AN ESTIMATE OF THE INTERACTION BETWEEN

AND FIBERS IN A UNIDIRECTIONAL COMPOSITE

MATERIAL UNDER TENSION

. S. Umanskii

MATRIX

UDC 678.029.46

1. A study of the s t r e s sed state of s t ructura l elements, including local effects at positions of s t r e ss t ransfer f rom matr ix to f ibers, is one of the important problems in connection with f iber - re in forced ma- ter ials .

In f iber - re inforced composite mater ia ls the greates t s trength of the matrix, other conditions remain- ing the same, is attained by providing s t rong elastic bonds between the fibers and the matrix. However such bonds may be weak or al together absent. Such is the case when there is poor wetting of the fibers, imperfect impregnation, incomplete polymerizat ion (in the case of res in binders), the presence of voids, bubbles, etc. it is well-known, for example, that oxide fibers, even when purified, are, in the majori ty of cases, not wettable by metals [2]. Similar sys tems are those involving soft plastics re inforced with a fibrous backing, where somet imes complete impregnation of the fibers with a plast icized polymeric coating cannot be provided. We r e m a r k that adhesion of fibers to binder is less effective at increased t empera - tures .

For cases in which a composi te material is under tension the t ransfer of toad f rom matr ix to fibers may be realized at the expense of frictional forces at the surface of separat ion between s t ruc tura l elements.

We give below an approximate t reatment for estimating the nature of the interaction of matr ix and fibers in the case of a uniaxial extension of a unidirectional composite material , wherein we assume that fr~ct~onal and weak elastic bonds operate simultaneously.

We assume that a tensional s t r e ss is applied at the end sections of the matrix. F o r the fibers we assume that they are continuous, have c i r cu la r c ross sections, are of the same dimensions, and that the binder is homogeneous, isotropic, and l inear ly-e las t ic . In addition we assume that the fibers in a layer a re hexagonally-distr ibuted (see Fig. la). Then as charac te r i s t i c repetit ive elements of the s t ruc ture we may take component cyl inders , consist ing of fibers surrounded by the la rges t possible nonintersecting cy- l indrical surfaces of the binder. When a specimen is put under tension in the fiber direction, all s imi la r cyl inders will be subjected to analogous conditions. Consequently, the problem reduces to a study of the s t r e s sed state of a component cyl inder (f iber-matrix), loaded at its ends by a uniform normal s t r e s s applied to the matr ix (see Fig. lb).

Let F F and F M be the c ross - sec t iona l a reas of f iber and matrix, which form a charac te r i s t ic e le- ment of the s t ructure; let d F and D M be their corresponding outside d iameters .

For a hexagonal distribution of fibers we have

D~ 0.952 dF 1/ v F

(v F is the fiber volume content in the material) .

F,~ 0.907 (1.1)

F p v F

The tension force P = ~FM applied to a charac te r i s t i c element at its end sections is taken up by the matrix. The end sections of the fibers are s t r e s s free. Starting f rom these sections, thetoad is t r ans fe r red

Institute of Problems of Strength, Academy of Sciences of the Ukrainian SSR. Translated f rom Pr ik - ladnaya Mekhanika, Vol. 6, No. 4, pp. 59-66, April, 1970. Original ar t ic le submitted July 4, 1968.

�9 1973 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 for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

382

~,,~'~ I ir.~.~/////~/////,v///.-,~. r

a b r

Fig. 1

f rom matr ix to f ibers at the expense of shear s t r e s ses on the surfaces of separation of fibers and matrix. These s t r e s se s are aroused at the expense of elastic bonds between the f ibers and the matrix, and also at the expense of fr ict ion on the surfaces separat ing them when the elastic bonds are weak or altogether ab- sent.

We cons ider the case in which the interaction between fibers and matr ix is real ized by means of e las - tic and frictional constraints simultaneously. This is possible, for example, whenever the elastic bonds are of an intermittent nature along the length of the fiber; however their number is large and they may be considered to be distributed over the whole length of the fiber.

We remark that cer ta in s t ruc tura l damping problems have been solved in a s imi lar setting [2].

We assume that the length a of the load- t rans fe r portion of the fiber is less than half the length l of the fiber, and that the tangential s t r e s s e s act only over the length of this portion. The middle portion of the fiber, of length l -2a , is free of tangential s t r e s ses over the surface separat ing it f rom the matrix. Over this portion the f iber and matr ix have the same elongations.

Up to the t ime of application of the tensional load the initial p r e s su re on the surface of separat ion is taken to be equal to P0- With loading of the composite this p re s su re changes and becomes equal to p(x) at a variable section of the f iber portion of length a. On the portion o f l en~h l -2a) the p ressu re stays con- stant at p(0). We denote the axial forces and s t r e s s e s in matr ix and fiber by NM(X), crM(x), NF(X), O'F(X), respect ively. We take the coordinate origin at the end of the portion of tangential s t r e s s concentrat ions (Fig. lb).

We calculate the radial displacements of the contact surfaces between matr ix and fiber, cha rac t e r - ized by a change in p r e s s u r e of magni tudep(x)-P0 and by the presence of axial forces NM(X ) and NF(X), and also by uniform heating to an excess temperature T. In our approximate treatment, neglecting the nonlineari ty in the distr ibution of p re s su res over the portion 0 _< x _< a and retaining the hypothesis of planar sections, we have

where #M, #F, EM, of l inear expansion for matr ix and fiber.

Express ion (1.3) has been written for a continuous fiber.

Since contact is maintained during loading, we have

u,, (x) = uF(x) .

Taking into account that on each section

NM (x ) -[- NF (x) = P = ~ F M =- ~av (FM + FF),

we obtain, f rom the formulas (1.2)-(1.5),

D M q- d~ ) N M (x) d F d F u,~(x) = [P0 - - P (x)l d F 2 2 ~-~ ~ - - - ~ q-~, --~,, 2E,~F------~q-~,T--~- , (1.2)

M F

~(x) =(1--~p)[P~ F ~F NF(x)dF2EF------~F + ~FTd~f2 ' (1.3)

EF, tiM, flF are, respect ively, Poisson ratios, moduli of elasticity, and coefficients

(1.4)

(1.5)

(1.6)

383

Here

- E~, F F . E,~ F~,

The re la t ive elongations of ma t r ix and f iber in the d i rec t ion of the x -ax i s , taking into account the s impl i f ica t ions agreed upon, a re given by

dwM NM (x) 8M(x) = dx --'-- EMF ~ 2 ,, M [p(x)--p(O)] q-[~,=T; (1.7)

dWF NF(X) PFFF (0)] + (1.8) ~F (x) = - 'dT- =-~--e-~F + 2 [p(x) - p

where w M and w F are axial d i sp lacements of va r iab le sec t ions .

If in the fo rmulas (1.7), (1.8} we neglect the influence on the elongation of the f iber and mat r ix of the change in the initial p r e s s u r e on the su r face of contact , we obtain

dw M N,,(x) d'WF -- NF(X) q-~}F T. (1.9) 8. (x) = ~ = ~ + ~ T; ~(x) = - ~ - -

The condition for the equilibrium of the matrix element cut off by two adjacent cross sections (see Fig. lb) has the form

dN~ (x) ~tx = t + q , (1.10)

where t = c(w M - WF) n/m is the intensi ty of reac t ions of the longitudinal e last ic const ra in ts ; c is the co- efficient of r igidi ty of these cons t ra in t s in n/m2; q = ~rdFfP(x ) is the intensi ty of the f r ic t ional fo rces b e - tween f iber and ma t r ix in zones where s l ippage occurs; f is the coefficient of d ry fr ict ion.

Substituting into Eq. (1.10) these express ions for t and q, and taking re la t ion (1.6)into account, we may wri te

dN.dx =c(w. --wF) + ~d~f Po- ~ [ eFe F P - ~--~. +(I~,,--ISF)Te,, . (1.11)

Different iat ing this equation with r e spec t to x and using Eqs. (1.7), (1.8), and (1.5), we obtain

Here

2=c(I--~')( E F M]. 2~ 2F F ,1 = % t , 1 + =

F r o m Eq. (1.12) we obtain

N,,(x) = e T (Dlchq~x+D2sh(px)+ " EFF F -~ E ~ F P - - E~,F~

1-F E . F . + EFF F

1/:r ~p = __s + r (1.13)

The constants D I and D2, and the length of the zone in which the tangential stresses are concentrated, may be determined from the following conditions:

N~ ----/~; (1.14)

w . = w F for x-----0; (1.15)

N ~ ( a ) = P for x = a . (1.16)

384

The axial force 1ql M at the section x = 0 of the matrix, on the boundary of the portion where the load is t ransfer red to the fiber, may be determined f rom the condition CM(0) = SF(0 ). We find, as a result , using expressions (1.7), (1.8), that

/ ~ = N. (0) = ~11 [( l l + k ~'~F')P--([t~*--[~F)TEJ.(I-~k + k + ~")] " (1.17)

Here

k = EFFF E, F ~, "

Consequently,

D1 = O;

(1 - - ~.),~ / k Po F~ ~ / 1 + k -}-+) (fs~, - - [~) TE,,F,,] ;

~q [ (Ixu--I~)P N~, (x) = ~ (1 -- X) ~ 1 + le + (I -- ~) XPo Fu 1 + k ([~,, --[{:) X TE~F MJ e'-C'shq;x

1 ~-/x F k TE M F t] + 1-'~1Z [(1 + k ~ " ) P -- ( 1 - - ~ + ~ - ) (~. -- ~F) J �9 (1.18)

The length of the zone of tangential s t r e s s concentrations is determined f rom formulas (1.16), (1.!8)"

[ ( ) -r- (I-~)r ~ . -~F]p ;k e shcpa= i T k ' | + 1 - - ~ -[-k ([~"--[~rJTE'*F*

,a t ~ . - ~ ) , k ; }. (1.1o) q~ 1 + k k-(1--~')xpoF--] ' -~ [l+tx~'+(l+~g)k](fj,* ~FJTE~,F*

If we neglect the relative elongations of fiber and matrix, character ized by the change in pressure on the surface separat ing them, i . e . , if in solving the problem we s ta r t out with the expressions (1.9)i then the formulas for NM(X ) and for a may be determined f rom Eqs. (1.18) and (1.19) as ~.--- 0. In the absence of elastic bonds the interaction of matr ix and fibers is realized at the expense of frictional forces on the surface of coat:act. In this case, c = 0, q~ = 1/2r?, and for T = 0 the expression for the length of the zone of concentration of tangential s t r e s se s takes the form

1 ~- l+'F~-. ~v +~po a ~--- ~ In (1.20)

' " + ~ P o (1 --x) 0 + k) The formula for the axial force in a fiber section on the portion (0 _< x _< a) may be determined from Eqs. (1.5), (1.18):

cp(1 ~,);[ l - i - -+~ +(1 ~,)xpoF~,-- [ ( l + A x ~ ' ) k - k - l + ~ 3 ( ~ " - - ~ r j r E " G - - 1 -[- k e shepx. (1.21)

We obtain the pressure on the surface of contact in the zone of tangential s t r ess concentration f rom the for - mulas (1.6), 0.18):

p (x) =po + (1 -- s " (1 + k) F M 1 -~ ~ (~. - - [~F) TE**

_~_.[F, __~it~ p (l_[_l~,)k+l_.[_ttF ] nx } + 1 + 5 F~, +( l - -Z)~po 1 + k :([~,--~)TE,, eTshcpx �9 (1.22)

385

60 60

0 ........ 0.1 0.2 0.a o.4 ~ .0 0.05 0.10 0.15 2c. E~ 6av

F ig . 2 F i g . 3

4 ,

illlr 2

o.1 0.2 p,,-/zF

F i g . 4 F ig . 5

2 !

4 6 aF/d F

1,0

a4

0 0.25 0.50

The l a r g e s t n o r m a l p r e s s u r e wi l l be at an end of a c h a r a c t e r i s t i c e l e m e n t of the c o m p o s i t e (x = a)

p(a) = p0 + r e . l . (1.23)

~ ) /i In a c c o r d with the s i m p l i f i c a t i o n s a g r e e d upon, the m a x i m u m ~ , ; r c i r c u m f e r e n t i a l s t r e s s e s in the m a t r i x at the s e c t i o n ind i ca t ed is g iven

.4 ~--~x~ by

The t ange n t i a l s t r e s s e s on the s u r f a c e of c on t a c t in the zone of t h e i r c o n c e n t r a t i o n a r e o b t a i n a b l e f r o m Eqs . (1.10) and (1.18):

0.75 x

F i g . 6 Z---- ~d F - - - ~ Po '+ ( 1 - - ~ , ) ( 1 - I - k ) u

) x e -r" T shq~x + ~ c h ~ x . (1.25)

The a p p r o x i m a t e s o l u t i o n g iven in 1 m a k e s i t p o s s i b l e to e s t i m a t e the in f luence of c e r t a i n f a c t o r s 2. on the c h a r a c t e r of the i n t e r a c t i o n be tween m a t r i x and f i b e r s in the un iax ia l e longa t ion of a u n i d i r e c t i o n a l c o m p o s i t e m a t e r i a l wi th weak e l a s t i c bonds .

In F i g . 2 we show the d e p e n d e n c e of the l eng th of the zone in which s t r e s s e s a r e t r a n s f e r r e d f r o m m a t r i x to f i b e r on the r i g i d i t y of the e l a s t i c bonds fo r two c o m p o s i t e s with f i b e r v o l u m e c o n c e n t r a t i o n s of 22.7% (Curve 1) and 56.7% (Curve 2).

In the c a l c u l a t i o n s we have used E F / E M = 4; f = 0.1; #M = 0.33; P F = 0,26; P0 = 0.

386

In the case where the elast ic constra ints opposing the longitudinal displacements are lacking, the length of the portion of f iber where load t ransfer takes place is compara t ive ly large, and for the composites in question amounts to about 70 fiber d iameters . As the rigidity of the bonds increases up to a value c / E M = 0.05, the length of the zone of tangential s t r e s s concentrat ion at the surface separat ing fiber and matr ix dec reases sharply, af ter which it s tabil izes, amounting to (6-10)d F.

The influence of the initial p r e s s u r e on the surface separat ing f ibers and matr ix on the length of the zone metnioned is especial ly large when the elast ic bonds are small or absent. As the initial p r e s su re grows, the portion of length a, varying according to a hyperbolic law, decreases rapidly. Thus for a com- posite with pa ramete r s as indicated (v F = 56.7%), the length of the portion where s t r e ss t r ans fe r occurs on the f iber dec reases a lmost fivefold in the presence of comparat ively small p res su res (on the order of 0.1- 0.15 of the magnitude of the tensile load applied to the specimen) [see Fig. 3, Curve 1 (c/E M = 0), and Curve 2 (c/E M = 0.001)]. Consequently, when ra the r small initial p re s su res exist between the fibers and the matrix, matr ix strengthening under tension of a composite can be achieved only at the expense of f r i c - tional const ra ints .

As the longitudinal elastic bonds increase in strength, the role of the frict ion forces diminishes and, c0nsequently, so does the influence of the initial p r e s su re on the magnitude a [Curve 3 (e/E M = 0.05) and Curve 4 ( c / E M = 0.5)1.

The effect on the length of the zone of tangential s t r e s s concentration obtained as a resul t of varying the Poisson coefficients of the matr ix and fiber mater ia ls is shown in Fig. 4. Here the Curves 1 and 2 c o r - respond to composi tes with v F = 22.7% and v F = 56.7% for P0 = 0, while the Curves 3, 4 correspond to composi tes of the same composit ions but with PJ~av = 0.05. Calculations were made for PM = 0.33 when frict ional const ra ints only were present .

In Fig. 5 curves showing the distribution of tangential s t r e s s e s on the surfaces separat ing fibers and matr ix as a function of the r igidity of the elast ic bonds are presented (VF = 56.7%; P0 = 0), where the ends of the portions of concentrat ion are located at the origin. The Curves 1, 2, 3, 4, 5 correspond to the values of c/Ely I = 0; 0.001; 0.05; 0.1; 0.2.

In Fig. 6 we presen t curves showing the distribution of axial s t r e s ses in f iber c ross sections (solid curves) and in matr ix c ros s sect ions (dashed curves) for the composi te mentioned (Curves 1, 2, 3, 4 c o r - respond to c /E M = 0; 0.001; 0.05, and 0.5).

Thus, with an increase in rigidity of the elastic bonds there is a substantial decrease in the length of the zone where a t r ans fe r of s t r e s s e s f rom matr ix to f iber takes place, and there is a sharp increase in the magnitude of the tangential s t r e s se s at the ends of the f ibers , resul t ing in a more uniform dis t r ibu- lion of normal s t r e s se s .

1.

2.

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

N. G. Kalinin, Yu. A. Lebedev, V. I. Lebedev, Ya. G. Panovko, and G. I. Strakhov, Structural Damping in Fixed Assembl ies [in Russian], Izd-vo Akad. Nauk Latv. SSR, Riga (1960). W. Sutton and J. Chorne, "Potential of oxide-f iber re inforced meta ls ," in: Fiber Composite Mater i- ais, Amer ican Society for Metals Seminar, Metals Park, Ohio (October, 1964).

387