Image Anal Stereol 2012;31:55-63
Original Research Paper
doi:10.5566/ias.v31.p.55-63
POROSIMETRY BY DOUBLE-RANDOM MULTIPLE TREE
STRUCTURING IN VIRTUAL CONCRETE
PIET STROEVEN1, NGHI LB LE
,1
, LAMBERTUS J SLUYS1 AND HUAN HE1,2
1
Faculty of Civil Engineering and Geosciences, Delft University of Technology, PO Box 5048, 2600 GA Delft,
the Netherlands; 2GeMMe, Minerals Engineering-Materials-Environment, University of Liège, Sart Tilman
B52, 4000 Liège, Belgium
e-mail: L.B.N.Le@tudelft.nl
(Received September 6, 2011; revised November 30, 2011; accepted January 11, 2012)
ABSTRACT
Two different porosimetry methods are presented in two successive papers. Inspiration for the development
came from the rapidly-exploring random tree (RRT) approach used in robotics. The novel methods are
applied to virtual cementitious materials produced by a modern concurrent algorithm-based discrete element
modeling system, HADES. This would render possible realistically simulating all aspects of particulate
matter that influence structure-sensitive features of the pore network structure in maturing concrete, namely
size, shape and dispersion of aggregate and cement particles. Pore space is a complex tortuous entity.
Practical methods conventionally applied for assessment of pore size distribution may fail or present biased
information. Among them, mercury intrusion porosimetry and 2D quantitative image analysis are popular.
The mathematical morphology operator “opening” can be applied to sections and even provide 3D
information on pore size distribution, provided isotropy is guaranteed. Unfortunately, aggregate grain
surfaces lead to pore anisotropy. The presented methods allow exploration of pore space in the virtual
material, after which pore size distribution is derived from star volume measurements. In addition to size of
pores their continuity is of crucial importance for durability estimation. Double-random multiple tree
structuring (DRaMuTS), presented herein, and random node structuring (RaNoS) provide such information.
The latter method will be introduced in a next issue of Image Anal Stereol.
Keywords: DEM, pore connectivity, porosimetry, star volume, virtual concrete.
alternative; the approach is basically simple and does
not require sophisticated equipment. The obtained
information is of two-dimensional (2D) nature, although
stereological methods exist for unbiased 3D interpretation. When structural isotropy cannot be guaranteed,
however, this may complicate the sampling procedure
and will increase labor-intensity of the QIA approach.
INTRODUCTION
Durability risks associated with cementitious materials can be due to transport of harmful substances
through the pore system in concrete, water included.
Of course, also micro-cracks can be instrumental in
this process. But this paper limits itself to porosimetry
aspects. Insight into the complex and highly tortuous
pore network structure would thus be of practical
significance. Various approaches have been applied
by researchers for this purpose that yielded quantitative
data on aspects of this three-dimensional (3D) network
system. A survey of the most practical approaches
can be found in Stroeven et al. (2010).
A relatively new way of approaching the problem
is based on modern computer facilities. Cementitious
materials can nowadays be realistically simulated by
discrete element modeling (DEM). This is fundamentally different from approaches by forerunners using
random generators (RG) for placing particles of
cementitious materials inside a container. As a sole
example, reference can be given to the development
by Wittmann et al. (1985) of so called “numerical
concrete” on the basis of Roelfstra’s RG-based system
(Roelfstra, 1989). A variety of such systems have
been and still is employed in concrete technology, a
selection of which is referred to in Stroeven et al.
(2009).
The most frequently used method is doubtlessly
mercury intrusion porosimetry (MIP). It offers 3D
information, however this could be significantly biased
because of the unrealistic schematization of the
geometry of the pore channels and the neglect of the
so called bottle necks in the pore system (Diamond,
2000). Quantitative image analysis (QIA) is a viable
55
STROEVEN P ET AL: Concrete Porosimetry
DEM far more realistically simulates particulate
materials as concrete, both on meso-level where
aggregate takes up roughly three-quarter of material
volume, and on micro-level where volume density of
the cement grains in the fresh material can get as high
as 60% for very low water to cement ratios (w/c)
relevant for (super) high performance concrete (S)HPC.
Dispersion and thus spacing of particles in virtual
material produced by DEM could be far closer to that
in the real material; this is the result of the incorporation
of the natural process of particle interference during
the “production” of the virtual material. This is of
crucial importance for reliable estimation of structuresensitive material properties (Stroeven et al., 2010).
The center-to-center spacing of the cement grains in
fresh concrete is maintained during maturation (Hu,
2004). Hence, the reliability of the simulated pore depercolation process also depends strongly on the
realistic dispersion of the cement grains. Significant
differences between process characteristics obtained
by (analogue) DEM- and RG-based systems and with
the digitized model in Garboczi and Bentz (2001)
have been shown in Chen et al. (2006).
past 10 to 15 years. Reference to such systems can be
found in Stroeven et al. (2009). Most have been developed for a specific target in concrete technology. The
Habanera's discrete element simulator (HADES) system
that will be introduced herein is probably an exception
to this rule because of its versatility. It can produce any
particulate material composed of particles with arbitrary
shape from the dilute up to the densest (crystallized)
packing state. Some illustrative examples are presented
in Fig. 1.
Two methods are in vogue for achieving dense
packing states using either static or dynamic algorithms,
respectively. HADES is an example of the second
category. A dynamic stage organizes particle interference during gradual reduction in the container size.
Particle overlap can be prevented by different force
systems, which come into action on tessellated surface
elements at neighboring sides of approaching particles
(Fig. 2). This renders possible packing particles of
arbitrary shape. As a consequence, the differences in
optimum packing capacity of concrete containing
river gravel and crushed rock as aggregate have been
investigated this way (He, 2010). This topic is also of
importance for the problem of pore network modeling
in compacted concrete (Vogel and Roth, 2001; Stroeven
and Guo, 2006) and is therefore of relevance for
estimating durability risks.
When cementitious materials are realistically simulated, the next problem confronting researchers in porosimetry is finding a practical way of detecting the
pores, and measuring their size and connectivity; a
complicated task. This basically methodological paper
will therefore deal with:
1.
2.
3.
4.
The somewhat non-spherical nature of cement
grains as found by Garboczi and Bullard (2004) can
equally be accounted for in the DEM-produced virtual
cement (He, 2010; He et al., 2010). The far more
complicated hydration algorithms are not yet available,
however. For details of the hydration algorithms in
the case of spherical cement grains, see Stroeven et al.
(2011). Since the impact on topology and geometry
of the pore network structure is expected not to be
significant, the porosimetry methods that will be
introduced have been applied to virtual Portland cement
and blended Portland cement pastes consisting of spherical particles.
Production of virtual material
Detection of pores
Assessment of pore connectivity
Measurement of pore size.
PRODUCTION OF VIRTUAL
CONCRETE
Various DEM systems have been developed the
Fig. 1. Examples of loose random packing states of differently shaped particles (He, 2010).
56
Image Anal Stereol 2012;31:55-63
Fig. 2. Mechanisms that come into action upon close encounter of neighboring particles to prevent overlap of
such particles. Only tessellated surface elements activated by local overlap of guard zones with thickness T are
participating. The intensity of reactive forces depends on local spacing.
red. In the second stage, pore features such as porosity,
gradient structures and connected fraction are evaluated.
This allows distinguishing not only between depercolated pores and pores that connected opposite
sites of a cube specimen, but also between continuous
pore channels and dead-end branches of such channels.
Finally, pore size distribution is assessed by star volume
measurements on the basis of the randomized point
system.
Earlier investigations on DEM-made concrete have
demonstrated the importance of aggregate grain surfaces for the pore de-percolation process during maturation. Chen et al. (2006) found the fraction of continuous pores (for w/c = 0.3) restricted to an inner
layer of the interfacial transition zone (ITZ), adjacent
to the aggregate grain surface. The virtual material was
produced by the DEM-system SPACE (Software Package for the Assessment of Compositional Evolution),
the predecessor of HADES that was based on spherical
particles (Stroeven, 1999). Continuity in porosity was
assessed by serial sectioning and 3D reconstruction, a
popular experimental method when dealing with soft
tissues. It was developed by Ye (2003) for virtual
concrete, however implemented in the RG-based
hydration, morphology and structure formation
(HYMOSTRUC3D) system (Breugel, 1991). Total and
continuous porosity values were determined as global
values by Chen et al. (2006) following this strategy.
By assessment of these values in each successive stage
of peeling off a layer from the cube specimen (like an
onion), the gradient structures of total and continuous
porosity were obtained. Steeply declining values were
found with zero values inside the ITZ (and in bulk
region). For results pertaining to different technological
para-meters, see Chen et al. (2004). The major drawback of this approach is its extremely time-consuming
character, even in case of virtual concrete. Its unpractical nature in case of physical experiments is obvious.
Inspiration for the present approach is derived
from the so called ‘rapidly-exploring random tree’
(RRT) algorithm in robotics developed in LaValle
and Kuffner (2001). This efficient path planning algorithm pursues finding a way from point A to point B,
avoiding any collision with dispersed obstacles. Path
planning is implemented by generating a ‘virtual tree’
system that includes sets of nodes (‘vertices’) and
lines (‘edges’) that connect pairs of nodes (like
branches of real trees). This tree grows incrementally
and randomly in 3D. Procedure starts by generating a
random point that is then moved towards the nearest
vertex, thereby defining a maximum incremental
distance. Next, a check for collision with an obstacle
is executed. No collision leads to addition of a new
point and a segment to the tree. At collision, a new
random point is generated and the procedure is
repeated until the goal is achieved. The expansion of
the tree is illustrated in Fig. 3 (top).
The RRT method needed significant upgrading
for porosimetry applications. Since a large number of
trials may be involved, generation of the complete
tree system would be laborious. In the present approach,
therefore, instead of boosting the no-collision trials,
intersections of the tested segment and obstacles (solid
phase in hydrated cement paste) are detected. Next, a
point is trimmed between the nearest intersection and
the considered vertex and then becomes a new vertex
as in Fig. 3 (bottom). This excludes making iterations,
DOUBLE-RANDOM MULTIPLE
TREE STRUCTURING
Pore characteristics in 3D virtual hydrated paste
are investigated by double-random multiple tree structuring (DRaMuTS) approach in which randomized
data structures are built incrementally in two stages.
In the first stage, the porous medium is rapidly explo57
STROEVEN P ET AL: Concrete Porosimetry
are generated in the model cement paste, which is
expressed by ‘double random’ in the name of the
approach. The first stage-produced tree vertices in
percolated as well as in de-percolated pores allow a
fast classification of the random points into three
groups, encompassing points inside percolated or depercolated pore phase or in solid phase, respectively.
Only the points inside pore space are considered in
further investigations. Counting points suffices for
the assessment of associated fractions of porosity. A
direct connection between opposite paste surfaces is
defined as a ‘main trunk’. This renders possible separating between main trunks and dead-end branches in
the fraction of percolated porosity.
because there is always one new vertex in the generation process. This speeds up the generation of the
whole tree system and constitutes therefore a significant improvement of the RRT algorithm. Moreover,
this allows investigating also dead-end branches of
continuous pore channels.
PORE SIZE IN 3D
In 2D sections of real or virtual concrete alike, the
most direct way of obtaining 3D local volume
information on irregularly shaped pores is by way of
the mathematical morphology operator “opening”
(Serra, 1982). This has been accomplished in Hu and
Stroeven (2005; 2006) and Stroeven et al. (2010).
The underlying requirement of structural isotropy can
be expected violated however inside the ITZ, so the
method should be applied to bulk cement only. A
next option is making star volume measurements; a
method employed in life sciences (Gundersen et al.,
1988; Smit et al., 1998).
A 3D pore structure is produced by HADES and
explored and visualized by the new porosimetry
methods introduced herein and in a next IA&S
publication. Hence, a direct 3D assessment method
for pore size would be attractive. Ye (2003) and Ye et
al. (2003) had the (by serial sectioning and 3D
reconstruction generated) pore network filled up by
spheres of successively increasing size, starting from
a pre-determined point. This has been indicated
leading to biased results (Hu, 2004; Hu and Stroeven,
2006). Instead, the earlier mentioned technique of star
volume measurements can be applied in 3D. The 3D
“stars” are positioned at random points inside the
pore system. Next, a large number of “pikes” extends
from the star center to the nearest pore surface. Their
length, li, is measured, whereupon local pore volume,
Vi, is obtained from (Gundersen et al., 1988)
Fig. 3. Expansion of tree system by RRT (top) and by
DRaMuTS, respectively (bottom).
Exploration by a single tree system seems not
appropriate in porosimetry. Parallel development at
the same time leads to the so called ‘multiple-tree’
network. Herein, tree systems grow incrementally
from a set of different points referred to as ‘seeds’,
which are randomly distributed in pore space.
Interestingly, sowing can be achieved on one side of
the paste model, alternatively, on both opposite sides;
sowing can be realized inside the paste model and it
can be a combination of the preceding procedures.
Additionally, efficiency can be improved by sowing
the seeds in the gaps between hydrated neighboring
cement grains. Upon completion of the generation
process, the connectivity between the tree systems is
checked. If so, trees will merge into a single one.
4
Vi = π li3 ,
3
After completion of pore exploration, the second
stage of the DRaMuTS approach starts, in which pore
features are investigated. To start with, random points
(1)
so that local pore size (i.e., diameter) equals 2 3 li3 .
The volume-based cumulative pore size distribution
58
Image Anal Stereol 2012;31:53-63
(4)
To obtain matured virtual cement paste, initially
the cement particles are randomly generated in a large
container. Next, the packing process is realized by
gradually reducing the container size while particles
move and collide among each other and with the rigid
container surfaces until the desired packing density is
obtained. Overlap is prevented by the mechanisms
indicated in Fig. 2. Six periodic boundaries are
assigned to the container for simulating bulk material
(PC1 and BPC1 samples). For simulating the ITZ zone
(PC2 and BPC2 samples), four periodic and two
opposite rigid boundaries are used. Gradient structures
are formed adjacent to the rigid surfaces due to the
wall effect (Stroeven and Hu, 2007). Also after
hydration such gradient structures are found. The
relevant cement hydration simulation (CemHydSim)
program, is described in detail in Stroeven et al.
(2011). This boundary zone denotes the interfacial
transition zone (ITZ) in concrete technology. As an
example, porosity in the ITZ is found exceeding that
in bulk material in agreement with experimental data.
In this research, the rigid surfaces are due to the
coarse aggregate like river gravel or crushed rock.
The order of magnitude size differences between
aggregate and cement grains allows simulating the
aggregate surface in the form of a flat container side.
For details, see Stroeven and Stroeven (2001).
Details of an example of the DRaMuTS approach
for the exploration of pore space in hydrated virtual
plain Portland cement paste and in gap-graded
blended Portland cement paste are presented in Table
1 (Stroeven et al., forthcoming 2012). Note that w/b
is the water to binder ratio (the binder encompasses
the Portland cement and the mineral admixture), and
P and R stand for periodic and rigid boundary conditions of the simulated pocket of material, respectively.
As mentioned above, the capillary pores are distinguished into de-percolated pores and pores that
connected opposite sites of a cube specimen. The
connected fraction in Table 2 is defined as the volume
ratio of percolated pores to that of all capillary pores.
Practically, this means that from any point in the
percolated pores the two external periodic surfaces of
the specimen can be reached. For estimation of transport-based durability properties, the very fine pores
detected at the high sensitivity level applied might
not be contributing significantly to transport through
the specimen. When they are removed for this
practical purpose the connected fraction will decline.
function is mathematically given by
F (s) =
∫
dV
Ω1 ( s )
,
∫ dV
(2)
Ω
where Ω is the total pore space and Ω1(s) is the space
of pores whose sizes are smaller than s . Since the
random points are uniformly distributed in the pore
space, the random points can be considered as a
discretized system of the pore space. Let pore size s
be discretized by a limited number of sizes sk (k =
1,2...,M), then the cumulative function is reflected at
sk as
N
F ( sk ) =
∑ B(s
i =1
k
> si )
N
,
(3)
where N is the total number of random points, and
B ( ) is a Boolean function that equals 1 if the condition between the brackets is satisfied and is 0 otherwise. Subsequently, the real volume-based pore size
distribution function, posd, is obtained through derivation of the cumulative functions as
f ( sk ) =
F ( sk +1 ) − F ( sk −1 )
sk +1 − sk −1
DRaMuTS EXAMPLE
Table 1. Characteristics of simulated cement samples.
Boundary
conditions
PC1
PC2
BPC1
BPC2
6P
4P + 2R
6P
4P + 2R
Diameter range (μm) and
volume percentage (%)
PC
blended
Specific surface area
Size of
Initial
Number of
(cm2/g)
pocket (μm) w/b ratio particles
PC
blended
3~30
(100 %)
-
1450
-
100
0.299
1090
3~30
(85.82 %)
1.5~2
(14.18 %)
1428
9994
100
0.299
18150
59
STROEVEN P ET AL: Concrete Porosimetry
Table 2. Experimental details.
PC1
PC2
BPC1
BPC2
Ultimate
DOH
Porosity
p (%)
0.715
0.715
0.715
0.715
5.50
5.44
5.26
5.20
The probability of detecting such tiny connections
(‘ink-bottles”) between initially dead-end pores increases indeed with the number of generated points or
edges. However, these very small pores may have
limited impact on transport capabilities. So, a lower
sensitivity level can be selected because of relevance
for durability issues. As a result, pore trees consisting
of continuous pores and dead-end pores branching off
such main trunks would be a realistic concept that
could underlie durability estimates.
Connected
fraction
of p (%)
99.78
99.66
99.80
99.78
The random tree structures in Fig. 4 are both
obtained at the level of 10,000 tree edges (shown later
to correspond to an intermediate sensitivity). Different
trees have different colors. Obviously, gap-graded
blending with a pozzolanic admixture (rice husk ash
served as reference) leads to fractionalization in the
pore network. However, at increasing sensitivity,
most of the pore system becomes continuous by pore
connections of decreasing diameters, as reflected by
Table 2. At this ‘optimum’ sensitivity, total porosity
is found declining somewhat in the blending case.
Hence, getting to the outside of the hardened
pocket from a random point in a pore is more difficult
or less straight forward in the blended case. As a
result, transport-based durability capacity of concrete
can be expected favored by gap-graded blending.
Fig. 5. Distribution of random points generated in the
second stage of the DRaMuTS approach and located
in pore space of sample PC2, at the top, and of sample
BPC2, at the bottom.
Of the points randomly dispersed in the second
stage of the DRaMuTS approach, those outside pore
space are removed. The result is displayed in Fig. 5.
Fig. 6 shows the gradient structures of porosity in the
four different samples. The zone of enhanced porosity
Fig. 4. Exploration by DRaMuTS in virtual hydrated
Portland cement paste (sample PC2) at the top and
gap-graded blended Portland cement paste (sample
BPC2), at the bottom, both with 10,000 tree edges.
60
Image Anal Stereol 2012;31:53-63
inside the ITZ around aggregate grains is reduced by
the gap-graded blending. Hence, the effect on global
transport will be reduced.
Fig. 7. Pore size distribution functions for plain and
blended Portland cement.
Fig. 6. Gradient structures in connected pore volume
at two different boundary conditions and for PC, at
the top, and gap-graded blended PC, at the bottom.
Fig. 8. Sensitivity analysis of the fraction of connected
edge lengths.
Finally, Fig. 7 presents volume-based pore size
distribution functions of plain and blended cases both
investigated at ‘highest’ sensitivity (i.e. 100,000 tree
edges in first stage and 1000,000 random points in
second stage are used in both cases). Data are obtained
by star volume measurements. The differences in pore
size distribution will increase at equal sensitivity
level. In addition to fractionalization and increased
pore tortuosity (Fig. 4), blending leads to refinement
in the pore structure; another factor that will positively
affect concrete durability due to hampered transport
through the material.
DISCUSSION
Present day computer technology renders possible
using a modern DEM system for simulating concrete.
It has been proven in the open literature (see, e.g.,
Stroeven et al., 2009)) that conventionally applied
random sequential addition (RSA) systems, which are
based on randomized particle addition, do not present
reliable information when structure-sensitive material
properties are at issue. Also fully randomized approaches to digitized material models such as proposed by
national institute of standards and technology (NIST)
researchers (Garboczi and Bentz, 2001) lead to biases
as also shown in the aforementioned paper. Reality is
between the too evenly distributed particle dispersion
in RSA systems and the chaotic dispersion state in the
latter approach. The development of the depercolating
pore system during hydration is such an example. The
HADES system on which this study is based offers a
modern concurrent algorithm-based dynamic approach,
The sensitivity analysis presented in Fig. 8 offers
insight into the number of tree edges (or number of
generated points) required for obtaining a stable value
of the ‘fraction of connected edges’. The latter is
defined as the total length of the tree edges connected
to both opposite surfaces of the sample pocket versus
total length of the generated tree edges. At very large
numbers, most of the pores are connected to the
outside surfaces, as also indicated in Table 2.
61
STROEVEN P ET AL: Concrete Porosimetry
simulating also the production conditions of the material
whereby particle interference is a major mechanism.
they confirm results obtained by the far more laborious
approach of serial sectioning and 3D reconstruction.
Experiments on gap-graded rice husk ash-blended
Portland cement concrete confirmed the improved
efficiency of gap-graded blending (Bui et al., 2005).
Earlier simulation results (obtained by the concurrent
algorithm-based dynamic SPACE system) revealed
this to be due to the improved packing density in the
ITZ. The present paper adds to these results the positive
effects on pore refinement that will have a favorable
impact on transport-based durability performance of
the material.
However, fundamentally different is the finite
value observed for connected pore volume in bulk.
This is due to the dead-end pores branching off the
trunks inside ITZs that will lead to connections
between pores of neighboring ITZs. Aggregate grains
have been demonstrated coming close enough to
cause ITZ percolation. In such percolated ITZ zones
of neighboring aggregate grains is the probability
highest for pore trees to get mutually connected
through their dead-end branches. The result is that the
dense random packing of the aggregate leads to
connected pathways (trunks) over the full extension
of concrete specimens or elements.
This paper deals with an ongoing study in the
field of porosimetry on virtual concrete. A novel
method is presented. This may open new horizons for
durability predictions. Popular methods for estimation
of durability properties are based on quantitative
image analysis results or involve practical approaches
discussed elsewhere (Stroeven et al., 2009), the evaluation of which depends on non-realistic assumptions
for pore geometry. Hence, the new topological and
geometric information on the spatial pore network
structure that comes available would probably also
ask for novel ways to approach durability issues.
Gap-graded blending of the PC leads to reduced
transport capacity of the hardened material. This is
due to increased tortuosity and refinement of the
continuous fraction of the pore system. This adds to
positive experimental findings on the mechanics of
gap-graded rice husk ash blended Portland cement
concrete that were supported by particle packing simulations with the concurrent algorithm-based SPACE
system (Bui et al., 2005). Also on the basis of gapgrading, strength is even found positively influenced
when an inert mineral admixture (carbon black) was
applied (Goldman and Bentur, 1993). This is the result
of high packing density favoring the development of
significant physical contributions (of Vander Waals
nature) to material strength. Investigations with
DraMuTS are underway aiming to reveal also positive
effects on durability resistance in this case.
The ITZ is shown to play an important role in the
transport process that may eventually lead to durability
problems. This dominant position of the ITZ is widely
accepted. To verify this by direct measurement would
be very complicated. The present authors are unaware
of such tests. Contrary, the presented computer simulation approach would offer at least an easier and far
more economical solution. Such a study is foreseen
for the near future.
REFERENCES
Breugel KV (1991). Simulation of cement hydration and
formation of structure in hardening cement based
materials. PhD Thesis, Delft University of Technology.
Delft.
Bui DD, Hu J, Stroeven P (2005). Particle size effect on
the strength of rice husk ash blended gap-graded Portland
cement concrete. Cem Concr Compos 27: 357–66.
Chen HS, Ye G, Stroeven P (2004). Computer simulation
of structure of hydrated cement paste enclosed by
inter-facial transition zone in concrete. In: Setzer MJ,
Palecki S, eds. Proceedings of International Conference
on Durability of High Performance Concrete and Final
Workshop of CONLIFE, 2004 September 23-24; Essen,
Germany. Freiburg: Aedificatio Publishers, 133–44.
Chen HS, Stroeven P, Ye G, Stroeven M (2006). Influence
of boundary conditions on pore percolation in model
cement paste. Key Eng Mater 302-303: 486–92.
Diamond S (2000). Mercury porosimetry: an inappropriate
method for the measurement of pore size distributions
CONCLUSIONS
A novel approach to porosimetry in virtual
concrete is presented that combines reliability and
economy. This method relies on realistic analogue
simulations of fresh cementitious materials by a
dynamic concurrent algorithm-based DEM system,
HADES. During simulated maturation, the process of
pore de-percolation can be numerically assessed. To
do so, the evolving pore network structure is analyzed
by DRaMuTS of which development is inspired by
experiences in robotics. The method renders possible
separating between continuous pores, dead-end pores
branching off such trunks, and isolated pores.
Tree-like pore network structures are obtained of
which volume density is highest adjacent to the aggregate grain surface inside the ITZ. As to this aspect,
62
Image Anal Stereol 2012;31:53-63
in cement-based materials. Cem Concr Res 30:1517–
25.
Garboczi EJ, Bentz DP (2001). The effect of statistical
fluctuation, finite size error, and digital resolution on
the phase percolation and transport properties of the
NIST cement hydration model. Cem Concr Res 31:
1501–14.
Garboczi EJ, Bullard JW (2004). Shape analysis of a reference cement. Cem Concr Res 34: 1933–7.
Goldman A, Bentur A (1993). The influence of microfillers
on enhancement of concrete strength. Cem Concr Res
23:962–72.
Gundersen HJG, Bagger P, Bendtsen TF, Evans SM,
Korbo L, Marcussen N, et al. (1988). The new stereological tools: Disector, fractionator, nucleator and
point sampled intercepts and their use in pathological
research and diagnosis. APMIS 96(7-12):857–81.
He H (2010). Computational Modelling of Particle Packing
in Concrete. PhD Thesis, Delft University of Technology.
Delft.
He H, Guo Z, Stroeven P, Stroeven M, Sluys LJ (2010).
Strategy on simulation of arbitrary-shaped cement grains
in concrete. Image Anal Stereol 29:79–84.
Hu J (2004). Porosity in concrete – morphological study of
model concrete. PhD Thesis, Delft University of
Technology. Delft.
Hu J, Stroeven P (2005). Depercolation threshold of porosity
in model cement: approach by morphological evolution
during hydration. Cem Concr Compos 27:19–25.
Hu J, Stroeven P (2006). Proper characterization of pore
size distribution in cementitious materials. Key Eng
Mater 302-303:479–85.
LaValle SM, Kuffner JJ (2001). Rapidly-exploring random
trees: Progress and prospects. In: Donald BR, Lynch
KM, Rus D, eds. Algorithmic and Computational Robotics: New Directions. Wellesley, MA: A K Peters, 293–
308.
Roelfstra PE (1989). A numerical approach to investigate the
properties of numerical concrete. PhD Thesis, Ecole
Polytechnique Fédérale de Lausanne. Lausanne.
Serra J (1982). Image analysis and mathematical morphology.
London: Academic Press.
Smit TH, Schneider E, Odgaard A (1998). Star length
distribution: a volume-based concept for the characterization of structural anisotropy. J Mircosc 191:249–57.
Stroeven M (1999). Discrete numerical modelling of Composite Materials - application to cementitious materials.
PhD Thesis, Delft University of Technology. Delft.
Stroeven P, Stroeven M (2001). Reconstructions by SPACE
of the Interfacial Transition Zone. Cem Concr Compos
23:189–200.
Stroeven P, Guo Z (2006). Modern routes to explore concrete’s complex pore space. Image Anal Stereol 25:
75–86.
Stroeven P, Hu J (2007). Gradient structures in cementitious
materials. Cem Concr Compos 29:313–23.
Stroeven P, Hu J, Stroeven M (2009). On the usefulness of
discrete element computer modeling of particle packing
for material characterization in concrete technology.
Comput Concr 6:133–53.
Stroeven P, Hu J, Koleva DA (2010). Concrete porosimetry:
Aspects of feasibility, reliability and economy. Cem
Concr Compos 32:291–9.
Stroeven P, Le NLB, Stroeven M, Sluys LJ (2011). Discrete
element modeling approach to porosimetry for durability
risk estimation of concrete. In: Oñate E, Owen DRJ,
eds. Proceedings of PARTICLES 2011, II International
Conference Particle-based Methods, Fun-damentals and
Applications (on CD), 2011 October 26-28; Barcelona,
Spain.
Stroeven P, Le NLB, He H (forthcoming 2012). Methodological approaches to 3D pore structure exploration
in cementitious materials. In: Proceedings of the 13th
International Conference on Non-conventional Materials
and Technologies, 2011 September 22-24; Changsha,
China.
Vogel HJ, Roth K (2001). Quantitative morphology and
network representation of soil pore structure. Adv Water
Resour 24:233–42.
Wittmann FH, Roelfstra PE, Sadouki H (1985). Simulation
and analysis of composite structures. Mater Sci Eng
68:239–48.
Ye G (2003). Experimental study and numerical simulation
of the development of the micro-structure and permeability of cementitious materials. PhD Thesis, Delft
University of Technology. Delft.
Ye G, Breugel Kv, Fraaij ALA (2003). Three-dimensional
microstructure analysis of numerically simulated cementitious materials. Cem Concr Compos 33:215–22.
63