T H E L O C A L E F F E C T S OF F I B E R - M A T R I X

I N T E R A C T I O N IN C O M P O S I T E S : 2

l~. S. U m a n s k i i , E . I . Z a l u z h n a y a , a n d S. M. M e d o v a y a

UDC 539.4.014

In the f i rs t communication [1] the s t r e s s state of a plate re inforced with a single, central ly situated fiber under the action of s tretching forces paral le l to the axis of the fiber was studied by a variat ional dif- ference method. We present below the resul ts of studying the local effects in the s t r e s s state of such a

1,6

2- I r

0,~ ~ ~ 3 "

. . . .

0

oo J " z( i ]

y (j') - 1 | I - ~

/ -/~2 / ' t i l t

/ /

/

t 5 9 tO t9 ~'

Fig. 1

N ' N 1,o

- 3 1,2

Y o,,

i/,

-I,2 - 1 2 Jt-

l J

1 5 7 9 77i

Fig. 2

-o,e

Fig. 1. Distribution of the normal s t r e s se s Cx (continuous lines) and ~y (dashed lines) in a plate re inforced with a single fiber under the action of a t r ansve r se load along lines paral le l to the fiber (i =1, 3, 5, 8).

Fig. 2. Distribution of the s t r e s se s cr_ (continuous lines) and a,, (dashed lines) x 3

in the plate considered, under the action of a t r ansve r se load along lines p e r - pencUcular to the fiber ( j= l , 7, 13, 15, 18).

Institute of Strength Problems, Academy of Sciences of the Ukrainian SSR. Trans la ted f rom P r o - blemy Prochnosti , No. 7, pp. 18-21, July, 1971. Original ar t ic le submitted June 23, 1970.

�9 1972 Consultants Bureau, a division o[ Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced [or any purpose whatsoever without permission o[ the publisher. A copy o[ this article is available [rom the publisher [or $15.00.

772

TABLE 1

Values ~ j equal to

l 3 5 7 9 11 12 13 15 17 21

--0,020 ---0.0'22 --0,023 --0,020 --0,012

--0,040 --0,046 --0,046 --0,039 ---0,023

---0,066 --0,074 --0,073 --0,057 --0.031

--0,I01 --0,I I0 ----0,106 --0,070 --0.031

--0,190 --0,172 --0,144 --0,062 --0,017

--0,316 --0,224 --0,153 --0,049 --0.010

--0.175 --0,218 --0.109 --0,027 --0,002

"oo66 --0,159 --0,017 - - 0 , 015 --0.001

--0.092 --0.I I0 --0,097 --0,046 --0.016

Not_...cc: For all values of i at j =i ~'xy/C0 =0.

--0,031 --0,037 --0,05 I --0,049 --0,029

40 J

0

-O0

-00 /

I 3 5 7 9 11 1// 17 Ig 21

Fig. 3 Fig. 3. D i s t r i b u t i o n of t e m p e r a t u r e s t r e s s e s x 3, 5, 8).

Fig. 4. D i s t r i b u t i o n of t e m p e r a t u r e s t r e s s e s cr (i=I, 3, 5, 8). Y

oy, kg/cm 2

,20a---?-" i ,ooo 'r , , ~ 800 ~ Y~*') I

6 0 0 ~ ~ 2 ~

,N 2OO

o _ _ , ~ ~ . ; ~ . I -20C 3

I 3 3 z 9 n /c u ~g 21 t

Fig. 4 in cross sections parallel to the fiber (i =i,

ac t ing p a r a l l e l to the f ibe r a long the l i nes

p la te when the s t r e t c h i n g load is p e r p e n d i c u l a r to the axis of the f iber , and the case of t h e r m a l s t r e s s e s with u n i f o r m hea t ing is a l so cons ide red . The d i m e n s i o n s of the p la te and f ibe r and a l so the phys ica l con- s t an t s a r e ind ica ted in [1]. In Eqs. (11) and (12) let us put qy; rx; Ty; T equal to zero , qx(ij) =q2.

F i g u r e 1 shows the d i s t r i b u t i o n of the s t r e s s e s ~x and Oy in v a r i o u s c r o s s sec t ions p a r a l l e l to the f iber . Cu rve 1 shows the s t r e s s d i s t r i bu t ion in the c r o s s s ec t ion a long the l ine i = 1, which p a s s e s a long the axis of the f ibe r ; cu rve 2 is a long the l ine i = 3, which co inc ides with the i n t e r f a c e between the m a t r i x and the f ibe r ; c u r v e s 3, 4 a r e a long the l ines i = 5 and 8, which pas s into the m a t r i x at d i s t a nc e s equal to 1 / 2 and 2.5 of the width of the f i be r f r o m the m a t r i x - f i b e r i n t e r f a c e .

As may be s een f r o m Fig. 1, an a p p r e c i a b l e c o n c e n t r a t i o n of o- x s t r e s s e s is o b s e r v e d in the zone a d - j o in ing the end of the f iber . The m a x i m u m va lues of crx at the end of the f ibe r r e a c h 1.8 a0, where cr0 a r e the s t r e s s e s in the u n d i s t u r b e d ma t r ix . The zone of high s t r e s s c o n c e n t r a t i o n c o m p r i s e s a reg ion of a p - p r o x i m a t e l y half the width of the f iber . Beyond the f ibe r t he re is a c e r t a i n load r e l i e f in the c r o s s sec t ions ind ica ted in the ma t r i x . The m i n i m u m va lues of cr x a r e 0.54 c'0, the zone of m a t r i x load r e l i e f having the s a m e extent as the c o n c e n t r a t i o n zone.

At the r e m a i n i n g poin ts of the c r o s s sec t ions drawn, the s t r e s s e s a r e c lose to the va lue of cr0. Along the ax is of the f ibe r a p p r e c i a b l e c o m p r e s s i v e fo rces ay (the dashed cu rve 1) act , the m a x i m u m va lues of which occur at the c r o s s sec t ions of the middle of the f iber , r e a c h i n g a va lue of Cry max = 1.7 a0. In the r e - gion of the end they d e c r e a s e to 0.2 cr 0. Here the d i s t r i bu t i on cu rve of a0 undergoes a b r e a k and, a f t e r a c e r t a i n i n c r e a s e , the s t r e s s e s O-y r ap id ly d e c r e a s e to zero .

773

O y , l ~ I g .I l l

t2OO t ,I 12~i' -2

gO0 ~

000

2O0

0

-206

Y

o x, kg/cm 2

"/.#

5 7 a .9 I l l

Fig . 5

O0 .7

-O0

-O0

q20

! 5 b 7

rxy, k g / c m 2

't/O~

00

50

?0

!

I I I

I - A - ~ ~

3 $ 7 g // 14 /7 /.q zI

Fig . 6

F ig . 5. D i s t r i b u t i o n of t e m p e r a t u r e s t r e s s e s (a) ey and (b) ax a long l i n e s p e r p e n d i c u l a r to the f i b e r ( j = l , 7, 13, 15, 18).

F ig . 6. D i s t r i b u t i o n of t e m p e r a t u r e t ang e n t i a l s t r e s s e s T in c r o s s s e c t i o n s p a r a l l e l to the f i b e r ( i = 3 , 4, 5, 7, 10). xy

A p p r o x i m a t e l y the s a m e q u a l i t a t i v e p i c t u r e of the d i s t r i b u t i o n is r e t a i n e d a t the p o i n t s of the l i ne d rawn a long the f i b e r - m a t r i x i n t e r f a c e (the d a s h e d c u r v e 2). The s t r e s s d i s t r i b u t i o n at po in t s on the l ine i =4, which p a s s e s into the m a t r i x to a d i s t a n c e of one q u a r t e r of the width of the f i b e r f r o m the m a t r i x - f i b e r i n t e r f a c e , i s shown a s the d a s h e d c u r v e 3. A l o n g th i s l i ne the s t r e s s e s ~y v a r y in s ign. In the f i b e r zone they a r e s t r e t c h i n g s t r e s s e s , ou t s ide the f i b e r they a r e c o m p r e s s i v e . A l o n g the l ine i = 8 , which p a s s e s at a d i s t a n c e of 2.5 t i m e s the width of the f i b e r , the s t r e s s e s ay a r e s t r e t c h i n g s t r e s s e s a t a l l po in t s .

The s t r e s s d i s t r i b u t i o n at po in t s of the l i ne s p e r p e n d i c u l a r to the f i b e r can be judged f r o m F ig . 2. C u r v e s 1, 2, 3, 4, 5 r e s p e c t i v e l y c h a r a c t e r i z e the s t r e s s d i s t r i b u t i o n a long the l i n e s j =1 , which p a s s e s in the midd l e of the f i b e r , i . e . , co inc id ing with the ox a x i s , j =7 , which i s d rawn at a d i s t a n c e of a q u a r t e r of the f i be r f r o m the ox a x i s , j = 13, p a s s i n g in the r e g i o n of the end o f the f i b e r a t a d i s t a n c e 1 / 4 of the width of the f i b e r f r o m i t s end, and j = 15, 18 p a s s i n g into the m a t r i x a t d i s t a n c e s of 1 / 4 and 1.5 of the width of the f i b e r f r o m i t s end.

We may note that the s t r e s s e s ~y in the c r o s s s e c t i o n s p e r p e n d i c u l a r to the f i b e r a r e s e l f - b a l a n c i n g . As has been noted, they a r e c o m p r e s s i v e s t r e s s e s in the f i be r , and they a r e s t r e t c h i n g s t r e s s e s in the m a t r i x .

The v a l u e s of the t a n g e n t i a l s t r e s s e s Wxy in c e r t a i n c h a r a c t e r i s t i c po in t s a r e p r e s e n t e d in T a b l e 1. The g r e a t e s t t a n g e n t i a l s t r e s s e s w a r e p r o d u c e d a long the m a t r i x - f i b e r b o u n d a r y (i = 3). The n a t u r e of t h e i r d i s t r i b u t i o n is the s a m e a s i ~ h e c a s e of a l ong i tud ina l l y a r r a n g e d f i b e r ( see our f i r s t c o m m u n i c a t i o n [1]) but in magn i tude they a r e a p p r e c i a b l y l ower . F o r e x a m p l e , Txy max = 0.316 e0, whi le fo r a l ong i t ud ina l a r r a n g e m e n t of the f i b e r Txy m a x = 0.94 a0.

Thus , the r e s u l t s of t h e s e i n v e s t i g a t i o n s showed tha t the p r e s e n c e of s h o r t f i b e r s in r e i n f o r c e d m a t e r i - a l s , a r r a n g e d p e r p e n d i c u l a r to the l ine of a c t i o n of the load , l e a d s to a s u b s t a n t i a l c o n c e n t r a t i o n of s t r e s s e s in the zones at the ends of the f i b r e s , which shou ld be t aken into accoun t .

Even by the ac t i on of a u n i f o r m t e m p e r a t u r e f i e ld , s t r e s s e s a r i s e in c o m p o s i t e s which a r e c a u s e d by the d i f f e r e n c e in the t h e r m a l expans ion coe f f i c i en t s of the f i b e r s and the m a t r i x . Consequen t ly , it i s of in - t e r e s t to s tudy the n a t u r e of the s t r e s s s t a t e of the c o m p o n e n t s in a c o m p o s i t e hav ing a v a r i a t i o n in i t s t e m - p e r a t u r e .

The so lu t ion to th i s p r o b l e m is a l s o i m p o r t a n t for e s t i m a t i n g the i n i t i a l s t r e s s e s p r o d u c e d in the p r e - p a r a t i o n of a c o m p o s i t e .

The r e s u l t s a r e p r e s e n t e d be low of an i n v e s t i g a t i o n into the s t r e s s s t a t e of the p l a t e d i s c u s s e d above , f r e e f r o m e x t e r n a l f o r c e s , which i s s u b j e c t e d to u n i f o r m h e a t i n g up to an e x c e s s t e m p e r a t u r e T = 150~

774

In Eqs . (11), (12) le t us put A q = C q = 0. Then , t a k i n g in to account tha t a l l T k = T , the e x p r e s s i o n s fo r B T and D T a c q u i r e the f o r m :

B r ----- Th / , l_ 1 [(I -{- vl)a,81 -{- (1 -1- V4) a484 - - (I ~ ~ ) ae88 - - (1 @ v11) r -I- Thl , l+ 1 [(I -!- v2)%82 :l". (1 -{- "v5)%8 s

D r = Thl.~_, (( 1 + v3) aa8 a -p ( 1 ~- ~4) a484 - - ( 1 +vs) a585-- ( 1 + v6) %8el -t- Thr.t+ l[( 1 -Pv7)%87

+ (1 + ~s) ~ 8 8 - - (1 + ~.) r - - (I + ~,0) a~08~0]. The d i s t r i b u t i o n of n o r m a l s t r e s s e s a x in c r o s s s e c t i o n s p a r a l l e l to the f i b e r i s shown in F ig . 3.

C u r v e s 1 , 2 , 3, 4 r e p r e s e n t the g r a p h s for the d i s t r i b u t i o n of a x r e s p e c t i v e l y in t h e s e c r o s s s e c t i o n s , v i z . , a long the a x i s of the f i b e r (i = 1), a long the m a t r i x - f i b e r i n t e r f a c e (i = 3), in the m a t r i x at a d i s t a n c e of haf t the width of the f i b e r f r o m the i n t e r f a c e (i = 5), and at a d i s t a n c e of 2.5 t i m e s the width of the f i b e r (i = 8).

In t h e s e c r o s s s e c t i o n s the s t r e s s e s ~x a r e , of c o u r s e , s e l f - b a l a n c i n g , the a r e a s of the c u r v e s for the c o m p r e s s i v e and s t r e t c h i n g s t r e s s e s be ing equal in a l l c r o s s s e c t i o n s . In t h e f i b e r zone the v a l u e s of a x a r e p o s i t i v e , o u t s i d e the f i b e r they a r e nega t i ve , i . e . , c o m p r e s s i v e . At the end of the f i b e r a c o n c e n t r a t i o n of ~x s t r e s s e s i s o b s e r v e d , which i s g r e a t e s t in c r o s s s e c t i o n s d rawn a long the ax i s of the f i b e r and a long the m a t r i x - f i b e r i n t e r f a c e ( c u r v e s 1, 2).

The d i s t r i b u t i o n of n o r m a l s t r e s s e s a p a r a l l e l to the ax i s of the f i b e r a long the l i n e s i = l , 3, 5, 8 Y a r e shown in F ig . 4. At the po in t s on the l i n e s d rawn a long the a x i s of the f i be r and the m a t r i x - f i b e r i n t e r -

f a ce ( c u r v e s 1, 2), the s t r e s s e s Cry a r e s t r e t c h i n g s t r e s s e s , and with s e l e c t e d p h y s i c o m e e h a n i c a l c h a r a c t e r - i s t i c s and d i m e n s i o n s of the f i b e r and m a t r i x r e a c h 1320 k g / c m 2 at the c e n t e r of ~he f i b e r , and 540 k g / c m 2 at po in t s of the i n t e r f a c e . A long the l ines p a s s i n g into the m a t r i x ( c u r v e s 3, 4) the s t r e s s e s Cry a r e c o m - p r e s s i v e in the f i b e r zone, b e i n g a p p r e c i a b i y l o w e r in a b s o l u t e va lue than the s t r e s s e s in the f i be r . Th is i s r e a d i l y s een f r o m F ig . 5a, which shows the s t r e s s e s a in c r o s s s e c t i o n s p e r p e n d i c u l a r to the f i be r , which a r e d rawn a long the l i ne s c o n s i d e r e d above (j =1, 7, 13 y 15, 18).

In a l l t h e s e c r o s s s e c t i o n s the s t r e s s e s (ry a r e s e l f - b a l a n c i n g . The g r e a t e s t s t r e t c h i n g s t r e s s e s ay at po in t s in the f i b e r a r e a l m o s t s ix t i m e s h i g h e r than t h o s e in the m a t r i x .

F u r t h e r m o r e , a c c o r d i n g to F ig . 5a and b, the a b s o l u t e va lue of the s t r e s s e s a a r e a p p r e c i a b l y h i g h e r �9 Y

than a x, the g r a p h s of which a r e shown a long the s a m e l i n e s a s for ~y. The maxzmum va lues of ay a r e h i g h e r than t h o s e of a x by a f a c t o r of a l m o s t 13.

The c u r v e s for the t a n g e n t i a l s t r e s s e s rxy a r e shown in F ig . 6. The h ighes t t a nge n t i a l s t r e s s e s a r e p r o d u c e d in the c r o s s s e c t i o n s c l o s e to the m a t x - i x - f i b e r i n t e r f a c e and a long the bounda ry i t s e l f . The tan. z g e n t i a l s t r e s s e s in the f i b e r a r e i n s ign i f i c an t .

The d i s t r i b u t i o n of t angen t i a l s t r e s s e s a long l i nes p a s s i n g a long the m a t r i x - f i b e r i n t e r f a c e and s i t u - a t ed f r o m it a t d i s t a n c e s of 1 /4 , 1 /2 , 3 /2 , and 3 t i m e s the width of the f i b e r is shown by c u r v e s 1, 2, 3, 4, 5

r e s p e c t i v e l y .

In the c r o s s s e c t i o n s at the m a t r i x - f i b e r i n t e r f a c e and in the c r o s s s e c t i o n c l o s e to i t , an a p p r e c i a b l e s t r e s s c o n c e n t r a t i o n i s o b s e r v e d at the end of the f iber �9 Owing to the i n a d e q u a c i e s of the d i f f e r e n c e method of so lv ing the p r o b l e m , it was not p o s s i b l e to c a l c u l a t e the s t r e s s peaks �9 As may be seen f r o m F ig . 6, the s t r e s s c o n c e n t r a t i o n d e c r e a s e s s u b s t a n t i a l l y a s the c r o s s s e c t i on i s moved away f r o m the f i b e r .

Thus, the p i c t u r e ob ta ined for the d i s t r i b u t i o n of t e m p e r a t u r e s t r e s s e s t e s t i f i e s that when c o m p o s i t e s in which the r e i n f o r c i n g e l e m e n t s a r e s h o r t f i b e r s a r e h e a t e d o r coo led , an a p p r e c i a b l e c o n c e n t r a t i o n of t e m p e r a t u r e s t r e s s e s is o b s e r v e d .

L I T E R A T U R E C I T E D

1. I~. S. U m a n s k i i , E. I. Za luzhnaya , and S. M. Medovaya , P r o b l . P r o c h n o s t i , No. 7, p . 764.

775