The influence of fibre discontinuities on the stress-strain behaviour of composites

THE INFLUENCE OF FIBRE DISCONTINUITIES ON THE STRESS-STRAIN BEHAVIOUR OF COMPOSITES B. SCHULTRICH

Sektion Physik, Technische Universitiit Dresden (DDR) and W. POMPE and H.-J. WEISS

Zentralinstitut fiir Festkiirperphysik und Werkstofforschung der Akademie der Wissenschaften der DDR, Dresden (DDR)

S UMMA R Y

It is attempted to calculate the a(e)-curve of short-fibre composites by considering regular arrays of plates in a ductile matrix. Several quantities of interest, such as stress variation along the fibre, Young's modulus, and yield stress, are calculated as functions of the parameters of material and structure of the composite. Among the latter, the overlap of the -fibres may affect the properties strongly. The change of composite behaviour from mainly elastic to yield may occur in several ways, depending on the parameters.

LIST OF SYMBOLS

u, tr, e, E = displacement, stress, strain and Young's m o d u l u s in the axial direction, respectively ~,, z, G = shear angle, shear stress, shear modulus, respectively g, h, 2 = auxiliary lengths Subscripts F, M, C o f fibre, matrix and composite, respectively v = volume fraction o f reinforcement k, x = inverse decay lengths l = length of fibre r, d, R, D = lateral dimensions o f the c o m p o n e n t s a = overlap length z = axial coordinate Bars denote cross-sectional average Asterisks denote yield 1 Fibre Science and Technology (11) (1978)--© Applied Science Publishers Ltd, England, 1978 Printed in Great Britain

2

B. SCHULTRICH, W. POMPE, H.-J. WEISS

INTRODUCTION

The behaviour of composites depends on a variety of parameters, such as the properties of the components, bonding between the components, alignment of fibres, statistical distribution of strength and so on. In order to optimise the parameters one should know, at least qualitatively, their part in the generation of internal stresses and the onset of local plastic flow, by which the gross properties of the composites are finally determined. The theoretical methods usually applied to tackle these problems can be divided into two complementary families, viz. the numerical and analytical methods. The numerical approach provides more or less exact solutions of the stress fields for special sets of parameters. Thus there are numerous detailed results for elastic or elastic-plastic fibrous composites. The information that one can get from them is limited, of course, due to the special geometry, etc. The analytical approach, on the other hand, makes use of comparatively crude models which enable the problem to be treated in a more general manner and to attain finally some mathematical expression which contains all the parameters of interest. The most simple and widely used models of this kind, following Cox I and Dow, 2 start from a characteristic composite element containing one fibre only embedded in a corresponding amount of matrix, which is envisaged as showing all the essential properties of the whole composite. The interaction between the fibres had been taken into account first by Hedgepeth and van D y k e ) - 5 They calculated stress increase in the fibres due to broken neighbouring fibres in order to obtain information on the mode of crack propagation.6 The only thing the matrix has to do in this model is to transfer the load from one fibre to another by means of shear stresses. The ability of the matrix to carry load by itself has been neglected. In this paper we investigate the elastic and plastic deformation behaviour of an aligned fibrous composite taking into account the interaction between neighbouring fibres. The one-fibre model is briefly treated to present the formalism which is applied to a two-dimensional multi-fibre model. The model consists of aligned discontinuous fibres embedded in a matrix, the load being applied in the direction of fibres. At first, no restrictions are imposed on the material properties of fibre, matrix and interface. Radial stresses and strains are neglected. The model is used to obtain results for the cases of elastic matrix and plastic matrix, the interface being perfectly bonded. The array of the fibres is characterised by the volume fraction and shape of the fibres, and further by an additional parameter called the overlap, which affects the stress concentrations at fibre ends and consequently the beginning of flow. ONE-FIBRE MODELS

These widely-used kinds of models are based on a representative composite element consisting of one fibre (radius r, length 1) embedded in a cylindrical piece of matrix

3

INFLUENCE OF FIBRE DISCONTINUITIES ON STRESS-STRAIN BEHAVIOUR

(radius R). The two radii determine the volume fraction of fibres: ( r / R ) 2 = v. Such a model of an elastic composite had been developed at first by Cox, 1 and later formulated more correctly by Dow. 2 The general formalism presented below aptly describes composites without any restrictions to the material laws of fibre, matrix and the properties of the interface. (Detailed calculations, however, are carried out for more special cases later.) The basic idea consists in a description of local deformation by average displacement on cross sections of fibre and matrix, fir and ffM, as well as by the displacement of fibre and matrix at the interface, uF and u M: t~tir g ~ = c~z

gM-

~uM c~z

Y~-

uF - fie hr

YM-

uM - uM hM

(1)

Here h r and h M are characteristic lengths comparable with r and R. Various expressions for h v and h M are given by Cox 1 and Dow; 2 they are approximately derived from an elastic calculation by Schultrich. 7 With the deformations known, one gets the stresses with the help of the material laws:

~r[gr], rr[~r], ~M[g~], ~ [ ~ ] ,

~[u~-uA

(2)

The bars above the symbols denote cross-sectional averages, z is the shear stress at the interface and u M - u r a possible discontinuity of displacement at the interface. The equilibrium conditions at each site along the composite element are in the absence of outside shear 3 3z 6v

--

2 rr

3 8z°U -

2r R2

_

rE z

2 r

v 1- v~

(3)

Obviously, these relations must then be compatible with the trivial balance of forces called the rule of mixtures: 6 ( z ) = v f v ( z ) + (1 - v ) 6 M ( z ) = a c

(4)

Besides the absence of shear there is another boundary condition for the outer boundary of the element: the end faces of the fibre are stress free, i.e. applied stress is acting on the matrix only.

This makes sense because of the fact that, in real composites, the fibre end faces are often debonded on account of stress concentrations due to sharp edges, or carry a negligible load because of the smallness of area. In the elastic perfectly bonded composite (i.e. up = u u ) the shear stress, because of: av = Evgv

zv = Gp?v

6 u = EMgM

can be expressed immediately by the mean displacements

zu = GuYu

(6)

4

B. SCHULTRICH, W. POMPE, H.-J. WEISS

G z = ~- (tiM - fie)

G GvGM h = GvhM + GuhF

(7)

The differential eqns. (3), together with the boundary conditions ofeqns. (4) and (5), then provide solutions of the well-known hyperbolic-function type: Ev( 6 v = ac E .1

coshKz ~ c~sh~cT2]

~¢r E v coshxz z = ~-a~ ~ cosh x//2

(S)

the characteristic length 1/K being defined by K =-

2

GL"

rh (I - v ) E u E F

E =_ vE v + (1 - v)E M

(9)

This length is in any case comparable with ( R - r), thus agreeing with St Venant's principle. The tensile stress in sufficiently long fibres has the value acEF/F, along nearly all of their length, only at a distance 1/x from the ends it begins to decrease towards zero. The shear stress, on the other hand, vanishes nearly everywhere, except within a range of length 1/x at either end, where it reaches its maximum value of: Z'max

xr a - = tanh 2 E

l ~~ 2

y

O'F,max

In order to describe the beginning of flow, Kelly and Tyson s suggest a useful and more simple approach: in the plastic regions which grow from the ends towards the middle of the fibre the interface shear stress can be approximately set equal to the critical shear stress %. In the remaining elastic region (0 < z < g) the shear stress decreases towards zero within an interval l/x, that is small compared with fibre length 1, provided the fibres are not too short. For this reason one can put z = 0 in the elastic region which corresponds to neglecting the strain differences of fibre and matrix, i.e. gr = gu. Hence we get for the stresses in the elastic and plastic regions: ar=a~Ev/E l-z 6F = 2%-t

for

O
for

l g _< z _< ~

(10)

The extent of plastic flow follows from the matching condition of stresses at z = g: l 2

g-2

rEvac ~Tzo

(11)

One-fibre models have been used to tackle more complicated problems as matrix stability during cold and hot working, 9 composite creep and relaxation 1° and interface slip during cyclic loading.*1

INFLUENCE OF FIBRE DISCONTINUITIES ON STRESS-STRAIN BEHAVIOUR

5

MULTI-FIBRE MODEL

In the one-fibre models, load redistribution between fibre and matrix only is taken into account. It is obvious, however, that the matrix should be able to transfer load from one fibre to another. Rosen 12 tried to take this interaction with the neighbourhood into consideration by placing the composite element into a homogeneous medium having the effective material properties of the composite. However, the formalism presented 12 is inconsistent, for it ascribes to the matrix the ability of carrying tensile load by assuming E e l f = v g F + ( 1 - V)EM, but neglects this ability within the composite element. If the material laws of the components are nonlinear, the method of placing the composite element into effective material leads to difficulties, because in that case the effective material law is not uniquely defined, if no additional assumptions are made. This is why we prefer to follow the course of Hedgepeth and van Dyke. 3-5 To this end, we apply the formalism already described to an arrangement of plates embedded in a matrix as shown in Fig. l(a), which may be regarded as a two-dimensional fibrous composite. This somewhat artificial arrangement may be regarded as a result of random fracture of continuous fibres. The deformation (see earlier) is described by the displacements of fibre and matrix, but here we need three figures for every layer: two for the upper and the a)

II

II ~Z

I

- \

/~2

d.

q~2

II

b)

U

II

I[ II

IE /'l"f

II -I/2 Fig. 1.

o

i/2

Arrays of fibres which our investigations are based on: (a) fibre discontinuities irregularly distributed; (b) regular array of fibres of length l.

6

I~. S C H U L T R I C H ,

W. POMPE,

H,-J. WEISS

lower side labelled by + and - , and one cross-sectional average: u~, u~, fir, uM, + u~, tiM. The displacement may be discontinuous at the interface. Disregarding radial displacements we get for the longitudinal tensile strain e = tnu/i'.:, and for the shear strain: y~

u~

_

÷ 7r

hM

y~

=

?i

-

u [ - ar -

(12)

hM fie - u ;

h~

hr

These deformations are related to stresses via the material laws of fibre, matrix and interface. The condition of,equilibrium takes the form: a

~-~ 6r =

r~

--

d

q

a

rM + --

~z 8~ --=

"~M

O

(13)

d and D denoting the thickness of fibre and matrix, respectively. The trivial balance of forces and the condition of stress-free fibre ends provide the boundary conditions of the problem: ~(z) = const. (external load)

a e = 0 at fibre ends

(14)

In a composite of high modulus ratio and sufficiently long fibres, where matrix tensile stress can be neglected, this model converges with that of Hedgepeth and van Dyke.3-5 ¢fM= 0 leads to -

rM = rM

+

2h M

In the special type of array with all the fibre ends at the same site z there is no load transfer between fibres for symmetry reasons, and the one-fibre model described earlier is adequate.

COMPOSITE

WITH

ELASTIC

MATRIX

In this chapter the formalism developed above is applied to a more regular arrangement of fibres (Fig. lb). Only one additional parameter is needed which describes the overlap of fibre ends. The overlap parameter a may assume values between l / 2 and 0, the case a = 0 corresponding to the one-fibre model without load transfer between fibres. Because of symmetry there are identical stresses versus z in every fibre. Putting the origin of the z-axis into the middle of the fibre number one, we get the stresses

INFLUENCE OF FIBRE DISCONTINUITIES ON STRESS--STRAIN BEHAV1OUR

~F,(Z) = ~,(z*)

~2(z) = Tdz*)

~M(z) = ~M(z*)

7

(15)

z*=_l-z-a If matrix and fibres are completely elastic and the interface perfectly bonding, we get the relations: O +2r~ OzaF~= -- d

6M = EMgM

-

T2 - - ~'1

Oz a M =

0

D

Z~ = ~ ( U M -- tir,)

GEl2

Zz =

(tie, -- tiM)

(16)

(the twofold sign + being related to the index numbers ,~, labelling the fibres) which lead immediately to a system of equations: 62 c~z2 gF~ + k2(gM -- eFt) = 0 62

OZ2 gM + ( ~c2 -- k2)

(gF' "~ ~Fa 2

gM) ~" O

(17)

Here l/~c and 1/k are characteristic lengths, 1/x being of the same order of magnitude as the fibre diameter, or rather the 'matrix diameter', whereas 1/k is of roughly the same size multiplied by the square root of the ratio of fibre and matrix moduli: k2

G .

.

.

2 .

E F hd

G 2 /£2 = k 2 + _ _ _ _ E M hD

(18a)

With h r ,~ d/2 and h M '~ D/2 after Schultrich and Weissbarth, 7 G/h from eqn. (7) reduces to: G = 2 vGMG v h d(l-v)G v+vG M

(18b)

The general solution of eqn. (17) is: gr~ = A exp(~cz) + Bexp ( - x z ) + Cz + D +_ E e x p ( k z ) +_ F e x p ( - k z ) gM -

v

Er

I-vEM

(A exp(xz) + B e x p ( - x z ) )

+ Cz + D

(.19)

The 'crack' in the neighbouring fibre number 2 at z = l/2 - a causes a discontinuity in tiF2 and as a consequence in (O/t~z)6 M. Therefore the solutions to eqn. (19) must be fitted together piecewise, with the condition that 6 M has to be continuous and 6F. has to be smooth at z = 1/2 - a.

8

B. SCHULTRICH, W. POMPE. H - J . WEISS

Together with the boundary conditions (14) and the symmetry (15) this leads to the solutions: K cosh ~(z + a/2) H sinhk(z + a/2)'~ H + K c o s h t c ( l - a)/2 + H + ~ s i n h k ( l - - a ) / 2 ] -I K cosh x(z + (a - •)/2) H sinhk(z + (a - l)/2)'~ H + K cosh xa/2 + H + K ~mhka/2 ) ac~-

I+

K2 - k 2 k2

K

coshK(z_+ a / 2 ) )

H+ Kcoshx(l

a)/21

~:2_k 2

K

coshK(z+(a-l)/2))

k2

H + K

cosh h'a/2

xd K sinhx(z + a/2) 2 H+ gcosh~c(l-a)/2 rd K 2 H+K

J

kd H coshk(z + a/2)'~ 2 n + K ~ S a ~

)

sinhx(z + ( a - l)/2) kd H coshka/2 + 2 H+K

coshk(z + ( a - l)/2)'~ ~mhka/72

)

(20) The upper of the two alternative lines of each formula refers to sites -l/2 < z < (1/2 - a), and the lower to (1/2 - a) < z < l/2. The twofold sign __+ is related to the index numbers ~,as in eqn. (16). The abbreviations H and K are defined by H - r(tanh Ka/2 + tanh x(l - a)/2) K = k ( c o t h k a / 2 + c o t h k ( l - a)/2)

(21)

The balance of forces, of course, is guaranteed by eqn. (20). For materials with Poisson's number v = I, which is approximately valid for most metals, x and k can be expressed by Young's moduli of fibre and matrix: 1 = 1 =

,

d/2 1 -v ~/3" v

(1-v)E v+vE M vE t . + ( 1 - v ) E M

dx/2 (1-v)E v+vE M --~ vE M

(22)

The meaning of k and x can be derived from the solutions (20). 1/K is undoubtedly the decay length of mean matrix tensile stress 6 M. Every change in matrix stress must go along with a corresponding change in fibre stress, therefore r can be met with in the formula for 6r too. The existence of an additional decay length 1/k in ~r hints at another type of stress concentration, the nature of which can be understood easily by looking at the limiting case EF/E M ---, oo:

INFLUENCE OF FIBRE DISCONTINUITIES ON STRESS-STRAIN BEHAVIOUR

=

k = X/3 "X/EMx//-D-~

for

E r / E M --* ~

9

(23)

If load is applied to such a composite, the fibres in the array of Fig. l(b) are pulled apart from each other by some small distance. Thus, in the vicinity of fibre ends the matrix is subjected to tensile strain. The spatial extent of this strain concentration should be comparable with the distance of fibres D, so we see immediately that 1/x is the right length for this purpose. Because of the mutual displacement of neighbouring fibres, there is non-decaying pure shear in the remaining part of the matrix. Thus it may be understood why one of the decay lengths, 1/k, can be much larger than the other and even tends to infinity. Several examples for mean tensile and shear stress and their variation along the fibre are illustrated in Fig. 2. If the overlap a is somewhat larger than 1/x, fibre stress peaks arise due to the discontinuities of neighbouring fibres. To discuss them we

|

i

.aM/~ ~

b~

cJ

/T

¥',,e

Rs

F -Q5

|

.|

i

Ior~fh of fibre

Fig. 2. Stresses along the fibre according to eqn. (20), dashed lines representing the one-fibre model: (a) mean axial fibre tensile stress; (b) mean axial matrix tensile stress; (c) interface shear stress.

10

B. SCHULTRICH, W. POMPE, H.-J. WEISS

look at the case of values a much larger than 1/~ and l/k, where H and K in eqns. (20) reduce to 2~c and 2k. Thus we get a maximum fibre stress of: Ep ar . . . . = ac ~ - d.

~:k

(24a)

K+k

and a m a x i m u m shear stress of: ma, = " < - f " 7" r + k

-

(24b)

The latter peak does not arise exactly in front of the discontinuity, where z equals zero, but at a distance of _ 1/(r - k). In (r/k) ~ _+(l/K). In the limit of a very low modulus matrix, i.e. 1/r ~ Ilk, and a moderately large overlap, so that ra ~> 1 and ka ~ l, we get for the artificial array of Fig. l(b) a m a x i m u m fibre tensile stress 6~,max = (2/v)a o provided v does not tend to zero. The result means that half the number of fibres carries the whole load, as expected in this special case. If a < 1/~, no additional stress peaks arise. Then the results agree with those of the one-fibre model. Equation (20), for instance, coincides for a = 0 with the twodimensional form of eqn. (8). The fibre overlap, as a typical composite parameter, becomes the more important the more the material is 'composite-like', i.e. the more fibre and matrix differ in their properties. Thus, large differences in the results of this model and the simple one-fibre model arise in cases of high modulus ratio, provided a is not too small. In our model the load transfer from the site of a fibre discontinuity to a neighbouring fibre, which is essential for the strength of the composite, can be seen immediately as a h u m p in the 6r(z)-plot of the latter fibre. In materials with fibres arranged so that a = 0, there is no load transfer between fibres, and such a material can hardly be called a composite. A situation similar to our case a = 0 might occur if neighbouring fibres break successively at the same site z under the stress transferred from the previously broken fibre. The overlap parameter a, of course, affects not only the internal stress fields, but the effective modulus as well. The effective Young's modulus is defined by the ratio between external load and mean strain in the direction of z-axis. The latter is obtained by averaging the matrix tensile strain from eqn. (20) along the fibre. Thus we get Ec, the effective Young's modulus of the composite

Ec=J~

I +7.

xZk2

"/-I+K

(25)

with the symbols defined by eqns. (21) and (22). In the limit of very long fibres, of course, we get the modulus of the continuous fibre composite /~. The modulus of the discontinuous fibre composite must obviously be lower than that, depending on the overlap parameter. This is visualised in Figs. 3 and 4, where the ratio of Young's moduli of short fibre composites and

INFLUENCE OF FIBRE DISCONTINUITIES ON STRESS-STRAIN BEHAVIOUR

T

11

2O

qe

ID

qs

_

q6

qs ot EM

2

q~

O2

I

I

I

I

q2

q~

0,6

08

~-E c) EF=,OOEN a.~

I

q2

d

,,~

I

qo

,

~]

I

qa

~,

Fig. 3. Ratio of short fibre composite modulus and continuous fibre composite modulus in axial tension, plotted against volume fraction of fibres, the aspect ratio being parameter of the curve family.

continuous fibre composites is plotted against the volume fraction of fibres. Figure 4 shows that the elastic properties are influenced by the overlap mainly in composites with large modulus ratio. The surprisingly small effect of poor overlap as seen in Fig. 4(a) does not mean that due overlap is not important for properties other than elastic, as will be shown in the following section.

COMPOSITE WITH ELASTIC-PLASTIC MATRIX

As soon as the matrix has been partly plastified by internal stresses due to external

12

B. S C H U L T R I C H , W. P O M P E , H.-J. WEISS

-

z5

-

E, =I0

b)

i

1=20

qe o,e

Er=3

o)

o,~

I ~---2o

o,2

o,2 I

I

q2

~

I

q6 Ec E

T

I

i

~e

o.2

c)

£e.m

I

oA

i

|

q~

qa

v

t

0,e O,6 0.~

i

q2

q~

o,6

!

0,8 v

Fig. 4. Effect of fibre overlap on composite modulus. load, the situation becomes rather involved.16 Nevertheless, we try to analyse it, at least in the case of low modulus ratio, which is realised with metallic composites. Then the decay length 1/k is nearly of the size of the distance of fibres, 1/K being even smaller than 1/k. The assumptions needed here are even more crude approximations than in the elastic case, but we try to keep them as least contradictory as possible. We do not consider all of the stress components and their contributions to matrix flow. Instead of this, we assume the matrix axial tensile stress to be predominant. With increasing load, the matrix tensile stress will reach the matrix flow stress in the vicinity of fibre ends, causing matrix flow. In these regions the matrix tensile stress keeps constant at tr*, and the-interface shear stress z cannot exceed 3" = a~12. Though it is not necessary, for reasons of simplicity we put z equal 3" at the interface

INFLUENCE OF FIBRE DISCONTINUITIES ON STRESS-STRAIN BEHAVIOUR

]3

between the plastified matrix and fibre. In the elastic region, ~ decreases rapidly along the fibre, and for this reason we make it equal to zero. The transfer of forces between the components is expressed by: d6F, dz

2Z~ d

d6 u dz

- -

=

ra - z 2 D

-

6r

=

Evgv

elastic matrix region: a u = E u e M

~= 0

plastic matrix region: a u = a *

~ = +~* = +__a~t/2

(26)

eM = gr

The index numbers, as in former chapter, refer to individual fibres. These equations have to be accompanied by the boundary conditions (14). 6u, as before, denotes the cross-sectional average of matrix tensile stress: 6~_ 6u =

D - 6(z) a~ +

(27)

D

6(z) being the lateral extent of the plastic region at the site z. The balance of forces at an elastic cross section, say, at z = l/4 in Fig. 5, relates aM to the overall composite stress:

~/2

=

(28)

a~

II I

I

I

I

Fig. 5. Plastic regions growing at fibre ends. Since a ~ is a constant, and eu as well as (d/dz)6 u are independent of z in our approximation (26), differentiation ofeqn. (28) provides the result that (d/dz)6(z) is independent of z, too: d z* -~z 6(z) = a * - a c E u / E

(29)

Thus, the boundary between elastic and plastic regions turns out to be a straight line in our model (Fig. 5). Now the situation has become tractable after all. The process of plastification seems to run through several stages: plastic zones originate at fibre ends and, spreading in each direction, meet the neighbouring fibres. They go on growing along the fibres, till the whole matrix is plastified. This can be described

14

B SCHULTRICH, W. POMPE. H.-J. WEISS

quantitatively as follows. The length 2 o of Fig. 5, which is required for the transfer of the load difference between elastic and plastic matrix cross section, is derived from eqn. (29): 20 = D a~r - r* a~EM/E

(30)

Load transfer between neighbouring fibres is effected by the length 2 of Fig. 5. Thus in the middle of the fibre, at z = 0, the stress 2r* av . . . . = d (20 + 2 2 )

(31)

has been built up. (For the sake of simplicity, the reasoning concerns the special case of maximum overlap a = 1/2, whereas the final result is given in a more general form.) The balance of forces at z = 0 relates 2 to the effective composite stress ac: (1 - v)a~ + vr*2/d crc = 2 1 + (1 - v ) E , / r .

(32)

Remember that v = d/(d + D) in our model. In order to set up a stress-strain relation, the corresponding effective strain has to be calculated. Since in our approximation the points P1 and P2 (Fig. 5) on neighbouring fibres, being opposite to each other at a c = 0, stay exactly opposite if the composite is loaded, the mean strain of that quarter of the fibre between z = 0 and z --- 1/4 is equal to the mean strain of the composite ec: ~c = ~4 f l 4 ~v dz

(33)

ee is constant and equal to the elastic matrix strain eM for 2 < z < l/2 - 2 - 2 o. From eqn. (28) we obtain in this region ev = a J ~ . At z < 2 the fibre strain varies linearly with z due to matrix shear stress: Gc

T* ,~2

(34)

~ = -~ + 4-EF ld Elimination of/l from eqns. (32) and (34) results in: ~c=~+4-~

---~)z-~Ev(a~0-o - l

ifa~>aco

(35)

This formula holds for a v~ 1/2, too (Fig. 6). Here a~o stands for the combination of symbols in eqn. (32) with 2 = 0, which is that stress where the plastic region spreading from the fibre end has just reached the neighbouring fibre. At lower stress a~ < a~o, the plastic regions are small areas embedded in the otherwise elastic

INFLUENCE OF FIBRE DISCONTINUITIES ON STRESS-STRAIN BEHAVIOUR

-lit

'b-*

o

...... II -,:::~,.'~;<'>:;'~' Fig. 6.

15

z

,

,~.~,r~,. II

~t' ~ : ' ' ' ! ~

Composite with partly plastified matrix to which our formulae (35) to (40) apply.

material. Their influence on the mechanical behaviour is neglected in our simplified approach, which means that the material shows linear elasticity: ~< = a ~ / E

if trc <

O'
(36)

Figure 7 indicates how the composite stress at a given strain is reduced if the continuous reinforcement is replaced by shorter and shorter fibres, according to eqns. (35) and (36). The results show clearly the consequences of local matrix flow, which begins far below the proper matrix yield strain e*. As has been discussed already, eqn. (35) is only valid at stresses above ac~. In addition to this lower limit of the region of validity of eqn. (35), there are several upper limits arising from different phenomena which destroy the structure represented in Fig. 6. Because of close relations to composite strength and failure modes, those phenomena will be mentioned in particular:

_Ailo

~'~g

"~~i~

oa

-

o,6 /

o/

,

,,

q2

-t/2 4~

Fig. 7.

z°lo //d

o,,c o,~; qe i

i

i

i

~,/~,

~(e)-plots according to eqn. (35), the aspect ratio being parameter.

]6

B. SCHULTRICH, W. POMPE, H.-J. WEISS

From the balance of forces at z = 0 we obtain the fibre stress peaks due to neighbouring fibre ends: O'F. . . .

"1"=-

2 - ( a c - (1 - v)a~)

i f a c > a~o

(37)

t'

Since the maximum fibre stress cannot exceed the fibre strength a~-, severe damage would occur at: /2

at1 = ~a~ + (1 - v)a*

(38)

In cases of small overlap a, neighbouring plastic regions may have met each other as a result of their spreading ere the fibre stress has reached a*. Since 2 o + 22 in eqn. (31) is confined to the overlap length a, the fibres will be stressed up to 2~*a/d. Then the composite carries the load" a c 2 --~" vr*a/d

+ (1 - v)a~

(39)

There is another possibility, namely, the elastic matrix stress a u may reach the yie.ld point a ~ before the plastic regions have met each other. This will occur at: ff*~_,/E M

0"c3 :

(40)

Since the matrix in its plastified state is still able to build up shear stress, the situation may come close to that described by eqn. (39). Thus, the lower of the two stresses ocl and a~2 will be the one at which the composite suffers severe damage. Details will depend on strain hardening and fracture characteristics of the components, but that

6~

eg. q8

~.

q~

lo

q~

vf I~-s , .tlz I

o,2 Fig. 8.

I

q~

|

q6

I

o,a

I

~,16~,

a(~)-plots constructed from Fig. ? by using information from Fig. 3(a).

INFLUENCE OF FIBRE DISCONTINUITIES ONSTRESS~STRAIN BEHAVIOUR

17

is not the subject of this paper, trc2 may get rather small due to short fibres or small overlap. Note that overlap is the essential parameter, the fibre length being important only as far as it confines the former to l/2. For the sake of simplicity, we have investigated separately a hypothetical purelyelastic composite, with the results given in eqns. (24) and (25), and a composite the behaviour of which is dominated by matrix yield, with the results (35) to (40). Since the elastic deformation of the latter had to be treated in a greatly simplified manner compared with the former, (25) is not contained in (35). This drawback may be overcome by replacing ~"by the effective composite modulus E c according to (25), as shown in Fig. 8. The stress comes out rather low for small aspect ratios, which is due to the absence of adhesion at the fibre end faces. Therefore, Fig. 8 is representing lower boundaries.

CONCLUDING REMARKS

We investigated in detail composites consisting of a regular array of plates embedded in a ductile matrix. Experimental work on such kinds of composites has been reported in literature in quite a few instances. Glavinchevski and Piggot 14 investigated polycarbonate reinforced with steel discs, for instance, whereas Rexer and Anderson 15 reported on aluminium reinforced with steel ribbons. The main aim of our paper, however, is to reach qualitative understanding of the behaviour of discontinuous fibre composites. We assume that our model composite has essential features in common with fibre composites. (Thus, for convenience, the plates have been called fibres in this paper.) In reality, the distances of fibres and overlap lengths are distributed randomly in most cases. Composite properties, as calculated in this paper, may be regarded then as properties of some composite region of small extent. Then the gross composite flow stress, for instance, might be obtained from the collective action of those composite regions with different local flow stress, analogous to the statistical theory of composite fracture. 6 This situation implies a size effect, especially for specimens with small cross-section. For further generalisation of the model, the classical concepts of fracture theory as well as the statistics of fibre strength should be taken into account. The results based on the completely periodic arrays with a = 0 and a = 1/2 allows conclusions to be drawn concerning minimum and maximum strength. On the other hand, the periodicity, as convenient as it may be for calculation purposes, results in an over-estimation of stress concentrations, with exception of the regular arrays of Glavinchevski and Piggot, 14 and Rexer and Anderson.15 One of the basic approximations of the model, the linear relation (12) for shear stress, requires a deeper explanation, which is given elsewhere/The lengths h e and hM, which have been replaced by d/2 and D/2 for simplicity, are essential for the

18

B. SCHULTRICH, W. POMPE, H.-J. WEISS

v a l u e s o f the d e c a y l e n g t h s 1/x a n d 1/k o f the elastic stress c o n c e n t r a t i o n s . A n o t h e r p e c u l i a r i t y o f o u r m o d e l , the l i n i n g u p o f the fibres in I n d i a n file, seems n o t to be a serious d r a w b a c k as l o n g as l/d is n o t t o o small. T h e n fibre e n d s are relatively rare, a n d the d e t a i l e d c o n f i g u r a t i o n at the e n d is o f m i n o r relevance. F r o m the great v a r i e t y o f p h e n o m e n a a n d m e c h a n i s m s w h i c h m a y be i n v o l v e d in the p r o b l e m o f c o m p o s i t e s t r e n g t h , we c h o o s e s o m e for d e t a i l e d i n v e s t i g a t i o n . O u r results, therefore, a p p l y to c o m p o s i t e s w h i c h e x h i b i t these p h e n o m e n a , as plastic m a t r i x flow in the a b s e n c e o f crack p r o p a g a t i o n , p r e d o m i n a n t l y .

REFERENCES 1. H. L. Cox, The elasticity and strength of paper and other fibrous materials, Brit. J. Appl. Phys., 3 (1952) p. 72. 2. N. F. D ow, G.E.C. Missile and Space Division Report No. R63SD61. Compare with G. S. Hollister and C. Thomas, Fibre Reinforced Materials, Elsevier, Amsterdam 1966, p. 23. 3. J. M. HEDGEPETH, Stress concentrations in filamentary structures, NASA T D-882 Langley Research Center (1961). 4. J. M. HEDGEPETHand P. VANDYKE,Local stress concentrations in imperfect filamentary composite materials, J. Comp. Mat., l (1967) p. 294. 5. P. VAN DYKE and J. M. HEDGEPETH, Stress concentrations from single-filament failures in composite materials, Textile Research J., 39 (1969) p. 618. 6. C. ZWEBEN,On the strength of notched composites, J. Mech. Phys. Sol., 19 (1971) p. 103. 7. B. SCHULTglCH,Faserverst~irkte Verbundwerkstoffe unter Zugbelastung Forschungsbericht, TU Dresden (1975); ZIID Berlin, FE 23911; J. Weissbarth, Ein ph~nomelogisches Modell zur Warmverformung faserverst~irkter Verbundwerkstoffe, Dissertation, TU Dresden (1972). 8. A. KELLYand W. R. Tvsor~, Tensile properties of fibre-reinforced metals: copper/tungsten and copper/molybdenium, J. Mech. Phys. Solids, 13 (1965) p. 329. 9. W. POMPE, H. G. SCHGPF, B. SCHULTRICH and J. WEISSBnRTH, Stabilit~itsverhalten yon faserverst/irkten Verbundwerkstoffen mi.t plastischer Matrix, Ann. Phys. (Leipzig), 30 (1973) p. 257. 10. A. KELLYand K. N. STREET,Creep of discontinuous fibre composites, Proc. R. Soc. Lond. (A), 328 (1972) p. 267. 11. W. POMPE and B. SCHtJLTglCH, Zur Grenzschichtd~impfung von Verbundwerkstoffen mit kurzfasrigen Einlagerungen, Ann. Phys. (Leipzig), 31 (1974) p. 101. 12. B. W. ROSEN, Mechanics of composite strengthening, Fiber Composite Materials, A.S.M. (1965) p. 42. 13. J. M. LIFSHITZand A. ROTEM, Time-dependent longitudinal strength of unidirectional fibrous composites, Fibre Sci. Techn., 3 (1970) p. 1. 14. B. GLAVINCHEVSKIand M, PIGGOT,Steel disc reinforced polycarbonate, J. Mater. Sci., 8 (1973) p. 1371. 15. F. REXERand E. ANDERSON,Die Festigkeitseigenschaften von bandverst/irkten Aluminium-StahlVerbundwerkstoffen, Proceedings of the conference 'Verbundwerkstoffe' Konstanz (1974). 16. B. D. AGARWAL,J. M. LIFSHITzand L. J. BROUTMAN,Elastic-plastic element analysis ofshort fibre composites, Fibre Science and Techn., 7 (1974) p. 45.