simple estimation on effective transport properties of a random composite material with cylindrical...

Z. angew. Math. Phys. 59 (2008) 889–9030044-2275/08/050889-15DOI 10.1007/s00033-007-6146-3c© 2008 Birkhauser Verlag, Basel

Zeitschrift fur angewandteMathematik und Physik ZAMP

Simple estimation on effective transport properties of a

random composite material with cylindrical fibres

I. V. Andrianov∗, V. V. Danishevs’kyy and D. Weichert

Abstract. Improved bounds for effective transport properties of a random non-percolated com-

posite with cylindrical fibres are developed by means of the security-spheres approach. The keypoint of the method is to obtain a solution for a regular composite that can be valid for all valuesof the volume fractions and properties of the components. For this aim we use the asymptotic

homogenization method; a cell problem is solved by a modified version of the boundary shapeperturbation technique.

Mathematics Subject Classification (2000). 74E30, 74Q05.

Keywords. Random composite, effective properties, asymptotic methods, homogenization.

1. Introduction

Theoretical prediction of effective properties of composite materials is an impor-tant problem in mechanics and engineering [1–4]. A lot of works are devoted toperiodic media. However, most of the real composites can possess random mi-crostructures. Hence, in order to obtain the exact solution we have to know,generally speaking, an infinite set of correlation functions, which are intended toprovide a statistical description of the microgeometry of the composite. Naturally,in many cases such a description is not available. On the other hand, variousbounding methods give a possibility to estimate the effective properties using onlya limiting amount of microstructural information.

It should be noted that many of commonly used bounds (e.g., the Hashin–Shtrikman variational bounds) exhibit divergence and become almost out of prac-tical use for densely packed (but not percolated) high-contrast composites. Then,in order to obtain a reasonable estimation on the effective properties one mustdevise improved bounding models that prohibit the appearance of cluster chainsdespite a volume fraction of one of the components may be beyond the percolationthreshold. In the present paper we propose such improved bounds for the effectivetransport properties of a random composite material with cylindrical fibres.

∗Corresponding author

890 I. V. Andrianov, V. V. Danishevs’kyy and D. Weichert ZAMP

a)

b)

Figure 1. Non-regular composite structure under consideration. a) General view of theshaking-geometry composite; b) deviation of the fibres about the regular square lattice.

Let us consider a non-regular fibre-reinforced composite consisting of a matrixand cylindrical inclusions (fibres) (Fig. 1). A centre of each fibre can randomlydeviate within a circle of diameter d whereas these circles themselves form a regu-lar square lattice of the period l. Such kind of a microstructure is usually referredas a shaking-geometry composite [5, 6]; from the engineering point of view it maycorrespond to a random “shaking” of the fibres about the periodic lattice caused,for example, by certain production or technological reasons. The deviation pa-rameter δ = d/l describes the rate of non-regularity of the structure; its maximumvalue δmax is determined by the case when neighbouring fibres can nearly toucheach other. Higher values of δ are not allowed since they would lead to penetra-tion of the fibres which is out of scope of the present paper. Trivial geometricalevaluations give:

δmax = 1 −√

c

cmax

, (1)

Vol. 59 (2008) Simple estimation on effective transport properties. . . 891

where c = πa2/

l2 is the volume fraction of the fibres, cmax = π/4 is the maximalvolume fraction of the fibres in the case of a perfectly regular material (δ = 0).We seek the effective transport coefficient (e.g., heat conductivity) 〈k〉; the con-ductivities of the components are kf and km respectively for the fibres and forthe matrix. Let us also suppose kf > km; the opposite case can be treated in thesame mathematical way using the well-known duality relation [2, 7–9]:

⟨

k(kf , km)⟩

=⟨

k(km, kf )⟩−1

. (2)

Kozlov [10] has shown that a regular lattice possesses the extreme effectiveproperties among the corresponding shaking-geometry random structures. Origi-nally this result has been proved for the case of dilute composites. Berlyand andMityushev [5, 6] have generalized the Kozlov’s theorem to the non-dilute cases.Therefore, a solution for the perfectly regular lattice can be considered as a lower

bound on the effective transport coefficient; below we will see that for dilute com-posites it almost coincides with the corresponding Hashin–Shtrikman bound [11].

On the other hand, an upper bound can be obtained by replacing the inputnon-regular assembly of fibres of radius a by the regular lattice of fibres of radiusa + d/2. Such estimation is also known as the security-spheres approach. It hasbeen originally proposed by Keller et al. [12] and further extended by Rubenfeldand Keller [13] and Torquato and Rubinshtein [14]; for the details and referencessee also an excellent book by Torquato [3]. In the case of high-contrast compositesthis bound appear to be essentially better than Hashin–Shtrikman results.

Thus, the starting point for the development of the improved bounds is toobtain a solution for the regular lattice. For this aim we utilize the asymptotichomogenization method [15–18]; a cell problem is solved by means of the boundaryshape perturbation technique [17–21].

Outline of the paper is as follows. In Section 2 we evaluate the effective trans-port coefficient for the perfectly regular case; the obtained results are valid for allvalues of the volume fractions and properties of the components. In Section 3 wepresent the improved bounds for the random shaking-geometry composite. Briefconcluding remarks are given in Section 4.

2. Solution for the perfectly regular composite

Let us consider a transverse transport process through the introduced above com-posite structure (Fig. 1) in the perfectly regular case (δ = 0), so the fibres arearranged in a periodic square lattice. The characteristic size l of heterogeneities issupposed to be much smaller than the size L of the whole sample of the material:l << L. Assuming the perfect bonding conditions at the interface ∂Ω between thecomponents the input boundary value problem (BVP) can be written as follows


ka

(

∂2ua

∂x22

+∂2ua

∂x23

)

= −fa, um = uf∣

∣

∂Ω, km ∂um

∂n= kf ∂uf

∂n

∣

∣

∣

∣

∂Ω

. (3)

Here and in the sequel variables indexed by “m” correspond to the matrix, in-dexed by “f” correspond to the fibres, the index “a” takes both of these references:“a” = “m”, “f”. Generally, problem (3) can have a number of different physicalinterpretations, but in the present paper it is discussed with a reference to theheat conduction. Then in the expressions above ka are the heat conductivitiesof the components, ua is the function of temperature distribution, fa is the den-sity of heat sources, ∂/∂n is the normal derivative to the interface ∂Ω. We shallseek the effective heat conductivity 〈k〉, however, all following results stay true forother transport coefficients such as electrical conductivity, diffusion, magnetic per-meability, etc. Due to the well-known longitudinal shear – transverse conduction

analogy [22] the elastic anti-plane shear modulus can also be evaluated in entirelythe same mathematical way.

Now we go on to study the input BVP (3) by means of the asymptotic homog-enization method [15–18]. Let us define a natural small parameter

ε = l/L, ε << 1, (4)

characterizing the rate of heterogeneity of the composite structure. In order toseparate macro- and microscale components of the solution we introduce so calledslow xs and fast ys co-ordinate variables

xs = xs, ys = xsε−1, s = 2, 3, (5)

and search the temperature field as an asymptotic expansion:

ua = u0(x) + εua1(x,y) + ε2ua

2(x,y) + ..., (6)

where x = x2e2 +x3e3, y = y2e2 + y3e3, e2 and e3 are the unit Cartesian vectors.The first term u0 = um

0 = uf0

of expansion (6) represents the homogeneous partof the solution; it changes slowly within the whole sample of the material anddoes not depend on fast co-ordinates. All next terms ua

i , i = 1, 2, 3, ..., describelocal variations of the temperature on the scale of heterogeneities. In the perfectlyregular case the microperiodicity of the medium induces the same periodicity forua

i with respect to fast co-ordinates:

uak(x,y) = ua

k(x,y + Lp), (7)

where Lp = ε−1lp, lp = p2l2+p3l3, ps = 0,±1,±2, ..., l2 and l3 are the fundamentaltranslation vectors of the square lattice. The spatial derivatives read

∂

∂xs

=∂

∂xs

+ ε−1∂

∂ys

. (8)


Substituting expressions (5)–(6), (8) into the input problem (3) and splittingit with respect to ε we come to a recurrent sequence of BVPs:

∂2ua0

∂y22

+∂2ua

0

∂y23

= 0, (9a)

um0 = uf

0

∣

∣

∣

∂Ω

, km ∂um0

∂n− kf ∂uf

0

∂n= 0

∣

∣

∣

∣

∣

∂Ω

; (9b)

∂2ua1

∂y22

+∂2ua

1

∂y23

= 0, (10a)

um1 = uf

1

∣

∣

∣

∂Ω

, km ∂um1

∂m− kf ∂uf

1

∂m=

(

kf − km) ∂u0

∂n

∣

∣

∣

∣

∣

∂Ω

; (10b)

ka

(

∂2u0

∂x22

+∂2u0

∂x23

+ 2∂2ua

1

∂x2y2

+ 2∂2ua

1

∂x3∂y3

+∂2ua

2

∂y22

+∂2ua

2

∂y23

)

= −fa, (11a)

um2 = uf

2

∣

∣

∣

∂Ω

, (11b)

km ∂um2

∂m− kf ∂uf

2

∂m= kf ∂uf

1

∂n− km ∂um

1

∂n

∣

∣

∣

∣

∣

∂Ω

; (11c)

and so on; here ∂/∂m is the normal derivative to the interface ∂Ω written in fastco-ordinates.

The first BVP (9) is satisfied trivially since u0 does not depend on fast co-ordinates ys, therefore ∂u0/∂ys = ∂u0/∂m = 0. The second BVP (10) allowsevaluation of the higher order component ua

1 of the temperature field; owing tothe periodicity condition (7) it can be considered within only one periodicallyrepeated unit cell.

In order to determine the effective heat conductivity, the BVP (11) should beconsidered. Let us apply to equation (11a) the homogenization operator

[

∫∫

Ωm0

(·) dy2dy3 +

∫∫

Ωin0

(·) dy2dy3

]

L−2

over the unit cell domain Ω0. Terms containing ua2 are eliminated by means of

the Green theorem taking into account the boundary condition (11c) and theperiodicity condition (7), which yields:


km

∫ ∫

Ωm0

(

∂2um1

∂x2∂y2

+∂2um

1

∂x3∂y3

+∂2um

2

∂y22

+∂2um

2

∂y23

)

dy2dy3

+kf

∫∫

Ωf0

(

∂2uf1

∂x2∂y2

+∂2uf

1

∂x3∂y3

+∂2uf

2

∂y22

+∂2uf

2

∂y23

)

dy2dy3 = 0.

Then after routine transformations we obtain

[

(1 − c)km + ckf]

(

∂2u0

∂x22

+∂2u0

∂x23

)

+km

L2

∫ ∫

Ωm0

(

∂2um1

∂x2∂y2

+∂2um

1

∂x3∂y3

)

dy2dy3

+kf

L2

∫ ∫

Ωin0

(

∂2uf1

∂x2∂y2

+∂2uf

1

∂x3∂y3

)

dy2dy3 = −[

(1 − c)fm + cff]

,

(12)where c = πa2/l2 is the volume fraction of the fibres.

Approximate analytical solution for ua1 is derived below. Substituting it to

equation (12), we shall come to the homogenized heat conduction equation

〈k〉(

∂2u0

∂x22

+∂2u0

∂x23

)

= −〈f〉 , (13)

where 〈k〉 is the effective heat conductivity, 〈f〉 = (1 − c) fm + cff is the effectivedensity of heat sources. 〈k〉 can be derived after evaluation of the integrals inequation (12). In the present paper we calculate these integrals numerically inprogram package Maple using standard in-built subroutines.

We obtain an approximate analytical solution of the cell problem (10) valid forall values of the volume fractions of the components and conductivities. For thisaim an asymptotic procedure based on the boundary shape perturbation technique[17–21] is utilized. Let us introduce in the unit cell (Fig. 2) the polar co-ordinatesr =

√

y22

+ y23, θ = arctan (y3/y2). Then equations (10a), (10b) are rewritten as

∂2ua1

∂r2+

1

r

∂ua1

∂r+

1

r2

∂2ua1

∂θ2= 0; (14)

um1 = uf

1

∣

∣

∣

r=A, km ∂um

1

∂r− kf ∂uf

1

∂r=

(

kf − km) ∂u0

∂n

∣

∣

∣

∣

∣

r=A

. (15)

Bakhvalov and Panasenko have shown (see [15], Chapter 6, §3) that for axiallysymmetric domains the periodicity condition (7) can be replaced by zero boundaryconditions at the centre and at the outer boundary ∂Ω0 of the unit cell:

uf1

= 0∣

∣

∣

r=0

, (16)


Figure 2. Periodically repeated unite cell of the regular square lattice of cylindrical fibres.

um1 = 0|r=R . (17)

In equation (17) the square shape of ∂Ω0 can be determined as

R(ξ) =R0

cos(ξ), (18)

where R0 = L/2 is the radius of the inscribed circle, ξ = θ − θ0 can be consideredas a small parameter (ξ < 1), θ = θ0 − π/4 ... θ0 + π/4, θ0 = πn/2, n = 0, 1, 2, ....

Solution of the cell problem (14)–(17) is searched as an asymptotic expansionin powers of ξ. This expansion should be invariant for the interchanging ξ ↔ −ξ,thus it contains only even powers of ξ:

ua1 = ua

1,0 + ξ2ua1,2 + ξ4ua

1,4.... (19)

The boundary condition (17) is formulated at r = R(ξ). Consequently, if we nowsubstitute series (19) directly into condition (17) then the parameter ξ will beinvolved in the arguments of the functions um

1,j , j = 0, 1, 2, ..., and splitting ofthe input problem with respect to ξ will not be possible. In order to eliminateξ from the arguments of um

1,j the boundary condition (17) should be transferredfrom the original contour r = R(ξ) to the inscribed circle r = R0 by means ofTaylor expansion

um1 |

r=R0

cos(ξ)= um

1,0

∣

∣

r=R0+ξ2

R0

2

∂um1,0

∂r

∣

∣

∣

∣

r=R0

+ξ4

(

5R0

24

∂um1,0

∂r+

R20

8

∂2um1,0

∂r2

)∣

∣

∣

∣

∣

r=R0

+... .

(20)Eventually we derive:


um1 =

(

C1r + C2r−1

) ∂u0

∂n, uf

1= C3r

∂u0

∂n, (21)

where

Cp = Cp,0 + ξ2Cp,2 + ξ4Cp,4 + O(ξ6), p = 1, 2, 3. (22)

The coefficients of series (22) are:

C1,0 =λχ

1 − λχ, C2,0 = − λ

1 − λχA2, C3,0 = −λ (1 − χ)

1 − λχ; (23)

C1,2 = − λχ

(1 − λχ)2, C2,2 =

λ2χ

(1 − λχ)2A2, C3,2 =

λχ (λ − 1)

(1 − λχ)2

;

C1,4 = −1

2

λχ (1 − 3λχ)

(1 − λχ)3

, C2,4 =1

2

λ2χ(1 − 3λχ)

(1 − λχ)3

A2,

C3,4 =1

2

λχ (λ − 1) (1 − 3λχ)

(1 − λχ)3

;

where λ =(

kf − km)/(

kf + km)

, χ = A2/

R20 = c/cmax, A = ε−1a.

Let us examine a practical convergence of expansion (22). The ratios of thethird-to-second constitutive terms are the same for every p:

ξ4Cp,4

ξ2Cp,2

=1 − 3λχ

2 (1 − λχ)ξ2. (24)

Ranges of the variables in expression (24) are: −1 ≤ λ ≤ 1, 0 ≤ χ ≤ 1,

0 ≤ ξ2 ≤ (π/4)2. It can be easily seen that series (22) diverges in the case of

perfectly conductive nearly touching fibres when λ → 1, χ → 1.How to remove this singularity? The first possibility may consist in application

of Pade approximants (PAs) [23]. By definition, the [M/N ] PAs to a functiong(z) is a polynomial of degree Mdivided by a polynomial of degree N which arechosen in such a way that the leading terms of the Maclaurin series expansionof the approximants match the first M + N + 1 terms of the Maclaurin powerseries expansion of g(z). In many problems of the singular perturbation analysisPAs allow to improve essentially the rate of convergence of obtained asymptoticsolutions and to extract accurate numerical results even from divergent series. Asusual, the most effective approximations are provided by so called diagonal PAswhen the numerator and the denominator are of the same degree (M = N). Inthe case under consideration the [2/2] PAs to series (22) is:

Cp =Cp,0 +

(

Cp,2 − Cp,0Cp,4

Cp,2

)

ξ2

1 − Cp,4

Cp,2ξ2

. (25)


As the second possibility to avoid divergence at λ → 1, χ → 1 we may proposethe following approximate estimation of the overall sum of series (22). The firstterm Cp,0 (zero-order approximation) represents a solution of the cell problem(14)–(17) when the outer boundary ∂Ω0 of the unit cell is replaced by a circle ofradius R0. At this step equation (14) as well as the boundary conditions (15), (16)are satisfied strictly, meanwhile there exists a discrepancy in condition (17). Allnext terms of the series attempt to reproduce the original square shape of ∂Ω0 sothat to satisfy condition (17) more and more precisely. From the other hand, theoriginal shape of ∂Ω0 can be restored exactly in zero-order approximation if in theexpression for χ = A2

/

R20 we will substitute R0 by R(ξ) (see equation (18)).

In this case the boundary conditions (15)–(17) are held strictly and the solutionappears to converge for all values of λ and χ. Thus we obtain:

C1 =λχ cos2(ξ)

1 − λχ cos2(ξ), C2 = − λ

1 − λχ cos2(ξ)A2, C3 = −λ

[

1 − χ cos2(ξ)]

1 − λχ cos2(ξ),

(26)The obtained solution satisfies equation (14) only approximately, but further

comparisons with known numerical results show that the error of this approxima-tion is not successive.

Let us check the developed solutions in the case of perfectly conductive fibreskf

/

km = ∞, which usually presents the main computational difficulties. Table 1displays numerical results for the effective conductivity 〈k〉 evaluated on the basisof series (22) and improved expressions (25) and (26). The obtained solutions arealso compared with data of Perrins et al. [24].

Table 1. Effective conductivity 〈k〉/km of the regular square lattice of perfectly conductive(kf /km = ∞) cylindrical fibres.

cThe present solutions; Cp are determined by

Perrins et al. [24]series (22) PA (25) formulas (26)

0.1 1.210 1.247 1.223 1.2220.2 1.470 1.544 1.506 1.5000.3 1.811 1.918 1.879 1.8600.4 2.306 2.417 2.395 2.3510.5 3.270 3.145 3.172 3.0800.6 7.106 4.386 4.517 4.3420.7 73.92 7.409 7.769 7.4330.74 — 10.91 11.46 11.010.76 — 15.29 15.99 15.440.77 — 20.18 21.04 20.430.78 — 35.01 36.60 35.93

Perrins’ approach is based on the idea inspired by Lord Rayleigh [25] to describepolarization of each inclusion in an external field by an infinite set of multipole


Figure 3. Effective conductivity 〈k〉 in the perfectly regular case. Solid curves – the presentsolution (by expressions (26)); circles – results of Perrins et al. [24].

moments. Then corresponding multipole coefficients can be calculated from aninfinite set of linear equations. Application of modern computers allows to takeinto account a large number of multipoles and to improve efficiently the compu-tational accuracy of the method. However, it should be pointed out that Perrins’method can not be applicable in the limiting case of perfectly conductive nearlytouching fibres (kf

/

km = ∞, c → cmax = 0.7853...) when rapid oscillations ofthe temperature field occur on microlevel. Both of the present solutions (25) and(26) predict this case correctly with a considerably small discrepancy between eachother. Finally we assume the solution of the cell problem in the form (21), (26).The behaviour of the effective conductivity at kf

/

km → ∞, c → cmax can beverified by comparison with an asymptotic formula of O’Brien [2, 26] shown inFig. 3. The case kf

/

km = ∞ should be checked separately; at this limit we havethe following results:

volume fracture c 〈k〉, formula of O’Brien [2, 26] 〈k〉, present solution0.784 74.41 73.340.785 139.5 138.40.7853 281.0 279.3

3. Improved bounds for the random shaking-geometry composite

Now let us go on to the non-regular case. Many studies in the theory of randommedia involve certain sets of statistical rules attempting to describe the micro-geometry of the composite. Such rules are usually expected to be determined a


priori, which obviously may not be possible for the most of real materials. Onthe other hand, for the shaking-geometry composites the maximal deviation ofinclusions about a basic periodic lattice can be measured experimentally relativelyeasy. For this reason, here we utilize the deviation parameter δ as the main factordescribing the rate of non-regularity of the composite structure.

Improved bounds on the effective conductivity 〈k〉 of the random shaking-geometry composite (Fig. 1) are deduced directly from the solution obtained inSection 2. Following the theoretical results of Berlyand and Mityushev [5, 6],solution for the perfectly regular lattice is assumed as the lower bound. The upperbound is obtained by the security-spheres approach [12–14]. In both cases wesuppose that in the composite under consideration there can be no clusters of thefibres. Study of structures with clusters is beyond the scope of the present paper.However, we should note that for a purely random distribution of cylindrical fibresthe percolation threshold is reached at the volume fraction of inclusions cp ≈ 0.41[27].

Let us introduce lower K1 and upper K2 bounds on 〈k〉 such that

K1 ≤ 〈k〉 ≤ K2, (27)

and denote the conductivity of the perfectly regular material (calculated in theprevious section) as a function K0 of the volume fraction c of the fibres, i.e.〈k〉|δ=0

≡ K0(c). We also suppose kf > km. Then the lower bound K1 is givenby the solution in the perfectly regular case at δ = 0 [8, 9]:

K1 = K0(c). (28)

In order to obtain the upper bound K2 we replace the input non-regular assemblyof fibres of radius a by the regular lattice of fibres of radius a+d/2. This estimationyields:

K2 = K0(c + 2δ√

ccmax + δ2cmax). (29)

For the comparison we also note the Hashin–Shtrikman variational bounds [11]:

K1 = km +c

1/

(kf − km) + (1 − c)/(2km), (30)

K2 = kf +1 − c

1/

(km − kf ) + c/

(2kf ). (31)

Numerical examples are presented in Fig. 5. We can observe that in the dilutecase the lower bound (28) almost coincides with the Hashin–Shtrikman estimation(30). Meantime, the situation for the upper bound is different. In the low-contrastcase (kf

/

km → 1) the Hashin–Shtrikman bound (31) is better. But in the high-

contrast case (kf/

km → ∞) the improved bound (29) provides essentially betterresults, while the Hashin–Shtrikman bound (31) becomes almost useless. A simple


Figure 4. Asymptotic behaviour of 〈k〉 in the perfectly regular case at kf /km → ∞, c → cmax.Solid curves – the present solution (by expressions (26)); dash curves – formula of O’Brien [2,26].

practical recommendation is that from two upper bounds (29) and (31) the lowestone should be chosen.

4. Concluding remarks

We have derived improved bounds for the effective conductivity of a randomshaking-geometry composite with cylindrical fibres. The considered microstruc-ture corresponds to a stochastic deviation of the fibres about a regular squarelattice. In the engineering practice such a deviation may take place due to varioustechnological reasons. The advantage of the proposed geometrical model is thatthe rate of the deviation is described by a single parameter δ, which in many casescan be easily measured experimentally.

We started with the evaluation of the effective conductivity for the perfectlyregular material. For this aim the asymptotic homogenization method [15–18] hasbeen utilized; the cell problem was solved using a nontrivial modification [17, 18]of the boundary shape perturbation technique [19–21].

Then, following the theoretical results of Kozlov [10] and Berlyand and Mityu-shev [5, 6], the lower bound has been determined by the solution for the regularcase. For dilute composites this estimation is very close to the correspondingHashin–Shtrikman bound [11]. The upper bound has been obtained by means ofthe security-spheres approach [3, 12–14]; for high-contrast materials (kf

/

km → ∞)


(a)

(b)

Figure 5. Bounds on 〈k〉 in the non-regular case. Solid curves – the present solution (the lowerbound (28) at δ = 0 and the upper bounds (29) for different values of δ). Dash curves – theHashin–Shtrikman [11] bounds (30), (31). a) Dilute example, c = 0.2; b) Densely-packedexample, c = 0.7.


it provides an essentially better estimation of the effective conductivity than theHashin–Shtrikman results.

The proposed asymptotic approach may be generalized to the study of variousshaking-geometry composite structures.

Acknowledgements

This work was supported by the German Research Foundation (DFG grant no.WE 736/25-1) and by the Alexander von Humboldt Foundation (Institutionalacademic co-operation programme, grant no. 3.4-Fokoop-UKR/1070297).

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Igor V. AndrianovInstitut fur Allgemeine MechanikRheinisch-Westfalische Technische Hochschule AachenTemplergraben 64Aachen 52062Germanye-mail: igor [email protected]

Vladyslav V. Danishevs’kyyPrydniprovska State Academy of Civil Engineering and Architecture

24-a Chernyshevskogo St.Dnipropetrovs’k 49600Ukrainee-mail: [email protected]

Dieter WeichertInstitut fur Allgemeine MechanikRheinisch-Westfalische Technische Hochschule AachenTemplergraben 64Aachen 52062Germanye-mail: [email protected]

(Received: December 20, 2006; revised: August 8, 2007)

Published Online First: March 26, 2008

To access this journal online:www.birkhauser.ch/zamp

simple estimation on effective transport properties of a random composite material with cylindrical...

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