Composite materials Lection_2

8/4/2019 Composite materials Lection_2

1/18

Method of determination of elastic constants of laminated

composite material

Laminated composite (or composite material of laminated structure) is compositematerial consist of pack of serially laid-up individual layers (monolayers), each of them

is characterized by individual thickness and stacking angle related to adopted coordinate

system. It is considered that the ideal adhesion exists between layers, as a result they

deform together at any pack loading, i.e. layers do not slip related to each other.

To research physical and mechanical properties of laminated composite material

V.V. Vasilievs model is used (fig. 1). An orthotropic strip is the representative elementin this theory. This strip has definite stiffness at tension, compression and shear. Elastic

constants of monolayers are defined theoretically by above-mentioned formulas or by

experimental way.

Vasilievs model of laminated composite material. Materials consist of any

orthotropic layers (for examples, unidirectional ones) can be analyzed by means of this

model. An orthotropic strip, possessing by definite stiffness at tension (compression)

along the axes 1, 2 and at shear in the layer plane, is the main seriated representative

element of composite structure. Strips are assumed to be uniform material, there is the

ideal adhesion between layers (so they are joined together).

Physical and mechanical characteristics of laminated composite can be expressed

by means of properties of layers (which in their turn can be determined by means of

previous model or by experimental way), reinforcing angle of each layer and layers

quantity. Now this model is widely used either prediction of laminated composites or

structure design, strength analysis etc.

Let consider composite material consist of any layers with thickness i ;

orthotropy axes of these layers apply angles i with axis of basic (global) coordinate

system (fig. 2). In general case equations of physical law for anisotropic material have

the following form:


2/18

123in x

2

z

y

1

1

1

2

2

Fig. 1 Laminated model of composite material (Vasilievs model)

,

xyG

xy

yE

yy,y

xE

xy,xxy

;

xG

,

xyy

;

xG

,

++=

++=

+=

(1)

where y,xy,x,xy,,,xy,x,yx,xy,xyG,y,x elastic constants, which should

be expressed by means of anisotropic layer characteristics;xy,y,x

- pack strain;xy,y,x - average stress along the pack thickness.

Let deform pack of layers up to strains xy,y,x , then define stresses xy,y,x ,which cause these strains.

12

i

x

x

y

xy Y

x

xyY

Fig. 2 Model of laminated composite material

The strains of each individual layer in local coordinate system can be determined

by well-known formulas because of compatible deformation of entire pack:


3/18

.2cos2sin)(

;cossincossin

;cossinsincos

ixyixyi21

iixyi2

yi2

xi2

iixyi2

yi2

xi1

+=

+=

++=

(2)

Generalized Hooks law for each orthotropic layer has the form:

.G

;EE

;EE

i12

i12i12

i1

i1i12

i2

i2i2

i2

i2i21

i1

i1i1

=

=

=

(3)

If we solve these equations related to stress, one can obtain:

,i12i12i12

i1i12i2i2i2

i2i21i1i1i1

G

);(E

);(E

=+=

+=

(4)

where

;1

EE

i21i12

i1i1 = .1

EE

i21i12

i2i2 (5)

Let substitute dependences (2) to (4) ones to express stresses i12i2i1 ,, by means

of pack strains x, y , xy . Then

( )( )

( )

2 21i 1i x i y i xy i i

2 221i x i y i xy i i

2 22i 2i x i y i xy i i


12i 12i y x i xy i

E cos cos sin cos

sin sin sin cos ;

E sin sin sin cos

cos cos sin cos ;

G sin2 cos 2 .

+ + ++ +

+ ++ + +

+

(6)

Let find projections of these stresses on, axes by known formulas of elasticity theory:

2 2xi 1i i 21 i 12i i

2 2yi 1i i 21 i 12i i

xyi 1i 2i i i 12i i

cos sin sin 2 ;

sin cos sin 2 ;

( )sin cos cos 2 ,

= + = + += +

(7)

or, taking into consideration expressions (6),


4/18

xi x 11i y 12i xy 13i

yi x 21i y 22i xy 23i

xyi x 31i y 32i xy 33i

;

;

.

= + += + += + +

(8)

Here4 2 2 4 2

1 1 i 1 i i 1 i 2 1 i i i 2 i i 1 2 i i

2 2 4 4 21 2 i 2 1 i 1 i 2 i i i 1 i 21 i i i 1 2 i

4 2 2 4 222 i 1 i i 1 i 21 i i i 2 i i 12 i i

1 3 i 3 1 i i

E co s 2 E sin cos E sin G sin 2 ;

( E E ) sin cos E (sin cos ) G sin

E sin 2 E sin cos E cos G s in 2 ;

sin co

= + + += = + + + = + + += = 2 2i 1i 2 1 i i 2 i 1 2 i i 1 2 i

2 2 233 i 1 i 2 i 1 i 2 1 i i i 12 i i

2 223 i 3 2 i i i 1 i 2 1 i i 2 i 1 2 i i

s E ( 1 ) cos E ( 1 ) sin 2G

( E E 2 E ) sin cos G cos 2 ;

sin cos E ( 1 ) sin E ( 1 ) cos 2G

= + +

= +

(9)

Let compose equilibrium equations onand axes:

=

=n

1ixixi ;

==

n

1iyiyi ;

==

n

1ixyixyi , (10)

where n total number of layers, ==n

1ii- total pack thickness.

After substitution of (8) dependences to (10) ones we can obtain formulas for

stresses xyyx ,, expressed by xyyx ,, strains:( )

( )

( ) .1

;1

;1

333231

232221

131211x

++=

++=

++=

(11)

Here

=

=n

1ikliil , (12)

where , l takes values 1, 2, 3.

Equations (11) are generalized Hooks law, which for design stage can be written

as following:

,g

;N

;N

333231xyx232221

131211xx

++==

++==

++==

(13)

where xyyx g,N,N - forces per unit width (force, acting on place with width of one linear


5/18

unit).

Let solve (11) equations system related to strains xyyx ,, :

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

2 22 33 23 12 33 13 23 12 23 22 1

2 12 33 13 23 11 33 13 12 13 11 2

2 12 23 13 22 12 13 11 23 11 22 1

= +

= + + = + +

(14)

where

22311

21322231312

212221133 2)( . (15)

If we compare coefficients at stresses in the equation systems (1) and (14) one can

obtain:

( ) ( )( )

y x22 2 3 3 2 3 1 2 3 3 1 3 2 3

y

, x y

1 2 2 3 2 2 1 3 1 2 3x y x

1 ; B ;

B B B B ; G B

= =

= =

- -

- ( )( ) ( )

( ) ( )

3 1 3 2 3

, 21 1 3 3 1 3 1 2 1 3 1 1 2 3

y x y

, ,

1 2 2 3 2 2 1 3 1 2 1 3 1 1 2 3x y

B ;

1 ; B B B B ;

G B

B B B B ; B B B B ;E B E B

= =

= =

-

- -

- -

( )21 1 2 2 1 2 y

1 .

G

= -

(16)

Formulas for determination of elastic properties of laminated composite materials

follow from these equations:

xy2 2 222 33 23 11 33 13 11 22 12

12 33 13 23 12 33 13 23 2 2

22 33 23 11 33 13

; ; G ;( ) ( ) ( )

; ;

= = = = =

(17)


6/18

12 23 22 13 12 13 11 23 , , 2 2

11 22 12 11 22 12

12 23 22 13 12 13 11 23, ,2 2

22 33 23 11 33 13

; ;

; .

= = = =

Following equations sequence from (16) expressions:

, y ,x y , , y xy yx

xy x xy y x y

; .G E G E E E

= = = = (18)Analysis of (11) equations shows that material is orthotropic in , axes in that case

when following conditions are fulfilled simultaneously:

0 32312313 ==== . (19)

In that case formulas (15) and (17) simplify to form:);( 212221133 = (20)

2 23311 22 12 11 22 12

xy22 11

12 12xy yx x,xy y,xy xy ,x xy,y

22 11

B ; ; G ;

B B; ; 0.

B B

= = == = = = = =

(21)

Let consider in detail some particular structures, which are widely used in practice.

1. The pack consists of one layer (n=1) with reinforcing angle and thickness= . Then

( )[ ]( )

( )[ ]( ) ( )[ ]( ) ( )[ ] .2cosG2cos1Esin1EossinBB

;2cosG2sin1Ecos1EossinBB

;2cosGcossinE2EEB

;2sinGcosEcossinE2sinEB

;2sinG)cos(sinEcossinEEB

;2sinGsinEcossinE2cosEB

122

1222

2113223

122

1222

2113113

212

222112133

212

42

22211

4122

212

44211

222112

212

42

22211

4111

==

==

++=

+++=

+++=

+++=

(22)

It is obvious, that application of these equations for determination of elastic properties

of composite material by means of (17) formulas leads to huge dependences, which are not

useful for qualitative analysis of results. Let derive formulas for elastic properties by

another way, taking into consideration that composite material is statically definable

system.

Let stresses xyyx ,, act in composite material element (fig. 3). Then in 1, 2 axeswe obtain:


7/18

x

x

12

1 122

xy

2 12 1

y

x

xyy

Fig. 3 To determination of elastic properties of composite material

with any reinforcing angle

( )

2 21 y xy

2 2

2 y xy

12 y x xy

cos sin sin 2 ;

sin cos sin 2 ;sin cos cos 2 .

= + += + = + (23)

These stresses stipulates strains 1221 ,, :2 2

1 21 21 x 21

1 2 1 2

2 221

y 21 xy1 2 1 2

cos sin

E E E E

sin cos 1

sin2 ; E E E E

= = +

+ + +

(24)

2 2 2 221

2 x 21 y 21 xy2 1 2 1 2 1

sin cos cos sin 1 sin2 ;

E E E E E E

= + + 12 x y xy

12 12 12

sin cos sin cos cos 2.

G G G + +

Strains in axes, can be calculated by the formulas:

( ) ,2cos2sin

;cossincossin

;cossinsincos

1221xy

122

22

1y

122

22

1

+=

++=

+=

(25)

which after substitution with expressions (24) and some transformations will obtain the

form:


8/18

4 42 2 12

1 2 12 1

2 2 12 12y

1 2 1 12 1

2 212 21xy

1 2 12

2 2 12y

1 2 1 12

cos sin 1 sin cos 2

E E G E

1 1 1 sin cos 2

E E E G E

1 1 cos2 sin2 cos sin ;

E E 2G

1 1 1 sin cos 2

E E E G

= + + + + + + +

++ = + +

12

1

4 42 2 12

y

1 2 12 1

E

sin cos 1 sin cos 2

E E G E

+ + + + +

2 212 21xy

1 2 12

2 212 21 xy x

1 2 12

2 212 21y

1 2 12

22 12

xy1 2 1 12

1 1 cos 2 sin2 sin cos ;

E E 2G

1 1 cos2 sin2 cos sin

E E 2G

1 1 cos2 sin2 sin cos

E E 2G

cos 21 1 sin 2 2

E E E G

++ += +

++ + + + + + + .

(26)

Following relationships can be obtained after comparison coefficients at stresses in

this equations system with general notation of physical law (1):

.G

1

G

1

E

1

E

12sin

G

1

;E

2

G

1cossin

E

cos

E

sin

1

;E

2

G

1cossin

E

sin

E

cos

1

12122

21

1

122

xy

1

12

12

22

2

4

1

4

y

1

12

12

22

2

4

1

4

x

+

++

+=

++=

++=

(27)


9/18

.G2

2coscosE

1sinE

12sinEG

;G2

2cossin

E

1cos

E

12sin

EG

;G

1

E

1

E

1cossin

EEE

122

2212

112

y

y,xy

xy

xy,y

12

2

2

212

1

12

x

x,xy

xy

xy,x

122

21

1

1222

1

12

y

yx

x

xy

+++==

+

+==

++

+==

(28)

Graphical dependences ofEx( ), Ey( ), Gxy( ), xy( ), x,xy( ), y,xy( )(fig. 4) shows that reinforcing angle change influences on material elastic properties

significantly. Formulas (27) and (28) are proved experimentally and show enough

validity for majority of composites.

y

/2 /4 x, xy

yx

yx

yx

xy

y

Gxy

Fig. 4 Dependence of unidirectional composite material

elastic properties on reinforcing direction

2. Pack consists of two layers of the same material with reinforcing along the axes

and.

This composite material are usually called orthogonal reinforced composite.Rigidity characteristics of package can be obtained from formulas (12), taking into

consideration (9) ones (n=2, 1= 1, 2= 2, 1=0, 2=0, 1 2 + ):( )

( ) .0BBBB;GB;EEB;EB;EEB

32233113122133

1221222112112221111 =+

(29)

Elastic constants can be obtained by formulas (21) and (29) because of material

orthotropy:


10/18

( )

( )

( ) .EE

EBB;GG

;EE

EEE

B

BB

1E

;EE

EEE

B

BB

1E

1221

21121

22

12xy12xy

2211

221

2121

21

1221

11

212

2221

y

1221

221

2121

21

2211

22

212

1121

x

++===

+

+

++

=

+=

+

+

++

=

+=

(30)

Let introduce the following notations:

11

1 2

= + - volume fraction of longitudinal layers,2

2 11 2 1 = = + - volume fraction of lateral layers.Taking into consideration these notations previous expressions will take the form:

( )( )

( )( )

( ) .E1EE

;GG

;E1E

EE1EE

;E1E

EE1EE

1121

211xy12xy

2111

221

21

1121y

1121

221

21

2111x

+==

++=

++=

(31)

These dependences evident about invariance of elastic constants of perpendicular

reinforced composites to absolute pack thickness, but elastic constants depend on relative

ratio of these layers thickness (fig. 5). In some cases to determine elasticity modulus of

structure [0,90] in practical calculations the well-known rule of mixture can be used, i..

( ) .E1EEE 21112211x (32)


11/18

1 1xy

0,5

x

Gx

y1

2G12

21

12

Fig. 5 Elastic constants of orthogonal reinforced composite

Error of such kind calculation lays in the range 510%, that is permissible for

design stage.

3. Pack consists of two layers of the same material and same thickness

221 == and reinforcing angles .21 == Such materials are called cross-plied

composites.

Let find rigidity characteristics of the pack by formulas (12):

( ) ( )[ ]( )( )[ ]

.0BBBB

;2cosGcossinE2EEB

;2sinGcosEcossinE2sinEB

;2sinGcossinEcossinEEB

;2sinGsinEcossinE2cosEB

32233113

212

222112133

212

42

22211

4122

212

44211

222112

212

42

22211

4111

====

++=

+++=

+++=

+++=

(33)

This material is orthotropic in axesand, for it coefficientsB13,B31,B23, B32 are

equal to zero.

Elastic constants are calculated by formulas (21). Comparison of dependences (22)

and (33) shows that coefficients B11, B22, B12, B33 are equal to zero for composites with

structures [ ] and [ ] . In connection with above-mentioned fact, it is interesting toknow, which of these materials has the larger elasticity modulus. Let consider, for example,

. The first formula of the system (17) one can write after definite transformations:


12/18

( )( )22

22 13 12 2312 11 2

22 22 22 33 23

1 .

= (34)

Comparison this expressions with following from (18) for composite with

reinforcing [ ]212

1122

1

(35)shows, that cross-plied composite material has large rigidity. Thus it is more efficient to

use structure [ ] instead of skew reinforcing [ ] or [ ] at the same structurethickness (i.e. structure mass).

4. Pack consists of four layers of the same material: n 4,

1 1 2 2 3 4 1 2 3 4 , , , 0 , 90 , - = = = = = = = .Such kind of structure is frequently used in composite constructions.

By formulas (12) we can obtain:

( )( ) ( ) ( )[ ]( ) ( )[ ]

.0BBBB

;2cosGcossinE2EE2GB

;2sinGcossinEcossinEE2EB

;2sinGcossinE2cosEsin2

;2sinGcossinE2sinEs2

32312313

212

2221121122133

212

44211

22212112112

212

22211

42

41122122

212

22211

42

41221111

====

++++=

+++++=

+++++=

+++++=

(36)

Find elastic constants from (20) equations. If we introduce notations

;11

= ,22

= (37)

where

,221 ++= (38)that

( ) .12 21 = (39)This permits to rewrite (36) and (20) in the form:


13/18

( ) ()

( ) ()( ) ( ) ( )

4 411 1 1 2 2 1 2 1 2

2 2 21 21 12 11

4 4

22 1 2 2 1 1 2 1 2

2 2 21 21 12 22

212 1 2 1 21 1 2 1 2

B E E 1 E cos E sin

2E sin cos G sin 2 B ;

B E E 1 E sin E cos

2E sin cos G sin 2 B ;

B E 1 E E sin

cos

+ + + ++ + =

+ + + + + + = + + +

-

-

-

( )( ) ( ) ( )

2 4 4 21 21 12 12

33 1 2 12 1 2 1 2 1 21

2 2 212 33

E sin cos G sin 2 B ;

B G 1 E E 2E

sin cos G cos 2 B ;

+ + = + + +

+ =

-

- - -

(40)

2 212 12

x 11 y 22 xy 3322 11

12 12 xy y

22 11

B BE B ; E B ; G B ;

B B

B B; .

B B

= = == =

- -

Thus, elasticity moduli, Poissons ratios of such kind composite material do not

depend on absolute pack thickness, but depend on layers thickness ratio.

Obtained above formulas for determination of set of elastic constants of laminated

composite material with any structure are classical now and all analysis and design of

composite structure can be provided by means of these formulas. This conclusion is based

on following fact: in local coordinate system each layer is orthotropic, i.e. this layer must

not be unidirectional. Examples of this structures are layers based on woven reinforcement;

groups of layers for which axes 1, 2 are orthotropy axes and properties of these groups of

layers are known in these axes; braided fabrics which frequently have reinforcement [ ] ,

nd isotropic materials, for example, metal sheets.

Thermomechanical characteristics of laminated composites

Linear temperature expansion coefficients of laminated composites


14/18

Composite pack of layers obtains temperature deformations

, , (fig. 6), which are sequence of temperature deformations oflayers at temperature change.

If all layers deform all together, that it is obvious, at pack of layers arbitrary

reinforcing layers restrict each other to deform free because of presence of individual

LCTE i1 and i2 . Because of this fact stresses appear in layers, for entire pack this

system of stresses is self-balanced.

12

x

y 1/2dxyT

1/2dxyT1

dyT

1 dxT

12

1/212i

2iT2i 1iT 1i b

Fig. 6 Temperature deformation of composite material

The following layers strains corresponds to pack deformations

, , :( )( )

( )



12i y x i xy i

cos sin sin cos ;

sin cos sin cos ;

sin2 cos 2 .

= + += +

+ (41)

Equations of generalized Hooks law for individual layer according to Duamel-Neumann hypothesis can be written:


15/18

1i 2i 1i 21i 1i

1i 2i

2i 1i 2i 12i 2i

2i 1i

12i12i

12i

;E E

;E E

.G

= += +=

(42)

One can obtain, solving this system related to stresses:

( ) ( )( ) ( )

1i 1i 1i 1i 21i 2i 2i

2i 2i 2i 2i 21i 1i 1i

12i 12i 12i

E ;

E ;

G .

+ + =

(43)

Differences ( )i1i1 and ( )i2i2 are deformations, which correspond tostresses i1 and i2 .

Let compose equilibrium equations of pack, taking into consideration formulas of

stresses rotation (7) and absence of internal loads:

( )( )( ) ] .02coscossinq

;02sincossinN

;02sinsincosN

n

1i

n

1iii12iii2i1iixyixy

n

1i

n

1iii12i

2i2i

2i1iiyiy

n

1i

n

1iii12i

2i2i

2i1iixix

==

=

= =

= =

= =

(44)

Substitute dependences (41) to (43), and obtained result to (44). Equilibrium

equations (44) obtain the following form after series of transformations:

x 11 y 12 xy 13 T1

x 21 y 22 xy 23 T 2

x 31 y 32 xy 33 T 3

B B B A ;

B B B A ;

B B B A ,

+ + =+ + =+ + =

(45)

where ijB coefficients are defined by (12), but coefficients T 1A , T 2A , T 3A - according

to formulas:


16/18

( ) ( )[ ]

( ) ( )[ ]

( ) ( )[ ]

= =

= =

= =

==

+++==

+++==

n

1i

n

1i i12i2i2i21i1i1iiii3Ti3T

n

1i

n

1ii2

i12i2

i2i22

i21i2

i1i1ii2Ti2T

n

1i

n

1ii2

i12i2

i2i22

i21i2

i1i1ii1Ti1T

.1E1EcossinaA

;sincosEcossinEaA

;cossinEsincosEaA

(46)

From system of equation (45) one can derive formulas for determination of LCTE:

( ) ( ) ( )[ ]

( ) ( ) ( )[ ]

( ) ( ) ( )[ ] .BBBABBBBABBBBAB

1

;BBBBABBBABBBBAB

1

;BBBBABBBBABBBAB

1

21222113T231113122T132223121Txy

131223113T33112132T231333121Ty

132223123T123323132T22333221Tx

++=

++=

++=

(47)

For orthotropic in axes, laminated composite:

13 31 23 32 T 3 B B B B A 0.= = = =Thus for orthotropic composites (47) formulas transform to form:

T 1 22 T 2 12x 2

11 22 12

A B A B;

B B B = T 2 11 T 1 12y 211 22 12

A B A B;

B B B = xy 0= . (48)

Equality 0xy = means, that orthotropic composite does not warp at heating, i..shear deformation does not appear. However, it does not mean absence of shear stresses

in separate layers. These stresses can be determined from (43).

Shrinkage coefficients of laminated composites

Scheme of composite material deformation at shrinkage of its components in

polymerization process is analogous to deformation scheme at temperature change as was

shown above. That is why we write the final results, skipping intermediate transformations.

Equation system for determination of shrinkage coefficients xyyx ,, has the form:

,ABBB

;ABBB

;ABBB

333xy32y31x

223xy22y21x

113xy12y11x

===

(49)

where


17/18

( )( ) ]

n n2 2

1 i y1i i 1i 1i i 21i i i 1 i 1

2 221 2i i 12i i

a E cos sin

E sin cos ;

= == = + +

+ +( )

( ) ]n n 2 2

2 i y2i i 1i 1i i 21i i i 1 i 1

2 22i 2i i 12i i

a E sin cos

E cos sin ;

= = = = + + + +

(50)

( ) ( )n n

3 i y3i i i i 1i 1i 21i 2i 2i 12ii 1 i 1

a sin cos E 1 E 1 .= =

= =

From the equations system (49) we find xyyx ,, :( ) ( ) ( )[ ]

( ) ( ) ( )[ ]

( ) ( ) ( )[ ] .BBBABBBBABBBBAB

1

;BBBBABBABBBBAB

1

;BBBBABBBBABBBAB

1

21222113231113122132223121xy

1312231133311132

2231333121y

13222311312332313222333221x

++=

++=

++=

(51)

Shrinkage coefficients of orthotropic composite are defined by formulas:

;

A2122211

122221x

= ;

A2122211

121112x

= 0 = . (52)In conclusion we write notation of physical law, taking into consideration

temperature and shrinkage deformations in accordance with Duamel-Neumann

hypothesis:

.GEE

;GEE

;GEE

xyxyxy

xy

y

yy,y

x

xx,xxy

yyxy

xyy,y

y

y

x

xxyy

xxxy

xyx,x

y

yyx

x

xx

+ +++=

+ +++=

+ ++=

(53)

For composite material orthotropic in axes , , formulas (53) transform to

following appearance:


18/18

.G

;EE

;EE

xy

xy

xy

yyy

y

x

xxyy

xxy

yyx

x

x

=+

+(54)

Reverse notations of these systems (after solution related to stresses) have the form:

( )

( )

( ),BBB1

;BBB1

;BBB1

xy33y32x31xy

xy23y22x21y

xy13y12x11x

++=

++=

++=

(55)

where

.

;

;

xyxyxyx

yyyy

xxxx

(56)

For orthotropic composite these formulas transform to:

( )

( )

,B1

;BB1

;BB1

xy33xy

y22x12y

y12x11x

=

+=

+=

(57)

where

.xyxy = (58)Thus, obtained dependences for determination of elastic constants of unidirectional

and laminated composite materials permit to express the single meaning of stress by

means of strain and vice versa. If metal elastic characteristics could be found in

guidebooks that for composite materials it is necessary to know to define these

properties by means of known physical and mechanical characteristics of composite

material components (for unidirectional materials) or by means of monolayers

characteristics obtained theoretically or by experimental way (for laminated

composites).

Date post:	07-Apr-2018
Category:	Documents
Upload:	yigitciftci
View:	230 times
Download:	0 times

Composite materials Lection_2

Documents