DSpace at VNU: Bending analysis of three-phase polymer composite plates reinforced by glass fibers and titanium oxide particles

Three-phase polymer composite which consists of poly-mer matrix, reinforced ﬁber and spherical particles are used to in-crease the mechanical and physical performance of material.. Altho

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Bending analysis of three-phase polymer composite plates reinforced by glass ﬁbers and titanium oxide particles

a

University of Engineering and Technology, Vietnam National University, Hanoi, Viet Nam

b

Shipbuilding Science and Technology Institute, Viet Nam

a r t i c l e i n f o

Article history:

Received 6 October 2009

Received in revised form 12 April 2010

Accepted 12 April 2010

Available online 10 May 2010

Keywords:

Three-phase composite

Bending analysis

Plate

Shear deformation

a b s t r a c t

This paper presents a method to determine bending deﬂection of three-phase polymer composite plates consisting of reinforced glass ﬁbers and titanium oxide (TiO2) particles This method analyzes bending with taking into account two important effects which are interaction between titanium oxide particles and polymer matrix and shear deformation Mechanical properties of the three-phase composite plates are clearly related to mechanical properties and volume fractions of constituent materials These explicit expressions play an important role in design of composite materials as well as optimal behavior of three-phase composite plates

1 Introduction

The composite material consists of two or more constituent

materials Three-phase polymer composite which consists of

poly-mer matrix, reinforced ﬁber and spherical particles are used to

in-crease the mechanical and physical performance of material There

have been some calculations determining the material constants

for three-phase composite, which is mainly calculated by the

numerical methods or only for YoungÕs modulus[1,2]

Although some researchers have treated problem of bending

analysis of composite rectangular plates[3,4], the bending analysis

for three-phase composite plate includes polymer matrix, ﬁbers and

particles, to the authorsÕ best knowledge, has not been considered

In this article, bending deﬂection of three-phase composite

plate consists of polymer matrix, reinforced glass ﬁbers and

tita-nium oxide (TiO2) particles is analyzed

A composite rectangular plate of length a, the width b, thickness

h, referred as a thin sheet and built from three-phase composite

consisting of the polymer matrix, reinforced glass ﬁbers and

tita-nium oxide particles, is considered (Fig 1) The purpose of present

study is to analyze bending of mentioned three-phase composite

plates taking into account effects of each phase (ﬁber and particle)

and shear deformation (e44,e55–0, ignoring the Kirchhoff–Love

hypothesis) on bending behavior of plates The relationship

between deformation and stress of plate in this case are

deter-mined as[4,5]:

r11¼ A11e11þ A12e22; r22¼ A22e22þ A12e11; r66¼ A66e66;

where

A11¼ E11

1 m12m23; A22¼ E22

1 m12m23;

A12¼ E11m23

1 m12m23¼

E22m12

1 m12m23; A66¼ G12;

Eijandmijare the YoungÕs module and Poisson ratios of anisotropic elastic materials[6]

The coefﬁcients A44 and A55 will be concerned in the below mention

2 Governing equations for bending analysis of composite plates

by the ﬁrst order shear deformation theory

Transverse shear stress components are deﬁned as[5]:

r44¼ f ðzÞwðx; yÞ;r55¼ f ðzÞuðx; yÞ ð2Þ

where

f ðzÞ ¼1

2 z

21 2

2

andw(x, y),u(x, y) denotes the rotations of the midplane on the (x, y) plane

From Eqs.(1) and (2)we have

e44¼ 1

A44 r44¼ f ðzÞ 1

A44wðx; yÞ; e55¼ 1

A55 r55¼ f ðzÞ 1

A55

uðx; yÞ ð3Þ

* Corresponding author.

E-mail address: ducnd@vnu.edu.vn (N Dinh Duc).

Contents lists available atScienceDirect

Computational Materials Science

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 m a t s c i

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The strain–displacement relations are given by:

e11¼@u

@x; e22¼@v

@y; e66¼@u

@yþ

@v

e44¼@w

@yþ wðx; yÞ; e55¼@w

where u, v are displacement components in the x, y directions,

respectively, and w is transverse displacement known as deﬂection

of the plate

Equilibrium equations of the plate in terms of three variables w,

w, anduare deﬁned as:

@ u

@xþ@w

@y¼ z

I 2 ðhÞ

Dx@

3 w

@x 3þ ðD1þ 2Dx;yÞ@3w

@x@y 212

h3I1 ADy 55

@ 2 u

@x 2þDxy

A 55

@ 2 u

@y 2

h

þðD1 þD xy Þ

A 44

@2u

@x@y

i

þ I2u¼ 0

Dy@

3 w

@y 3þ ðD1þ 2Dx;yÞ@ 3 w

@x 2 @y12

h3I1 ADy 44

@ 2 w

@y 2þDxy

A 44

@ 2 w

@x 2

h

þðD1 þD xy Þ

A 55

@2w

@x@y

i

þ I2w¼ 0

8

>

ð6Þ

where

I1ðhÞ ¼

Z 0:5h

0:5h

zI0ðzÞdz ¼ 1

120h

3

; I2ðhÞ ¼

Z 0:5h

0:5h

f ðzÞdz ¼ 1

12h

3

;

Dx¼h

3

12A11; D1¼

h3

12A12; Dy¼

h3

12A22; Dxy¼

h3

12A66

ð7Þ

Eq.(6)is a set of basic equations for determining deﬂection of the

three-phase composite plates with shear deformation is considered

The coefﬁcients A11, A22, A12, A66, A44, A55are used to determine

coefﬁcients Dijin Eq.(7) Also, Aijquantities can be calculated from

the elastic coefﬁcients of the composite material

3 Determining the elastic coefﬁcients of the three-phase

polymer composite material

In this work, the elastic moduli of the three-phase composite

are estimated based on the two-phase composite material

theoret-ical models Firstly the elastic moduli of two-phase composite

which consists of particles and original polymer matrix is

calcu-lates[7–9] This two-phase material is assumed to be a ‘‘effective”

matrix material And then a new two-phase composite consists of

‘‘effective” matrix material and reinforced ﬁber is considered[8]

Refs.[8,10–16]have been proposed such approach and solve a

ser-ies of issues for composite materials

In this work, stress components in the composite plate are

de-ﬁned as follows[8]:

Considering two ﬁrst terms on the right side of Eq.(8), which accounts for the interaction between matrix and particles, the moduli of the effective matrix material are deﬁned as[8]:

G ¼ Gm

1 ncð7 5mmÞH

1 þ ncð8 10mmÞH; K ¼ Km

1 þ 4ncGmLð3KmÞ1

1 4ncGmLð3KmÞ1 ð9Þ

where

L ¼Kc Km

Kcþ4G m 3

; H ¼ Gm=Gc 1

8 10mmþ ð7 5mmÞG m

G c

;

E ¼ 9KG 3K þ G; m¼3K 2G

E,m, G, K, n are the YoungÕs modulus, Poisson ratio, shear modulus, bulk modulus, and the volume ratio of the component materials in composite, respectively The subscriptm,a,cindicates the polymer material, ﬁber, and particle, respectively

The YoungÕs modulus of the polymer composite reinforced tita-nium oxide particles is presented in[9] The YoungÕs modulus in the Eqs.(9), (10)is consistent with the developed expression and results in[7,9]

Elastic moduli of the three-phase composite material is then calculated based on the effective matrix material and ﬁber with six elastic constants[17] It is given by:

E11¼ naEaþ ð1 naÞE þ 8Gnað1 naÞðma mÞ

2 naþ xnaþ ð1 naÞðxa 1ÞG

a

;

E22¼ m2

21

E11þ 1 8G

2ð1 naÞðv 1Þ þ ðva 1Þðv 1 þ 2naÞG

a

2 naþ vnaþ ð1 naÞðva 1ÞG

a

"

(

þ2vð1 naÞ þ ð1 þ navÞG

a

vþ naþ ð1 naÞG

a

#)1

;

m21¼ m ðvþ 1ÞðmmaÞna

a

m23¼ E22m 2

21

E11

þE22 8G 2

ð1 naÞvþ ð1 þ navÞG

a

vþ naþ ð1 naÞG

a

"

2ð1 naÞðv 1Þ þ ðva 1Þðv 1 þ 2naÞG

a

#

;

G12¼ G1 naþ ð1 naÞ

G a

1 naþ ð1 þ naÞG

a

; G23¼ G vþ naþ ð1 naÞG

a

ð1 naÞvþ ð1 þ vnaÞG

a

;

where

Ga¼ 0:5Ea ð1 þmaÞ; Gc¼

0:5Ec

ð1 þmcÞ; E ¼

9KG 3K þ G;

m¼3K 2G

4 Some numerical results and discussions

To illustrate proposed approach, a three-phase composite square plate (a = b) consists of polymer matrix, particles reinforced, and ﬁbers reinforced along the x axis is considered

Table 1 Characteristics of investigated three-phase composite.

Material component YoungÕs modulus Poisson ratio

m

Fig 1 Conﬁguration and coordinate of a three-phase composite plate.

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The shear deformation is considered in this calculation,

there-fore, a plate with thickness is 15 times smaller than itÕs width

and length (h/a = h/b = 1/15) is considered The plate is assumed

to be simply supported on the all edges and subjected to a

trans-verse load distributed on the surface as

q ¼ q0sinpx

a sin

py

a :

Three-phase composite material is a transversely isotropic

medium with Oxy We have following relations

A11¼ A22¼ E11¼ E

A12¼ E11m12¼ Em

A44¼ A55¼ G23¼ G0

A66¼ G12¼ E11

2ð1 þm12Þ¼

E 2ð1 þmÞ¼ G

ð13Þ

The simply supported boundary conditions are deﬁned as

w ¼ 0; Mx¼ 0; w¼ 0jx¼0;x¼a;

To satisfy the simply supported boundary condition, the follow-ing approximate functions are assumed

w ¼ A sinpx

a sin

py

a ; u¼ B cospx

a sin

py

a ; w¼ C sin

px

a cos

py

a : ð15Þ

The bending deﬂection of square plate can be calculated from

Eq.(6):

w ¼ w0 1 þ1

5k

; where k ¼ p2E

G0

ð1 mÞ

h a

2

;

w0¼ q0a

4

4Dp4sinpx

a sin

py

w ¼ w0 1 þ1

5k

¼ q0a

4

4Dp4sinpx

a sin

py

a 1 þ

1 5

p2E

G0

ð1 mÞ

h a

2

: ð17Þ

The maximum bending deﬂection occurs at the center of plate

x = y = a/2 and is given by:

wMax¼3q0að1 m2

12Þ

E11n3p4 1 þ1

5

p2E11n2

G23ð1 m12Þ

¼3q0a

p4

ð1 m2

12Þ

E11n3 þp2ð1 þm12Þ

5G23n

¼3q0a

Table 2

The deﬂection w 1 (n a , n c ).

Fig 2 The relationship between the deﬂection w 1 and n a , n c

Table 3

h/a ratio when n a = 0.

Fig 3 h/a ratio when n a = 0 versus volume ratio n c of particle component.

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w1¼ ð1 m2

12Þ

E11n3 þp2ð1 þm12Þ

5G23n

and

n ¼ h=a;m12¼m21E11

E22

:

In this work, a three-phase composite material with

character-istics inTable 1is investigated

According to Eqs.(11), (13), and (16), bending deﬂection of the

three-phase composite plate versus variation of the volume

frac-tion of ﬁber phase na, and particle phase ncwhen accounting the

shear deformation is presented inTable 2andFig 2

increasing volume ratio of either ﬁber or particle component The

deﬂection changes due to the variation of ﬁber volume ratio larger

than the particle volume ratio

5 The condition for considering the shear deformation

In this section, the difference between w (accounted shear

deformation) and w0(without shear deformation) is considered

This difference is depended on the factor k (see Eq.(16)) Factor k

is investigated in the ﬁrst case (C1) of (w–w0)/w0< 5%, and the

sec-ond one (C2) of (w–w0)/w0< 10% Eq.(16)shows that the factor k is

depended on h/a ratio, naand nc

For the ﬁrst case C1, Eq.(16)shows that the difference (w–w0)/

w0< 5% when k 6 0.25 The h/a ratio is then given by:

h

a6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:25G0ð1 mÞ

p2E

s

¼ 1

2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 mÞG0

E

s

¼ 1

2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 m12ÞG23

E11

s

:

The h/a ratio for the second case C2 (k 6 0.5) is given by:

h

a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:5G0

ð1 mÞ

p2E

s

¼1

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 m12ÞG23

2E11

s

:

The ratio h/a will be investigated in several speciﬁc composites when changing volume ratios of the component materials

composite of matrix and particles without ﬁber component) It shows that the h/a value is not much changed when changing vol-ume ratio of particle component in the two-phase composite Therefore, the h/a ratio is main factor when plate bending analysis

Table 4andFig 4shows the h/a ratio when na= 0.2 (three-phase composite with 20% reinforced ﬁber) with volume ratio ncchanges from zero to 50% The thin plate (h/a 6 1/20) theory can be used when ncsmaller than 0.35, the thick plate theory with accounting

of shear deformation has to be accounted when nclarger than 0.35 for the ﬁrst case C1 The thin plate theory cannot be used for the case C2

Table 5andFig 5shows the h/a ratio when nc= 0.2 (three-phase composite with 20% reinforced particle) with volume ratio na

changes from zero to 50% It shows that the shear deformation decreases when increasing the volume ratio of ﬁber component

Table 4

h/a ratio when n a = 0.2.

Fig 4 h/a ratio when n a = 0.2 versus volume ratio n c of particle component.

Table 5

h/a ratio when n c = 0.2.

Fig 5 h/a ratio when n c = 0.2 versus volume ratio n a of ﬁber component.

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The thin plate theory can be used for the case C1 when volume

ra-tio of ﬁber component nalarger than 0.15

6 Conclusion

A bending analysis of three-phase composite plate with

accounting shear deformation and particle–matrix interaction is

proposed This analysis shows that the deﬂection is decreased

when increasing volume ratio of either ﬁber or particle component

The deﬂection changes due to the variation of ﬁber volume ratio

larger than the particle volume ratio Bending deﬂection and elastic

moduli of three-phase composite are explicitly expressed on

com-ponent material mechanical properties and geometry of plate

Therefore, the composite material characteristic and also structure

design can be effectively optimized

Acknowledgment

This work has been supported by ‘‘Vietnam National University–

Hanoi key science research project”, coded QGTÐ 09.01

References

[1] B Jaing, R.C Batra, Intel Mater Syst Struct 12 (2001) 165–182.

[2] J.C Afonso, G Ranalli, Compos Sci Technol 65 (2005) 1264–1275 [3] S Timoshenko, S Krieger, Theory of Plates and Shells, Mc Graw-Hill Book Company, New York, 1959.

[4] J.N Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press, 2004.

[5] A.K Malmeicter, V.P Tamuz, G.A Teterc, Strength of Composite Materials,

‘‘Zinatie”, Riga, 1980.

[6] S.G Lekhnitskii, Theory of Elasticity of an Anisotropic Elastic body, Holden-Day, Inc, San Francisco, 1981.

[7] R.M Christensen, Mechanics of Composite Materials, Wiley, New York, 1979 [8] G.A Vanin, Nguyen Dinh Duc, Mech Compos Mater 32 (3) (1996) 291–305 [9] P Thomas, Selvin, J Kuruvilla, T Sabu, Mater Lett 58 (2004) 281–289 [10] G.A Vanin, Mech Compos Mater 32 (5) (1996) 668–675.

[11] G.A Vanin, Mech Compos Mater 32 (6) (1996) 380–386.

[12] Nguyen Dinh Duc, Mech Compos Mater 32 (6) (1996) 370–379.

[13] G.A Vanin, Nguyen Dinh Duc, Mech Compos Mater 33 (2) (1997) 155–160 [14] Nguyen Dinh Duc, Mech Compos Mater 33 (2) (1997) 251–257.

[17] G.A Vanin, Micro-Mechanics of Composite Materials, Nauka Dumka, Kiev, 1985.

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