- Email: [email protected]
Mechanical behavior of laminated CNT-reinforced composite skew plates subjected to dynamic loading
Accepted Manuscript Mechanical behavior of laminated CNT-reinforced composite skew plates subjected to dynamic loading L.W. Zhang PII:
S1359-8368(16)32776-7
DOI:
10.1016/j.compositesb.2017.03.041
Reference:
JCOMB 4975
To appear in:
Composites Part B
Received Date: 21 November 2016 Revised Date:
18 March 2017
Accepted Date: 22 March 2017
Please cite this article as: Zhang LW, Mechanical behavior of laminated CNT-reinforced composite skew plates subjected to dynamic loading, Composites Part B (2017), doi: 10.1016/ j.compositesb.2017.03.041. 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.
ACCEPTED MANUSCRIPT
Mechanical behavior of laminated CNT-reinforced composite skew plates subjected to dynamic loading
L. W. Zhang*
RI PT
School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
SC
Abstract
This paper presents the first known mechanical behavior of laminated CNT-reinforced
M AN U
composite skew plates subjected to a transverse sudden dynamic load. The plate is composed of multilayers of nanocomposite reinforced with single-walled carbon nanotubes (SWCNTs). The problem is formulated using the first-order shear deformation theory (FSDT), and solution to the problem is obtained through the element-free IMLS-Ritz method. The elastodynamic behavior is furnished by employing the Newmark-β method. Material properties of CNT-reinforced
TE D
composites are predicted through the Mori–Tanaka approach. The stability and precision of the IMLS-Ritz method are validated by convergence and comparison studies. The effects of skew angles, width-to-thickness ratio, CNT-volume fraction, CNT-distribution along the layer thickness, CNT-fiber orientation and boundary
EP
conditions on the elastodynamic responses of laminated CNT-reinforced composite skew plates are examined.
AC C
Keywords: CNT-reinforced composites; elastodynamic; dynamic loading; laminated skew plates; IMLS-Ritz method
*
Corresponding author. E-mail address: [email protected]
ACCEPTED MANUSCRIPT 1. Introduction Nanocomposites, comprising nanoparticle reinforced composites and nanotube reinforced composites, have been the subject of increasing attention because of their excellent mechanical, optical and electronic applications [1]. Being one of the advanced nanomaterials, carbon nanotube (CNT) has been utilized as the
RI PT
reinforcement materials for nanocomposite structures [2]. Because of their excellent strength, CNT-reinforced composites can be exploited as an alternative substitute for the available functional materials [3-8]. However, research exploration of this material is still far away from its wide application and safe use. This aspiration has
SC
attracted many researchers to make efforts on deeply understanding of the mechanical behaviors of CNT-reinforced composites [9-12]. Mohammadimehr et al. [13] studied
M AN U
the vibration of viscoelastic double-bonded polymeric nanocomposite plate. Mantari and Granados [14] presented the bending and free vibration analyses of functionally graded plates resting on elastic foundation. Mantari and Granados [15] further presented the static analysis of functionally graded plates by using a new first shear deformation theory. Kiani [16] studied the free vibration behavior of CNT-reinforced functionally graded composite skew plates.
TE D
Due to the increasing needs of laminated composite structures in aerospace, mechanical, marine, and automotive industries, they become the subject of increasing research over decades [17] [18]. Reddy [19] used the finite element method to study the forced motions of laminated composite plates. The transverse shear strains, rotary
EP
inertia and large rotations were taken into accounted. Fan and Wang [20] discussed the effects of matrix cracks on the nonlinear bending and thermal postbuckling of
AC C
CNT-reinforced composite beams. Kumar et al. [21] studied the buckling of laminated composite skew plates with various boundary conditions under linearly varying in-plane loads. Kavousi et al. [22] studied the dynamic response of laminated composite beams subjected to multiple masses low-velocity impacts. Taehyo et al. [23] used a 4-node quasi-conforming shell element to study the static and dynamic responses of laminated composite plates and shells. The Mori-Tanaka approach [24] are well used to estimate the effective elastic parameters of composite materials. Mori-Tanaka [24] proposed a rational apprimation to correlate averaged stresses and strains of the constituent fiber with those of the
ACCEPTED MANUSCRIPT matrix in a composite. Benveniste [25] found that the Mori-Tanaka approach can be reformulated by making use of the equivalent inclusion idea in terms of a more compacted tensor [26]. Xin et al. [27] used the Mori-Tanaka approach to study the functionally graded thick-walled tube subjected to internal pressure.
RI PT
A proven approximation technique, the Ritz method [28] which is a generalized Rayleigh method, has been used in computational analyses. Employing the moving least-squares (MLS) shape function in the Ritz method was used by Zhou. et al. [29] for vibration analysis of skew plates. The MLS has been employed as an
SC
approximation function in the meshless/element-free method; however, the drawback of the MLS approximation has made it difficult to obtain an accurate numerical
M AN U
solution when the algebraic equation system becomes ill-conditioned or presence of singularity. To overcome this drawback, the IMLS approximation was proposed for construction of the shape functions [30-33].
In this paper, the Mori–Tanaka approach is employed to estimate the material properties of CNT-reinforced composites. The attention is only limited to the modeling of the dynamic responses of laminated CNT-reinforced composite skew
TE D
plates subjected to a transverse sudden dynamic loading. The element-free IMLS-Ritz procedure is employed to derive the discretized equations system for the elastodynamic problem. The elastodynamic behavior is furnished by employing the Newmark-β method. The effects of skew angles, width-to-thickness ratio, CNT-volume fraction,
EP
CNT-distribution along the layer thickness, CNT-fiber orientation and boundary conditions on the elastodynamic responses of laminated CNT-reinforced composite
AC C
skew plates are examined. 2. Problem description
A laminated plate which is perfectly bonded with CNT-reinforced composite layers of thickness h and the CNT-orientation is shown in Fig. 1. β is the angle of CNT-orientation. The plate, as illustrated in Fig. 2, is considered with UD, FG-O and FG-X CNT-distributions. The CNT-content by volume VCNT can be approximated by * UD : VCNT ( z ) = VCNT
(1)
ACCEPTED MANUSCRIPT 2z * FG-O : VCNT ( z ) = 2 1− V h CNT
(2)
z * FG-X : VCNT ( z ) = 4 VCNT h
(3)
* VCNT =
wCNT
wCNT + (
, ρ CNT )(1− wCNT ) m ρ
RI PT
where
(4)
SC
in which, wCNT is the mass fraction of CNTs, ρ m and ρ CNT are the densities of the matrix and carbon nanotube, respectively; z is a local thickness coordinate variable of
M AN U
* a typical layer, and VCNT ( z ) = VCNT corresponds to the uniformly distributed
CNT-composite layer (UD). The UD, FG-O and FG-X CNT-reinforced composite plates have the same mass and volume of CNTs.
2.1. Energy formulation of elastodynamic problem
On the basis of first order shear deformation laminated plate theory [34], the
TE D
displacement field of the CNT reinforced composite plate can be expressed as (5)
v ( x, y , z ) = v0 ( x, y ) + zθ y ( x, y ) ,
(6)
w( x, y, z ) = w0 ( x, y ) ,
(7)
EP
u ( x, y, z ) = u0 ( x, y) + zθ x ( x, y) ,
AC C
in which, (u, v, w) are the displacements ( x, y , z ) , (u0 , v0 , w0 ) are the displacement projections on the mid-plane, and the rotation functions should approach the respective slopes of the transverse deformation:
θx =
∂u ∂v ,θ y = ∂z ∂z
(8)
where θ x , θ y are the transverse normal rotations about the positive y and negative x axes. The nonzero strain tensor can be expressed as
ε xx γ yz ε yy = ε 0 + zκ , = γ 0 , γ xz γ xy
(9)
ACCEPTED MANUSCRIPT where
(10)
RI PT
∂u0 1 ∂w0 2 ∂θ x + ∂ x 2 ∂ x ∂x ∂w θy + 0 2 θ ∂ ∂y ∂v0 1 ∂w0 y ε0 = + ,κ = , γ0 = . ∂y ∂y 2 ∂y θ + ∂w0 x ∂θ ∂θ y ∂u0 ∂v0 ∂w0 ∂w0 ∂x x + + + ∂x ∂y ∂y ∂x ∂x ∂y
The moment resultants and external force of a composite plate can be derived as
SC
follows
(11)
N xx M xx Qy N = N yy , M = M yy , Qs = , Qx N xy M xy
(12)
M AN U
T N A B 0 ε0 N T M = B D 0 κ − M , Qs 0 0 A s γ 0 0
where
A ,
B ,
As are the matrices of extensional stiffness,
D and
TE D
in which
bending-extensional coupling stiffness, bending stiffness and transverse shearing stiffness, respectively. They can be defined as
AC C
EP
A11 A = A12 A16
A16 B11 A26 , B = B12 B16 A66
A12
A22 A26
D11 D = D12 D16
D12 D22 D26
B12 B22 B26
D16 As D26 , As = 44s A45 D66
B16 B26 , B66
A45s , A55s
(13)
(14)
where
( Aij , Bij , Dij ) = ∫
h /2
− h /2
Qij (1, z, z 2 )dz ,
Aijs = K ∫
h /2
− h /2
Qij dz ,
(15)
in which Qij (i , j = 1, 2, 6) are the plane-stress stiffness coefficients, Qij (i , j = 4, 5) are the shear stiffness coefficients and K is the shear correction factor which is taken as
K = 5 / (6 − ( v1V1 + v2V2 )) for functionally graded materials [35].
ACCEPTED MANUSCRIPT For a two-dimensional elastodynamic problem on a CNT-reinforced skew composite plate, we have
σ ij , j ( x , t ) + bi ( x , t ) = cu& ( x , t ) + ρ u&&( x , t ) , x = ( x1 , x2 ) ∈Ω ,
(16)
where ui is the displacement, ρ and c are the mass density and damping coefficient,
problem domain. In Eq. (16):
∂ui ∂ 2u , u&&i = 2i , ∂t ∂t
The boundary conditions are
M AN U
in which u&i is the velocity, u&&i is the acceleration.
SC
u&i =
RI PT
respectively. σ ij is to the stress tensor, bi is the body force and Ω refers to the
ui ( x) = u%i ( x), x ∈Γu , ti ( x ) = σ ij ( x ) n = t%i ( x ), x ∈ Γσ ,
(17)
(18)
where ti is the traction vector, and u%i and t% are the prescribed displacement and
respectively.
TE D
traction vector on the displacement boundary Γu and traction boundary Γσ ,
The initial conditions are taken to be
EP
ui ( x,0) = ui 0 ( x) , u&i ( x,0) = u&i 0 ( x) , x ∈ Ω .
(19)
AC C
The strain energy of the CNT-composite plate can be expressed as ,
(20)
where
A B 0 ε0 ε = κ , S= B D 0 . γ 0 0 As 0
(21)
and the external work can be expressed as We = ∫ u T f dΩ + ∫ u T t dΓ , Ω
Γ
(22)
ACCEPTED MANUSCRIPT where f represents the external load and t is the prescribed traction on the natural boundary. The total energy functional can be written in the following equation
RI PT
(23)
2.2. Geometric mapping
For a skew plate, the orthogonal coordinates x-y can be converted to an oblique
expressed as N
N
SC
coordinates ξ − η . As shown in Fig. 4, the transformation equations can be
M AN U
x = ∑ x i N i ( ξ , η ) , y = ∑ y i N i (ξ , η ) , i=1
( i =1,2,3,4),
(24)
i=1
1 N i = (1+ ξiξ )(1+ ηiη ) 4 where
N i (ξ ,η ) is the transformation shape function,
(25)
xi
and
yi
are the
TE D
co-ordinates of corner point i in the physical domain, and ξ i and ηi are the co-ordinates of corner point i in the computational domain. Using the chain rules in geometric transformation so as to obtain the first and second-order derivatives of an arbitrary function with respect to x and y coordinate
AC C
EP
variables to ξ and η coordinate varibales, i.e.
where
() () ()
,xx
,xy
, yy
() ()
−1 = J 22
,x
,y
= J −1 11
() () ()
,ξξ
,ξη
,ηη
() ()
,ξ
,η
,
−1 −1 − J 22 J 21 J11
(26)
() ()
,ξ
,η
,
(27)
ACCEPTED MANUSCRIPT y,ξ , y,η
(28)
x ,ξξ J 21 = x,ξη x,ηη
y,ξξ y,ξη , y,ηη
(29)
in which
J ij
y,ξ 2 y,ξ y,η y,η 2
(x,ξ x,η + x,η y,ξ ) , 2x,η y,η 2x,ξ y,ξ
(30)
SC
x 2 ,ξ J 22 = x,ξ x,η x 2 ,η
RI PT
x ,ξ J11 = x,η
is the transformation Jacobian matrix. Based on the above
M AN U
transformation Jacobian matrix, the governing equations can be transformed from the physical domain to a computational domain.
2.3. The Mori-Tanaka Approach
We employed the Mori-Tanaka approach for predicting the material properties of
TE D
CNT-reinforced composites. For a two phase composite, the effective elastic module tensor L of CNT-reinforced composites can be written in the following expression [36]:
（31）
EP
L = Lm + VCNT [( LCNT − Lm ) ⋅ A]⋅[Vm I + VCNT (A)]−1 ,
where LCNT and Lm are the stiffness tensors of CNT and matrix, I being a unit
AC C
tensor, and the angled brackets denote the average of an overall possible orientation of inclusions. A is given by
A = [I + S ⋅ L−1 (LCNT − Lm )]−1 , m
（32）
which is the dilute mechanical strain concentration tensor and S is the Eshelby tensor. Considering a polymer elastic matrix reinforced with aligned and straight CNTs. Each CNT is modeled as a long straight fiber with transversely isotropic elastic properties. The Hill’s elastic moduli of the CNTs and its inter-phase [37] are:
ACCEPTED MANUSCRIPT
nr
lr
lr
0
lr
kr + mr
kr − mr
0
lr
kr − mr
kr + mr
0
0
0
0
pr
−
v CNT E11CNT
−
v CNT E11CNT
0
1
−
v CNT CNT E22
0
E22CNT v CNT − CNT E22 0
1 E22CNT
0
(33)
RI PT
1 CNT E11 CNT −v E11CNT = CNT −v E CNT 11 0
0
1
G12CNT
Once we have the Hill’s elastic moduli for the CNTs, the Hill’s elastic moduli of the
k=
Em (Em f m + 2kr (1+ vm ) + fr [1− 2vm ]) 2(1+ vm )[Em (1+ fr − 2vm ) + 2 f m kr (1− vm − 2vm2 )]
l=
Em (vm f m [Em + 2kr (1+ vm )] + 2 f r kr [1− vm2 ]) (1+ vm )[2 f m kr (1− vm − 2vm2 ) + Em (1+ f r − 2vm )]
M AN U
n=
SC
naocomposite can be obtained according to [38] and expressed as:
Em2 f m (1+ f r − vm f m ) + 2 f m f r (kr nr − lr2 )(1− vm )2 (1− 2vm ) (1+ vm )[2 f m kr (1− vm − 2vm2 ) + Em (1+ f r − 2vm )]
TE D
Em [2 f m2 kr (1− vm ) + f r nr (1+ f r − 2vm ) + 4 f m f r lr vm ] + 2 f m kr (1− vm − 2vm2 ) + Em (1+ f r − 2vm ) p=
Em [Em f m + 2mr (1+ vm )(3+ f r − 4vm )] 2(1+ vm ){Em [ f m + 4 f r (1− vm )] + 2 f m mr (3− vm − 4vm2 )}
EP
m=
Em [Em f m + 2(1+ f r ) pr (1+ vm )] 2(1+ vm )[Em (1+ f r ) + 2 f m pm (1+ vm )]
(34)
(35)
(36)
(37)
(38)
where k is the plane-strain bulk modulus normal to the fiber direction, n is the uniaxial
AC C
tension modulus in the fiber direction, l is the associated cross modulus, m and p are the shear moduli in planes normal and parallel to the fiber direction, respectively. Consequently, the elastic moduli E11 and E22 of the composite parallel and normal to CNTs and Poisson’s ratio v12 can be expressed as:
l2 4m(kn − l 2 ) l E11 = n − , E22 = ,v12 = 2 k 2k kn − l
(39)
In this study, poly(methyl methacrylate), denoted as (PMMA) [39, 40] is selected for the polymer matrix material, whereas, the armchair (10, 10) SWCNTs are selected to be the reinforcement material. The material properties of the PMMA matrix are
ACCEPTED MANUSCRIPT v m = 0.34, ρ m = 1150kg / m3 ,v m = 0.34 , and E m = 2.5 GPa. The SWCNTs material properties are assumed to be G12CNT = 1.9455, E11CNT = 5.6466 TPa,
E22CNT = 7.0800 TPa
and G12CNT = 1.9455.
RI PT
2.4. Discrete system equations
The displacement field of the plates can be defined over the computational domain as m
h u ( x ) = ∑ p i ( x )ai ( x) = p ( x)a( x) , T
(40)
SC
i =1
where pi ( x )(i = 1, 2,…, m) are the basis function, m is the number of terms in the
M AN U
basis, and coefficients ai ( x ) are functions of the spatial coordinates x. Define a weighted discrete norm as n
J = ∑ w( x − x I )[u h ( x, x I ) − u( x I )]2 I =1
2
m = ∑ w( x − x I ) ∑ p i ( x I ) ⋅ a i ( x ) − u ( x I ) , I =1 i =1 w( x − x I )
(42b)
TE D
n
where
(41a)
is a weight function with the compact support, and xI ( I = 1, 2 ,..., n )
are the nodes with domains of influence that cover the point x.
EP
Moreover, the cubic spline function is a weight function which is given by
AC C
2 2 3 3 − 4 zI + 4 zI 4 4 w( x − x I ) = wI ( x ) = − 4 z I + 4 z I2 − z I3 3 3 0
zI =
x − xI , dI
1 2 1 for < zI ≤ 1 , 2 otherwise for 0 ≤ z I ≤
d I = d max cI ,
(43)
(44)
where d I is the size of the support, d max is a scaling factor (2.0 ≤ dmax ≤4.0) and cI is the average distance between nodes in the influenced domain of the point x. It is obvious that a( x) can be determined subjects to
ACCEPTED MANUSCRIPT ∂J = 0, ∂a
(45)
A( x ) a ( x ) = B ( x ) u ,
(46)
which leads to
RI PT
such that
A( x) = P TW ( x) P ,
(47)
B( x) = P TW ( x) ,
(48)
u = ( u1 , u 2 , L , u n ) T ,
(49)
SC
On the Hilbert space span( p ), the following inner product n
M AN U
( f , g) = ∑ w(x − x I ) f (x I )g(x I )
(50)
I =1
can be defined, and ∀f ( x), g( x) ∈ span( p) such that
p1 ( x1 ) 0 P= M 0
TE D
0 p2 ( x 2 ) M 0
L 0 L 0 , O M L pm ( x n )
EP
0 w( x − x1 ) 0 w( x − x 2 ) W ( x) = M M 0 0
L 0 L 0 . O M L w( x − x n )
(51)
(52)
AC C
The local approximation u h ( x ) is
,
(53)
where
% ( x ) = (Φ % ( x ), Φ % ( x ),L , Φ % ( x )) = p T ( x ) A% ( x ) B ( x ) . Φ 1 2 n
(54)
and m
% ( x ) = ∑ p ( x )[ A% ( x ) B( x )] , Φ I j jI j =1
(55)
ACCEPTED MANUSCRIPT which represents the shape function of the IMLS approximation corresponding to node I. The weighted orthogonal basis function set can be furnished according to the Schmidt method as
(56)
RI PT
p1 = 1,... p1 = r
(z i−1 , pk ) −∑ pk ,i = 2,3... k =1 ( pk , pk ) i−1
i−1
The weighted orthogonal basis function set p = ( pi ) , however, can be formed in terms of the polynomial function in the following fashion
SC
p% = ( p%i ) = (1, x, x2 , x3, x1x2 , x1x3, x2 x3, x12 , x22 , x32...) ,
(57)
Alternatively, the weighted orthogonal basis function set can be written in the
i −1
M AN U
following form ( p% i , pk ) pk , k =1 ( pk , pk )
pi = p% i − ∑
i = 1, 2,3,...
(58)
Substituting the approximation function into the functional defined in Eq. (23) and applying the Ritz procedure
TE D
∂Π * ∂u (t ) = 0, ∆ = u1 (t ), 1 , I = 1, 2, ⋅⋅⋅, n . ∂∆ ∂t
(59)
The discrete control equation (DCE) is
MU&& (t ) + CU& (t ) + KU (t ) = F (t ) ,
EP
(60)
where M is the global mass matrix, C is the damping matrix, F is the global tttt
AC C
external force vector and K is the global stiffness matrix, in which T T % T cΦ % dΩ , F = M = ∫ G T mGJ d Ω , C = ∫ Φ I J ∫Γ Φ% %d Γ + ∫Ω Φ% bd Ω ,
(61)
K = Kb + Km + Ks ,
(62)
Ω
Ω
σ
and
T
K IJb = ∫ BIb DBJb d Ω ,
(63)
Ω
T
T
T
K IJm = ∫ BIm DBJm d Ω + ∫ BIm DBJb d Ω + ∫ BIb DBJm d Ω , Ω
Ω
Ω
T
K IJs = ∫ BIs As BJs d Ω , Ω
(64) (65)
ACCEPTED MANUSCRIPT where
0
M AN U
∂Φ I ∂η ∂Φ I ∂ζ
0 0 0 0 0 0 , 0 0 0
(66)
∂Φ I ∂ζ ∂Φ I ∂η
0 0 s BI = 0 0
0 ΦI 0 0 0
0 0 ΦI 0 0
0 I0 0 0 I1
0 0 I0 0 0
TE D
Φ I 0 GI = 0 0 0
AC C
EP
I0 0 m =0 I1 0
(67)
SC
∂Φ I ∂ζ m BI = 0 ∂Φ I ∂η
0 ∂Φ I , ∂η ∂Φ I ∂ζ
RI PT
∂Φ I 0 0 0 ∂ζ BIb = 0 0 0 0 ∂Φ I 0 0 0 ∂η
ΦI 0
0 0 0 ΦI 0
I1 0 0 I2 0
0 , ΦI
0 0 0 , 0 Φ I
0 I1 0. 0 I 2
(68)
(69)
(70)
We can rewrite Eq. (48) without damping in the following expression
MU&& (t ) + KU (t ) = F (t ) .
(71)
2.5. Newmark-β method The Newmark-β method is employed for time discretization of equations of motion. The method has demonstrated to furnish accurate solution and provide good numerical
ACCEPTED MANUSCRIPT stability. The Taylor expansion is applied for displacement and velocity at a time step
t + ∆t , i.e. (72)
U& t +∆t = U& t + ∆tU&&t − β1∆tU&&t + β1∆tU&&t +∆t ,
(73)
U&&t +∆t =
RI PT
1 1 1 Ut +∆t = U t + ∆tU& t + ∆t 2U&&t − β2 ∆t 2U&&t + β2 ∆t 2U&&t +∆t , 2 2 2
2 2 & 1 U − Ut ) − U t − U&&t + U&&t . 2 ( t +∆t β2 ∆t β 2 ∆t β2
(74)
.
(75)
M AN U
3. Numerical examples and discussion
SC
Thus, Eq. (66) can be rewritten according to the Newmark-β method as
In this study, laminated CNT-composite skew plates are composed two parts: one part is the matrix is made of PMMA (polymethyl methacrylate) and the other part is the reinforcement is made of (10,10) SWCNTs.
The boundary conditions used in the part are defined as follows:
TE D
Simple supported ( S ): v0 = w0 = θ y = 0 for sides b and c,
u0 = w0 = θ x = 0 for sides a and d.
Clamped (C) :
u0 = v0 = w0 = θ x = θ y = 0 for sides b and c,
EP
u0 = v0 = w0 = θ x = θ y = 0 for sides a and d.
3.1. Validation studies
AC C
Comparison studies are conducted for a square three-layer [0o/90o/0o] clamped plate subjected to a sudden transverse dynamic load. The stability and accuracy of the IMLS-Ritz method are demonstrated by varying the number of nodes and scaling factors. Some geometrical properties of the laminated plates are: a = b = 25cm ,
h = 5cm , ρ = 8 × 10−6 N / cm2 , E2 = 2.1 × 106 N / cm 2 , E1 / E2 = 25 , v = 0.25 , G12 = G13 = G23 = 0.5E2 and q0 = 10 N / cm 2 as shown in Fig. 3. Different scaling factors d max are considered in this convergence study. It was concluded that the scaling factors d max , ranging from 2.1 to 3.4, has very little influence on the results. It is decided to use a scaling factor d max = 2.1 for the
ACCEPTED MANUSCRIPT following studies. As shown in Fig. 5(a), it can be found that the numbers of nodes ranging from 7 × 7 to 15 ×15 also have very little effects on the results. Therefore, the number of nodes 15 ×15 will be used in the following studies. Fig. 5(b) shows a comparison plot with Mindlin’s solution [41] and the HOSD solution [42]. The
well with the Mindlin solution [41] and HOSD solution [42].
RI PT
IMLS-Ritz results obtained using d max = 2.1 and number of nodes 15 ×15 agree
We have further carried out a convergence study to demonstrate the stability and accuracy of the IMLS-Ritz method for a fully clamped three-layer [0o/90o/0o] laminated CNT-composite skew plate with a uniform distribution of CNTs and a skew
SC
angle 60o subjected to a sudden dynamic load q0 = 10 N / cm 2 with different quantities of nodes and different scaling factor. The geometric properties of this
h = 2cm
CNT-composite ,
E11CN = 5.6466TPa
plate
are:
M AN U
laminated
,
E22CN = E33CN = 7.0800TPa
a = b = 20cm ,
v12CN = 0.175
, ,
E m = 2.5 × 10 9 N / m 2 , v m = 0.34 , and q0 = 10 5 N / m 2 . The volume fraction ratio of * = 0.11 . CNTs is taken as VCNT
As displayed in Figs. 6(a) and (b), different scaling factor d max and different
TE D
numbers of nodes considered in this convergence study for the laminated CNT-composite skew plate are considered. This study shows that the scaling factors
d max ranging from 2.1 to 3.4 and the numbers of nodes ranging from 9 × 9 to 15 × 15 have very small influence on the convergence of results. Therefore, it is
EP
decided to use the support size d max = 2.1 and number of nodes 15 × 15 for the
AC C
following parametric studies.
3.2 Parametric studies In this section, parametric studies are carried out to investigate the dynamic behaviors of laminated CNT-composite skew plates by considering various influenced factors. The geometrical properties of the laminated CNT-composite plate are: width and height a = b = 20cm , thickness h = 2cm and density ρ for the matrix and CNT are 1150 and 1400 kg / m 3 . The transverse sudden dynamic load q0 = 10 N / cm 2 . We consider three types of laminated CNT-composite plates with the following arrangements: (a) three-layer [0o/90o/0o], (b) four-layer [-45o/45o/45o/-45o] and (c)
ACCEPTED MANUSCRIPT four-layer [-45o/45o/-45o/45o]. The plate is having a skew angle α ranging from 30° to 90° with UD, FG-O and FG-X CNT-distributions. The CNT-volume fraction * =0.11. The plate is subjected to a fully clamped boundary condition. The results VCNT
obtained are shown in Figs. 7-9. It can be seen from the figures that the values of central deflection decreases dramatically with decreasing of the skew angle α.
RI PT
We also study the influence of width-to-thickness ratios on the central deflections of the laminated CNT-composite plates. The skew angle α selected for this study is 60o. Fig. 10 depicts the results for a fully clamped two-layer [0°/90°] laminated plate with UD, FG-O and FG-X CNT-distributions under different width-to-thickness ratios
SC
a / h = 10, 20,50 . The CNT-volume fraction is 0.11. It is evident that the FG-O and FG-X laminated CNT-composite plates are having the greatest and smallest of central
M AN U
deflections, respectively. The results concluded that when the plate is having FG-X CNT-distribution which means the CNT-distributions are close to top and bottom surfaces of the plate, the stiffness is the strongest. On the other way, when a plate is having the FG-O CNT-distribution which means the CNT-distributions are concentrated at the plate’s mid-plane, the stiffness is the weakest. Besides, we also study the influence of CNT-volume fraction on a fully clamped
TE D
four-layer [0°/90°/90°/0°] laminated CNT-composite plate subjected to a transverse sudden dynamic load. The skew angle α is chosen to be 60o for this study. Fig. 11 shows the central deflections for CNT-composite laminated plates which have the CNT-volume fractions of 0.11, 0.14, and 0.17, respectively. It is evident that the
EP
central deflection decreases with the increasing of CNT-volume fraction. It shows that the plate’s stiffness is larger when the CNT-volume fraction is higher.
AC C
The dynamic responses of full clamped angle-ply [-β/β/β/-β] laminated CNT-composite plates with UD, FG-O and FG-X CNT-distributions are examined. Fig. 12 presents the results of angle-ply laminated CNT-composite plates with β = 90o, 75 o, 60 o and 45o. With β varying from 90o to 45o, the central deflection decreases. Figs. 13-15 depicted the vertical central deflections for 4-ply [0°/90°/0°/90°], [-45o/45o/45o/-45o] and [-45o/45o/-45o/45o] laminated CNT-composite plates with UD, FG-O and FG-X CNT-distributions. The plate is subjected to four different types of boundary conditions, i.e. simply supported (SSSS), two edges simply supported and the other two free (SFSF), fully clamped (CCCC), and two edges clamped and the * other two free (CFCF). The CNT-volume fraction VCNT = 0.11 , width-to-thickness
ACCEPTED MANUSCRIPT ratios a / h =10 and skew angle α = 60o are chosen for this study. It is found that the vertical central deflection of the laminated CNT-composite plates with (a) SFSF boundary condition is the largest, and (b) CCCC boundary condition is the smallest. Besides, it is also observed that the vertical central deflection of SSSS and CFCF plates lied between SFSF and CCCC plates. This is expected and can be explained
RI PT
that the clamped edge has the strongest constraint, followed by simply-supported edge and then free edge.
4. Conclusion
The elastodynamic behavior of laminated CNT-reinforced composite skew plates
SC
subjected to a transverse sudden dynamic load is studied. The governing equations of plates are derived based on the FSDT. The IMLS-Ritz method is employed for
M AN U
solution. The numerical time discretization used to determine the dynamic responses of plate is carried out using the Newmark-β method. The Mori-Tanaka approach is employed to estimate the material properties of CNT-reinforced composites. Different types of CNT-distributions along the thickness of the layers are considered, i.e. uniform distributed (UD) and functionally graded distributions (FG-O and FG-X). The effects of CNT-volume fraction, CNT-distributions, geometrical shape
TE D
parameters and width-to-thickness ratio on the dynamic behavior of skew laminated plates with different boundary conditions are examined. These results, unknown
EP
before, can be used for future references.
Acknowledgement
AC C
The work described in this paper was fully supported by grants from the National Natural Science Foundation of China (Grant no. 11402142).
References
[1] Sang HC, Lee YH. Carbon nanotubes and graphene towards soft electronics. Nano Convergence. 2014;1(1):1-26. [2] Liew KM, Lei ZX, Zhang LW. Mechanical analysis of functionally graded carbon nanotube reinforced composites: A review. Composite Structures. 2015;120:90-7. [3] Ansari R, Torabi J, Shojaei MF. Buckling and vibration analysis of embedded functionally graded carbon nanotube-reinforced composite annular sector plates under thermal loading. Composites Part B: Engineering. 2017;109:197-213. [4] Jooybar N, Malekzadeh P, Fiouz A. Vibration of functionally graded carbon nanotubes reinforced composite truncated conical panels with elastically restrained
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
against rotation edges in thermal environment. Composites Part B: Engineering. 2016;106:242-61. [5] Pourasghar A, Chen Z. Thermoelastic response of CNT reinforced cylindrical panel resting on elastic foundation using theory of elasticity. Composites Part B: Engineering. 2016;99:436-44. [6] Hui D. Effects of shear loads on vibration and buckling of antisymmetric cross-ply cylindrical panels. International Journal of Non-Linear Mechanics. 1988;23(3):177-87. [7] Dong S, Zhou J, Hui D, Pang X, Wang Q, Zhang S, et al. Interaction between edge dislocations and amorphous interphase in carbon nanotubes reinforced metal matrix nanocomposites incorporating interface effect. International Journal of Solids and Structures. 2014;51(5):1149-63. [8] Tornabene F, Fantuzzi N, Bacciocchi M. Linear static response of nanocomposite plates and shells reinforced by agglomerated carbon nanotubes. Composites Part B: Engineering. 2017, http://dx.doi.org/10.1016/j.compositesb.2016.07.011, In press. [9] Zhang LW. Geometrically nonlinear large deformation of CNT-reinforced composite plates with internal column supports. Journal of Modeling in Mechanics and Materials. 2017;1(1):20160154. [10] Viet NV, Wang Q, Kuo WS. A studying on load transfer in carbon nanotube/epoxy composites under tension. Journal of Modeling in Mechanics and Materials. 2017;1(1):20160153. [11] Ashour AS. Vibration of variable thickness plates with edges elastically restrained against translation and rotation. Thin-Walled Structures. 2004;42(1):1-24. [12] Fantuzzi N, Tornabene F, Bacciocchi M, Dimitri R. Free vibration analysis of arbitrarily shaped Functionally Graded Carbon Nanotube-reinforced plates. Composites Part B: Engineering. 2017, http://dx.doi.org/10.1016/j.compositesb.2016.09.021, In press. [13] Mohammadimehr M, Rousta Navi B, Ghorbanpour Arani A. Free vibration of viscoelastic double-bonded polymeric nanocomposite plates reinforced by FG-SWCNTs using MSGT, sinusoidal shear deformation theory and meshless method. Composite Structures. 2015;131:654-71. [14] Mantari JL, Granados EV. An original FSDT to study advanced composites on elastic foundation. Thin-Walled Structures. 2016;107:80-9. [15] Mantari JL, Granados EV. A refined FSDT for the static analysis of functionally graded sandwich plates. Thin-Walled Structures. 2015;90:150-8. [16] Kiani Y. Free vibration of FG-CNT reinforced composite spherical shell panels using Gram-Schmidt shape functions. Composite Structures. 2017;159:368-81. [17] Casimir JB, Khadimallah MA, Nguyen MC. Formulation of the dynamic stiffness of a cross-ply laminated circular cylindrical shell subjected to distributed loads. Computers & Structures. 2016;166:42-50. [18] Tornabene F, Fantuzzi N, Bacciocchi M, Viola E. Effect of agglomeration on the natural frequencies of functionally graded carbon nanotube-reinforced laminated composite doubly-curved shells. Composites Part B: Engineering. 2016;89:187-218. [19] Reddy JN. Geometrically nonlinear transient analysis of laminated composite plates. Aiaa Journal. 1982;21(4):621-9. [20] Fan Y, Wang H. The effects of matrix cracks on the nonlinear bending and thermal postbuckling of shear deformable laminated beams containing carbon nanotube reinforced composite layers and piezoelectric fiber reinforced composite layers. Composites Part B: Engineering. 2016;106:28-41. [21] Kumar A, Panda SK, Kumar R. Buckling behaviour of laminated composite skew plates with various boundary conditions subjected to linearly varying in-plane edge loading. International Journal of Mechanical Sciences. 2015;100:136-44.
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
[22] Latifi M, Kharazi M, Ovesy HR. Nonlinear dynamic response of symmetric laminated composite beams under combined in-plane and lateral loadings using full layerwise theory. Thin-Walled Structures. 2016;104:62-70. [23] Park T, Kim K, Han S. Linear static and dynamic analysis of laminated composite plates and shells using a 4-node quasi-conforming shell element. Composites Part B Engineering. 2005;37(s 2–3):237-48. [24] Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica. 1973;21(5):571-4. [25] Benveniste Y. A new approach to the application of Mori-Tanaka's theory in composite materials. Mechanics of Materials. 1987;6(2):147-57. [26] Liu L, Huang Z. A Note on mori-tanaka's method. Acta Mechanica Solida Sinica. 2014;27(3):234-44. [27] Xin L, Yang S, Zhou D, Dui G. An approximate analytical solution based on the Mori-Tanaka method for functionally graded thick-walled tube subjected to internal pressure. Composite Structures. 2015(May):74–82. [28] Ritz W. Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik. Journal Für Die Reine Und Angewandte Mathematik. 1909;1909(135):1-61. [29] Zhou L, Zheng WX. Vibration of skew plates by the MLS-Ritz method. International Journal of Mechanical Sciences. 2008;50(7):1133-41. [30] Cheng RJ, Liew KM. Analyzing modified equal width (MEW) wave equation using the improved element-free Galerkin method. Engineering Analysis with Boundary Elements. 2012;36(9):1322-30. [31] Zhang Z, Liew KM, Cheng Y, Lee YY. Analyzing 2D fracture problems with the improved element-free Galerkin method. Engineering Analysis with Boundary Elements. 2008;32(3):241-50. [32] Li D, Bai F, Cheng Y, Liew KM. A novel complex variable element-free Galerkin method for two-dimensional large deformation problems. Computer Methods in Applied Mechanics and Engineering. 2012;233–236:1-10. [33] Ren H, Cheng Y. The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems. Engineering Analysis with Boundary Elements. 2012;36(5):873-80. [34] Reddy JN. Mechanics of laminated composite plates and shells: Theory and analysis, Second ed.,: CRC Press, Boca Raton, FL; 2004. [35] Efraim E, Eisenberger M. Exact vibration analysis of variable thickness thick annular isotropic and FGM plates. Journal of Sound and Vibration. 2007;299(4–5):720-38. [36] Eshelby JD. The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 1957;241(1226):376-96. [37] Malekzadeh P, Dehbozorgi M. Low velocity impact analysis of functionally graded carbon nanotubes reinforced composite skew plates. Composite Structures. 2016;140:728-48. [38] Shi D-L, Feng X-Q, Huang YY, Hwang K-C, Gao H. The Effect of Nanotube Waviness and Agglomeration on the Elastic Property of Carbon Nanotube-Reinforced Composites. Journal of Engineering Materials and Technology. 2004;126(3):250-7. [39] Han Y, Elliott J. Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites. Computational Materials Science. 2007;39(2):315-23. [40] Shen HS. Nonlinear bending of shear deformable laminated plates under transverse and in-plane loads and resting on elastic foundations. Compos Struct. 2000;50(2):131-42.
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
[41] Reddy JN. Dynamic (transient) analysis of layered anisotropic composite‐material plates. International Journal for Numerical Methods in Engineering. 1983;19(2):237-55. [42] Mallikarjuna, Kant T. Dynamics of laminated composite plates with a higher order theory and finite element discretization. Journal of Sound & Vibration. 1988;126(3):463-75.
ACCEPTED MANUSCRIPT Figures z
z
x h
y
y
x
RI PT
h/2
β
α
(a) UD
M AN U
SC
Fig. 1 Geometry of laminated CNT-composite skew plate and the CNT orientation.
(b) FG-O
(c) FG-X
Fig. 2 Cross section of the CNT-reinforced composite plates with three types of
AC C
EP
TE D
CNT-distributions.
P(t)
qoP
O
t
Fig. 3 The plate is subjected to a transverse sudden dynamic loading.
ACCEPTED MANUSCRIPT η
y
(1,1) 3
(-1,1) 4 3
d
b 1
ζ
c
α
1
x a
(-1,-1)
2
2 (1,-1)
RI PT
4
Fig. 4 Transformation of a skew plate from the physical domain to a computational
M AN U
SC
domain.
dmax=2.1, 15×15 Ref. [38] Ref.[32] Ref. [39] Ref.[33]
3
-6
-6
Central deflection, w (10 m)
3
Central deflection, w (10 m)
7×7 11×11 15×15
0
0.5 1.0 -4 Time(10 s)
1.5
2
1
0 -0.4 0
AC C
-0.4 0
EP
1
TE D
2
(a)
0.5 1.0 -4 Time(10 s)
1.5
(b)
Fig. 5 Convergence and comparison studies for a square laminated three-layer [0o/90o/0o] clamped plate on: (a) different number of nodes and (b) comparison with the existing the Mindlin solutions and HOSD results.
ACCEPTED MANUSCRIPT
present:9×9 present:11×11
4 3 2 1 0 0
1
2 3 -4 Time(10 s)
(a)
5
M AN U dmax=2.1
4
dmax=2.5
3 2
dmax=2.9
1 0
dmax=3.4
TE D
-5
Central deflection, w (10 m)
5
4
RI PT
present:13×13 present:15×15
SC
-5
Central deflection, w (10 m)
5
0
1
2 3 Time(10-4s)
4
5
EP
(b) Fig. 6 Convergence study for a four-layer [0o/90o/90o/0o] laminated CNT-composite skew plate with UD CNT-distribution on: (a) different numbers of nodes ranging
AC C
from 9 × 9 to 15 × 15 and (b) different support sizes d max ranging from 2.1 to 3.4.
ACCEPTED MANUSCRIPT UD 6
α =90
o
α =60
o
α =75
o
α =45
o
4 3
1 0 2
4 6 -4 Time(10 s)
8
FG-O
6
α =90
o
α =75
o
5 4 3 2 1 0 2
o
α =45
o
4 6 -4 Time(10 s)
TE D
-1 0
α =60
10
α =30
o
M AN U
7
RI PT
2
8 -5
o
5
-1 0
Central deflection,w (10 m)
α =30
SC
-5
Central deflection, w (10 m)
7
8
10
FG-X
7 6
α =90
o
α =75
o
α =60
o
α =45
o
α =30
o
EP
-5
Central deflection, w (10 m)
8
5 4
AC C
3 2 1 0
-1 0
2
4 6 -4 Time(10 s)
8
10
Fig. 7 The central deflection versus time for a three-layer [0o/90o/0o] laminated CNT-composite skew plate having skew angle α ranging from 30° to 90° with UD, * FG-O and FG-X CNT-distributions (CCCC boundary condition and VCNT =0.11).
ACCEPTED MANUSCRIPT UD 6
α =90
o
α =60
o
α =75
o
α =45
o
α =30
5 4 3
1 0 2
4 6 -4 Time(10 s)
8
FG-O 7 o
α =60
o
α =75
o
α =45
o
5 4 3 2 1 0 -1 0
10
α =30
o
M AN U
α =90
TE D
-5
6
RI PT
2
-1 0
Central deflection, w (10 m)
o
SC
-5
Central deflection, w (10 m)
7
2
4 6 Time(10-4s)
8
10
FG-X
7
o
α =75
o
EP
α =90
α =60
o
α =45
o
α =30
o
5 4
AC C
-5
Central deflection, w (10 m)
6
3 2 1 0
-1 0
2
4 6 -4 Time(10 s)
8
10
Fig. 8 The central deflection versus time for a four-layer [-45o/45o/45o/-45o] laminated CNT-composite skew plate having skew angle α ranging from 30° to 90° with UD, * =0.11). FG-O and FG-X CNT-distributions (CCCC boundary condition and VCNT
ACCEPTED MANUSCRIPT UD 6
α=90o
α=60o
α=75o
α=45o
5 4 3
1 0 2
4 6 Time(10-4s) FG-O
-5
6
α=90o
α=60o
α=75o
α=45o
5 4 3 2 1 0
TE D
-1 0
10
α=30o
M AN U
7
8
RI PT
2
-1 0
Central deflection, w (10 m)
α=30o
SC
-5
Central deflection, w (10 m)
7
2
4 6 -4 Time(10 s)
8
10
FG-X
7
α=90o
α=60o
α=75o
α=45o
5
α=30o
4
AC C
-5
Central deflection, w (10 m)
EP
6
3 2 1 0
-1 0
2
4 6 Time(10-4s)
8
10
Fig. 9 The central deflection versus time for a four-layer [-45o/45o/-45o/45o] laminated CNT-composite skew plate having skew angle α ranging from 30° to 90° with UD, * =0.11). FG-O and FG-X CNT-distributions (CCCC boundary condition and VCNT
ACCEPTED MANUSCRIPT UD:a/th=10 FG-O:a/th=10 FG-X:a/th=10
6 5 4 3
1 0 -1 0
2
4 6 -4 Time(10 s) UD:a/th=20 FG-O:a/th=20 FG-X:a/th=20
20 15 10 5 0 2
4 6 Time(10-4s)
TE D
-5 0
M AN U
-5
Central deflection, w (10 m)
25
250 200
10
8
10
UD:a/th=50 FG-O:a/th=50 FG-X:a/th=50
EP
-5
Central deflection, w (10 m)
8
RI PT
2
SC
-5
Central deflection, w (10 m)
7
150
AC C
100
50
0
-50 0
2
4 6 -4 Time(10 s)
8
10
Fig. 10 The central deflection versus time for a fully clamped two-layer [0°/90°] laminated CNT-composite skew plates having a/h = 10, 20 and 50 with UD, FG-O * and FG-X CNT-distributions ( VCNT =0.11).
ACCEPTED MANUSCRIPT *
VCNT =0.11
5
*
VCNT =0.14
*
VCNT =0.17
4 3 2 1 0 -1 0
2
4 6 -4 Time(10 s)
8
*
VCNT =0.14
3 2 1 0 -1 0
2
*
VCNT =0.17
M AN U
4
4 6 -4 Time(10 s)
10
SC
*
VCNT =0.11
5
TE D
-5
Central deflection, w (10 m)
FG-O 6
RI PT
-5
Central deflection, w (10 m)
UD 6
8
10
5 4 3
*
VCNT =0.11
*
VCNT =0.14
*
VCNT =0.17
EP
-5
Central deflection, w (10 m)
FG-X
6
2 1
AC C
0
-1 0
2
4 6 -4 Time(10 s)
8
10
Fig. 11 The central deflections versus time for a fully clamped four-layer * [0°/90°/90°/0°] laminated plates having VCNT =0.11, 0.14 and 0.17 with UD, FG-O
and FG-X CNT-distributions.
ACCEPTED MANUSCRIPT o
o
o
o
o
o
o
o
(-90 /90 /90 /-90 )
9
-5
(-60 /60 /60 /-60 )
7
o
o
o
o
o
o
o
(-45 /45 /45 /-45 )
5
RI PT
3 1 -1 0
2
4 6 -4 Time(10 s) FG-O o
o
o
o
8
10
o
o
o
o
9
(-90 /90 /90 /-90 )
(-75 /75 /75 /-75 )
7
(-60o/60o/60o/-60o)
(-45o/45o/45o/-45o)
M AN U
-5
Central deflection, w (10 m)
o
(-75 /75 /75 /-75 )
SC
Central deflection, w (10 m)
UD
5 3 1 -1 0
4 6 -4 Time(10 s)
TE D
2
8
10
9
-5
Central deflection, w (10 m)
FG-X
o
o
o
o
o
o
o
(-60 /60 /60 /-60 )
o
o
o
o
(-75 /75 /75 /-75 ) o
o
o
o
(-45 /45 /45 /-45 )
EP
7
o
(-90 /90 /90 /-90 )
5
AC C
3 1
-1 0
2
4 6 -4 Time(10 s)
8
10
Fig. 12 The central deflections versus time for angle-ply [-β/β/β/-β] laminated CNT-composite skew plates having β = 45o, 60o, 75o and 90o with UD, FG-O and FG-X CNT-distributions.
ACCEPTED MANUSCRIPT
2
4 6 -4 Time(10 s)
8
10
SFSF CFCF
M AN U
SSSS CCCC
2
4 6 -4 Time(10 s)
TE D
-5
Central deflection, w (10 m)
FG-O 9 8 7 6 5 4 3 2 1 0 -1 0
SFSF CFCF
RI PT
SSSS CCCC
SC
-5
Central deflection, w (10 m)
UD 9 8 7 6 5 4 3 2 1 0 -1 0
8
10
SSSS CCCC
SFSF CFCF
AC C
-5
EP
Central deflection, w (10 m)
FG-X
9 8 7 6 5 4 3 2 1 0 -1 0
2
4 6 Time(10-4s)
8
10
Fig. 13 The central deflections for a four-layer [0°/90°/0°/90°] laminated CNT-composite skew plate having UD, FG-O and FG-X CNT-distributions subjected to SSSS, CCCC, SFSF, CFCF boundary conditions.
ACCEPTED MANUSCRIPT
2
4 6 -4 Time(10 s)
8
10
M AN U
SSSS CCCC SFSF CFCF
2
4 6 -4 Time(10 s)
TE D
-5
Central deflection, w (10 m)
FG-O 10 9 8 7 6 5 4 3 2 1 0 -1 0
RI PT
SSSS CCCC SFSF CFCF
SC
-5
Central deflection, w (10 m)
UD 10 9 8 7 6 5 4 3 2 1 0 -1 0
8
10
8
10
SSSS CCCC SFSF CFCF
AC C
-5
EP
Central deflection, w (10 m)
FG-X
10 9 8 7 6 5 4 3 2 1 0 -1 0
2
4 6 Time(10-4s)
Fig. 14 The central deflections for a four-layer [-45o/45o/45o/-45o] laminated CNT-composite skew plate having UD, FG-O and FG-X CNT-distributions subjected to SSSS, CCCC, SFSF, CFCF boundary conditions.
ACCEPTED MANUSCRIPT SSSS CCCC
2
4 6 -4 Time(10 s)
10
M AN U
SSSS CCCC SFSF CFCF
2
4 6 -4 Time(10 s)
TE D
-5
Central deflection, w (10 m)
FG-O 10 9 8 7 6 5 4 3 2 1 0 -1 0
8
RI PT
SFSF CFCF
SC
-5
Central deflection, w (10 m)
UD 10 9 8 7 6 5 4 3 2 1 0 -1 0
8
10
8
10
SSSS CCCC SFSF CFCF
AC C
-5
EP
Central deflection, w (10 m)
FG-X
10 9 8 7 6 5 4 3 2 1 0 -1 0
2
4 6 Time(10-4s)
Fig. 15 The central deflections for a four-layer [-45o/45o/-45o/45o] laminated CNT-composite skew plates having UD, FG-O and FG-X CNT-distributions subjected to SSSS, CCCC, SFSF, CFCF boundary conditions.
S1359-8368(16)32776-7
DOI:
10.1016/j.compositesb.2017.03.041
Reference:
JCOMB 4975
To appear in:
Composites Part B
Received Date: 21 November 2016 Revised Date:
18 March 2017
Accepted Date: 22 March 2017
Please cite this article as: Zhang LW, Mechanical behavior of laminated CNT-reinforced composite skew plates subjected to dynamic loading, Composites Part B (2017), doi: 10.1016/ j.compositesb.2017.03.041. 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.
ACCEPTED MANUSCRIPT
Mechanical behavior of laminated CNT-reinforced composite skew plates subjected to dynamic loading
L. W. Zhang*
RI PT
School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
SC
Abstract
This paper presents the first known mechanical behavior of laminated CNT-reinforced
M AN U
composite skew plates subjected to a transverse sudden dynamic load. The plate is composed of multilayers of nanocomposite reinforced with single-walled carbon nanotubes (SWCNTs). The problem is formulated using the first-order shear deformation theory (FSDT), and solution to the problem is obtained through the element-free IMLS-Ritz method. The elastodynamic behavior is furnished by employing the Newmark-β method. Material properties of CNT-reinforced
TE D
composites are predicted through the Mori–Tanaka approach. The stability and precision of the IMLS-Ritz method are validated by convergence and comparison studies. The effects of skew angles, width-to-thickness ratio, CNT-volume fraction, CNT-distribution along the layer thickness, CNT-fiber orientation and boundary
EP
conditions on the elastodynamic responses of laminated CNT-reinforced composite skew plates are examined.
AC C
Keywords: CNT-reinforced composites; elastodynamic; dynamic loading; laminated skew plates; IMLS-Ritz method
*
Corresponding author. E-mail address: [email protected]
ACCEPTED MANUSCRIPT 1. Introduction Nanocomposites, comprising nanoparticle reinforced composites and nanotube reinforced composites, have been the subject of increasing attention because of their excellent mechanical, optical and electronic applications [1]. Being one of the advanced nanomaterials, carbon nanotube (CNT) has been utilized as the
RI PT
reinforcement materials for nanocomposite structures [2]. Because of their excellent strength, CNT-reinforced composites can be exploited as an alternative substitute for the available functional materials [3-8]. However, research exploration of this material is still far away from its wide application and safe use. This aspiration has
SC
attracted many researchers to make efforts on deeply understanding of the mechanical behaviors of CNT-reinforced composites [9-12]. Mohammadimehr et al. [13] studied
M AN U
the vibration of viscoelastic double-bonded polymeric nanocomposite plate. Mantari and Granados [14] presented the bending and free vibration analyses of functionally graded plates resting on elastic foundation. Mantari and Granados [15] further presented the static analysis of functionally graded plates by using a new first shear deformation theory. Kiani [16] studied the free vibration behavior of CNT-reinforced functionally graded composite skew plates.
TE D
Due to the increasing needs of laminated composite structures in aerospace, mechanical, marine, and automotive industries, they become the subject of increasing research over decades [17] [18]. Reddy [19] used the finite element method to study the forced motions of laminated composite plates. The transverse shear strains, rotary
EP
inertia and large rotations were taken into accounted. Fan and Wang [20] discussed the effects of matrix cracks on the nonlinear bending and thermal postbuckling of
AC C
CNT-reinforced composite beams. Kumar et al. [21] studied the buckling of laminated composite skew plates with various boundary conditions under linearly varying in-plane loads. Kavousi et al. [22] studied the dynamic response of laminated composite beams subjected to multiple masses low-velocity impacts. Taehyo et al. [23] used a 4-node quasi-conforming shell element to study the static and dynamic responses of laminated composite plates and shells. The Mori-Tanaka approach [24] are well used to estimate the effective elastic parameters of composite materials. Mori-Tanaka [24] proposed a rational apprimation to correlate averaged stresses and strains of the constituent fiber with those of the
ACCEPTED MANUSCRIPT matrix in a composite. Benveniste [25] found that the Mori-Tanaka approach can be reformulated by making use of the equivalent inclusion idea in terms of a more compacted tensor [26]. Xin et al. [27] used the Mori-Tanaka approach to study the functionally graded thick-walled tube subjected to internal pressure.
RI PT
A proven approximation technique, the Ritz method [28] which is a generalized Rayleigh method, has been used in computational analyses. Employing the moving least-squares (MLS) shape function in the Ritz method was used by Zhou. et al. [29] for vibration analysis of skew plates. The MLS has been employed as an
SC
approximation function in the meshless/element-free method; however, the drawback of the MLS approximation has made it difficult to obtain an accurate numerical
M AN U
solution when the algebraic equation system becomes ill-conditioned or presence of singularity. To overcome this drawback, the IMLS approximation was proposed for construction of the shape functions [30-33].
In this paper, the Mori–Tanaka approach is employed to estimate the material properties of CNT-reinforced composites. The attention is only limited to the modeling of the dynamic responses of laminated CNT-reinforced composite skew
TE D
plates subjected to a transverse sudden dynamic loading. The element-free IMLS-Ritz procedure is employed to derive the discretized equations system for the elastodynamic problem. The elastodynamic behavior is furnished by employing the Newmark-β method. The effects of skew angles, width-to-thickness ratio, CNT-volume fraction,
EP
CNT-distribution along the layer thickness, CNT-fiber orientation and boundary conditions on the elastodynamic responses of laminated CNT-reinforced composite
AC C
skew plates are examined. 2. Problem description
A laminated plate which is perfectly bonded with CNT-reinforced composite layers of thickness h and the CNT-orientation is shown in Fig. 1. β is the angle of CNT-orientation. The plate, as illustrated in Fig. 2, is considered with UD, FG-O and FG-X CNT-distributions. The CNT-content by volume VCNT can be approximated by * UD : VCNT ( z ) = VCNT
(1)
ACCEPTED MANUSCRIPT 2z * FG-O : VCNT ( z ) = 2 1− V h CNT
(2)
z * FG-X : VCNT ( z ) = 4 VCNT h
(3)
* VCNT =
wCNT
wCNT + (
, ρ CNT )(1− wCNT ) m ρ
RI PT
where
(4)
SC
in which, wCNT is the mass fraction of CNTs, ρ m and ρ CNT are the densities of the matrix and carbon nanotube, respectively; z is a local thickness coordinate variable of
M AN U
* a typical layer, and VCNT ( z ) = VCNT corresponds to the uniformly distributed
CNT-composite layer (UD). The UD, FG-O and FG-X CNT-reinforced composite plates have the same mass and volume of CNTs.
2.1. Energy formulation of elastodynamic problem
On the basis of first order shear deformation laminated plate theory [34], the
TE D
displacement field of the CNT reinforced composite plate can be expressed as (5)
v ( x, y , z ) = v0 ( x, y ) + zθ y ( x, y ) ,
(6)
w( x, y, z ) = w0 ( x, y ) ,
(7)
EP
u ( x, y, z ) = u0 ( x, y) + zθ x ( x, y) ,
AC C
in which, (u, v, w) are the displacements ( x, y , z ) , (u0 , v0 , w0 ) are the displacement projections on the mid-plane, and the rotation functions should approach the respective slopes of the transverse deformation:
θx =
∂u ∂v ,θ y = ∂z ∂z
(8)
where θ x , θ y are the transverse normal rotations about the positive y and negative x axes. The nonzero strain tensor can be expressed as
ε xx γ yz ε yy = ε 0 + zκ , = γ 0 , γ xz γ xy
(9)
ACCEPTED MANUSCRIPT where
(10)
RI PT
∂u0 1 ∂w0 2 ∂θ x + ∂ x 2 ∂ x ∂x ∂w θy + 0 2 θ ∂ ∂y ∂v0 1 ∂w0 y ε0 = + ,κ = , γ0 = . ∂y ∂y 2 ∂y θ + ∂w0 x ∂θ ∂θ y ∂u0 ∂v0 ∂w0 ∂w0 ∂x x + + + ∂x ∂y ∂y ∂x ∂x ∂y
The moment resultants and external force of a composite plate can be derived as
SC
follows
(11)
N xx M xx Qy N = N yy , M = M yy , Qs = , Qx N xy M xy
(12)
M AN U
T N A B 0 ε0 N T M = B D 0 κ − M , Qs 0 0 A s γ 0 0
where
A ,
B ,
As are the matrices of extensional stiffness,
D and
TE D
in which
bending-extensional coupling stiffness, bending stiffness and transverse shearing stiffness, respectively. They can be defined as
AC C
EP
A11 A = A12 A16
A16 B11 A26 , B = B12 B16 A66
A12
A22 A26
D11 D = D12 D16
D12 D22 D26
B12 B22 B26
D16 As D26 , As = 44s A45 D66
B16 B26 , B66
A45s , A55s
(13)
(14)
where
( Aij , Bij , Dij ) = ∫
h /2
− h /2
Qij (1, z, z 2 )dz ,
Aijs = K ∫
h /2
− h /2
Qij dz ,
(15)
in which Qij (i , j = 1, 2, 6) are the plane-stress stiffness coefficients, Qij (i , j = 4, 5) are the shear stiffness coefficients and K is the shear correction factor which is taken as
K = 5 / (6 − ( v1V1 + v2V2 )) for functionally graded materials [35].
ACCEPTED MANUSCRIPT For a two-dimensional elastodynamic problem on a CNT-reinforced skew composite plate, we have
σ ij , j ( x , t ) + bi ( x , t ) = cu& ( x , t ) + ρ u&&( x , t ) , x = ( x1 , x2 ) ∈Ω ,
(16)
where ui is the displacement, ρ and c are the mass density and damping coefficient,
problem domain. In Eq. (16):
∂ui ∂ 2u , u&&i = 2i , ∂t ∂t
The boundary conditions are
M AN U
in which u&i is the velocity, u&&i is the acceleration.
SC
u&i =
RI PT
respectively. σ ij is to the stress tensor, bi is the body force and Ω refers to the
ui ( x) = u%i ( x), x ∈Γu , ti ( x ) = σ ij ( x ) n = t%i ( x ), x ∈ Γσ ,
(17)
(18)
where ti is the traction vector, and u%i and t% are the prescribed displacement and
respectively.
TE D
traction vector on the displacement boundary Γu and traction boundary Γσ ,
The initial conditions are taken to be
EP
ui ( x,0) = ui 0 ( x) , u&i ( x,0) = u&i 0 ( x) , x ∈ Ω .
(19)
AC C
The strain energy of the CNT-composite plate can be expressed as ,
(20)
where
A B 0 ε0 ε = κ , S= B D 0 . γ 0 0 As 0
(21)
and the external work can be expressed as We = ∫ u T f dΩ + ∫ u T t dΓ , Ω
Γ
(22)
ACCEPTED MANUSCRIPT where f represents the external load and t is the prescribed traction on the natural boundary. The total energy functional can be written in the following equation
RI PT
(23)
2.2. Geometric mapping
For a skew plate, the orthogonal coordinates x-y can be converted to an oblique
expressed as N
N
SC
coordinates ξ − η . As shown in Fig. 4, the transformation equations can be
M AN U
x = ∑ x i N i ( ξ , η ) , y = ∑ y i N i (ξ , η ) , i=1
( i =1,2,3,4),
(24)
i=1
1 N i = (1+ ξiξ )(1+ ηiη ) 4 where
N i (ξ ,η ) is the transformation shape function,
(25)
xi
and
yi
are the
TE D
co-ordinates of corner point i in the physical domain, and ξ i and ηi are the co-ordinates of corner point i in the computational domain. Using the chain rules in geometric transformation so as to obtain the first and second-order derivatives of an arbitrary function with respect to x and y coordinate
AC C
EP
variables to ξ and η coordinate varibales, i.e.
where
() () ()
,xx
,xy
, yy
() ()
−1 = J 22
,x
,y
= J −1 11
() () ()
,ξξ
,ξη
,ηη
() ()
,ξ
,η
,
−1 −1 − J 22 J 21 J11
(26)
() ()
,ξ
,η
,
(27)
ACCEPTED MANUSCRIPT y,ξ , y,η
(28)
x ,ξξ J 21 = x,ξη x,ηη
y,ξξ y,ξη , y,ηη
(29)
in which
J ij
y,ξ 2 y,ξ y,η y,η 2
(x,ξ x,η + x,η y,ξ ) , 2x,η y,η 2x,ξ y,ξ
(30)
SC
x 2 ,ξ J 22 = x,ξ x,η x 2 ,η
RI PT
x ,ξ J11 = x,η
is the transformation Jacobian matrix. Based on the above
M AN U
transformation Jacobian matrix, the governing equations can be transformed from the physical domain to a computational domain.
2.3. The Mori-Tanaka Approach
We employed the Mori-Tanaka approach for predicting the material properties of
TE D
CNT-reinforced composites. For a two phase composite, the effective elastic module tensor L of CNT-reinforced composites can be written in the following expression [36]:
（31）
EP
L = Lm + VCNT [( LCNT − Lm ) ⋅ A]⋅[Vm I + VCNT (A)]−1 ,
where LCNT and Lm are the stiffness tensors of CNT and matrix, I being a unit
AC C
tensor, and the angled brackets denote the average of an overall possible orientation of inclusions. A is given by
A = [I + S ⋅ L−1 (LCNT − Lm )]−1 , m
（32）
which is the dilute mechanical strain concentration tensor and S is the Eshelby tensor. Considering a polymer elastic matrix reinforced with aligned and straight CNTs. Each CNT is modeled as a long straight fiber with transversely isotropic elastic properties. The Hill’s elastic moduli of the CNTs and its inter-phase [37] are:
ACCEPTED MANUSCRIPT
nr
lr
lr
0
lr
kr + mr
kr − mr
0
lr
kr − mr
kr + mr
0
0
0
0
pr
−
v CNT E11CNT
−
v CNT E11CNT
0
1
−
v CNT CNT E22
0
E22CNT v CNT − CNT E22 0
1 E22CNT
0
(33)
RI PT
1 CNT E11 CNT −v E11CNT = CNT −v E CNT 11 0
0
1
G12CNT
Once we have the Hill’s elastic moduli for the CNTs, the Hill’s elastic moduli of the
k=
Em (Em f m + 2kr (1+ vm ) + fr [1− 2vm ]) 2(1+ vm )[Em (1+ fr − 2vm ) + 2 f m kr (1− vm − 2vm2 )]
l=
Em (vm f m [Em + 2kr (1+ vm )] + 2 f r kr [1− vm2 ]) (1+ vm )[2 f m kr (1− vm − 2vm2 ) + Em (1+ f r − 2vm )]
M AN U
n=
SC
naocomposite can be obtained according to [38] and expressed as:
Em2 f m (1+ f r − vm f m ) + 2 f m f r (kr nr − lr2 )(1− vm )2 (1− 2vm ) (1+ vm )[2 f m kr (1− vm − 2vm2 ) + Em (1+ f r − 2vm )]
TE D
Em [2 f m2 kr (1− vm ) + f r nr (1+ f r − 2vm ) + 4 f m f r lr vm ] + 2 f m kr (1− vm − 2vm2 ) + Em (1+ f r − 2vm ) p=
Em [Em f m + 2mr (1+ vm )(3+ f r − 4vm )] 2(1+ vm ){Em [ f m + 4 f r (1− vm )] + 2 f m mr (3− vm − 4vm2 )}
EP
m=
Em [Em f m + 2(1+ f r ) pr (1+ vm )] 2(1+ vm )[Em (1+ f r ) + 2 f m pm (1+ vm )]
(34)
(35)
(36)
(37)
(38)
where k is the plane-strain bulk modulus normal to the fiber direction, n is the uniaxial
AC C
tension modulus in the fiber direction, l is the associated cross modulus, m and p are the shear moduli in planes normal and parallel to the fiber direction, respectively. Consequently, the elastic moduli E11 and E22 of the composite parallel and normal to CNTs and Poisson’s ratio v12 can be expressed as:
l2 4m(kn − l 2 ) l E11 = n − , E22 = ,v12 = 2 k 2k kn − l
(39)
In this study, poly(methyl methacrylate), denoted as (PMMA) [39, 40] is selected for the polymer matrix material, whereas, the armchair (10, 10) SWCNTs are selected to be the reinforcement material. The material properties of the PMMA matrix are
ACCEPTED MANUSCRIPT v m = 0.34, ρ m = 1150kg / m3 ,v m = 0.34 , and E m = 2.5 GPa. The SWCNTs material properties are assumed to be G12CNT = 1.9455, E11CNT = 5.6466 TPa,
E22CNT = 7.0800 TPa
and G12CNT = 1.9455.
RI PT
2.4. Discrete system equations
The displacement field of the plates can be defined over the computational domain as m
h u ( x ) = ∑ p i ( x )ai ( x) = p ( x)a( x) , T
(40)
SC
i =1
where pi ( x )(i = 1, 2,…, m) are the basis function, m is the number of terms in the
M AN U
basis, and coefficients ai ( x ) are functions of the spatial coordinates x. Define a weighted discrete norm as n
J = ∑ w( x − x I )[u h ( x, x I ) − u( x I )]2 I =1
2
m = ∑ w( x − x I ) ∑ p i ( x I ) ⋅ a i ( x ) − u ( x I ) , I =1 i =1 w( x − x I )
(42b)
TE D
n
where
(41a)
is a weight function with the compact support, and xI ( I = 1, 2 ,..., n )
are the nodes with domains of influence that cover the point x.
EP
Moreover, the cubic spline function is a weight function which is given by
AC C
2 2 3 3 − 4 zI + 4 zI 4 4 w( x − x I ) = wI ( x ) = − 4 z I + 4 z I2 − z I3 3 3 0
zI =
x − xI , dI
1 2 1 for < zI ≤ 1 , 2 otherwise for 0 ≤ z I ≤
d I = d max cI ,
(43)
(44)
where d I is the size of the support, d max is a scaling factor (2.0 ≤ dmax ≤4.0) and cI is the average distance between nodes in the influenced domain of the point x. It is obvious that a( x) can be determined subjects to
ACCEPTED MANUSCRIPT ∂J = 0, ∂a
(45)
A( x ) a ( x ) = B ( x ) u ,
(46)
which leads to
RI PT
such that
A( x) = P TW ( x) P ,
(47)
B( x) = P TW ( x) ,
(48)
u = ( u1 , u 2 , L , u n ) T ,
(49)
SC
On the Hilbert space span( p ), the following inner product n
M AN U
( f , g) = ∑ w(x − x I ) f (x I )g(x I )
(50)
I =1
can be defined, and ∀f ( x), g( x) ∈ span( p) such that
p1 ( x1 ) 0 P= M 0
TE D
0 p2 ( x 2 ) M 0
L 0 L 0 , O M L pm ( x n )
EP
0 w( x − x1 ) 0 w( x − x 2 ) W ( x) = M M 0 0
L 0 L 0 . O M L w( x − x n )
(51)
(52)
AC C
The local approximation u h ( x ) is
,
(53)
where
% ( x ) = (Φ % ( x ), Φ % ( x ),L , Φ % ( x )) = p T ( x ) A% ( x ) B ( x ) . Φ 1 2 n
(54)
and m
% ( x ) = ∑ p ( x )[ A% ( x ) B( x )] , Φ I j jI j =1
(55)
ACCEPTED MANUSCRIPT which represents the shape function of the IMLS approximation corresponding to node I. The weighted orthogonal basis function set can be furnished according to the Schmidt method as
(56)
RI PT
p1 = 1,... p1 = r
(z i−1 , pk ) −∑ pk ,i = 2,3... k =1 ( pk , pk ) i−1
i−1
The weighted orthogonal basis function set p = ( pi ) , however, can be formed in terms of the polynomial function in the following fashion
SC
p% = ( p%i ) = (1, x, x2 , x3, x1x2 , x1x3, x2 x3, x12 , x22 , x32...) ,
(57)
Alternatively, the weighted orthogonal basis function set can be written in the
i −1
M AN U
following form ( p% i , pk ) pk , k =1 ( pk , pk )
pi = p% i − ∑
i = 1, 2,3,...
(58)
Substituting the approximation function into the functional defined in Eq. (23) and applying the Ritz procedure
TE D
∂Π * ∂u (t ) = 0, ∆ = u1 (t ), 1 , I = 1, 2, ⋅⋅⋅, n . ∂∆ ∂t
(59)
The discrete control equation (DCE) is
MU&& (t ) + CU& (t ) + KU (t ) = F (t ) ,
EP
(60)
where M is the global mass matrix, C is the damping matrix, F is the global tttt
AC C
external force vector and K is the global stiffness matrix, in which T T % T cΦ % dΩ , F = M = ∫ G T mGJ d Ω , C = ∫ Φ I J ∫Γ Φ% %d Γ + ∫Ω Φ% bd Ω ,
(61)
K = Kb + Km + Ks ,
(62)
Ω
Ω
σ
and
T
K IJb = ∫ BIb DBJb d Ω ,
(63)
Ω
T
T
T
K IJm = ∫ BIm DBJm d Ω + ∫ BIm DBJb d Ω + ∫ BIb DBJm d Ω , Ω
Ω
Ω
T
K IJs = ∫ BIs As BJs d Ω , Ω
(64) (65)
ACCEPTED MANUSCRIPT where
0
M AN U
∂Φ I ∂η ∂Φ I ∂ζ
0 0 0 0 0 0 , 0 0 0
(66)
∂Φ I ∂ζ ∂Φ I ∂η
0 0 s BI = 0 0
0 ΦI 0 0 0
0 0 ΦI 0 0
0 I0 0 0 I1
0 0 I0 0 0
TE D
Φ I 0 GI = 0 0 0
AC C
EP
I0 0 m =0 I1 0
(67)
SC
∂Φ I ∂ζ m BI = 0 ∂Φ I ∂η
0 ∂Φ I , ∂η ∂Φ I ∂ζ
RI PT
∂Φ I 0 0 0 ∂ζ BIb = 0 0 0 0 ∂Φ I 0 0 0 ∂η
ΦI 0
0 0 0 ΦI 0
I1 0 0 I2 0
0 , ΦI
0 0 0 , 0 Φ I
0 I1 0. 0 I 2
(68)
(69)
(70)
We can rewrite Eq. (48) without damping in the following expression
MU&& (t ) + KU (t ) = F (t ) .
(71)
2.5. Newmark-β method The Newmark-β method is employed for time discretization of equations of motion. The method has demonstrated to furnish accurate solution and provide good numerical
ACCEPTED MANUSCRIPT stability. The Taylor expansion is applied for displacement and velocity at a time step
t + ∆t , i.e. (72)
U& t +∆t = U& t + ∆tU&&t − β1∆tU&&t + β1∆tU&&t +∆t ,
(73)
U&&t +∆t =
RI PT
1 1 1 Ut +∆t = U t + ∆tU& t + ∆t 2U&&t − β2 ∆t 2U&&t + β2 ∆t 2U&&t +∆t , 2 2 2
2 2 & 1 U − Ut ) − U t − U&&t + U&&t . 2 ( t +∆t β2 ∆t β 2 ∆t β2
(74)
.
(75)
M AN U
3. Numerical examples and discussion
SC
Thus, Eq. (66) can be rewritten according to the Newmark-β method as
In this study, laminated CNT-composite skew plates are composed two parts: one part is the matrix is made of PMMA (polymethyl methacrylate) and the other part is the reinforcement is made of (10,10) SWCNTs.
The boundary conditions used in the part are defined as follows:
TE D
Simple supported ( S ): v0 = w0 = θ y = 0 for sides b and c,
u0 = w0 = θ x = 0 for sides a and d.
Clamped (C) :
u0 = v0 = w0 = θ x = θ y = 0 for sides b and c,
EP
u0 = v0 = w0 = θ x = θ y = 0 for sides a and d.
3.1. Validation studies
AC C
Comparison studies are conducted for a square three-layer [0o/90o/0o] clamped plate subjected to a sudden transverse dynamic load. The stability and accuracy of the IMLS-Ritz method are demonstrated by varying the number of nodes and scaling factors. Some geometrical properties of the laminated plates are: a = b = 25cm ,
h = 5cm , ρ = 8 × 10−6 N / cm2 , E2 = 2.1 × 106 N / cm 2 , E1 / E2 = 25 , v = 0.25 , G12 = G13 = G23 = 0.5E2 and q0 = 10 N / cm 2 as shown in Fig. 3. Different scaling factors d max are considered in this convergence study. It was concluded that the scaling factors d max , ranging from 2.1 to 3.4, has very little influence on the results. It is decided to use a scaling factor d max = 2.1 for the
ACCEPTED MANUSCRIPT following studies. As shown in Fig. 5(a), it can be found that the numbers of nodes ranging from 7 × 7 to 15 ×15 also have very little effects on the results. Therefore, the number of nodes 15 ×15 will be used in the following studies. Fig. 5(b) shows a comparison plot with Mindlin’s solution [41] and the HOSD solution [42]. The
well with the Mindlin solution [41] and HOSD solution [42].
RI PT
IMLS-Ritz results obtained using d max = 2.1 and number of nodes 15 ×15 agree
We have further carried out a convergence study to demonstrate the stability and accuracy of the IMLS-Ritz method for a fully clamped three-layer [0o/90o/0o] laminated CNT-composite skew plate with a uniform distribution of CNTs and a skew
SC
angle 60o subjected to a sudden dynamic load q0 = 10 N / cm 2 with different quantities of nodes and different scaling factor. The geometric properties of this
h = 2cm
CNT-composite ,
E11CN = 5.6466TPa
plate
are:
M AN U
laminated
,
E22CN = E33CN = 7.0800TPa
a = b = 20cm ,
v12CN = 0.175
, ,
E m = 2.5 × 10 9 N / m 2 , v m = 0.34 , and q0 = 10 5 N / m 2 . The volume fraction ratio of * = 0.11 . CNTs is taken as VCNT
As displayed in Figs. 6(a) and (b), different scaling factor d max and different
TE D
numbers of nodes considered in this convergence study for the laminated CNT-composite skew plate are considered. This study shows that the scaling factors
d max ranging from 2.1 to 3.4 and the numbers of nodes ranging from 9 × 9 to 15 × 15 have very small influence on the convergence of results. Therefore, it is
EP
decided to use the support size d max = 2.1 and number of nodes 15 × 15 for the
AC C
following parametric studies.
3.2 Parametric studies In this section, parametric studies are carried out to investigate the dynamic behaviors of laminated CNT-composite skew plates by considering various influenced factors. The geometrical properties of the laminated CNT-composite plate are: width and height a = b = 20cm , thickness h = 2cm and density ρ for the matrix and CNT are 1150 and 1400 kg / m 3 . The transverse sudden dynamic load q0 = 10 N / cm 2 . We consider three types of laminated CNT-composite plates with the following arrangements: (a) three-layer [0o/90o/0o], (b) four-layer [-45o/45o/45o/-45o] and (c)
ACCEPTED MANUSCRIPT four-layer [-45o/45o/-45o/45o]. The plate is having a skew angle α ranging from 30° to 90° with UD, FG-O and FG-X CNT-distributions. The CNT-volume fraction * =0.11. The plate is subjected to a fully clamped boundary condition. The results VCNT
obtained are shown in Figs. 7-9. It can be seen from the figures that the values of central deflection decreases dramatically with decreasing of the skew angle α.
RI PT
We also study the influence of width-to-thickness ratios on the central deflections of the laminated CNT-composite plates. The skew angle α selected for this study is 60o. Fig. 10 depicts the results for a fully clamped two-layer [0°/90°] laminated plate with UD, FG-O and FG-X CNT-distributions under different width-to-thickness ratios
SC
a / h = 10, 20,50 . The CNT-volume fraction is 0.11. It is evident that the FG-O and FG-X laminated CNT-composite plates are having the greatest and smallest of central
M AN U
deflections, respectively. The results concluded that when the plate is having FG-X CNT-distribution which means the CNT-distributions are close to top and bottom surfaces of the plate, the stiffness is the strongest. On the other way, when a plate is having the FG-O CNT-distribution which means the CNT-distributions are concentrated at the plate’s mid-plane, the stiffness is the weakest. Besides, we also study the influence of CNT-volume fraction on a fully clamped
TE D
four-layer [0°/90°/90°/0°] laminated CNT-composite plate subjected to a transverse sudden dynamic load. The skew angle α is chosen to be 60o for this study. Fig. 11 shows the central deflections for CNT-composite laminated plates which have the CNT-volume fractions of 0.11, 0.14, and 0.17, respectively. It is evident that the
EP
central deflection decreases with the increasing of CNT-volume fraction. It shows that the plate’s stiffness is larger when the CNT-volume fraction is higher.
AC C
The dynamic responses of full clamped angle-ply [-β/β/β/-β] laminated CNT-composite plates with UD, FG-O and FG-X CNT-distributions are examined. Fig. 12 presents the results of angle-ply laminated CNT-composite plates with β = 90o, 75 o, 60 o and 45o. With β varying from 90o to 45o, the central deflection decreases. Figs. 13-15 depicted the vertical central deflections for 4-ply [0°/90°/0°/90°], [-45o/45o/45o/-45o] and [-45o/45o/-45o/45o] laminated CNT-composite plates with UD, FG-O and FG-X CNT-distributions. The plate is subjected to four different types of boundary conditions, i.e. simply supported (SSSS), two edges simply supported and the other two free (SFSF), fully clamped (CCCC), and two edges clamped and the * other two free (CFCF). The CNT-volume fraction VCNT = 0.11 , width-to-thickness
ACCEPTED MANUSCRIPT ratios a / h =10 and skew angle α = 60o are chosen for this study. It is found that the vertical central deflection of the laminated CNT-composite plates with (a) SFSF boundary condition is the largest, and (b) CCCC boundary condition is the smallest. Besides, it is also observed that the vertical central deflection of SSSS and CFCF plates lied between SFSF and CCCC plates. This is expected and can be explained
RI PT
that the clamped edge has the strongest constraint, followed by simply-supported edge and then free edge.
4. Conclusion
The elastodynamic behavior of laminated CNT-reinforced composite skew plates
SC
subjected to a transverse sudden dynamic load is studied. The governing equations of plates are derived based on the FSDT. The IMLS-Ritz method is employed for
M AN U
solution. The numerical time discretization used to determine the dynamic responses of plate is carried out using the Newmark-β method. The Mori-Tanaka approach is employed to estimate the material properties of CNT-reinforced composites. Different types of CNT-distributions along the thickness of the layers are considered, i.e. uniform distributed (UD) and functionally graded distributions (FG-O and FG-X). The effects of CNT-volume fraction, CNT-distributions, geometrical shape
TE D
parameters and width-to-thickness ratio on the dynamic behavior of skew laminated plates with different boundary conditions are examined. These results, unknown
EP
before, can be used for future references.
Acknowledgement
AC C
The work described in this paper was fully supported by grants from the National Natural Science Foundation of China (Grant no. 11402142).
References
[1] Sang HC, Lee YH. Carbon nanotubes and graphene towards soft electronics. Nano Convergence. 2014;1(1):1-26. [2] Liew KM, Lei ZX, Zhang LW. Mechanical analysis of functionally graded carbon nanotube reinforced composites: A review. Composite Structures. 2015;120:90-7. [3] Ansari R, Torabi J, Shojaei MF. Buckling and vibration analysis of embedded functionally graded carbon nanotube-reinforced composite annular sector plates under thermal loading. Composites Part B: Engineering. 2017;109:197-213. [4] Jooybar N, Malekzadeh P, Fiouz A. Vibration of functionally graded carbon nanotubes reinforced composite truncated conical panels with elastically restrained
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
against rotation edges in thermal environment. Composites Part B: Engineering. 2016;106:242-61. [5] Pourasghar A, Chen Z. Thermoelastic response of CNT reinforced cylindrical panel resting on elastic foundation using theory of elasticity. Composites Part B: Engineering. 2016;99:436-44. [6] Hui D. Effects of shear loads on vibration and buckling of antisymmetric cross-ply cylindrical panels. International Journal of Non-Linear Mechanics. 1988;23(3):177-87. [7] Dong S, Zhou J, Hui D, Pang X, Wang Q, Zhang S, et al. Interaction between edge dislocations and amorphous interphase in carbon nanotubes reinforced metal matrix nanocomposites incorporating interface effect. International Journal of Solids and Structures. 2014;51(5):1149-63. [8] Tornabene F, Fantuzzi N, Bacciocchi M. Linear static response of nanocomposite plates and shells reinforced by agglomerated carbon nanotubes. Composites Part B: Engineering. 2017, http://dx.doi.org/10.1016/j.compositesb.2016.07.011, In press. [9] Zhang LW. Geometrically nonlinear large deformation of CNT-reinforced composite plates with internal column supports. Journal of Modeling in Mechanics and Materials. 2017;1(1):20160154. [10] Viet NV, Wang Q, Kuo WS. A studying on load transfer in carbon nanotube/epoxy composites under tension. Journal of Modeling in Mechanics and Materials. 2017;1(1):20160153. [11] Ashour AS. Vibration of variable thickness plates with edges elastically restrained against translation and rotation. Thin-Walled Structures. 2004;42(1):1-24. [12] Fantuzzi N, Tornabene F, Bacciocchi M, Dimitri R. Free vibration analysis of arbitrarily shaped Functionally Graded Carbon Nanotube-reinforced plates. Composites Part B: Engineering. 2017, http://dx.doi.org/10.1016/j.compositesb.2016.09.021, In press. [13] Mohammadimehr M, Rousta Navi B, Ghorbanpour Arani A. Free vibration of viscoelastic double-bonded polymeric nanocomposite plates reinforced by FG-SWCNTs using MSGT, sinusoidal shear deformation theory and meshless method. Composite Structures. 2015;131:654-71. [14] Mantari JL, Granados EV. An original FSDT to study advanced composites on elastic foundation. Thin-Walled Structures. 2016;107:80-9. [15] Mantari JL, Granados EV. A refined FSDT for the static analysis of functionally graded sandwich plates. Thin-Walled Structures. 2015;90:150-8. [16] Kiani Y. Free vibration of FG-CNT reinforced composite spherical shell panels using Gram-Schmidt shape functions. Composite Structures. 2017;159:368-81. [17] Casimir JB, Khadimallah MA, Nguyen MC. Formulation of the dynamic stiffness of a cross-ply laminated circular cylindrical shell subjected to distributed loads. Computers & Structures. 2016;166:42-50. [18] Tornabene F, Fantuzzi N, Bacciocchi M, Viola E. Effect of agglomeration on the natural frequencies of functionally graded carbon nanotube-reinforced laminated composite doubly-curved shells. Composites Part B: Engineering. 2016;89:187-218. [19] Reddy JN. Geometrically nonlinear transient analysis of laminated composite plates. Aiaa Journal. 1982;21(4):621-9. [20] Fan Y, Wang H. The effects of matrix cracks on the nonlinear bending and thermal postbuckling of shear deformable laminated beams containing carbon nanotube reinforced composite layers and piezoelectric fiber reinforced composite layers. Composites Part B: Engineering. 2016;106:28-41. [21] Kumar A, Panda SK, Kumar R. Buckling behaviour of laminated composite skew plates with various boundary conditions subjected to linearly varying in-plane edge loading. International Journal of Mechanical Sciences. 2015;100:136-44.
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
[22] Latifi M, Kharazi M, Ovesy HR. Nonlinear dynamic response of symmetric laminated composite beams under combined in-plane and lateral loadings using full layerwise theory. Thin-Walled Structures. 2016;104:62-70. [23] Park T, Kim K, Han S. Linear static and dynamic analysis of laminated composite plates and shells using a 4-node quasi-conforming shell element. Composites Part B Engineering. 2005;37(s 2–3):237-48. [24] Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica. 1973;21(5):571-4. [25] Benveniste Y. A new approach to the application of Mori-Tanaka's theory in composite materials. Mechanics of Materials. 1987;6(2):147-57. [26] Liu L, Huang Z. A Note on mori-tanaka's method. Acta Mechanica Solida Sinica. 2014;27(3):234-44. [27] Xin L, Yang S, Zhou D, Dui G. An approximate analytical solution based on the Mori-Tanaka method for functionally graded thick-walled tube subjected to internal pressure. Composite Structures. 2015(May):74–82. [28] Ritz W. Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik. Journal Für Die Reine Und Angewandte Mathematik. 1909;1909(135):1-61. [29] Zhou L, Zheng WX. Vibration of skew plates by the MLS-Ritz method. International Journal of Mechanical Sciences. 2008;50(7):1133-41. [30] Cheng RJ, Liew KM. Analyzing modified equal width (MEW) wave equation using the improved element-free Galerkin method. Engineering Analysis with Boundary Elements. 2012;36(9):1322-30. [31] Zhang Z, Liew KM, Cheng Y, Lee YY. Analyzing 2D fracture problems with the improved element-free Galerkin method. Engineering Analysis with Boundary Elements. 2008;32(3):241-50. [32] Li D, Bai F, Cheng Y, Liew KM. A novel complex variable element-free Galerkin method for two-dimensional large deformation problems. Computer Methods in Applied Mechanics and Engineering. 2012;233–236:1-10. [33] Ren H, Cheng Y. The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems. Engineering Analysis with Boundary Elements. 2012;36(5):873-80. [34] Reddy JN. Mechanics of laminated composite plates and shells: Theory and analysis, Second ed.,: CRC Press, Boca Raton, FL; 2004. [35] Efraim E, Eisenberger M. Exact vibration analysis of variable thickness thick annular isotropic and FGM plates. Journal of Sound and Vibration. 2007;299(4–5):720-38. [36] Eshelby JD. The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 1957;241(1226):376-96. [37] Malekzadeh P, Dehbozorgi M. Low velocity impact analysis of functionally graded carbon nanotubes reinforced composite skew plates. Composite Structures. 2016;140:728-48. [38] Shi D-L, Feng X-Q, Huang YY, Hwang K-C, Gao H. The Effect of Nanotube Waviness and Agglomeration on the Elastic Property of Carbon Nanotube-Reinforced Composites. Journal of Engineering Materials and Technology. 2004;126(3):250-7. [39] Han Y, Elliott J. Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites. Computational Materials Science. 2007;39(2):315-23. [40] Shen HS. Nonlinear bending of shear deformable laminated plates under transverse and in-plane loads and resting on elastic foundations. Compos Struct. 2000;50(2):131-42.
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
[41] Reddy JN. Dynamic (transient) analysis of layered anisotropic composite‐material plates. International Journal for Numerical Methods in Engineering. 1983;19(2):237-55. [42] Mallikarjuna, Kant T. Dynamics of laminated composite plates with a higher order theory and finite element discretization. Journal of Sound & Vibration. 1988;126(3):463-75.
ACCEPTED MANUSCRIPT Figures z
z
x h
y
y
x
RI PT
h/2
β
α
(a) UD
M AN U
SC
Fig. 1 Geometry of laminated CNT-composite skew plate and the CNT orientation.
(b) FG-O
(c) FG-X
Fig. 2 Cross section of the CNT-reinforced composite plates with three types of
AC C
EP
TE D
CNT-distributions.
P(t)
qoP
O
t
Fig. 3 The plate is subjected to a transverse sudden dynamic loading.
ACCEPTED MANUSCRIPT η
y
(1,1) 3
(-1,1) 4 3
d
b 1
ζ
c
α
1
x a
(-1,-1)
2
2 (1,-1)
RI PT
4
Fig. 4 Transformation of a skew plate from the physical domain to a computational
M AN U
SC
domain.
dmax=2.1, 15×15 Ref. [38] Ref.[32] Ref. [39] Ref.[33]
3
-6
-6
Central deflection, w (10 m)
3
Central deflection, w (10 m)
7×7 11×11 15×15
0
0.5 1.0 -4 Time(10 s)
1.5
2
1
0 -0.4 0
AC C
-0.4 0
EP
1
TE D
2
(a)
0.5 1.0 -4 Time(10 s)
1.5
(b)
Fig. 5 Convergence and comparison studies for a square laminated three-layer [0o/90o/0o] clamped plate on: (a) different number of nodes and (b) comparison with the existing the Mindlin solutions and HOSD results.
ACCEPTED MANUSCRIPT
present:9×9 present:11×11
4 3 2 1 0 0
1
2 3 -4 Time(10 s)
(a)
5
M AN U dmax=2.1
4
dmax=2.5
3 2
dmax=2.9
1 0
dmax=3.4
TE D
-5
Central deflection, w (10 m)
5
4
RI PT
present:13×13 present:15×15
SC
-5
Central deflection, w (10 m)
5
0
1
2 3 Time(10-4s)
4
5
EP
(b) Fig. 6 Convergence study for a four-layer [0o/90o/90o/0o] laminated CNT-composite skew plate with UD CNT-distribution on: (a) different numbers of nodes ranging
AC C
from 9 × 9 to 15 × 15 and (b) different support sizes d max ranging from 2.1 to 3.4.
ACCEPTED MANUSCRIPT UD 6
α =90
o
α =60
o
α =75
o
α =45
o
4 3
1 0 2
4 6 -4 Time(10 s)
8
FG-O
6
α =90
o
α =75
o
5 4 3 2 1 0 2
o
α =45
o
4 6 -4 Time(10 s)
TE D
-1 0
α =60
10
α =30
o
M AN U
7
RI PT
2
8 -5
o
5
-1 0
Central deflection,w (10 m)
α =30
SC
-5
Central deflection, w (10 m)
7
8
10
FG-X
7 6
α =90
o
α =75
o
α =60
o
α =45
o
α =30
o
EP
-5
Central deflection, w (10 m)
8
5 4
AC C
3 2 1 0
-1 0
2
4 6 -4 Time(10 s)
8
10
Fig. 7 The central deflection versus time for a three-layer [0o/90o/0o] laminated CNT-composite skew plate having skew angle α ranging from 30° to 90° with UD, * FG-O and FG-X CNT-distributions (CCCC boundary condition and VCNT =0.11).
ACCEPTED MANUSCRIPT UD 6
α =90
o
α =60
o
α =75
o
α =45
o
α =30
5 4 3
1 0 2
4 6 -4 Time(10 s)
8
FG-O 7 o
α =60
o
α =75
o
α =45
o
5 4 3 2 1 0 -1 0
10
α =30
o
M AN U
α =90
TE D
-5
6
RI PT
2
-1 0
Central deflection, w (10 m)
o
SC
-5
Central deflection, w (10 m)
7
2
4 6 Time(10-4s)
8
10
FG-X
7
o
α =75
o
EP
α =90
α =60
o
α =45
o
α =30
o
5 4
AC C
-5
Central deflection, w (10 m)
6
3 2 1 0
-1 0
2
4 6 -4 Time(10 s)
8
10
Fig. 8 The central deflection versus time for a four-layer [-45o/45o/45o/-45o] laminated CNT-composite skew plate having skew angle α ranging from 30° to 90° with UD, * =0.11). FG-O and FG-X CNT-distributions (CCCC boundary condition and VCNT
ACCEPTED MANUSCRIPT UD 6
α=90o
α=60o
α=75o
α=45o
5 4 3
1 0 2
4 6 Time(10-4s) FG-O
-5
6
α=90o
α=60o
α=75o
α=45o
5 4 3 2 1 0
TE D
-1 0
10
α=30o
M AN U
7
8
RI PT
2
-1 0
Central deflection, w (10 m)
α=30o
SC
-5
Central deflection, w (10 m)
7
2
4 6 -4 Time(10 s)
8
10
FG-X
7
α=90o
α=60o
α=75o
α=45o
5
α=30o
4
AC C
-5
Central deflection, w (10 m)
EP
6
3 2 1 0
-1 0
2
4 6 Time(10-4s)
8
10
Fig. 9 The central deflection versus time for a four-layer [-45o/45o/-45o/45o] laminated CNT-composite skew plate having skew angle α ranging from 30° to 90° with UD, * =0.11). FG-O and FG-X CNT-distributions (CCCC boundary condition and VCNT
ACCEPTED MANUSCRIPT UD:a/th=10 FG-O:a/th=10 FG-X:a/th=10
6 5 4 3
1 0 -1 0
2
4 6 -4 Time(10 s) UD:a/th=20 FG-O:a/th=20 FG-X:a/th=20
20 15 10 5 0 2
4 6 Time(10-4s)
TE D
-5 0
M AN U
-5
Central deflection, w (10 m)
25
250 200
10
8
10
UD:a/th=50 FG-O:a/th=50 FG-X:a/th=50
EP
-5
Central deflection, w (10 m)
8
RI PT
2
SC
-5
Central deflection, w (10 m)
7
150
AC C
100
50
0
-50 0
2
4 6 -4 Time(10 s)
8
10
Fig. 10 The central deflection versus time for a fully clamped two-layer [0°/90°] laminated CNT-composite skew plates having a/h = 10, 20 and 50 with UD, FG-O * and FG-X CNT-distributions ( VCNT =0.11).
ACCEPTED MANUSCRIPT *
VCNT =0.11
5
*
VCNT =0.14
*
VCNT =0.17
4 3 2 1 0 -1 0
2
4 6 -4 Time(10 s)
8
*
VCNT =0.14
3 2 1 0 -1 0
2
*
VCNT =0.17
M AN U
4
4 6 -4 Time(10 s)
10
SC
*
VCNT =0.11
5
TE D
-5
Central deflection, w (10 m)
FG-O 6
RI PT
-5
Central deflection, w (10 m)
UD 6
8
10
5 4 3
*
VCNT =0.11
*
VCNT =0.14
*
VCNT =0.17
EP
-5
Central deflection, w (10 m)
FG-X
6
2 1
AC C
0
-1 0
2
4 6 -4 Time(10 s)
8
10
Fig. 11 The central deflections versus time for a fully clamped four-layer * [0°/90°/90°/0°] laminated plates having VCNT =0.11, 0.14 and 0.17 with UD, FG-O
and FG-X CNT-distributions.
ACCEPTED MANUSCRIPT o
o
o
o
o
o
o
o
(-90 /90 /90 /-90 )
9
-5
(-60 /60 /60 /-60 )
7
o
o
o
o
o
o
o
(-45 /45 /45 /-45 )
5
RI PT
3 1 -1 0
2
4 6 -4 Time(10 s) FG-O o
o
o
o
8
10
o
o
o
o
9
(-90 /90 /90 /-90 )
(-75 /75 /75 /-75 )
7
(-60o/60o/60o/-60o)
(-45o/45o/45o/-45o)
M AN U
-5
Central deflection, w (10 m)
o
(-75 /75 /75 /-75 )
SC
Central deflection, w (10 m)
UD
5 3 1 -1 0
4 6 -4 Time(10 s)
TE D
2
8
10
9
-5
Central deflection, w (10 m)
FG-X
o
o
o
o
o
o
o
(-60 /60 /60 /-60 )
o
o
o
o
(-75 /75 /75 /-75 ) o
o
o
o
(-45 /45 /45 /-45 )
EP
7
o
(-90 /90 /90 /-90 )
5
AC C
3 1
-1 0
2
4 6 -4 Time(10 s)
8
10
Fig. 12 The central deflections versus time for angle-ply [-β/β/β/-β] laminated CNT-composite skew plates having β = 45o, 60o, 75o and 90o with UD, FG-O and FG-X CNT-distributions.
ACCEPTED MANUSCRIPT
2
4 6 -4 Time(10 s)
8
10
SFSF CFCF
M AN U
SSSS CCCC
2
4 6 -4 Time(10 s)
TE D
-5
Central deflection, w (10 m)
FG-O 9 8 7 6 5 4 3 2 1 0 -1 0
SFSF CFCF
RI PT
SSSS CCCC
SC
-5
Central deflection, w (10 m)
UD 9 8 7 6 5 4 3 2 1 0 -1 0
8
10
SSSS CCCC
SFSF CFCF
AC C
-5
EP
Central deflection, w (10 m)
FG-X
9 8 7 6 5 4 3 2 1 0 -1 0
2
4 6 Time(10-4s)
8
10
Fig. 13 The central deflections for a four-layer [0°/90°/0°/90°] laminated CNT-composite skew plate having UD, FG-O and FG-X CNT-distributions subjected to SSSS, CCCC, SFSF, CFCF boundary conditions.
ACCEPTED MANUSCRIPT
2
4 6 -4 Time(10 s)
8
10
M AN U
SSSS CCCC SFSF CFCF
2
4 6 -4 Time(10 s)
TE D
-5
Central deflection, w (10 m)
FG-O 10 9 8 7 6 5 4 3 2 1 0 -1 0
RI PT
SSSS CCCC SFSF CFCF
SC
-5
Central deflection, w (10 m)
UD 10 9 8 7 6 5 4 3 2 1 0 -1 0
8
10
8
10
SSSS CCCC SFSF CFCF
AC C
-5
EP
Central deflection, w (10 m)
FG-X
10 9 8 7 6 5 4 3 2 1 0 -1 0
2
4 6 Time(10-4s)
Fig. 14 The central deflections for a four-layer [-45o/45o/45o/-45o] laminated CNT-composite skew plate having UD, FG-O and FG-X CNT-distributions subjected to SSSS, CCCC, SFSF, CFCF boundary conditions.
ACCEPTED MANUSCRIPT SSSS CCCC
2
4 6 -4 Time(10 s)
10
M AN U
SSSS CCCC SFSF CFCF
2
4 6 -4 Time(10 s)
TE D
-5
Central deflection, w (10 m)
FG-O 10 9 8 7 6 5 4 3 2 1 0 -1 0
8
RI PT
SFSF CFCF
SC
-5
Central deflection, w (10 m)
UD 10 9 8 7 6 5 4 3 2 1 0 -1 0
8
10
8
10
SSSS CCCC SFSF CFCF
AC C
-5
EP
Central deflection, w (10 m)
FG-X
10 9 8 7 6 5 4 3 2 1 0 -1 0
2
4 6 Time(10-4s)
Fig. 15 The central deflections for a four-layer [-45o/45o/-45o/45o] laminated CNT-composite skew plates having UD, FG-O and FG-X CNT-distributions subjected to SSSS, CCCC, SFSF, CFCF boundary conditions.