Mechanics of Composite Materials, Vol. 49, No. 4, September, 2013 (Russian Original Vol. 49, No. 4, July-August, 2013)

A Nonlinear stability analysis of imperfect
three-phase polymer composite plates

Nguyen Dinh Duc,1* Tran Quoc Quan,1 and Do Nam2
In memory of G. A. Vanin
Keywords: nonlinear stability analysis, laminated 3-phase composite plate, imperfection
An analytical investigation into the nonlinear response of a thin imperfect laminated three-phase polymer
composite plate consisting of a matrix and reinforcing fibers and particles and subjected to mechanical loads
is presented. All formulations are based on the classical theory of plates with account of interaction between
the matrix and reinforcement, the geometrical nonlinearity, and an initial geometrical imperfection. By using
the Galerkin method, explicit relations for the load–deflection relationships are determined. The effects of
reinforcing fibers and particles, material and geometrical properties, and imperfections on the buckling and
postbuckling load-carrying capacities of a 3-phase composite plate are analyzed and discussed.

1. Introduction
Three-phase composites are materials consisting of a matrix and reinforcing fibers and particles. They have been investigated by Vanin G. A and Duc N. D since 1996 [1, 2], who determined the elastic modulus for various 3-phase composites
[3, 4]. Their findings have shown that fibers are able to improve the elastic modulus, but particles can retard the penetration
of heat [5], reduce the creep [6], and hinder the formation of defects in materials [7].
Despite the large number of applications, our understanding of their structure is inadequate. A general overview of
3-phase composites can be found in [8]. Recently, the deflection [9] and the creep of three-phase composite plates in bending
have been studied. These investigations have shown that an optimal 3-phase composite can be obtained by controlling the
volume ratios of fibers and particles.

Vietnam National University,Hanoi, 144 Xuan Thuy-Cau Giay- Hanoi-Viet Nam
University of Engineering and Technology, Vietnam National University, Hanoi
*
Corresponding author; tel: 84-4-37547978; fax: 84-4-37547724; e-mail: [email protected]
1
2



G = Gm



K = Km

where



L=

1 −ψ c ( 7 − 5ν m ) H

1 + ψ c ( 8 − 10ν m ) H

,

1 + 4ψ c Gm L ( 3K m )

−1

1 − 4ψ c Gm L ( 3K m )

−1

(1)


The elastic moduli for the 3-phase composite reinforced with unidirectional fibers are calculated as in [24]:



346

E11 = ψ a Ea + (1 −ψ a ) E +

8Gψ a (1 −ψ a ) (ν a −ν

)

2 −ψ a + xψ a + (1 −ψ a ) ( xa − 1)

G
Ga

,


TABLE 1. Elastic Moduli of 3-Phase Composite Materials
E1, GPa

Case

E2, GPa

ν12

G12, GPa

2.0103
1.8382
1.6843
1.5461

ψ a = const , y c increases
1
2
3
4
5
6
7

18.2019
17.8415
17.5209
17.2338
16.9751
16.7404
16.5273

8.0967
7.4411
6.8385
6.2829
5.7687
5.2916
4.8474



where



1.0513
1.0116
0.9457
0.8496
0.7190
0.5486
0.3327



G
G
χ (1 −ψ a ) + 1 + ψ a χ
 2
 2 (1 −ψ a ) ( χ − 1) + ( χ a − 1)( χ − 1 + 2ψ a )
G
G
ν
1

a
a

+ 2
=  21 +

G
χ + ψ a + (1 −ψ a )
Ga
Ga
G12 = G
, G23 = G
,
G
G
1 −ψ a + (1 + ψ a )
(1 −ψ a ) χ + 1 + χψ a
Ga
Ga
1 + ψ a + (1 −ψ a )





7.9971
7.3880
6.8385
6.3402
5.8860
5.4702
5.0880

(

ν 23


ν 21 = ν −

( χ + 1) (ν −ν

a

)ψ a

G
2 −ψ a + χψ a + (1 −ψ a ) ( χ a − 1)
Ga

(3)


,




,

x = 3 − 4ν .

For numerical calculations, we chose a 3-phase polymer composite made of a polyester AKAVINA matrix (made
in Vietnam), glass fibers (made in Korea), and titanium oxide (made in Australia) with the following properties: AKAVINA —
E = 1.43 GPa and n = 0.45; glass fibers — E = 22 GPa and n = 0.24; TiO2 — E = 5.58 GPa and n = 0.20 [25].
347



0.55
0.2
0.25

0.6
0.2
0.2

0.65
0.2
0.15

0.7
0.2
0.1

0.75
0.2
0.05

0.8
0.2
0.0

8
0.5
0.3
0.2


a

b

Fig 1. Black-and-white SEM images of 1D (a) and 2D (b) 2-phase composite materials (25% fibers,
without particles).
The elastic moduli of the composite materials calculated by formulas (3) for different volume fractions of components
are given in Table 1.
The 14 cases considered are detailed in Table 2.
Figure 1 illustrates SEM images of the structure of 2-phase polymer composites with glassy polyester fibers (without
particles). In the 1D composite, all fibers are oriented in one direction, but in the 2D one — in two perpendicular directions.
Figure. 2 illustrates the structure of 3-phase 1Dm and 2Dm composites, respectively, containing TiO2 particles.
From these images, which were taken using SEM in the Laboratory of Micro-Nano Technology of the University of
Engineering and Technology, Vietnam National University, Hanoi, it is seen that the greater amount of particles is introduced,
the finer is the resulting material.
3. Governing Equations
Consider a 3-phase midplane-symmetric composite plate. The plate is referred to a Cartesian coordinate system
( x, y,z ), where xy is the midplane of the plate and z is the thickness coordinate, − h 2 £ z £ h 2 . The length, width, and
total thickness of the plate are a , b and h , respectively.
In this study, the classical theory of shells is used to establish the governing equations and to determine the nonlinear
response of composite plates [12, 15]. According to this theory,

348


b

a

Fig. 2. Black-and-white SEM images of 1Dm (a) and 2Dm (b) 3-phase composite materials (25%

,x , y 

  , y ,x

 k   −w 
, xx
 x  

 k y  =  − w, yy  ,

 

 k xy   −2 w, xy 



with u and v the displacement components along the x and y directions, respectively.
Hooke’s law for the composite plate is expressed as
σ 
′
 Q11
 x 
 ′
 σ y  =  Q12


 Q′
 σ xy  k  16
where





′ = (Q12 − Q22 + 2Q66 ) sin 3 θ cos θ + (Q11 − Q12 − 2Q66 ) sin θ cos3 θ ,
Q16



′ = Q11 sin 4 θ + Q22 cos 4 θ + 2(Q12 + 2Q66 ) sin 2 θ cos 2 θ ,
Q22



′ = (Q11 − Q12 − 2Q66 ) sin 3 θ cos θ + (Q12 − Q22 + 2Q66 ) sin θ cos3 θ ,
Q26


with


(5)

′ = Q66 (sin 4 θ + cos 4θ ) + Q11 + Q22 − 2(Q12 + Q66 ) sin 2 θ cos 2 θ
Q66
Q11 =

E1
E1
,
=

Here, θ is the angle between fibers and the coordinate system. The force and moment resultants of the composite plates are
determined by [13]
n

Ni = ∑



hk

∫

k =1 hk −1
n

Mi = ∑

hk

∫

k =1 hk −1

σ i k dz , i = x, y, xy,

(6)
z σ i k dz , i = x, y, xy.

Insertion of Eqs. (4) and (5) into Eq. (6) gives the constitutive relations



Dij =

1 n
∑ (Qij′ )k (hk2 − hk2−1 ),
2 k =1

1 n
∑ (Qij′ )k (hk3 − hk3−1 ), i, j = 1, 2, 6.
3 k =1

The nonlinear equilibrium equations of the composite plate are [13]:




N x, x + N xy , y = 0

N xy , x + N y , y = 0 ,

M x, xx + 2 M xy , xy + M y , yy + N x w, xx + 2 N xy w, xy + N y w, yy = 0 .

(8)

From relations (7), it follows that



*
*



350

*
A11
=

2
A22 A66 − A26
A A − A12 A66
A A − A22 A16
*
*
, A12
= 16 26
, A16
= 12 26
,
∆
∆
∆

(9)




*
A22

*
*
*
*
*
*
F11 = A11
B11 + A12
B12 + A16
B16 , F12 = A11
B12 + A12
B22 + A16
B26 ,



*
*
*
*
*
*
F16 = A11
B16 + A12
B26 + A16
B66 , F21 = A12
B11 + A22
B12 + A26
B16 ,


B26 ,



*
*
*
F66 = A16
B16 + A26
B26 + A66
B66 .

Introducing Eq. (9) into the expression of M ij in relations (7) and then the results into Eq. (8), we have



N x, x + N xy , y = 0 , N xy , x + N y , y = 0 ,



P1 f, xxxx + P2 f, yyyy + P3 w, xxyy + P4 w, xxxy + P5 w, xyyy + P6 w, xxxx + P7 w, yyyy



+ P8 w, xxyy + P9 w, xxxy + P10 w, xyyy + N x w, xx + 2 N xy w, xy + N y w, yy = 0

with




N x = f, yy , N y = f, xx , N xy = − f, xy .

(11)

For an imperfect composite plate, Eqs. (10) are modified to the form [22, 23]

P1 f, xxxx + P2 f, yyyy + P3 w, xxyy + P4 w, xxxy + P5 w, xyyy + P6 w, xxxx + P7 w, yyyy + P8 w, xxyy


(

)

(

)

(

)

+ P9 w, xxxy + P10 w, xyyy + f, yy w, xx + w,*xx − 2 f, xy w, xy + w,*xy + f, xx w, yy + w,*yy = 0,

(12)

where w* ( x, y ) is a known function representing an initial small imperfection of the plate. The geometrical compatibility
equation for an imperfect composite plate is written as




f, xy − F21k x − F22 k y − F26 k xy ,



0
*
*
*
ε xy
= A16
f, yy + A26
f, xx − A66
f, xy − F61k x − F62 k y − F66 k xy .

(14)

351


Inserting Eqs. (14) into Eq. (13) gives the compatibility equation for an imperfect composite plate [16, 21-23]



*
*
A11
f, xxxx + E1 f, xxyy + A22
f, yyyy + F21 w , xxxx + F12 w , yyyy + E2 w , xxyy




=
w N=
M y = 0 , N y = N y 0 at y = 0, b .
xy

(16)

Case 2. All four edges of the plate are simply supported and immovable (IM). In this case, the boundary conditions are

w= u= M x = 0 , N x = N x 0 at x = 0, a ,
w= v= M y = 0 , N y = N y 0 at y = 0, b .

(17)

Case 3. All edges are simply supported. The edges x = 0, a are freely movable, whereas the edges y = 0, b are immovable. In this case, the boundary conditions are defined as
w N=
M x = 0 , N x = N x 0 at x = 0, a ,
=
xy



w= v= M y = 0 , N y = N y 0 at y = 0, b ,

(18)

where N x0 and N y0 are in-plane compressive loads at the movable edges (i.e., Case 1 and the first of Case 3) or fictitious
compressive edge loads at the immovable edges (i.e., Case 2 and the second of Case 3).
The approximate solutions w and f satisfying boundary conditions (16)-(18) are assumed to be [22, 23]


352

A1 =

1

⋅

δ n2

*
λm2
32 A22

W (W + 2 µ h) , A2 =

A3 =

1

⋅

λm2

*
δ n2
32 A11

−( F21λm4 + F12δ n4 + E2 λm2 δ n2 )



F λm4 + F12δ n4 + E2 λm2 δ n2 4
F21λm4 + F12δ n4 + E2 λm2 δ n2 4
ab
[ P1 21
+
P
λ
δn
2 * 4
m
* 4
* 4
* 4
4
A22
λm + A11
δ n + E1λm2 δ n2
A22 λm + A11
δ n + E1λm2 δ n2



* 4
* 4
A22
λm + A11
δ n + E1λm2 δ n2


δ n + E1λm2 δ n2

+ P2

λm4 δ n2 ( E3λm2 + E4δ n2 )

* 4
* 4
A22
λm + A11
δ n + E1λm2 δ n2



− P6 λm4 − P7δ n4 − P8 λm2 δ n2 ]W

1 
1 4
ab  1 4
 W (W + 2 µ h ) −
 * δ n + * λm  W (W + µ h ) (W + 2 µ h )
* 
A11 
64  A22
A11


8λmδ n ( F21λm4 + F12δ n4 + E2 λm2 δ n2 )
* 4
* 4


b11 =






b21 = − P1

− P3

W
W (W + 2 µ ) 1
+ b31
+ b4W W + 2 µ ,
W +µ
W +µ

(

−32mBa2 ( F21m 4 Ba4 + F12 n 4 + E2 m 2 n 2 Ba2 )
*
* 4
m 4 Ba4 + A11
n + E1m 2 n 2 Ba2 )
3nBh 2 ( A22

m 4π 2 Ba4 ( F21m 4 Ba4 + F12 n 4 + E2 m 2 n 2 Ba2 )
*


n 2π 2 ( F21m 4 Ba4 + F12 n 4 + E2 m 2 n 2 Ba2 )
*
* 4
Bh2 ( A22
m 4 Ba4 + A11
n + E1m 2 n 2 Ba2 )

m 4π 6 Ba4 ( E3 m 2 Ba2 + E4 n 2 )
*
* 4
Bh6 ( A22
m 4 Ba4 + A11
n + E1m 2 n 2 Ba2 )

m 4π 2 Ba4
n 2 Bh2

+ P7

n 2π 2
Bh2

+ P8

m 2π 2 Ba2
Bh2




* 
*
*
n
B
3Bh2  A22
16
A
A
A
h
22
11
11 


, P2 =

P2
h

, P3 =

P3
h

3

, P4 =


h

3

, P6 =

P6
h

3

, P7 =

P7
h

3

, P8 =

P8

h3

,

E
E2
F
F

*
* 4
n 2 Bh2 ( A22
m 4 Ba4 + A11
n + E1m 2 n 2 Ba2 )

m 2π 2 Ba2 ( F21m 4 Ba4 + F12 n 4 + E2 m 2 n 2 Ba2 )
*
* 4
Bh2 ( A22
m 4 Ba4 + A11
n + E1m 2 n 2 Ba2 )

m 2 n 2π 6 Ba2 ( E3 m 2 Ba2 + E4 n 2 )

− P5

*
* 4
Bh6 ( A22
m 4 Ba4 + A11
n + E1m 2 n 2 Ba2 )

− P4

+ P6

− P2

n 2π 2 ( F21m 4 Ba4 + F12 n 4 + E2 m 2 n 2 Ba2 )

.

We are also interested in finding the lower buckling load. To this end, we consider Fx = Fx (W ) at odd numbers m
dFx
and n, and from equation
= 0 , we get that
dW
4mn3 Ba2  P1
P 

+ 2 
*
* 4
* 
 A*
3
m 4 Ba4 + A11
n + E1m 2 n 2 Ba2 )
3( A22
 22 A11  .
−π 2  1 4
1 4 4 
n +
m Ba
*

8  A*
A
11
 22

+ b21 .

4.2. Numerical results and discussion of 3-phase composite plates

The results presented in this section, which are found from Eq. (21), correspond to the deformation mode with the
half-wave numbers m= n= 1.
The numerical calculation were performed for two following cases: 6-layer asymmetric plates with the stacking sequence 0/45/–45/45/–45/90 and 5-layer symmetric plates with the stacking sequence 45/–45/0/45/–45.
354


a
5

b

F .109, GPa

3

x

8

F .109, GPa

3

x

7

W/h

1

3

0

W/h
1

2

3

4

5

Fig. 3. Pressure–deflection curves Fx–W/h of 6-layer 3-phase polymer composite plates with ya = 0.2
at yc = 0 (1), 0.1 (2), and 0.2 (3) (a) and with yc = 0.2 at ya = 0 (1), 0.1 (2), and 0.2 (3); a/b = 1,
b/h = 30, and m = 0.1.

а
12

b

F .109, GPa


0.5

2

W/h
0

1

2

3

4

5

W/h
0

1

3

5

7

9



4
1
0

1

2

W/h

1

3

0

4

W/h
1

2

3

Fig. 5. Pressure–deflection curves Fx–W/h of 6-layer (a) and 5-layer (b) 3-phase polymer composites
plates with an initial imperfection m = 0 (1), 0.1 (2), 0.2 (3), and 0.3 (4); m = n = 1, b/a = 1, b/h = 30,
ya = 0.2, and yc = 0.15.
Figure 3 illustrates the effect of particles and fibers on the buckling of 6-layers 3-phase plates and Fig. 4 — the same


1

3
W/h 4

0

1

2

3

3

2

1

3
4

W/h
0

1

2



2

6

3
2
1

8
6

1

4

4

2

W/h
0

0.5

1.0

1.5

2.0


5. Concluding Remarks
The paper presents an analytical investigation of the nonlinear response of 3-phase polymer composite plates subjected
to mechanical loadings. All formulations are based on the classical theory of plates with account of a geometrical nonlinearity and initial imperfections. The Galerkin method is used to obtain explicit expressions for the load–deflection relationships.
The study reveals the effects of fibers and particles and the geometrical parameters and imperfections of the plates on their
nonlinear response to mechanical loadings.
Acknowledgment. In loving memory of Professor Vanin G. A. , it is our greatest hounor to devote our findings to
him on the 5th anniversary (2008-2012) of his death and the 82th anniversary of his birth (1930-2012).
This work was supported by Project QGDA.12.03 of Foundation for the Science and Technology Development of
Vietnam National University, Hanoi. The authors are grateful for this financial support.
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