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

A NoNliNeAr stAbility ANAlysis of imperfect three-phAse polymer composite plAtes Nguyen Dinh Duc, 1* tran Quoc Quan, 1 and Do Nam 2 in memory of G.. Vanin Keywords: nonlinear stability a

A NoNliNeAr stAbility ANAlysis of imperfect

three-phAse polymer composite plAtes

Nguyen Dinh Duc, 1* tran Quoc Quan, 1 and Do Nam 2

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 in-vestigated 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

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

1Vietnam National University,Hanoi, 144 Xuan Thuy-Cau Giay- Hanoi-Viet Nam

2University of Engineering and Technology, Vietnam National University, Hanoi

*Corresponding author; tel: 84-4-37547978; fax: 84-4-37547724; e-mail: ducnd@vnu.edu.vn

Russian translation published in Mekhanika Kompozitnykh Materialov, Vol 49, No 4, pp 519-536 , July-August,

2013 Original article submitted October 31, 2012

Plates, shells, and panels are the basic structures used in engineering and industry These structures play an important role as the main supporting components in all kinds of structures in machinery, civil engineering, shipbuilding, flight vehicle manufacturing, etc The stability of composite plates and shells is the first and most important problem in an optimal design

In fact, many researchers are interested in this problem [8-24] Therefore the investigation on 3-phase composite plates and shells is important both from the scientifical and practical points of view

Among the most fundamental references on the composite materials, we can mention monographs [12] (on laminated orthotropic composite plates and shells) and [20] (on composite FGM) Some information on the stability of laminated com-posite and FGM plates can be obtained in [11-24]

In the present paper, we study the stability of a 3-phase polymer composite with an imperfection on an elastic foundation The paper focuses on deriving of an algorithm for calculating the stability of the composites by analyzing the load–deflection relationship obtained from the basic equations for laminated composites The effects of the material proper-ties of components, geometrical properproper-ties, imperfections, and of the elastic foundation on the stability of 3-phase polymer composite plates are also examined

2 Determination of the elastic moduli of a 3-phase composite

The elastic moduli of 3-phase composites are estimated by consecutively using two theoretical models of a 2-phase

composite: nD m = O m + nD [1-10] This paper considers a 3-phase composite reinforced with particles and unidirectional fibers, so the model of the problem will be 1D m = O m + 1D First, the moduli of the effective matrix O m, called the “effective moduli,” are calculated At this step, the effective matrix consists of the original matrix and particles and is considered to be homogeneous and isotropic The next step is estimating the elastic moduli for a composite material consisting of the effective matrix and unidirectional reinforcing fibers

Assuming that all the components (matrix, fibers, and particles) are homogeneous and isotropic, we denote by

E E E m, , , n n na c m, , , y ya c m, , and ya c Young’s moduli, the Poisson ratios, and the volume fractions of matrix, fibers, and particles, respectively As in [1, 2], we obtain the moduli for the effective composite in the form

H

K K G L K

G L K

−

−

1 1

ψ

where

L K K

K G

+ 4 3

G

c

8 10ν 7 5ν

E and n can be calculate from G and K :

E KG

K G

= +

9

3 , ν = +

−

K G

K G.

We should note that formulas (1) and (2) take into account the nonlinear effects and the interaction between particles and the base and differ from the well-known formulas

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

G

− + + −( ) ( − )

,

E G

G G

a

2 11

1 8

−

( ψψa χa

a

G G

)( − )

+

























2

1

1

−

+ + −( )



























−

a

a

G G G G

,

G G

G G G G

a

a

12

= + + −( )

− + +( )

, G G

G G G G

a

a

23

1

=

+ + −( )

−

,

(3)

23 22

212 11

1

1

E E G

x x G

G

G

a

a

= − +

−













− ( ) ( − )+( − ) ( − + )

− + + −( ) ( − )







a

a

G

G





 ,

21

1

− + + −( ) ( − )

a a

a

G G

,

where

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].

TABLE 1 Elastic Moduli of 3-Phase Composite Materials

ψa =const , ycincreases

ψc =const , ya increases

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 1D m and 2D m 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,

Fig 1 Black-and-white SEM images of 1D (a) and 2D (b) 2-phase composite materials (25% fibers,

without particles)

TABLE 2 Cases of Different Volume Fractions of Fibers and Particles

ε ε γ

ε ε γ

x y xy

x y xy

x y xy

z

k k k



















=























 +















0 0 0











where

ε ε γ

x y xy

u w

v w

u v w w

0 0 0

2 2

2 2

























=

+ +





















 ,

k k k

w w w

x y xy

xx yy xy

























=

−

−

−





















, , ,

, 2

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 σ σ

x y

xy k

Q Q Q

Q Q Q

Q Q Q



















 =

 11 12 16







































k

x y

xy k

ε ε γ

where

Q11 Q11 4 Q Q Q

cos θ sin θ ( )sin θcos ,θ

Q12 Q12 4 4 Q Q Q

(cos θ sin ) (θ )sin θcos ,θ

Q16 Q12 Q22 Q66 3 Q Q Q

Q22 Q11 4 Q Q Q

sin θ cos θ ( )sin θcos ,θ

Q26 Q11 Q12 Q66 3 Q Q Q

Q66 Q66 4 c 4 Q Q Q Q

with

Q E E

E

E

2

1 12 2

1

12 21

=

−

=

−

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

fibers, without particles)

Q E E

E

E

E Q

2

1 12 2

2

1 11

1

=

−

=

Q E E

E Q

2

1 12 2

12 22

1

=

−

= ν

ν , Q66=G12

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 i i k dz

h

h

k

n

k

k

=  

−

∫

∑

=

σ

1

1

, i x y xy= , , ,

h

h

k

n

k

k

−

∫

∑

=

σ

1

1

, i x y xy= , ,

(6)

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

N N N x, y, xy ( ,A A A, ) x (A A A, , ) y (A A A, ,

+( ,B B B k11 12, 16) x+(B B B k12, 22, 26) y+(B B B k16, 26, 66) xy (7)

M M M x, y, xy (B B B, , ) x (B B B, , ) y (B B B, ,

+(D D D k11, 12, 16) x+(D D D k12, 22, 26) y+(D D D k16, 26, 66) xy,

where

A ij Q ij k h k h k

k

n

=

∑( ) ( 1),

B ij Q ij k h k h k

k

n

=

∑

1

2 1( ) ( 2 21),

D ij Q ij k h k h k

k

n

=

∑

1

3 1( ) ( 3 31), i j, =1 2 6, , 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, +2M xy xy, +M y yy, +N w x ,xx+2N w xy ,xy+N w y ,yy =0 (8) From relations (7), it follows that

εx0 A N x A N y A N xy F k x F k y F k xy

εy0 A N x A N y A N xy F k x F k y F k xy

εxy0 A N x A N y A N xy F k x F k y F k xy

where

A11* = A A22 66−A262

∆ , A12* = A A16 26−A A12 66

∆ , A16* = A A12 26−A A22 16

A22* = A A11 66−A162

∆ , A26* = A A12 16−A A11 26

∆ , A66* = A A11 22−A122

∆ = A A A11 22 66−A A11 262 + A A A −A A −A A

12 16 26 122 66 162 22

F11=A B11 11* +A B12 12* +A B16 16* , F12=A B11 12* +A B12 22* +A B16 26* ,

F16=A B11 16* +A B12 26* +A B16 66* , F21=A B12 11* +A B22 12* +A B26 16* ,

F22=A B12 12* +A B22 22* +A B26 26* , F26=A B12 16* +A B22 26* +A B26 66* ,

F61=A B16 11* +A B26 12* +A B66 16* , F62=A B16 12* +A B26 22* +A B66 26* ,

F66=A B16 16* +A B26 26* +A B66 66*

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,

P f1 ,xxxx+P f2 ,yyyy+P w3 ,xxyy+P w4 ,xxxy+P w5 ,xyyy+P w6 ,xxxx+P w7 ,yyyyy

+P w8 ,xxyy+P w9 ,xxxy+P w10 ,xyyy +N w x ,xx+2N w xy ,xy+N w y ,yy =0 (10) with

P E1= 12, P2 =E21, P3 =E11+E22−E33, P4 =E32−E13, P5 =E31−E23,

P6 =E14, P7 =E25, P8=E15+E24+2E36, P9 =2E16−E34, P10=2E26−E35,

E11=F11, E12=F21, E13=F61, E21=F12, E22=F22, E23=F62, E31=F16, E32=F26, E33=F66,

E14=B F11 11+B F12 21+B F16 61−D11, E15=B F11 12+B F12 22+B F16 62−D12,

E16=B F11 16+B F12 26+B F16 66−D16, E24=B F12 11+B F22 21+B F26 61−D12,

E25=B F12 12+B F22 22+B F26 62−D22, E26=B F12 16+B F22 26+B F26 66−D26,

E34=B F16 11+B F26 21+B F66 61−D16, E35=B F16 12+B F26 22+B F66 62−D26,

E36=B F16 16+B F26 26+B F66 66−D66,

where f(x,y) is the stress function defined by

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]

P f1 ,xxxx+P f2 ,yyyy+P w3 ,xxyy+P w4 ,xxxy+P w5 ,xyyy+P w6 ,xxxx+P w7 ,yyyyy+P w8 ,xxyy

+P w9 ,xxxy +P w10 ,xyyy+ f,yy(w,xx+w,*xx)−2f xy(w xy+w xy)+ f x

, , ,* , 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

εx yy, εy xx, γxy xy, w,xy w w,xx ,yy w w,xy ,*xy w w xx yy

, ,*

From constitutive relations (9), in conjunction with Eq (11), one can obtain that

εx0 A f yy A f xx A f xy F k x F k y F k xy

εy0 A f yy A f xx A f xy F k x F k y F k xy

εxy0 A f yy A f xx A f xy F k x F k y F k xy

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

A f11* xxxx E f1 xxyy A f22 yyyy F21 xxxx F12 yyyy E2 xxy

−E3w,xxxy+E4w,xyyy−(w,2xy−w w,xx ,yy+2w w,xy ,*xy−w w,xx ,*yy−w , yy y w,*xx)=0, (15) where

E1=2A12* −A66* ,E2=F11+F22+2F66, E3=2F26+F61, E4=2F16+F62

Equations (12) and (15) are nonlinear in the variables w and f and are used to investigate the stability of thin

com-posite plates subjected to mechanical loads

In the present study, the edges of composite plates are assumed to be simply supported Depending on in-plane restraints

at the edges, three cases of boundary conditions, labeled as Cases 1, 2, and 3, will be considered [13, 6,21]

Case 1 All four edges of the plate are simply supported and freely movable (FM) The associated boundary

condi-tions are

w N= xy =M x=0 , N x=N x0 at x=0,a,

w N= xy =M y=0 , N y=N y0 aty=0,b (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 x0 at x=0,a,

w v M= = y =0 , N y=N y0 aty=0,b (17)

Case 3 All edges are simply supported The edges x=0, are freely movable, whereas the edges y a =0, are im-b

movable In this case, the boundary conditions are defined as

w v M= = y=0 , N y=N y0 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]

f = A1cos2λm x A+ 2cos2δn y A+ 3sinλm xsinδn y

0 2

1 2

1 2

where λm=m aπ/ , δn =n bπ/ , W is the deflection amplitude, and m is the imperfection parameter The coefficients

A i i ( = −1 3 are determined by inserting Eqs (19) into Eq (15):)

A

A n m W W h

1 22

2 2

1

A m n W W h

2 11

2 2

1

A F F E

A A E W

4

12 4 2 2 2

22 4 11 4 1 2 2

A A E W

2

4 2

22 4 11 4 1 2 2

Introducing Eqs (19) into Eq (12) and applying the Galerkin procedure to the resulting equation yield

ab P F F E

4 1 21

4

12 4 2 2 2

4

4

F F E

A A E

P F F E

3 21

4

12 4 2 2 2

2 2 4

E E

A A E

4 2

+

P E E

A A E P P

5

2 4

4

7 4



1 3

1

A* P A* W W h ab

A n A m W W h W h

64

22

4 11

4





8 3

21 4 12 4 2 2 2

22 4 11 4 1 2 2

F F E

A A E

( * * ) W(W+µh)−ab N( x λm+N y δn) (W +µh)

where m and n are odd numbers This is the basic governing equation for the nonlinear response of 3-phase polymer composite

plates under mechanical loading conditions

4 Nonlinear stability Analysis

Consider a simply supported polymer composite plates whose all edges are movable Let us analyze two cases of mechanical loads

4.1 polymer composite plates under axial compressive loads

Let a polymer composite plate be subjected to an axial compressive load F x uniformly distributed along its two curved

edges x = 0 In this case, q=0,N y0 =0, and N x0= −F h x , and Eq (20) leads to

W b W W W b W W

+ +

+

µ

µ

where

b mB F m B F n E m n B

nB A m B A n

11

2

21 4 4 12 4 2 2 2 2 2

22 4 4 11

32 3

+

1 2 2 2

+E m n B a) ,

b P m B F m B F n E m n B

n B A m B A

2

1 1

4 2 4

21 4 4 12 4 2 2 2 2

2 2

22 4 4 1

+

( *

11 4 1 2 2 2

*n +E m n B a) − P n F m B F n E m n B

2

2 2

2

P m B F m B F n E m n B

B A m B A n E

3

2 2 2

21 4 4 12 4 2 2 2 2 2

22 4 4 11 4

11m n B2 2 2a) − P m B E m B E n

4

4 6 4

6

+

P m n B E m B E n

B A m B A n E m n B

5

2 2 6 2

6

22 4 4 11 4 1 2 2 2

n B P n B P

m B B

a

a h

6

4 2 4

2 2

2 2 2 2

b mnB B

P A

P A

a h

3

22

2 11

4 3















* * , b

n B A h n A m B a

41

2

2 2 22

4 11

4 4

16

and

h

1= 1 , P P

h

2 = 2 ,P P

h

3 = 33 ,P P

h

4 = 43 ,P P

h

5= 53 , P P

h

6 = 63, P P

h

7= 73 , P P

h

8 = 83 ,

A11* =hA11*, A22* =hA22* , E1=hE1, E E

h

2 = 2 , F F

h

21= 21 , F F

h

12= 12, E E

h

3 = 33 ,

E E h

4 = 43 , W W

h

= , B b

h

h = , B b

a

a = For a perfect composite plate (µ = 0 subjected only to an axial compressive load F) x, Eq (21) leads to

F x =(b b11+ ) +b b W+

3 1 2 1 4

1 2

from which the upper buckling compressive load may be obtained at W → 0 :

F b P m B F m B F n E m n B

n B A

h

1

4 2 4

21 4 4 12 4 2 2 2 2

2 2 22

( * m m B4 4a A n E m n B a

11 4 1 2 2 2

2

2 2

2

P m B F m B F n E m n B

B A m B A n E

3

2 2 2

21 4 4 12 4 2 2 2 2 2

22 4 4 11 4

( * * 11m n B2 2 2a) − P m B E m B E n

4

4 6 4

6

+

P m n B E m B E n

B A m B A n E m n B

5

2 2 6 2

6

22 4 4 11 4 1 2 2 2

n B P n B P

m B B

a

a h

6

4 2 4

2 2

2 2 2 2

We are also interested in finding the lower buckling load To this end, we consider F x=F W x ( ) at odd numbers m and n, and from equation dF

dW

x =0 , we get that

b

th = −( 11+ 31)

4 1

32 3

2

21 4 4 12 4 2 2 2 2

22 4 4 11 4 1

mnB F m B F n E m n B

A m B A n E m

a

3 2 1 22

2 11 2

22

4 11 4

4 3 8

n B

mn B P

A

P A

A n A m B

a

a















aa4

















Since dF

dW

x

W W th

=

> , the lower buckling load is

F b b

b b

xlower = −( + ) +

11 31 2 4

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 se-quence 0/45/–45/45/–45/90 and 5-layer symmetric plates with the stacking sese-quence 45/–45/0/45/–45