Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

49, 4, pp. 987-1001, Warsaw 2011

THE INVERSE DETERMINATION OF VOLUME FRACTION

OF FIBRES IN REINFORCED COMPOSITE WITH

IMPERFECT THERMAL CONTACT BETWEEN

CONSTITUENTS

Magdalena Mierzwiczak

Jan A. Kołodziej

Poznan University of Technology, Institute of Applied Mechanics, Poznań, Poland

e-mail: magdalena.mierzwiczak@wp.pl; jan.kolodziej@put.poznan.pl

Thepaper considers the problemof determination of the volume fraction
of fibres in an unidirectionally reinforced composite in order to provide
the appropriate effective thermal conductivity. The problem formulated
in such a way should be treated as an inverse heat transfer problem.
The thermal conductivities of constituents (fibres andmatrix) and fibres
arrangementareknown.The calculationsarecarriedout for an imperfect
thermal contact between the fibres andmatrix.

Keywords: effective thermal conductivity, inverse heat transfer problem,
boundary collocationmethod, Newtonmethod

1. Introduction

In the literature, the following problems are considered to be the classical
inverse heat conduction problems:

i) determination of heat sources (i.e. Yan et al., 2008),

ii) determination of heat transfer coefficient (i.e. Hon andWei, 2004),

iii) Cauchy’s problem (i.e. Marin, 2005),

iv) determination of temperature-dependent thermal conductivity (i.e.
Chantasiriwan, 2002),

v) determination of the unknown initial temperature (i.e. Hon, 2009).

The above mentioned problems usually apply to homogeneous media. In the
case of the composite materials (nonhomogeneous media) other practically
important issues might have to be considered. One of them is the inverse



988 M. Mierzwiczak, J.A. Kołodziej

problem of determination of the volume fraction of constituents in order to
obtain the appropriate effective thermal conductivity.

Let us consider a unidirectional fibrous composite with a regular arrange-
ment of fibres (Fig.1a). If the thermal conductivity coefficients of constituents
and their volume fractions are known then the composite can be treated as a
homogeneous region forwhich the effective thermal conductivity can be deter-
mined as a function of knownparameters. Currently, there aremany papers in
which the effective thermal conductivity coefficient is determined for a regular
arrangement of fibres for the given thermal conductivity of constituents and
volume fraction of fibres (direct problem). The method of determination is
usually based on the solution to the heat transfer equation at themicrostruc-
ture level in repeated elements of the array (i.e. Han and Cosner, 1981). In
almost all these papers, the thermal analysis of fibrous composites is based on
the assumption that thermal contact betweenfibres andmatrix is perfect.This
assumption implies continuity of temperature and heat flux on the boundary
of fibres and matrix, which simplifies the theoretical analysis of the problem.
However, in real composites, there is always a thin layer between the purefibre
and purematrixmaterial that participates in thermal interaction between the
composite constituents. To the best knowledge of the authors, the problem
of the effective thermal conductivity of the composite with imperfect thermal
contact was considered for the first time in the paper byBenveniste andMiloh
(1986), in which the Kapitza thermal resistance boundary condition was used
(direct problem). Benveniste andMiloh (1986) developed theHashimmethod
and applied it to the analysis of composites with different types of inclusions,
but not fibres. For fibrous composites with imperfect thermal contact between
constituents the determination of the effective thermal conductivity (direct
problem) was considered in the paper by Goncalves andKołodziej (1993).

The purpose of this paper is to propose an analytic-numerical algorithm
for determination of the volume fraction of fibres in order to obtain a given
value of the transverse effective thermal conductivity λz. Such a problem can
be treated as one kind of the inverse heat conduction problem for a composite
material. Because the algorithm for the inverse problem is, in some sense,
based on the solution to the direct problem, a brief outline of this paper is
as follows: in Section 2 the direct problem is considered in a similar way as
First Model in the paper by Goncalves and Kołodziej (1993). The inverse
problem of determination of the volume fraction of fibres in the composite for
a given effective thermal conductivity is considered in Section 3. In Section 4,
we present numerical results for both direct and inverse problems. Finally,
concluding remarks are given in Section 5.



The inverse determination of volume fraction... 989

2. Direct problem: determination of the effective thermal

conductivity for the composite with an imperfect thermal

contact between the components

In real composite materials, the contact surface between the components is a
transitional layer of a certain thicknesswithmechanical, thermal and chemical
interactions between the components. The transition layer of a finite thickness
is taken into account in the formulation of contact condition of thefibre-matrix
in the form of so-called Kapitza condition

λf
∂T̃f
∂r
=β(T̃m− T̃f) (2.1)

where T̃f, T̃m are the temperature field of the fibre and matrix respectively,
λf is the thermal conductivity offibres, β is the factor determining the thermal
contact resistance components of the composite.When β→ 0, we have a case
of ideal insulation, and when β → ∞ there is perfect contact between the
components.

Fig. 1. A unidirectional reinforced fibrous composite with fibre arrangement
according to a square array for the imperfect thermal contact between the fibre and
matrix: (a) general view, (b) formulation of the nondimensional boundary value

problem in a repeated element

Let us consider a unidirectional compositewith fibres arranged in amatrix
in a regular, square arraywith an imperfect thermal contact between the fibre
and matrix (Fig.1a), where a is the radius of the fibres, 2b is the distance
between neighbouring fibres, E = a/b, and ϕ=πE2/4 is the volume fraction



990 M. Mierzwiczak, J.A. Kołodziej

offibres.Theratio of thermal conductivity offibres λf tomatrix λm is denoted
as F = λf/λm, R= r/b is the dimensionless radius, (X = x/b,Y = y/b) are

dimensionless Cartesian coordinates, Tf/m = (T̃f/m − T̃R)/(T̃L − T̃R) is the

dimensionless temperature field, T̃L, T̃R are the temperature on the left and
on the right boundary in a repeated element, respectively.

In order to solve the nondimensional boundary value problem in the re-
peated element of the composite (Fig.1b), the boundary collocation method
is used (Kołodziej and Zieliński, 2009). The general solution to the Laplace
equation ∇2T =0 in the polar coordinate system (R,θ) has the form

T =A0+A1θ+A2θ lnR+A3 lnR

+
∞∑

k=1

[(BkR
k+CkR

−k)cos(kθ)+(DkR
k+EkR

−k)sin(kθ)]
(2.2)

where A0,A1,A2,A3,Bk,Ck,Dk,Ek are integral constants.

Given the repeated element of the square array Ω=Ωf∪Ωm in the region
of the fibre Ωf and thematrix Ωm, the solution is predicted to has form (2.2).
Some of the constants must be determined strictly by the conditions at the
bottom and on the left side of the repeated element

∂Tf
∂θ
=
∂Tm
∂θ
=0 for θ=0

Tf =Tm =1 for θ=
π

2

(2.3)

and the contact conditions of fibre-matrix

F
∂Tf
∂R
=
∂Tm
∂R

for R=E

∂Tf
∂R
=Bi(Tm−Tf) for R=E

(2.4)

where Bi= bβ/λf is the dimensionless resistance number.

After determining the constants from boundary conditions (2.3) and from
contact conditions of fibre-matrix (2.4), marking the remaining constants
as wk, (w1 = B1,w2 = B3, . . . ,wk = B2k−1, . . .) and cutting off an infinite
number of the test functions to N expressions, we obtain a solution for the
temperature field of the fibre andmatrix



The inverse determination of volume fraction... 991

Tf =1+
N∑

k=1

wkR
(2k−1)cos[(2k−1)θ]

(2.5)

Tm =1+
N∑

k=1

wk
2

[
(F +1+γk)R

(2k−1)− (F −1−γk)
E2(2k−1)

R(2k−1)

]
cos[(2k−1)θ]

where γk =(2k−1)/(BiE).

The constants wk are determined by fulfilment of the condition of col-
location points on the upper Γ2 and on the right Γ1 edge of the concerned
region

Tm =0 for X =1

∂Tm
∂Y
=0 for Y =1

(2.6)

Condition (2.6)2 can be written for polar coordinates

∂Tm
∂Y
=
∂Tm
∂R
sinθ+

1

R

∂Tm
∂θ
cosθ=0 (2.7)

Choosing N1 points on the right boundary Γ1 and N2 points on the upper
boundary Γ2 and collocating conditions (2.6)1 and (2.7), we obtain a system
of linear equations N1+N2 with N unknown constants wk, k=1, . . . ,N:

— (Rj,θj)∈Γ1, j=1, . . . ,N1

N∑

k=1

wk
[
(F+1+γk)R

(2k−1)
j −(F−1−γk)

E2(2k−1)

R
(2k−1)
j

]
cos[(2k−1)θj] =−2 (2.8)

— (Rj,θj)∈Γ2, j=1, . . . ,N2

N∑

k=1

wk(2k−1)
[
(F +1+γk)R

2(k−1)
j sin[2(k−1)θj]

− (F −1−γk)
E2(2k−1)

R2kj
sin(2kθj)

]
=0

(2.9)

The constants wk obtained by the Gaussian elimination method provide es-
timates of the value of the global heat flux through the unit region of the
considered element

q=
1

b

(
−λf

a∫

0

∂T̃f
∂x

∣∣∣
x=0
dy−λm

b∫

a

∂T̃m
∂x

∣∣∣
x=0
dy

)
(2.10)



992 M. Mierzwiczak, J.A. Kołodziej

The transverse effective thermal conductivity is defined by the formula

λz =
qb

∆T̃
(2.11)

where b is the distance between the isothermal boundaries, ∆T̃ = T̃L− T̃R is
the difference of temperature at the isothermal edges.

After taking into consideration in formula (2.11) the definition of the non-
dimensional temperature and coordinates, the value of effective thermal con-
ductivity in relation to the thermal conductivity of the matrix can be calcu-
lated from the relationship

λz
λm
=−F

E∫

0

1

R

∂Tf
∂θ

∣∣∣
θ=π
2

dR−

1∫

E

1

R

∂Tm
∂θ

∣∣∣
θ=π
2

dR (2.12)

or

λz
λm
=
N∑

k=1

wk
2
(−1)k[(F +1+γk)+(F −1−γk)E

2(2k−1)] (2.13)

3. Inverse problem: determination of the volume fraction of fibres

in composite for a given effective thermal conductivity

When designing composites with specific properties of fibres and matrix, we
must estimate the fraction of volume of fibres to obtain effective thermal con-
ductivity values. Assuming that E = a/b is unknown,we use the known value
of effective thermal conductivity in relation to the thermal conductivity of the
matrix λz/λm. From the collocation of the boundary conditions in N1+N2
points on the right Γ1 andupper Γ2 edge of the consideration region and from
condition (2.13), we obtain a systemof N1+N2+1 non-linear equationswith
N+1 the unknowns wk,E:

— (Rj,θj)∈Γ1, j=1, . . . ,N1

fj(wk,E)= 1+
N∑

k=1

wk
2

[
(F +1+γk)R

(2k−1)
j

− (F −1−γk)
E2(2k−1)

R
(2k−1)
j

]
cos[(2k−1)θj] = 0

(3.1)



The inverse determination of volume fraction... 993

— (Rj,θj)∈Γ2, j=1, . . . ,N2

fN1+j(wk,E)=
N∑

k=1

wk
2
(2k−1)

[
(F +1+γk)R

2(k−1)
j sin[2(k−1)θj]

(3.2)

−(F −1−γk)
E2(2k−1)

R2kj
sin(2kθj)

]
=0

and

fN1+N2+1(wk,E)=
N∑

k=1

wk
2
(−1)k[(F +1+γk)

+(F −1−γk)E
2(2k−1)−

λz
λm
=0

(3.3)

Thenon-linear systemof N+1equations f(wk,E)with N1+N2+1unknowns
w= [w1, . . . ,wN,E]

⊤ is solved by using the Newton iteration method





w1
...
wN
E





(i+1)

=





w1
...
wN
E





(i)

−




f
(i)
1
...

f
(i)
N1+N2

f
(i)
N1+N2+1







J
(i)
1,1 · · · J

(i)
1,N+1

... · · ·
...

J
(i)
N1+N2,1 · · ·

...

J
(i)
N1+N2+1,1 · · · J

(i)
N1+N2+1,N+1




−1

w
(i+1) =w(i)− f(w(i))J(w(i))−1 →w(i+1) =w(i)−Y(w(i)) (3.4)

Y(w(i))= f(w(i))J(w(i))−1 →J(w(i))Y(w(i))= f(w(i))

The functions fi are described by equations (3.1)-(3.3), while Jacobi elements
have the following form:
— k=1, . . . ,N, j=1, . . . ,N1

Jj,k =
∂fj
∂wk
=
1

2

[
(F +1+γk)R

(2k−1)
j − (F −1−γk)

E2(2k−1)

R
(2k−1)
j

]
cos[(2k−1)θj]

(3.5)
— j=1, . . . ,N1

Jj,N+1 =
∂fj
∂E
=
N∑

k=1

wk(1−2k)

2

·
[R(2k−1)j
E2Bi

+
(
2(F −1)−

4k−3

EBi

)E(4k−3)

R
(2k−1)
j

]
cos[(2k−1)θj]

(3.6)



994 M. Mierzwiczak, J.A. Kołodziej

— k=1, . . . ,N, j=1, . . . ,N2, i= j+N1

Ji,k =
∂fi
∂wk
=
(2k−1)

2

[
(F +1+γk)R

2(k−1)
j sin[2(1−k)θj]

+(F −1−γk)
E2(2k−1)

R2kj
sin(2kθj)

] (3.7)

— j=1, . . . ,N2, i= j+N1

Ji,N+1 =
∂fi
∂E
=
N∑

k=1

wk
(2k−1)2

2

[R2(k−1)j
E2Bi

sin[2(k−1)θj]

+
(
2(F −1)−

4k−3

EBi

)E4k−3

R2kj
sin(2kθj)

]
(3.8)

— k=1, . . . ,N, i=N1+N2+1

Ji,k =
∂fi
∂wk
=
1

2
(−1)k[(1+F +γk)+(F −1−γk)E

2(2k−1)] (3.9)

— i=N1+N2+1

Ji,N+1 =
∂fi
∂E
=
N∑

k=1

wk
2k−1

2
(−1)k

[(
2(F −1)−

4k−3

EBi

)
E4k−3−

1

E2Bi

]

(3.10)
To start the Newton iteration, we need to know w(0) = [w01, . . . ,w

0
N,E

0]⊤ as
the initial condition.As the initial value of the constants {w0k}

N
k=1, the solution

to the linear problem for E = 0.1 has been adopted. The condition for the
end of iteration has been set at: δNewton = ‖w

(i+1)−w(i)‖max ¬ 10
−7, where

‖ ·‖max means themaximum norm.

4. Results of the numerical experiment

For the direct problem Fig.2 shows the values of the effective thermal con-
ductivity in relation to the thermal conductivity of the matrix λz/λm =
=λz/λm(ϕ)|F,Bi as a function of the volume fraction of fibres ϕ for the assu-
med value of the coefficient F ∈ {0.5,1,2,10,20} and the resistance number
Bi∈{0.001,0.01,0.1,1,10,1000}.
For a very small value of the resistance number Bi= 0.001 and Bi=0.01

with the increasing volume fraction of fibres ϕ in the matrix for all tested



The inverse determination of volume fraction... 995

Fig. 2. The effective thermal conductivity in relation to the thermal conductivity of
the matrix as a function of the volume fraction of fibres in the matrix for different
values of the ratio of thermal conductivity fibres to the matrix and different values

of resistance number

values of F, the effective thermal conductivity decreases and the composite
becomes a perfect insulator. For Bi= 1000 the graph obtained corresponds to
the conditions close to ideal thermal contact between the components of the
composite.

For the components of the composite with the identical values of the ther-
mal conductivity F = 1, one can notice that for Bi = 10 the obtained value
of the effective thermal conductivity is close to unity, which indicates the cor-
rectness of the results.

The value of the effective thermal conductivity in relation to the thermal
conductivity of the matrix λz/λm, except for the constants characterizing
the composite F,E,Bi, also depends on the constants wk arising from the
approximate fulfilment of the boundary conditions at the collocation points.



996 M. Mierzwiczak, J.A. Kołodziej

Table 1 shows the influence of the number of the collocation points on the
maximum error fulfilling the collocation boundary conditions calculated at
the control points (between collocation points).

Table 1. Impact of the number of collocation points on the effective thermal
conductivity of the composite and themaximal error of fulfilling the boundary
conditions at control points

N1 N2
E =0.5, F =10, Bi= 1 E =0.9, F =10, Bi= 0.01

λz/λm δmax
∣∣
Tm=0

δmax
∣∣
∂Tm

∂Y
=0
λz/λm δmax

∣∣
Tm=0

δmax
∣∣
∂Tm

∂Y
=0

5 4 1.2365 7.084E-05 8.834E-04 0.2857 4.238E-03 9.794E-02

6 5 1.2365 1.542E-05 2.613E-04 0.2865 2.940E-03 8.168E-02

7 6 1.2365 2.888E-07 4.599E-05 0.2861 9.246E-04 2.783E-02

8 7 1.2365 5.729E-06 2.352E-04 0.2862 6.949E-05 5.838E-03

9 8 1.2365 1.216E-05 5.404E-04 0.2863 1.475E-03 6.737E-02

10 9 1.2365 3.057E-05 1.524E-03 0.2860 4.651E-03 2.332E-01

11 10 1.2365 4.499E-05 2.455E-03 0.2867 7.300E-03 3.991E-01

12 11 1.2365 1.398E-04 8.409E-03 0.2851 2.287E-02 1.376E+00

13 12 1.2365 1.154E-04 7.488E-03 0.2872 1.924E-02 1.249E+00

14 13 1.2366 1.222E-03 8.621E-02 0.2778 1.918E-01 1.354E+01

15 14 1.2364 2.727E-04 2.055E-02 0.2883 4.592E-02 3.460E+00

N1 N2
E=0.5, F =0.5, Bi= 10 E =0.9, F =0.5, Bi= 0.1

λz/λm δmax
∣∣
Tm=0

δmax
∣∣
∂Tm

∂Y
=0
λz/λm δmax

∣∣
Tm=0

δmax
∣∣
∂Tm

∂Y
=0

5 4 0.8504 4.570E-05 6.018E-04 0.2381 4.694E-03 1.082E-01

6 5 0.8504 1.009E-05 1.822E-04 0.2389 3.259E-03 9.036E-02

7 6 0.8504 1.204E-07 2.532E-05 0.2386 1.024E-03 3.074E-02

8 7 0.8504 3.599E-06 1.474E-04 0.2386 8.080E-05 6.646E-03

9 8 0.8504 7.689E-06 3.417E-04 0.2388 1.642E-03 7.498E-02

10 9 0.8504 1.935E-05 9.645E-04 0.2384 5.171E-03 2.593E-01

11 10 0.8504 2.848E-05 1.554E-03 0.2391 8.117E-03 4.438E-01

12 11 0.8504 8.848E-05 5.322E-03 0.2375 2.541E-02 1.529E+00

13 12 0.8504 7.302E-05 4.739E-03 0.2397 2.140E-02 1.390E+00

14 13 0.8503 7.732E-04 5.456E-02 0.2300 2.122E-01 1.498E+01

15 14 0.8504 1.726E-04 1.301E-02 0.2408 5.110E-02 3.851E+00



The inverse determination of volume fraction... 997

The best results are obtained for 7 collocation points at the right edge Γ1
and for 6 points at the upper edge Γ2 of the repeated element for E = 0.5,
8 points on the right-hand side and 7 points at the upper edge for E = 0.9.
The increase in the number of the collocation points does not improve the
quality of the results.

The results of the iterative calculation of the volume fraction of fibres for
the composite are shown in Fig.3. The value of volume fraction of fibres ϕ
in composites is presented as a function of the effective thermal conductivity
in relation to the thermal conductivity of the matrix ϕ=ϕ(λz/λm)

∣∣
F,Bi for

specific values of the coefficient F ∈ {0.5,1,2,10,20} and resistance number
Bi= {0.001,0.01,0.1,1,10,1000}.

Fig. 3. The volume fraction of fibres in the matrix as a function of the effective
thermal conductivity in relation to the thermal conductivity of the matrix of the
composite for different relative values of the thermal conductivity of the fibre and

matrix and different values of the resistance number



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λz/λm =1.24, F =10, Bi=1 λz/λm =0.285, F =10, Bi= 0.01

ϕ E δmax
∣∣
Tm=0

δmax
∣∣
∂Tm

∂Y
=0

ϕ E δmax
∣∣
Tm=0

δmax
∣∣
∂Tm

∂Y
=0

5 4 0.1985 0.5027 7.19E-05 8.96E-04 0.6370 0.9006 4.30E-03 9.95E-02

6 5 0.1985 0.5027 1.56E-05 2.65E-04 0.6381 0.9013 3.07E-03 8.52E-02

7 6 – – – – 0.6376 0.9010 9.61E-04 2.89E-02

8 7 – – – – 97.0018 11.1133 3.31E-01 1.35E+01

9 8 – – – – – – – –

N1 N2
λz/λm =0.85, F =0.5, Bi= 10 λz/λm =0.24, F =0.5, Bi= 0.1

ϕ E δmax
∣∣
Tm=0

δmax
∣∣
∂Tm

∂Y
=0

ϕ E δmax
∣∣
Tm=0

δmax
∣∣
∂Tm

∂Y
=0

5 4 0.1969 0.5007 4.58E-05 6.04E-04 0.6340 0.8985 4.51E-03 1.04E-01

6 5 0.1969 0.5008 1.01E-05 1.83E-04 0.6349 0.8991 3.17E-03 8.79E-02

7 6 0.1969 0.5008 1.20E-07 2.54E-05 0.6345 0.8988 9.80E-04 2.94E-02

8 7 0.1969 0.5008 3.61E-06 1.48E-04 0.9252 1.0854 1.14E+00 4.94E+01

9 8 0.1969 0.5008 7.71E-06 3.43E-04 – – – –

10 9 0.1969 0.5007 1.94E-05 9.67E-04 – – – –

11 10 0.1969 0.5008 2.86E-05 1.56E-03 – – – –



The inverse determination of volume fraction... 999

Similarly, as in the problem of identification of λz/λm, the iterative iden-
tification of the volume fraction of fibres ϕ in a composite, the number of
collocation points N1+N2, where the boundary condition is fulfilled in an
approximate manner, affects the accuracy of the calculations. Table 2 shows
the impact of the number of collocation points on the value of volumetric
fraction of fibres ϕ in the composite and the maximum error of fulfilment of
boundary conditions at the control points.

In the inverse problem, the impact of the number of collocation points on
thequality of the results is greater than in thedirect problem.Exceeding a cer-
tain value of the number of collocation points (different for different examples
– see Table 2) leads to lack of convergence of the calculations (bad conditio-
ning of the collocation matrix) or to obtaining a very large error of boundary
conditions.

While solving the inverse problemdetermining thevolume fraction of fibres
using iterative methods, the convergence of the method should be examined.
Table 3 shows the convergence of the used Newton iterative method for four
test examples. For the composite with a surface resistance of the fibre-matrix
boundary used in determining the volume fraction of fibres, the Newton me-
thod is not fast.Nevertheless it should be noted that themethod is acceptably
convergent because after amaximumof 10 iterations for the test examples the
correct results were obtained with an error less than 10−7.

Table 3.Convergence of the Newton method for the four test examples

It
er
a
ti
o
n F =10 F =0.5

λz/λm =1.24 λz/λm =0.285 λz/λm =0.85 λz/λm =0.24

Bi= 1 Bi=0.01 Bi=10 Bi=0.1

δNewton E δNewton E δNewton E δNewton E

1 4.28E+00 0.1000 2.33E+00 0.1000 2.71E+00 0.1000 2.44E+00 0.1000

2 1.84E+00 4.3809 1.22E+00 2.4306 2.38E+00 0.9689 9.92E-01 2.5430

3 1.83E+00 4.1689 5.21E-01 1.2077 2.85E-01 0.8492 1.16E+00 1.5510

4 1.11E+00 2.3362 1.81E-01 0.6864 6.01E-02 0.5646 2.54E-01 0.3907

5 6.11E-01 1.2267 3.23E-02 0.8670 3.80E-03 0.5046 1.67E-01 0.6444

6 8.16E-02 0.6153 1.67E-03 0.8993 1.69E-05 0.5008 8.18E-02 0.8112

7 3.00E-02 0.5337 2.62E-06 0.9010 3.35E-10 0.5008 5.83E-03 0.8930

8 9.83E-04 0.5037 9.04E-12 0.9010 – – 2.53E-06 0.8988

9 1.64E-06 0.5027 – – – – 2.91E-10 0.8988

10 3.87E-12 0.5027 – – – – – –



1000 M. Mierzwiczak, J.A. Kołodziej

5. Conclusions

Thepresentedmethod of determining the volume fraction of fibres of a compo-
site or the effective thermal conductivity, except for the cases of themaximum
fibre density and the excellent thermal resistance at the border of the com-
ponents, is easy to implement and efficient. It can be easily applied to other
configurations of the regular arrangement of fibres in the matrix, for example
to a triangular or hexagonal mesh. The study compared the influence of the
ratio of the thermal conductivity of fibres to the thermal conductivity of the
matrix F and the influence of the resistance number Bi – a parameter of the
surface resistance on the border of fibre-matrix, on the value of the volumetric
fraction of fibres and the value of the effective thermal conductivity of the
composite. It was also shown that the increasing of the number of collocation
points does not reduce the error of approximation of the boundary conditions
while it leads to the ill-conditioning of the system of equations.

Acknowledgments

The second author made this work in frame a Grant 4917/B/T02/2010/39 from

the PolishMinistry of High Education.

References

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Odwrotne określenie objętościowego udziału włókien we wzmocnionym

kompozycie z niedoskonałym kontaktem termicznym pomiędzy

składnikami

Streszczenie

Wpracy rozważa się problemokreślenia objętościowegoudziałuwłókienw jedno-
kierunkowowzmocnionym kompozycie w celu uzyskania odpowiedniego efektywnego
współczynnika przewodzenia ciepła. Problem sformułowany w ten sposób jest trak-
towany jako odwrotny problem przewodzenia ciepła. Współczynniki przewodzenia
ciepła składników (włókien i matrycy) oraz sposób ułożenia włókien są znane. Ob-
liczenia są wykonane dla niedoskonałego kontaktu termicznego pomiędzy włóknami
i matrycą.

Manuscript received September 27, 2010; accepted for print December 16, 2010