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Composites Science and Technology

45 (1992) 135-152

Prediction of on-axes elastic properties of

plain weave fabric composites

N. K. Na ik V. K. Ganes h

Aerospace Engineering Department, Indian Institute of Technology, Po wai , Bombay 400 076, India

(Received 14 February 1991; revised version received 4 September 1991; accepted 2 October 1991)

Two fabric composite models are presented for the on-axes elastic analysis of

two-dimensional orthogonal plain weave fabric lamina. These are two-

dimensional models taking into account the actual strand cross-section

geometry, possible gap between two adjacent strands and undulation and

continuity of strands along both warp and fill directions. The shape functions

considered to define the geometry of the woven fabric lamina compare well

with the photomicrographs of actual woven fabric lamina cross-sections. There

is a good correlation between the predicted results and the experimental

values. Certain modifications are suggested to the simple models available in

the literature so that these models can also be used to predict the elastic

properties of woven fabric laminae under specific conditions. Some design

studies have been carried ou t for graphi te/epoxy woven fabric laminae. Effects

of woven fabric geometrical parameters on the elastic properties of the

laminae have been investigated.

Keywords:

woven fabric lamina, prediction, two-dimensional, plain weave,

elastic constants

NOTATI ON

a

a~*, b~*, d~*

i , j = 1 , 2 , 6

ax,, Zx,

ay,, z r`

Ao B j Dij

i , j = l , 2 , 6

EL, ET, ~/LT, GET,

G rr

eL(o), eT(o), vLT(o),

CLT(O), C (O)

Ex, vxy,

St rand wid th

Extens iona l , coupl ing and

b e n d i n g c o m p l i a n c e

cons t an t s

Param eters as def ined in

Fig. 6

Param eters as def ined in

Fig. 5

Extens iona l , coupl ing and

bend ing s t if fness cons t an t s

UD compo si t e e las t ic p ro-

per t i es a long the f ib re and

t ransverse f ibre di rect ions

Loca l red uced e l as ti c con-

s t an ts for undu la t ion angle

0

Elast ic constants of uni t

c e l l / WF l a m i n a

Composites Science and Technology 0266-3538/92/$05.00

1992 Elsevier Science Publishers Ltd.

135

g

h

hm

hx,(x, y) ,

i = 1 , 2 , 3 , 4

hyi (y), i= 1, 2, 3, 4

H

k

Qij, i , j = 1, 2, 6

%,

i , j = 1 , 2 , 6

V

Vm

x, y, z

G a p b e t w e e n t h e a d j a c e n t

st rands

Maximum s t rand th i ckness

Thickness of mat r ix a t

x = 0 , y = 0

Thickness of mat r ix and

s t rands in X -Z p l ane a t a

poin t as d ef ined by co-

ordin ates x and y (Fig. 6)

Thickness of mat r ix and

s t ra n d s i n Y -Z p l a n e a t

x = 0 (Fig. 5)

Tota l t h i ckness of WF

lamina

Transverse bu lk modulus

Re duc ed s t if fness of

l amina

Loca l reduced and aver -

aged comp l i ance cons t an ts

V o l u m e

Fibre vo lume f rac t ion

Mat r ix vo lum e f rac t ion

Car t es i an co-ord ina t es

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136 N. K. Naik, V. K. Ganesh

ZX (x, y) , i = 1, 2

zy i(y ) , i= 1, 2

O(x ) , O(y )

O

Shape func t ions o f s t r and

u n d u l a t i o n in X - Z p l a n e

(Fig. 6)

Shape func t ions o f s t r and

u n d u l a ti o n in Y - Z p l a ne

(Fig. 5)

Local of f -axis angle of the

u n d u l a t e d s t r a n d

Maximum of f - ax i s ang le o f

t h e u n d u l a t e d s t r a n d

Subscr ipts

f

L

T

W

Quant i t ies in f i l l di rect ion

Quan t i t i es in f ib re

d i r ec t ion

Quant i t i es in t r ansver se

f ib re d i r ec t ion

Q u a n t i t i e s i n w a r p

di r ec t ion

Superscripts

el

f

m

O

Q u a n t i t i e s o f e l e m e n t

Quan t i t i es o f f ib re

Quant i t i es o f mat r ix

W F c o m p o s i t e o v e r a l l

p r o p e r t i e s

p m Q u a n t i t i e s o f p u r e m a t r i x

s Qua nt i t i es o f s t r and

s l Qu ant i t ies of s l ice

* Qua nt i t i es o f U D cros sp ly

l a m i n a t e

Overbar s ind ica te average va lues /quan t i t i es t r ans -

formed to g loba l d i r ec t ion

Abbre v ia t ions

C C A

C L T

E A M

M K M

M M P M

PS

S A M

SP

U D

W F

1-D

2-D

3-D

Compos i t e cy l inder as -

s e m b l a g e ( m o d e l )

C las s ica l l amina te theory

E l e m e n t a r r a y m o d e l

M o d i f i e d K a b e l k a ' s m o d e l

Modi f i ed mosa ic para l l e l

m o d e l

P a r a l l e l - s e r i e s ( m o d e l )

S l i ce a r r ay model

S e r i e s - p a r a l l e l ( m o d e l )

U n i d i r e c t i o n a l

W o v e n f a b ri c

O n e - d i m e n s i o n a l

T w o - d i m e n s i o n a l

T h r e e - d i m e n s i o n a l

1 I N T R O D U C T I O N

The increas ing use o f compos i t e mater i a l s has

revolu t ion i sed the aerospace indus t ry over the

pas t two decades . The ab i l i ty to vary the

p r o p e r t i e s a n d p e r f o r m a n c e o f c o m p o s i t e m a t e -

r i a l s has been in l a rge measure r espons ib le fo r

the g rea t impac t tha t these mater i a l s have had .

Trad i t iona l ly , advanced compos i t e s t ruc tures

have been f abr i ca ted f rom tape prepregs which

were sys temat ica l ly s t acked to fo rm a l amina te .

Thi s type of cons t ruc t ion t ends to g ive op t imal

in-plane s t i f fness and s t rength. Since the pr imary

loads usual ly are in-plane, the use of such

c o m p o s i t e s a p p e a r e d l o g i c a l . H o w e v e r , t h e r e a r e

m a n y s i t u a t i o n s w h e r e n e i t h e r p r i m a r y n o r

secondary loads a re in -p lane . In such s i tua t ions

tape prepreg l amina tes may no t be the mos t

appropr ia t e .

The fu ture fo r compos i t es i s undergo ing a

t r ans i t ion . The aerospace per formance c r i t e r i a

cons is t ing of high specif ic s t i f fness and high

spec i f i c s t r eng th a re be ing supplemented wi th

h igh toughnes s and e f f i c i en t manufac turab i l i ty .

Wi th th i s , t ex t i l e s t ruc tura l compos i t es in genera l

and woven f abr i c (WF) compos i t es in par t i cu la r

are f inding increas ing use in pr imary as wel l as

secondary s t ruc tura l app l i ca t ions a long wi th

unid i r ec t iona l (UD) t ape compos i t es . Making use

of the un ique combina t ion of l igh t weigh t ,

f lexibi l i ty , s t rength and toughness , text i le s t ruc-

tu res l ike wovens , kn i t s , b ra ids and nonwovens

have now been r ecogni sed as a t t r ac t ive r e in for -

cements fo r s t ruc tura l app l i ca t ions .

Woven f abr i c i s fo rmed by in te r l ac ing two

mutua l ly perpendicu la r s e t s o f yarns . The

lengthwise th reads a re ca l l ed warp and the

crosswise threads f i l l or wef t . The inter lacing

pat tern of the warp and f i l l i s known as the

w e a v e . T w o - d i m e n s i o n a l ( 2 - D ) f u n d a m e n t a l

weaves are plain, twi l l and sat in . The micro-

m e c h a n i c a l b e h a v i o u r o f w o v e n f a b r i c l a m i n a t e s

depends on the f abr i c p roper t i es , which in tu rn

d e p e n d o n t h e f a b r i c s t r u c t u r e . T h e p a r a m e t e r s

involved in de te rmin ing the f abr i c s t ruc ture a re

weave , f abr i c count , f inenes s o f yarn , f ib re

charac te r i s t i cs , yarn s t ruc ture , degree o f undula-

t ion , e t c . The a rch i t ec ture o f a WF lamina i s

c o m p l e x a n d t h e r e f o r e t h e p a r a m e t e r s c o n t r o l l i n g

t h e m e c h a n i c a l a n d t h e r m a l p r o p e r t i e s o f W F

compos i t es a r e too numerous . Thi s makes i t

imprac t i ca l to charac te r i s e the WF compos i t es

th rough t es t s a lone , neces s i t a t ing ana ly t i ca l

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Prediction of on-axes elast ic propert ies o f plain weave fabric composi tes 137

m o d e l s w h i c h c a n p r e d i c t t h e m e c h a n i c a l a n d

t h e r m a l p r o p e r t i e s o f t h e W F c o m p o s i t e s .

A var i e ty o f ana ly t i ca l model s (Raju et al. ~)

h a s b e e n p r o p o s e d f o r t h e p r e d i c t i o n o f t h e

t h e r m o - e l a s t i c p r o p e r t i e s o f W F l a m i n a e . T h e

model s a r e based on the c l as s i ca l l amina te theory

(CLT ) 2-4 or f ini te ele m en t analys is , s-7 Ha lph in

et

al. 2

e x t e n d e d t h e l a m i n a t e a n a l o g y d e v e l o p e d t o

predict the elas t ic s t i f fness of a randomly-

or i en ted , shor t - f ib re compos i t e to 2 -D and 3-D

w o v e n f a b r i c c o m p o s i t e s . T h e w e a v e g e o m e t r y

cons idered here r epresen t s the f abr i c in 1 -D on ly

and a l so the c i r cu la r geomet ry o f the s t r and

cros s - sec t ion cons idered here i s no t r ea l i s t i c .

C h o u & I s h i k a w a 3 h a v e p r e s e n t e d t h r e e m o d e l s

to p red ic t the e l as t i c p roper t i es o f WF lamina .

These a re the mosa ic model ,S the f ib re

u n d u l a t i o n m o d e l s a n d t h e c o m b i n a t i o n o f t h e

a b o v e t w o , t h e b r i d gi n g m o d e l . 9 T h e m o s a i c

model idea l i s es the WF lamina as an

as semblage of asymmet r i c c ros sp ly l amina tes .

D e p e n d i n g o n w h e t h e r t h e p i e c e s o f t h e c r o s s p l y

laminate are in paral lel or in ser ies , i .e . i sos t rain

or i sos t r es s condi t ion , r espec t ive ly , the bounds of

the s ti ffnes s as p red ic ted by the m osa ic mod el can

b e e v a l u a t e d . T h i s m o d e l d o e s n o t c o n s i d e r t h e

s t r and con t inu i ty and s t r es s d i s tu rbance a t the

in te r f ace o f the as semblage . The f ib re undula t ion

model cons ider s the s t r and con t inu i ty and

undula t ion , bu t i t i s a 1 -D model as i t cons ider s

t h e u n d u l a t i o n o f t h e s t r a n d i n t h e l o a d i n g

d i r e c t i o n o n l y . T h e c o m b i n a t i o n o f m o s a i c a n d

f ib re undula t ion model s , ca l l ed the b r idg ing

m o d e l , w a s p r o p o s e d t o a n a l y s e s a t i n w e a v e

fabr ics . The model cons ider s the b r idg ing e f f ec t

presen t in the s a t in weave f abr i c due to the

presence of non- in te r l ac ing r eg ions . The br idg ing

model cons ider s the f ib re con t inu i ty and i s a 2 -D

model fo r s a t in weave , bu t r educes to f ib re

u n d u l a t i o n m o d e l i n t h e c a s e o f p l a i n w e a v e . T h i s

m o d e l c o n s i d e r s t h e u n d u l a t i o n i n t h e l o a d i n g

di r ec t ion , as in the case o f the f ib re undula t ion

m o d e l , b u t t h e s t r a n d u n d u l a t i o n i n t h e

t r ansver se d i r ec t ion and i t s ac tua l c ros s - sec t iona l

g e o m e t r y a r e n o t c o n s i d e r e d . T h e s e m o d e l s w e r e

l a t e r e x t e n d e d t o e v a l u a t e t h e t h e r m a l p r o p e r t i e s

and to analyse hyb r id W F lam inae . ~-~2 In

genera l , t he ana ly t i ca l p red ic t ions d id no t

cor re la t e wel l wi th the expe r imen ta l r esu l t s ~3 for

p la in weave f abr i c compos i t es .

K a b e l k a 4 s u g g e s te d a m e t h o d o f e v a l u a t i n g t h e

e las t i c and thermal p roper t i es o f a p la in weave

fabr ic l amina . Thi s i s a 2 -D model t ak ing in to

cons idera t ion the undula t ion in bo th warp and f i l l

d i r ec t ions , bu t the ac tua l s t r and c ros s - sec t iona l

g e o m e t r y w a s n o t c o n s i d e r e d . T h e p r o p e r t i e s o f

the undula ted warp and f i l l s t r ands were

eva lua ted under the cons tan t s t r es s condi t ion in

the s t r and and then the c l as s i ca l l amina te theory

was used to p red ic t the overa l l p roper t i es .

A 3-D f in i t e e l ement ana lys i s was p resen ted by

R a j u et al. ~ t o p r e d i c t t h e t h e r m a l e x p a n s i o n

coef fi c ien t s o f the W F lam ina . He re , aga in the

WF lamina was idea l i s ed as an as semblage of

asymm et r i c c ros sp ly l am ina tes . Wh i tcom b 6 a l so

used the 3 -D f in i t e e l ement ana lys i s to ana lyse

W F l a m i n a . H e r e , t h e u n d u l a t i o n a n d c o n t i n u i t y

o f t h e s t r a n d s w e r e c o n s i d e r e d i n o r d e r t o s t u d y

t h e e f f e c t o f v a r i o u s w e a v e p a r a m e t e r s o n t h e

m e c h a n i c a l p r o p e r t i e s o f t h e W F l a m i n a . T h e

undula t ion shapes a t the in te r l ac ing cons idered in

t h e a b o v e s t u d i e s w e r e v e r y a p p r o x i m a t e a n d

m a y n o t p r e s e n t a n a c c u r a t e b e h a v i o u r o f a p la i n

weave f abr i c l amina .

Z h a n g & H a r d i n g 7 an d D o w & R a m n a t h 14

p r e s e n t e d f a b r i c m o d e l s b a s e d o n e n e r g y

pr inc ip les . Z han g & Ha rd ing 7 used the s t r a in

energy equ iva lence pr inc ip le to p red ic t the e l as t i c

proper t i es o f a p la in weave l amina . The f in i t e

e l e m e n t m e t h o d w a s u s e d t o e v a l u a t e t h e s t r a i n

energ ies o f the cons t i tuen t phases fo r the

analys is . In these s tudies (Refs 7 and 14) , the

u n d u l a t i o n o f t h e s t r a n d w a s c o n s i d e r e d i n t h e

load ing d i r ec t ion on ly and there fore a l l t he

inheren t d i s c repanc ies p resen t in 1 -D model s

would a l so be p resen t .

1 . 1 T h e c h o i c e o f a 2 - D m o d e l

A s ing le l ayer WF compos i t e i s des igna ted as WF

l a m i n a . T h e w o v e n f a b r i c c a n b e i n t h e f o r m o f

a n o p e n w e a v e o r a c l o s e w e a v e . I n t h e c a s e o f

t h e o p e n w e a v e , t h e r e m a y b e g a p s b e t w e e n t w o

adjacen t s t r ands , whereas c lose weave f abr i cs a r e

t i g h t l y w o v e n w i t h o u t a n y g a p b e t w e e n t w o

adjacen t s t r ands . There can a l so be cer t a in

fabr ics made of twis t ed s t r ands which would

invar iab ly have a cer t a in amount o f gap even i f

they a re t igh t ly woven . I t i s obv ious tha t the

p r e s e n c e o f a g a p b e t w e e n t h e a d j a c e n t s t r a n d s

would a f f ec t the s t i f fnes s o f the WF lamina and

h e n c e s h o u l d b e a c c o u n t e d f o r w h i l e e v a l u a t i n g

t h e t h e r m o - m e c h a n i c a l p r o p e r t i e s .

T h e e x p e r i m e n t a l l y d e t e r m i n e d f i b r e v o l u m e

fract ion, Vf , of the WF lamina is the overal l Vf ,

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z

h o

h o

SECTI ON S o- S O

h 0 = h m/2

z

~

, :~ : :~ ,

I

SECTION S~ - S~

z . ?z . ,

~ . . i

I

SECTION Sz -S z SECTION S 3- S 1

t 1 1 - t - -

o~ 2

SEC TION S 4 - S~ j~_~11"

I I I J l I F

h4=( hm+ hf ) /2 S 0 E PLAN

WARF

lllfl .---_54

5a

Ii ---5Z

. - - -5 1

,----~o x

Fig. 1. Pla in we ave fabric lamina structure--cross-sections at different intervals.

V~', bu t fo r the ana lys is o f the W F lamina the

s t rand Vf, V~, form s the input . I t is the ref ore

neces sary to eva lua te the s t r and Vf f rom the

o v e r a ll Vf d e t e r m i n e d e x p e r i m e n t a l l y . T h e

a v a i l a b l e m e t h o d o l o g i e s d o n o t t a k e i n t o a c c o u n t

t h e g a p b e t w e e n t h e a d j a c e n t s t r a n d s , t h e a c t u a l

c ros s - sec t iona l geomet ry o f the s t r and , and

s t r and undula t ion t r ansver se to the load ing

di r ec t ion . Ma them at ica l ly , the s e r ies mo del 3

should g ive the lower bound of s t i f fnes s due to

the as sumpt ion of the i sos t r es s condi t ion and

t h e r e b y h i g h e r c o m p l e m e n t a r y e n e r g y . B u t

owing to the g ros s s impl i f i ca t ion of no t

c o n s i d e r in g t h e a b o v e m e n t i o n e d p a r a m e t e r s , t h e

1-D ser i es model p red ic t s h igher s t i f fnes s than

t h e e x p e c t e d l o w e r b o u n d . A l s o , f o r t h e

eva luat io n of the s t ra nd Vf f rom the over al l Vf,

t h e i n f o r m a t i o n a b o u t t h e g a p a n d s t r a n d

undula t ion in bo th warp and f i l l d i r ec t ions i s

neces sary .

F igure 1 p resen t s the c ros s - sec t ions o f a p la in

weave f abr i c l amina a t d i f f e ren t s ec t ions f rom the

midpoint of the f i l l s t rand (So-S0) to the midpoint

Fig. 2. Optical micrograph---cross-sectionalview of a plain

weave fabric laminate.

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Prediction of on-axes elastic properties of plain weave fabric composites

139

Fig. 3. Plain weave fabric structure.

i s ca l l ed a s l i ce a r r ay model , abbrev ia ted SAM.

In the s econd model , the un i t ce l l i s d i s c re t i s ed

in to s l i ces e i ther a long or ac ros s the load ing

di r ec t ion . The s l i ces a r e fu r ther subdiv ided in to

e l e m e n t s . T h e i n d i v i d u a l e l e m e n t s a r e a n a l y s e d

s e p a r a t e l y . T h e e l e m e n t s a r e t h e n a s s e m b l e d i n

paral lel or ser ies to obtain the s l ice elas t ic

cons tan t s . Fur ther , the s l i ces a r e as sembled

e i ther in s e r i es o r para l l e l to ob ta in the e l as t i c

cons tan t s o f the un i t ce l l . Th i s s cheme of

discret is ing the uni t cel l into s l ices and fur ther

i n t o e l e m e n t s i s c a l l e d a n e l e m e n t a r r a y m o d e l ,

a b b r e vi a te d E A M .

of the gap ($4-$4) . Figure 2 is an opt ical

micrograph showing the typ ica l c ros s - sec t ions o f

the p la in weave f abr i c l amina a t d i f f e ren t

sec t ions . A typ ica l p la in weave f abr i c s t ruc ture i s

shown in F ig . 3 . I t i s s een tha t the th icknes s o f

the f i l l s t r and decreases g radua l ly f rom the

midpoin t o f the s t r and to zero in the gap r eg ion .

Thi s r educ t ion due to the s t r and c ros s - sec t iona l

geomet ry would r educe the overa l l s t i f fnes s o f

t h e W F l a m i n a . T h e r e f o r e , t h e g e o m e t r y o f t h e

s t r and c ros s - sec t ion should be cons idered whi le

evaluat ing the s t i f fness and this requires a 2-D

m o d e l . T h e a v a i l a b l e 1 - D m o d e l s p r e d i c t h i g h e r

s t i f fnes s as the maximum s t r and th icknes s i s

c o n s i d e r e d i n t h e s e m o d e l s .

2 F A B R I C C O M P O S I T E M O D E L S

T h e p l a i n w e a v e f a b r i c c o m p o s i t e m o d e l s

p r e s e n t e d h e r e a r e 2 - D i n t h e s e n s e t h a t t h e y

c o n s i d e r t h e u n d u l a t i o n a n d c o n t i n u i t y o f t h e

s t r and in bo th the w arp a nd f il l d i r ec t ions . Th e

mode l s a l so account , fo r the p r esenc e o f the gap

b e t w e e n a d j a c e n t s t r a n d s a n d d i f f e r e n t m a t e r i a l

and geomet r i ca l p roper t i es o f the warp and f i l l

s t rands .

2.1 Refined mod els

Two re f ined model s a r e p resen ted in th i s s ec t ion .

In the f i rs t model , the uni t cel l i s discret ised into

s l i ces a long the load ing d i r ec t ion . The ind iv idua l

s l i ces a r e ana lysed s epara te ly and the un i t ce l l

e l as t i c p roper t i es a r e eva lua ted by as sembl ing the

s l i ces under the i sos t r a in condi t ion . Such a model

2. 1.1 Slice array mode l (S AM )

In the ana lys i s , the s t r and i s t aken to be

t r ansver se ly i so t rop ic and i t s e l as t i c p roper t i es

are eva lua ted f rom the t r ansver se ly i so t rop ic

f ibre and matr ix proper t ies at s t rand Vf . I t should

b e n o t e d t h a t o w i n g t o t h e p r e s e n c e o f p u r e

mat r ix pocke t s in the WF lamina , the s t r and Vf

w o u l d b e m u c h h i g h e r t h a n t h e c o m p o s i t e o v e r a ll

Vf. The s t r and prop er t i es a r e eva lua ted us ing the

c o m p o s i t e c y li n d e r a s s e m b l a g e ( C C A ) m o d e l

(Refs 15 and 16) which is br ief ly presented in the

Appendix . The de ta i l s o f eva lua t ion of s t r and Vf

from c om posi te ov eral l Vf is discussed late r .

The r epresen ta t ive un i t ce l l o f a WF lamina i s

t aken as shown in F ig . 4 (a ) . By v i r tue o f the

symmet ry of the in te r l ac ing r eg ion in p la in weave

fabr ic , on ly one quar te r o f the in te r l ac ing r eg ion

is analysed. The analys is of the uni t cel l i s then

per formed by d iv id ing the un i t ce l l in to a number

of s l ices as shown in Fig. 4(b) . These s l ices are

then idea l i s ed in the fo rm of a four - l ayered

lamina te i . e . an asymmet r i c c ros sp ly s andwiched

be tween two pure mat r ix l ayer s as shown in F ig .

4 (c ) . The e f f ec t ive p roper t i es o f the ind iv idua l

l a y e r c o n s i d e r i n g t h e p r e s e n c e o f u n d u l a t i o n a r e

used to eva lua te the e l as t i c cons tan t s o f the

idea l i s ed l amina te . Th i s , in tu rn , i s used to

eva lua te the e l as t i c cons tan t s o f the un i t ce l l /WF

lamina .

I n o r d e r t o d e f in e t h e u n d u l a t i o n a n d g e o m e t r y

of the s t r and c ros s - sec t ion the fo l lowing shape

f u n c t i o n s a r e u s e d . T h e b e l o w m e n t i o n e d

expres s ions a re wi th r e fe rence to F igs 5 and 6 .

I n t h e Y - Z p l a n e , i . e . a l o n g t h e w a r p d i r e c t i o n

(Fig. 5)

hf :~y

zy , ( y ) = - ~- cos - - (1)

ayt

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140 N. K . Na i k , V . K . Gane sh

. j J ' '

" ~ h f ~

- ~

(a ) UNIT CELL

c

(b) ACTUAL SLICES

2

* FACTORED h w

(c) IOEALISED SLICES

Fig.

a nd

hm/2

i

hw

hf 0

hm/2

5 . P la in

kZ

_ _ _ ~ ~ ( y ) . . . . . . . . . . . . . .

WARP

hY2 [Y)

L _ _ _

$

- - o f f 2 - - -~ g f l2

w e a v e f a b r i c l a m i n a c r o s s - s e c t i o n :

d i r e c t i o n .

w a r p

hyl( y) hf + hm

- ~ zy2(y )

hy2(y) = hw

hy3(y) = zy2(y ) - zy~ (y) y = O--~afl2

= 0 y = ae/2----~ (af + g,)12

hy4 (y) hf + hm

= ~ zy~(y)

(3 )

In the X- Z plane , i . e . a long the fi ll d i rec t ion

(Fig. 6)

hw

: rx h m

zx l ( x , y) = -~- cos -- - hy~(y) + - - (4)

ax, 2

hw ~rx hm

zx2(x, y) = - -~- cos (aw + gw) h y j y ) + ~ (5)

Fig . 4 . P la in we ave fabr ic l a mina un i t ce l l and i t s

idea l i sa t ion .

a nd

whe re

zy2(y ) hf :ry

= ~ co s ia r + gf)

J t a f

ay t =

2[ ~ - cos-,(2zY '~ ]

\ h f / J

h, ( ~ae ~

zy, = + ~- cos \2 (a f + &) /

(2)

FILL

I

hx3 (x,y)

hf

r

i ~ . . . . axt/2---

hm/2

4 aw/2

Fig . 6 . P la in w eave fabr ic l amina

d i r e c t i o n .

J

J

cross - sec t ion : fill

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Prediction o f on-axe s elastic properties o f plain weav e fabric composites 141

w h e r e

a n d

$'t'aw

axt =

_~[2z~;~

2 cos t~

\ h w /

h~ [ n:a .

= - - ~ C O S ~ , - - ~ /

Zx, 2 2 ( a . + g . ) /

hXl(X, y ) = hw + h.,_ _ zx l( x, y)

2

hxz(x , y ) = zx , ( x , y ) - z x2(x, y )

x = 0---~ aw l2 (6)

= 0 x = aw/2--->(aw + gw )/2

hx3(x, y) = hy3(y)

hx4(x, y) = zxz(x, y) - hx3(x, y)

+ (hw +

h,,)/2 + hf

The va l id i ty o f the above expres s ions can be

a s c e r t a i n e d b y c o m p a r i n g t h e o p t i c a l m i c r o g r a p h

of the ac tua l WF lamina c ros s - sec t ion a long the

f i l l di rect ion (Fig. 7) and the s imulated plot

m a k i n g u s e o f t h e s a m e s t r a n d p a r a m e t e r s ( F i g .

8).

I t can be s een in the above expres s ions tha t the

para me ter z~, wo uld r educ e to zero and ax, to aw

w h e n t h e g a p b e t w e e n t h e a d j a c e n t w a r p s t r a n d s

i s z e r o ( X - Z p l a n e ) . S i m i l a r l y , i n t h e Y - Z p l a n e ,

zy, wou ld r ed uce to zero and ay, to a~. Th e idea o f

i n t r o d u c i n g t h e s e p a r a m e t e r s i n t h e s h a p e

func t ions i s to s imula te the gap be tween the

F i g . 7 . A c t u a l g e o m e t r y o f t h e p l a i n w e a v e f a b r i c l a m i n a

c r o s s - s e c t i o n : s c a n n i n g e l e c t r o n m i c r o g r a p h .

0.09 rnrn--=~ O-&8mm

Fig. 8. Actual geometry of the plain weave fabric lamina

cross-section: simulated.

strands mathematically, which was otherwise not

poss ib le i f the s am e expres s ion i s used for zx~ a n d

zx2 and s imilar ly for zy~ a n d zyz. T h e s e

p a r a m e t e r s o n l y s t e e p e n t h e o u t e r c o n t o u r o f t h e

s t r and c ros s - sec t ion wi thout d i s tu rb ing the

overa l l undula t ion of the s t r and . The s lope of the

f i l l s t rand is so maintained that at a given point in

a l l the s ec t ions acros s the load ing d i r ec t ion the

s lope of the s t rand is the s am e, i .e . the loca l of f -axis

angle of the f i l l s t rand, Of , i s not a funct ion of y.

Similar ly, the local of f -axis angle of the warp

s t r and , Ow, i s no t a func t ion of x . The s t eep en ing

o f t h e o u t e r c o n t o u r a n d m a i n t a i n i n g t h e s a m e

off-axis angle in al l the planes across the loading

di r ec t ion a t a g iven po in t make the c ros s - sec t ion

o f t h e W F l a m i n a u n i t c e l l u n s y m m e t r i c a l a b o u t

i t s midplane . Thi s can be s een in F ig . 1 which

presen t s the c ros s - sec t ions o f two ad jacen t un i t

ce l l s . Only the c ros s - sec t ions a t the midpoin t o f

the s t rand (S0-S0) and the gap ($4-$4) are

s y m m e t r i c a b o u t t h e i r m i d p l a n e s . H e r e , a s y m -

m e t r y o r s y m m e t r y i n d i c a t e s t h e p r e s e n c e o r

absence of averaged coupl ing s t i f fnes s t e rms of

tha t c ros s - sec t ion , r espec t ive ly . In a l l t he o ther

sec t ions i t i s s een tha t the th icknes s o f the top

pure mat r ix l ayer i s l es s than tha t o f the bo t tom

pure mat r ix l ayer . Wi th th i s , h~>h~ a n d h~'>h~

and ha '> h~. Thi s i s the beh aviou r in the r eg ion

AB (F ig . 1 ) , whereas the behaviour i s as sumed to

b e t h e r e v e r s e i n t h e r e g i o n B C . I n o t h e r w o r d s ,

the th icknes s o f the pure mat r ix l ayer would be

more a t the top than a t the bo t tom in the r eg ion

BC. Mathemat ica l ly , i t means tha t the coupl ing

ef fec t o f r eg ion AB and BC are ba lanc ing each

other . Th i s exerc i s e i s done to s ee tha t the

averaged coupl ing t e rms a re zero fo r the un i t

ACDE and th i s i s t rue as the p la in weave f abr i c

compos i t es do no t twis t g loba l ly on ex tens ion .

Thi s apparen t twis t ing of the f abr i c on ex tens ion

w a s s e e n b e c a u s e o f t h e s h a p e f u n c t i o n

c o n s i d e r e d . T h e o t h e r w a y o f e l i m i n a t i n g t h e

coupl ing t e rms i s to pu l l the en t i r e f abr i c in such

I t __ ? t f ~ t

a way tha t h ~ -h l in s ec t ion S~-S1 , h 2 - h 2 in

sect ion Sz-S2 and so on.

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142

N. K. Naik, V. K. Ganesh

The volume of t he pure mat r ix reg ion in t he

uni t cel l can be evaluated by calculat ing the

th i cknesses i n t he pure mat r ix reg ion for t he

shape func t ions cons idered and then in t egra t ing

to ge t t he vo lume of t he mat r ix in t he pure

mat r ix reg ion . The th i ckness ord ina t es i n t he

pure mat r ix reg ion a re g iven by

hx,(x, y)

a n d

hx4(x, y)

as g iven in express ion (6). Know ing the

overal l Vf of the WF lamina the st rand Vf can be

calculated.

The st rand f ibre volume fract ion i s given by

vw o

V~- ~- W - V pm ( 7 )

The t ransversely i sot ropic st rand elast ic con-

stants can be evalua ted from V~ and th e f ibre and

mat r ix proper t i es . I t should be no ted here tha t

these proper t i es a re t he proper t i es o f t he s t ra igh t

s t rand , i . e . t he proper t i es o f t he equiva l en t UD

lamina.

The e l as t i c cons t an t s o f t he undula t ed s t rands

a long the g loba l axes a re t o be de t e rmined in

order t o eva lua t e t he g loba l e l as t i c cons t an t s o f

the WF l amina . In t he case of warp s t rands (F ig .

4) , i t i s done by t ransforming the compl i ance of

the warp st rand for the off-axis angle at the

midpoint of that s l ice. In the case of f i l l s t rands,

the e f fec t ive mean va lue of t he compl i ance i s

calculated by considering sect ions of infini tesimal

thicknesses along the f i l l s t rand and t ransforming

the compl i ance of t hese in f in i t es imal sec t ions

a long the g loba l d i rec t ion and then in t egra t ing

them in the in t e rva l (0 -~ O 0, Her e , O~ is t he

off-axis a ngle at x = (aw+gw)/2 i .e . the maxi-

mum off-axis angle.

The local off-axis angle in the fi l l strand

Of(x)

is

expressed as

0f(x) = tan -l d

[zx2(x,

y)]

= t a n - / _ - - - s in } (8)

\2(a w + gw) (aw + gw)

and in the warp st rand i t i s expressed as

d

Ow(y) = tan-'-d-y [Zy2(y)]

1[ :rh f ~y

= t a n - / - - - s in ] ( 9)

\2(af + g0 (ae--+ f)

The respect ive off-axis angle reduces the

effect ive elast ic constants in the global X and Y

di rec t ions . The reduced compl i ance can be

writ ten as (Ref. 17)

1 m 4

s , , (8 ) - - -

E L ( S ) E l .

n4

1 2VLT]m2n2+

I i

2 ( o ) =

E T ( 0 ) E T

S1 2( 0 ) ~--- -VTL(0)

_

VTLm_ + --V'r'rn

ET(0 ) ET ET

1

m 2 n 2

_ _ - - ..[_ _ _

s (o) =

GLT(0) GET Gvr

where m = cos 8, n = sin 0.

For the f i l l s t rand the mean

compl i ance i s expressed as

(lO)

value of t he

afo

~j =~ Sij(O) dO (11)

In an actual WF lamina O is very smal l , and

there fore the fu nct ions sin 0 and cos 0 ca n b e

subst i tuted by the f i rs t term of thei r Taylor series

in the integrat ion of eqn (11).

Integrat ing eqn (11), the effect ive elast ic

constants of the f i l l s t rand are

E L

1+5-

O z

GET

(~fLT = (~2( G L T -

1)

1+ 3 \GTr

(12)

Afte r eva lua t ing the reduced e l as t i c cons t an t s

of t he warp s t rand as expla ined ear l ie r and of t he

fi l l s t rand by using eqn (12) in the sl ice, and also

cons ider ing the presence of pure mat r ix l ayers ,

the extensional s t i ffness of that s l ice can be

expressed as

1 4

A~ (y) = ~ ~ hXk(X, Y)(Q_.ij)k (13)

k = l

where , hxk(x, y) and (Qi j )k are the thicknesses

and mean t ransformed s t i f fness of t he k th l ayer

in the nth sl ice. Her e, hxk(x, y) i s eva luate d at

constan t x, for di fferen t values of y.

The th i ckness of t he warp s t rand i s maximum

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Prediction of on-axes elastic properties of plain weave fabric composites

143

a t x = 0 a n d z e r o f r o m x = a , / 2 t o x = ( a , +

gw)/2.

T h e r e f o r e t h e m i d t h i c k n e s s o f t h e w a r p

s t r and i s t aken for the ex tens iona l s t i f fnes s

ca lcu la t ions i . e . t he th icknes s hw i s mul t ip l i ed by

a factor

[0.71a, / (a, +

gw)] . The ba lance of the

th icknes s i s as sumed to be f i l l ed wi th pure

matr ix.

From the ex tens iona l s t i f fnes s o f the s l i ces the

e las t i c cons tan t s o f the un i t ce l l a r e eva lua ted by

as sembl ing the s l i ces toge ther under the i sos t r a in

condi t ion in a l l t he s l i ces , i . e . t he averaged

in-p lane ex tens iona l s t i f fnes s i s eva lua ted . The

averaged in -p lane ex tens iona l s t i f fnes ses o f the

u n i t c e l l / W F l a m i n a c a n b e e x p r e s s e d a s

2 ~(.,+go/2

= A~ (y)dy (14)

Aij (af + gf) Jo

I t can be s een f rom F ig . 4 tha t the un i t ce l l i s

n o t s y m m e t r i c a b o u t i t s m i d p l a n e a n d t h e r e f o r e

the coupl ing s t i f fnes s t e rms a re p resen t . But

owing to the na ture o f in te r l ac ing of the s t r ands

in the p la in weave f abr i c the coupl ing t e rms in

t w o a d j a c e n t u n i t c e l l s o f t h e W F l a m i n a w o u l d

have oppos i t e s igns and there fore a re zero fo r the

WF lamina as a whole . The e las t i c cons tan t s o f

t h e u n i t c e l l / W F l a m i n a c a n t h e n b e o b t a i n e d

f rom the expres s ions : TM

E x=A ~, (1 A~2 )

Gxy=A66

(15)

A~2

Vyx A22

In the case o f ba lanced p la in weave f abr i cs the

Young ' s modul i in bo th f i l l and warp d i r ec t ions ,

i . e . Ex and Ey , a r e the s ame. For an unba lanced

p l a i n w e a v e f a b r i c , t h e Y o u n g ' s m o d u l u s i n t h e

w a r p d i r e c t i o n s h o u l d b e c a l c u l a t e d b y t h e s a m e

proc edu re as in the f il l d i r ec t ion .

2. 1.2 Element cirray mod el (E AM )

T h e l i m i t a t i o n s o f S A M a r e t h a t t h i s m e t h o d

approximates the s t i f fnes s con t r ibu t ion of the

w a r p s t r a n d a n d a c c o u n t s f o r t h e g a p b e t w e e n

t h e a d j a c e n t w a r p s t r a n d s a p p r o x i m a t e l y . I t

s h o u l d a l s o b e n o t e d t h a t w h e n t h e m a x i m u m

off-axis angle, O, is subs tant ial ly high such that

the f i r s t t e rm of the Taylor s e r i es would no t be

accura te enough to def ine the s ine and cos ine

func t ions , SAM would f a i l t o g ive accura te

results .

I n E A M t h e s e c o n s t r a i n t s a r e o v e r c o m e b y

c

(a) SERIES-PARALLEL COMBINATION

C'

S

Z X

A"

z

(b) PARALLEL-SERIES COMBINATION

Fig. 9. Plain weave fabric lamina unit cei l d iscret ised into

slices and elements.

subdividing the s l ices into ele me nts (1, 2 , 3) of

inf ini tes imal thickness (Fig. 9) . Then, wi thin

these e l ements , the e l as t i c cons tan t s o f the warp

and f i l l s t r ands a re t r ans formed for the loca l

off -axis angle (Fig. 9) and CLT is used to

eva lua te the s t i f fnes s o f tha t e l ement . The

average in -p lane compl iance of the s l i ces a r e

eva lua ted under the cons tan t s t r es s condi t ion in

every e lement o f tha t s l i ce , i . e . t he mean in tegra l

v a l u e o f t h e e l e m e n t c o m p l i a n c e o v e r t h e l e n g t h

of the s l i ce a long the f i l l s t r and a re eva lua ted .

From the compl iances o f the s l i ces the s t i f fnes ses

of the s l i ces a r e ca lcu la ted and then the e l as t i c

cons tan t s o f . the un i t ce l l a r e eva lua ted

cons ider ing a cons tant s t rain s tate in al l the

s l i ces . Th i s p rocedure where the e l ements in the

s l ices are assembled in ser ies ( isos t ress condi t ion)

and then the s l i ces a r e cons idered in para l l e l

( i sos t r a in condi t ion) i s one way of eva lua t ing the

overal l s t i f fness (Fig. 9(a) ) . Such a scheme is

r e fe r r ed to as a s e r i es -para l l e l (SP) combina t ion .

The o ther way i s to make the s l i ces acros s the

loading direct ion as shown in Fig. 9(b) . The s l ices

A ' , B ' a n d C ' a r e s u b d i v i d e d i n t o e l e m e n t s .

T h e n t h e e l e m e n t s i n t h e s li ce s A ' , B ' a n d C ' a r e

as sembled wi th i sos t r a in condi t ion to ob ta in the

s l ice s t i f fness . The s l ice s t i f fnesses are inver ted to

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144 N. K . Naik , V . K . Ganesh

obta in the s l i ce compl iances . The s l i ces A ' , B '

and C ' a r e p laced in s e r i es a long the load ing

di r ec t ion . The un i t ce l l compl iance i s ob ta ined by

the in tegra ted average of the s l i ce compl iances .

The un i t ce l l s t i f fnes ses a re ob ta ined by inver t ing

the un i t ce l l compl iances . Thus i s the para l l e l -

s e r i es (PS) combina t ion .

Here , the expres s ions used to def ine the

u n d u l a t i o n a n d t h e g e o m e t r y o f t h e s t r a n d

cros s - sec t ion a re the s ame as the ones used in

SAM , i . e . eqns (1 ) - (6 ) . The s t r and Vf and the

local of f -axis angle in f i l l and warp direct ions are

calcu lated f rom eqns (7) , (8) and (9) , respec-

t ively. The elas t ic cons tants of the warp and f i l l

s t r ands wi th in the e l ement a r e t r ans formed us ing

eqn (10) . Then the s t i f fnes ses o f the e l ements a r e

ca lcu la ted f rom CLT. The e las t i c cons tan t s o f the

u n i t c e l l / W F l a m i n a a r e t h e n e v a l u a t e d a s

descr ibed ear l i e r , i . e . by e i ther a s e r i es -para l l e l

o r p a r a l l e l - s e r ie s c o m b i n a t i o n .

In the SP combina t ion (F ig . 9 (a ) ) , t he average

(b*~ ~ in the

f the s l i ce coupl ing compl iance , , a , ,

n t h s l ice would be nul l i f ied by a s imilar s l ice in

the ad jacen t un i t ce l l . But , the e l ement coupl ing

stiffness, (Bij) e~, and be nd in g stiffnes s, (D~j) ~,

would increase the va lue o f the e l ement

ex tens iona l com pl iance , * '~

axi) ,

on inver s ion . Thi s

w o u l d a m o u n t t o l o c a l s o f t e n i n g o f t h e e l e m e n t

and hence a r educ t ion in the s t i f fnes s o f the s l i ce

and f ina lly the W F lam ina . But in a PS

combina t ion (F ig . 9 (b) ) , t he average of the s l i ce

coupling stiffness in the nth slice, (B~j)S~, w ou ld be

nul l i f ied by a s imilar s l ice in the adjacent uni t

ce l l . I n a PS combina t ion , s ince the s l i ce coupl ing

s t i f fnesses are zero, the s l ice extens ional com-

p l i a n c e s , * ' ~

aij) , are no t a f f ec ted by the coupl ing

and bending s t i f fness terms on invers ion. A PS

c o m b i n a t i o n w o u l d t h e r e f o r e p r e d i c t a h i g h e r

va lue of s t i f fnes s compared to a SP combina t ion .

I n a W F l a m i n a , l o c a l ly i n d u c e d m o m e n t

resu l t an t s would be p resen t as a r esu l t o f the

appl icat ion of the in-plane s t ress resul tants . For a

p la in weave f abr i c l amina , owing to the na ture o f

i n t e r l a c i n g t h e i n d u c e d m o m e n t w o u l d b e s u c h

tha t i t cons t r a ins the loca l bending deformat ion .

T h i s w o u l d a m o u n t t o s e t t i n g t h e e l e m e n t

c u r v a t u r e t e r m s t o z e r o . W h e n t h i s i s d o n e , b o t h

S P a n d P S c o m b i n a t i o n s w o u l d g i v e t h e s a m e

results .

I n S A M t h e m e a n i n t e g ra l v a l u e w a s c a l c u l a te d

by us ing an exac t in tegra t ion . But in EAM the

i n t e g r a t i o n b e c o m e s c o m p l e x a n d t h e i n t e g r a l

s h o u l d t h e r e f o r e b e e v a l u a t e d n u m e r i c a l l y .

3 M O D I F I E D S I M P L E M O D E L S

The s imple model s ava i l ab le in the l i t e r a tu re a re

no t accura te fo r the p red ic t ion of the e l as t i c

cons tan t s o f 2 -D p la in weave f abr i c l aminae .

Here , modi f i ca t ions a re sugges ted to the ex i s t ing

s imple m ode ls , 3'4 whic h m ake the resul ts of these

m o d e l s c o m p a r a b l e w i t h t h e r e f i n e d m o d e l

pred ic t ions .

3.1 Modif ied mosaic paral le l model (MMPM)

In the 1 -D para l l e l mod el , 3 the f abr i c i s idea l i s ed

as an as semblage of un i t s o f an t i symmet r i c

cros sp ly l amina tes p laced in para l l e l ac ros s the

load ing d i r ec t ion . Here , the con t inu i ty and

undula t ion of the s t r ands a re no t cons idered . A

cons tan t midplane s t r a in i s as sumed in o rder to

eva lua te the s t i f fnes s o f the WF lamina . F rom

th i s as sumpt ion the equa t ions fo r the in -p lane

s t i f fness for a plain weave fabr ic lamina reduce to

Aij = A~ (16)

In the above model the c ros sp ly s t i f fnes ses a re

ca lcu la ted f rom the e las t i c p roper t i es o f the UD

lamina at the s t rand Vt . Therefore the s t i f fnesses

pred ic ted by the mosa ic para l l e l model a r e much

higher than the exper imenta l r esu l t s . I f the

overa l l Vf o f the W F lamina e xper im enta l ly

de te rmined i s used to eva lua te the e l as t i c

p r o p e r t i e s o f t h e U D l a m i n a a n d t h e n t h e m o s a i c

para l l e l model i s used to p red ic t the WF lamina

s t i f fness proper t ies , the resul ts are in good

agreement wi th the p red ic t ions o f the r e f ined

model s as wel l as the exper imenta l r esu l t s .

3.2 Modif ied Kabeika's model (MKM)

In th i s mo del (Ref . 4 ) , t he W F lam ina i s idea l i s ed

as a th ree- l ayered l amina te cons i s t ing of two

undula ted l aminae in the c ros sp ly conf igura t ion

(F ig . 10) and one pure mat r ix l ayer . Here ,

the UD e las ti c p roper t i es o f the und ula ted

laminae in the c ros sp ly conf igura t ion a re r educed

for undula t ion and then CLT i s used to eva lua te

the e l as t i c p roper t i es o f the WF lamina .

The local of f -axis angle of the f i l l and warp

s t r ands a re expres sed a s :

d [h w z~x~

0f(x) = tan -1 ~ t - - cos - - t (17)

\ 2 a w /

Ow(y) = ta n- l - v - cos (18)

u y

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145

Y

J

J

Y

~X

I Z h.,,, ,- ,,,, y)

~

, y

Fig. 10. Plain we av e fab ric lamina--representation of

interlacing.

The ra t ios

hw/aw

a n d

hf/ae

c a n b e c o n s i d e re d t o

be very smal l for the actual s t rand configurat ions.

Hence the maximum off-axis angles in the f i l l and

warp d i rec t ions a re

:rhw

Of =

2aw (19)

:thf

O w ~ - -

2ae

The reduced e l as t i c p roper t i es o f t he equiv-

a l en t warp and f i l l l aminae a re eva lua t ed by

f inding the mea n in t egra l va lue of t he loca l

compl i ance of t he respec t ive l amina . Thi s i s done

by t ransforming the compl i ance for t he loca l

off-axis angle and then integrat ing the t rans-

formed compl i ance .

The average compl i ance may be expressed as

S0= ~ S0(t~) d 0 (20)

The above express ion can be used for bo th

warp and f i l l s t rands by insert ing the respect ive

s t rand geomet r i ca l and e l as t i c parameters .

Inver t ing the compl i ance , t he e f fec t ive e l as t i c

proper t i es can be found . Kno wing the e f fec t ive

e l as t i c p roper t i es o f t he equiva l en t l aminae , t he

st i ffness of the WF lamina i s evaluated using

CLT.

In thi s metho d , t he th i cknesses of t he warp and

fi l l laminae are taken as the thicknesses of the

respect ive st rands. But , in an actual case,

normal ly the st rands are el l ipt ical ly shaped wi th

maximum th i ckness a t t he mid sec t ion . Thi s

model would therefore g ive a h igher s t i f fness

because the maximum s t rand th i ckness i s

cons idered for ca l cu la t ions . Secondly , t he pre -

sence of a gap i s not accounted for .

The presence of a gap can be approximate ly

t aken in to account by rep lac ing the s t rand wid th

by s t rand wid th p lus t he gap be tween the

corresponding ad jacent s t rands in eqns (17)-

(19).

In order t o account for t he e l l i p t i ca l shape of

the st rand, the st rand thickness i s factored to i t s

mid value whi le calculat ing the in-plane st i ffness.

The remain ing th i ckness i s t aken as pure mat r ix

l ayer . The ord ina t e of t he s t rans th i ckness

fo l lows the s ine func t ion , t here fore the mid

thickness of the warp and f i l l s t rand would be

hf = 0.707hf

(21)

/~w = 0-707hw

If t he gap i s p resen t , t he average th i ckness can be

approximated as

ae

/~f = 0 707 h'((af + gf) ) (22)

hw= 0 707hw((aw+ gw))

4 E X P E R I M E N T A L W O R K

T h e e x p e r i m e n t a l p ro g ra m m e w a s d e s i g n e d t o

de te rmine the e l as t i c modul i o f t he WF l amina

along the warp and f i l l di rect ions. The experi -

ment s were car r i ed ou t on E-g lass / epoxy and

carbon/epoxy l aminae . The th i ckness of t he

E-glass fabric was 0.2 mm , a nd th e wa rp and f i ll

t h read count s were 15 per cm, whi l e t he

th i ckness of t he c arbon fabr ic was 0 .16 mm and

i ts warp and f i l l thread counts were 8-8 per cm. I t

m a y b e n o t e d t h a t e v e n t h o u g h t h e n u m b e r o f

count s a re t he same a long the warp and f i l l

d i rec t ions , t he fabr i cs a re no t ba l anced because

of di fferent gaps along the warp and f i l l di rect ions

and hence a d i f fe ren t degree of undula t ion . The

epoxy res in LY556 wi th hardener HY951

suppl i ed by Ciba tu l , Ind ia , was used and the

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l a m i n a e w e r e p r e p a r e d a t r o o m t e m p e r a t u r e i n a

s p e c i a l l y d e s i g n e d m a t c h e d d i e m o u l d . T h e

overa l l f ib re vo lume f r ac t ions o f the l aminae

w e r e d e t e r m i n e d a s d e s c r i b e d i n t h e A S T M

specification D 3171.

S ta t i c t ens i l e t es t spec imens were p repared

accord ing to A ST M spec i f ica t ion D 3039 . The

l a m i n a e t h i c k n e s s e s o f t h e E - g l a s s / e p o x y a n d

c a r b o n / e p o x y c o m p o s i t e s w e r e l e s s t h a n t h e

m i n i m u m r e q u ir e d b y A S T M D 3 0 3 9 . S in c e n o

other s t andards a re ava i l ab le fo r such t es t ing , the

s a m e s t a n d a r d w a s u s e d f o r t h e s p e c i m e n s m a d e

f r o m W F l a m i n a e . T h e t e s t s w e r e p e r f o r m e d o n a

L l o y d M 5 0 K m a c h i n e . T h e s p e c i m e n s w e r e

tes ted a t room tempera ture (27C) a t a c ros shead

speed of 1 m m/ mi n . A to ta l o f 40 spec ime ns was

tes ted . The s ca t t e r r ange fo r ca rbon/epoxy for Ey

was 56-61 GPa and for Ex i t was 47-50 GPa. For

E-g las s /epoxy , the s ca t t e r r ange was 17-21 GPa.

The mean va lues o f the t es t r esu l t s a r e p resen ted

in the next sect ion.

T h e g e o m e t r i c a l p a r a m e t e r s o f t h e f a b r i c w e r e

d e t e r m i n e d b y m e a n s o f a n o p t i c a l m i c r o s c o p e a t

a magnif icat ion of 20.

5 R E S U L T S A N D D I S C U S S I O N

T w o f a b r i c c o m p o s i t e m o d e l s h a v e b e e n p r e -

sen ted fo r the on-axes e l as t i c ana lys i s o f 2 -D

o r t h o g o n a l p l a i n w e a v e f a b r i c l a m i n a e . T h e

model s cons ider the ac tua l s t r and c ros s - sec t iona l

g e o m e t r y a n d t h e p r e s e n c e o f a g a p b e t w e e n t h e

ad jacen t s t r ands . An ana ly t i ca l t echn ique to

eva lua te VI f rom V~' de t e rm ine d exp er ime nta l ly

i s a l so p resen ted . The shape func t ions cons idered

a r e c o m p a r e d w i t h a s c a n n i n g e l e c t r o n m i c r o -

graph . Th e shape func t ions agree wel l wi th the

a c t u a l g e o m e t r y o f t h e W F l a m i n a .

S o m e a p p r o x i m a t i o n s a r e i n c o r p o r a t e d i n

S A M i n o r d e r t o r e d u c e t h e c o m p u t a t i o n a l

c o m p l e x i t y w i t h o u t c o m p r o m i s i n g o n t h e f i n a l

r esu l t s fo r ac tua l WF lamina conf igura t ions .

These approx imat ions would pred ic t s l igh t ly

h i g h e r s t i f f n e s s c o m p a r e d t o E A M . I n E A M t w o

combina t ions o f as sembl ing the e l ement s t i f fnes s

are p resen ted . In the SP combina t ion , the loca l

b e n d i n g d e f o r m a t i o n s c a n b e c o n s i d e r e d o r t h e y

can be as sumed to be cons t r a ined by loca l ly

i n d u c e d m o m e n t s . T h e a s s u m p t i o n t h a t t h e l oc a l

bending deformat ions a re cons t r a ined i s r ea l i s t i c

cons ider ing the na ture o f in te r l ac ing of the p la in

weave f abr i c compos i t es .

Table 1. Elastic properties of fibre and matrix

Material

(GPa) (OPa) (GPa) (GPa)

Fibre

Car bon t9 230-0 40-0 24-0 14-3 0-26

E-glass~ 72.0 72.0 27-7 27.7 0-30

Gra phi te 19 388.0 7.2 6.8 2-4 0-23

Matrix

epoxy~ 3.5 3.5 1.3 1.3 0-35

Isotropic.

I n o r d e r t o e x a m i n e t h e m i c r o m e c h a n i c a l

approaches fo r the p red ic t ion of the e l as t i c

cons tan t s o f a WF lamina , th ree mater i a l sys tems

w i t h d i ff e r en t w e a v e g e o m e t r i e s w e r e c o n s i d e r e d .

The e las t i c p roper t i es o f the f ib res and mat r ix a re

g i v e n i n T a b l e 1 . A s s u m e d g e o m e t r y w i t h i n t h e

p r a c t i c a l r a n g e w a s t a k e n f o r t h e g r a p h i t e / e p o x y

mater i a l sys tem in o rder to s tudy the s ens i t iv i ty

of the f abr i c geomet ry on the e l as t i c p roper t i es o f

t h e W F l a m i n a . T h e f a b r i c g e o m e t r i c a l p a r a -

m e t e r s o f c a r b o n / e p o x y a n d E - g l a s s / e p o x y W F

l a m i n a e a r e t h e a c t u a l d i m e n s i o n s m e a s u r e d w i t h

an op t i ca l microscope (Table 2 ) . These two

m a t e r ia l s y s te m s w e r e c o n s i d e r e d t o c o m p a r e t h e

resu l t s o f the p roposed model s wi th the

exper imenta l r esu l t s . Tab le 3 p resen t s the

me asure d V~' and the cor resp ondin g ca lcu la ted

V~. In the case o f the g raphi t e / epoxy WF lamina ,

V~' was ca lcula ted f rom a V~ of 0.8. Th e

max imum V~ was as sume d to be 0 .8 to ensure

tha t the f ib res do no t become cont iguous . In the

above ca lcu la t ions the V~ va lues o f the w arp and

f i l l s t r ands were as sumed to be the s ame and th i s

as sumpt ion i s va l id as the d iameter o f the f ib res

in the warp and f il l s t r ands a re the s am e a nd the

proces s ing condi t ions a re the s ame.

I t i s seen f rom Tab le 3, tha t V~' i s a lmo st ha lf

of V~. This is poss ible in the case of a plain

w e a v e f a b r i c l a m i n a b e c a u s e o f t h e n u m b e r a n d

s ize o f pure r es in pocke t s p resen t . I t may no t be

poss ible to achiev e a V~' of abo ut 0.5 a nd a bov e

in the case o f p la in weave f abr i c l aminae . Wi th

th is the maxim um V~ of 0 -7-0 .8 wou ld have been

a t t a ined .

Table 4 p resen t s the e l as t i c p roper t i es o f the

UD lamina a t VI and V~' ca lcu la ted f rom the

com posi te cyl in der assem blag e ( CC A ) mo del . 15,16

W i t h t h e s e U D l a m i n a p r o p e r t i e s a n d d i f f e r e n t

m o d e l s d e s c r i b e d i n t h e p r e c e d i n g s e c t i o n s a n d

Ref . 4 , the e l as t i c p roper t i es o f the WF laminae

c o n s i d e r e d w e r e p r e d i c t e d . I n t h e c a s e o f t h e

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Table 2. Plain weave fabric lamina strand and weav e geometrical parameters

Mat e r i a l F i l l s t r and Wa r p s t r and H V ~

( m m )

af hf gf aw hw gw

( m m ) ( m m ) ( m m ) ( m m ) ( m m ) ( m m )

C ar bon / epoxy 0 .96 0 .08

E- G l a s s / epox y 0- 62 0 - 10

G r ap h i t e / ep oxy 2 .00 0 - 50

0-18 1.10 0-08 0.04 0.16 0.44 a

0-05 0.62 0.10 0.05 0-20 0.42 a

0,50 2-00 0.50 0.50 1-00 0.41 b

a D e t e r m i ned expe r i m en t a l l y .

b Calcula ted f rom s t rand Vf of 0 .80.

147

Table 3. Overall Vf and the correspon ding strand Vf

Ma teri al Ov era l l Vf (Vf ) Str and Vf (V~)

C ar b on / ep oxy 0 - 44 0 .78

E- G l a s s / epox y 0 .42 0 -70

G r ap h i t e / ep oxy 0 -41 0 .80

E-glass/epoxy WF lamina, the strand appeared,

when seen through an optical microscope, to

have a slight twist. Ideally, the UD lamina

properties evaluated using the CCA model

should be multiplied by the fibre-to-strand

property translation efficiency factor and then

these properties should be used for further

Table 4. Elastic properties of UD lamina using CCA m odel

Material EL ET GLT Gaa- VLT Vf

(GPa) (GVa) (GPa) (GPa)

Carbon /epoxy 18 2. 50 18-50 7-55 6.70 0.28 0.78

105.40 8.60 3.00 3.00 0.40 0- 44

E-Glass/epoxy 5 1. 5 0 17-50 5.80 6.60 0.31 0.70

32.25 8.55 2.85 3.10 0.39 0.42

Graph ite/epox y 311.00 6.30 4-40 2,10 0.25 0.80

161.00 5.00 2-27 1,70 0- 30 0. 41

calculations. But, in this case, as the angle of

twist was small the translation efficiency factor

was taken as unity. The predicted results are

tabulated in Table 5.

Table 5. Elastic properties of plain weav e fabric lamina: Comparison o f predicted and

experimental results

Mat e r i a l Mode l

Er Ex Gxy Wx V7

( G P a ) ( G P a ) ( G P a )

C a r bo n / epo xy SA M 58 .9 52 -4 5 .1 0 .07

EA M - - PS 57-1 51 -2 4 -7 0 .10

- - S P 35.6 35.4 4 .7 0-10

MM PM 57.6 57.6 3 .0 0 .07

MK M 64.8 58.5 5 .3 0 .04 0 .44

Expe r i m en t 60 .3 49 .3 - - - -

K a b e l k a ' s m e t h o d

(Ref, 4) 89.3 89-3 7,0 0.04

E- G l a s s / epoxy SA M 20-3 20 .3 3 ,7 0 -23

EA M - - PS 19 .6 19 .6 3 .7 0 .20

ra SP 17-9 17.9 3-7 0.20

MM PM 20.7 20-7 2 .9 0 .17

MK M 21.7 21.5 3 .9 0 .13 0 .42

Expe r i m en t 19 -3 19- 3 - - - -

K a b e i k a ' s m e t h o d

(Ref . 4) 29.2 29.2 4 .9 0-12

G r ap h i t e / epo xy SA M 28 .8 28 -8 2 .8 0 .08

E A M - - PS 23-7 23 .7 2 .8 0 - 12

- - S P 16.2 16.2 2 .8 0 .12

MM PM 80.1 80-1 2-3 0 .02

M KM 32.2 32.2 3 .0 0-04 0 .41

E x p e r im e n t . . . .

K a b e l k a ' s m e t h o d

(Ref . 4) 44-3 44.3 3.7 0.03

M M P M - - M o d i f i e d m o s a ic p a r a l l el m o d e l ,

M K M - - M o d i f i e d K a b e i k a ' s m o d e l .

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The un i t ce l l was subd iv ided in to 50 s l i ces

pa ra l l e l t o t he l oad ing d i rec t i on in SAM. In

EAM, the un i t ce l l was subd iv ided in to 50 s l i ces .

The s l i ces were a long the l oad ing d i rec t i on in t he

S P c o m b i n a t i o n w h e r e a s t h e y w e r e a c r o s s t h e

l o a d i n g d i r e c t i o n i n t h e P S c o m b i n a t i o n . F o r

bo th t he SP and PS com bina t ions , each s l i ce was

f u r t h e r s u b d i v i d e d i n t o 5 0 e l e m e n t s . H e n c e , t h e

to t a l num ber o f e l em en t s was 2500 . Th i s was

a r r ived a t a f t e r a convergence s tudy . In t he case

o f c a r b o n / e p o x y a n d E - g l a s s / e p o x y , t h e e x-

p e r i m e n t a l v a l u e s o f Y o u n g ' s m o d u l i a l o n g t h e

warp and f i l l d i rec t i ons a re a l so p resen t ed in

T a b l e 5 . T h e c o m p a r i s o n o f t h e p r e d i c t e d r e s u l t s

b y S A M , P S , t h e m o d i f i e d m o s a i c p a r al l el m o d e l

( M M P M ) a n d t he m o d i f i ed K a be l k a 's m o d e l

( M K M ) e x h i b i t g o o d a g r e e m e n t w i t h t h e

e x p e r i m e n t a l r e s u l t s i n t h e c a s e o f c a r b o n / e p o x y

a n d E - g l a s s / e p o x y .

T h e l o ca l b e n d i n g d e f o r m a t i o n is c o n s i d e r e d i n

SP and the re fo re t he re su l t s o f SP a re l ower

com pared to t he re su l t s o f PS . Bu t , i n an ac tua l

p l a in weave fab r i c l am ina , l oca l bend ing

defo rm a t ions due to t he coup l ing e f fec t i n each

un i t ce l l can be a ssum ed to be cons t ra ined . The

resu l t s o f PS in wh ich the e l em en t coup l ing t e rm s

do no t a f fec t t he s l i ce com pl i ance a re t he re fo re

t aken a s rea l i s t i c . I t shou ld be no t ed tha t t he

resu l t s o f PS wi th o r w i thou t t he coup l ing t e rm s

wou ld be t he sam e and the re su l t s o f SP wi thou t

t h e c o u p l i n g t e r m s w o u l d b e t h e s a m e a s t h e

resu l t s o f PS . The re su l t s p re sen t ed in Tab le 5

aga ins t SP com bina t ion a re t he re su l t s ob t a ined

c o n s i d e ri n g t h e l o c al b e n d i n g d e f o r m a t i o n s .

C o m p a r i n g t h e r e s u l t s o f P S a n d S P o f E A M ,

i t can be seen tha t t he coup l ing t e rm s have

a f f e c t e d t h e c a r b o n / e p o x y W F l a m i n a r e s u l t s

m o r e t h a n t h e E - g l a s s / e p o x y o r g r a p h i t e / e p o x y

WF l am ina re su l t s . Th i s i s due t o l e ss undu la t i on

i n c a r b o n / e p o x y W F l a m i n a a n d l a r g e r EL/ET

r a ti o o f t h e e q u i v a l e n t c a r b o n / e p o x y U D l a m i n a .

Th i s wou ld i nc rease t he abso lu t e va lue o f t he

coup l ing t e rm s an d the r eby l ead to g rea t e r l oca l

s o f t e n i n g . I n t h e c a s e o f t h e g r a p h i t e / e p o x y W F

lam ina , a l t hough i t s equ iva l en t UD EL/E.r ra t io

i s h igh , t he e f fec t o f coup l ing t e rm s i s sm a l l e r , by

c o m p a r i s o n w i th t h e c a r b o n / e p o x y W F l a m i n a ,

b e c a u s e o f t h e g r e a t e r u n d u l a t i o n o f t h e s t r a n d s .

I t m ay be no t ed tha t t he E-g l a ss and g raph i t e

p l a in weave fab r i c s cons ide red a re ba l anced

whereas t he ca rbon p l a in weave fab r i c i s

unba l anced (Tab le 2 ) . Hence , t he e l a s t i c m odu l i

a long the warp and f i l l d i rec t i ons a re t he sam e fo r

t h e E - g l a s s / e p o x y a n d g r a p h i t e / e p o x y W F

lam inae whereas t hey a re d i f fe ren t fo r t he

c a r b o n / e p o x y W F l a m i n a . F o r t h e c a r b o n / e p o x y

WF lamin a, s ince aw > af and gw < g~ , one wou ld

expec t E , . t o be g rea t e r t han

Ex.

Such re su l t s a re

o b t a i n e d f r o m S A M a n d P S . F o r S P , t h e t r e n d

can be d i f fe ren t depend ing upon the e f fec t o f

c o u p l i n g t e r m s . T h e c o n t r i b u t i o n o f t h e a b s o l u t e

va lues o f t he coup l ing t e rm s i s m ore i n t he

ove r l ap reg ion than in t he gap reg ion .

I n t h e c a se o f t h e c a r b o n / e p o x y a n d

E-g lass / epoxy WF l am inae t he d i f fe rence i n

re su l t s o f SAM and PS i s l e s s , bu t t he re su l t s o f

PS in t he case o f a g raph i t e / epoxy WF l am ina a re

cons ide rab ly l ower t han the re su l t s p red i c t ed by

SAM. Th i s i s due t o t he l ower s t rand th i ckness t o

s t rand wid th

(h/a)

ra t io in the case of

c a r b o n / e p o x y a n d E - g l a s s / e p o x y W F l a m i n a e

a n d h i g h e r

h/a

f o r g r a p h i t e / e p x o y W F l a m i n a .

R e c a l l i n g S A M , i n o r d e r t o c o n s i d e r t h e n e t

e f fec t o f t he warp s t rand , warp s t rand th i ckness

was fac to red to i t s m id va lue and the e f fec t o f t he

g a p w a s t a k e n i n t o a c c o u n t a p p r o x i m a t e l y . T h e

d i f fe rence i n t he re su l t s o f SAM and PS i s due t o

t h i s a p p r o x i m a t i o n i n S A M . T h e a p p r o x i m a t i o n

seem s to be va l id when the h/a rat io is low as the

s l o p e o f t h e s t r a n d o u t e r c o n t o u r w o u l d b e s m a l l

and the t h i ckness o f t he s t rand wou ld be nea r ly

un i fo rm a long the s t rand wid th . In PS , t he s l i ces

a r e s u b d i v i d e d f u r t h e r i n t o e l e m e n t s a n d t h e

th i ckness o f t he warp s t rand a t t he m idpo in t o f

tha t e l em en t i s cons ide red wh i l e ca l cu l a t i ng the

s t i f fness . The re fo re , t h i s m e thod wou ld g ive

consis tent resul t s for a l l h/a ra t i o s and the re su l t s

wou ld a lways be l e ss t han tha t o f SAM. The on ly

d rawback o f PS i s t ha t i t i nvo lves m ore

c a l c u l a t i o n s a n d t h e r e f o r e c o n s u m e s m o r e c o m -

pu ta t i ona l t im e .

G e n e r a l e v a l u a t i o n o f a p p r o x i m a t e m e t h o d s i s ,

o f cou rse , im poss ib l e . T he va l id i t y o f the

m o d i f i e d s i m p l e m o d e l s , i .e . M M P M a n d M K M ,

i s t he re fo re a ssessed on ly on the bas i s o f

ind iv idua l cases . Two ac tua l WF l am inae

c o n f i g u r a t i o n s a n d o n e a s s u m e d g e o m e t r y c a s e

were cons ide red to a ssess t he m od i f i ed s im p le

m o d e l s . T h e a s s u m e d g e o m e t r y c a s e w a s

c o n s i d e r e d t o e v a l u a t e t h e m o d i f i e d s i m p l e

m ode l s a t a h igh l eve l o f undu la t i on and fo r t he

sake o f gene ra l i t y .

T h e r e s u l t s o b t a i n e d f r o m t h e r e f i n e d m o d e l s ,

m o d i f i e d s i m p l e m o d e l s a n d a s i m p l e m o d e l ( R e f .

4 ) a re t abu la t ed i n Tab le 5 . The s im p le m ode l

p resen t ed in Ref . 4 i s on ly fo r c lo se weave and

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Prediction of on-axes elastic properties of pla in weave abric composites 149

h e n c e i t d o e s n o t t a k e t h e p r e s e n c e o f a g a p i n t o

a c c o u n t . F o r t h e p r e s e n t c a l c u l a t i o n s , t h e r e f o r e ,

t h e g a p w a s a s s u m e d t o b e e q u a l t o z e r o . T h e

r e s u l t s o f M M P M c o m p a r e w e l l w i t h t h e r e s u l t s

o f r e f i n e d m o d e l s i n t h e c a s e o f c a r b o n / e p o x y

and E-g las s / epoxy WF laminae and i s g ros s ly

i n a c c u r a t e f o r a g r a p h i t e / e p o x y W F l a m i n a . B u t

t h e r e s u l t s o f M K M c o m p a r e w e l l f o r t h e

E - g l a s s / e p o x y W F l a m i n a a n d n o t s o w e l l f o r

g r a p h i te / e p o x y a n d c a r b o n / e p o x y W F l a m i na e .

M M P M g i v e s a n a c c u r a t e p r e d i c t i o n f o r

c a r b o n / e p o x y a n d E - g l a s s / e p o x y W F l a m i n a e ,

b e c a u s e t h e s e W F l a m i n a e c o n f i g u r a t i o n s h a v e

l o w e r h/a r a t ios . I t may be no ted tha t whi l e

f o r m u l a t i n g t h i s m o d e l t h e p r e s e n c e o f u n d u l a -

t i o n w a s i g n o r e d a n d l o w e r h/a a m o u n t s t o

u n d u l a t i o n a n g l e t e n d i n g t o z e r o . T h e a u t h e n -

t i c i ty o f th i s can be ver i f i ed by compar ing the

resu l t s o f MMPM wi th the r esu l t s o f the r e f ined

m o d e l s i n t h e c a s e o f a g r a p h i t e / e p o x y W F

lamina . He re , i t i s s een tha t MM PM gives very

h igh modulus va lues as i t does no t cons ider the

u n d u l a t i o n w h e r e a s t h e u n d u l a t i o n i s t h e

pr inc ipa l parameter which r educes the s t i f fnes s in

t h e c a s e o f a g r a p h i t e / e p o x y W F l a m i n a .

T h e p r e d i c t i o n o f M K M c o m p a r e s w e l l w i t h

t h e r e s u l t s o f r e f i n e d m o d e l s c o m p a r e d t o t h e

resu l t s o f MMPM for the f abr i c s t ruc tures hav ing

higher h/a, i . e . f o r g r a p h i t e / e p o x y . F o r t h e c a s e

o f a c a r b o n / e p o x y W F l a m i n a , t h o u g h t h e

pred ic t ion i s no t as accura te as tha t o f MMPM,

the resul ts are also not gross ly inaccurate . This is

because MKM main ly cons ider s the s t i f fnes s

r e d u c t i o n d u e t o t h e p r e s e n c e o f u n d u l a t i o n a n d

t h e u n d u l a t i o n i s q u i t e s m a l l f o r a c a r b o n / e p o x y

W F l a m i n a .

The s t i f fnes s r educ t ion in a WF lamina i s

main ly due to the lowe r V~' and th e p resenc e of

u n d u l a t i o n i n t h e s t r a n d s a s c o m p a r e d t o t h e U D

cros sp ly l amina tes . MMPM cons ider s the r e -

duced V~' by cons ider in g the s t r and prope r t i es a t

V~', bu t doe s no t c ons ide r the un dula t ion of

s t r ands . Thi s model i s there fore app l i cab le fo r

f abr i c s t ruc tures hav ing very much l es s undula-

t ion . I t i s wor th no t ing here tha t mos t o f the

ac tua l woven f abr i cs used in s t ruc tura l app l i ca-

t ions and made of h igh modulus f ib res have lower

h/a r a t ios . Th ere c an be a co mb ina t io n of Vf and

u n d u l a t i o n w h i c h w o u l d g i v e p r a c ti c a l ly t h e s a m e

r e s u l t s b y M M P M a n d M K M . T h i s c a n b e s e e n i n

the case o f E-g las s / e poxy WF lam ina . F igu re 11

shows the var iat io n of V~' and V~ as a funct ion o f

h/a

rat io . He re, i t i s seen t hat b oth V~' and V~

are cons tan t fo r a l l h/a r a t ios and a g iven gap .

Thi s i s t rue because the var i a t ion of h/a w o u l d

c o r r e s p o n d i n g l y re d u c e t h e t o t a l t h i c k n es s o f t h e

l a m i n a a n d t h e v o l u m e o f p u r e m a t r i x r e g i o n s ,

thereby keep ing V~' cons tan t . Th i s c l ear ly

ind ica tes tha t MMPM gives the s ame r esu l t fo r

all

h/a

r a t ios fo r a g iven mater i a l sys tem and gap .

Figure 12 shows the var ia t ion of V~' and V~ as a

func t ion of the gap wid th to s t r and wid th (g/a)

r a t io . Here , i t i s s een tha t V~ reduces wi th the

same V~ as the gap increases . N ow , wi th th i s

o b s e r v a t i o n i t c a n b e s h o w n t h a t M M P M

cons ider s the gap ind i r ec t ly , i . e . wi th the

presence of the gap , V~' f a ll s and c or respo nding ly

a l te r s th e e q u i v a l e n t U D e l a st ic c o n s ta n t s . M K M

cons ider s the e f f ect o f the r ed uce d Vf by

c o n s i d e r i n g t h e b a l a n c e o f t h e f a c t o r e d w a r p a n d

f i l l layer thicknesses as a pure matr ix layer .

The pre dic ted values of Gxy and vxy are also

p r e s e n t e d i n T a b l e 5 . I n g e n e r a l , t h e r e f i n e d a n d

modi f i ed s imple model s g ive lower va lues o f WF

l a m i n a G~y, w h e r e a s t h e s e m o d e l s g i v e h i g h e r

va lues o f Vxy c o m p a r e d t o t h e s i m p l e m o d e l . T h e

lower va lues o f G~y are due to the lower va lu e o f

V~' and the p resenc e of undula t ion . The h igher

values of Vxy o c c u r f o r t h e s a m e r e a s o n s . T h e

va lues o f Gxy we re ev a lua te d by c ar ry ing ou t the

ana lys i s a long the warp and f i l l d i r ec t ions

s e p a r a te l y i n t h e c a s e o f c a r b o n / e p o x y W F

l a m i n a . T h e s a m e r e s u l t s w e r e o b t a i n e d i n b o t h

cases.

T h e d e g r e e o f u n d u l a t i o n d e p e n d s o n t h e

h/(a + g) r a t i o . L o w e r h/(a + g) r a t io ind ica tes a

l o w e r d e g r e e o f u n d u l a t i o n a n d vice versa. F i g u r e

11 presents the ef fect of the hw/aw rat io on E~ as

a func t ion of gap (gw) fo r a ba lanc ed , p la in wea ve

fabr ic lam ina with aw = af , hw = hf an d gw = gf. I t

i s obv ious f rom the p lo t tha t as the hw/aw rat io

increases for a given gap, Ex reduces . This is

a t t r ibu ted to the f ac t tha t as the hw/aw rat io

increases , the e f f ec t o f undula t ion i s increased .

The r educ t ion in E~ i s s t eeper fo r l a rger va lues o f

gw. As s een f rom F ig . 11 , the p resence of l a rger

gw c a n f u r t h e r r e d u c e t h e u n d u l a t i o n a n d

consequent ly the h igher va lue o f E~ i s ob ta ined

than w i th lower va lues o f gw unt i l an op t im um

hw/aw v a l u e i s r e a c h e d . T h e t r e n d w o u l d b e t h e

r e v e r s e a b o v e t h e o p t i m u m v a l u e o f t h e

hw/aw

r a t io . In o ther words , in th i s r ange , lower va lues

o f Ex w o u l d b e o b t a i n e d w i t h h i g h e r v a l u e s o f g w

than wi th low er va lues o f gw.

The var i a t ion of Ex as a func t ion of gw/aw for

dif ferent

hw/aw

rat ios is presented in Fig. 12 for a

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150


o

O.

t,L,l

100

90

80

70

60

50

40

30

20

10

GRAPHITE /EPOXY

k

~, ,B-- - - - gw law = 0-5 Ow= 2'0 mm

-- _ ~ ~'S T R AND Vf

I

~ gw = 0 5 mm

- o / ) ' \

\~. --OVERALL

Vf

_ ~ N ~ gw = 0 5

mm

1 I l I

0"0 0-1 0"2 0"3 0"4

h w / a w

]Fig. 1]. Var iatio n of K~ and V as a function o f h~/a,~.

1-0

0.8

0'6

0.4

0 2

O-0

05

balanced, plain weave fabric lamina. The effect

of gwiS twofold. As the gap is increased,

obviously V~' would decrease with the same V~,

in turn the elastic moduli would reduce. On the

other hand, the presence of a gap would reduce

the degree of undulation and hence the elastic

moduli would increase. From this it is obvious

A

o

(1.

X

uJ

4 0 -

hw/aw= 0.15

STRAND V~

3 s = r -~

h w /a W = 0-2

__~_ . c~

h w / a w = 0 3

1 I I I

0'0 0 2 0,4 0 6 0 8

gw/aw

that the optimum gap would give the maximum

possible elastic moduli. In addition to this, the

fabrics with gaps between adjacent strands, i.e.

open weave fabrics, provide better wettability

and in turn better performance of the WF

lamina/laminate. It is seen from Fig. 12 that as

gw/aw

increases, Ex increases until an optimum gw

- 1 - 0

GRAPHITE /EPOXY

aw= 2"0 mrn

0 8

0 ' 6

0'4

0'2

0.0

1 0

Fig. 12. Variatio n of Ex and Vt as a functio n of

gw/aw.

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Prediction of on-axes elastic properties o f plain weave fabric composites 151

is reached, and thereafter it decreases. For larger

h/a

rat ios , the numerical value of the o ptim um

gap is greater than for lower values of

h/a

ratios.

For certain combinations of fabric geometrical

parameters , the optimum gap can be zero, as

seen for

hw/aw=O.15.

Even though the per-

centage gain in Ex due to optim um gw may not be

considerable , the magnitude of gap achieved can

be significant enough to facilitate better wet-

tability and formability. The gain in Ex as

presented in Fig. 12 is the absolute value. The

gain in terms of specif ic modulus would be much

higher owing to the difference in densities of fibre

and matrix. It may be noted that Figs 11 and 12

are plotted by using SAM .

For the balanced plain weave fabric lamina,

the properties along the warp and fill directions

are the same. Hence, the discussion relating to

Ex along the fill direction and Ey along the warp

direct ion are the same. For the unb alanced, plain

weave fabric lamina, the same analysis can be

used along the warp direction to obtain Ey.

6 C O N C L U S I O N S

T h e p r e d i c t i o n s o f t h e r e f i n e d m o d e l s h a v e b e e n

e v a l u a t e d b y c o m p a r i s o n w i t h t h e r e s u l t s o f a n

e x p e r i m e n t a l p r o g r a m m e . I t i s s e e n t h a t t h e

pred ic t ions o f the r e f ined model s match wel l wi th

the exper imenta l r esu l t s . I t should be no ted tha t

cer t a in l imi ta t ions a re inheren t in the use o f

modi f i ed s imple model s in t e rms of the r ange of

appl i cab i l i ty . The r esu l t s ob ta ined f rom the

m o d i f i e d s i m p l e m o d e l s , h o w e v e r , c e r t a i n t l y

i n d i c a t e t h a t t h e s e t e c h n i q u e s , w h e n u s e d w i t h

s o m e j u d g m e n t , a r e v e r y s at i sf a c to r y e n g i n e e r i n g

tools.

T h e r e f i n e d m o d e l , S A M , w a s u s e d t o s t u d y

the ef fect of h /a a n d g /a o n t h e W F l a m i n a

longi tud ina l mod ulus and V~'. I t i s s een tha t there

is a significant effect of the

h /a

r a t io on the

longi tu dinal mo dul us , b ut V~' i s con s tan t for al l

h /a r a t i o s . W i t h t h e o p t i m u m g a p b e t w e e n t h e

two ad jacen t s t r ands , the spec if i c s t if fnes s would

be the h ighes t . Th e o vera l l Vf o f the WF lamina

r e d u c e s w i t h t h e i n c r e a s e i n t h e g a p b e t w e e n t h e

ad jacen t s t r ands wi th the s ame s t r and Vf .

A C K N O W L E D G E M E N T

This work was supported by the Structures Panel,

Aeronaut ic s Research Deve lo pme nt Board,

Ministry of Defence , Government of India,

Grant No. Aero/RD -134/10 0/10/90 -91/659 .

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152


A P P E N D I X

T h e c o m p o s i t e c y l i n d e r a s se m b l a g e (C C A )

model (Refs 15 and 16) gives simple closed form

analyt ical expressions for the effect ive composi te

mo duli EL, GeT , VET an d k , wh ile the mo du li GTr

and ET are bracke ted by c lose bounds . Here , t he

UD composi t e i s model l ed as an assemblage of

long composi t e cy l inders cons i s t i ng of t he inner

c i rcu la r f i b re and the ou te r concent r i c mat r ix

she l l . The f ib re and mat r ix a re cons idered to be

t ransversely i sot ropic.

The t ransverse bu lk modulus of t he UD

composi te i s given by

km(k e + G ~ ) (1 - G ) + U ( k m + G ~ ) V f

k

H e r e

(k ' + G~'-r)(1 - Vr) + (k m + G~r )Vf

1 4 4V~T 1

k e E~- E [ G ~

1 4 4VLmT 1

k m E ~ E ~ G ~

T h e l o n g i t u d i n a l Y o u n g ' s m o d u l u s o f t h e U D

composi t e i s g iven by

EL = EfL Vf + E'~Vm

4(VII

- V ~ T ) 2 V m

Vf

+

V . , / U + V f /k m + 1 / G ~

The longi tud ina l Poi sson ' s ra t i o and shear

modu lus a re g iven by

VLT = v~TV,+ v~TVm

( v [ T - v ' ~ O ( 1 / k ~ - 1 / U )V W , , ,

+

Vm/k f + Vf/ k m + |/G~ -r

G~V~. + G ~T(1 + Vd

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