (FRC) layers. The effect of matrix cracks is considered and a refined self-consistent model (SCM) is selected to describe the degraded stiffness of the plate. The material properties of both FRC layers and CNTRC layers are assumed to be temperature-dependent. The plate rests on a visco-Pasternak foundation in thermal environments. A modified Hertz model is utilized to describe the contact force between the

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impactor and the plate. Based on a higher order shear deformation theory and von Kármán nonlinear strain-displacement relationships, the motion equations of the plate are established and solved by means of a two-step perturbation approach. The effects of the crack density,

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CNT volume fraction, temperature variation and the foundation stiffness on the nonlinear low-velocity impact response of hybrid laminated plates with multiple matrix cracks are

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discussed in detail.

Keywords: hybrid laminated plates; carbon nanotube reinforced composite; matrix cracks;

nonlinear low-velocity impact; visco-Pasternak foundation

_______________ *Corresponding author. E-mail address: [email protected] (H. Wang)

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1. Introduction Presently, laminated structures made of fiber reinforced composite (FRC) have been widely used in engineering application. During manufacturing or working conditions,

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low-velocity impact may occur for the composite laminates in the presence of thermal environment [1]. Since low-velocity impact on composite laminates may generate the damage such as matrix cracking and delamination, it causes significant safety risks and should be paid

enough attention to. Compared with traditional fiber reinforced composite (FRC),

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functionally graded material (FGM) whose stiffness could be optimized in the direction of thickness may provide more satisfying impact damage resistance [2]. Numerous investigations [2-12] have been conducted on low-velocity impact of FGM stuctures. Among

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these investigations, some are focused on circle plate [8-11] while the other are focused on rectangle ones [3-7]. Khalili et al. [3] studied the low-velocity impact phenomenon of FGM plate based on the classical plate theory. They employed a linearized Hertz contact model to describe the contact behavior between the impactor and the plate and found that the temperature increase made greater horizontal stress in the plate. Shariyat and Nasab [4]

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studied the effects of hierarchical viscoelastic properties of materials on the low-velocity impact response of FGM plates. A modified Hertz contact law was used in their paper and the governing equations were built based on an explicit shear-bending decomposition theory and solved by a differential quadrature method (DQM). An investigation on the low-velocity

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impact response of FGM plates resting on a three-parameter elastic foundation was carried out by Najafi et al. [5]. In their study, a higher order shear deformation theory was used to

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establish the motion equations of the plate and a modified Hertz model was utilized to simulate the contact process. Shariyat and Farzan [6, 7] also employed a refined contact law to analyze the low-velocity impact of a rectangular plate with the impact point was not at the center of the plate. In their research, the first order shear deformation plate theory was employed and the effect of pre-stresses and the elastic foundation stiffnesses on the low-velocity impact was detailed discussed. Kiani et al. [12] studied the low-velocity impact response of a thick FGM beam with four kinds of boundary conditions in thermal environment. In addition, they also analyzed the effect of impact position. The conception of functionally graded (FG) can also be applied on carbon nanotube 2

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reinforced composite (CNTRC) [13]. In fact, a lot of research work [13-16] has been implemented on the mechanical behaviors of FG-CNTRC structures. These studies have already been reviewed and summarized by Shen [17]. However, to the best of authors’

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knowledge, only a little literature paid attention to the low-velocity impact response of FG-CNTRC plates. Wang et al. [18] firstly investigated the low-velocity impact response of

FG-CNTRC plates and sandwiches with FG-CNTRC face sheets. Later, only Malekzadeh and Dehbozorgi [19] and Song et al. [20] did similar research works to enrich the study of

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low-velocity impact response of FG-CNTRC plates. It is worth mentioning that the study of low-velocity impact of FG-CNTRC beam was first carried out by Jam and Kiani [21].

As we know, FG-CNTRC structures always possess better mechanical behavior than that

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of those made of traditional FRC. Due to outstanding design performance, the laminated plate is widely used in engineering practice. Hence, in this paper, we only replace some FRC layers by FG-CNTRC layers in the laminated plate and study the low-velocity impact response of this novel hybrid laminated plate. In our previous works [22, 23], we presented two novel FG distributions for CNTRC layers in a laminated structure, one of them named FG-1 has been

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proved to be efficiency for improving the mechanical properties. The plate is assumed to be simply supported and rest on a visco-Pasternak foundation while the impactor is assumed to be an isotropic ball. A modified Hertz model is proposed to describe the contact force between the impactor and the plate. The dynamic equations of the plates are based on a higher

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order shear deformation plate theory and von Kármán stress-strain relationship, and the solutions are derived by a two-step perturbation technique [24]. The effect of matrix cracking

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which is assumed to appear only in FRC layers is also included. Furthermore, the effect of temperature variation is also investigated through a mixed micromechanical model. The numerical results show the effects of matrix cracks and viscous foundation on the contact force and the centre deflection of hybrid laminated plates.

2. Governing Equations A hybrid laminated plate with length a and width b which consists of n plies is considered. Each ply has a constant thickness hp, and the total thickness of the plate is h =

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n×hp. An impactor of Mass mi, initial velocity V0 and radius Ri is impacted to the center point on the surface of the plate whose four edges are all assumed to be simply supported and

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in-plane immovable. The schematic of the impactor and plate is shown in Fig.1.

2.1 Modified contact model

In a case of quasi-static approximation, the Hertz contact law is acceptable [25]. Hence, the total contact force Fc is assumed to be related to the local contact indentation δ(t):

Fc (t ) = K c [δ (t )]r The local contact indentation δ(t) is defined by

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δ (t ) = W i (t ) − W (t )

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(1)

i

(2)

where W (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 exponent 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 [26]. Kc is contact

Kc =

4 ∗ i E R 3

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stiffness and defined by (3)

where Ri is the radius of the impactor, and

 1 − ν iν i 1  E =  +  i Ez   E

−1

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∗

(4)

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where Ei and νi are the Young’s modulus and Poisson’s ratio of the impactor, respectively, and Ez is the transverse Young’s modulus at the surface of the plate, which can be approximated the same as E22. We assume that Ez is not a fixed value during impact process but is dependent

on indentation. A newly definition of Ez can be written as

E z (t ) =

1 tu +δ ( t ) E22 ( Z ) dZ δ (t ) ∫tu

(5)

in which Z = tu is the top surface of the plate. During the unloading phase, the contact force Fc can be defined as

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 δ (t ) − δ 0  Fc (t ) = Fmax    δ max − δ 0 

s

(6)

where Fmax and δmax are the maximum contact force and indentation. The local indentation δ0

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equals to zero when δmax remains below a critical indentation during the loading phase [27]. It shows that exponent s = 2.5 provides a good fit to the experimental data [28].

2.2 Nonlinear dynamics

As shown in Fig.1, a three-dimensional coordinate system (X, Y, Z) is used, in which X,

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Y are, respectively, in the length and width directions of the plate and Z is in the direction of

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the downward normal to the middle surface. Let W ∗ and W be the initial and additional deflections of the plate, respectively, and Ψ x and Ψ y be the mid-plane rotation of the normal about the X axis and Y axis. The plate rests on a visco-elastic foundation and the foundation is assumed to be bonded well with the plate in the large deflection region. The load-displacement

relationship

of

the

foundation

can

be

expressed

by

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p = K1W − K 2∇2W + CdW& , where p is the force per unit area, K1 , K2 and Cd are, respectively, the Winkler foundation stiffness, the shearing layer stiffness of the foundation and the damping coefficient for viscoelastic foundation. Based on a higher order shear deformation plate theory and von-Kármán type nonlinear strain-displacement relationships,

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the motion equations of a shear deformable laminated plate, which includes the plate-foundation interaction and thermal effect, can be expressed by

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~ ~ ~ ~ ~ ~ L11 (W ) − L12 (Ψ x ) − L13 (Ψ y ) + L14 ( F ) − L16 ( M T ) = L (W + W ∗ , F ) ∂Ψ&&y ∂Ψ&& ~ + L17 (W&& ) + I8 ( x + β ) − ( K1W − K 2∇ 2W + CdW& ) + q ∂X ∂y

(7)

~ ~ ~ ~ 1~ L21 ( F ) + L22 (Ψ x ) + L23 (Ψ y ) − L24 (W ) = − L (W + 2W ∗ ,W ) 2

(8)

~ ~ ~ ~ ~ ~ ∂W&& L31 (W ) + L32 (Ψ x ) + L33 (Ψ y ) + L34 ( F ) − L35 ( N T ) − L36 ( S T ) = I 9 + I10Ψ&&x ∂X

(9)

∂W&& ~ ~ ~ ~ + I10Ψ&&y L41 (W ) + L42 (Ψ x ) + L43 (Ψ y ) + L44 ( F ) − L45 ( N T ) − L46 ( S T ) = I 9 ∂Y

(10)

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~ ~ where the linear operators Lij ( ) and the nonlinear operators L ( ) are defined as in [24]. In Eqs. (7) – (10), the superposed dots indicate differentiation with respect to time t . N T ,

temperature change ∆T (X, Y, Z) and defined by

 N xT  T Ny  N xyT 

M xT M yT M xyT

PxT  N hk  Ax     PyT  = ∑ ∫  Ay  (1, Z , Z 2 )∆TdZ i =1 hk −1  Axy  PxyT   k

 PxT   T  Py   PxyT   

In Eq. (11),

 Ax  Q11 Q12     Ay  = − Q12 Q22  Axy  Q16 Q26  

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 S xT   M xT  4  T  T  S y  =  M y  − 3h 2  S xyT   M xyT     

(11)

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while S T is defined by

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M T and P T are the thermal forces, moments and higher order moments caused by the

Q16   c 2 s2   α   Q26   s 2 c 2   11  α Q66  2cs − 2cs   22 

(12)

(13)

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in which α11 and α22 are thermal expansion coefficients of kth layer in longitudinal and transverse directions. Qij is the transformed stiffness coefficient for a single ply. Since we assume that four edges of the plate are all simply supported and in-plane

X = 0, a;

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immovable, the expression of the boundary conditions can be written as

(14a)

M x = Px = 0

(14b)

U =0

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W =Ψy = 0

(14c)

W =Ψx = 0

(14d)

M y = Py = 0

(14e)

V =0

(14f)

Y = 0, b;

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in which U and V are the displacements of the plate in the X direction and Y direction, respectively. In the following analysis, the longitudinal vibration of impactor is neglected. Hence, the

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motion equation of the impactor and the corresponding initial conditions can be written as

&& i (t ) + F (t ) = 0 , W i (0) = 0 , W& i (0) = V miW c 0

(15)

3. Micromechanical models

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It is assumed that for both FRC and CNTRC, the matrix is the same. In micro scale, the only difference for two materials is the reinforcement. From [13] and [29], the properties of

E11 = η1Vr E11r + Vm E m ,

η2 E 22

η3

=

Vr Vm Vr2 E m E 22r + Vm2 E 22r E m − 2vr vm , η + − V V 4 r m E22r Em Vr E 22r + Vm E m

Vr Vm + , G12r G m

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G12

=

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FRC and CNTRC can be gotten together and written as

ν 12 = Vrν 12r + Vmν m

(16)

where E11, E22, G12 and ν12 are elastic moduli, shear modulus and Poisson’s ratio, respectively. The superscript or subscript r denotes reinforcement, while m denotes matrix. V is the volume

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fraction of component (reinforcement or matrix) in materials and the relationship Vr + Vm =1 is satisfied. It is worth noting that efficiency parameters ηi (i=1, 2, 3, 4) introduced here to

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consider the bonding between reinforcement and matrix depend on the type of material. For FRC, η1 = η2 = η3 = η4 = 1, while for CNTRC, η4 = 0 and the values of η1, η2 and η3 will be detailed given in Section 5. Usually, fibers are uniformly distributed in the thickness direction in FRC and the

volume fraction of fiber is independent of position Z. However, the condition is different for CNTs, which can be functionally graded distributed in the thickness direction. Hence, the CNT volume fraction is the function of Z. In the present study, three regular types of FG-CNTRC, i.e. FG-V, FG-Λ and FG-X, are employed for a single layer. The volume

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fractions corresponding to those types can be expressed as (FG-V)

(17a)

t − Z  ∗ VCN VCN = 2 1   t1 − t0 

(FG-Λ)

(17b)

VCN = 2

2Z − t1 − t0 ∗ VCN t1 − t0

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 Z − t0  ∗ VCN VCN = 2  t1 − t0 

(FG-X)

(17c)

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where the subscript CN denotes CNT. Z = t1 and Z = t0 are, respectively, the top surface and

* bottom surface of a CNTRC ply, and VCN depends on the mass densities of CNTs and

* VCN =

wCN + ( ρ

CN

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matrix, and can be written as

wCN ρ ) − ( ρ CN ρ m ) wCN m

(18)

where wCN is the mass fraction of CNTs. It is worth noting that for uniformly distributed * CNTRC ply VCN = VCN .

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At the early age of service, transverse matrix cracking always presents in a FRC laminated structures due to tensile or fatigue loading [30, 31]. The overall elastic moduli of the composite may change if matrix cracks arise. According to [32], the self-consistent model (SCM) for the overall compliance matrices S of matrix-cracked composite can be expressed

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by

S = S0 + ρcrk Λ

(19)

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where the subscript 0 denotes intact composite and the damage parameter ρcrk can be written as

1 4

ρ crk = πρ crk

(20)

in which the detailed expression of the crack density parameter ρcrk is defined as [33]

ρcrk = 4ηl 2

(21)

where η is the number of cracks per unite area and l is the half length of two adjacent cracks. It must be emphasized that the surface layer containing cracks may be regarded as half of a

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layer of the thickness. This means the crack density of a surface layer is twice as much as that of interior layer with the same angle-ply. The coordinate system presented in Fig.1 is different from that used in [32], in which the fiber is aligned in the Z-direction, whereas in the present

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work the fiber is aligned in the X-direction. Accordingly, the 6×6 matrix Λ can be derived under present coordinate system. However, only three nonzero components can be obtained, and for the sake of convenience, they are expressed in terms of compliances Sij of effective medium as

S11S 22 − S122 ( α1 + α 2 ) S11 ( S11S 22 − S122 )( S11S33 − S132 ) ( α1 + α 2 ) S11

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Λ44 =

(22a)

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Λ22 =

Λ66 = S55 S66 where α1 and α2 are roots of

(S11S22 − S122 )α 2 − [S11S44 + 2(S11S23 − S12S13 )]α + S11S33 − S132 = 0

(22b)

(22c)

(23)

These results imply that only three compliance coefficients S22, S44 and S66 are affected

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by the cracks. S66 can be obtained directly from Eqs. (19) and (22c) while Eq. (23) can be solved by Newton-Raphson method as reported in [33]. The remaining unknowns S22 and S44 are then solved from Eqs. (22a) and (22b). Based on the laminated plate theory, the reduced

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compliance matrix S can be expressed as

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 S11   c 4    2 2  S12   s c  S 22   s 4  = 3  S16   2 sc  S   2s 3c  26   2 2  S 66  4s c

2s 2c 2 s4 s 2c 2   1 − 2 s 2c 2 s 2c 2 − s 2 c 2   S11    2s 2c 2 c4 s 2c 2   S12   2 sc ( s 2 − c 2 ) − 2s 3c sc ( s 2 − c 2 )  S 22    2 sc (c 2 − s 2 ) − 2sc 3 sc (c 2 − s 2 )  S66   − 8s 2 c 2 4 s 2c 2 1 − 4s 2 c 2 

(24)

The relationships between stiffness coefficients Qij and compliance coefficients S ij are

Q11 =

S 22 S12 S11 , Q12 = − , Q22 = , 2 2 S11S 22 − S12 S11S 22 − S12 S11S 22 − S122

Q44 =

1 1 1 , Q55 = , Q66 = S 44 S55 S66

(25)

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4. Solution method A two-step perturbation technique [24] is adopted herein to solve the motion equations

are introduced

x =π

(

(

)

∗

)

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obtained in Section 2. For the sake of convenience, the following dimensionless parameters

(

)

W ,W Y a F,F∗ X ∗ , y = π , β = , W ,W = ∗ ∗ ∗ * 1 4 , F , F ∗ = , ∗ 12 [ D11D22 A11 A22 ] a [ D11∗ D22 ] b b

(

)

12

12

4    M x , 2 Px  πt a 3h ( M x , Px ) = 2 ∗  ∗ ∗ * * 1 4 , t = π D11[ D11D22 A11 A22 ] a (γ T 3 , γ T 4 , γ T 6 , γ T 7 ) ∆T = V0 =

T T a 2 ( Ax , Ay )∆T , (γ T 1 , γ T 2 )∆T = 2 , ∗ 12 ρ0 π [ D11∗ D22 ]

E0

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2

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∗ (Ψ x ,Ψ y )  A11∗   D22  A12∗ γ = γ = , , , , γ = − (Ψ x ,Ψ y ) = 24 14 5  ∗   D∗  ∗ ∗ 14 * A11∗ A22 π [ D11∗ D22 ] A22  11   A22 

a

a a 2 ∆T  T T 4 T 4 T  D , D y , 2 Fx , 2 Fy  , ωL = ΩL 2 ∗  x π π hD11  3h 3h 

aV0

ρ0

∗ ∗ π D11∗ D22 A11∗ A22

E0

, γ 170 = −

ρ0 E0

4 E0 ( I 5 I1 − I 4 I 2 ) I1 E0 a 2 , γ 171 = , 2 ∗ 3ρ 0 h 2 I1D11∗ π ρ 0 D11

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 a4  a2 b4  b2  a3    ( K1 , k1 ) = K1  4 ∗ , ( K , k ) = K , C = C , , 2 2 2 2 ∗ d d 3  3  π 3 D11∗  π D11 E0 h   π D11 E0 h 

E0 Qa 4 , = λ q ∗ ∗ 14 ρ 0 D11∗ π 4 D11∗ [ D11∗ D22 A11∗ A22 ]

E0

ρ0

,

(26)

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(γ 80 , γ 90 , γ 10 ) = ( I 8 , I 9 , I10 )

,

where ρ 0 and E0 are herein the reference values of ρ m and Em at the room temperature (T0

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T T T = 300 K). Ax , Dx , Fx , etc., are defined as

 AxT  T  Ay

DxT DTy

n hk  A  FxT  x 3 = −  A  (1, Z , Z )∆T ( Z , T )dZ ∑ T ∫ Fy  k =1 hk −1  y 

(27)

where Ax and Ay are given in Eq. (13). By employing Eq. (26), the boundary condition of Eq. (13) may then be rewritten in the

following dimensionless form x = 0, π;

W =Ψy = 0

(28a)

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M x = Px = 0 π

π

0

0

∫∫

(28b)

2  2 2 ∂2F ∂Ψ y    ∂2F ∂Ψ x ∂ 2W 2 ∂ W    γ β γ γ γ γ β γ γ γ β − + + − +  24 5 24  511 233 24  611 244 ∂y 2 ∂x 2 ∂x ∂y  ∂x 2 ∂y 2   

2  1  ∂W  2 − γ 24   + (γ 24γ T 1 − γ 5γ T 2 ) ∆T  dxdy = 0 2  ∂x  

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(28c)

y = 0, π;

W =Ψx = 0

(28d)

0

0

∫∫

(28e)

2 2  ∂2F ∂Ψ y    ∂Ψ x ∂ 2W 2 ∂ F 2 ∂ W    γ β γ γ γ β γ γ γ β − + + − +  2 5 24  220 522 24  240 622 ∂y 2 ∂x ∂y  ∂x 2 ∂y 2    ∂x

 ∂W 1 − γ 24 β 2  2  ∂y

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π

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M y = Py = 0 π

2    + (γ T 2 − γ 5γ T 1 )∆T  dydx = 0  

  

(28f)

Then the corresponding motion equations of the plate can be expressed as

L11 (W ) − L12 (Ψ x ) − L13 (Ψ y ) + γ 14 L14 ( F ) = γ 14 β 2 L(W + W ∗ , F )

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&& && && ) + γ ( ∂Ψ x + β ∂Ψ y ) − ( K W − K ∇ 2W + C W& ) + λ + L17 (W 80 1 2 d q ∂x ∂y 1 L21 ( F ) + γ 24 L22 (Ψ x ) + γ 24 L23 (Ψ y ) − γ 24 L24 (W ) = − γ 24 β 2 L(W + 2W ∗ , W ) 2

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L31 (W ) + L32 (Ψ x ) − L33 (Ψ y ) + γ 14 L34 ( F ) = γ 90

∂W&& + γ 10Ψ&&x ∂x

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L41 (W ) − L42 (Ψ x ) + L43 (Ψ y ) + γ 14 L44 ( F ) = γ 90 β

&& ∂W + γ 10Ψ&& y ∂y

(29)

(30)

(31)

(32)

Note that W* = 0 at room temperature. Using a two-step perturbation technique, the solutions can be written in the following

forms

W ( x, y,τ , ε ) = ∑ ε j w j ( x, y,τ ) , j =1

F ( x, y,τ , ε ) = ∑ ε j f j ( x, y,τ ) , j =0

Ψ x ( x, y,τ , ε ) = ∑ ε ψ xj ( x, y,τ ) , Ψ y ( x, y,τ , ε ) = ∑ ε jψ yj ( x, y,τ ) j

j =1

j =1

λq = ∑ ε λ j ( x, y,τ ) j

(33)

j =1

11

  

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where ε is a small perturbation parameter without any physical meaning and τ = εt is introduced to improve perturbation procedure for solving nonlinear vibration problem. Using Eqs. (28)-(32), a set of perturbation equations for the different order of ε can be obtained and solved order by order. Finally, we obtain the asymptotic solutions up to the third order of ε:

F =−

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W = εA11(1) sin mx sin ny + ε 3 A31(3) sin 3mx sin ny + ε 3 A13(3) sin mx sin 3ny + O(ε 4 ) (34a) ( 0) (0) B00(0 ) 2 b00( 0 ) 2 &&(3) ] sin mx sin ny − ε 2 ( B00 y 2 + b00 x 2 ) y − x + ε [ B11(1) + B 11 2 2 2 2

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(2) (2) + ε 2 B02 cos 2ny + ε 2 B20 cos 2mx + ε 3 B31(3) sin 3mx cos ny + ε 3 B13(3) sin mx cos 3ny

+ O (ε 4 )

(34b)

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( 2) Ψ x = ε [C11(1) + ε 2C&&11(1) ] cos mx sin ny + ε 2C20 sin2mx + ε 3C31(3) cos 3mx sin ny

+ ε 3C13( 3) cos mx sin 3ny + O (ε 4 )

(34c)

&& (1) ] sin mx cos ny + ε 2 D( 2) sin 2ny + ε 3C (3) sin 3mx cos ny Ψ y = ε [ D11(1) + ε 2 D 11 02 31 + ε 3C13( 3)sinmx cos 3ny + O (ε 4 )

(34d)

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&&(1) + g A& (1) + g A(1) ] sin mx sin ny + [εA(1) ]2 ( g cos 2mx + g cos 2ny) λq = ε [ g 40 A 10 c 10 41 10 10 421 422 + [εA10(1) ]3 g 43 sin mx sin ny + O(ε 4 )

(34e)

We take (x, y) = (π/2, π/2) which means the center point of the plate. Then, the second

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(1) perturbation parameter εA10 in Eq. (34e) can be replaced by the maximum dimensionless

deflection of the plate Wm through Eq. (34a). Applying Galerkin process, Eq. (34e) can be

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rewritten as

g 40

d 2 (Wm ) d (Wm ) 2 3 + gc + g 41 (Wm ) + g 42 (Wm ) + g 43 (Wm ) = g q (W i − Wm ) 2 2 dτ dτ 3

(35)

After the process of non-dimension, Eq. (15) can be rewritten as

W&& i = g i (W i − Wm ) 3 2

(36)

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

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5. Parametric studies and discussion In this section, the numerical results of low-velocity impact response of hybrid plates with various parameters are presented and discussed. The impactor is made of steel with the mechanical properties: Ei = 207 GPa, νi = 0.3 and ρi = 7960 kg/m3. The geometry of the

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impactor is spherical and the radius is Ri = 1mm. Unless otherwise statement, the initial velocity of the impactor is taken to be 3 m/s. The square hybrid laminated plate has 20mm width and 1mm thickness, and each ply in the laminated plate has the same thickness. As is mentioned before, the FRC and CNTRC have the same matrix material, whose mechanical

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properties are assumed to be νm = 0.34, αm = 45(1+0.0005∆T)×10-6 K-1 , Em = (3.52-0.0034T) GPa and ρm = 1150 kg/m3. For FRC, the volume fraction of carbon fiber is fixed at 0.6 and the

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detailed material properties of the fiber are [34]: E11f = 233.05 GPa, E22f = 23.1 GPa, G12f = 8.96 GPa, νf = 0.2, ρf = 1750 kg/m3, α11f = -0.54×10-6 K-1, and α 22f =10.08×10-6 K-1. While for CNTRC, the (10, 10) single walled carbon nanotubes (SWCNTs) are selected to be reinforcements and the temperature dependent material properties are listed in Table 1 [35]. In

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* =0.12, 0.17 and 0.28) of CNTs are considered and computation, three volume fractions ( VCN

the corresponding efficiency parameters are given by * VCN =0.12: η1 =0.137, η 2 =1.022, η 3 =0.715,

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* VCN =0.17: η1 =0.142, η 2 =1.626, η 3 =1.138,

* VCN =0.28: η1 =0.141, η 2 =1.585, η 3 =1.109.

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In addition, we assume that out-plane shear moduli G13 = G12 and G23 = 1.2G12. It has been proved in our previous works [22, 23] that FG-1 grading profile could significantly improve the mechanical properties of plates. Hence, we only take FG-1 into account. For FG-1, FG-V distribution is used for CNTRC layers above the middle plane, while CNTRC layers below the middle plane, FG-Λ distribution is used. If the CNTRC layer is crossed by the middle

plane, we use FG-X distribution for the layer. It is assumed that the matrix cracks occur only in FRC layers and never propagate during the impact process. Unless otherwise statement, the properties of materials constituted of the hybrid plate and assumptions mentioned above are used in the following examples. 13

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5.1. Comparison Studies Before carrying out the parametric studies, we have to test the effectiveness and accuracy of the present method. The comparisons between present results and other existing results on

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low-velocity impact of plate constituted of different kind of material are shown in three validated examples.

Figs. 2 and 3 illustrate contact force for low-velocity impact of isotropic plates. In Fig. 2,

the present results are compared with the results of Wang et al. [18] and Khalili et al. [3]. A

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modified Hertz contact model was employed by Wang et al. [18] while a spring-mass contact model was utilized by Khalili et al. [3]. The material properties of the plate and the impactor

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are the same. The mass density, Young’s modulus and Poisson’s ratio of the impactor are 7971.8 kg/m3, 200 GPa and 0.3, respectively. The radius of the impactor is 10mm and the initial velocity of the impactor is 1m/s. The size of the square plate is 200×200×8 mm3. It can be seen from Fig. 2, the results obtained by present method agree very well with those of Wang et al. [18] and Khalili et al. [3]. The other validated example of an isotropic plate impacted by the spherical impactor with various initial velocities, i.e. 10 m/s, 15 m/s and 20

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m/s, is plotted in Fig. 3. The plate is constituted of high-strength low-alloy (HSLA) steel with length 1000 mm, width 500 mm and thickness 10 mm. The material properties of HSLA steel are 7900 kg/m3 for mass density, 206 GPa for Young’s modulus and 0.3 for Poisson’s ratio.

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The diameter of the impactor is 12.7 mm. The existing results for contact force in Dai and Jiang [36] were acquired by finite difference method on the basis of classical plate theory. In

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this example, the increase of the initial velocity means the larger impact energy, which certainly causes the greater contact force. It can be found that the present results are slightly greater than the results of Dai and Jiang [36] at the stage of loading, but slightly lower than the results of Dai and Jiang [36] at the stage of unloading. For a CNTRC plate, the comparison between the present results and Wang et al. [18] is

depicted in Fig. 4. Except the mass density is 7960 kg/m3, other mechanical properties of the impactor are the same as those used in Fig. 3. The length and width of the plate are both 200 mm and the ratio of length to thickness is 10. The initial velocity of the impactor is 3 m/s. Obviously, the present results are in good agreement with the results of Wang et al. [18]. 14

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5.2. Parametric studies To compare the effect of the material for the outer layers on low-velocity impact

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response of the hybrid plate, two stack sequences [0C/90F/0C/90F/0C] and [90F/0C/0C/0C/90F], where superscripts C and F, respectively, represent CNTRC and FRC, are both taken into account

∗ , is taken to be 0.28. The uniform as shown in Fig. 5. The CNT volume fraction, VCN

distribution (UD) of CNTs is also included as a comparator. It is clearly observed that the

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plate with [0C/90F/0C/90F/0C], where the FG-1 distribution for CNTs is used, has highest

contact force but the lowest deflection. This is because FG-1 CNTRC layer provides the

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highest contact stiffness. No matter which stack sequence is, the plate with FG-1 CNTRC layers may acquire higher overall stiffness but lower center deflection. Kundalwal and his co-authors [37, 38] did the similar research work on hybrid laminated structures with the various lay-ups and they found the remarkable effect of lay-up on dynamic properties of laminated structures, which can also be observed in this example. The plate with [0C/90F/0C/90F/0C] layup and FG-1 grading profile layers is only considered in the following

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examples, due to its best behavior on deflection of the plate in this example. The effect of matrix cracking on contact force and deflection of the hybrid plate is illustrated in Fig. 6. Two matrix crack densities (0.2 and 0.5) are chosen and the CNT volume fraction for each CNTRC layer is 0.17. The FG distribution for CNTRC layers is out of

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consideration in this example. Because the transverse matrix cracks only appear in the interior FRC layers, the contact stiffness between the impactor and the plate is affected hardly, but the

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overall stiffness of the plate is still degraded. It can be found that the presence of matrix cracks can slightly increase the peak value of deflection but has almost no effect on the contact force. Because the effect of matrix cracking on low-velocity impact can be ignored, this factor is not considered in the following examples. Fig. 7 presents the effect of CNT volume fraction (0.12, 0.17 and 0.28) in a single CNTRC layer on low-velocity impact response of the hybrid plate. Obviously, the stiffness of CNTRC layers will be increased with higher volume fraction of CNT reinforcements. As seen the results from this figure, the contact force is increased with improving the CNT volume

15

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fraction. However, under the same condition, the curve of deflection of the plate declines. Fig. 8 shows the low-velocity impact response of the hybrid plate under different temperatures (300K, 400K and 500K). In this example, the volume fraction of CNT is fixed to be 0.17 and

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the initial velocity of the impactor is set to be 9 m/s. In fact, the increase of temperature will cause the stiffness degradation of both the matrix and CNT. That means the stiffness of each

layer in the plate is reduced and it can be seen from Fig. 8 that increasing the temperature may reduce both contact force and plate deflection.

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The effect of various initial velocities (3, 6 and 9 m/s) on contact force and deflection of the hybrid plate is shown in Fig. 9. The CNT volume fraction in each CNTRC layer is taken to be 0.28. As expected, the larger velocity may cause the greater contact force and higher

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curve of deflection. Fig. 10 demonstrates the effect of viscous foundation stiffness on low-velocity impact response of the hybrid plate. The foundation stiffness are (k1, k2) = (1000, 100) for the Pasternak foundation, (k1, k2) = (0, 0) for the plate without elastic foundation, and (k1, k2) = (1000, 100) with Cd = 1 (or 3) for the visco-Pasternak foundation. The CNT volume fraction of each CNTRC layer is the same as the value used in Fig. 9. It can be observed that

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the peak value of the contact force is almost invariable, due to no effect of the foundation on contact stiffness. However, the impact history could be extended by increasing the stiffness of elastic foundation, while the center deflection of the plate may be reduced with the increase of

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foundation stiffness and viscidity.

6. Conclusions

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The nonlinear low-velocity impact response of matrix cracked hybrid plates containing both FRC layers and CNTRC layers resting on a visco-Pasternak foundation is investigated in this paper. The matrix cracking is modeled by a refined SCM and a modified Hertz model is utilized to describe the contact force between the impactor and the plate. The numerical results show the following main conclusions. 1. The results illustrate that the FG-1 distribution of CNTRC layers for laminated plate has a significant influence on the low-velocity impact response. Obviously, the plate possesses a lower deflection when the outer layers are CNTRC. 2. The peak value of deflection for the plate is increased slightly when the matrix cracking is 16

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occurred. However, the presence of matrix cracks plays little or no role on the contact force. 3. The history of the low-velocity impact could be extended because of the foundation. The

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center deflection of the plate is reduced significantly with the increase of foundation stiffness and viscidity.

Based on the above points, it is advised that the surface layers of the hybrid laminated

plate applied in engineering is selected to be CNTRC. The viscous foundation considered in

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this paper is also suggested in practical application.

Acknowledgments

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The authors wish to thank Professor H-S Shen of Shanghai Jiao Tong University for his considerable support.

Appendix

In Eqs. (35) and (36) (with others are defined as in Ref.[24]) ∗ g 05 g 07 − γ 170 + γ 171 (m 2 + n 2 β 2) g 06

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∗ g 40 = − g 08 − γ 14γ 24

 m 2 g 04 + n 2 β 2 g 03 g m 2 g 02 + n 2 β 2 g 01  − γ 80  − γ 14γ 24 05  g 00 g 06 g 00  

g 05 g 07 + K1 + K 2 (m 2 + n 2 β 2 ) + 2G42Φ − γ 14 (γ T 1m 2 + γ T 2 n 2 β 2 )∆T g 06

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g 41 = g 08 + γ 14γ 24

g 42 = −

g 43 =

2

3π

2

 γ8

γ 14γ 24 mnβ 2 

γ6

+

(A.2)

γ9 g  + 4 05 (1 − cos mx)(1 − cos nx) + 3 g 43Φ γ7 g 06 

2 4 4 γ 14γ 24  m 4 n 4 β 4 m 4 + γ 24 n β + 2γ 5 m 2 n 2 β 2  + +2   2 16  γ 7 γ6 (γ 24 − γ 52 ) 

gc = Cd

gq =

(A.1)

(A.3)

(A.4)

(A.5)

∗ ∗ 18 4a 2 K c [ D11∗ D22 A11∗ A22 ] m  n  sin π  sin π  4 ∗ π D11  2  2 

17

(A.6)

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∗ ∗ 18 a 2 K c ρ0[ D11∗ D22 A11∗ A22 ] 2 i π E0 m

G42 = −

 γ8

2 3π

2

γ 14γ 24mnβ 2 

γ6

+

(A.7)

γ9 g  + 4 05 (1 − cos mx)(1 − cos nx) γ7 g 06 

Φ (T ) = λ + Θ2λ2 + Θ2λ2 + L

λ=

16 2 π mnG08

(A.8)

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gi =

(A.9)

 (γ T 3 − γ T 6 )m 2 g102 + (γ T 4 − γ T 7 )n 2 β 2 g101  h∆T 2 2 2 m + n − γ γ β  T3 T4  ∗ ∗ ∗ ∗ 14 g 00   [ D11D22 A11 A22 ]

γ γ 8 g  γ 14γ 24m 2n 2 β 2  8 + 9 + 4 05  3π G08 g 06  γ6 γ7 2

Θ3 = 2Θ22 −

2 4 4  m4 n4 β 4 1 m 4 + γ 24 n β + 2γ 5m 2n 2 β 2   + +2 γ 14γ 24  16G08 γ6 γ 242 − γ 52  γ7 

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Θ2 =

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(A.10)

(A.11)

(A.12)

g101 = (γ 31 + γ 320m 2 + γ 322n 2 β 2 )(γ 131m 2 + γ 133n 2 β 2 ) − γ 331m 2 (γ 120m 2 + γ 122n 2 β 2 )

(A.13)

g102 = (γ 42 + γ 430m 2 + γ 432n 2 β 2 )(γ 120m 2 + γ 122n 2 β 2 ) − γ 331n 2 β 2 (γ 131m 2 + γ 133n 2 β 2 )

(A.14)

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References

g 05 g 07 + K1 + K 2 (m 2 + n 2 β 2 ) − γ 14 (γ T 1m 2 + γ T 2 n 2 β 2 )∆T g 06

(A.15)

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G08 = g 08 + γ 14γ 24

[1] Boroujerdy MS, Kiani Y. Low velocity impact analysis of composite laminated beams

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subjected to multiple impacts in thermal field. ZAMM 2016; 96: 843-56. [2] Yalamanchili VK, Sankar BV. Indentation of functionally graded beams and its application to low-velocity impact response. Compos Sci Technol 2012; 72: 1989-94.

[3] Khalili SMR, Malekzadeh K, Gorgabad AV. Low velocity transverse impact response of functionally graded plates with temperature dependent properties Compos Struct 2013; 96: 64-74. [4] Shariyat M, Farzan F. Low-velocity impact analysis of the hierarchical viscoelastic FGM plates, using an explicit shear-bending decomposition theory and the new DQ method. Compos Struct 2014; 113: 63-73. 18

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[5] Najafi F, Shojaeefard MH, Saeidi H. Nonlinear low-velocity impact response of functionally graded plate with nonlinear three-parameter elastic foundation in thermal field. Compos Part B 2016; 107: 123-40.

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[6] Shariyat M, Farzan F. Nonlinear eccentric low-velocity impact analysis of a highly prestressed FGM rectangular plate, using a refined contact law. Arch Appl Mech 2013; 83: 623-41.

[7] Shariyat M, Farzan F. Eccentric low-velocity impact analysis of transversely graded plates

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with Winkler-type elastic foundations and fully or partially supported edges. Thin-Walled Struct 2014; 84: 112-22

[8] Gunes R, Aydin M, Kemal M, Reddy JN. The elasto-plastic impact analysis of

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functionally graded circular plates under low-velocities. Compos Struct 2011; 93: 860-9. [9] Dai H-L, Guo Z-Y, Yang L. Nonlinear dynamic response of functionally graded materials circular plates subject to low-velocity impact. J Compo Mater 2012; 47: 2797-807. [10] Shariyat M, Jafari R. Nonlinear low-velocity impact response analysis of a radially preloaded two-directional-functionally graded circular plate: A refined contact stiffness

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approach. Compos Part B 2013; 45: 981-94.

[11] Gunes R, Aydin M, Kemal M, Reddy JN. Experimental and numerical investigations of low velocity impact on functionally graded circular plates. Compos Part B 2014; 59: 21-32.

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[12] Kiani Y, Sadighi M, Salami SJ, Eslami MR. Low velocity impact response of thick FGM beams with general boundary conditions in thermal field. Compos Struct 2013; 104:

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293-303.

[13] Shen H-S. Nonlinear bending of functionally graded carbon nanotube reinforced composite plates in thermal environments. Compos Struct 2009; 91: 9-19.

[14] Shen H-S, Xiang Y. Postbuckling of nanotube-reinforced composite cylindrical shells under combined axial and radial mechanical loads in thermal environment. Compos Part B 2013; 52: 311-22. [15] Mehrabadi SJ, Aragh BS, Khoshkhahesh V, Taherpour A. Mechanical buckling of nanocomposite rectangular plate reinforced by aligned and straight single walled carbon nanotubes. Compos Part B 2012; 43: 2031-40. 19

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[16] Wu HL, Yang J, Kitipornchai S. Nonlinear vibration of functionally graded carbon nanotube reinforced composite beams with geometric imperfections. Compos Part B 2016; 90: 86-96.

composite structures: a review. Adv Mech 2016; 46: 201611. [18]

Wang

Z-X,

Xu

J,

Qiao

P.

Nonlinear

low-velocity

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[17] Shen H-S. Modeling and analysis of functionally graded carbon nanotube reinforced

impact

analysis

of

temperature-dependent nanotube-reinforced composite plates. Compos Struct 2014; 108:

SC

423-34.

[19] Malekzadeh P, Dehbozorgi M. Low velocity impact analysis of functionally graded carbon nanotubes reinforced composite skew plates. Compos Struct 2016; 140: 728-48.

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[20] Song ZG, Zhang LW, Liew KM. Dynamic responses of CNT reinforced composite plates subjected to impact loading. Compos Part B 2016; 99: 154-61.

[21] Jam JE, Kiani Y. Low velocity impact response of functionally graded carbon nanotube reinforced composite beams in thermal environment. Compos Struct 2015; 132: 35-43. [22] Fan Y, Wang H. Thermal postbuckling and vibration of postbuckled matrix cracked

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hybrid laminated plates containing carbon nanotube reinforced composite layers on elastic foundation. Compos Struct 2016; 157: 386-97. [23] Fan Y, Wang H. The effects of matrix cracks on the nonlinear bending and thermal postbuckling of shear deformable laminated plates containing carbon nanotube reinforced

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composite layers and piezoelectric fiber reinforced composite layers. Compos Part B 2016; 106: 28-41.

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[24] Shen H-S. A two-step perturbation method in nonlinear analysis of beams, plates and shells. Singapore: John Wiley & Sons Inc; 2013.

[25] Abrate S. Impact engineering of composite structures. Springer Press; 2011. [26] 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.

[27] 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. [28] Sun CT, Chen JK. On the impact of initially stressed composite laminates. J Comput 20

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Math 1985; 19: 490-504. [29] Shen H-S. Hygrothermal effects on the postbuckling of shear deformable laminated plates. Int J Mech Sci 2001; 43: 1259-81.

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[30] Tong J, Guild Fj, Ogin SL, Smith PA. On matrix crack growth in quasi isotropic laminates I. Experimental investigation. Compos Sci Technol 1997; 57: 1527-35.

[31] Varna J, Joffe R, Akshantala NV, Talreja R. Damage in composite laminates with off-axis plies. Compos Sci Technol 1999; 59: 2139-47.

crack systems. Mech Mater 1983; 2: 123-37.

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[32] Laws N, Dvorak GJ, Hejazi M. Stiffness changes in unidirectional composites caused by

[33] Dvorak GJ, Laws N, Hejazi M. Analysis of progressive matrix cracking in composite

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laminates I. Thermoelastic properties of a ply with cracks. J Compos Mater 1985; 19: 216-34.

[34] Bowles DE, Tompkins SS. Prediction of coefficients of thermal expansion for unidirectional composite. J Compos Mater 1989; 23: 370-88.

[35] Shen H-S, Zhang C-L. Thermal buckling and postbuckling behavior of functionally

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graded carbon nanotube-reinfroced composite plates. Mater Des 2010; 31: 3403-11. [36] Dai H-L, Jiang H-J. Nonlinear dynamic analysis for a rectangular HSLA steel plate subjected to low velocity impact. J Vib Control 2016; 22: 4062-73. [37] Kundalwal SI, Kumar RS, Ray MC. Smart damping of laminated fuzzy fiber reinforced

105001.

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composite shells using 1-3 piezoelectric composites. Smart Mater Struct 2013; 22:

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[38] Kundalwal SI, Ray MC. Smart damping of fuzzy fiber reinforced composite plates using 1-3 piezoelectric composites. J Vib control 2016, 22: 1526-46.

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Table 1 Temperature-dependent material properties for (10, 10) SWCNT [35]. (ν 12 = 0.175 , ρCN = CN

1750 kg/m3)

5.6466 5.5679 5.5308

CN (TPa) E 22

7.0800 6.9814 6.9348

-6 -6 CN CN G12CN (TPa) α11 ( × 10 /K) α 22 ( × 10 /K)

1.9445 1.9703 1.9643

3.4584 4.1496 4.5361

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300 400 500

CN (TPa) E11

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T (K)

22

5.1682 5.0905 5.0189

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Fig. 1. Geometry and coordinate system of a hybrid laminated plate.

23

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2000

Present Wang et al. [18] Khalili et al. [3]

1000

500

0.02

0.04

0.06

0.08

Time (ms)

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EP

0 0.00

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Contact Force (N)

1500

Fig. 2. Low-velocity impact response of isotropic plates with four edges simply supported

24

-7

1.0x10

-7

8.0x10

-8

6.0x10

-8

4.0x10

-8

2.0x10

-8

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1.2x10

V0 = 10 m/s (Dai and Jiang [36]) V0 = 15 m/s (Dai and Jiang [36]) V0 = 20 m/s (Dai and Jiang [36]) V0 = 10 m/s (Present) V0 = 15 m/s (Present)

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V0 = 20 m/s (Present)

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Dimensionless Contact Force

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0.0

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0

20

40

60

80

100

Dimensionless Time

Fig. 3. Low-velocity impact response of isotropic plates under various initial impact velocities

25

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1000

Present Wang et al. [18]

600

400

200

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Contact Force (N)

800

UD FG-Λ

EP

FG-X

0.04

0.08

0.12

0.16

Time (ms)

AC C

0 0.00

Fig. 4. Comparisons of impact response for CNTRC plates with both ends simply supported.

26

25 i

20

F

C

F

C

[0 /90 /0 /90 /0 ] & UD C F C F C [0 /90 /0 /90 /0 ] & FG-1 F C C C F [90 /0 /0 /0 /90 ] & UD F C C C F [90 /0 /0 /0 /90 ] & FG-1

a = b, b/h = 20 * = 0.28 VCN

15

10

5

0 0.000

0.005

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Contact Force (N)

C

R = 1 mm, V0 = 3 m/s

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(a)

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0.010

0.015

0.020

(b)

0.015

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Time (ms)

i

R = 1 mm, V0 = 3 m/s a = b, b/h = 20

* = 0.28 VCN

0.009

EP

Centre Deflection (mm)

0.012

AC C

0.006

C

0.003

0.000 0.00

F

C

F

C

[0 /90 /0 /90 /0 ] & UD C F C F C [0 /90 /0 /90 /0 ] & FG-1 F C C C F [90 /0 /0 /0 /90 ] & UD F C C C F [90 /0 /0 /0 /90 ] & FG-1 0.01

0.02

0.03

0.04

Time (ms)

Fig. 5. Low-velocity impact response of hybrid laminated plates with different lay-up: (a) Contact force history and (b) deflection of plate.

27

(a) 18

ρcrk = 0

F

C

F

C

UD, [0 /90 /0 /90 /0 ] i R = 1mm, V0 = 3 m/s

ρcrk = 0.2

15

ρcrk = 0.5

a = b, b/h = 20 ∗ VCN = 0.17

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12

9

6

3

0 0.000

0.005

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Contact Force (N)

C

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0.010

0.015

0.020

0.025

(b) 0.016

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Time (ms)

ρcrk = 0

0.012

ρcrk = 0.5

EP

Centre Deflection (mm)

ρcrk = 0.2

AC C

0.008

C

0.000 0.00

F

C

F

C

UD, [0 /90 /0 /90 /0 ] i R = 1mm, V0 = 3 m/s

0.004

a = b, b/h = 20 ∗ VCN = 0.17 0.01

0.02

0.03

0.04

Time (ms) Fig. 6. The effects of matrix cracks used in outer layers on low-velocity impact response of hybrid plates with both ends simply supported: (a) Contact force history and (b) deflection of plate 28

(a)

24 C

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F

C

F

C

FG-1, [0 /90 /0 /90 /0 ] i R = 1mm, V0 = 3 m/s

20

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16

12

8

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Contact Force (N)

a = b, b/h = 20

∗ VCN = 0.12 ∗ VCN = 0.17 ∗ VCN = 0.28

4

0 0.000

0.005

0.010

0.015

0.020

0.025

(b)

0.018

0.012

∗ VCN = 0.12 ∗ VCN = 0.17 ∗ VCN = 0.28

EP

Contact Force (N)

0.015

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Time (ms)

AC C

0.009

0.006

C

0.003

0.000 0.000

F

C

F

C

FG-1, [0 /90 /0 /90 /0 ] i R = 1mm, V0 = 3 m/s a = b, b/h = 20 0.005

0.010

0.015

0.020

0.025

0.030

0.035

Time (ms)

Fig. 7. The effects of CNT volume fraction on low-velocity impact response of hybrid plates with both ends simply supported: (a) Contact force history and (b) deflection of plate. 29

(a)

80 C

F

C

F

C

FG-1, [0 /90 /0 /90 /0 ] i R = 1mm, V0 = 9m/s a = b, b/h = 20 ∗ VCN = 0.17

SC

60

Contact Force (N)

RI PT

ACCEPTED MANUSCRIPT

20

M AN U

40

T = 300 K T = 400 K T = 500 K

0 0.000

0.005

0.010

0.015

0.020

0.06

(b)

0.04

T = 300 K T = 400 K T = 500 K

EP

Center Deflection (mm)

0.05

TE D

Time (ms)

AC C

0.03

0.02

C

0.01

0.00 0.00

F

C

F

C

FG-1, [0 /90 /0 /90 /0 ] i R = 1mm, V0 = 9m/s a = b, b/h = 20 ∗ VCN = 0.17 0.01

0.02

0.03

0.04

Time (ms)

Fig. 8. The effects of temperature variation on low-velocity impact response of hybrid plates with both ends simply supported: (a) Contact force history and (b) deflection of plate. 30

(a)

V0 = 3 m/s

80

C

C

F

C

FG-1, [0 /90 /0 /90 /0 ] i R = 1mm a = b, b/h = 20 ∗ VCN = 0.28

V0 = 6 m/s V0 = 9 m/s

SC

60

40

M AN U

Contact Force (N)

F

RI PT

ACCEPTED MANUSCRIPT

20

0 0.000

0.004

0.008

0.012

0.016

Time (ms)

0.030

C

F

C

F

C

FG-1, [0 /90 /0 /90 /0 ] i R = 1mm a = b, b/h = 20 ∗ VCN = 0.28

EP

0.025 0.020 0.015

AC C

Center Deflection (mm)

0.035

TE D

(b) 0.040

0.010

V0 = 3 m/s V0 = 6 m/s

0.005

0.000 0.00

V0 = 9 m/s 0.01

0.02

0.03

0.04

Time (ms)

Fig. 9. The effects of initial velocity on low-velocity impact response of hybrid plates with both ends simply supported: (a) Contact force history and (b) deflection of plate.

31

24 C

C

F

C

16

SC

Contact Force (N)

20

F

FG-1, [0 /90 /0 /90 /0 ] i R = 1 mm a = b, b/h = 20 * VCN = 0.28

12

8

M AN U

(a)

RI PT

ACCEPTED MANUSCRIPT

without elastic foundation With Pasternak foundation with visco-Pasternak foundation (Cd = 1)

4

with visco-Pasternak foundation (Cd = 3)

0 0.000

0.003

0.006

0.009

0.012

0.015

0.018

Time (ms)

0.012 C

C

F

C

EP

0.003

TE D

0.006

0.000

AC C

Centre Deflection (mm)

0.009

F

FG-1, [0 /90 /0 /90 /0 ] i R = 1 mm a = b, b/h = 20 * VCN = 0.28

-0.003

-0.006 0.00

without elastic foundation with Pasternak foundation with visco-Pasternak foundation (Cd = 1) with visco-Pasternak foundation (Cd = 3) 0.01

0.02

0.03

0.04

Time (ms)

Fig. 10. The effects of elastic foundation on low-velocity impact response of hybrid plates with both ends simply supported: (a) Contact force history and (b) deflection of plate.

32