S T U D Y O F L O C A L E F F E C T S I N V O L V I N G I N T E R A C T I O N

B E T W E E N T H E F I L A M E N T S A N D T H E M A T R I X IN

C O M P O S I T E M A T E R I A L S : 1

I~. 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

On applying s t r e s s e s to composi t ion m a t e r i a l s r e in fo rced with discont inuous (discrete) f i laments o r whiskers , a high s t r e s s concentra t ion tends to develop; this a l so happens at b reaks in in i t ia l ly continuous f i laments . A study of the s t r e s s e d s t a l e in composi t ion e lements is t h e r e f o r e an impor tan t p rob lem in the mechanics of r e in fo rced media .

We note that, without de t e rmin ing the s t r e s s e d s ta te of the e lements in a s t ruc tu re , such p r o b l e m s as that of e s t ab l i sh ing the opt imum propor t ions of the components , the opt imum re l a t ionsh ip between t h e i r p r o p e r t i e s , the d imensions of the r e in fo rc ing components , and so forth cannot be p r o p e r l y solved. At the same t ime, the study of local effects such as the s t r e s s d i s t r ibu t ion in the reg ions in which s t r e s s is t r a n s - mit ted f rom the m a t r i x to the f i laments const i tu tes an e x t r e m e l y compl ica ted problem; at the p r e s e n t t ime an exact ana ly t ica l solution is comple te ly imposs ib le .

Fo r the s imp les t cases the re a r e a number of approx imate solut ions based on e x t r e m e l y idea l ized models [1].

In this p a p e r we shall cons ide r the p l a n e - s t r e s s e d s ta te of a p la te r e in fo rced with a s ingle cen t ra l f i lament and subjected to the action of an a r b i t r a r y contour load in an a r b i t r a r y t e m p e r a t u r e field (Fig. la) , in o r d e r to e s t ima te the local effects involved in the t r a n s m i s s i o n of s t r e s s e s f rom the m a t r i x to the f i l a - ments . We shal l a s sume that s t rong bonding fo rces act between the f i lament and ma t r ix .

The edge (boundary) p rob lem for a compos i t e body may be solved in t e r m s o f d i sp l acemen t s by the f in i t e -d i f fe rence method. The d i f ference equations a r e obtained on the bas i s of the Lagrange va r i a t iona l p r inc ip le .

Let us denote the components of the d i sp lacement vec tor by ff and ~. We shall int roduce the d imen- s ion less coord ina tes x= (Y/L); y= (~/L) and the d imens ion les s d i s p l a c e m e n t - v e c t o r components u = ~ / L ; p-- ~ / L .

F o r a p l a n e - s t r e s s e d s ta te the total potent ia l energy of the sys t em, const i tut ing the d i f ference be- tween the work of the in ternal fo rces and that of the external loads, t akes the form

2(l--v,) [kJx] ~Oy] + 2 ~ , ~ - - } - v,~Oy + ax ] 2(1-{-

(1)

where E is the e las t i c modulus; u is the Poisson coefficient , ~ is the coeff ic ient of l i nea r t h e r m a l expan- sion; T is the t e m p e r a t u r e ; Xv, Yv a re the s t r e s s components appl ied to the contour (kg/cm2); s is a d imen- s ion less sect ion of the contour, s = ~ ' / L ; and t is the th ickness of the plate .

The contour in tegra l is taken along the boundary of the plate in the x, y plane.

Accord ing to the p r inc ip le of v i r tua l d i sp l acemen t s , the actual d i s p l a c e m e n t s u and • wil l be such as will ensure a s t a t iona ry s ta te of the potent ia l energy:

6~ = O. (2) Inst i tute of Strength P r o b l e m s , Academy of Sciences of the Ukrainian SSR, Kiev. T rans l a t ed f rom

P r o b l e m y Prochnos t i , No. 7, pp. 11-17, July, 1971. Original a r t i c l e submit ted June 23, 1970.

�9 1972 Consultants Bureau, a division of 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 from the publisher [or $15.00.

764

B

Matrix

a

2C . i - 4 l - z

3

�9 ! O

i-7, 2 5

5

f%,,

i . i - r i.1,/~-z I

7 r- "'~I

0 I tz I i . / ,2

9 I 12 ...:.,

- - - -~ - - - - - ""%1

I ' " , , ' "

h i , [ P l _ I

Fig. 1. Plate re inforced with one filament and ca r ry ing a contour load (a); quar te r of the plate with an a rb i t ra r i ly -drawn rectangular mesh (b); a r rangement of meshes around the point i, j (c).

It is well known that Eq. (2) contains the conditions of equilibrium and the boundary conditions; thus the displacements obtained f rom the s tat ionary (steady-state) condition anti the s t r e s ses deduced therefrom constitute the solution to the problem in hand.

The s t r e s s components are expressed in t e rms of the displacements in the following manner:

2G [au ~ 0v--(1 + v ) a T ] ;

20 fc)v Ou 1 o'u = ~ [ ~ -k- v ~x - - ( 1 -I- v) aT., ; (3)

r.u = 6 ~ + N .

The condition of s tat ionary potential energy is satisfied if

06 0; O~ a-Y = ~ = 0, 14)

f rom which the values of u and v may be determined.

For s implici ty of solution we suppose that the load is symmetr ica l with respect to the coordinates of the axes. In this way we need only consider the quar ter of the plate lying in the f i rs t quadrant.

The potential energy ~ may be calculated by an approximate method described ear l ie r [2]. The con- tour of the region (Fig. lb) is divided into a finite number of sections by drawing a rectangular mesh on the region in question. The points of intersection of this grid or mesh we call nodes. We have also drawn an auxiliary (broken-line) mesh, the lines of this dividing the sections of the main mesh into equal parts .

Let us consider the node formed by the lines i and j (Fig. lc). The regions formed by the lines of the main and auxiliary meshes are called cells and numbered as shown in Fig. lc.

In the potent ia l -energy expression (1), the integration over the whole a rea of the plate is replaced by summation of the energies over all the cells composing the plate; the f i rs t derivatives are replaced by the corresponding difference expressions, while condition (4), originally formulated relative to the continuous variables u(x, y) and v(x, y), is replaced by a corresponding set of conditions applied to the discrete values of uij, ~ij at the nodes of the mesh.

We thus obtain 2n conditions

~6' 06' 0,fis - 0; Ov:i - O, (5)

where n is the number of nodes.

765

T A B L E 1

0.134 O,L2L 0.083 0,032

0,136 0,122 0,083 0.031

0,139 0,123 Q.080 0,046

0,141 0,121 0.067 0,002

Values ofox/%for j, equal to

tl 13

0,147 0,230 ----0,571 0,110 0,112 --0,097 0,003 -'0.039 --0. I01

--0,005 --0,026 --0,007

15

0,311 0.298 0,108 0,027

17

--0,066 --0,028

0.062 0,039

--0,138 --0,120 --0,055 --0,005

21

--0,203 --0,203 - - 0 , 1 4 9

--0,058

Read the program and initial data into the arithmetical unit

Regenerate variable orders

Write a "1" into the iine-number counter O)

Write a "1" into the column-num- ber counter (i)

Increased i by "1"

No [ Check condition i > ima x

Choose conditions from Pl-Pl0 for i , j

I Calculate size of steps for point i, j using a logical scale

Calculate the coefficients of the system of equations and write into the matrix

l q.

]

Clear counter J

[ Increase by "1"

[ Fig. 3.

.Check condition j >Jmax

i Write matrix and column of free I

I terms on magnetic tape

Write program for solving the system of equations by the Gauss method from magne-

tic tape into arithmetical un it

I 8olve system of equations by the Gauses method

l, Print results

, Calculate stresses

Print remits, Stop. [ J

B l o c k d i a g r a m o f t h e p r o g r a m .

~ o

766

2;

1

b

5

~PrAII

~ 1 1 Ic~l I I

g l ~ l l I g,t~t I I

ltPl~t I I ~,tll

~/I r.,rA i i

I I I I

'e i i i i to IIII

IJll IIII IIII

t

7 Y t! i I I

O , q

c

Fig. 2. E x a m p l e s of the d ispos i t ion of a node: a) The point i, j l ies at the boundary between the m a t r i x and the f i lament; b) the point l ies on the con tour of the plate y = cons t , c) Q u a r t e r of the plate with one p a r t i c u l a r type of mesh .

~ 8 " " - -

On di f fe ren t ia t ing the e x p r e s s i o n s in Eqs . (5) non- z e r o t e r m s will only a r i s e f r o m the numbered ce i ls adjacent to the point under cons ide ra t ion (Fig. lc) . We shal l t h e r e f o r e only cons ide r these ce l ls in subsequent

12 ca lcu ln t ions , us ing e ' t o denote the ene rgy sum e= ~ ek

te-~--t

Since the de r iva t ive of a sum is equal to the sum of the de r iva t ives , Eq. (5) may be r e w r i t t e n as

12 12 "

. . -= o . ( 6 )

k ~ | # = 1

In o r d e r to c o m p a r e the d i f fe rence equat ions, we mus t c o m p a r e the e n e r g y e x p r e s s i o n s for each cell , r e - m e m b e r i n g that within the cel t art funct ions and d e r i v a - t i ve s r e m a i n cons tan t . The de r iva t ives a r e r ep laced by d i f f e rence exp re s s ions . Thus , fo r example , in the eighth cel l

Ou ui4-I,i - - ul I . Ov vii - - v i , j - I

Ox h i , / t . I ' Oy h i _ l , i

The e x p r e s s i o n s for the potential ene rgy of the eighth cel l and its de r iva t ives with r e s p e c t to uij and vii take the f o r m

)' z O - , ~ ) I.\ h~.~+t / -I- ~ -I-2~,~ h~.i_~ ~ nj..j_~ I (I-: . vs) u~i--ui'J--~l-V~+Li--v~

vii - - vi . i ._l " ] - - " l 'Ui '4"l ' l - -ul i "1" ~ 4 ; --2(I-l-Vs) I. ~ ) asTs -F 2(I 'F vs) (anTs) ' hi'i+' h i ' i ' (7)

0r s EsL ~t [

5u~s - 20-v~ , ) I 1 u i + J , i - - u I i 1 1 2 h~+, l'ii-~ - 7 vs(vi i--vi . i -~) + - 4 -(1 -- ~a)(uiJ--u~.J -I)

_ _ _ I ) ~'~+~ t- "4- ( 1 -- '~a) tv~+~./-- vii) + --~ ( 1 I- v8) %T~hjj_ ~ ; (8) hi . i -- 1

Oe~ e,L~t [ + ~h~.i+, I - - - + ( 1 Va) -----~ (1 - - v )h;.i_~ a~'i~ 2 ~ ) (v. - v:.;_ ,. h;j_ L + T vs(u.-, .J-- %) -- ( % - - u~.j_:) - - '~s)(%,.i ." h~.i+,

21 (1 - - Va) asTsh i,i ~.l ] . ('9)

Since the s u m m a t i o n is c a r r i e d out o v e r individual ce l l s , we may suppose that the mechan ica l c h a r a c - t e r i s t i c s of the m a t e r i a l change on pass ing f r o m one cell to another ; the method in ques t ion may thus be s u c c e s s f u l l y applied to s t r u c t u r e s composed of in_homogeneous m a t e r i a l s such as the compos i t ion m a t e r i a l s of p r e sen t in te res t , in p a r t i c u l a r the p la te r e in fo rced by a single f i lament .

We shal l t h e r e f o r e h e n c e f o r w a r d a t tach indices indicat ing the number of the cell to the e las t ic modu- lus, P o i s s o n coeff ic ient , and t h e r m a l - e x p a n s i o n coeff ic ient , r e g a r d i n g these charactelZist ics as only r e - main ing cons tant within the bounds of one p a r t i c u l a r cell .

In the p r o b l e m under cons ide ra t ion the phys i comechan i ca l c h a r a c t e r i s t i c s may take e i the r one of two va lues , r e s p e c t i v e l y r e l a t ing to the f i lament and ma t r i x ma te r i a l . We shall a t tach the index f to the f o r m e r (El, uf, a f), and leave the second without indices (E, u, ~).

In Eq. (6) we divide all the t e r m s by the quant i ty EL2t /2(1 - v2). Then in the exp re s s ions for the d e r i - va t ives such as (8) and (9) we shal l have a f ac to r 6 k, in each cell capable of taking the fol lowing va lues : 6 k = 1, if the cell belongs to the m a t r i x ; 6k = 4, if the cel l belongs to the f i lament ; and 6 k = 0, if the cell t ies outs ide the reg ion in quest ion, whe re

767

o,a

0,0

az

ol

/ 3 .~ 7 9 // lZ~ t7 lg 21 ;"

F i g . 4 . D i s t r i b u t i o n o f t a n g e n t i a l s t r e s s e s r x y a l o n g l i n e s p a r a l l e l t o t h e f i l a m e n t .

X -- Ef( I --v=) _ �9 (1 o)

A f t e r s u m m a t i o n a n d t h e c o l l e c t i o n o f s i m i l a r t e r m s , E q s . (6) m a y be w r i t t e n in t h e f o l l o w i n g m a n n e r in d i f f e r e n c e f o r m :

uiiAii - ul,i_xA~.i_ ~ - - ul, i+lAt,l+ t - -U~_l , iA i_ t , i

- - utq_LiAi.H,i - - o t_L i_ lB t_ t , i_ t Jr v t_LiB t_l, I

-]-- v i_Li+tBr_ i , i+ t -~ v i , i_ tBi , t_ t -1- oqBq

[-- oi,l+lBi,i+ t Jr v i+t , i_~B,+t , i_t Jr ot+t,iBi+L i

- - oi+t,i+~Bl+Li+t - - Aq - - B r = O. (11)

viiCii - - ~ - - ~ - - r ')t-t , iCi-l, i - - vi+t. iCt+Li - - u z -~ , i - tD ~-] ,i-~ + u i - t , i D i - I ,I Jr t l i--l , iq-IDi-l, i+t

-~ Ul , i_ lDi j_ l "Jr u q D q - - b l i , / t.iDi,i+l-~ - Ui4.1d_lD i4_lj_ I "Jr- t t i+idDi+l. i - - u l t_Ij4_IDi+I,i+I - - Cq - - D r = 0, (12)

w h e r e

Aq = Ai . i_ ~ F Ai,i+ I ~ Ai_l , i 4- A~+Li;

- - 1 - - v,t p~ 1 - - "v 7 1 - - v 8 A i , i _ l : : 1 ~av36a+ ~ 4 J r " ~ 7 Jr- ~ 6 a ;

I - - v ~ , l - - ' v . ~ 6 o + l - - v z o , ~ Ai,i_i. I -- ,-~ -6~ n ~ 80 q- ~ v l o ;

Ai_t. i = a(61 + 64 ) + b(62 + 85);

A,4_Li = c(6 s 4- 6it) 4- d(6 o + 612);

, I - -v3 =v2&, j r ~ ~6, LJi 14 I : V l ( 5 1 ~ " ~ 6a; B~-t. l+l

B,. xi .gt6L - - v~ 6', 1 - - ",,4 J r ~ �9 . .- . ~ 64 65;

I - - ",.':3 Bi.i_ l - -~ - 6 a - v 4 6 a - ~ - ~ 6 7 J r v a 6 a ;

, 4-,., 5 l +,'~. L . ~ 6 s + . L ~ , . ' Bii = Di i" " ~ a ~ O a - -

l - -v s ,~ l - - v i e ,~ . Bi.i4 t vS6~ - - -......"'~-- v~ - - Va6ojr ~ "-'to,

I - - 'v7 67 _}_ ~lx6xl; 1 - - rio 61 ~ Jr V 1 ~ 8 1 2 ; Bi+l . i_ l ~ o - Bi+t;i+t - - 2

.__ l - - v a & _ _ l - - v 9 Bi.t_l, j -- ~ ~a ~ 69 --- "911611 -]- Vaz612;

Aq == 2 ~ - ~ [qx(q) (hi , i - t64 "-}- hi,i+t65) -11- Ty(q) (hLf:_164 + ht,~+16e)l;

B r : [(1 + "90 atTt61 Jr (1 + 'v 4) %T464 - - (1 + "%) aaT86 s - - (1 + 'vtt) auTt t6t t l hi , l_ j

-4- [( 1 -4- v2) a.,T,.,6~. Jr (I -~- "gs) %T~6e, - - (1 4- '%) agT~6 e - - ( 1 + 'v~2 ) a~2Tr9612] hi,i+l;

Cq = Ct,i_ 1 4- Ci, i+ t Jr- Cl_l, I Jr Cl+~,l;

I C~a_~ =~ -~.(63 + 64) + -;- ((57 + 6~);

' - " + + bs, Jr C~-t,I ~- ~ a61 1 - - %

I - - v s 1 - - V x l 1 - - v g 1 - - v 1 Ct+l,I = " - ' - E ' - c6s + ~ c 6 . + ~ d69 4- ~ d6~;

768

2

1 3 5 6 7 o 9 I0 a

0

2

0 t . . .v 3 ~ 6 7 d 9 lO

b

Fig . 5. D i s t r i b u t i o n of the t a n g e n - t i a l s t r e s s e s Zxy (a ) ; and n o r m a l s t r e s s e s 6y (b) ; a long l i n e s p e r p e n d i - c u l a r to the f i l a m e n t .

~z = 68 = ~9 = 610 : = 6i t ~ 612 :~- I ; v I = v,~ :-= Va = v4 = v5 = vo ~ ~'/;

| - - 'V 1 D l - t , i - t = 2 6t + vaSa;

I - - ~ 2

Dt_, /+t ~ ~ 62 + v J S ~ ;

1 - - v 1 l - - ~ 2

D i - t , l ~ "2 61 2 82__ .V l64 .qt_ .~585;

D t , l _ l = ~383 I --v 4 1 --v s 2 6 4 - - v 7 6 7 -~ ~ . ~ 8 ;

1 - - v 5 1 - - vg' Dr,l+ L = "%6~ - - ~ 65 - - "v m6to % ~ 69;

I - - "v lz

D i 4_l,i_ 1 ~ 'Vr67 + - ' T - - - 611;

1 - - - v l ~ Di+l . i+ l = v10610 ' 2 612;

I - - v H 1 - - vl2 Dt+t.I ---- v868-- v969 ~ 611 + ~ 61~-;

C q = ~ lqv(ti}(hi_t ~6.1 -~- ht.i+168) "q- "~x{ti) (hl . i-~4 -~ hi.l+,~)5)];

D r =: [( 1 + v3) r 3 �9 T a �9 63 + (I -1- v4) a 4 �9 T.t6 ~

- - ( l q- "%) a 5 �9 Z 5 . 65 - - (1 -+- v 6) (16' Z6" ~6] h i , i - t

+ [(l + v7) a 7 .T767 -~- (1 -{- V8)(18 .T8.68

- - (1 -]- v~) (19 .T 9 .60 - - (1 -!- rio ) (110"Tm '61ol ht,t~_,;

0~1 ~ El2 ~ (/3 - - (14 ~ (15 == f~6 ~ (~]~ (17 ~ (18 "~-- (~9 ~ (110 --- (111 :~"0~12 ~ (1"

a ~ h i ' l - I " b = hi'i4-1" c = hi'l--I " d-- - -hi ' i+t h i , i_ 1 ' h i , l_ I ' hi,t-hi ' hi,l+ 1 �9

We note tha t the quan t i t i e s Aq, Cq accoun t i ng fo r the con tour load van i sh fo r a l l i n t e r n a l nodes .

T h e s e equa t ions a r e a p p l i c a b l e to a l l the nodes . It is only n e c e s s a r y to s p e c i f y the a p p r o p r i a t e v a l - ues of the c o e f f i c i e n t s 5k, v. , at- fo r each node pos i t ion . F o r e x a m p l e , if a node l i e s at the m a t r i x - f i l a - men t i n t e r f a c e (Fig . 2a), then in Eqs. (11) and (12) we mus t put 6 1 = 6 2 = 5 3 = 6 4 = 6 s = 5 6 = ' ~ ; 6 t =68=69=6to=6"~I

= 5 1 2 = 1 ; vl = v2 = P3 = /)4 = /)5 = /)6 = / ) f ; u7 = v8 = v9 = ulo = v i i = /)12 = /); ~1 = ~ 2 = O/3 = o r 4 = ~ = o r 6 = o g f ; Ol7 = c e s = C e g = ~ 1 9

= Otl l = O~12 = O~.

If the node l i e s at one of the b o u n d a r i e s of the r e g i o n , for e x a m p l e , y = cons t (F ig . 2b), then we have

61=63=54=?i7=68=<~11=1; 62=65=66=69=610=612=0. A l l /)k=/); Otk=~.

In the p r e s e n t p r o b l e m we have e ight d i f f e r en t t y p e s of cond i t i ons , depend ing on the l o c a t i o n of the noda l po in t s (at the f i l a m e n t - m a t r i x i n t e r f a c e , in the f i l a m e n t r e g i o n , in the m a t r i x r e g ion , a t the edge of the p l a t e y = cons t , and so on). We m a y denote t h e s e by the s y m b o l s P I - P 8 . In add i t i on to th i s , t h e r e a r e s o m e e x t r a cond i t ions fo r nodes on the s y m m e t r y a x e s . F o r the ox ax i s (denote by Pg)

Vii = Ot._l,] = Oi+l, i = 0; h i , l _ l = h ld + l ;

r ) t_L i_ l = - - v t_ t , i+ t ; U~--LI-] = u , _ L i + z ;

Ol,]_ 1 ~ - - Ut,]+l; U i , l - - I = Ili,i+!;

'O' l+ l , ] - - I = - - O t + l , / + l ; U i + I , / - - I ==- / ' / i + 1 , 1 + 1 '

F o r the oy ax i s (P l0 )

u q = u t , l _ ~ ~ u t , l+ ~ = O; h~,t_ ~ = h i j + ] ;

u l _ t , i _ l = - - u t + L / _ l ; v l - t , i _ t = o t+t , l_ t ;

ul._l,i ~ - - ut+l , i ; O~-i,i ~- Oi+l,/;

u " - I , / + I ~ - - u t " H . / + I ; U i - - t . / + l ---- [ ~ i + , / + "

769

L

3

2

!

0

3 5 7 g 11 I# 17 Ig 21 1

Fig. 6. Dis t r ibut ion of the no rma l s t r e s s e s 6 a long l ines para l l e l to the filameYt.

By va ry ing i and j and choos ing the a p p r o p r i a t e condi t ions f r o m P 1 - P 1 0 we obtain a s y s t e m of a l g e b r a i c equat ions with a s y m m e t r i c a l s t r ip m a t r i x of coeff ic ients .

The fo rma t ion of the mat r ix , the solut ion of the s y s t e m of equat ions , and the de te rmina t ion of the s t r e s s e s at each node in the mesh f r o m the d i sp l acemen t s so found were r ead i ly ach ieved by means of a Minsk 22 compute r . An en la rged block d i a g r a m of the p r o g r a m for de t e rmin ing the s t r e s s e s and s t r a i n s in m a t e r i a l s r e i n f o r c e d with individual f i l aments is p r e sen t ed in Fig. 3.

The s y s t e m of l inea r a l g e b r a i c equat ions with a s y m m e - t r i ca l s t r ip ma t r i x was solved by the Gauss method, us ing a p r o g r a m developed in the Ukra in ian S c i e n t i f i c - R e s e a r c h De - sign Ins t i tu te of the Coal Minis t ry .

The s t r e s s e s were d e t e r m i n e d a s w e i g h t e d means over the four ce l l s ad jacent to each point i, j. g a r d i n g t h e s t r e s s e s in each cel l as constant , we may wri te

R e -

o'*F4 -~- osF6 + osF, + osF9 a t i = F , + F 6 + F a + F ~ (13)

where F4, Fs, F8, F9 a r e the a r e a s of ce l l s 4, 5, 8, and 9, and a 4, as, as, a s a r e the s t r e s s e s in these ce l ls .

Star t ing f r o m e x p r e s s i o n s (3) and (13) and choos ing the a p p r o p r i a t e va lues of the P o i s s o n coeff ic ient v k, shea r modulus Gk, and t h e r m a l - e x p a n s i o n coeff ic ient a k in a c c o r d a n c e with the pos i t ion of the node i, j, we may der ive f o r m u l a s for ca lcula t ing the s t r e s s e s f r o m the d i sp l acemen t s found p rev ious ly . Fo r e x a m - ple, in the case of points lying on the v e r t i c a l boundary between the ma t r ix and f i lament ,

2G [uq(•215 / -',- (vt.i+t--vl./_O(• , +vai+,. i) T[a(l +v)h~d+t + a t ( l +vf) • axlI= l - - v [ I Z i _ _ l , i J F h l , i 4 _ l " ( h t _ l , t - J r h l , t + l ) ( h l , J _ l . J r h i , j + t ) hi_l,i-~-hid+ 1 (14)

Put t ing • =1, vf= v, we obtain a f o rm u l a for the s t r e s s e s in the un i fo rm reg ion :

26 [ u'+~'i+--ui-''i !- v t ' 'a '+ ' -v~' i - t (1 - - v )aT] . OxiJ ~ I - - v ~hi , l_ | ~ .hi , i4_ I h i , i _ I "-t-hi,l+ I (15) J

F o r m u l a s for Cry and Txy a r e de r ived ana logous ly .

We shal l now p re se n t some ca lcu la t ions r e l a t ing to a p la te r e i n f o r c e d with a s ingle f i l ament under the act ion of a longitudinal (with r e s p e c t to the f i lament) load of in tens i ty ql (Fig. la ) , put t ing qx(ij); rx( i j ) ; Ty(ij); T equal to z e r o in Eqs. (11) and (12) and qy(ij) =ql.

The s t r e s s e d s ta te of a plate r e i n f o r c e d with a s ingle f i lament and subjec ted to a t r a n s v e r s e load in a un i form t e m p e r a t u r e field will be cons ide red in a l a t e r publ icat ion (communica t ion 2).

The ma te r i a l of the plate is ha rdened epoxy r e s i n with an e las t ic modulus of E = 3.5.104 k g / c m 2, a Po i s son coeff ic ient of v = 0.4, and a coeff ic ient of l inear t h e r m a l expansion oz = 65 �9 10 -6 deg -1. The f i lament ma te r i a l is s teel (Ef= 2.106 k g / c m 2 ; v f= 0.3; cef=12.5. 10 -6 deg-1).

One qua r t e r of the plate, with the mesh and d imens ions a t tached , is shown in Fig . 2c. The sma l l s tep is 0.025 and the l a rge step 0.05. This kind of division leads to 472 equat ions (we note that fo r nodes lying on the s y m m e t r y axes t he re is one equation each).

A compute r ca lcula t ion gives the s t r a in s uij and vij and the s t r e s s e s a ; a ; T at each node of the x y xy mesh; the d is t r ibut ion of these is i l lus t ra ted in F igs . 4 -6 and in Tab le 1 for c e r t a i n c r o s s s ec t ions .

In Fig. 4 the cu rves 1-4 r e p r e s e n t the d is t r ibut ion of tangent ia l s t r e s s e s r x a long the l ine i = 3, which p a s s e s through the m a t r i x - f i l a m e n t in t e r face (Fig. 2c), and a l so a long the l ines i :=Y4, 6, 10, p a s s i n g th rough the ma t r ix at d i s tances of 1 /4 of the width of the f i lament , one f i l ament width, and t h r e e f i lament widths, r e spec t ive ly , f r o m the m a t r i x - f i l a m e n t in t e r f ace .

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We s e e f r o m the f i g u r e tha t the g r e a t e s t c o n c e n t r a t i o n o f t a nge n t i a l s t r e s s e s o c c u r s n e a r the end of the f i l a m e n t a t the i n t e r f a c e be tween the m a t r i x and the f i l a m e n t ( cu rve 1). In view of the l i m i t a - t ions of the d i f f e r e n c e method , the s t r e s s peak could , un fo r tuna t e ly , not be d e t e r m i n e d . The m e s h would have to be g r e a t l y r e f i n e d in o r d e r to do t h i s .

The zone of c o n c e n t r a t i o n of the t a n g e n t i a l s t r e s s e s o c c u p i e s 1 .5 -2 f i l a m e n t widths , which i s c o n s i d - e r a b l y s m a l l e r than tha t d e r i v e d f r o m the op t i ca l me thod [3]. Th i s i s p e r f e c t l y r e a s o n a b l e , s i nce i dea l coupl ing was c o n s i d e r e d to t ake p l a c e at the m a t r i x - f i l a m e n t i n t e r f a c e in the p r e s e n t c a l c u l a t i o n s , whi le the op t i ca l me thod c o n s i d e r e d the r e a l bond f o r c e s . On moving away f r o m the f i l a m e n t the c o n c e n t r a t i o n of t angen t i a l s t r e s s e s f a l l s s h a r p l y ; the c h a r a c t e r of the c u r v e s a l s o changes , and they b e c o m e s m o o t h e r (F ig . 4, c u r v e s 3 and 4). At the end of the f i l a m e n t a l l the c u r v e s have a s h a r p b r e a k . The d i s t r i b u t i o n of the t angen t i a l s t r e s s e s a long l i nes p e r p e n d i c u l a r to the f i l a m e n t m a y be judged f r o m F ig . 5a.

F i g u r e 5a p r e s e n t s c u r v e s r e l a t i n g to the l i n e s j =8 (1), 14 (2), 16 (3), 20 (4). The l ine j =8 p a s s e s at a d i s t a n c e of 1 /4 of the f i l a m e n t length f r o m the end of the f i l a m e n t , the l ine j = 14 at the end of the f i l amen t , whi le the l i n e s j = 16 and j = 20 p a s s t h rough the m a t r i x at d i s t a n c e s of 1 /2 and 2.5 f i l a m e n t widths f r o m the end.

We s e e f r o m the f i g u r e that the s t r e s s m a x i m u m l i e s not at t h e m a t r i x - f i l a m e n t i n t e r f a c e but a s h o r t d i s t a n c e away f r o m th i s , a l though on moving t o w a r d s t h e e n d of the f i l a m e n t the s t r e s s peak c o m e s c l o s e r to the a c t u a l i n t e r f a c e .

At a d i s t a n c e of the o r d e r of t h r e e f i l a m e n t widths f r o m t h e m a t r i x - f i l a m e n t i n t e r f a c e the t a nge n t i a l s t r e s s e s a r e i n s i g n i f i c a n t , and t end to z e r o on moving away f r o m the f i l a m e n t .

The d i s t r i b u t i o n of the n o r m a l s t r e s s e s a y 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 t he f i l a m e n t a r e shown in F ig . 5b. C u r v e 1 c h a r a c t e r i z e s the s t r e s s d i s t r i b u t i o n a long the l ine j = 1, i . e . , the ox ax i s , c u r v e 2 the l ine j =8, c u r v e 3 the l i n e j =14, and c u r v e 4 the l ine j =20.

The g r e a t e s t d e l o a d i n g of the m a t r i x o c c u r s a long the ox ax i s , i . e . , in the m i d d l e of the f i l a m e n t . The g r e a t e s t s t r e s s e s in the f i l a m e n t e x c e e d the g r e a t e s t s t r e s s e s on the con tour of the p l a t e in the s a m e s e c - t ion by a f a c t o r of 10, and the s t r e s s e s in the m a t r i x at the po in t s c l o s e to the f i l a m e n t by a l m o s t 20 t i m e s .

We m a y judge 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 ~y a long l i nes p a r a l l e l to the f i l a m e n t f r o m F ig . 6. The g r e a t e s t s t r e s s e s o c c u r a long the l ine p a s s i n g a long- the ax i s of the f i l a m e n t ( c u r v e 1, i =1) .

On p a s s i n g t h rough t h e f i l a m e n t - m a t r i x b o u n d a r y t h e r e is a b r e a k in the cu rve . C u r v e 2 d e s c r i b e s the s t r e s s d i s t r i b u t i o n ay a long the f i l a m e n t - m a t r i x i n t e r f a c e . At the end of the f i l a m e n t t h e r e i s a sub - s t a n t i a l s t r e s s c o n c e n t r a t i o n .

C u r v e s 3 and 4 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 ey in the m a t r i x a long the l i ne s i = 5 and 8, p a s s - ing a t a d i s t a n c e of 1 /2 and 2.5 f i l a m e n t widths f r o m the i n t e r f a c e . In the f i l a m e n t zone the s t r e s s e s a r e c o n s i d e r a b l y s m a l l e r than the n o m i n a l s t r e s s a0, and at a po in t on the ox ax i s we have a y = 0.2 e0, f~r i = 5 and ~y = 0.37 o- 0, for i = 8. Beyond the f i l a m e n t the s t r e s s e s i n c r e a s e , t end ing t o w a r d e0.

The a b s o l u t e va lue of the s t r e s s e s a x is c o n s i d e r a b l y s m a l l e r than tha t of a y and Txy. The va lues of cr x a r e g iven in T a b l e 1 for 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 . In the s e c t i o n s i= cons t t h e s e a r e s e l f - b a l a n c e d .

We no te tha t the s t r e s s d i s t r i b u t i o n in a p l a t e r e i n f o r c e d with a s i ng l e f i l a m e n t a g r e e s qua l i t a t i ve ly with that ob ta ined by the op t i c a l me thod for m o d e l s with o the r p l a t e and f i l a m e n t d i m e n s i o n s .

LITERATURE CITED

I. G.S. Hollister and K. Thomas, Filament-Strengthened Materials [Russian translation], Metallurgiya, Moscow (1969).

2. D.S. Griffin and R. B. Kellog, "Numerical solution of axially-symmetrical and plane elasticity pro- blems," in: Mechanics [Russian translation],No. 2, Mir, Moscow (1968).

3. E . I . Z a l u z h n a y a and t~. S. U m a n s k i i , P r o b l e m y P r o c h n o s t i , No. 1 (1969).

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