AP_07_6.vp


1 Introduction
Fracture surfaces are valuable sources of information on

the structural composition and physical properties of materi-
als. For these reasons they are a subject of interest for many
research laboratories. Since the publication of the basic work
by Mandelbrot and his co-workers [1] many authors have
tried to correlate the fractal dimensions of fracture surfaces
with the mechanical properties of materials. This effort has
been impacted by the great complexity of these surfaces,
especially in the case of composite porous materials. Ce-
mentitious materials have complex fracture surfaces that have
been extensively studied [2–4].

The values of the fractal dimensions of a range of materi-
als show only a narrow scatter, ranging from ~2.0 to ~2.2.
Repeatedly determined dimensions of different fractured
samples of the same materials have often resulted in identical
values and this has led some authors [5] to the idea of a
universal co-dimension (Hurst exponent) H � 08. that charac-
terizes fracture surfaces as a whole. Though this idea may
invoke certain doubts at first sight [6], it should be carefully
considered before it is rejected or accepted.

The aim of this paper is to investigate the concept of a uni-
versal co-dimension of fracture surfaces. As will be shown, this
concept may be verified experimentally using fracture sur-
faces of porous materials in connection with their compressive
strength. For this purpose, it is necessary first to derive corre-
sponding relations for fractal porosity and fractal strength,
and then to apply them to a particular material, in our case
cement gel.

2 Fractal porosity
The large class of porous materials possesses at least one

common feature, namely, they are composed of grains (parti-
cles, globules, etc.) of microscopic size l. The grains are usually
arranged fractally with number distribution N(l)

N l
L
l

D
( ) � �

�
�

�
�
� , l<L. (1)

The porosity P of the cluster

P
l
L

D
� 	 �

�
�

�
�
�

	
1

3
(2)

modifies its form if the porous material consists of more than
one (n 
 1) fractal cluster

P
l
L
i

i

D

i

n i

� 	
�

�
��

�

�
��

	

�
�1

3

0

. (3)

However, relation (3) does not take into account the case
of a composite material in which the fractal clusters of charac-
teristic sizes Li can be stochastically scattered and mixed with
other phases so that the size � of the investigated sample may
considerably exceed the cluster sizes Li � �. In order to gen-
eralize relation (3), let use suppose that there are mi fractal
clusters with dimension Di in the sample. Their volume frac-
tions �i m Li� 3 3� enable us to calculate the porosity of the
whole sample, as follows

P
l
Li
i

i

D

i

n i

� 	
�

�
��

�

�
��

	

�
�1

3

0

� . (4)

Eq. (4) includes all possibilities of fractal, non-fractal
( )D � 3 or mixed arrangements of the solid environment
surrounding the pores. Provided there are n 
 1components
distributed over the whole sample (Li � �, mi �1, � i �1),
Eq. (4) then converts back to (3).

3 Fractal compressive strength
One of the most widely used relations for compressive

strength � of a porous material is that of Balshin [7], though
other relations [8] have also been proposed for this purpose.
Balshin considered an ideal case when pores are not filled
with an incompressible liquid and compressive strength � is
directly dependent on the compactness ( )1 	 P of the material
� �� 	0 1* ( )P k. However, as soon as the virtual incompres-
sibility of the pore liquid is included together with some other
factors, certain remaining strength s0 appears as a constant
when the porosity reaches a critical value Pcr, i.e. �( )P scr � 0.
The generalized Balshin function then reads

� � �� 	
�

�
��

�

�
�� 
 � 	 	 


� � 	 �

0 0 0 01 1

0 1 1

* ( ) ,

,

P
P

s P b s

b P

k
k

cr

cr 0 0 0
0� � �� �

�*
*

,
Pk

cr

, (5)

where P is total porosity and � is compressive strength of the
sample with a virtually incompressible fluid filling at least a
part of its pore space.

Combining (4) and (5), the compressive strength of po-
rous matter appears as a function of the fractal structure

� � ��
�

�
��

�

�
�� 	



�

�
�

�

�

�
�



	

�
�0

3

0
0i

i

i

D

i

n k
l
L

b s
i

. (6)

©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 27

Acta Polytechnica Vol. 47  No. 6/2007

Dimension of Fracture Surfaces
T. Ficker

The question of the universality of a dimension of fracture surfaces is discussed, and it is shown that such a general parameter may exist at
least for a particular class of materials.

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



4 Dimensions of a fracture surface
Generally, in the case of a mixed structure containing both

fractal and non-fractal regions some of the dimensions Di are
associated with volume (mass) fractals (0<Di<3) and some
with non-fractal solid phases (Di � 3). Similarly, when analyz-
ing the fracture surfaces of such multicomponent materials,
projections (Di

*) of the volume components (Di) appear on
these surfaces. These projected surface “patterns” preserve
the fractal or non-fractal characters of their volume parent
objects, but their dimensions Di

* are smaller than those of the
original objects. Provided a fracture surface has its own di-
mension S and its morphology is “typical” rather than “spe-
cial”, the relation between Di

* and Di can be expressed [9] as
follows

� �D D S D Si i i
* *max , ( ,� 	 	 � �0 3 3 , (7)

where 3 	 S is the codimension of the fracture surface. If
the fracture surface were a perfect Euclidean plane (S � 2),
expression (7) would lead to the well-known equation
D Di i

* � 	1. Using (7), the exponent 3 	 Di in Eq. (6) can be

replaced by S Di	 * and the generalized Balshin strength

function now reads

� � ��
�

�
��

�

�
�� 	



�

�
�
�

�

�

�
�
�



	

�
�0

0
0i

i

i

S D

i

n
k

l
L

b s
i
*

. (8)

This function may contain many parameters, so that it is
very difficult to fit it to the experimental data, because there

may be more than one “reliable” set of parameters �0,� �� i i
n
�1,

li, Li, S, Di
*, b, k, s0. Fortunately, the structure of a porous ma-

terial often contains only one type of grain or – at least –
one grain type of fractal arrangement (i �1) dominates over
the solid remainder (i � 0), which is usually of a non-fractal
character (D D S0 03� � �* )

� � � �

� �

�
�

�
��

�

�
�� 
 	



�

�
�
�

�

�

�
�
�


 �
	

0 1
1

1
0 0

0 1

1l
L

b s
S D

k*

( )

exp ,

ln
,

*S D
A

s

A
L
l

k
	�

�
�
�

�

�
�
� 	



�
�
�

�

�
�
�




�
�

�
��

�

�
��

1
0

1

1

1

�

� � 	( ) .b �0

(9)

To our knowledge, the fractal analyses of fracture surfaces
published so far have not distinguished between the dimen-
sions Di

* of the projected objects and the dimension S of the
fracture surface itself. Here, for the first time clear differences
between these two kinds of dimensions are specified and
discussed. Although the fracture surface dimension S as a
separate parameter and independent of the inner fractal
structure may seem to be rather vague, it nevertheless has a

clear interpretation as the fracture surface dimension that
would be directly measurable if the sample were fully compact
(non-porous, i.e. b � 0, s0 0� ), and, therefore, non-fractal
( )*D D Si i� � �3 , which means a perfect Euclidean body.
Nevertheless, the surface itself may be a fractal with non-inte-
ger S, which should even be expected in most cases.

The foregoing equations and discussion have shown that
fracture surfaces bear information about the multi-compo-
nent volume structure that is projected on the surface as the

spectrum of surface patterns � �Di i

n*
�0

that preserve their

fractal or non-fractal origins. The fracture surface has its
own dimension independent of from the dimensions of the
projected surface patterns. These facts require a strict distinc-
tion between dimensions Di

* and S, and also among Di
*

themselves. This can be done either locally (e.g. using a mi-
croscopic technique) or globally within restricted fractal scales
l Li i, if, of course, there are no overlaps between the scales.

5 Experimental results
To illustrate the soundness of the concept of fractal com-

pressive strength (9) and the concept of, their application to
hydrated Portland cement is presented in the following para-
graphs. Seventy-two samples of hydrated ordinary Portland
cement paste of various water-to-cement ratios 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 frac-
tured. The fracture surfaces were used for further fractal anal-
ysis. The 3-D digital reconstruction of the fracture surfaces
(Fig. 1) was performed using a confocal microscope and
then a series of horizontal sections (contours) were analyzed
with resolution 0.04�m2/pixel by means of the standard box-
-counting method [10–12] to obtain a representative D* for
the particular self-affine surface. The box-counting analyses
were performed in the length interval 02 30. ,� �m m . The

28 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/

Acta Polytechnica Vol. 47  No. 6/2007

Fig. 1: Digital 3-D reconstruction of fracture surface (confocal mi-
croscopy: 0.34 mm × 0.21 mm; total magnification 600×)



second parts of the fractured samples were cut into small
cubes and subjected to destructive tests to determine their
compressive strength values.

Samples prepared with different water-to-cement ratios
r possess different porosity. From cement technology it is
well-known that with increasing ratio r the porosity increases
exponentially. Naturally, this will change the dimensions of
the projected patterns D*. Six groups of samples with differ-
ent r means six different D* at which we are able to measure
the dependence �( )*D and check it according to Eq. (9). The
result is shown in Fig. 2, along with all the fitting parameters.
Since the assumed analytical form (9) of dependence �( )*D
has been reproduced well, we may conclude that compressive
strength is one of those mechanical quantities whose value is
“coded” in the surface arrangement of the fractured samples
of porous materials.

Dimension S of the fracture surfaces of hydrated cement
paste proved to be near to 2.24, indicating a fractally cor-
rugated surface onto which another fractal component
D* . ; .� 2075 2170 is projected (� �1). The range of values of
D* is in full accord with other authors [2–6]. The contribution
of the non-fractal solid phase (�0 0� ) is negligible (���0).
As far as dimension D of the original volume (mass) fractal of
the cement paste used is concerned, its value is rather high
D � 2835 2930. ; . but this was expected due to the low dimen-
sion D* of the corresponding projection and the high value of
the co-dimension, i.e. Hurst exponent H, of the fracture surface
itself H S� 	 �3 076. . The value of D near to 2.90 agrees with
the dimension of the cement gel within the hydrated cement
paste having r � 0 4. and exposed to ~55 % RH [13], [14].

6 Conclusion
It has been shown that the fracture surface possesses its

own dimension S which is independent on the dimensions of

the patterns projected on this surface. Dimension S seems to
depend more on the fracture process itself than on structural
components. This feature may partly support the claims of
several authors [5] that there is a universal Hurst exponent
H � 08. for a wide class of materials. Now it is clear that if such
a universal exponent really exists, it is not as a co-dimension
of the surface projected patterns3 	 D* but as a co-dimension
of the fracture surface itself H S� 	3 . In our case H � 076. ,
which is almost identical with the previously anticipated [5]
value 0.8.

Dimension S can be determined not only as the difference
3 	 	( )*D D but also as a “byproduct” when measuring a con-
venient physical quantity, in our case compressive strength.
Dimension S of fracture surfaces may be interpreted as the di-
mension that would be directly measurable if the sample were
fully compact (non-porous and non-fractal), i.e. a perfect
Euclidean body. Nevertheless, the fracture surface itself may
be a fractal with non-integer S, which should be expected
most cases.

7 Acknowledgment
This work was supported by an Internal Grant assigned by

the Faculty of Civil Engineering, Brno University of Technol-
ogy, in 2007.

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[3] Wang, Y., Diamond, S.: A Fractal Study of the Fracture
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©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 29

Acta Polytechnica Vol. 47  No. 6/2007

Fig. 2: Dependence of compressive strength on fractal dimension with cement paste



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[9] Falconer, K. J., Professor of Mathematics, University of
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[13] Winslow, D., Bukowski, J. M., Young, J. F.: The Fractal
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[14] Boddoe, R. E., Lang, K.: Effect of Moisture on Fractal
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Prof. RNDr. Tomáš Ficker, DrSc.
phone: +420 541 147 661
e-mail: ficker.t@fce.vutbr.cz

Department of Physics

University of Technology
Faculty of Civil Engineering
Žižkova 17
662 37 Brno, Czech Republic

30 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/

Acta Polytechnica Vol. 47  No. 6/2007