I N T E R A C T I O N O F T H E E L E M E N T S O F

M A T E R I A L R E I N F O R C E D W I T H F I B E R

]~. S. U m a n s k i i

A COMPOSITE

One way to increase dras t ica l ly the specific strength and the rigidity of mater ia ls is the creat ion of composit ions consist ing of h igh-s t rength fibers and binders (matrices) of different types and with different propert ies~

Besides glass plast ics, widely used in construction, in recent yea r s there has been great interest in composit ions consist ing of metallic and ce ramic ma t r i ces re inforced with high-strength thin f ibers and, in par t icular , with whisker c rys ta l s whose strength, as is well known, approaches the theoret ical .

Study of the factors affecting the mechanical behavior of these composit ions is an important task in the general problem of obtaining mater ia ls with given proper t ies .

The present ar t icle , in an approximate formulation, investigates cer ta in questions concerned with the interaction of f ibers and mat r i ces in a unidirectional composite mater ia l . In the elast ic region, the matr ix is re inforced with fibers, due to the substantial difference in the elastic moduli of the components of the mater ia l . In the case of plastic deformation of the matr ix, the mater ia l is re inforced by the f ibers, without any essential difference in the moduli. With elongation of the sample along the f ibers, the load, in both cases , is t ransmi t ted from the mat r ix to the fibers as a resul t of tangential forces ar is ing at their contact surface.

These forces develop mainly due to the elastic bonds between the fiber and the matr ix. In sys tems with poor wettability and poor impregnation, the elastic bonds are weak or do not exist at allo Then, t r ans - fer of forces f rom the matr ix to the f ibers can occur as a resul t of the frict ion forces at the i r separating surface. This is the more t rue since, in many composite mater ia ls , after impregnation there ar ise initial p r e s s u r e s at the surface separating the fiber and the binder, connected with a difference in the Poisson coefficients and in the coefficients of thermal expansion and, in the case of mat r ices made of synthetic res ins , as a resul t of shrinkage during polymerizat ion.

Below, assuming elastic work of the f ibers and the matr ix, and the presence of only frict ion forces between them, an evaluation is made of the s t r e ss distr ibution in the elements of a composi te mater ia l , as a function of the content of f ibers in the mater ia l and of the elastic proper t ies of its components.

Starting f rom different p remises , s imi lar investigations were made by Day; the resul ts are given brief ly by Satton in [1].

1. We consider the uniaxial elongation of a composi te mater ia l re inforced with unidirectional fibers. Such a layer is the main element of many complex composi te mater ia ls . It is assumed that the fibers have identical dimensions, that they are continuous, that they have a round t r ansve r se c r o s s section, and that, like the binder, they are homogeneous, isotropic, and l inear ly elastic. It is assumed also that the f ibers in the layer are distributed in a hexagonal pat tern (Fig. 1, a). Consequently, as charac te r i s t i c repeating elements of the s t ructure , there can be assumed cyl indrical construct ions, consist ing of f ibers surrounded 5y the la rges t possible non-intersect ing round cylindrical surfaces of the binder (Fig. 1). With elongation of the sample in the direct ion of the f ibers, all the s imi lar cyl inders act in an analogous manner.

Thus, the problem reduces to an investigation of the state of s t r e ss of the cyl indrical s t ruc ture (fiber- matrix), loaded at the ends with a uniform normal s t ress , a, applied to the matr ix (Fig. 1, b). We denote by Ff and F M the a reas of the t r ansve r se c ross sections of the fiber and the matr ix making up a cha rac - te r i s t ic element of the s t ructure; df and D M are their outside d iameters .

Institute for Prob lems of Strength, Academy of Sciences of the USSR. Transla ted from Poroshkovaya Metallurgiya, No. 1 (73), pp. 101-107, January, 1969. Original ar t ic le submitted July 3, 1967.

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_M.D . . .

a

"-_ ,//////////z///~ ~ ~ ] ~ v , ~

~.VI/I1711/AIIIIIIIIIIIItlII/qlIZ~ N ,+ dN M

~ a ~ Z - - ~ a

b c

Fig. 1. Hexagonal d is t r ibut ion of f ibe r s and cha rac t e r i s t i c e l e - men t of compos i t e m a t e r i a l .

af/df

5

3 2 t

' b . . . . . . b o o,/ o,3 0,4 r 6,6 0,7 o,0 p

Fig. 2. Dependence of the length of the l o a d - t r a n s m i s - sion sect ion to the f iber on the initial p r e s s u r e .

hal f the length of the f iber , l~

The re la t ionsh ips between the d imens ions of the e l emen t s of a s t ruc tu ra l cy l inder with hexagonal dis t r ibut ion of the f ibe r s a r e eas i ly de te rmined . Thus:

0,907 ,,

where Vf is the vo lumet r i c content of f ibe rs in the m a t e r i a l .

The elongating force P = KFM in the end c r o s s ~ections is r ece ived by the ma t r ix . The end c r o s s sect ions of the f ibers a re f ree of fo rces . Starting with these c r o s s sect ions , the load is t r a n s m i t t e d f rom the m a - t r i x to the f ibe r s by shear ing fo rces at the f i l a m e n t - m a t r i x sepa ra t ion su r f aces .

We a s s u m e that the length of the in te rac t ion sect ion, a, is l e s s than Within the l imi t s of sec t ion a, t he re is mutual s l ippage of the m a t r i x and the

f iber . In the middle pa r t of a f iber of l e n g t h / - 2 a , t he r e a re no tangent ia l s t r e s s e s at the separa t ion s u r - face. In th is section, the f ibers and the m a t r i x a re subject to identical elongations.

With such a formulat ion, the p rob lem is analogous to an invest igat ion of the dis t r ibut ion of fo rces in molded compounds [2, 3].

Let the init ial p r e s s u r e at the sepa ra t ion su r face be fo re the applicat ion of the elongating load be equal to P0. After loading of the compos i t e m a t e r i a l , the p r e s s u r e changes and, in an instantaneous c r o s s sec t ion of the s l ippage sect ion, is equal to p (x). In the sect ion ( / - 2 a ) , where the re is no sl ippage, the p r e s - sure is constant and equal to p (0). Correspondingly , the axial fo rces and the s t r e s s e s in the m a t r i x and the f iber a re designated by N M (x), a M (x), Nf(x), af(x) .

The or ig in of coordina tes is taken at the end of the sl ippage section. We calcula te the radia l d i sp l ace - ments of the contact su r faces of the f iber (uf) and the m a t r i x (UM) connected with a change in the p r e s s u r e by an amount equal to p (x) -P0, and the act ions of the axia l fo rces .

Neglect ing the nonl inear i ty of the d is t r ibut ion of the p r e s s u r e s p (x) within the l imi t s of the s l ippage sec t ion (0_< x -< a), we have:

,% (x) df u (x) = - - 2E M ~ IDa-- d i 2E~,F~, ' (2)

[P0 - - P (x)ldf Nf (x) d f uf(x) = (1 - - ~ 0 2El - - ~f 2F~F~ ' (3)

where ~z M,/~fl EM, Ef a r e the Po i s son coeff ic ients and the e las t ic moduli of the m a t r i x and the f ibers .

E x p r e s s i o n (3) is wr i t ten for a continous f iber .

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af/df 24

2O

14

a

#/[Po/6 -- o

2 ~:~05

2 # • 2025 50 lO0 Ef/E M b

Fig. 3. Dependence of the length of the sect ion of load- t r a n s m i s s i o n to the f iber on the vo lumet r ic content of f ibers in the compos i te ma t e r i a l (a) and the ra t io of the e las t ic moduli of the m a t e r i a l of the f ibers and the m a - t r ix (b).

7

J,O 03. 0,8- 0,7.

0 j o,o;. /,

o,l

o 1 2 3 4 5 • 7 xf/df

Fig. 4. Dis t r ibut ion of tangential s t r e s s e s and no rma l p r e s s u r e s in sect ions of l o a d - t r a n s m i s - sion to the f iber ( E f / E M = 4, Vf= 18.1%): Lines 1, 2, 3, 4, 5, 6, 7 co r respond to P o / ~ = 0, 0.05, 0.1, 0.2, 0.3, 0.5, 1.0.

Since the contact is p r e s e r v e d , then

~M = gf" (4)

Taking into account that , in each c r o s s sect ion

NM(x ) + Nf(x) = P = gF~. (5)

f rom (1)-(5) we find:

p ( x ) = p o - -

EMF,, E.FM ] NM(x) nS - ( + r<f E--Fff-/ e

E M Ff (6)

The re la t ive elongations of the m a t r i x and the f iber along the x axis a re equal to:

dw N M (x) 21-t ~ Ff s (x) = - ~ - E M F . E M F~, [p (x) - - Po],

dwf Nf (x) + 2~fFf el(X) = ~ x = Ef Ff Ef FI [p (x) - - Po].

(7)

(8)

where w M and wf a re the axial d i sp lacemen t s of instantaneous c r o s s sect ions .

F r o m the condition of equi l ibr ium of an e lement of the m a t r i x cut by two adjacent t r a n s v e r s e c r o s s sect ions (Fig. 1, c), neglecting the tangential fo rces of the elast ic bonds in compar i son with the f r ic t ion fo rces at the separa t ion sur face , we have:

d~r (x) dx = q = ~ df fp (x), (9)

where f is the coeff icient of d ry fr ict ion.

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Nf(x) P

.0,6

~e

7

~2 ~$ ~5 ~8 50 x/a i

Taking account of (6), Eq. (9) can be t rans formed thus:

dN~, (X) dx ~lN~(x) ~ ( EMF~' P) = zpoF~ -- ~f ~ �9

Here, for brevity, the following designations are made:

(lo)

E~ Ff E F ;ndf: z = ( l - - v f ) - ~ f + l + ~ t +2~-~-; ; = ~ M + , a f e f F f ; ~ l - • "(11)

Fig. 5. Curves of the distr ibution of the axial forces in t r a n s v e r s e c ro s s sect ions of a f iber inthe slip- page section, at different f iber con- tents in the composi te mater ia l (E f /EM= 4, p0 = 0). Lines 1, 2, 3, 4, 5, 6, 7 cor respond to Vf= 4.3, 8.25, 15.1, 30.2, 45.3, 56.7, 86~

From Eq. (10) we have:

1 E~F. p) N M(x)=Ce n ' - - ~ ( • F --~f ~ . (12)

The constant C is determined from the condition at the bound- a ry of the section, where there is no slippage:

and x = 0; N,, (0) = N,, (13)

1 C N.+--~(upoF M- E F p) = PfE--E-~ " (14)

The force NM is eas i ly found from the condition of the equality of the re la t ive elongations of the fiber and the matr ix in the above section:

~ (0) = ~ # )

From (5)-(8) we find:

EfFf " ~ E--'~ P' N , , = l - - ~ 1 + E~,F, (15)

where the designation is made

2;~e e EM FM / (16)

Introducing (15) into (14), we have:

Thus,

EfFf ] + • 1 7 6 . (1--~,) 1+ E---~]

1 [ (~t'~-~f)P ] Ef Ff I + • e "~ N " ( x ) = T ( 1 - - X ) ( I - [ - E - - - ~ ]

1 EM FM p].

(17)

(1 s)

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The length of the section, a, of the t r ansmiss ion of forces f rom the matr ix to the fiber is determined from the condition at the end of the matrix:

a t x = a

N . (a) = P = ~r . = %(F~ + Ff),

where a m is the mean stress, referred to unit area of a transverse cross section of the sample.

Hence we find:

(19)

af nF,, I*,, + z P-=2

df -- 4~Ff In ~r ~ - - ~f Po (2 0) ( efPf /

The normal s t r e s se s in t r ansve r se c r o s s sections of the matr ix and the fiber in the section 0 _ x _< a are determined from (19) and (5):

. . . . ~- - - - E f F f \ + ~ - enX

~- (1 --~) ~1 + ~ ) (21)

1 [ Po E.F. ] .

; N-~---[~f efFf J' [ _ a,(x) Np (x) (x) (22)

a t (x) = 1 �9 -- 1 . -

The normal p r e s s u r e s and the tangential s t r e s ses at the separat ion surface in the zone of t r a n s m i s - sion of force from the mat r ix to the fiber are found from (21) and (6) or (9):

x 2 3

It must be noted that the quantity (1-M [see (16)] for composi te mater ia ls with more extended com- posit ions is equal to about 0.95-0.97 and, in calculation of the length of the fo rce - t r ansmis s ion section, as well as of the s t r e s se s in the matr ix and the fiber, can be taken equal to unity. This cor responds to ne- glecting the second t e r m s in (7) and (8), that is, the effect of the relat ive elongations along the fiber and the matrix, due to a change in the initial p res su re .

2. The formulas obtained permit evaluating the effect of some factors on the nature of the in te rac- tion between the f ibers and the matr ix with elongation of a sample of a unidirectional mater ia l cut along the f ibers, in the presence of fr ict ion forces only.

The dependence of the length of the section of t r ansmiss ion of the load from the matr ix to the fiber on the relat ive value of the initial p r e s s u r e (formula 20), for a composi te mater ia l with #M = 0.83,/~f= 0.26, Ef/EM = 4,and a volumetric content of f ibers Vf = 18.1%, is shown in Fig. 2. With the absence of an initial p r e s s u r e at the separat ion surface between the mat r ix and the fiber, and of elastic bonds between them, the length of the section of t r ansmiss ion of s t r e s se s f rom the mat r ix to the fiber is compara t ive ly great . For the mater ia l under consideration, with f= 0.15, it is about 50 d iameters of the fiber. With a r i se in the initial p re s su re , the length of the section dec reases rapidly by a hyperbolic law. Thus, with an initial p r e s s u r e amounting to 0.3 of the value of the s t r e s s applied to the sample, the length of the l oad - t r ansmis - sion section is 8.5 df. Thus, in the presence of re la t ively large initial p r e s s u r e s between the fiber and the matr ix, the main par t of the load is t ransmi t ted to the fibers in small sections.

The effect of the volumetr ic content in a composi te mater ia l with the above p a r a m e t e r s on the length of the load- t ransmiss ion section, at different values of the relat ive initial p res su re , can be evaluated from Fig. 3, a. In the absence of an initial p ressure , a change in the volumetr ic content of f ibers has only a slight effect on the length of the slippage section. This effect becomes already ve ry marked at re la t ively

84

small initial p r e s su re s ; with an increase in the content of f ibers in the composi te mater ia l , the length of the load- t r ansmiss ion section dec reases noticeably.

With an increase in the rigidity of the f ibers, the length of the load- t ransmiss ion section increases ; this effect is sharpes t at P0 = 0 (Fig. 3, b). In this case, for a composi te mater ia l with the above pa r ame- te rs , with a change in the rat io of the elastic moduli of the f ibers and the matr ix f rom 1 to 100, the length of the section increases by almost 8 t imes. With a r i se in the initial p res su re , a change in E f / E M affects the value of a less c lear ly . We note that, with an increase in the length of the slippage section, the com- posite mater ia l behaves as a more viscous mater ia l .

The nature of the distr ibution of the normal p r e s s u r e s and tangential s t r e s se s at points of the sec- tion of t r ansmiss ion of forces to the fiber can be evaluated from Fig. 4, on which the ends of the slippage sections at different p r e s s u r e s are located at the origin of coordinates. The curves of the distr ibution of the normal s t r e s se s in t r ansve r se c r o s s sections of a fiber in the slippage section, ai, at different volu- met r ic contents of fibers in the composi te mater ia l are given in Fig. 5.

The effects of the elastic bonds and of uniform heating on the interaction of the elements of a com- posite mater ia l will be examined in subsequent communicat ions.

I.

2.

3.

LITERATURE CITED

V. Satton, Rocket Techniques, Journal Amer . Rocket Soc., 4, (1962). V. I. Feodos 'ev, Selected Prob lems and Questions on the Resis tance of Materials [in Russian], Izd-vo "Nauka," Moscow (1967). N. G. Kalinin, Yu. A. Lebedev, V. I. Lebedeva, et al., Constructional Damping in Fixed Compounds [in Russian], Izd-vo An Latv. SSR (1960).

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