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Acta Polytechnica Vol. 51 No. 3/2011

Fracture Surfaces of Porous Materials

T. Ficker

Abstract

A three-dimensional absolute profile parameter was used to characterize the height irregularities of the fracture surfaces
of cement pastes. The dependence of these irregularities on porosity was studied and its non-linear character was proved.
Ananalytical form for thedetectednon-linearitywas suggested and then experimentally tested. The surface irregularities
manifest scale-invariance properties.

Keywords: roughness analysis, fracture surfaces, cement-based materials, confocal microscopy.

1 Introduction
The morphology of fracture surfaces have been stud-
ied for a long time to reveal the details of fracture
processes. However, in the research field of cementi-
tiousmaterials therehavebeenonlya restrictednum-
ber of studies that deal with surface features of frac-
tured specimens. Some of the early surface studies
of hydrated cement materials were focused on frac-
tal properties [1,2] whereas others [3–5] investigated
roughness numbers (RN) or similar surface charac-
teristics [6–9].
When dealing with the roughness of the fracture

surfaces of porousmaterials like cementitiousmateri-
als, an important question arises, namely, what type
of relationship is therebetweensurface roughnessand
porosity. Recently, Ponson at al [10,11] have pointed
to a close relationship between these two quantities.
The authors studied the roughness of fracture sur-
faces with glass ceramics made of small glass beads
sintered in bulk with porosity P that could be var-
ied in the interval (0,0.3). They observed that the
roughness of the fracture surfaces increases linearly
with increasing porosity (see their figure 3.3 in [10]).
It would be valuable to know whether such a lin-
ear behavior is a general property of all porous ma-
terials, or only a specific feature of glass ceramics.
For this reasonwe performed a large series of experi-
ments with cement pastes. This material was chosen
because its porosity can be easily controlled within
a broad porosity interval by means of the water-to-
cement ratio r = w/c. Correctknowledgeof the func-
tional dependence of roughness on porosity may be
useful for further surface studies of fractured porous
materials.

2 Experimental arrangement

Ordinary Portland cement CEM 42,5 I R-sc of do-
mestic provenance was used to create 108 specimens

of hydrated pasteswith six differentwater-to-cement
ratios r (0.3, 0.4, 0.5, 0.6, 0.7, 0.8). The specimens
were rotated during hydration to achieve better ho-
mogeneity. All specimens were stored for the whole
time of hydration at 100 % RH and 20◦C. After
60 days of hydration the specimens were fractured
in three-point bending tests and the fracture sur-
faceswere immediately used formicroscopic analysis.
Other parts of the specimens were used for porosity
measurements and for further mechanical tests.
Porosity was determined by the common weight-

volume method. The wet specimens were weighed
and their volume wasmeasured, then they were sub-
jected to 105◦C for one week until their weight no
longer changed, and the dry specimens wereweighed
again.
The 3D profile parameter Ha was used to char-

acterize the roughness of the fracture surfaces of
the hydrated cement pastes. In fact, Ha represents
the averaged ‘absolute’ height of the fracture relief
z = f(x, y)

Ha =
1

L · M

∫ ∫
(LM)

|f(x, y)|dxdy (1)

where L × M is the area of the vertical projection
of the 3D fracture profile f(x, y) into the plane xy.
Parameter Ha has great statistical relevancy, since
it is a global averaged characteristic covering the en-
tire tested surface L × M. The 3D profiles f(x, y)
were created using the Olympus Lext 3100 confo-
cal microscope. One of these profiles is shown in
Figure 1. The profiles are formed by the software
that processed a series of optical sections created
by the confocal microscope at various heights of the
fracture surfaces. Approximately 150 image sections
were taken for each measured surface site, starting
from the very bottom of the surface depressions (val-
leys) and proceeding to the very top of the surface
protrusions (peaks). The investigated area L × M =
1280 μm×1280 μm(1024 pixels×1024 pixels) was

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Acta Polytechnica Vol. 51 No. 3/2011

Fig. 1: 3D confocal relief of fractured cement paste

chosen in five different places of each fracture surface
(in the center, and in four positions near the cor-
ners of the rectangular area), i.e. each plotted point
on the graphs of the profile parameters corresponds
to an average value composed of 90 measurements
(18 samples×5 surface measurements). Each mea-
surementwasperformed for threedifferentmagnifica-
tions, namely 5×, 20× and 50×, giving 270measure-
ments thatwereperformed for the particular r-value.
Since each site measurement amounts to about 150
optical sections (digital files), 40500 files had to be
processed to create 270 digital maps per one r-value.
This resulted in 1620 digital maps for all r-values
altogether (6 r-values×270 maps for each r-value).
These 1620 digital fracture surfaces were then sub-
jected to 3D profile surface analysis.
In this way an extensive statistical ensemble was

created to provide a sufficiently reliable basis for
drawing relevant conclusions.

3 Results and discussion
Figure 2presents threedependences of Ha(P) formed
within threemagnifications: 5×, 20×and50×. Their

graphs do notmanifest linear behavior. All the three
graphs show very similar shapes, which are well de-
scribed by non-linear functions (curves in Figure 2)
that can be expressed as follows

Ha(P)=
Ho

(Po − P)β
+ ho (2)

This is in fact a power-law function pointing to
fractal-like properties. Relation (2) contains fourpos-
itive fitting parameters Ho, Po, β, and ho, the mean-
ings ofwhich canbe explained on the basis of asymp-
totic patterns. Firstly, Po must always be greater
than variable P , otherwise function (2) would be a
decreasing function, and this would contradict the
experimental data in Figure 2. This means that Po
is the limiting value of porosity P when the water-
to-cement ratio r goes to infinity

Po = lim
r→∞

P =1 (3)

Assuming Po = 1 and P → 0 (material with ‘zero
porosity’), the limiting roughness of the correspond-
ing fracture surfaces reads

lim
P → 0
Po = 1

Ha(P)= Ho + ho (4)

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Acta Polytechnica Vol. 51 No. 3/2011

Fig. 2: The dependence Ha(P) between the roughness of the fracture surfaces and the porosity of the cement paste

Fig. 3: Normalized scale-invariant data Ha/M ax(Ha) in dependence on porosity P

From (4) it follows that the sum Ho + ho represents
the height irregularityat ‘zero porosity’. Since differ-
entmagnifications provide different resolutions, ‘zero
porosity’ will be determined differently for different
magnifications, and thus the sum Ho + ho will also
vary with magnification. The asymptotic form of
Ha(P) when the material contains a very small but
non-zero porosity value, e.g. 0 < P < 0.30 can also
be useful for further consideration. In this case, the
Taylor expansion of (2) may be utilized

Ha(P) ≈
Ho

P
β
o

(
1+ β

P

Po

)
+ ho = aP + b (5)

a =
βHo

P
1+β
0

, b =
Ho

P
β
o

+ ho. (6)

It is obvious from(5) that at sufficiently small porosi-
ties the roughness of the fracture surfaces can be ap-
proximated by a linear function Ha(P) ≈ aP + b,
which is in agreement with the observation of Pon-
son [10],whose experimentswith glass ceramics (P <
0.3) indicated such behavior (figure 3.3 in Ref [10]).
Our data in Figure 2 does not contain a suffi-

cient number of experimental points in the region
of small porosities (P < 0.35), within which linear

behavior can be observed. However, the clearly non-
linear behavior of function Ha(P) at higher porosi-
ties is a real fact illustrated by all the graphs in
Figure 2. The interval P ∈ (0.35,0.50) is char-
acterized by an abrupt non-linear increase, and at
P > 0.50 rapid growth unmistakably continues fur-
ther. With our specimens, these three intervals of
porosities P < 0.35, P ∈ (0.35,0.50) and P > 0.50
correspond to the following water-to-cement ratios:
r1 < 0.4, r2 ∈ (0.4,0.6) and r3 > 0.6, respec-
tively. It is well known that at sufficiently small
water-to-cement ratio values (r1 < 0.4), gel poros-
ity dominates over capillary porosity in hydrated ce-
ment pastes. Within the specimens mixed with in-
termediate r-values r2 ∈ (0.4,0.6), capillary porosity
starts to assume a governing role, and at still higher
r-values r3 > 0.6 the porosity of the specimens is
prevalently formed by capillary porosity. The over-
all non-linear surface irregularity (2) of cementitious
materials seems to be a result of the interplay be-
tween gel and capillary porosities. As soon as the
onset of capillary porosity appears, the height irreg-
ularities of the fracture surfaces increase their values
abruptly, as can be seen with all the graphs in Fi-
gure 2 and still more straightforwardly with the two
graphs in Figure 3, where the boundary between the

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Acta Polytechnica Vol. 51 No. 3/2011

gel and the capillary porosities can be recognized at
P ≈ 0.3. This effect can be characterized as a cross-
over to larger scale porosities.
It is interesting to note that the shapes of the

graphs of functions Ha(P) shown inFigure 2 aremu-
tually very similar, regardless of the magnifications
that are used. In this connection, the question of
their scale invariance arises. Thismeans that the de-
pendence Ha(P) can be described by the same func-
tional type (2), but with the fitting parameters Ho,
Po, β, ho adapted to the results of the particular
magnification. However, when these three functions
are normalized by using the highest measured val-
ues M ax(Ha), a unified function results (Figure 3A).
Using the pattern of the unified function, the linear
behavior of surface roughness at small porosities can
be illustrated straightforwardly— see Figure 3B.
The graph of the unified function shown in Fi-

gure 3A represents the results of all three magnifica-
tions used here— 5×, 20× and 50× — and it simul-
taneously manifests the scale invariant properties of
the height irregularities of the fracture surfaces.

4 Conclusion
The experiments have proved a close relationship be-
tween the roughness (height irregularities) of fracture
surfacesandtheporosityofmaterials. The functional
relationof these twoquantities is generallynon-linear
and may be described by a power law function (2).
When normalizing this function, it becomes indepen-
dentof themagnification that is used. Thenon-linear
behavior of Ha(P) with cement pastes is the result
of the influence of gel and capillary porosities. The
region of small gel porosities is characterized by a
moderate (almost linear) increase in surface rough-
ness Ha(P), whereas the region of capillary porosity
is a domain with an abrupt increase in this quantity.
The presentedproperties of the surface roughness

of fracturedcementpastesmayalsobeuseful formor-
phological and structural studies of other porousma-
terials.

Acknowledgement

This work was supported by the Ministry of the
Czech Republic under Contract no. ME09046 (Kon-
takt).

References

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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
Brno University of Technology
Veveř́ı 95, 662 37 Brno, Czech Republic

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