Study - An Investigation of Single Sand Particle Fracture Using X

Study - An Investigation of Single Sand Particle Fracture Using X
Zhao, B. et al. (2015). Géotechnique 65, No. 8, 625–641 [http://dx.doi.org/10.1680=geot.4.P.157]
An investigation of single sand particle fracture using X-ray
micro-tomography
B. ZHAO , J. WANG , M. R. COOP , G. VIGGIANI† and M. JIANG‡
Particle breakage is of fundamental importance for understanding the mechanical behaviour of sands
and is relevant to many geotechnical engineering problems. In order to gain new insights into the
mechanism of breakage of individual sand particles under single-particle compression, this study
combines mechanical tests with three-dimensional X-ray micro-computed tomography (μCT)
performed ‘in situ’, that is, during loading. A novel mini-loading apparatus was developed to
perform in-situ compression tests within a laboratory nanofocus X-ray CT. The tests were performed on
eight particles, four Leighton Buzzard sand (LBS) particles and four highly decomposed granite
(HDG) particles, to study their different fracture mechanisms. A series of image processing and
analysing techniques was utilised to obtain both qualitative and quantitative results. The most
important factors in determining the fracture patterns of the LBS and HDG particles were found to be
particle morphology and initial microstructure, respectively. Versatile fracture patterns deviating from
simple vertical splitting were observed, particularly in HDG particles. The change of morphology
parameters during loading was found to depend on the fracture mechanisms and material properties,
independently of their initial values. The fragments of both the LBS and HDG particles satisfy the
fractal distribution, which indicates that the fragmentation is scale invariant. Different energy
dissipation mechanisms were found. The energy dissipation by friction gradually prevails against the
energy dissipated in generating new surfaces.
KEYWORDS: fractals; laboratory tests; particle crushing=crushability; particle-scale behaviour; sands
has also been investigated by analytical models (e.g.
Cavarretta & O’Sullivan, 2012). This test provides important
information for calibrating discrete-element method (DEM)
modelling of crushable sand particles. Some DEM researchers have used crushable aggregates that were composed of
spherical particles cemented at their contact points
(Robertson, 2000; McDowell & Harireche, 2002; Bolton
et al., 2008; Hanley et al., 2011; Wang & Yan, 2012, 2013; Cil
& Alshibli, 2012, 2014), whereas Lobo-Guerrero & Vallejo
(2005) implemented a different particle fracture approach
involving the replacement of a rigid particle with a group of
smaller particles based on a certain tensile failure criteria.
Physical tests, analytical methods and numerical simulations
have all been used to understand the relationship between the
load–deformation response of individual particles and the
overall stress–strain response of a granular material
(McDowell & Bolton, 1998; Nakata et al., 1999, 2001;
Cheng et al., 2003; Russell & Einav, 2013). However, limited
studies have been conducted to understand the relationship
between single-particle fracture and particle external morphologies and internal microstructure (e.g. cleavage, impurities and voids) owing to experimental difficulty in
examining the evolution of particle microstructure during
the fracture process.
X-ray micro-tomography provides a powerful tool for
dealing with the above difficulty. A rich body of studies have
been conducted on the deformation and fracture mechanisms
of a wide range of materials including biological materials
(e.g. tissues and bones), construction materials (e.g. ceramics,
asphalt and rock), composite materials, synthetics and
cellular solids (Stock, 2008). For granular soils, X-ray
tomography was first used to investigate strain localisation
in triaxial tests (Desrues et al., 1996; Alshibli et al., 2000).
These studies demonstrated that X-ray tomography is an
effective tool for observing the localisation patterns and
quantifying the evolution of void ratio inside a shear band.
INTRODUCTION
Particle breakage plays an important role in determining the
constitutive behaviour and deformation characteristics of
granular materials, especially at high stress levels. For
one-dimensional compression, particle breakage was found
to be the principal source of plastic volumetric compression
on the normal compression line (McDowell & Bolton, 1998),
and a strong correlation was found between single-particle
crushing strength and the macroscopic yield stress of sand
(McDowell, 2002; Yoshimoto et al., 2012). In triaxial tests,
the dilatant behaviour of crushable soils was found to be
governed by the tensile strength of particles (McDowell &
Bolton, 1998; Yoshimoto et al., 2012). The successive
breakage of soil particles tends to form a fractal particle
size distribution, which gives a linear distribution on a double
logarithmic graph (McDowell et al., 1996; McDowell &
Bolton, 1998; Coop et al., 2004; McDowell, 2005; Tarantino
& Hyde, 2005; Russell, 2011).
Displacement-controlled single-particle compression tests,
in which individual particles are compressed between two
rigid platens, are often used to measure the strength of sand
particles and other granular materials. Single-particle compression tests have been performed on sand particles with a
wide range of particle sizes and mineral types (Lee, 1992;
Nakata et al., 1999; McDowell, 2002; Cil & Alshibli, 2012;
Cavarretta et al., 2010). The particle fracture phenomenon
Manuscript received 18 August 2014; revised manuscript accepted
16 April 2015.
Discussion on this paper closes on 1 January 2016, for further details
see p. ii.
Department of Architecture and Civil Engineering, City
University of Hong Kong, Hong Kong, China.
† University Grenoble Alpes, Grenoble, France.
‡ Department of Geotechnical Engineering, Tongji University,
Shanghai, China.
625
626
ZHAO, WANG, COOP, VIGGIANI AND JIANG
Early studies used a voxel size of hundreds of microns that did
not accurately resolve the particles. More recently, a number
of studies on granular materials have been conducted with
much better resolution (a few microns), allowing grain-scale
characterisation of sand deformation under load (e.g. Andò
et al., 2013). Grain-scale observations enrich our understanding with the full kinematics (i.e. three-dimensional (3D)
displacements and rotations) of all the individual sand grains
in a specimen (Hall et al., 2010), the evolution of 3D particle
morphology and fabric of a real sand under loading (Fonseca
et al., 2012, 2013), and the evolution and distribution of
particle breakage (Andò et al., 2013; Cil & Alshibli, 2014).
The fracture of silica sand particles has been examined
experimentally with 3D synchrotron tomography and
numerically with 3D DEM simulations simultaneously (Cil
& Alshibli, 2012, 2014).
The objective of this paper is to explore the fracture
process of the sand particles under single-particle compression using micro-computed tomography (μCT) and
crushing tests. A novel in-situ single-particle compression
apparatus was developed and used to carry out crushing
tests on two kinds of sand particles, namely Leighton
Buzzard sand (LBS) particles and highly decomposed
granite (HDG) particles. The μCT scanning of the particles
was carried out while they were loaded. A series of
image-processing and analysis techniques were then applied
to the 3D CT images of the particles to yield knowledge
of the particle fracture behaviour at both qualitative and
quantitative levels, the former containing information of
fracture patterns and microstructure evolution and the
latter giving statistics of fragment morphology. This information offers novel insights into the relationship between
single-particle fracture behaviour and the intrinsic particle
microstructure. The breakage energy, which was evaluated
by using the load–displacement curve and the measured surface area of the particle fragments, provides further valuable
information of the fracture mechanics of sand particles.
EXPERIMENTAL SET-UP
A mini-loading apparatus was designed to perform
single-particle crushing tests on natural sand particles in a
laboratory μCT system (v|tome|x m, phoenix|X-ray, General
Electric Company (GE)) located at Shanghai Yinghua NDT
Equipment Trade Co., Ltd. A 3D scheme and a photograph
of the apparatus in operation are shown in Fig. 1. The apparatus consists of three major parts: the sample chamber,
the loading and data acquisition system and the bearing
frame.
The sample chamber, located in the top part of the
apparatus, is a high-strength radiolucent tube made of
polyetherimide (PEI). With its high strength, the PEI tube
can be long and thin enough to let the GE nanofocus X-ray
tube get close to the sand particle to reach the highest CT
resolution. The PEI tube is 105 mm long, and has a 22 mm
outer diameter and a 10 mm inner diameter. Inside the PEI
tube, the sand particle is loaded between two ceramic platens.
The loading platens are made of ceramic owing to its low
density (3·85 g=cm3), whereas carbon steel has a density of
7·85 g=cm3. A sharp change of linear attenuation coefficient
from the particle to the loading platens will greatly increase
the cone beam errors (Davis & Elliott, 2006). Therefore,
lower density platens can reduce the amount of blurring on
the CT images, especially around the contact points.
According to the theoretical work by Russell & Muir
Wood (2009), Young’s modulus, Poisson ratio, hardness
and roughness of the loading platens can all have effects on
the actual contact area and thus the stress distribution and
fracture mechanisms of the particle. In an effort to minimise
X-ray
tube
Sand
particle
GE nanofocus
X-ray tube
Sand
particle
Mini-loading
apparatus
Load cell
LVDT
Stepping
motor
Precision
rotation unit
(a)
(b)
Fig. 1. Close-up of the mini-loading apparatus: (a) 3D view of the
apparatus; (b) photograph of apparatus in operation inside X-ray
machine (GE, General Electric Company)
any such effect, the present authors selected a ceramic platen
with a high Young’s modulus of 380 GPa, a high hardness of
1700 HV (HV is the Vickers pyramid number), a low Poisson
ratio of 0·22 and a low roughness of Ra ¼ 0·2 μm. After the
test, no plastic deformation (i.e. breaking of asperities) was
identified from the CT images. Therefore, it is believed that
the ceramic platens had little influence on the particle
fracture and energy dissipation mechanisms. In order to
prevent fragments from jumping out of the scanning area and
to maintain the fracture patterns during scanning, the sand
particle is immersed in high-viscosity silicone grease during
the test.
The force is exerted on particles by a microstep linear
actuator (NA14B16-T4-MC04, Zaber Technologies Inc.),
which is capable of exerting a force up to 222 N with a
microstep size of 0·09525 μm. The exerted force is measured
by a miniature load cell (Model 111, Sensorwerks) with
dimensions small enough to fit inside the apparatus. This
load cell is rated for 222 N with a non-linearity of 1·33 N and
non-repeatability of 2·66 N. A linear variable differential
transformer (LVDT) (DFg 2·5, Solartron Metrology), rated
for a range of ±2·5 mm with an infinite resolution and
non-linearity of ±7·5 μm, is used to measure the displacement of the loading platen. In order to review the load and
displacement signals, a model 7220 (Measurement Systems
Ltd) direct current (DC) measurement processor with 16-bit
resolution is used. The linear actuator and measurement
processor are controlled by specially written in-house software that allows for running both displacement- and
load-controlled tests.
The PEI tube is screwed onto the bearing frame, the base
of which is fixed onto a precision rotation unit by way of a
three-jaw chuck (Fig. 1(b)). The precision rotation unit
rotates the apparatus and the sample 360° to finish one
scan. During each scan, the linear actuator stops while the
load cell and the LVDT keep recording the force and
displacement.
SINGLE SAND PARTICLE FRACTURE USING X-RAY MICRO-TOMOGRAPHY
MATERIALS TESTED
In this study, two types of natural sand particles with
different morphology, mineralogy and microstructure were
tested. Considering the load capacity of the apparatus (linear
actuator and load cell) and the resolution of the CT scanning,
particles with dimensions between 1·2 and 2·0 mm were used,
which were selected randomly. LBS particles are mainly
composed of quartz, which is chemically stable. Their shape
and surface texture are mainly modified by the geological
transportation process. Fig. 2(a) shows a microscopic view of
a typical LBS particle with a high degree of roundness and a
smooth surface. The Geotechnical Control Office (1984)
proposed six granite weathering grades, namely fresh rock,
slightly decomposed rock, moderately decomposed rock,
highly decomposed rock, completely decomposed rock and
residual soil. HDG particles, corresponding to the fourth
weathering level, were used in this study. It is granitic rock
weathered to the point that it readily fractures into pieces that
can be broken by hand. They are composed of feldspar,
quartz, mica and other minerals. At this weathering grade,
plagioclase has been almost completely altered to clay
minerals and the formation of dissolution features, such as
pits, trenches and channels, could be observed in the alkali
feldspar. Kaolin and micaceous minerals represent 10 to 30%
in mass. Fig. 2(b) shows a microscopic view of a typical HDG
particle, which has a larger angularity and more complex
mineralogy than the LBS particles.
Although μCT imaging is a very powerful tool for
visualising 3D structures, it can be difficult to identify
1 mm
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different phases because it is based on X-ray linear
attenuation coefficients. Scanning electron microscopy
(SEM) and energy-dispersive X-ray spectroscopy (EDS)
were used to identify the different minerals from the
surface of the HDG particles, and this information was
then used to interpret the μCT images. Fig. 3(a) shows two
minerals (quartz and feldspar) and a micro-crack between
these two phases, while Fig. 3(b) shows both feldspar and its
weathering products.
IMAGE ACQUISITION, PROCESSING AND
ANALYSIS
The image acquisition was conducted using a Phoenix v|
tome|x m, in which the samples are radiated with a
high-resolution nanofocus X-ray beam (180 kV=15 W).
Each sample was scanned rotating the apparatus 360°
around its central axis between an X-ray tube and a detector.
The detector has 2000 2000 pixels. A series of absorption
radiographs of the sample were acquired during the step-wise
rotation of the apparatus. The 3D volume reconstruction was
achieved using Phoenix Datos|x CT, a software using the
Feldkamp filtered back projection algorithm. The voxel size
of these reconstructed images is 3·3 μm when the X-ray tube
is closest to the PEI tube. The effective scanning volume is a
cylinder with both height and diameter being 6·6 mm (the
voxel size number of pixels). This means that the particles
and fragments should stay inside this volume if it is desired to
keep the same voxel size during the test. The CT data can be
1 mm
(a)
(b)
Fig. 2. Microscopic view: (a) typical LBS particle; (b) typical HDG particle
Weathering products
of feldspar
Quartz
Feldspar
Feldspar
Quartz
300 µm
(a)
Fig. 3. SEM images of typical HDG particles: (a) initial micro-crack; (b) weathering products
100 µm
(b)
ZHAO, WANG, COOP, VIGGIANI AND JIANG
628
visualised as a stack of images, each of which is a crosssection of the scanned sample with the thickness of one voxel
size. An example of a slice is shown in Fig. 4(a).
The reconstructed 3D CT images were put through a series
of image-processing and analysis procedures to obtain a
qualitative and quantitative description of the particle
fracture. The objective of image processing is to reduce
noise, to segment different phases and to identify individual
fragments. Then, image analysis techniques were
implemented to measure the morphology parameters of
each component. Many techniques implemented in this
study have been described by Fonseca (2011) and Fonseca
et al. (2012).
To reduce noise typically found in raw CT data, two
filtering algorithms, a 3D median filter and a 3D adaptive
Gaussian filter, were applied for LBS particles and HDG
particles, respectively. The median filter is the best-known
non-linear spatial filter, which replaces the intensity value of
a voxel by the median of the intensity values in its
neighbourhood (Gonzalez & Woods, 2010). The 3D
median filter has a size of 3 3 3, and it is repeated three
times for each CT image. This method was chosen because of
its excellent noise-reduction capability with considerably less
blurring. However, for the CT images of HDG particles, the
3D median filter could not achieve satisfactory smoothing
due to its limited filter size. In this case, the adaptive
Gaussian filter, which uses a larger neighbourhood and
reduces the blurring at the sharp edges, could be used
(Hodson et al., 1981; Deng & Cahill, 1993; Fonseca, 2011).
Fig. 4(b) shows the filtered image by the adaptive Gaussian
filter with the parameters set as smoothing ¼ 1·8 and edge
threshold ¼ 0·08. It can be seen that the image was nicely
smoothed while the edges between different materials were
preserved. The histogram of original image and filtered
image in Fig. 4(c) shows that the intra-class variances were
reduced after the filtering, which is helpful in performing the
thresholding segmentation.
Then, thresholding segmentation separates the filtered
images into different regions, namely, grains, silicone grease
and voids, based on their representative intensity levels.
Because there are more than two phases in the CT images,
choosing proper thresholds based on histograms (Fig. 4(c))
was done using the intrinsic function, ‘multithresh’, in
Matlab (Mathworks) based on the multilevel Otsu threshold
method (Otsu, 1979). Generally speaking, the method
divides the histogram into different classes and minimises
the intra-class variances. For the filtered image (Fig. 4(b)),
two threshold values were found to be 2561 and 6134, as
marked in Fig. 4(c). Fig. 4(d) shows the segmented image
based on these two threshold values; the three grey levels
represent sand particle, silicone grease and voids, respectively.
In this study, only the domain of sand particles and their
fragments after fracture are of interest.
Thresholding segmentation separates the sand particles
and fragments from other materials, but individual fragments cannot be identified when they touch. So the
LBS
particle
Silicone
grease
Voids
1 mm
1 mm
(a)
(b)
6
Original image
Filtered image
Number of voxels: ×103
5
Silicone
grease
4
LBS
particle
3
Voids
2
2561
1
6134
0
0
2
4
6
Intensity level: ×103
(c)
8
10
1 mm
(d)
Fig. 4. Illustration of image processing: (a) raw grey image; (b) image after the 3D adaptive Gaussian filter; (c) histogram before and after
filtering; (d) image after multilevel thresholding
SINGLE SAND PARTICLE FRACTURE USING X-RAY MICRO-TOMOGRAPHY
morphological watershed algorithm (Beucher & Lantuejoul,
1979) was implemented to separate them. To illustrate the
method, the segmentation process is presented here on a
two-dimensional (2D) slice of a fractured particle (Fig. 5(a))
as an example, but it should be noted that the algorithms
implemented in this study are 3D. Before applying the
watershed algorithm, a distance map which represents the
distance between each pixel and its nearest surface boundary
was calculated (Fig. 5(b)). The distance map was inverted so
that the maximum regions changed into minimum regions. If
watershed segmentation is applied directly to the inverted
distance map, it will almost always result in oversegmentation, due to the intensity variations within both
the objects and background (Gonzalez & Woods, 2010). An
approach based on the concept of markers was therefore used
to control the over-segmentation (Wählby et al., 2004).
The watershed lines were determined by means of a
marker-based watershed from the specified markers. Fig. 5
(c) shows the watershed lines and the binary image after
removing the watershed lines. Then the fragments could be
separated and labelled as different components (Fig. 5(d)).
The results are the 3D labelled images, each integer in these
images representing an individual fragment.
The shape of a particle can be decomposed into components in three scales, sphericity (or form), roundness (or
angularity) and roughness (or smoothness) (Barrett, 1980;
Cho et al., 2006). The 3D digitised image data of each
component allow quantitative measurement of morphology.
Specifically, volume is obtained by multiplying the number of
voxels defining the particle by the size of each voxel, that is,
‘resol3’, where resol is the spatial resolution of the image.
629
Surface area is estimated by a method based on the local
configuration of binary valued voxels (Lindblad, 2005). In
this method, each configuration is assigned a surface area
weight and the total area is calculated as the sum of the local
area contributions.
Principal component analysis (PCA) is applied to determine the orientation of the major, minor and intermediate
axes for each component. This method has been applied to
the analysis of tomographic data in different fields (Phillion
et al., 2008; Fonseca et al., 2012). The components are
rotated so that their principal axes are parallel to the
Cartesian axes. The major (a), intermediate (b) and minor
(c) dimensions of a component are then calculated by
multiplying the component’s voxel coordinates after rotation
and the voxel size. As the data for each component are 3D,
two aspect ratios are calculated, the elongation index (EI)
and the flatness index (FI), which are defined as EI ¼ b=a,
and FI ¼ c=b (a . b . c). Two more parameters, namely
sphericity (S) and convexity (CX), are also used here to
provide measures of compactness. Sphericity describes how
closely a component resembles a sphere. It was originally
defined as SAsphere=SA, where SA is the surface area of the
component, and SAsphere is the surface area of a sphere
having the same volume as the component (Wadell, 1932).
Here,
the ffi sphericity S is calculated by way of
ffiffiffiffiffiffiffiffiffiffiffiffiffi
p
3
S¼ 36πV 2 =SA, where V is the volume of the component.
Convexity is defined as CX ¼ V=VCH, where VCH is the
volume of the convex hull enclosing the component.
Referring to Fig. 6, a convex hull is defined as the minimal
convex surface containing all voxels of the component. A
surface is convex if it contains all the line segments
High
Low
1 mm
(a)
1 mm
(b)
1 mm
(c)
1 mm
(d)
Fig. 5. Illustration of morphological watershed process on a 2D slice of fractured particle: (a) binary image after thresholding segmentation; (b)
distance map; (c) watershed lines and binary image removed watershed lines; (d) labelled fragments
ZHAO, WANG, COOP, VIGGIANI AND JIANG
630
Component
+
Convex hull of
component
Fig. 6. Two-dimensional illustration of convex hull
connecting each pair of points inside this surface. It should be
noted that the aspect ratios only reflect the form of particle
shape, while S and CX are sensitive to both roundness and
form (Barrett, 1980).
RESULTS
Initial particle morphology
In the present study, in total, eight particles – four LBS
particles and four HDG particles – were tested. Their
morphology information is summarised in Table 1. Fig. 7
shows 3D visualisations of the tested particles. The width of a
particle is defined as its minor dimension (c). The volumes of
the LBS particles tested vary from 2·18 mm3 to 3·30 mm3,
while the HDG particles have a wider size variation from
3·06 mm3 to 11·30 mm3. The aspect ratios of these two kinds
of particles are very similar. The HDG particles are more
angular than the LBS particles, which results in lower
sphericity and convexity values. The sphericity values of all
the particles are slightly smaller than their convexity values.
Fracture process and fracture patterns
Each particle was scanned three times, including the initial
condition and two damaged conditions with the same X-ray
scanning settings and the same voxel sizes. Fig. 8 shows the
load–displacement curves for LBS-1 and HDG-1 and their
three scanning points. Hereafter, the scans will be denoted by
a code in which the test number is followed by the number of
the scan (e.g. LBS-1-S2 is the second scan of test LBS-1). For
the LBS particles, the first scan was taken after a compression
force of around 10 N was applied to the particle to examine
their initial loading positions. For HDG particles, the first
scan was performed before applying any compression,
because this would have altered the initial microstructure of
the particles. The second and third scans were taken after the
first fracture and at a large displacement, respectively. The
force–displacement curves show that during each scan (which
took about 1·5 h), the changes of force and displacement
were very small (e.g. the compression force reductions of
LBS-1 and HDG-1 during scan 2 are both 0·95 N).
LBS-1 is selected as an example to illustrate the fracture
process observed from the CT images (Fig. 9). The three
columns correspond to LBS-1-S1, LBS-1-S2 and LBS-1-S3,
respectively. Figs 9(a)–9(c) present the typical 2D grey-level
slices of the CT images after processing by a 3D median
filter. In these images, brightness is proportional to X-ray
absorption. Therefore, dark regions correspond to lowdensity phases (voids and silicone grease), while bright
regions correspond to high-density phases (ceramic platens
and sand particles). It is found that the small radius of
curvature at the top contact point (Fig. 9(a)) caused stress
concentration and extensive fragmentation (Fig. 9(b)). The
two main tensile fracture planes were parallel to the
compression direction (Fig. 9(b)). The fragmentation
observed in LBS-1-S3 may be caused by direct compression
by the loading platens or the interaction between fragments
and different fracture mechanisms; for example, tension and
bending may happen here (Fig. 9(c)) (Quinn, 2007).
The CT images from LBS-1-S2 and LBS-1-S3 demonstrated that the silicon grease could successfully keep the
fragments inside the scanning volume and preserve the
fracture patterns. It should be noted that the fracture patterns
in scan 3 might be different without silicone grease. The
fractures usually have a complex spatial distribution, which is
difficult to examine only from the 2D slices. The fragments
are labelled with different colours and made semi-transparent
so that the fracture patterns can be examined more easily
(Figs 9(e) and 9(f)). Fig. 9(d) shows the condition of the
contact point at the top (noted by an arrow), where there was
a small radius of curvature and a concave feature nearby. The
middle sections of the thresholding segmented images
perpendicular to the loading direction are shown in Figs 9
(g)–9(i). There were many intersecting and branching cracks,
which are helpful to interpret the sequence and propagation
direction of the cracks. Three cracks stopping at the
intersection have been identified and marked in Fig. 9(h).
The first crack of the intersecting cracks passes through the
undisturbed material, and the second crack approaches and
is stopped at the intersection since it is unable to traverse the
previously cleaved material. Similar observations were made
by Quinn (2007).
Generally speaking, sand particle fracture is a brittle
fracture consisting of relatively rapid propagation of multiple
cracks through a stressed material. The patterns of crack
propagation and branching not only point the way back to
the point where cracking starts, but can also provide
information about the origin of fracture, fracture energy
Table 1. Summary of the morphology of particles tested
Particle ref.
LBS-1
LBS-2
LBS-3
LBS-4
HDG-1
HDG-2
HDG-3
HDG-4
Voxel size: μm
Width: mm
Volume: mm3
Surface area: mm2
S
CX
EI
FI
3·89
3·67
3·31
3·31
3·57
3·31
3·31
3·31
1·36
1·43
1·43
1·60
1·44
2·78
2·09
1·81
2·18
2·24
3·16
3·30
3·06
11·30
5·46
6·38
10·11
10·05
12·24
13·38
15·59
49·32
21·86
27·87
0·80
0·82
0·94
0·89
0·67
0·50
0·68
0·60
0·90
0·89
0·98
0·97
0·82
0·78
0·84
0·77
0·87
0·86
0·90
0·85
0·86
0·86
0·90
0·86
0·70
0·82
0·74
0·79
0·68
0·83
0·84
0·63
SINGLE SAND PARTICLE FRACTURE USING X-RAY MICRO-TOMOGRAPHY
1000 µm
1000 µm
(a)
(b)
1000 µm
1000 µm
(c)
1000 µm
(e)
(f)
631
1000 µm
(d)
1000 µm
(g)
1000 µm
(h)
Fig. 7. Three-dimensional CT reconstruction of tested particles: (a–d) LBS-1 to LBS-4; (e–h) HDG-1 to HDG-4
120
LBS-1
HDG-1
LBS scan points
Force: N
90
HDG scan points
60
S1
30
S3
S2
0
0
0·1
0·2
0·3
0·4
Displacement: mm
Fig. 8. Load–displacement curves from tests LBS-1 and HDG-1
and stress state (Quinn, 2007). Based on the observations of
fracture patterns from CT images, it is possible either to
confirm or question many hypotheses researchers have made
over the years about single-particle fracture mechanisms.
The irregular morphology of natural sand particles,
together with loading condition, defines the boundary conditions, influences the stress distribution inside the particle
and leads to different fracture patterns. For example, the
small radii of curvature near the contact points of LBS-1 and
LBS-2, as shown in Figs 7(a) and 7(b), are likely to create
stress concentrations. The high elastic energy stored by the
stress concentrations resulted in extensive fragmentation
around the contact points (Fig. 9(b)). The resulting fragments have sizes varying from around 10 μm to several
100 μm. The crushing on contact surfaces effectively
increased the contact area and reduced the stress concentration at the surface. LBS-3 and LBS-4 had quite similar
fracture patterns, as shown in Fig. 10. Two planes were nearly
perpendicular to each other, and separated the particle in
four major fragments, as a result of tensile stress on the
planes. Severe damage occurred close to the contact surfaces
where the stress was mainly compressive before the fracture
occurred.
The analytical solution for the stress field within an elastic
sphere subject to diametrically opposite normal forces given
by Russell & Muir Wood (2009) indicates that the failure will
initiate somewhere directly below and close to the loading
areas. This initial failure is a type of shear failure because the
failure criterion used is a function of the second invariant of
the deviatoric stress tensor. Although the real stress fields
within the sand particles tested in the present study would
differ significantly from the idealised solution given by
Russell & Muir Wood (2009), mainly due to the irregular
particle shapes, microstructure within particles and loading
contact geometries, the validity of their solution can be
roughly confirmed based on careful observation of the CT
results. For example, the tensile splitting failure of LBS-1
shown in Fig. 9(b) could have started with a shear failure at a
short distance below the upper loading contact, where a
small fragment is found, and then propagated rapidly along
the vertical plane containing the loading contacts. But this
event was preceded by the severe fragmentation of the upper
local contact area due to large stress concentrations, which
may then make the entire stress field and hence failure
mechanism within the particle drastically different from the
analytical solution of Russell & Muir Wood (2009). However,
it should be pointed out that the accurate identification of the
initial fracture of a single particle is beyond the capability of
current X-ray CT information.
X-ray CT revealed that there were some initial voids inside
the LBS particles, especially in LBS-3 and LBS-4. These
initial voids have the effect of reducing the stress needed for
the development of the fracture, which results in creating a
size effect in terms of particle strength (McDowell & Bolton,
1998; Nakata et al., 1999; McDowell & Amon, 2000). Figs
10(c) and 10(d) show that the fracture went through some of
these voids. However, it is difficult to determine, only from
the CT images, whether the fracture actually started from
these voids. There were also several big initial voids inside
ZHAO, WANG, COOP, VIGGIANI AND JIANG
632
1000 µm
(a)
1000 µm
(b)
1000 µm
(d)
1000 µm
(e)
1000 µm
(g)
1000 µm
(c)
1000 µm
(f)
1000 µm
(h)
1000 µm
(i)
Fig. 9. Visualisation of fracture process of LBS-1 from three scans: (a–c) 2D slices parallel to the loading direction; (d–f) 3D labelled images from
top view; (g–i) sections of the thresholding segmented images from top view
particle LBS-2, but the flaws remained intact after the
fracture. It is possible that the stress concentrations near
the contact points caused by small radii of curvature reduce
the probability of fractures passing through the initial voids.
Conchoidal fractures that do not follow natural planes of
separation and often are curved were observed in Fig. 10(d).
The above observations of the single particle fracture process
are potentially valuable for the development of novel fracture
mechanics models for single particle failure (e.g. Brzesowsky
et al., 2011).
Different kinds of initial microstructures were observed
inside the HDG particles, as shown in Figs 11(a)–11(d). The
two major minerals of HDG-2 shown in Fig. 11(b) are quartz
(darker one) and feldspar (brighter one). This heterogeneity
makes the particles much easier to break under local stress
concentrations, as indicated by fractures passing through
impurities of very high density (Fig. 11(e)), phases boundaries (Figs 11(f) and 11(g)) and weathering products
(Fig. 11(h)). The fractures inside the feldspar show a cleavage
pattern, which is caused by crystallographic planes with low
bonding. For particles HDG-3 and HDG-4, the fracture
disappeared in the porous weathering products from feldspar
(Figs 11(g) and 11(h)).
Figure 12 and Fig. 13 present 2D slices parallel and
perpendicular to the loading directions of HDG particles at
three scanning points. It is found that the fracture patterns
are strongly influenced by the initial microstructures that
cause the structural weakness. Gallagher (1987) classified
structural weakness into original structures (e.g. cleavage,
remnant grain boundaries and impurities) and younger
structures (e.g. fractures and solution pits). The fracture
patterns shown in Fig. 12 could be divided into three groups,
original fabric (OF) related, younger fabric (YF) related or
independent of fabric (IF). Fracture patterns of an original
fabric nature are those that are dependent on the inherent
mineralogical and crystallographic structure, such as the
fracture through an impurity (Fig. 13(a)) and the fractures
along the cleavage of feldspar (Fig. 13(b)). Feldspar has two
cleavage planes that intersect at 90°, which probably resulted
in the almost perpendicular fracture planes in Fig. 13(b).
Younger fabric features arise during the geological history of
sand grains, such as pits and fractures that are generated by
the mechanical or chemical weathering process. The younger
weakness will greatly reduce the strength of the particles and
completely control the induced fractures (Figs 12(c), 12(f),
12(h) and 12(k)). These types of microstructural fractures are
SINGLE SAND PARTICLE FRACTURE USING X-RAY MICRO-TOMOGRAPHY
1000 µm
633
1000 µm
(a)
(b)
C
C
C
1000 µm
1000 µm
(c)
(d)
Fig. 10. Fracture patterns of LBS-3 and LBS-4: (a, b) top view of the fragments of LBS-3 and LBS-4; (c, d) fracture surface of LBS-3 and LBS-4
(C: conchoidal fractures)
1000 µm
(a)
1000 µm
1000 µm
(b)
(c)
D
D
D
M
M
CL
1000 µm
(e)
D
Feldspar
Quartz
1000 µm
(d)
1000 µm
1000 µm
(f)
(g)
1000 µm
(h)
Fig. 11. Two-dimensional grey-level slices of HDG particles perpendicular to the loading directions: (a–d) HDG-1 to HDG-4 at scan 1; (e–h)
HDG-1 to HDG-4 at scan 2 (CL: fractures following cleavage; D: fractures along an initial defect; M: fractures following mineral boundaries)
more frequently found in HDG that has been created by
weathering, leading to particles that are more complex in
mineralogy and often have internal microstructures (Lee &
Coop, 1995). Fractures that are independent of structural
weakness need relatively high stresses for them to be created.
For example, a conchoidal fracture is a sign of fabric
independent fracture (Fig. 13(a)). Conchoidal fractures are
generated in quartz because it has no cleavage. The new
fractures propagating in quartz that has not been influenced
by microstructural weakness are in this category and
predominate in LBS, which has an almost purely quartz
mineralogy from its sedimentary origin.
A bending fracture indicated by the double cantilever curl
that formed when the fracture propagated from the tensile
ZHAO, WANG, COOP, VIGGIANI AND JIANG
634
YF
B
IF
YF
1000 µm
1000 µm
(a)
1000 µm
(b)
(c)
YF
OF
YF
500 µm
500 µm
(d)
500 µm
(e)
(f)
IF
S
IF
YF
1000 µm
1000 µm
(g)
1000 µm
(h)
(i)
B
YF
OF
IF
IF
1000 µm
(j)
1000 µm
(k)
1000 µm
(l)
Fig. 12. Two-dimensional grey-level slices of HDG particles parallel to the loading directions: (a–c) HDG-1 at three scanning points; (d–f)
HDG-2 at three scanning points; (g–i) HDG-3 at three scanning points; ( j–l) HDG-4 at three scanning points (OF: original fabric related fracture;
YF: young fabric related fracture; IF: fracture independent of fabric; B: bending fracture; S: shear fracture)
side (lower side) of the particle into the compression side
(upper side) is marked in Fig. 12(b) (Quinn, 2007). This
could be caused by the multiple lower contact points.
Another example of a bending fracture is shown in Fig. 12(k).
It seems that the internal weakness formed a cantilever beam,
which resulted in this bending fracture. Fig. 12(h) shows a
shear fracture, which made an angle of about 43° from the
loading direction. The main cause for this shear fracture
could be the internal weakness located at the middle of the
particle. Many studies on the crushing of single sand particles
between two platens are based on the hypothesis of tensile
failure mode with fractures perpendicular to the loading
direction, which follows the model proposed by Jaeger (1967)
for the diametrical compression of spherical rocks (Lee,
1992; McDowell & Bolton, 1998; Nakata et al., 1999). This
hypothesis seems not valid for some of the tested particles,
especially for HDG particles with more complex external
morphologies and internal microstructures.
Morphology evolution
Next, results are presented from quantitative morphological analysis of the particles and their fragments from the 3D
labelled images. To present the evolution of morphology
SINGLE SAND PARTICLE FRACTURE USING X-RAY MICRO-TOMOGRAPHY
635
CL
92·1°
C
IM
500 µm
500 µm
(a)
(b)
Fig. 13. Two-dimensional grey-level slices of HDG particles: (a) HDG-1 at scan 2, parallel to the loading direction; (b) HDG-3 at scan 2,
perpendicular to the loading direction (C: conchoidal feature; IM: impurity; CL: fractures following cleavage)
during the fracture process, the scan results of the four
particles of LBS and HDG are combined and denoted using
uniform symbols. For example, ‘LBS-S2’ denotes all the
fragments from scan 2 of the four LBS particles. Only
fragments larger than 103 mm3 (about 17 000 voxels for the
largest voxel size in our test) are analysed because of the
difficulties in segmenting small fragments and accurately
evaluating their shape parameters. Note that each of the
smallest fragments counts less than 0·05% of the total volume
of the particle.
Figure 14 shows the evolution of the shape parameters for
the LBS particles. Sphericity, convexity and aspect ratios tend
to decrease during the particle fracture process, except for the
slight increase of the flatness index (FI) from LBS-S2 to
LBS-S3. This means that the newly generated fragments are
less spherical, less convex and with lower aspect ratio than the
original particles. Similar results have been reported by
Altuhafi & Coop (2011) from one-dimensional compression
on uniformly graded samples in which they used a 2D laser
image analysis instrument to measure particle morphology
after the tests. In spite of severe fragmentation occurring
between scan 3 and scan 2, the distributions of shape
parameters do not change much. The distributions of the
elongation index (EI) and the flatness index (FI) for LBS-S3
are very similar, which is not surprising, given the similar
physical meaning of these two parameters. It is interesting
that the distributions of aspect ratios after crushing are in
agreement with the results from Takei et al. (2001), although
they tested more angular particles. The evolution of shape
parameters for the HDG particles is plotted in Fig. 15. Each
line in this figure includes the morphology information of all
four HDG particles. Although the HDG particles have
smaller initial sphericity values than LBS particles, most
fragments from both materials have a sphericity value
between 0·25 and 0·65. However, the distributions of all the
shape parameters of HDG particles display less difference
between scan 3 and scan 2 than those of LBS particles,
suggesting less variance of the fracture mechanisms of HDG
particles from scan 2 to scan 3 owing to their more highly
heterogeneous microstructures.
Figure 16 shows the evolution of median values of the
shape parameters for the LBS and HDG particles. The
median values of sphericity and convexity for both LBS and
HDG particles decrease during the fracture process.
Although the HDG particles have much smaller initial
sphericity and convexity values than the LBS particles, they
converge to the similar sphericity value of 0·51 and convexity
value of 0·55. This indicates that, at the severe fragmentation
level (i.e. scan 3), the median values of sphericity and
convexity of the fragments are largely independent of the
initial particle morphology, which is mainly resulted from the
similar fracture mechanisms at this stage of fragmentation. In
terms of the aspect ratios, the LBS and HDG particles have
almost the same initial values of EI and FI, but the final
LBS-S1 S
100
100
LBS-S1 EI
LBS-S2 S
LBS-S2 EI
LBS-S3 S
LBS-S1 CX
80
LBS-S3 EI
LBS-S1 FI
80
LBS-S2 FI
LBS-S3 CX
60
% Smaller
% Smaller
LBS-S2 CX
40
20
LBS-S3 FI
60
40
20
0
0
0
0·2
0·4
0·6
0·8
Sphericity and convexity
(a)
1·0
0
0·2
0·4
0·6
Aspect ratio
0·8
1·0
(b)
Fig. 14. Distributions of shape parameters for LBS particles at different scan stages: (a) sphericity and convexity; (b) aspect ratio. (S: sphericity;
CX: convexity; EI: elongation index; FI: flatness index)
ZHAO, WANG, COOP, VIGGIANI AND JIANG
636
HDG-S1 S
100
100
HDG-S1 EI
HDG-S2 S
HDG-S2 EI
HDG-S3 S
HDG-S1 CX
80
HDG-S3 EI
HDG-S1 FI
80
HDG-S2 FI
HDG-S3 CX
60
% Smaller
% Smaller
HDG-S2 CX
40
40
20
20
0
0
0
0·2
0·4
0·6
0·8
Sphericity and convexity
(a)
HDG-S3 FI
60
1·0
0
0·2
0·4
0·6
Aspect ratio
(b)
0·8
1·0
Fig. 15. Distributions of shape parameters for HDG particles at different scan stages: (a) sphericity and convexity; (b) aspect ratio (S: sphericity;
CX: convexity; EI: elongation index; FI: flatness index)
1·0
1·0
LBS S50
LBS EI50
HDG S50
HDG EI50
0·9
LBS CX50
HDG CX50
Median values
Median values
0·9
0·8
0·7
0·6
LBS FI50
HDG FI50
0·8
0·7
0·6
0·5
0·5
1
2
Scan no.
(a)
3
1
2
Scan no.
(b)
3
Fig. 16. Variation of median values of shape parameters during fracture: (a) sphericity and convexity; (b) aspect ratios
median values of the LBS particles are smaller than those of
the HDG particles. According to the classification proposed
by Zingg (1935), the median aspect ratios of the LBS
fragments are ‘blade’, and those of the HDG fragments are
‘spheroid’. Again, this result may be attributed to the more
heterogeneous microstructure of the HDG particles.
By comparing the shape parameters of all the components
from three scans, a strong correlation between sphericity and
convexity was found in Fig. 17(a), which is in agreement with
the results from Fonseca et al. (2012). To explore further the
relationship between S and CX, S 1·5 is calculated as the ratio
between the volume of the component and the volume of the
sphere having the same surface area as the component. It was
found in Fig. 17(b) that the whole set of data and the
correlation line shift upward above the equality line,
indicating that the volume of the sphere having the same
surface area as the component is larger than the volume of its
convex hull. When this was not the case it was found to be
caused by unsuccessfully separated fragments.
Fractal fragmentation
It has been found by many researchers that fragmentation
is a scale-invariant process (Turcotte, 1986). The fractal size
distributions arising from the fragmentation process have
been taken as evidence of such scale invariance. A natural
material may be fragmented in a variety of ways. Weathering,
explosions, impact and geological loading (e.g. basal tills
sheared by glaciers) can all generate fragments that satisfy a
fractal condition (Turcotte, 1986; Altuhafi & Baudet, 2011).
Using the 3D morphology of the fragments from the
single-particle compression tests, the present authors now
examine if the fragmentation satisfies a fractal condition.
A fractal can be defined by the relationship between
number and size. There are a variety of ways to represent the
size–frequency distribution of fragments. Many experimental
results indicate that the size–frequency distribution of
fragments is given by N ~ r D, where N is the cumulative
number of fragments with a characteristic dimension larger
than r, and D is the fractal dimension corresponding to the
negative slope in a log N(.r) plotted against log r plot
(Turcotte, 1986).
Figure 18 shows two forms of fractal distribution of
fragment size for LBS and HDG particles in scan 2 and
scan 3. The particle size was described by both characteristic
radius and volume. The characteristic radius is defined as the
radius of a sphere that has the same volume as the fragment.
It appears that the fragmentation at an extensive crushing
state (scan 3) satisfies a fractal condition with a fractal
dimension (D) of about 2·0 for both LBS and HDG particles.
SINGLE SAND PARTICLE FRACTURE USING X-RAY MICRO-TOMOGRAPHY
1·0
1·0
y = 1·1054x –0·0102
637
y = 1·0166x + 0·1765
0·8
0·8
0·6
0·6
CX
CX
Equality
line
0·4
0·4
0·2
0·2
Equality
line
0
0
0
0·2
0·4
0·6
0·8
1·0
0
0·2
0·4
S
(a)
0·6
0·8
1·0
S1·5
(b)
Fig. 17. Correlation between sphericity and convexity
V: mm3
0·01
–D = –1·98
100
V: mm3
0·1
0·001
9
0·1
3 – D= 1·02
End of fractal
N(> r)
0·01
End of fractal
LBS-S2
LBS-S3
HDG-S2
HDG-S3
10
–D = –2·04
3log (r) + log N(> r)
0·001
1000
8
3 – D = 0·96
LBS-S2
LBS-S3
HDG-S2
HDG-S3
1
0·06
0·60
7
0·06
0·60
r: mm
r: mm
(a)
(b)
Fig. 18. The fractal distribution of fragment size: (a) log N(> r) plotted against log r; (b) 3log r + log N(> r) plotted against log r
Some large fragments that had not experienced significant
fragmentation defined the upper end of the fractal region
(Fig. 18). The lower end of the fractal range was determined
by the minimum detectable volume of 103 mm3. Owing to
the possible systematic errors in the determination of fractal
dimension using the log (N ) against log (r) plot (Thompson,
1991; Sufian and Russell, 2013), a second form of plotting
using 3log (r) þ log (N ) against log (r) is given in Fig. 18(b).
It is found that both plots could clearly identify the upper end
of the fractal region for scan 3. For scan 2, the fractal
distribution is less obvious and well defined because of the
limited degree of fragmentation. However, no systematic
error is found using the form of fractal plotting in Fig. 18(a).
It should be mentioned that, since the volume and number of
fragments were directly measured from CT data in the
present study, there is no need to adopt the grain size
cumulative distribution by mass such as Md( d )=MT ¼
D
D
D
(d 3D d3min
)=(d3max
d3min
) suggested by Einav
(2007), which can be used to back-calculate the number of
fragments by assuming all particles are spherical (e.g. Zhang
& Baudet, 2013).
The fractal distribution in Fig. 18 is based on the radii of
equivalent spheres. This is not intended to suggest that the
particle fragments are spherical, because the values of
sphericity of the various sizes in Fig. 19 clearly show that
they are not. Instead this is just a ‘characteristic’ dimension.
This approach has been used since it is not possible to
reconstruct the true shape and dimension of the particles
from a shape parameter such as sphericity. From Fig. 19 it is,
however, clear that the mean sphericity is constant for
fragment sizes below about 0·06 mm and so, whatever
shape were assumed, the characteristic dimension would
simply be proportional to the radius that has been used, so
that the fractal dimension would not change. It is possible
that the change of shape that is evident in Fig. 19 at 0·06 mm3
may contribute to the end of the fractal domain at the same
size in Fig. 18(a).
Breakage energy
The 3D geometrical information of fragments is particularly useful when coupled with load–deformation response in
measuring bulk material behaviour, for example, breakage
energy. There are two basic steps to measure breakage energy
from the recorded data (Landis et al., 2003). First, the work
done by the load is calculated from load–displacement
curves. The energy input is calculated as the area of the
load–displacement curve during each loading period.
Second, the change of surface area resulted from crackforming processes is calculated based on the actual surface
area of particles and their fragments calculated directly from
the CT images. The breakage energy at any instant during the
ZHAO, WANG, COOP, VIGGIANI AND JIANG
638
1·0
0·8
S
0·6
0·4
0·2
LBS-S2
LBS-S3
0
0·001
0·01
1
0·1
Volume: mm3
Fig. 19. Correlation between the averaged sphericity values (S) and
particle size for LBS fragments at scan 2 and scan 3 in log-scale
volume intervals (e.g. (10−3, 10−2·5) mm3, (10−2·5, 10−2) mm3)
loading process can then be calculated as the ratio of the
incremental energy input to the incremental surface area
change. It should be mentioned that the energy so calculated
includes not only the energy dissipated by the creation of new
surfaces during the fracture process, but also energy
dissipated through other possible mechanisms like frictional
dissipation between the loading platens and the particle,
frictional dissipation between newly created fragments,
acoustic energy. These are possible reasons why the breakage
energy measured in the present study was much larger than
the surface energy that Griffith used to account purely for
new surface generation (Griffith, 1921). Such a definition of
breakage energy is necessary because of the difficulty in
distinguishing the surface energy from other forms of energy
dissipation in the present experiment.
Figure 20 shows the relationship between the cumulative
net work done by the load and the crack surface area
measured for the LBS and HDG particles. It should be
mentioned that the calculation of the cumulative net work
from the load–displacement curve has excluded the amount
of work consumed by the loading system, which was
40
Net work of load: N mm
LBS-1
LBS-2
LBS-3
LBS-4
HDG-1
30
HDG-2
HDG-3
20
HDG-4
Concrete (from
Landis et al., 2003)
10
0
0
50
100
150
200
250
300
Cumulative change in fracture surface area: mm2
Fig. 20. Relationship between net work of load and cumulative change
in surface area
calculated using an elastic stiffness of 8·07 N=μm for the
apparatus measured in a compliance test using a brass
cylindrical specimen. This stiffness value is much higher than
the value of 1·7 N=μm reported by Cavarretta (2009) because
the deformation of the load cell is excluded from the LVDT
measurement in the current apparatus during the loading
process (Fig. 1(a)). However, in an unloading event, the
relaxation of the load cell and the other components of the
apparatus will result in a very low unloading stiffness and
the possible transfer of the energy from the loading system to
the sample, which is responsible for the dynamic stress drop
after the peak load seen in Fig. 8. Given this defect, the
integral under the curves in Fig. 8 was taken as the upper
bound of input energy. The lower bound of input energy was
calculated excluding all the unloading work (i.e. areas under
all unloading curves in Fig. 8). It is found that the upper
bound of input work is usually twice the lower bound value.
Because of the brittle failures observed in these tests, the
present authors adopted the lower bound input work for the
calculation of breakage energy, assuming a vertical unloading behaviour. In Fig. 20, results obtained by Landis et al.
(2003) on concrete specimens in uniaxial compression are
also plotted for comparison. Note that the concrete specimens tested by Landis et al. (2003) were cylinders with
diameter and height both equal to 4 mm. The net work done
and fracture surface area vary considerably from particle to
particle. The breakage energy is calculated as the slope of
each curve. The breakage energy of all the tested particles
between scan 1 and scan 2 varies from 43·06 N=m to
154·49 N=m, while that between scan 2 and scan 3 is much
higher, from 207·64 N=m to 484·76 N=m. The average values
increase from 84·99 N=m to 321·60 N=m, which means that
more energy was dissipated in generating new fractures
during the second loading period. In comparison, the
concrete specimens of Landis et al. (2003) show a larger
change of the breakage energy from 68 N=m to 460 N=m.
This indicates that the concrete, as a porous composite
material made of aggregates and cement, is initially more
fragile than natural sand particles, but becomes equally
energy-dissipative at a large strain stage where severe
fragmentation has occurred and energy dissipation mechanisms become more versatile.
The measured breakage energy is much larger than the
surface energy, which is usually between 0·28 and 11·5 N=m
for quartz and sandstone (Parks, 1984; Sufian & Russell,
2013). There are two main reasons for this large breakage
energy. First, it is because the breakage energy measured
includes energy dissipated through a variety of ways, as
mentioned above. Second, the surface area could be greatly
underestimated owing to a significant number of undetected
micro-fragments and micro-cracks (Hoagland et al., 1973).
To illustrate the influence of the undetected small fragments
in the surface area calculation, the total surface area is
estimated from a fractal distribution using the same method
as was used by Chester et al. (2005) and Russell (2014). Based
on the observed fractal size distributions, a fractal dimension
of 2·0 was assumed here. The number of fragments in each
size interval was determined from the fractal distributions for
LBS in scan 2 and scan 3. as shown in Fig. 18(a). The
sphericity was used as the shape parameter to correlate
volume and surface area. From Fig. 19 this was taken as 0·5,
since this value appears to be constant below about 0·06 mm3
and so it seems reasonable to assume the same value for the
undetected fragments. Then the total surface area of LBS
fragments was calculated by adding up all the fragments in
each size interval. It should be noted that for an assumed
fractal dimension of 2·0, the total surface area of the
fragments with different characteristic sizes remains constant. Therefore, if the minimum fragment size continues to
SINGLE SAND PARTICLE FRACTURE USING X-RAY MICRO-TOMOGRAPHY
be reduced, the estimated total surface area keeps increasing
the same amount for each characteristic size. If a minimum
fragment size of 1012 mm3 was assumed, the resulting total
surface area of LBS fragments at scan 2 and scan 3 increased
79·9% and 150·1% from that measured from X-ray CT
images, respectively. Accordingly, the average breakage
energy of LBS would reduce from 85·0 N=m to 36·9 N=m
between scan 1 and scan 2 and from 321·6 N=m to 90·6 N=m
between scan 2 and scan 3, respectively.
CONCLUSIONS
This paper has investigated the single-particle fracture
behaviour of two different types of natural sand particles
using a nanofocus X-ray CT. By virtue of a novel miniloading apparatus and image-processing and analysis techniques, a detailed study has been presented on the fracture
patterns, evolution of shape parameters, fractal conditions
and breakage energy of a single-particle fracture event. The
in-situ testing technique was particularly used in the current
study to monitor closely the crushing process of sand
particles and the evolving micro-morphologies of particle
fragments. This has the obvious advantage over the postmortem observation approach, which can only provide the
information at the initial and the end conditions, and cannot
allow the type of analysis made in this paper. The principal
conclusions of this work are given below.
The initial particle morphology, heterogeneity and mineralogy are important factors that influence the fracture
patterns. For the LBS particles, the fracture planes were
mainly parallel to the loading direction. Local stress
concentrations can occur at contact points with small radii
of curvature, leading to extensive fragmentation. The initial
microstructures made the particles easier to break, especially
in the HDG particles. These were identified as internal voids,
impurities, phase boundaries and weathering products. The
fracture inside the feldspar shows a cleavage pattern, but
quartz has no cleavage, so giving a conchoidal fracture when
no imperfection intervenes.
The more complex external morphologies and internal
microstructures led to a much richer array of fracture patterns
in HDG particles. Initial structural weaknesses including
internal voids, impurities, phase boundaries and weathering
products resulted in the generation of different forms of
tensile, shear and bending cracks through cleavage planes,
mineral boundaries, voids or impurities. Obviously, these
versatile fracture behaviours do not conform to the hypothesis of simple vertical splitting along the loading direction
and attest to the predominant influence of particle morphology and microstructure.
Although the LBS particles had much larger sphericity
and convexity values than the HDG particles, fracture
caused their values to decrease and converge to the same
level. In contrast, while the initial aspect ratios were similar,
the final values were lower for LBS particles. The stronger
LBS particles tended to produce lower aspect ratio fragments
with lower angularity, but resulted in the same level of
sphericity and convexity as the HDG fragments.
The fragments of both LBS and HDG particles were found
to have a fractal distribution with the fractal dimension equal
to about 2·0.
For both LBS and HDG particles, increasing breakage
energy indicates secondary toughening mechanisms, such as
frictional dissipation, gradually prevail over the energy
dissipation due to crack formation. The results were found
to agree well with the experimental results on a very small
concrete specimen reported in the literature.
The merits of the novel insights into the single-particle
fracture acquired from this study lie in the statistical
639
information of fragmentation morphology and its relationship with the fracture strength and energy during the fracture
process. This information advances our understanding of
the grain-scale fracture mechanics of natural sand particles.
Although the results presented are based on only eight
sand particles, many observations and insights have broad
implications on the fracture behaviour of quasi-brittle
geomaterials over a range of initial morphology, intrinsic
microstructure and length scale, and may also shed light on
the investigation of the fracture mechanics of other synthetic
quasi-brittle materials such as concrete and ceramics. These
results serve as the basis for further exploration of the
quantitative relationship between the fracture strength of
sand particles and their initial and evolving morphologies,
microstructures and length scales during the crushing process
in the next phase of the authors’ research. Last, it should be
mentioned that the major findings of this study also bear
significance for numerical modelling workers, especially
DEM modellers, in that the experimental observations of
single-particle fracture, particularly the statistical information of the evolution of fragmentation morphology,
should be considered and incorporated into the DEM
model for more sophisticated modelling of sand crushing
and the development of physically more robust and rigorous
constitutive models for sands in future.
ACKNOWLEDGEMENTS
The study presented in this paper was supported by
General Research Fund No. CityU 120512 from the
Research Grant Council of the Hong Kong SAR and
Research Grant No. 51379180 from the National Science
Foundation of China. The authors also appreciate Dr
Beatrice Baudet’s discussion about the fractal plotting of
particle fracture.
NOTATION
a
b
CX
c
D
dmax
dmin
Md
MT
N
Ra
S
V
VCH
major dimension of component
intermediate dimension of component
convexity ratio between the volume of the component and its
convex hull
minor dimension of component
fractal dimension corresponding to the negative slope in a log
N(. r) against log r plot
largest particle size
smallest particle size
mass of all the particles that are smaller than d
total mass of sample
cumulative number of fragments with a characteristic
dimension larger than r
parameter of surface roughness
sphericity
volume of component
volume of the convex hull enclosing component
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