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Composites Science and Technology 45 (1992) 221-228 The fibre skeleton structure and transport properties of stochastically reinforced composites A. K. Stepanov, V. V. Tvardovsky* & A. A. Khvostunkov Institute of Solid State Physics of the Academy of Sciences of the USSR, Chernogolovka, Moscow district 142432, Russia (Received 30 September 1991; accepted 21 October 1991) A percolation approach for the description of effective properties of stochasti- cally reinforced fibrous composites is proposed. Composites with conductive fibres and non-conductive matrix are considered. It is shown that the insulator-conductor transition may be observed at low fibre volume fraction. Introducing a value of the contact resistance, two cluster models are involved in the analysis which yields a linear dependence of the effective electrical conductivity on fibre volume fraction. The results of theoretical calculations are in good agreement with experimental measurements of the electrical conductivity of the composites. Keywords: stochastically reinforced composite, short fibres, resistance, con- ductivity, percolation threshold, infinite cluster, transport properties 1 INTRODUCTION The problem of effective properties of non- homogeneous solids has been considered in numerous works: good reviews can be found, for example, in Refs 1 and 2. Most well-known theories do not yield satisfactory results because the difference of corresponding phase properties tends to infinity. Furthermore some phenomena (e.g. the conductivity threshold) cannot be described in the framework of the classical models of non-homogeneous solids, the reason being the change of current flow mechanism from the diffusive to the percolative type (see, for example, Ref. 3) when the current does not flow through each point of a sample but finds preferable conductive paths. Experimental data published do evidently testify that the conduc- tivity threshold depends on the inclusion shape, 4 the inclusion size distributionfl and the size ratio of conductive and non-conductive particles. 6'7 The analysis of some of these relationships may be found in Ref. 4. * To whom correspondence should be addressed. Composites Science and Technology 0266-3538/92/$05.00 © 1992 Elsevier Science Publishers Ltd. 221 The percolation theory describes threshold phenomena in non-homogeneous solids: various approaches and models of the theory are collected in Refs 8 and 9. The change of conductivity mechanism from the diffusive to the percolative type is connected with the ap- pearance of an infinite conducting path, a so-called infinite cluster. It is obvious that the formation of the infinite cluster of intersecting inclusions depends upon their volume fraction, size, shape and orientation parameters. An important problem of the percolation theory is the determination of critical values of the parameters mentioned,s,9 A dependence of the percolation threshold on the fibre volume fraction of composites contain- ing a stochastic skeleton of conductive fibres in a non-conductive matrix on the fibre aspect ratio is to be considered here. The fibre volume fraction, o p, at which the percolation threshold is observed is referred to as the critical value. A method of evaluation of the value of vf p for a 3D-model of an orthogonally reinforced fibrous composite is suggested. The 3D-model is also used to describe transport properties of stochas- tically reinforced composites.

The fibre skeleton structure and transport properties of stochastically reinforced composites

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Page 1: The fibre skeleton structure and transport properties of stochastically reinforced composites

Composites Science and Technology 45 (1992) 221-228

The fibre skeleton structure and transport properties of stochastically reinforced

composites

A. K. Stepanov, V. V. Tvardovsky* & A. A. Khvostunkov Institute of Solid State Physics of the Academy of Sciences of the USSR, Chernogolovka, Moscow district 142432, Russia

(Received 30 September 1991; accepted 21 October 1991)

A percolation approach for the description of effective properties of stochasti- cally reinforced fibrous composites is proposed. Composites with conductive fibres and non-conductive matrix are considered. It is shown that the insulator-conductor transition may be observed at low fibre volume fraction. Introducing a value of the contact resistance, two cluster models are involved in the analysis which yields a linear dependence of the effective electrical conductivity on fibre volume fraction. The results of theoretical calculations are in good agreement with experimental measurements of the electrical conductivity of the composites.

Keywords: stochastically reinforced composite, short fibres, resistance, con- ductivity, percolation threshold, infinite cluster, transport properties

1 INTRODUCTION

The problem of effective properties of non- homogeneous solids has been considered in numerous works: good reviews can be found, for example, in Refs 1 and 2. Most well-known theories do not yield satisfactory results because the difference of corresponding phase properties tends to infinity. Furthermore some phenomena (e.g. the conductivity threshold) cannot be described in the framework of the classical models of non-homogeneous solids, the reason being the change of current flow mechanism from the diffusive to the percolative type (see, for example, Ref. 3) when the current does not flow through each point of a sample but finds preferable conductive paths. Experimental data published do evidently testify that the conduc- tivity threshold depends on the inclusion shape, 4 the inclusion size distributionfl and the size ratio of conductive and non-conductive particles. 6'7 The analysis of some of these relationships may be found in Ref. 4. * To whom correspondence should be addressed.

Composites Science and Technology 0266-3538/92/$05.00 © 1992 Elsevier Science Publishers Ltd.

221

The percolation theory describes threshold phenomena in non-homogeneous solids: various approaches and models of the theory are collected in Refs 8 and 9. The change of conductivity mechanism from the diffusive to the percolative type is connected with the ap- pearance of an infinite conducting path, a so-called infinite cluster. It is obvious that the formation of the infinite cluster of intersecting inclusions depends upon their volume fraction, size, shape and orientation parameters. An important problem of the percolation theory is the determination of critical values of the parameters mentioned, s,9

A dependence of the percolation threshold on the fibre volume fraction of composites contain- ing a stochastic skeleton of conductive fibres in a non-conductive matrix on the fibre aspect ratio is to be considered here. The fibre volume fraction, o p, at which the percolation threshold is observed is referred to as the critical value.

A method of evaluation of the value of vf p for a 3D-model of an orthogonally reinforced fibrous composite is suggested. The 3D-model is also used to describe transport properties of stochas- tically reinforced composites.

Page 2: The fibre skeleton structure and transport properties of stochastically reinforced composites

222 A. K. Stepanov, V. V. Tvardovsky, A. A. Khvostunkov

Two models of the infinite conductive cluster are then introduced to describe the effective conductivity of a non-conductive matrix rein- forced with conductive fibres when vf> v~. The value of the contact resistance r0 is used here essentially.

Finally, a comparison of the theory and experimental data is presented. The data on carbon-fibre/gypsum-plaster composites were found in Ref. 10, those on carbon-fibre/alumina- matrix composites were made specially for the purpose of the comparison.

2 PERCOLATION THRESHOLD

We are to consider here a non-conductive medium stochastically reinforced with conductive fibres of diameter d and length l, assuming the fibres to be long, i.e. that

d<<l (1)

To calculate the percolation threshold and effective transport properties we introduce a simple model.

2.1 The basic geometry of the 3D-model of an orthogonally reinforced fibrous composite

(a) We let the distribution of fibre centres within the space be Poissonian:

(Xv) ~ * - k--~--" exp(-XV) (2)

where ~k is the probability of the presence of exactly k centres in volume V, and X is the mean number of fibre centres in the unit volume.

Obviously the fibre volume fraction, vf, can be expressed in terms of X as

~rd2l vf= X . - - ~ (3)

(b) We let each fibre be aligned along one of the co-ordinate axes (x, y, z) only, and the probability for a fibre to belong to one of three subsets is therefore equal to 1/3.

2.2 Distribution of the numbers of fibre intersections

We now introduce a cartesian (x, y, z ) co-ordinate system associated with the centre and the direction of a fibre as shown in Fig. 1. We are

/ -

/ × J

/

/ 2d

1. f

y

)

Fig. 1. A fibre with the system of co-ordinates.

seeking, now, the probability, P, , of the number of intersections of a given fibre with k neighbours. Since in the vicinity of the percolation threshold the mean number, m, of intersections is quite small and because of eqn (1), we can neglect the intersections of fibres which have the same alignment. In accordance with the Poisson distribution given by eqn (2) the probability of having zero (k = O) intersections of a given fibre by fibres from the xth and yth subsets is

Po = e x p ( - ~ . . V) (4)

Here V is a characteristic volume, shown in Fig. 1 as a 3D cross

V = 4dl z (5)

A given fibre can be intersected by another fibre only if the centre of the second one lies within this volume. The intersections of a given fibre by fibres belong to two subsets (i.e. xth and yth) are obviously mutually independent. Therefore the probability, Pk, of any given fibre having k points of the intersections can be written as

(,~, V3) k P* - k---~" exp(-~.V3) k i> 0 (6)

Here 4dl 2

V3 = ~V = 3 (7)

Equation (6) is, in fact, the Poisson distribution with the parameter XV3 equal to the mean number, m, of intersections on a fibre

m - - ~ k .pk=XV3 (8) k=O

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Stochastically reinforced composites 223

From eqns (3) and (7) the mean number of intersections can be expressed in terms of of and l/d as

16 m = ~ vfl/d (9)

It follows immediately that if l/d >> 1 the value of m can be large even when v f is small.

2.3 Lower limit for the percolation threshold on a random lattice of fibres

To estimate lower limit, v[, for fibre volume fraction, or, at which the infinite cluster appears we shall model the formation of such a cluster by a stochastic branching process of the Hal ton- Watson type. ll It means that we see the infinite cluster as a random tree which has 1 , . . . , N , . . . generations (Fig. 2) and the number of branches in each generation is stochastically independent of each other. The probability, qk, of having k branches generated by a single branch in each generation is

Pk+l k >t 0 (10) qk - 1 - po

Note that the multiplier ( 1 - p o ) -~ accounts for the existence of fibres with no intersections. A branching tree becomes infinite if the Hal ton- Watson process is over-critical. It takes place when the mean number, M, of branches generated by a single branch in previous generation is larger than 1. From eqns (6)-(8) and (10) we get

M=- ~ kqk = m ~ 1 (11) 1 - p o

If M > I then we have an infinite tree, so a

Fig. 2. A branching tree of intersecting fibres.

solution of equation

M = 1 (12)

yields a lower limit of the percolation threshold. Substituting eqns (11) and (6) into eqn (12) yields

m = 2(1 - e x p ( - m ) ) (13)

Equation (13) has a single root for m > 0:

rnl = 1.595 (14)

Hence, from eqn (9) a lower limit for the percolation threshold can be written as

v~ = d 3:trn~ = 0.94d/1 (15) 16l

2.4 Upper limit for the percolation threshold

In the above analysis there are no closed cycles in the infinite cluster. In fact there may be some, and to take these into account requires an upper limit for the percolation threshold.

Now let us consider the cluster formation in a layer of thickness 2d normal, say, to the z axis. We subdivide the layer into rectangular cells 1/4 x I/4 × 2d. Centres of these cells form the regular square lattice. Let P be the probability that any fixed cell contains at least one fibre centre aligned along the x axis and at least one fibre centre aligned along the y axis

( ( Z 2dP] }2 (16)

This means that there is fibre intersection in the volume of this cell. So with the aid of the probability, P, we may reduce the problem of the appearance of an infinite cluster in the layer to the well-known site percolation problem on a two-dimensional lattice, s'9 Namely, we say that each centre of the cell, independent of others, is occupied with a probability P and is vacant with a probability ( 1 - P). When P > Pc~ then with the probability of 1 there is an infinite cluster of occupied sites on the square lattice. It is easy to see that the infinite cluster of occupied sites induces infinite clusters of intersecting fibres in our layer. This, in turn, immediately gives us an infinite cluster of conducting fibres in the whole space. Thus, by substituting Per--~ 0"59 of the site problem on the square lattice in eqn (16) we obtain the upper bound of A, or the upper bound, v~, of the critical volume fraction, v p, in

Page 4: The fibre skeleton structure and transport properties of stochastically reinforced composites

224 A. K. Stepanov, V. V. Tvardovsky, A. A. Khvostunkov

accordance with eqn (3):

24 /]'cr = d-~ l n ( l -- V~cr)

v~' = - 6 ~ ln(1 - vrP-~,) -/-~ 27.55 (17)

Equation (17) gives the same dependence of v~' on d/ l as eqn (15). It is clear that these simple considerations may give a rather rough estimate of the ratio d/ l in eqn (17). A more complicated method of estimating v~ which decreases the value of v~' by more than a factor of ten will be published elsewhere.

Our simple model therefore permits us to state the equivalence of the critical volume fraction of fibres to the shape ratio factor d/ l as

v~ = Cd/ l (18)

Thus the critical volume fraction, v p, of conducting fibres which is sufficient for the formation of an infinite conducting cluster decreases (and may be diluted) as the fibre length to diameter ratio tends to infinity. It seems to be true under these circumstances that at the critical point vf= v~ the infinite cluster has a reduced number of closed cycles. For this reason, below subdivisions 3-4 we may let the constant C-~ 0.94 (see eqn (15)). And for the case of an isotropic, stochastically-reinforced composite we obtain C - 0 . 8 (corresponding results to be published elsewhere). This value is in a good agreement with the previous one, so we shall adopt it for the following analysis.

3 EFFECTIVE CONDUCTIVITY OF COMPOSITE

We shall analyse here the transport properties of the composites under consideration, calculating values of the effective electrical conductivity. Depending on structural parameters and the ratio of conductivities of the fibre and matrix, there can obviously be observed at least two extreme types of current transport. The diffusion type takes place when the current flows through both the fibres and the matrix. If the conditions are such that the current avoids the matrix volume and only flows along an infinite cluster, we observe the second type of transport.

In the latter case it is important to take into

account the contact resistance (let it be ro). t The value of r0 depends on some factors (e.g. the longitudinal (a~ I) and transverse (o( ) fibre conductivities, the matrix conductivity, Om, the size d and the shape of the fibre cross-section, the structure of the fibre/matrix interface, etc.). It seems to be impossible to measure the value of ro in a direct experiment.

Considering further the contact conductivity model, we suppose that the inequality

( v f - vP)/v p >> 1 (19) holds.

Note that we now consider the fibre intersec- tions defined above as fibre contacts. So using the results obtained above we obtain the mean number of contacts, m, equal to

Uf m = mp v~' (20)

Here mp is the critical value of m at the percolation threshold. For this value, in accord- ance with eqns (14) and (18), the quantity mp -~ 1.6 may be adopted.

One can see that it is possible to have m >> 1 if the inequality (19) is fulfilled.

Now assuming

Orn •< Of (21) and denoting

Al = l / m = lvU(mpVO

as the mean distance between contacts on a fibre, we can write the following set of relationships

AI < Om (22a)

rod 2

Al O m << rod2 << O'f (22b)

Al of rod2 (22c)

Al of << rod2 (22d)

These relationships are to be used to identify the type of transport mechanisms.

3.1 Diffusion type of composite conductivity

If eqn (22a) is fufilled then the current enters a fibre through the matrix by the diffusion

t The idea of introducing this parameter has been expressed by Dr V. M. Kijko.

Page 5: The fibre skeleton structure and transport properties of stochastically reinforced composites

Stochastically reinforced composites

Table 1. Inemcient lengths of conductive fibre

225

am/O f 10 -2 10 - 3 1 0 - 4 1 0 - 5 1 0 - 6 1 0 - 7 10-8 10-9 d/25 9 × 1 0 - 2 2.27 X 1 0 - 2 6-2 X 10 - 3 1.8 X 1 0 - 3 5.1 X 1 0 - 4 1.5 x 10 4 4-45 x 10 -5 1-3 x 10 -5

mechanism. In this case there is no threshold on the effective conductivity versus fibre volume fraction curve, and the case can therefore be treated by well-known theories.

Averaging the current over celestial angles yields the conductivity of an isotropic, stochastically-reinforced composite, namely

o , = ~(o~ + 2 o , ~) (23)

where o+, o~ are the transverse and longitudinal effective conductivities of the corresponding unidirectionally reinforced composite. The poly- disperse model 2 gives the simple formula for the value of the transverse conductivity

a m "[- Of + l)f(O'f - - O'm) O'~ Z = Om ( 2 4 )

O" m "JI- Of - - Uf(O'f - - O'm)

Unlike the transverse conductivity, the lon- gitudinal conductivity, o4, depends more strong- ly on the fibre length. For infinite fibres we have

05 = (1 - Vf)O" m "Jr- VfO'f ( 2 5 )

If the fibre length is l then one can derive the estimate

1 ) (26) a~ = (1 - v f ) o m + u f a f 1 - cosh(I/2s)

The 'ineffective' length s depends on am~Of as well as on yr. For sufficiently small values of vf the value of s can be obtained as the solution of the equation

J,(dl2s) r,(dl2s) Of Jo (d /2 s ) -- Om Yo(d /2~) ( 2 7 )

where J~(x), Yn(x) ( n = 0 , 1) are Bessel functions. The dependence of s on Orm/a f (see Table 1) shows that the ineffective length, s, can be much greater than the fibre length, l.

3.2 The contact conductivity

Let us consider the contact conductivity in a 3D model of an orthogonally reinforced composite for the case when eqn (22b) holds. Hence the current through the matrix may be neglected.

While the number of contacts m >> 1 (see eqn (20)), we may assume that each fibre belongs to

an infinite cluster. This means that each fibre takes part in the macroscopic current transport. Because the right hand part of the inequality given by eqn (22b) is satisfied, the contacts provide a main contribution to the cluster resistance. So the fibre resistance

41 rf - ~ d 2 o . f

may be neglected. Let the mean electric field intensity vector,

( E ) , be directed along the z axis, and let the fibre be aligned in the same direction. Assuming that the current flows along a path corresponding to a minimum of ohmic losses, we conclude that the current enters into the fibre through the lowest contact and goes out through an upper contact (see Fig. 3). If the mean distance between the extreme contacts is 10 then the mean current will be:

(i) =lo(E) 2r0 (28)

Lt A II

Fig. 3. A possible current path. The current enters into the fibre through a lowest contact and goes out through an

upper one, missing the intermediate ones.

Page 6: The fibre skeleton structure and transport properties of stochastically reinforced composites

226 A. K. Stepanov, V. V. Tvardovsky, A. A. Khvostunkov

The mean number, N, of fibres intersecting a unit area normal to the z axis is

4vf N -

3~rd 2

Hence the mean current density, axis direction is written as

(29)

( j ) , in the z

(j) = ( i ) N (30)

Finally, defining

( j) = a , ( E ) (31)

we obtain the effective conductivity

2Vdo (32)

O. -- 3~rd2r °

Introducing the mean number of contacts per a unit length of fibre as Cm = m / l and using the Poisson distribution (eqn (6)) for the number of contacts per unit length, ~,

(Cm~)k. exp(-cm~ e) k/> 0 Pk - k!

we obtain the exponential distribution of distances, A, between neighbouring contacts

P{A ~< ~} = 1 - exp(-cm~)

with the mean value

(A) = Cm' = l / m

Hence the average distance between extreme contacts is

lo= l - 2 (A) = l ( 1 - 2/m)

Substituting eqn (20) into the above equation we have

/ 0 = / ( 1 _ 2vP) mpVf/

So eqn (32) can now be rewritten as

2vfl . ( 1 2vP) (33) o . -- 3:td2r ° mpVf/

Thus, the effective conductivity, o . , is proportional to the fibre volume fraction, vf. The model works up to the maximum density of 3D orthogonally-paeked fibres which is at of = 0-58.

While eqn (22c) holds, the resistances of the fibres and the contacts are comparable. So in this case the numerical calculations of effective conductivity must be done taking into account all the possible paths of the current on a random lattice of the resistances.

3.3 The weak contact resistance model

We now consider the case defined by the inequality given by eqn (22d). The current passes between two contacts without ohmic loss. Let us again use the 3D model. The fibre skeleton in this case may be thought of as a random cubic lattice with the mean period equal to the mean distance between contacts, l /m. This lattice has link breaks at mean distances I. The number of breaks in a volume V >>/3 coincides with the number of fibres in V. So we may assume that the distribution of the number of breaks in V is the same Poisson function as the number of fibre centres.

Consider a single layer of lattice of thickness l /m in the z axis direction. A mean number, N, of fibres intersecting a unit area normal to the z axis direction is given by formula (29) which may be presented as

3. N = 7 " 1 × 1 × 1 (34)

The mean number of z link breaks in a single layer of volume l /m x 1 × 1 is

3.1 AN = - - x 1 x 1 (35)

3m

Here 3./3 is the mean number of z link breaks in unit volume. So the mean number of fibres, No, which are aligned along the z axis and intersect a single layer of volume l /m × 1 × 1 is

No = N - AN (36)

The current, i, flowing through the fibre directed along the z axis can be obtained as

i - ( E ) l / m (37) Art

where

41 A r f - srd2mo f (38)

is the resistance of fibre of length l /m. The mean current density in the z axis direction is now

(j) = iNo (39)

Equation (39), together with eqns (31) and (35)-(38), yields the effective conductivity

a , = i3v,a,(1 - 1/m) (40)

Page 7: The fibre skeleton structure and transport properties of stochastically reinforced composites

Stochastically reinforced composites 227

4 COMPARISON OF THEORY AND EXPERIMENT

We shall compare theoretical prediction of both the percolation threshold and the conductivity versus fibre volume fraction curve with cor- responding experimental data.

4.1 The percolation threshold

We use here the experimental data on the dependence of resistivity, p , , of gypsum plaster reinforced with carbon fibres on fibre weight fraction published in Ref. 10. The fibres had a length l = 3 m m and diameter d = 12#m. The ratio of conductivities was O'm/O'f~ 10 -9. The experimental data '° are depicted in Fig. 4, where the weight fractions are converted to fibre volume fractions. Between vf = 0.26% and vf= 0.43% there is a jump of resistivities. The calculation of percolation threshold according to eqn (18) with C = 0.8 gives the value v p - 0 . 3 2 % which is in a good agreement with the quantities observed.

It should be mentioned that experimental data for the resistivities of a polymer matrix reinforced with short aluminium f i b r e s 4 a r e also described by the percolation approach.

4.2 The contact conductivity

To make the comparison in this case a special experiment was performed. Alumina-

3.5

~ '3 .0

~ 2 . 5

~2.o

~°1,0

0.5

0.0

Fig.

o

o

o o

i o

0.0 0.1 0.2 0.3 0.4 F I B R E V O L U M E F R A C T I O N

5. Conduc t i v i t y o f an a l um ina -ma t r i x / ca rbon - f i b re composite versus fibre content (vol. %).

matrix/carbon-fibre composites were made by the powder metallurgy fabrication route de- scribed in detail in Ref. 12. The initial length of the fibres was 2 - 3 m m and their diameter d = 6 # m .

The specimens had a rectangular cross-section of 1.8mm × 4 .0mm and length about 20mm. Four indium contacts were placed on each specimen. The distance between the potential contacts was 10-15 mm.

The experimental data recalculated as conduc- tivities are shown in Fig. 5. The resistivities for small fibre volume fractions are depicted in Fig. 6. The experiment provides evidence of the

~ 1 0 71 (.3

I ~ 1 0 '

)-

~ 1 0 ~ tD

o 1 0 ~ L_ S Q. mlO s

lOl~.O ' L O

i i .I i i L I L i i I i i i

0 5 1.0 1 5 2.0 F I B R E V O L U M E F R A C T I O N (~)

Fig. 4. Resistivity of a gypsum plaster reinforced with carbon fibres (l = 3 ram, d = 12 #m) versus fibre content (vol. %).10 An arrow indicates the percolation threshold

o p = 0.32% (eqn (18)).

~ 1 0 g (.3

:~I0' I 0

~ 1 0 7 I -

> 1 0 ~

~ 1 0 ~ _

o

~-10 ~ _

10

o o 0 o

o

I L I I t I t I I I I I t ) J

0 1 2 3 4 5 6 7 8 FIBRE VOLUME FRACTION (Z)

Fig. 6. Resistivity of an alumina-matrix/carbon-fibre com- posite versus fibre volume fraction in the region of low fibre

volume fractions.

Page 8: The fibre skeleton structure and transport properties of stochastically reinforced composites

228 A. K. Stepanov, V. V. Tvardovsky, A. A. Khvostunkov

presence of the percolat ion threshold at be tween 3 and 4% of of. The solid curve in Fig. 5 is calculated according to eqn (33) with parameters l = 0 . 1 2 m m and ro = 89.15 f2 being obtained by the least squares method. Arrows in Figs 5 and 6 show the value of the percolation threshold, vf p = 0.0389, according to eqn (18).

The adequacy of the theory to describe the experimental data was established by the F ratio (variance ratio) '3 for the probability level a" = 0.001. Thus we may state that the effective conductivity of the composite under considera- tion is defined by contact resistance.

ACKNOWLEDGEMENTS

We thank Professors S. T. Mileiko and I. L. Aptekar and Dr V. M. Kijko for numerous and fruitful discussions and for their help in preparing this paper. We are also grateful to Mr L. S. Kojevnikov, Mr A. I. Ivanov and Mrs T. A. Tchernoff for their assistance with the experiments.

REFERENCES

5 C O N C L U S I O N S

An approach for the description of a jump in the effective transport propert ies of fibrous compos- ites reinforced with short fibres is proposed and has been successfully used to interpret cor- responding experimental data on conductivity. It is shown that the insu la tor -conductor transition is observed at low fibre volume fraction, re, of reasonably long conductive fibres embedded in a dielectric matrix.

On the basis of the proposed idea of contact resistance, ro, the two cluster models were incorporated into discussion and the linear dependence of effective electrical conductivity on vf was found. From our point of view this effect is related to the conductivity of the system on an 'effective' sublattice, by which we mean the 'effective' sublattice whose characteristic period along the electric field density direction is lo. The quantity lo is changed weakly with increase in volume fraction, vf, above the percolat ion threshold. In this region of of the parameters of a real lattice of intersecting fibres (infinite cluster) became nonlinear with increase in of. Hence the increase in effective conductivity should also presumably be nonlinear.

1. Hale, D. K., The physical properties of composite materials. J. Mat. Sci., U (1976) 2105-41.

2. Christensen, R. M., Mechanics of Composite Materials. J. Wiley & Sons, New York, Chichester, Brisbane, Toronto, 1979.

3. Sherman, R. D., Middleman, L. M. & Jacobs, S. M., Electron transport processes in conductor filled polymers. Polym. Engng Sci., 23 (1983) 36-46.

4. Bigg, D. M., Mechanical and conductive properties of metal fibre-filled polymer composites. Composites, 10 (1979) 95-100.

5. Bhattacharya, S. K. & Chaklader, A. C. D., Review on metal filled plastics. Part 1: Electrical conductivity. Polym. Plast. Technol. Eng., 19 (1982) 21-51.

6. Malliaris, A. & Turner, D. T., Influence of particle size on the electrical resistivity of compact mixtures of polymeric and and metallic powder. J. Appl. Phys., 42 (1971) 614-18.

7. Aharoni, S. M., Electrical resistivity of a composite of conducting particles in an insulating matrix. J. Appl. Phys., 43 (1972) 2463-5.

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9. Kirkpatrick, S., Percolation and conduction. Rev. Mod. Phys., 43 (1973) 574-89.

10. Chung, D. D. L. & Zheng, Q. J., Electronic properties of carbon fibre reinforced gypsum plaster. Compos. Sci. and Technol., 36 (1989) 1-6.

11. Sevostianov, B. A., Branching Processes. Nauka, Moscow, 1971 (in Russian).

12. Mileiko, S. T. & Khvostunkov, A. A., Strength, fracture toughness and electrical conductivity of carbon fibre-carbide matrix composites. Mekhanika Kom- positnyh Materialov (in press).

13. Hald, A., Statistical Theory with Engineering Applica- tions. New York, London, 1957.