ap-6-10.dvi


Acta Polytechnica Vol. 50 No. 6/2010

Fractality of Fracture Surfaces

T. Ficker

Abstract

A recently published fractal model of the fracture surfaces of porous materials is discussed, and a series of explanatory
remarks are added. The model has revealed a functional dependence of the compressive strength of porous materials on
the fractal dimension of fracture surfaces. This dependence has also been confirmed experimentally. The explanatory
remarks provide a basis for better establishing the model.

Keywords: fractal dimension, fracture surfaces, porous materials, compressive strength.

1 Introduction
Mandelbrot and his co-workers [1] started fractal re-
search of fracture surfaces of solid materials. Af-
ter their pioneering work had been published, many
other authors [2, 3, 4, 5] tried to correlate the frac-
tal properties of fracture surfaces with the mechan-
ical quantities of materials. Due to the complexity
of these surfaces, especially with composite materi-
als, the results of such studies were not consistent,
sometimes even contradictory. Good examples of
such complicated surfaces are the fracture surfaces
of cementitious materials. Nevertheless, they are ex-
tensively studied [6, 7].
Recently a fractal model of the fracture surfaces

of porous materials has been published [8, 9, 10]. Its
functionality has been tested andprovedwith porous
cementitious materials. One of the most important
results of themodel consists in the relation σ = f(D)
which is the dependence of the compressive strength
σ ofmaterials on the fractal dimension D of fracture
surfaces. This finding may be of practical impor-
tance, since it indicates the possibility of estimating
compressive strength on the basis of the fractal ge-
ometry of fracture surfaces.
The aim of this paper is to provide necessary

explanations and comments on particular steps per-
formed within the derivation of the model [8–10] in
order tomake transparent all its parts. After a short
overview of the basic relations of the model (sec-
tion 2), a series of explanatory sections 3.1–3.3 fol-
lows.

2 Outline of the model
A short sketch of the fractal model [8, 9, 10] of
fracture surfaces of porous materials is presented
here. The content of this section has its source in
Ref. [10], which is the most recent presentation of
the model.

2.1 Fractal porosity

The large class of porousmaterials possesses at least
one common feature, namely, they are composed of
grains (particles, globules, etc.) ofmicroscopic size l.
The grains are usually arranged fractally with num-
ber distribution N(l) and fractal dimension D

N(l)=

(
L

l

)D
, l < L. (1)

Assuming the volume of a globule to be v =const · l3
and the volumeof a sample V =const·L3, the poros-
ity P of the cluster may be derived as follows

P =
V − N · v

V
=1− N

v

V
=1− N

(
l

L

)3
=

1−
(

L

l

)D (
l

L

)3
=1−

(
l

L

)3−D
.

(2)

In general, the porosity P of a sample with a charac-
teristic size Λ, stochastically scattered fractal clus-
ters of sizes {Li}i=0,1,2,...,n � Λ and dimensions
{Di}i=0,1,2,...,n reads

P =1−
n∑

i=0

ξi

(
li
Li

)3−Di
, ξi = miL

3/Λ3 (3)

where mi is the number of fractal clusters with di-
mension Di.

2.2 Fractal compressive strength

The relations for estimating the compressive strength
of porous materials published earlier [11, 12] rely on
porosity P as a main decisive factor. In the frac-
tal model under discussion the Balshin [11] relation
σ = σ∗o(1 − P)

k was used as a starting point. The
relation was developed [10] into a more general form

σ = σ∗o

(
1−

P

Pcr

)k
+so = σo(1− P − b)k +so, (4)

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Acta Polytechnica Vol. 50 No. 6/2010

0 ≤ b =1− Pcr ≤ 1, 0 ≤ σ∗o ≤ σo =
σ∗o
P kcr

(5)

where so is the remaining strength which may be
caused, among others, by the virtual incompressibil-
ity of pore liquids.
Combining (3) and (4), the compressive strength

of porous matter appears as a function of the fractal
structure

σ = σo

[
n∑

i=0

ξi

(
li
Li

)3−Di
− b

]k
+ so. (6)

2.3 Dimension of fracture surface

When performing the fracture of a porous material
whose inner (volume) structure has fractal dimen-
sion Di, this structure is projected onto the fracture
surface with a lower dimension D∗i < Di. Provided a
fracture surface has its owndimension S and itsmor-
phology is ‘typical’ rather than ‘special’, the relation
between D∗i and Di can be expressed [13] as follows

Di =max{0, D∗i +(3− S)} , D
∗
i ≤ S < 3 (7)

where 3 − S is the co-dimension of the fracture sur-
face. Using (7), the exponent 3 − Di in Eq. (6) can
be replaced by S − D∗i and the generalized strength
function now reads

σ = σo

[
n∑

i=0

ξi

(
li
Li

)S−D∗i
− b

]k
+ so. (8)

This function may containmany parameters, so that
it is difficult to fit it to the experimental data, be-
cause there may be more than one ‘reliable’ set of
parameters σo, {ξi}ni=1, li, Li, S, D

∗
i , b, k, so. For-

tunately, the structure of porous material often con-
tains only one type of grain, i.e. one type of frac-
tal arrangement (i = 1) dominates over the solid
rest (i = 0) which is usually of non-fractal charac-
ter (Do =3)

σ =σo

[
ξ1

(
l1
L1

)S−D∗1
+(ξo − b)

]k
+ so =

σo [ξ1 exp((S − D∗1)/A) − γ]
k
+ so,

(9)

A =
1

ln(L1/l1)
, γ =(b − ξo).

2.4 Experimental tests

Relation (9) is directly applicable to samples of hy-
dratedPortland cement paste, since it is a composite
whose main component (Calcium-Silicate-Hydrate
gel) is known for its inner fractal structure. Other
components can be assigned to a non-fractal rem-
nant. Therefore, this material was used [8, 9, 10] to
test the functionality of Eq. (9).

Fig. 1: Dependence of compressive strength on fractal
dimension with cement paste [8]

Seventy-two samples of hydrated ordinary Port-
land cement paste of various water-to-cement ratio
r (0.4, 0.6, 0.8, 1.0, 1.2, 1.4) were prepared. After
28 days of hydration the samples were subjected to
three-point bending and were fractured. The frac-
ture surfaces were used for further fractal analysis.
The 3-D digital reconstruction of the fracture sur-
faceswasperformedusinga confocalmicroscope, and
then a series of horizontal sections (contours) were
analyzed with resolution 0.2 μm2/pixel by means of
the standardbox-countingmethod [8, 9, 10] to obtain
a representative D∗ for the particular surface. The
box-counting analyses were performed in the length
interval 〈2 μm,300 μm〉. The second parts of the
fractured sampleswere cut into small cubes and sub-
jected to destructive tests to determine their com-
pressive strength values σ.
It is known that cement pastes of higher water-

to-cement ratio suffer from sedimentation of cement
clinker grains and bleeding, which may lead to lower
homogeneity and modified w/c ratio. The influ-
ences of these effects on strengthmeasurements have
been partly suppressed by the fact that sampleswith
higher w/c are localized on the strength curve in the
less sensitive region in which the curve is bent and
asymptotically approaches the horizontal direction.
The surface inhomogeneity has been partly compen-
sated by taking microscopic images from different
sites on the surface.
Samples prepared with different water-to-cement

ratio r possess different porosity. From cement tech-
nology it is well-known that with increasing ratio r
the porosity increases. Naturally, this will change
the dimensions of the projected patterns D∗. Six
groups of sampleswith different r means six different
D∗ at which we are able to measure the dependence
σ(D∗) and check it according to Eq. (9). The re-
sult can be seen in Fig. 1 along with all the fitting

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Acta Polytechnica Vol. 50 No. 6/2010

parameters. Since the assumed analytical form (9)
of the dependence σ(D∗) has been reproduced well,
onemay conclude that compressive strength is one of
thosemechanical quantities whose value is ‘coded’ in
the surface arrangement of the fractured samples of
porous materials.

3 Explanatory remarks and
discussion

The following paragraphs provide comments on the
proposed concept of fractal compressive strength in
order to clarify all its crucial points.

3.1 Derivation of fractal porosity

Thederivationof fractal porosity startswithEq. (1),
which determines the number N of fractal elements
on the length scale l. The length L has been taken
as a reference scale standing for the largest possible
scale on which only one fractal element is present
(the so-called initiator, to use Mandelbrot’s nomen-
clature [14]). In short, the object under discussion
shows power law behavior (1) only within a limited
length interval (l, L) whose borders l and L were
coined by Mandelbrot [1] as the ‘inner’ and ‘outer’
cutoffs. Beyondthis interval the objectbehaves asan
ordinary non-fractal Euclidean body. On the small-
est length scale l we can ‘see’ a great number (N(l))
of basic building elements of size l, and as we go to
larger length scales, the number of corresponding el-
ements decreases. At the length scale l = L there
is only one element, i.e. No = 1. This is a com-
mon property of all self-similar fractals and can be
demonstrated very instructively with all determin-
istic fractals [15], e.g., the Cantor set, Koch curve,
Menger sponge, etc.
To explain the origin of Eq. (1), it is nec-

essary to go to the definition of a fractal mea-
sure and a fractal dimension. The most general
definitions of these quantities are those of Haus-
dorff [16]. However, his definitions are rather sophis-
ticated and not convenient for computer implemen-
tation. For software processing there are some mod-
ifications, among which the box-counting procedure
is frequently used [17, 18, 19, 20].
The box-countingmeasure M is given as a sumof

d-dimensional ‘boxes’ (ld) needed to cover the frac-
tal objects embedded in the E-dimensionalEuclidean
space. The boxes are parts of the d-dimensional net-
work created in the Euclidean space:

M =
N∑

i=1

ld = N · ld =exp(lnN) · ld =

[exp(ln l)]
ln N
ln l · ld = l

ln N
ln l · ld = ld−

ln N
ln(1/l) =

ld−D → lim
l→0

ld−D =

{
∞, d < D
0, d > D

}
. (10)

The fractal box-counting dimension is defined by the
point of discontinuity of the function M(d). Accord-
ing to Eq. (10), this is just the point

d = D =
lnN
ln(1/l)

(11)

where themeasure M abruptly changes its value from
infinity to zero. From such a defined dimension D it
is easy to express the number N of fractal elements
whose size is equal to l or L

N(l) = l−D, (12)

N(L) = L−D = No. (13)

Combining (12) and (13), we obtain

N(l)= No

(
L

l

)D
. (14)

Bearing in mind that No belongs to the ‘initiator’ of
the fractal object, i.e. No =1, we obtain

N(l)=

(
L

l

)D
(15)

with a total interval of fractality (l, L). Relation (15)
is in fact Eq. (1), which was a starting point in
deriving fractal porosity in section 2.1. The func-
tionality of (15) can be easily verified using deter-
ministic fractals [15]. For example, the Koch curve
(D = ln4/ ln3) in its thirdgenerationhas N3 =4

3 el-
ementswith the length l3 =1/3

3. The number N3 =
43 can be obtained fromEq. (15) by inserting L =1,
l3 = 1/3

3 and D = ln4/ ln3 which give the follow-

ing result N3 =
[
1/(1/33)

]D
= 33D = 33ln4/ ln3 =

e3ln3·ln4/ ln3 = e3ln4 = 43 in full agreement with
whathadbeen expected. If the length L of the initia-
tor is different from one, however, the result remains

unchanged, i.e. N3 =
[
L/

(
L/33

)]D
=33D. . .

As far asEq.(2) is concerned,wemayraise aques-
tion about its validity if the basic building elements
of size l are small spheres tightly packed in the Eu-
clidean space. Due to the compactness of the struc-
ture it holds D = 3. Eq. (2) then yields P = 0
instead of P > 0, which would be expected since
there are always gaps between spheres, regardless of
their type of space arrangement. Herewe shouldbear
in mind that spheres of finite diameter cannot gen-
erate a true fractal since it requires the presence of
an infinitely fine structure, i.e. l → 0. In such a
case Eq. (2) provides P = 1 − 00 which is, how-
ever, an uncertain expression that allows no mathe-
matical decision to be made. Nevertheless, the con-
dition l → 0 ensures that with tight arrangement

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Acta Polytechnica Vol. 50 No. 6/2010

there are no gaps between ‘spheres’, since ‘point-like
spheres’ completely fill in the Euclidean space and,
thus, porosity must be zero. This means that the
uncertain expression P = 1 − 00 should also con-
verge to zero. In addition, performing the same pro-
cedurewith small cubes instead of small spheres, the
value D = 3 (tight arrangement) can be attained
even with cubes of finite size (l > 0), and the cor-
responding porosity is then exactly zero (P = 0),
as required. Therefore, the non-zero porosity in the
case of ‘tightly’ packed spheres offinite size is a con-
sequence of a shape artifact and not of erroneous be-
havior of Eq. (2).
Whendealingwitha compositematerial, not each

of its components is a fractal and not each fractal
cluster is delocalized over the whole sample. For this
reason, it is necessary to assume that the sample con-
sists of sets (i = 0,1,2, . . . , n) of both fractal and
non-fractal clusters whose characteristic sizes Li are
smaller than or at most equal to the size Λ of the
sample (Fig. 2). If mi denotes a number of clusters
of the i-th type (either fractal or non-fractal), the
porosity P can be derived by analogy with Eq. (2)

P =
V −

∑n
i=0 mi · Ni · vi

V
=

Λ3 −
∑n

i=0 mi ·
(

Li
li

)Di
· l3i

Λ3
=

1−
n∑

i=0

mi ·
(

Li
li

)Di
· l3i

Λ3
= (16)

1−
n∑

i=0

(
miL

3
i

Λ3

)
·
(

li
Li

)3−Di
=

1−
n∑

i=0

ξi ·
(

li
Li

)3−Di

where ξi = miL
3
i /Λ

3. Eq. (16) includes all pos-
sibilities of fractal, non-fractal or mixed arrange-
ments. For example, when only n+1 fractal compo-
nents exist and are delocalized over thewhole sample
(Li =Λ, mi =1, ξi =1), Eq. (16) reads

P =1−
n∑

i=0

(
li
Li

)3−Di
=1−

n∑
i=0

(
li
Λ

)3−Di
(17)

If there are only n + 1 non-fractal compact com-
ponents localized inside the sample (Li < Λ), then
porosity assumes a common ‘non-fractal’ form

P =1−
n∑

i=0

ξi, ξi =
miL

3
i

Λ3
=

vi
V

. (18)

If fully delocalized compact (i.e., non-fractal) compo-
nents (Li = Λ) are considered, then, naturally, only
one of them can be taken into account since no more
than a single fully compact component can fill in the
volume of the sample, i.e. Do = 3, Lo = Λ, mo = 1,
ξo =1, n =0

P =1− ξo =0 (fully compact body). (19)

Finally, in cases when one fractal component (D1)
and one non-fractal component (compact Do = 3)
are present, either of them localized in several sites
of the sample (m1 – fractal clusters, mo – non-fractal
clusters), the porosity can be expressed as follows

P = 1− ξ1
(

l1
L1

)3−D1
− ξo, (20)

ξ1 =
m1L

3
1

Λ3
, ξ0 =

moL
3
o

Λ3
.

Fig. 2: A scheme of a porous composite material (3 com-
ponents)

3.2 Derivation of compressive
strength

In technical literature, many relations have been in-
troduced by various authors. Most of these rela-
tions use porosity P as a main governing factor.
Let us discuss some relations concerning compres-
sive strength. Balshin [11] suggested a power func-
tion σ = σ∗o(1 − P)

k that was related to porous
metallic ceramic materials. His relation is equiv-
alent to the well-known expression of Powers [21].
Ryshkewitch [22] recommended the exponential func-
tion σ = σ∗o exp(−bP), which is in fact an asymp-
totic form1 (P → 0) of the Balshin power function.
Schiller [23] presented an expression similar to that
of Balshin in the form σ = σ∗o[1 − (P/Pcr)

1/a], cor-
rected for a critical porosity Pcr atwhich compressive
strength approaches zero. Several other relations are
summarized in Ref. [12].

1The Balshin relation can be rewritten in the Ryshkewitch fomula considering (1 − P)k = exp[k · ln(1 − P)] and restricting to
the first term of the Taylor series of the logarithmic function.

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Acta Polytechnica Vol. 50 No. 6/2010

There are two important points that should be
taken intoaccountwhendealingwith the compressive
strength of porous materials, namely, the so-called
critical porosity Pcr and partly incompressible pore
liquids. Schiller [23] considered the critical porosity
Pcr as a limiting factor for compressive strength, i.e.
σ(Pcr) = 0. But the limit may also be somewhat
influenced by the virtual incompressibility of porous
liquids. Liquids are displaced in the porous network
under the action of an imposed external mechani-
cal load, but narrow pores hinder the liquid move-
ment [24] and due to the virtual incompressibility of
the liquid the strength of the structuremaybe some-
what modified. It is natural that this effect concerns
especially quite narrowpores, andwith their increas-
ing diameters this effect weakens. However, let us
term themodified strength as the remaining strength
so. Now it is clear that Pcr and so should be corre-
lated to fulfill the condition σ(Pcr)= so. Taking the
Balshin relation σ = σ∗o(1 − P)

k as a good starting
point, his form may be generalized by taking into
account Pcr and so, as follows

σ = σ∗o(1 −
P

Pcr
)k + so, σ(Pcr)= so. (21)

Now the critical porosity Pcr does not represent the
absolute limit of strengthbut it onlydefinesa limit at
which the influence of incompressible liquids begins
to play a role.

3.3 Universal exponent of fracture
surfaces

It is important to realize that the fractality of porous
materials is determined by their solid skeleton and
notbytheirpores,whichareaconsequenceof thevol-
ume arrangement of material components possessing
dimensions {Di}. As soon as the volume structure is
brokenanda fracture surfaceappears, a newtopolog-
ical situation occurs. The volume components {Di}
create surface patterns {D∗i } with lower dimensions
D∗i < Di. The decrease of the dimensions {Di} can
easily be found when the fracture surface is a plane
(intersection of the Euclidean plane and the volume
fractal component). In this case D∗i = Di − 1, as
is well-known. In general, the value of the dimen-
sional shift of a fractal that has originally been em-
bedded in theEuclideanspace (E)andthenprojected
onto a subspace (S < E) is called the co-dimension
(E−S). Whentheoriginal space is three-dimensional
(E = 3) and the subspace two dimensional (S = 2),
the co-dimension is one (E − S = 1), as in the case
of intersection of the Euclidean plane with a volume
fractal. However, fracture surfaces are not smooth
Euclidean planes but rather irregular wavy surfaces.
Let us consider the simplest case of fracture of a non-
porous fully compact solid, e.g. a puremetal. Such a
solid has no fractal volume component, but its frac-

ture surface is fractal (2 < D∗o < 3), as has been
shown elsewhere [25]. On the other hand, when a
solid consisting of one delocalized fractal component
(D) is broken, the dimension D∗ of the correspond-
ing fracture surface is equal to the dimension D∗ of
the fractal projection onto an ‘imaginary’ subspace
S, i.e. D∗ = D − (3 − S). The dimension S of the
subspace can be calculated from the dimensions of
the volume fractal D and its surface projection D∗,
i.e. S =3−(D − D∗), provided there are techniques
for determining D and D∗. Similarly, if a solid is
composed of more than one fractal component {Di},
the dimensions of their surface projections {D∗i } are
given as follows

D∗i = Di − (3− S) ⇒ 3− Di = S − D
∗
i (22)

This relation has been usedwhen going fromEq. (6)
to Eq. (8), i.e. from volume fractals to their surface
projections. The dimensions of surface projections
D∗i are ‘measurable’ e.g. using the confocal tech-
nique, if these components are extended over differ-
ent length scales and do not overlap each other. In
our case of hydrated cement paste there is one fractal
component (Calcium-Silicate-Hydrate gel) that dom-
inates over the other non-fractal components, and
this simplifies the computations according toEq. (9).
There is no reason why the mentioned subspace

has to be of the Euclidean type. It can also be of
fractal type, i.e. its dimension S can be not only an
integer but also a non-integer number. And this is
the case of compactmetals possessing the dimension
Di =3, forwhichEq. (22) gives D

∗
i = S. Bouchaud,

Lapasset and Planès [25], when investigated metal-
lic fractures, found D∗i = 2.2 which means S = 2.2.
In our case of porous Calcium-Silicate-Hydrates the
dimension S has also been found in the same rank
S ≈ 2.2, although these two materials are quite dif-
ferent. The idea that S may be a universal exponent
related to fracture surfaces has been indicated previ-
ously [25], and our results seem to support it, nev-
ertheless, this concept should be studied further. If
future experiments confirm the concept, then it will
be necessary to distinguish carefully between the two
types of exponents S and D∗i . The former exponent
S is a relatively stable and probably universal expo-
nent which would be directly measurable if the sam-
ple were fully compact (non-porous and non-fractal),
i.e. aperfectEuclideanbody. Thedimension S seems
to depend more on the fracture process itself than on
structural components. The latter exponents are the
dimensions {D∗i } of surface projections. They vary
with the properties of materials, which has been il-
lustrated in the previous studies [9, 10] by using a
series of samples of different compressive strength.
These studies have simultaneously confirmed that
fracture surfaces bear information on the compres-
sive strength of porous materials (Fig. 2).

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Acta Polytechnica Vol. 50 No. 6/2010

4 Conclusion
The fractalmodel of compressivestrengthmaybeap-
plicable to all fractal porous materials. If the partic-
ular material is composed of a single fractal compo-
nent, the model contains only a few parameters that
can be easily fitted to experimental data. However,
whenmore fractal components are present,manypa-
rametershave tobefitted andtheremayarisenumer-
ical problems in selecting their ‘right’ values among
all the options, each of which satisfies the optimal-
izing criteria equally well. There is no general nu-
merical procedure that will guarantee such a right
selection of values. In these cases an intuitive and
heuristic approach, supported by physical reasoning,
may be instrumental in finding an optimum solution.

Acknowledgement

This work was supported by Grant no. ME09046
provided by the Ministry of Education, Youth and
Sports of the Czech Republic.

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[25] Bouchaud, E., Lapasset, G., Planès, J.: Euro-
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Prof. RNDr. Tomáš Ficker, DrSc.
Phone: +420 541 147 661
E-mail: ficker.t@fce.vutbr.cz
Department of Physics
Faculty of Civil Engineering
University of Technology
Žižkova 17, 662 37 Brno, Czech Republic

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