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
Trang 1Nonlinear stability analysis of imperfect three-phase polymer composite
plates in thermal environments
Nguyen Dinh Duca,⇑, Pham Van Thub
a
Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam
b
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 clas-sical 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
Va-nin 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
sup-porting component in all kind of structure in machinery, civil
engi-neering, ship building, flight vehicle manufacturing, etc The
stability of composite plate and shell is the first and most
impor-tant problem in optimal design In fact, many researchers are
inter-ested in this problem including the studies of the composite plates
[4–12] However, researches on the stability of three-phases
com-posite plates and shells are very few Whereas, the choice of a
suit-able ratio of components materials in three-phases composite is
very important in designing new composite materials and predict
mechanical and physical properties of advanced designed materi-als 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 ob-tained in[8–12]
Recently, in[7]we have studied nonlinear stability of the three-phase polymer composite plate under mechanical loads In the present paper, we have studied the nonlinear stability of three-phase polymer composite with imperfections in thermal environ-ments 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 mod-ules of three-phase composite material was determined by theo-retical 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 in-clude the effects of temperature Thereby, we can calculate
nonlin-0263-8223/$ - see front matter Ó 2013 Elsevier Ltd All rights reserved.
⇑Corresponding author Tel.: +84 4 37547978; fax: +84 4 37547724.
E-mail address: ducnd@vnu.edu.vn (N.D Duc).
Contents lists available atScienceDirect
Composite Structures
j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / c o m p s t r u c t
Trang 2ear 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
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
consec-utively: 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 Omwhich called ‘‘effective modules’’ are
cal-culated 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
elas-tic modules for a composite material consists of the effective
ma-trix 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;wcto 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
1 wcð7 5mmÞH
K ¼ Km
1 þ 4wcGmLð3KmÞ1
here
L ¼Kc Km
Kcþ4Gm3 ; H ¼
Gm=Gc 1
8 10mmþ ð7 5mmÞGmGc: ð3Þ
E; mcan be calculate from ðG; KÞ as
E ¼ 9KG
3K þ G; m¼3K 2G
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
uni-directional fiber are chosen to be calculated using Vanin’s formulas
[17]with six independent elastic modulus
E1¼ waEaþ ð1 waÞE þ 8Gwað1 waÞðma mÞ
2 waþ xwaþ ð1 waÞðxa 1ÞG
a
;
E2¼ m2
21
E1þ 1
8G
2ð1 waÞðv 1Þ þ ðva 1Þðv 1 þ 2waÞG
a
2 waþ vwaþ ð1 waÞðva 1ÞG
a
"
(
þ2vð1 waÞ þ ð1 þ wavÞG
a
vþ waþ ð1 waÞG
a
#)1
;
G12¼ G1 þ waþ ð1 waÞ
G a
1 waþ ð1 þ waÞG
a
; G23¼ G vþ waþ ð1 waÞG
a
ð1 waÞvþ ð1 þ vwaÞG
a
;
m23
E22
¼ m2
21
E11
þ 1
8G 2
ð1 waÞx þ ð1 þ waxÞG
a
x þ waþ ð1 waÞG
a
"
2ð1 waÞðx 1Þ þ ðxa 1Þðx 1 þ 2waÞ
G a
2 waþ xwaþ ð1 waÞðxa 1ÞG
a
#
;
m21¼ m ðvþ 1ÞðmmaÞwa
2 waþ vwaþ ð1 waÞðva 1ÞG
a
;
ð5Þ
in which
v¼ 3 4m;
2.2 Determine the effective thermal expansion coefficient of composite
Similar to the elastic modulus, the thermal expansion coeffi-cient of the three-phase composite materials were also identified
in two steps: First, to determine the coefficient of thermal expan-sion of the effective matrix The current paper uses the Duc’s result from[18]for calculating the thermal expansion coefficient of effec-tive matrix
a¼amþ ðacamÞ Kcð3Kmþ 4GmÞwc
Kmð3Kcþ 4GmÞ þ 4ðKc KmÞGmxc
in whicha⁄is the effective thermal expansion coefficient of effec-tive matrix;am,acare 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
a1¼a ðaaaÞwaE1
1 Eaþ 8GaðmamÞð1 waÞð1 þmaÞ
2 waþ xwaþ ð1 waÞðxa 1ÞG
a
;
a2¼aþ ðaa1Þm21 ðaaaÞð1 þmaÞmm21
mma
:
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 inTable 1 [16]
The results of elastic modules of composite materials for differ-ent volume ratios of compondiffer-ent materials written in(5)are given
inTable 2 [7], in which 14 variant cases of different volume ratios
of component three-phase composite materials respectively are gi-ven inTable 3 [7]
Fig 1illustrates the SEM images of the structure of the two-phase composite polymer with the based two-phase composed of the glassy polyester fiber (without the particles) in 1D (all the compos-ite fibers are reinforced in one direction) Fig 2 illustrates the structure of three-phase 1Dm in the presence of the TiO2particles 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 resis-tance of the materials by doping the particles.Figs 1 and 2show 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 Engi-neering 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-sym-metric 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 coor-dinator, h/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]
Trang 3ey
cxy
0 B
1 C
A ¼
e0 x
e0 y
c0 xy
0 B
1 C
A þ z
kx
ky
kxy
0 B
1 C
where
e0 x
e0 y
c0 xy
0 B
1 C
A ¼
u;xþ w2
;x=2
v;yþ w2
;y=2
u;yþv;xþ w;xw;y
0 B
1 C A;
kx
ky
kxy
0 B
1 C
A ¼
w;xx
w;yy
2w;xy
0 B
1 C A; ð10Þ
in which u,vare the displacement components along the x, y direc-tions, respectively
Hooke law for a composite plate is defined as[19]
rx
ry
rxy
0 B
1 C
k
¼
Q0
11 Q0
12 Q0 16
Q012 Q022 Q026
Q016 Q026 Q066
0 B
1 C
k
exa1DT
eya2DT
cxy
0 B
1 C
k
in which
Q0
11¼ Q11cos4
hþ Q22sin4hþ 2ðQ12þ 2Q66Þsin2hcos2
h;
Q0
12¼ Q12ðcos4hþ sin4hÞ þ ðQ11þ Q22 4Q66Þsin2hcos2
h;
Q016¼ ðQ12 Q22þ 2Q66Þsin3hcosh þ ðQ11 Q12 2Q66Þsin hcos3h;
Q0
22¼ Q11sin4hþ Q22cos4
hþ 2ðQ12þ 2Q66Þsin2hcos2
h;
Q0
26¼ ðQ11 Q12 2Q66Þsin3hcosh þ ðQ12 Q22þ 2Q66Þsin hcos3h;
Q066¼ Q66ðsin4hþ cos4hÞ þ ½Q11þ Q22 2ðQ12þ Q66Þsin2hcos2
h; ð12Þ
and
Q11¼ E1
1 E2 E1m2 12
¼ E1
1 m12m21
;
Q22¼ E2
1 E2 E1m2 12
¼E2
E1
Q11;
Q12¼ E1
1 E2
E1m2 12
¼m12
Q22
;
Q66¼ G12;
ð13Þ
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) 1.43 0.345
Titanium oxide TiO 2 (Australia) 5.58 0.20
Table 2
Elastic modules for three-phase composite materials.
E 1 (GPa) E 2 (GPa) m12 G 12 (GPa) G 23 (GPa)
wa = constwc – Particle’s ratio increase
Case 1 18.2019 8.0967 0.8043 1.8616 2.7174
Case 2 17.8415 7.4411 0.8722 1.6747 2.4451
Case 3 17.5209 6.8385 0.9457 1.5093 2.2044
Case 4 17.2338 6.2829 1.0256 1.3618 1.9900
Case 5 16.9751 5.7687 1.1132 1.2296 1.7978
Case 6 16.7404 5.2916 1.2097 1.1103 1.6244
Case 7 16.5273 4.8474 1.3167 1.0021 1.4672
wc = constwa – Fiber’s ratio increase
Case 8 24.2929 7.9971 1.0513 1.4840 2.6771
Case 9 20.9035 7.3880 1.0116 1.4974 2.4247
Case 10 17.5209 6.8385 0.9457 1.5093 2.2044
Case 11 14.1451 6.3402 0.8496 1.5199 2.0103
Case 12 10.7762 5.8860 0.7190 1.5295 1.8382
Case 13 7.4144 5.4702 0.5486 1.5382 1.6843
Case 14 4.0598 5.0880 0.3327 1.5461 1.5461
Table 3
Variant cases of different volume ratios of matrix, fibers and particles.
wm 0.5 0.55 0.6 0.65 0.7 0.75 0.8
wa 0.2 0.2 0.2 0.2 0.2 0.2 0.2
wc 0.3 0.25 0.2 0.15 0.1 0.05 0.0
wm 0.5 0.55 0.6 0.65 0.7 0.75 0.8
wa 0.3 0.25 0.2 0.15 0.1 0.05 0.0
wc 0.2 0.2 0.2 0.2 0.2 0.2 0.2
Fig 1 Black–White SEM image of 1D composite two-phase material (25% of fibers
Fig 2 Black–White SEM image of 1Dm composite three-phase material (25% of fibers and 10% of particles).
Trang 4where h is an angle between the fiber and the coordinate system.
The force and moment resultants of the composite plates are
deter-mined by[13,14]
Ni¼Xn
k¼1
Zhk
hk1
½rikdz; i ¼ x; y; xy;
Mi¼Xn
k¼1
Z hk
hk1
z½rikdz; i ¼ x; y; xy:
ð14Þ
Substitution of Eqs.(9) and (11)into Eq.(14)and the result into
Eq.(14)give the constitutive relations as
ðNx;Ny;NxyÞ ¼ ðA11;A12;A16Þe0
xþ ðA12;A22;A26Þe0
yþ ðA16;A26;A66Þc0
xy
þ ðB11;B12;B16Þkxþ ðB12;B22;B26Þkyþ ðB16;B26;B66Þkxy
DT½a1ðA11;A12;A16Þ þa2ðA12;A22;A26Þ;
ðMx;My;MxyÞ ¼ ðB11;B12;B16Þe0
xþ ðB12;B22;B26Þe0
yþ ðB16;B26;B66Þc0
xy
þ ðD11;D12;D16Þkxþ ðD12;D22;D26Þkyþ ðD16;D26;D66Þkxy
DT½a1ðB11;B12;B16Þ þa2ðB12;B22;B26Þ;
ð15Þ
where
Aij¼Xn
k¼1
Q0ij
kðhk hk1Þ; i; j ¼ 1; 2; 6;
Bij¼1
2
Xn
k¼1
Q0
ij
kh2k h2k1
; i; j ¼ 1; 2; 6;
Dij¼1
3
Xn
k¼1
Q0
ij
kh3k h3k1
; i; j ¼ 1; 2; 6:
ð16Þ
The nonlinear equilibrium equations of a composite plate based
on the classical theory are[13–15]
Mx;xxþ 2Mxy;xyþ My;yyþ Nxw;xxþ 2Nxyw;xyþ Nyw;yy¼ 0: ð17cÞ
Calculated from Eq.(15)
e0
x¼ A11Nxþ A12Nyþ A16Nxy B
11kx B
12ky B
16kxy
þDT a1D
11þa2D
12
;
e0
y¼ A12Nxþ A22Nyþ A26Nxy B21kx B22ky B26kxy
þDTða1D
21þa2D
22Þ;
e0
xy¼ A16Nxþ A26Nyþ A66Nxy B16kx B26ky B66kxy
þDT a1D
16þa2D
26
;
ð18Þ
where
A
11¼A22A66 A
2
26
D ; A
12¼A16A26 A12A66
D ; A
16¼A12A26 A22A16
A22¼A11A66 A
2
16
D ; A26¼A12A16 A11A26
D : A66¼A11A22 A
2 12
D¼ A11A22A66 A11A226þ 2A12A16A26 A212A66 A216A22; ð19Þ
B
11¼ A11B11þ A12B12þ A16B16; B
12¼ A11B12þ A12B22þ A16B26;
B
16¼ A11B16þ A12B26þ A16B66; B
21¼ A12B11þ A22B12þ A26B16;
B
22¼ A12B12þ A22B22þ A26B26; B
26¼ A12B16þ A22B26þ A26B66;
B
61¼ A16B11þ A26B12þ A66B16; B
62¼ A16B12þ A26B22þ A66B26;
B
66¼ A16B16þ A26B26þ A66B66; D
11¼ A11A11þ A12A12þ A16A16;
D
12¼ A11A12þ A12A22þ A16A26; D
21¼ A12A11þ A22A12þ A26A16;
D22¼ A12A12þ A22A22þ A26A26; D16¼ A16A11þ A26A12þ A66A16;
D26¼ A16A12þ A26A22þ A66A26:
Substituting once again Eq (18)into the expression of Mijin
(15), then M into Eq.(17c)leads to
Nx;xþ Nxy;y¼ 0;
Nxy;xþ Ny;y¼ 0;
P1f;xxxxþ P2f;yyyyþ P3w;xxyyþ P4w;xxxyþ P5w;xyyyþ P6w;xxxx
þ P7w;yyyyþ P8w;xxyyþ P9w;xxxyþ P10w;xyyyþ Nxw;xx
þ 2Nxyw;xyþ Nyw;yy¼ 0;
ð20Þ
where
P1¼ B21; P2¼ B12; P3¼ B11þ B22 2B66; P4¼ 2B26 B61;
P5¼ 2B16 B62;
P6¼ B11B
11þ B12B
21þ B16B
61; P7¼ B12B
12þ B22B
22þ B26B
62;
P8¼ B11B
12þ B12B
22þ B16B
62þ B12B
11þ B22B
21þ B26B 61
þ 4B16B
16þ 4B26B
26þ 4B66B
66;
P9¼ 2 B11B
16þ B12B
26þ B16B
66þ B16B
11þ B26B
21þ B66B 61
;
P10¼ 2 B12B
16þ B22B
26þ B26B
66þ B16B
12þ B26B
22þ B66B 62
; ð21Þ
f(x, y) is stress function defined by
Nx¼ f;yy; Ny¼ f;xx; Nxy¼ f;xy: ð22Þ
For an imperfect composite plate, Eq.(20) are modified into form as[7,8]
P1f;xxxxþ P2f;yyyyþ P3w;xxyyþ P4w;xxxyþ P5w;xyyyþ P6w;xxxx
þ P7w;yyyyþ P8w;xxyyþ P9w;xxxyþ P10w;xyyy
þ f;yy w;xxþ w
;xx
2f;xy w;xyþ w
;xy
þ f;xx w;yyþ w
;yy
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
e0 x;yyþe0 y;xxc0 xy;xy¼ w2
;xy w;xxw;yyþ 2w;xyw
;xy w;xxw
;yy
w;yyw
From the constitutive relations(18)in conjunction with Eq.(22)
one can write
e0
x¼ A11f;yyþ A12f;xx A16f;xy B11kx B12ky B16kxy
þDTa1D11þa2D12
;
e0
y¼ A12f;yyþ A22f;xx A26f;xy B21kx B22ky B26kxy
þDT a1D
21þa2D 22
;
e0
xy¼ A16f;yyþ A26f;xx A66f;xy B16kx B26ky B66kxy
þDT a1D
16þa2D 26
:
ð25Þ
Setting Eq.(25)into Eq.(24)gives the compatibility equation of
an imperfect composite plate as[7,8]
A22f;xxxxþ E1f;xxyyþ A11f;yyyy 2A26f;xxxy 2A16f;xyyyþ B21w;xxxx
þ B12w;yyyyþ E2w;xxyyþ E3w;xxxyþ E4w;xyyy
w2
;xy w;xxw;yyþ 2w;xyw
;xy w;xxw
;yy w;yyw
;xx
where
E1¼ 2A12þ A66;
E2¼ B11þ B22 2B66;
E3¼ 2B26 B61;
E4¼ 2B16þ B62:
ð27Þ
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
Trang 5In the present study, the edges of composite plates are assumed
to be simply supported and four edges of the plate are simply
sup-ported and immovable (IM) In this case, boundary conditions are
w ¼ u ¼ Mx¼ 0; Nx¼ Nx0 at x ¼ 0; a;
w ¼v¼ My¼ 0; Ny¼ Ny0 at y ¼ 0; b; ð28Þ
where Nx0, Ny0are fictitious compressive edge loads at immovable
edges
The approximate solutions of w and f satisfying boundary
con-ditions(28)are assumed to be[7,8]
ðw; wÞ ¼ ðW;lhÞ sin kmx sin dny; ð29aÞ
f ¼ A1cos 2kmx þ A2cos 2dny þ A3sin kmx sin dny þ A4
cos kmx cos dny þ1
2Nx0y
2
þ1
2Ny0x
km= mp/a, dn= np/b W is amplitude of the deflection and l is
imperfection parameter The coefficients Ai(i = 14) are determined
by substitution of Eqs.(29a) and (29b)into Eq.(26)as
A1¼ 1
32A22
d2n
k2mWðW þ 2lhÞ;
A2¼ 1
32A
11
k2m
d2WðW þ 2lhÞ;
A3¼ðQ2Q4 Q1Q3Þ
Q22 Q21 W;
A4¼ðQ2Q3 Q1Q4Þ
Q22 Q21
W;
ð30Þ
where Qi(i = 14) = Qi(km, dn) was mentioned inAppendix A
Subsequently, substitution of Eqs.(29a) and (29b)into Eq.(23)
and applying the Galerkin procedure for the resulting equation
yield
ab
4 P1
ðQ2Q4 Q1Q3Þ
Q22 Q21
k4mþ P2
ðQ2Q4 Q1Q3Þ
Q22 Q21
d4
"
þP3ðQ2Q4 Q1Q3Þ
Q22 Q21 k
2
md2 P4ðQ2Q3 Q1Q4Þ
Q22 Q21
P5
ðQ2Q3 Q1Q4Þ
Q2
Q2 þ P6k
4
mþ P7d4þ P8k2md2
# W
þ1
3kmdn P1
1
A22þ P2
1
A11
WðW þ 2lhÞ
ab
64
1
A22d
4þ 1
A11k
4 m
WðW þlhÞðW þ 2lhÞ
þ8
3
ðQ2Q4 Q1Q3Þ
Q22 Q21
kmdnWðW þlhÞ
ab
4 Nx0k
2
mþ Ny0d2
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
sub-jected to temperature environments uniformly raised from stress
free initial state Ti to final value Tf and temperature difference
DT = Tf Tiis constant
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]
Zb 0
Z a 0
@u
@xdxdy ¼ 0;
Z a 0
Zb 0
@v
From Eqs.(10) and (18)one can obtain the following expres-sions in which Eq.(22)and imperfection have been included
@u
@x¼ A
11f;yyþ A12f;xx A16f;xyþ B11w;xxþ B12w;yyþ 2B16w;xy
þDT D
11a1þ D12a2
1
2w
2
;x w;xw
;x;
@v
@y¼ A
12f;yyþ A22f;xx A26f;xyþ B21w;xxþ B22w;yyþ 2B26w;xy
þDT D
21a1þ D
22a2
1
2w
2
;y w;yw
;y:
ð33Þ
Substitution of Eqs.(29a) and (29b)into Eq.(33)and then the result into Eq.(32)give fictitious edge compressive loads as
Nx0¼ J1W þ J2WðW þ 2lhÞ þ J3DT;
Ny0¼ J4W þ J5WðW þ 2lhÞ þ J6DT; ð34Þ
where Ji(i = 16) = Ji(km, dn) are give inAppendix A
Subsequently, setting Eq.(34)into Eq.(31)give
DT ¼ b11W þ b12 W
W þlþ b
1 3
;ineWðW þ 2lÞ
W þl þ b
1
4WðW
where
b11¼32hmnp2
3a2b2X
ðQ2Q4 Q1Q3Þ
Q2
Q2 ;
b12¼1 X
P1ðQ2Q4Q1Q3ÞQ2
Q2 þ P6
m 4p4
a 4 þ P2ðQ2Q4Q1Q3ÞQ2
Q2 þ P7
n 4p4
b4
þ P3ðQ2QQ4Q12 Q3Þ
Q2 þ P8
m 2 n 2p4
a 2 b2 P4ðQ2QQ3Q12 Q4Þ
Q2
m 3 np4
a 3 b
P5ðQ2Q3Q1Q4ÞQ2 Q 2
mn 3p4
ab 3
2 6 6 6
3 7 7
7;
b13¼4mnp2
3a2b2X
P1
A22þ
P2
A11
!
;
b14¼p4
16X
1
A22
n4
b4þ
1
A11
m4
a4
!
;
ð36Þ
where
X ¼ J3k2mþ J6d2n; W ¼ W
For a perfect composite plate (l= 0), Eq.(37)leads to equations that determining buckling temperatureDTbof the plate It can be obtained by taking the limit functionDTðWÞ with W ! 0
DTb¼32hmnp2
3a2b2X
ðQ2Q4 Q1Q3Þ
Q2
4.2 Numerical results and discussion The asymmetric plate with simply supported boundary condi-tion 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 fi-ber angle 45/45/0/45/45 and four-layers plate with the twisted an-gle 0/45/45/90
Trang 6Fistly, 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 4shows the effect of
param-eters (m, n) on nonlinear buckling of five-layers symmetric plate
with the fiber angle 0/90/0/90/0 We can see that thermal
buck-ling 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
geom-etry 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 0/45/45/90
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 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).
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 6 Effects of fiber’s ratiowa on the nonlinear response of three-phase polymer composite five-layers plate under uniform temperature rise (immovable edges).
Fig 7 Effects of ratio particle’s ratiowc on the nonlinear response of three-phase polymer composite four-layers plate under uniform temperature rise (immovable
Trang 7Figs 5 and 6represent the effect of the particles and the fibers
on buckling of the five-layers three phase plate Similarly,Figs 7
three-phase plate
InFigs 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 ratiowc= 0.1) and curve 2 inFig 6(fiber ratiowa= 0.1;
particles ratiowc= 0.2), easily notice that with the same reinforced
component ratiowa+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
expen-sive fibers by particles of titanium oxide in order to reduce the
price However, the five-layers symmetric plate has a better
ther-mal loading ability than the four-layers symmetric one The same
conclusion was also found in study case of the three phase
lami-nated composite plates under mechanical loads[7]
Figs 9 and 10represent the imperfection effects on buckling of
five-layers and four-layers symmetric plates The results show the
subtle change with the increase of the imperfection degree How-ever, 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 bet-ter 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 expres-sions of load–deflection curves
From the obtained results in this paper, we make the following conclusions:
Fig 8 Effects of fiber’s ratiowa on the nonlinear response of 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).
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 11 Effects of ratio b/h on the nonlinear response of three-phase polymer composite five-layers plate under uniform temperature rise (immovable edges).
Trang 8– 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
stron-ger than those of the glass fibers
– With the same thickness and size, the thermal loading ability of the symmetric five-layers plate is better than that of the sym-metric four-layers plates
– The geometry affects significantly on the stability of the com-posite plates
The advantage of study approach in this paper is the nonlinear response of three-phase polymer composite plates which is pre-sented explicitly on parameters of polymer matrix, fibers and particles and also with the geometrical parameters, imperfec-tion and ratio of components in composite, so when adjusting the phase component, we can design and structural new mate-rials meet the technical requirements
Acknowledgment 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
Q1¼ A22k4mþ A11d4þ E1k2md2;
Q2¼ 2A26k3mdnþ 2A16kmd3;
Q3¼ B 21k4mþ B12d4nþ E2k2md2n
;
Q4¼ E3k3mdnþ E4kmd3;
J1¼A
22L1 A12L3
A11A22 A212 ;
J2¼A
22L2 A12L4
A
11A
22 A2 ;
J3¼ A
22H1 A12H2
A
11A
22 A2 ;
J4¼A
11L3 A12L1
A
11A
22 A2 ;
J5¼A
11L4 A12L2
A11A22 A212 ;
J6¼ A
11H2 A12H1
A11A22 A212 ;
where
L1¼ 1 ab
A
11d2þ A12k2m
kmdn
ðQ2Q4 Q1Q3Þ
Q22 Q21 þ 4A
16
ðQ2Q3 Q1Q4Þ
Q22 Q21
þ 4 B 11k2mþ B12d2
;
L2¼k
2 m
8;
L3¼ 1 ab
A
12d2þ A22k2m
kmdn
ðQ2Q4 Q1Q3Þ
Q2 Q2 þ 4A
26
ðQ2Q3 Q1Q4Þ
Q2 Q2
þ 4 B 21k2mþ B22d2
;
L4¼d
2
8;
H1¼ D11a1þ D12a2; H2¼ D12a1þ D22a2:
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