DSpace at VNU: A nonlinear stability analysis of imperfect three-phase polymer composite plates

Composite Structures 109 (2014) 130–138

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Composite Structures
journal homepage: www.elsevier.com/locate/compstruct

Nonlinear stability analysis of imperfect three-phase polymer composite
plates in thermal environments
Nguyen Dinh Duc a,⇑, Pham Van Thu b
a
b

Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam
Institute of Shipbuilding, Nha Trang University, 44 Hon Ro, Nha Trang, Khanh Hoa, Viet Nam

a r t i c l e

i n f o

Article history:
Available online 7 November 2013
Keywords:
Nonlinear stability
Laminated three-phase composite plate
Thermal environments
Imperfection

a b s t r a c t
This paper presents an analytical investigation on the nonlinear response of the thin imperfect laminated
three-phase polymer composite plate in thermal environments. The formulations are based on the classical plate theory taking into account the interaction between the matrix and the particles, geometrical

nonlinearity, initial geometrical imperfection. By applying Galerkin method, explicit relations of load–
deflection curves are determined. Obtained results show effects of the fibers and the particles, material,
geometrical properties and temperature on the buckling and post-buckling loading capacity of the three
phase composite plate, therefore we can proactively design materials and structural composite meet the
technical requirements as desired when adjustment components.
Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction
Three phase composite is a material consisted of matrix of the
reinforced fibers and particles which have been investigated by Vanin and Duc since 1996. They have determined the elastic modulus
for three phases composite 3Dm [1] and 4Dm [2]. Their findings
have shown that the fibers are able to improve the elastic modulus,
the particles can resist to the penetration and the heat, reduce the
creep deformations and the defects in materials.
Despite of a large number of applications, our understanding on
the structure of three phase composite materials (plate and shell)
is not much. The general view of three-phase composite can be
found in [3]. Recently, there are several claims on the deflection
and the creep for the three phase composite plate in the bending
state [4]. These findings have shown that optimal three-phase
composite can be obtained by controlling the volume ratios of fiber
and particles.
Plate, shell and panel are basic structures used in engineering
and industry. These structures play an important role as main supporting component in all kind of structure in machinery, civil engineering, ship building, flight vehicle manufacturing, etc. The
stability of composite plate and shell is the first and most important problem in optimal design. In fact, many researchers are interested in this problem including the studies of the composite plates
[4–12]. However, researches on the stability of three-phases composite plates and shells are very few. Whereas, the choice of a suitable ratio of components materials in three-phases composite is
very important in designing new composite materials and predict
⇑ Corresponding author. Tel.: +84 4 37547978; fax: +84 4 37547724.
E-mail address: (N.D. Duc).
0263-8223/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved.

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mechanical and physical properties of advanced designed materials. Therefore, from scientific and practical point of view, it is,
therefore, very important and meaningful to carry an investigation
on the three phase composite plate and shell. Actively choosing
material components and ratio of its mixing allows us to decide
the advance materials and forecasting its physic-mechanical
characteristics.
Several fundamental references on composite plates and shells
are Brush and Almroth [13], Reddy [14] (for laminated composite
plate and shell) and Shen [15] (for composite FGM). Some research
on the stability of laminated composite and FGM plates can be obtained in [8–12].
Recently, in [7] we have studied nonlinear stability of the threephase polymer composite plate under mechanical loads. In the
present paper, we have studied the nonlinear stability of threephase polymer composite with imperfections in thermal environments. The paper focuses on deriving the algorithm for calculating
the stability of three-phase composite by analyzing the load–
deflection relationship base on the basic equations of laminated
composite, while also studies the effect of component material
properties, temperature, geometrical properties and imperfection
on the stability of three-phase polymer composite plate.
A special point of the results is to show the algorithms explicit
determine the coefficients of thermal expansion of the three-phase
composite material on the elastic modulus, coefficients of thermal
expansion and the ratio of component material (the elastic modules of three-phase composite material was determined by theoretical and experimental methods are published in [7,16]).
Moreover, the first time the article showed performances Hooke’s
law relationship of stress–strain three-phase composite plate include the effects of temperature. Thereby, we can calculate nonlin-


131

N.D. Duc, P. Van Thu / Composite Structures 109 (2014) 130–138

ear behavior of three-phase composite plates under temperature
loads and determine the effect of the component elements and
structure of materials on thermal stability of the plate.
2. Determine the elastic modules and the effective thermal
expansion coefficients of three-phase composite

in which

v ¼ 3 À 4m;
va ¼ 3 À 4ma :

ð6Þ

2.2. Determine the effective thermal expansion coefficient of composite
2.1. Determine the elastic modules of composite
The elastic modules of three-phase composites are estimated
using two theoretical models of the two-phase composite consecutively: nDm = Om + nD [1,2,7]. This paper considers three-phase
composite reinforced with particles and unidirectional fibers, so
the problem’s model will be: 1Dm = Om + 1D. Firstly, the modules
of the effective matrix Om which called ‘‘effective modules’’ are calculated. In this step, the effective matrix consists of the original
matrix and particles, it is considered to be homogeneous, isotropic
and have two elastic modules. The next step is estimating the elastic modules for a composite material consists of the effective matrix and unidirectional reinforced fibers.
Assume that all the component phases (matrix, fiber and particle)
are homogeneous and isotropic, we will use Em, Ea,Ec; mm, ma, mc; wm,
wa; wc to denote Young modulus and Poisson ratio and volume ratio
for matrix, fiber and particle, respectively. According to Vanin and
Duc in [1,2], we can obtain the modules for the effective composite as

G ¼ Gm
K ¼ Km

1 À wc ð7 À 5mm ÞH
;
1 þ wc ð8 À 10mm ÞH
1 þ 4wc Gm Lð3K m ÞÀ1
1 À 4wc Gm Lð3K m ÞÀ1

ð1Þ
ð2Þ

;

here

L¼

Kc À Km
;
K c þ 4G3m

H¼

Gm =Gc À 1
:
8 À 10mm þ ð7 À 5mm Þ GGmc

ð3Þ

 can be calculate from ðG; KÞ as
E; m

E¼

9KG
3K þ G

m ¼

;

3K À 2G
6K À 2G

ð4Þ

:

We should note that formulas (1) and (2) take into account the
nonlinear effects and the interaction between the particles and the
base. These are different from the other well-known formulas.
The elastic modules for 3-phase composite reinforced with unidirectional fiber are chosen to be calculated using Vanin’s formulas
[17] with six independent elastic modulus

E1 ¼ wa Ea þ ð1 À wa ÞE þ
(
E2 ¼

m221
E1

þ

1

"

Þ
8Gwa ð1 À wa Þðma À m
2 À wa þ xwa þ ð1 À wa Þðxa À 1Þ GGa

;

 À 1Þ þ ðva À 1Þðv
 À 1 þ 2wa Þ GG
2ð1 À wa Þðv
a

8G

 wa þ ð1 À wa Þðva À 1Þ GG
2 À wa þ v
a
#)À1
G


vð1 À wa Þ þ ð1 þ wa vÞ Ga
þ2
;
v þ wa þ ð1 À wa Þ GGa

G12 ¼ G

1 þ wa þ ð1 À wa Þ GGa

; G23 ¼ G

v þ wa þ ð1 À wa Þ GGa
; ð5Þ
 þ ð1 þ v
 wa Þ GG
ð1 À wa Þv
a

1 À wa þ ð1 þ wa Þ GGa
"
ð1 À wa Þx þ ð1 þ wa xÞ GGa
m23
m2
1
¼ À 21 þ
2
E22
E11 8G
x þ wa þ ð1 À wa Þ GG
a
À

2ð1 À wa Þðx À 1Þ þ ðxa À 1Þðx À 1 þ 2wa Þ GGa

2 À wa þ xwa þ ð1 À wa Þðxa À 1Þ GGa
 þ 1Þðm
 À ma Þwa
ðv
m21 ¼ m À
;
 wa þ ð1 À wa Þðva À 1Þ GG
2 À wa þ v
a

#
;

Similar to the elastic modulus, the thermal expansion coefficient of the three-phase composite materials were also identified
in two steps: First, to determine the coefficient of thermal expansion of the effective matrix. The current paper uses the Duc’s result
from [18] for calculating the thermal expansion coefficient of effective matrix

aà ¼ am þ ðac À am Þ

K c ð3K m þ 4Gm Þwc
;
K m ð3K c þ 4Gm Þ þ 4ðK c À K m ÞGm xc

ð7Þ

in which a⁄ is the effective thermal expansion coefficient of effective matrix; am, ac are the thermal expansion coefficients of original
matrix and particle, respectively.
Then, determining two coefficients of thermal expansion of the
three-phase composite using formulas from [17] of Vanin

"
Ã

Ã

a1 ¼ a À ða À a

À1
a Þwa E1

Ea þ

8Ga ðma À mÞð1 À wa Þð1 þ ma Þ

#

;
2 À wa þ xwa þ ð1 À wa Þðxa À 1Þ GGa
m À m21
a2 ¼ aà þ ðaà À a1 Þm21 À ða À aa Þð1 þ ma Þ
:
m À ma
ð8Þ
To numerical calculating, we chosen three phase composite
polymer made of polyester AKAVINA (made in Vietnam), fibers
(made in Korea) and titanium oxide (made in Australia) with the
properties as in Table 1 [16].
The results of elastic modules of composite materials for different volume ratios of component materials written in (5) are given
in Table 2 [7], in which 14 variant cases of different volume ratios
of component three-phase composite materials respectively are given in Table 3 [7].

Fig. 1 illustrates the SEM images of the structure of the twophase composite polymer with the based phase composed of the
glassy polyester fiber (without the particles) in 1D (all the composite fibers are reinforced in one direction). Fig. 2 illustrates the
structure of three-phase 1Dm in the presence of the TiO2 particles.
From the above illustrations, it is obviously that the more the
particles are doped, the finer the material is, in other words, the
less the holes are. These results also mean that we can increase
the elastic modulus as well as strengthens the penetration resistance of the materials by doping the particles. Figs. 1 and 2 show
the SEM pictures of our proposed fabricated composite structures.
These pictures are taken by ourself using the SEM instrumentation
at the Laboratory for Micro-Nano Technology, University of Engineering and Technology, Vietnam National University, Hanoi. Also,
we made these composite material samples in the Institute of Ship
building, Nha Trang University.

3. Governing equations
Consider a three phase composite plate with midplane-symmetric. The plate is referred to a Cartesian coordinate system x, y,
z, where xy is the mid-plane of the plate and z is the thickness coordinator, Àh/2 6 z 6 h/2. The length, width, and total thickness of
the plate are a, b and h, respectively.
In this study, the classical theory is used to establish governing
equations and determine the nonlinear response of composite
plates [13–15].


132

N.D. Duc, P. Van Thu / Composite Structures 109 (2014) 130–138

Table 1
Properties of the component phases for three-phase considered composite.
Component phase

Young modulus,
E (GPa)

Poisson ratio, m

Matrix polyester AKAVINA (Vietnam)
Glass fiber (Korea)
Titanium oxide TiO2 (Australia)

1.43
22
5.58

0.345
0.24
0.20

Table 2
Elastic modules for three-phase composite materials.
E1 (GPa)

E2 (GPa)

m12

G12 (GPa)

G23 (GPa)

0.8043

0.8722
0.9457
1.0256
1.1132
1.2097
1.3167

1.8616
1.6747
1.5093
1.3618
1.2296
1.1103
1.0021

2.7174
2.4451
2.2044
1.9900
1.7978
1.6244
1.4672

1.0513
1.0116
0.9457
0.8496
0.7190
0.5486
0.3327

1.4840
1.4974
1.5093
1.5199
1.5295
1.5382
1.5461

2.6771
2.4247
2.2044
2.0103
1.8382
1.6843
1.5461

wa = const wc – Particle’s ratio increase
Case
Case
Case
Case
Case
Case
Case

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

wc = const wa – Fiber’s ratio increase
Case
Case
Case
Case
Case
Case
Case

8

9
10
11
12
13
14

24.2929
20.9035
17.5209
14.1451
10.7762
7.4144
4.0598

7.9971
7.3880
6.8385
6.3402
5.8860
5.4702
5.0880

Fig. 2. Black–White SEM image of 1Dm composite three-phase material (25% of
fibers and 10% of particles).

0
B
@

1

0

1

0

1

e0x
ex
kx
C
B
B
C
ey A ¼ @ e0y C
A þ z@ ky A;
cxy
kxy
c0xy

ð9Þ

where

0
Table 3
Variant cases of different volume ratios of matrix, fibers and particles.

B
@

Case

1

2

3

4

5

6

7

wm
wa
wc

0.5
0.2
0.3

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

9

10

11

12

13

14

0.5
0.3
0.2

0.55
0.25
0.2

0.6
0.2
0.2

0.65
0.15
0.2

0.7
0.1
0.2

0.75
0.05
0.2

0.8
0.0
0.2

wm
wa
wc

1

0

1

0

1

0

1

e0x
u;x þ w2;x =2
Àw;xx
kx
B
B
C B
C

0 C
ey A ¼ @ v ;y þ w2;y =2 C
Àw
;
k
¼
A @ yA @
;yy A;
À2w;xy
kxy
c0xy
u;y þ v ;x þ w;x w;y

ð10Þ

in which u, v are the displacement components along the x, y directions, respectively.
Hooke law for a composite plate is defined as [19]

0

1

0

10

1

ex À a1 DT
rx

Q 011 Q 012 Q 016
B
C
B 0
C
0
0 C B
@ ry A ¼ @ Q 12 Q 22 Q 26 A @ ey À a2 DT A ;
0
0
0
cxy
rxy k
Q 16 Q 26 Q 66 k
k

ð11Þ

in which
4

2

Q 011 ¼ Q 11 cos4 h þ Q 22 sin h þ 2ðQ 12 þ 2Q 66 Þsin hcos2 h;
4

2

Q 012 ¼ Q 12 ðcos4 h þ sin hÞ þ ðQ 11 þ Q 22 À 4Q 66 Þsin hcos2 h;
3

Q 016 ¼ ðQ 12 À Q 22 þ 2Q 66 Þsin h cosh þ ðQ 11 À Q 12 À 2Q 66 Þsinhcos3 h;
4

2

Q 022 ¼ Q 11 sin h þ Q 22 cos4 h þ 2ðQ 12 þ 2Q 66 Þsin hcos2 h;
3

Q 026 ¼ ðQ 11 À Q 12 À 2Q 66 Þsin h cosh þ ðQ 12 À Q 22 þ 2Q 66 Þsinhcos3 h;
4

2

Q 066 ¼ Q 66 ðsin h þ cos4 hÞ þ ½Q 11 þ Q 22 À 2ðQ 12 þ Q 66 ފsin h cos2 h;
ð12Þ
and

Q 11 ¼
Q 22 ¼
Q 12 ¼
Fig. 1. Black–White SEM image of 1D composite two-phase material (25% of fibers
without particles).

E1
1 À EE21 m212

¼

E1

;
1 À m12 m21

E2
E2
¼ Q 11 ;
1 À EE21 m212 E1
E1
1 À EE21 m212

Q 66 ¼ G12 ;

¼

m12
Q 22

;

ð13Þ


N.D. Duc, P. Van Thu / Composite Structures 109 (2014) 130–138

where h is an angle between the fiber and the coordinate system.
The force and moment resultants of the composite plates are determined by [13,14]

Ni ¼
Mi ¼

n Z
X

hk

k¼1 hkÀ1
n Z hk
X
k¼1

hkÀ1

½ri Šk dz;

Nx;x þ Nxy;y ¼ 0;
Nxy;x þ Ny;y ¼ 0;
P1 f;xxxx þ P2 f;yyyy þ P3 w;xxyy þ P4 w;xxxy þ P5 w;xyyy þ P6 w;xxxx

i ¼ x; y; xy;

þ 2Nxy w;xy þ Ny w;yy ¼ 0;
where

i ¼ x; y; xy:

P1 ¼ BÃ21 ;

Substitution of Eqs. (9) and (11) into Eq. (14) and the result into
Eq. (14) give the constitutive relations as
0

x

0
y

0
xy

ðNx ; Ny ; Nxy Þ ¼ ðA11 ; A12 ; A16 Þe þ ðA12 ; A22 ; A26 Þe þ ðA16 ; A26 ; A66 Þc
þ ðB11 ; B12 ; B16 Þkx þ ðB12 ; B22 ; B26 Þky þ ðB16 ; B26 ; B66 Þkxy
À DT½a1 ðA11 ; A12 ; A16 Þ þ a2 ðA12 ; A22 ; A26 ފ;

ðM x ; My ; M xy Þ ¼ ðB11 ; B12 ; B16 Þe0x þ ðB12 ; B22 ; B26 Þe0y þ ðB16 ; B26 ; B66 Þc0xy

P2 ¼ BÃ12 ; P3 ¼ BÃ11 þ BÃ22 À 2BÃ66 ; P4 ¼ 2BÃ26 À BÃ61 ;

2BÃ16

P5 ¼
À BÃ62 ;
Ã
P6 ¼ B11 B11 þ B12 BÃ21 þ B16 BÃ61 ;

P 7 ¼ B12 BÃ12 þ B22 BÃ22 þ B26 BÃ62 ;
Ã
P8 ¼
þ
þ B16 B62 þ B12 BÃ11 þ B22 BÃ21 þ B26 BÃ61
Ã
þ 4B16 B16 þ 4B26 BÃ26 þ 4B66 BÃ66 ;

À
Á
P9 ¼ 2 B11 BÃ16 þ B12 BÃ26 þ B16 BÃ66 þ B16 BÃ11 þ B26 BÃ21 þ B66 BÃ61 ;
À
Á
P10 ¼ 2 B12 BÃ16 þ B22 BÃ26 þ B26 BÃ66 þ B16 BÃ12 þ B26 BÃ22 þ B66 BÃ62 ;
B11 BÃ12

B12 BÃ22

ð21Þ

þ ðD11 ; D12 ; D16 Þkx þ ðD12 ; D22 ; D26 Þky þ ðD16 ; D26 ; D66 Þkxy
À DT½a1 ðB11 ; B12 ; B16 Þ þ a2 ðB12 ; B22 ; B26 ފ;

f(x, y) is stress function defined by

ð15Þ
where
n 

X
Aij ¼
Q 0ij ðhk À hkÀ1 Þ;
k¼1
n 
X

k

ð20Þ

þ P7 w;yyyy þ P8 w;xxyy þ P9 w;xxxy þ P10 w;xyyy þ Nx w;xx
ð14Þ

z½ri Šk dz;

133

Nx ¼ f;yy ;

N y ¼ f;xx ;

Nxy ¼ Àf;xy :

ð22Þ

For an imperfect composite plate, Eq. (20) are modified into
form as [7,8]

i; j ¼ 1; 2; 6;



2
2
Q 0ij
hk À hkÀ1 ;

Bij ¼

1
2 k¼1

Dij ¼

n 


1X
3
3
Q 0ij
hk À hkÀ1 ;
k
3 k¼1

k

P1 f;xxxx þ P2 f;yyyy þ P3 w;xxyy þ P4 w;xxxy þ P 5 w;xyyy þ P 6 w;xxxx

i; j ¼ 1; 2; 6;

ð16Þ

þ P7 w;yyyy þ P8 w;xxyy þ P9 w;xxxy þ P10 w;xyyy







þ f;yy w;xx þ wÃ;xx À 2f ;xy w;xy þ wÃ;xy þ f;xx w;yy þ wÃ;yy
¼ 0;

i; j ¼ 1; 2; 6:

ð23Þ

The nonlinear equilibrium equations of a composite plate based
on the classical theory are [13–15]

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

Nx;x þ Nxy;y ¼ 0;

ð17aÞ

Nxy;x þ Ny;y ¼ 0;

ð17bÞ

e0x;yy þ e0y;xx À c0xy;xy ¼ w2;xy À w;xx w;yy þ 2w;xy wÃ;xy À w;xx wÃ;yy

M x;xx þ 2M xy;xy þ M y;yy þ Nx w;xx þ 2N xy w;xy þ Ny w;yy ¼ 0:

ð17cÞ

ð24Þ

From the constitutive relations (18) in conjunction with Eq. (22)
one can write

Calculated from Eq. (15)

e0x ¼ AÃ11 Nx þ AÃ12 Ny þ AÃ16 Nxy À BÃ11 kx À BÃ12 ky À BÃ16 kxy
À
Á
þ DT a1 DÃ11 þ a2 DÃ12 ;
e0y ¼ AÃ12 Nx þ AÃ22 Ny þ AÃ26 Nxy À BÃ21 kx À BÃ22 ky À BÃ26 kxy
þ DTða1 DÃ21 þ a2 DÃ22 Þ;
0
exy ¼ AÃ16 Nx þ AÃ26 Ny þ AÃ66 Nxy À BÃ16 kx À BÃ26 ky À BÃ66 kxy
À
Á
þ DT a1 DÃ16 þ a2 DÃ26 ;

ð18Þ

where

A22 A66 À A226
A16 A26 À A12 A66
A12 A26 À A22 A16
¼
; AÃ12 ¼
; AÃ16 ¼
;

D
D
D
A11 A66 À A216
A12 A16 À A11 A26
A11 A22 À A212
AÃ22 ¼
; AÃ26 ¼
: AÃ66 ¼
;
D
D
D
2
2
2
D ¼ A11 A22 A66 À A11 A26 þ 2A12 A16 A26 À A12 A66 À A16 A22 ;
ð19Þ
AÃ11

BÃ11 ¼ AÃ11 B11 þ AÃ12 B12 þ AÃ16 B16 ;

BÃ12 ¼ AÃ11 B12 þ AÃ12 B22 þ AÃ16 B26 ;

¼ AÃ11 B16 þ AÃ12 B26 þ AÃ16 B66 ;
¼ AÃ12 B12 þ AÃ22 B22 þ AÃ26 B26 ;
¼ AÃ16 B11 þ AÃ26 B12 þ AÃ66 B16 ;
¼ AÃ16 B16 þ AÃ26 B26 þ AÃ66 B66 ;
¼ AÃ11 A12 þ AÃ12 A22 þ AÃ16 A26 ;
¼ AÃ12 A12 þ AÃ22 A22 þ AÃ26 A26 ;

¼ AÃ16 A12 þ AÃ26 A22 þ AÃ66 A26 :

BÃ21 ¼ AÃ12 B11 þ AÃ22 B12 þ AÃ26 B16 ;

BÃ16
BÃ22
BÃ61
BÃ66
DÃ12
DÃ22
DÃ26

À w;yy wÃ;xx :

BÃ26 ¼ AÃ12 B16 þ AÃ22 B26 þ AÃ26 B66 ;
BÃ62 ¼ AÃ16 B12 þ AÃ26 B22 þ AÃ66 B26 ;
DÃ11 ¼ AÃ11 A11 þ AÃ12 A12 þ AÃ16 A16 ;
DÃ21
DÃ16

¼
¼

AÃ12 A11
AÃ16 A11

þ
þ

AÃ22 A12

AÃ26 A12

þ
þ

AÃ26 A16 ;
AÃ66 A16 ;

Substituting once again Eq. (18) into the expression of Mij in
(15), then Mij into Eq. (17c) leads to

e0x ¼ AÃ11 f;yy þ AÃ12 f;xx À AÃ16 f;xy À BÃ11 kx À BÃ12 ky À BÃ16 kxy
À
Á
þ DT a1 DÃ11 þ a2 DÃ12 ;
e0y ¼ AÃ12 f;yy þ AÃ22 f;xx À AÃ26 f;xy À BÃ21 kx À BÃ22 ky À BÃ26 kxy
À
Á
þ DT a1 DÃ21 þ a2 DÃ22 ;
e0xy ¼ AÃ16 f;yy þ AÃ26 f;xx À AÃ66 f;xy À BÃ16 kx À BÃ26 ky À BÃ66 kxy
À
Á
þ DT a1 DÃ16 þ a2 DÃ26 :

ð25Þ

Setting Eq. (25) into Eq. (24) gives the compatibility equation of
an imperfect composite plate as [7,8]

AÃ22 f;xxxx þ E1 f;xxyy þ AÃ11 f;yyyy À 2AÃ26 f;xxxy À 2AÃ16 f;xyyy þ BÃ21 w;xxxx

þ BÃ12 w;yyyy þ E2 w;xxyy þ E3 w;xxxy þ E4 w;xyyy


À w2;xy À w;xx w;yy þ 2w;xy wÃ;xy À w;xx wÃ;yy À w;yy wÃ;xx
¼ 0;

ð26Þ

where

E1 ¼ 2AÃ12 þ AÃ66 ;
E2 ¼ BÃ11 þ BÃ22 À 2BÃ66 ;
E3 ¼ 2BÃ26 À BÃ61 ;

ð27Þ

E4 ¼ 2BÃ16 þ BÃ62 :
Eqs. (23) and (26) are nonlinear equations in terms of variables
w and f and used to investigate the stability of thin composite
plates subjected to thermal loads.


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N.D. Duc, P. Van Thu / Composite Structures 109 (2014) 130–138

In the present study, the edges of composite plates are assumed
to be simply supported and four edges of the plate are simply supported and immovable (IM). In this case, boundary conditions are

The in-plane condition on immovability at all edges, i.e. u = 0 at

x = 0, a and v = 0 at y = 0, b, is fulfilled in an average sense as
[8,11,12]

w ¼ u ¼ Mx ¼ 0;

Nx ¼ Nx0

at x ¼ 0; a;

Z

w ¼ v ¼ M y ¼ 0;

Ny ¼ Ny0

at y ¼ 0; b;

ð28Þ

b

Z

0

where Nx0, Ny0 are fictitious compressive edge loads at immovable
edges.
The approximate solutions of w and f satisfying boundary conditions (28) are assumed to be [7,8]

ðw; wÃ Þ ¼ ðW; lhÞ sin km x sin dn y;

ð29aÞ

f ¼ A1 cos 2km x þ A2 cos 2dn y þ A3 sin km x sin dn y þ A4
1
1
 cos km x cos dn y þ Nx0 y2 þ Ny0 x2 ;
2
2

ð29bÞ

a

0

Z

@u
dxdy ¼ 0;
@x

a

Z

0

0

b

@v
dydx ¼ 0:
@y

ð32Þ

From Eqs. (10) and (18) one can obtain the following expressions in which Eq. (22) and imperfection have been included

@u
¼ AÃ11 f;yy þ AÃ12 f;xx À AÃ16 f;xy þ BÃ11 w;xx þ BÃ12 w;yy þ 2BÃ16 w;xy
@x
À
Á 1
þ DT DÃ11 a1 þ DÃ12 a2 À w2;x À w;x wÃ;x ;
2
@v
Ã
Ã
Ã
¼ A12 f;yy þ A22 f;xx À A26 f;xy þ BÃ21 w;xx þ BÃ22 w;yy þ 2BÃ26 w;xy
@y
À
Á 1
þ DT DÃ21 a1 þ DÃ22 a2 À w2;y À w;y wÃ;y :
2

ð33Þ

km = mp/a, dn = np/b. W is amplitude of the deflection and l is
imperfection parameter. The coefficients Ai(i = 1Ä4) are determined
by substitution of Eqs. (29a) and (29b) into Eq. (26) as

Substitution of Eqs. (29a) and (29b) into Eq. (33) and then the
result into Eq. (32) give fictitious edge compressive loads as

1 d2n
A1 ¼
WðW þ 2lhÞ;
32AÃ22 k2m

Ny0 ¼ J 4 W þ J 5 WðW þ 2lhÞ þ J 6 DT;

A3 ¼
A4 ¼

Q 22 À Q 21
ðQ 2 Q 3 À Q 1 Q 4 Þ
Q 22 À Q 21

ð30Þ

Q 22 À Q 21

1

ðQ 2 Q 3 À Q 1 Q 4 Þ

k2m d2n À P4

ðQ 2 Q 3 À Q 1 Q 4 Þ
Q 22 À Q 21

#

þ P6 k4m þ P7 d4n þ P8 k2m d2n W

Q 22 À Q 21


1
1
1
þ km dn P1 Ã þ P2 Ã WðW þ 2lhÞ
3
A22
A11


ab 1 4
1 4
d þ
k WðW þ lhÞðW þ 2lhÞ
À
64 AÃ22 n AÃ11 m

W þl

; ineWðW þ 2lÞ

W þl

1

þ b4 WðW
ð35Þ

Á
ab À
Nx0 k2m þ Ny0 d2n ðW þ lhÞ ¼ 0;
4

b1 ¼

32hmnp2 ðQ 2 Q 4 À Q 1 Q 3 Þ

;
2
Q 22 À Q 21
3a2 b X
3
2



ðQ 2 Q 4 ÀQ 1 Q 3 Þ
m4 p 4
n4 p 4
1Q3Þ
P 1 ðQ 2 QQ 42 ÀQ

þ
P
þ
P
þ
P
6
2
7
2
4
2
2
4
a
Q 2 ÀQ 1
b 7
6
2 ÀQ 1
7


16
2 n2 p4
7
6
1
ðQ 2 Q 4 ÀQ 1 Q 3 Þ
ðQ 2 Q 3 ÀQ 1 Q 4 Þ m3 np4
m

b2 ¼ 6 þ P 3
þ
P
À
P
7;
8
4
2
2
2
2
2
3b
2
a
Q 2 ÀQ 1
Q 2 ÀQ 1
a b
7
X6
5
4
ðQ 2 Q 3 ÀQ 1 Q 4 Þ mn3 p4
ÀP5 Q 2 ÀQ 2
3
ab
2
1
!

4mnp2 P1
P2
1
þ Ã ;
b3 ¼
2
Ã
3a2 b X A22 A11
!
Àp4 1 n4
1 m4
1
b4 ¼
þ
;
16X AÃ22 b4 AÃ11 a4
ð36Þ

where

X ¼ J 3 k2m þ J 6 d2n ;

 ¼ W:
W
h

ð37Þ

For a perfect composite plate (l = 0), Eq. (37) leads to equations
that determining buckling temperature DTb of the plate. It can be

obtained by taking the limit function DTðWÞ with W ! 0

8 ðQ 2 Q 4 À Q 1 Q 3 Þ
þ
km dn WðW þ lhÞ
3
Q 22 À Q 21
À

1

þ b3

þ 2lÞ;

W;

ðQ 2 Q 4 À Q 1 Q 3 Þ

W

1

where

"
ab
ðQ Q À Q 1 Q 3 Þ 4
ðQ Q À Q 1 Q 3 Þ 4
P1 2 42

km þ P2 2 42
dn
2
4
Q2 À Q1
Q 2 À Q 21

ÀP 5

1

DT ¼ b1 W þ b2

W;

where Qi(i = 1Ä4) = Qi(km, dn) was mentioned in Appendix A
Subsequently, substitution of Eqs. (29a) and (29b) into Eq. (23)
and applying the Galerkin procedure for the resulting equation
yield

þP 3

ð34Þ

where Ji(i = 1Ä6) = Ji(km, dn) are give in Appendix A
Subsequently, setting Eq. (34) into Eq. (31) give

1 k2m
A2 ¼
WðW þ 2lhÞ;

32AÃ11 d2n
ðQ 2 Q 4 À Q 1 Q 3 Þ

Nx0 ¼ J 1 W þ J 2 WðW þ 2lhÞ þ J 3 DT;

ð31Þ

where m, n are odd numbers. This is basic equation governing the
nonlinear response of three-phase polymer composite plates under
thermal loads.
4. Nonlinear stability analysis
4.1. Plate three-phase under uniform temperature rise
Consider a simply supported polymer composite plates subjected to temperature environments uniformly raised from stress
free initial state Ti to final value Tf and temperature difference
DT = Tf À Ti is constant.

DT b ¼

32hmnp2 ðQ 2 Q 4 À Q 1 Q 3 Þ
2

3a2 b X

Q 22 À Q 21

:

ð38Þ

4.2. Numerical results and discussion

The asymmetric plate with simply supported boundary condition subjected to in-plane compressive loads as well as a uniform
temperature rise, the bifurcation buckling load (temperature) does
not exist [20]. Hence, the numerical calculation will be calculated
for three following cases of symmetric plates [19,20]: five-layers
plate with the fiber angle 0/90/0/90/0, five-layers plate with the fiber angle 45/45/0/45/45 and four-layers plate with the twisted angle 0/45/À45/90.


N.D. Duc, P. Van Thu / Composite Structures 109 (2014) 130–138

135

Fig. 5. Effects of particle’s ratio wc on the nonlinear response of three-phase
polymer composite five-layers plate under uniform temperature rise (immovable
edges).
Fig. 3. Effects of fiber angles góc quâ´n on the nonlinear response of symmetric
three-phase polymer composite plates with five-layers 0/90/0/90/0 and 45/45/0/
45/45 under uniform temperature rise (immovable edges).

Fig. 6. Effects of fiber’s ratio wa on the nonlinear response of three-phase polymer
composite five-layers plate under uniform temperature rise (immovable edges).

Fig. 4. Effects of mode (m, n) on the nonlinear response of symmetric three-phase
polymer composite plate with five layers 0/90/0/90/0 under uniform temperature
rise (immovable edges).

Fistly, we consider the effects of fiber angle on buckling of
five-layers plate with two case of the fiber angle: 0/90/0/90/0
and 45/45/0/45/45. From Fig. 3, easily see that the plate with
fiber angle 0/90/0/90/0 has a better thermal loading ability than
the other one.

We also calculate thermal buckling loads with different modes
(m, n) = (1, 1), (1, 3), (3, 1), (3, 3). Fig. 4 shows the effect of parameters (m, n) on nonlinear buckling of five-layers symmetric plate
with the fiber angle 0/90/0/90/0. We can see that thermal buckling load is at minimum when (m, n) = (1, 1). In another word,
buckling happens at first with (m, n) = (1, 1). Therefore, from
now on, all of our figures and calculations are based only on
(m, n) = (1, 1).
Then, we consider the effects of material parameter and geometry on nonlinear response of the symmetric five-layers three
phase plate with the fiber angle 0/90/0/90/0 and the symmetric
four-layers three phase plate with the fiber angle 0/45/À45/90.

Fig. 7. Effects of ratio particle’s ratio wc on the nonlinear response of three-phase
polymer composite four-layers plate under uniform temperature rise (immovable
edges).


136

N.D. Duc, P. Van Thu / Composite Structures 109 (2014) 130–138

Fig. 8. Effects of fiber’s ratio wa on the nonlinear response of three-phase polymer
composite four-layers plate under uniform temperature rise (immovable edges).

Fig. 10. Effects of imperfection on the nonlinear response of symmetric three-phase
polymer composite four-layers plate under uniform temperature rise (immovable
edges).

Fig. 9. Effects of imperfection on the nonlinear response of symmetric three-phase
polymer composite five-layers plate under uniform temperature rise (immovable
edges).

Figs. 5 and 6 represent the effect of the particles and the fibers
on buckling of the five-layers three phase plate. Similarly, Figs. 7
and 8 represent the effects of those on buckling of four-layers
three-phase plate.
In Figs. 5–8, we can realize that the increase of the particles and
fibers density will increase the thermal loading ability of the plates.
Especially notice on the curve 2 in Fig. 5 (fiber ratio wa = 0.2;
particles ratio wc = 0.1) and curve 2 in Fig. 6 (fiber ratio wa = 0.1;
particles ratio wc = 0.2), easily notice that with the same reinforced
component ratio wa + wc = 0.3, the titanium oxide particles make
the thermal capacity of the composite plate better and stronger.
The titanium oxide particles play an important role in thermal
resistance of the polymer matrix so we can replace a part of expensive fibers by particles of titanium oxide in order to reduce the
price. However, the five-layers symmetric plate has a better thermal loading ability than the four-layers symmetric one. The same
conclusion was also found in study case of the three phase laminated composite plates under mechanical loads [7].
Figs. 9 and 10 represent the imperfection effects on buckling of
five-layers and four-layers symmetric plates. The results show the

Fig. 11. Effects of ratio b/h on the nonlinear response of three-phase polymer
composite five-layers plate under uniform temperature rise (immovable edges).

subtle change with the increase of the imperfection degree. However, with the same geometrical size of the plates, the imperfect
five-layers imperfect plate has a better thermal loading ability than
the four-layers imperfect plate.
Figs. 11–14 illustrates the geometrical effects b/h and b/a on
buckling of plates. It is not surprising that the thicker the plate
(b/h) is large) and the larger the length/width (b/a) ratio is, the better thermal loading ability on buckling of plates is.
5. Concluding remarks
The paper presents an analytical investigation on the nonlinear
response of three-phase polymer composite plates subjected to

thermal loads. The formulations are based on the classical theory
of plates taking into account geometrical nonlinearity and initial
imperfection. Galerkin method is used to obtain explicit expressions of load–deflection curves.
From the obtained results in this paper, we make the following
conclusions:


137

N.D. Duc, P. Van Thu / Composite Structures 109 (2014) 130–138

– With the same thickness and size, the thermal loading ability of
the symmetric five-layers plate is better than that of the symmetric four-layers plates.
– The geometry affects significantly on the stability of the composite plates.
The advantage of study approach in this paper is the nonlinear
response of three-phase polymer composite plates which is presented explicitly on parameters of polymer matrix, fibers and
particles and also with the geometrical parameters, imperfection and ratio of components in composite, so when adjusting
the phase component, we can design and structural new materials meet the technical requirements.

Acknowledgment

Fig. 12. Effects of ratio b/h on the nonlinear response of three-phase polymer
composite four-layers plate under uniform temperature rise (immovable edges).

This work was supported by Project code 107.02-2013.06 in
Mechanics of the National Foundation for Science and Technology
Development of Vietnam – NAFOSTED. The authors are grateful for
this financial support.
Appendix A

Q 1 ¼ AÃ22 k4m þ AÃ11 d4n þ E1 k2m d2n ;
Q 2 ¼ 2AÃ26 k3m dn þ 2AÃ16 km d3n ;
À
Á
Q 3 ¼ À BÃ21 k4m þ BÃ12 d4n þ E2 k2m d2n ;
Q 4 ¼ E3 k3m dn þ E4 km d3n ;
J1 ¼
J2 ¼
J3
J4
Fig. 13. Effects of ratio b/a on the nonlinear response of three-phase polymer
composite five-layers plate under uniform temperature rise (immovable edges).

J5
J6

AÃ22 L1 À AÃ12 L3
AÃ11 AÃ22 À AÃ2
12
AÃ22 L2 À AÃ12 L4

;

;
AÃ11 AÃ22 À AÃ2
12
AÃ H1 À AÃ12 H2
¼ À 22Ã Ã
;
A11 A22 À AÃ2

12
AÃ L3 À AÃ12 L1
¼ 11
;
AÃ11 AÃ22 À AÃ2
12
AÃ L4 À AÃ12 L2
¼ 11
;
AÃ11 AÃ22 À AÃ2
12
AÃ H2 À AÃ12 H1
¼ À 11Ã Ã
;
A11 A22 À AÃ2
12

where

L1 ¼

L2 ¼

L3 ¼

Fig. 14. Effects of ratio b/a on the nonlinear response of three-phase polymer
composite four-layers plate under uniform temperature rise (immovable edges).

– Increasing the density of fibers and particles in three phase
composite polymer improves the thermal loading ability of

the composite plates. However, the effects of the titanium oxide
particles on thermal capacity of the composite plates are stronger than those of the glass fibers.

L4 ¼

À Ã 2
Á
Ã
2
1 A11 dn þ A12 km ðQ 2 Q 4 À Q 1 Q 3 Þ
ðQ Q À Q 1 Q 4 Þ
þ 4AÃ16 2 32
ab
km dn
Q 22 À Q 21
Q 2 À Q 21
À Ã 2
Á
Ã
2
þ 4 B11 km þ B12 dn ;
k2m
;
8
À Ã 2
Á
Ã
2
1 A12 dn þ A22 km ðQ 2 Q 4 À Q 1 Q 3 Þ
ðQ Q À Q 1 Q 4 Þ

þ 4AÃ26 2 32
ab
km dn
Q 22 À Q 21
Q 2 À Q 21
À Ã 2
Á
þ 4 B21 km þ BÃ22 d2n ;
d2n
;
8

H1 ¼ DÃ11 a1 þ DÃ12 a2 ;

H2 ¼ DÃ12 a1 þ DÃ22 a2 :

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