Revisiting localized deformation in sand with complex systems

Transcription

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Revisiting localized
deformation in sand with
complex systems
rspa.royalsocietypublishing.org
Antoinette Tordesillas1 , David M. Walker1 ,
Edward Andò2 and Gioacchino Viggiani2
1 Department of Mathematics and Statistics, University of
Research
Cite this article: Tordesillas A, Walker DM,
Andò E, Viggiani G. 2013 Revisiting localized
deformation in sand with complex systems.
Proc R Soc A 469: 20120606.
http://dx.doi.org/10.1098/rspa.2012.0606
Received: 15 October 2012
Accepted: 3 January 2013
Subject Areas:
geophysics, materials science, complexity
Keywords:
complex networks, granular materials, X-ray
microtomography
Author for correspondence:
Antoinette Tordesillas
e-mail: [email protected]
Melbourne, Victoria 3010, Australia
2 Grenoble-INP/UJF-Grenoble 1/CNRS UMR 5521, Laboratoire 3SR,
Grenoble 38041, France
Complex systems techniques are used to analyse
X-ray micro-CT measurements of grain kinematics
in Hostun sand under triaxial compression. Network
nodes with the least mean shortest path length to all
other nodes, or highest relative closeness centrality,
reside in the region where the persistent shear band
ultimately develops. This trend, whereby a group
of grains distinguishes themselves from the rest in
the sample, remarkably manifests from the onset of
loading. The shear band’s boundaries and thickness,
evident from the network communities’ borders and
essentially constant mean size, provide corroborating
evidence of early detection of strain localization.
Our findings raise the possibility that the formation
and the location of the persistent shear band may
be decided in the nascent stages of loading, well
before peak shear stress. Grain-scale digital image
correlation strain measurements and statistical tests
confirm the results are robust. Moreover, the trends
are unambiguously reproduced in a discrete element
simulation of plane strain compression.
1. Introduction
Electronic supplementary material is available
at http://dx.doi.org/10.1098/rspa.2012.0606 or
via http://rspa.royalsocietypublishing.org.
The development of a robust theory for granular
materials (e.g. soil, powders, grains, etc.) is arguably
one of the most formidable challenges in mechanics.
Terzaghi [1] reflected on the long history of this problem
in reference to soil mechanics: ‘(Coulomb) purposely
ignored the fact that sand consists of individual grains.
Coulomb’s idea proved very useful as a working
hypothesis, but it developed into an obstacle against
further progress as soon as its hypothetical character
c 2013 The Author(s) Published by the Royal Society. All rights reserved.
Our strategy for material characterization is to analyse material behaviour in two distinct spaces:
the physical and the abstract ‘state space’ of graphs or complex networks. Analysis proceeds
in three phases. Phase 1 is undertaken in the physical space: we use X-ray micro-tomography
to establish the position and kinematics (displacements and rotations) of individual grains in
a deforming Hostun sand sample, over many stages of a triaxial compression test. Phase 2 is
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2. Strategy for material characterization
2
rspa.royalsocietypublishing.org Proc R Soc A 469: 20120606
came to be forgotten by Coulomb’s successors. [. . .] The way out of the difficulty lies in dropping
the old fundamental principles and starting again from the elementary fact that sand consists of
individual grains’. In some respects, Terzaghi’s words did not go unheeded. The consideration
of the behaviour of geomaterials from the most basic units, i.e. the grains, has in fact been the
raison d’être of micromechanics of granular materials. This discipline owes much of its success
to the discrete element method (DEM), introduced in the 1970s by Cundall & Strack [2]. DEM’s
dominance in micromechanical analysis remains unsurpassed, with all but a few datasets at the
particle scale arising from DEM simulations [3–9]. As aptly enunciated by Sibille & Froiio [10]:
‘This has led to the paradox of micromechanics of granular materials as a science based almost
entirely on “virtual evidence”’.
Recent developments from laboratory experiments, however, herald a new era in
micromechanics of granular materials. Measurements of contact forces, contacts and grain
kinematics in two-dimensional idealized assemblies of photoelastic discs have been achieved and
analysed [11–14]. The past few years have also witnessed advances in the measurement of basic
properties for natural geomaterials at ever-increasing spatial resolution: existing datasets include
information on kinematics at the grain-scale [15–20]. With the measurement of individual grain
properties now within experimental reach, researchers are facing an unprecedented opportunity
to integrate or complement these measurements with data from DEM simulations [12,21] to probe
the rheology of geomaterials, just as Terzaghi envisioned—from observations of behaviour of
individual grains.
Experimental access to information at the grain-scale allows us to answer existing questions
as well as to discover new mechanisms operating across the spatial scales, from the grain to the
bulk. Of course, this new capability throws down the gauntlet to mathematics and statistics to
mine these copious data (kinematics of tens of thousands of grains in different stages of a loading
test) for new and useful insights, so that granular behaviour can be better understood, predicted
and ultimately controlled. Our objective is to revisit the phenomenon of strain localization (shear
banding) using a complex systems analysis of kinematical data for Hostun sand from X-ray
micro-CT measurements and for an analogue from DEM simulations. We employ concepts and
techniques from complex networks inspired in part by progress in the study of structure and
function of brain networks from neuronal data, obtained using magnetic resonance imaging
(MRI) [22]. The central question we seek to answer in this study is: can measures of complex
networks, suitably constructed from individual grain kinematics, detect strain localization? If so,
do such network properties provide early detection of failure and offer new and useful insights
on material behaviour?
This paper is organized as follows. Our proposed three-phase strategy for material
characterization is presented in §2. In §3, we discuss phase 1 for the two systems under study: the
triaxial compression test on Hostun sand and a kinematical analogue from a DEM simulation of
plane strain compression of a polydisperse assembly of spherical particles. Section 4 presents the
method for constructing the functional complex networks from the kinematical data (the initial
part of phase 2). To extract as much physical insight from the network analysis and, to the extent
possible, render transparent the physical meaning of the network properties, we combine in §5:
the network analysis in the abstract state space forged around the question posed above (the
latter part of phase 2); and the mapping of network properties from state space to the physical
domain, along with the battery of statistical and randomization tests to establish whether or not
the findings are robust (phase 3). We conclude in §6.
In brief, we describe here the two datasets for phase 1. The first is from a triaxial compression
test on Hostun sand, undertaken with in situ X-ray micro-tomography scanning. The second,
referred to hereafter as ‘baseline’, comes from a discrete element simulation of a plane strain
compression test. This baseline system should not be considered as a model for the real Hostun
sample; however, it can be thought of as a kinematical analogue of Hostun in two dimensions,
since the samples’ response to loading exhibits broadly similar rheological features, including that
of a single persistent shear band. Salient aspects of each system’s rheology are summarized here
for completeness, with further details reserved in the electronic supplementary material. This
information on rheology serves as a reference point for phase 3, against which the consistency
and novelty of the findings from phase 2 can be objectively judged.
(a) Kinematics of Hostun sand under triaxial compression
The experimental dataset is from a triaxial compression test on a dense sample of dry Hostun
sand, comprising 54 889 grains. The key novelty of this test is that it is performed in an X-ray
tomograph. The sample, measuring 11 mm in diameter by 22 mm in height, is prepared by
pluviation into a latex membrane stretched in a mould. The cell pressure is kept constant, while
the so-called stress deviator (i.e. the difference between axial stress and cell pressure) is applied
by advancing a piston which compresses the sample axially, at a constant speed. Further details
concerning the material and the experimental technique are given in Andò et al. [15]. The response
is typical of dense sand as shown in the electronic supplementary material: there is a peak stress
deviator at around 5.5 per cent imposed axial strain, followed by a reduction in stress, tending
towards a ‘residual’ value at the end of the test. The volumetric response is slightly contractive
then dilative throughout, as expected. Note that these global measurements are the only ones
typically possible in a standard triaxial test. However, in this test, X-ray tomography scans were
carried out at 16 levels of imposed axial strain. This is evident in the drops of axial stress seen in
figure 1a which correspond to stress relaxations at constant imposed axial strains. For each scan,
a three-dimensional image of the sample is reconstructed. Coming from the 16 scans, kinematical
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3. Hostun sand and baseline system: current state of knowledge
3
undertaken in the abstract state space: we map the kinematical data from phase 1 to a state space
where the deforming sample is represented by an evolving complex network, before examining
the evolution of key properties of this network. In this study, the network is constructed solely
from individual grain kinematics (i.e. grain positions do not enter the analysis in state space).
Finally, in phase 3, we map the information gained from the state space into the physical domain
of the sample, using grain positions, to extract spatial–temporal information on material rheology.
This study focuses mainly on phases 2 and 3; phase 1 is reported elsewhere but is summarized
here for completeness. Several key elements of phases 2 and 3 warrant attention. Phase 2 has
been deliberately designed to enable an additional layer of exploration of the physical system
within a pure mathematical framework—without appeal to physical laws or need for the network
properties to bear an explicit physical meaning. This abstraction to a state space in phase 2 enables
us to: (i) fully exploit key developments in the related disciplines of graph theory and complex
networks and (ii) forge a method of network construction as well as judiciously select network
properties based on the questions motivating the study. It is not until phase 3, when information
from the state space is mapped back to the physical domain, that the interpretation of the physical
significance of the results from phase 2 is established. A critical element of phase 3 is the series of
tests devised using modern statistics to probe the veracity of the findings from phase 2.
In support of the above characterization of Hostun sand rheology, two more independent
studies are performed to establish comprehensively the robustness of the results. Specifically, we
test the reproducibility of trends uncovered for Hostun sand from: (i) calculations of continuum
strain at grain-scale resolution using digital image correlation (DIC) and (ii) implementation of
the proposed method of characterization for a kinematical analogue from DEM simulations.
(a)
04
5 02
08 09
4
15 16
slices oriented to contain
axis of sample and normal
to shear band
3
2
volumetric
strain, ev
1 scan 01
–8
–6
–4
–2
0
5
axial strain (%)
10
15
load applied from bottom
(b)
porosity (%)
£ 30% 01
02
04
06
08
09
10
11
12
13
14
15
16
≥ 60%
(c)
component of grain displacement in direction perpendicular to cut (cut = red arrow in (d))
0.52 mm
0.52 mm
0 mm
0 mm
01–02
vertical displacement (mm)
(d)
02–04
08–09
15–16
zoom into shear band
0
7–
8
D
50
displacement (mm)
0
0.4
0.4 mm
01–02
0°
02–04
08–09
15–16
zoom into shear band
rotation (°)
(e)
10
–1
2
D
50
rotation (°)
0
20°
≥ 20°
Figure 1. (a) Stress ratio and volumetric strain versus axial strain for the Hostun sand experiment (scan 09 coincides with
peak stress ratio); (b) porosity maps displaying a distinct band of localized dilation only after peak stress ratio (scans 10–16);
(c) component of grain displacement normal to cut versus grain position along cut (shear band boundaries in 15–16 are used
to identify cut direction); (d) vertical displacement; and (e) rotation fields from ID-Track over four strain intervals, with slices
oriented in order to contain both the axis of the sample and the normal to the shear band that eventually forms. Tracked grains
in the displacement (rotation) field are coloured by the norm of their displacement vector (value of rotation); the axis of rotation
is not shown. The sample is compressed from the bottom. (Online version in colour.)
data for 12 strain intervals are analysed in the present study: 01–02, 02–04, 04–05, 05–08, 08–09,
09–10, 10–11, 11–12, 12–13, 13–14, 14–15, 15–16. The peak stress deviator coincides with scan 09.
A recently developed technique (called ID-Track) is then employed in tracking the grains in
four dimensions, i.e. three-dimensional space and time [15]. Each grain is assigned a unique label
(an ID), and measurable grain features (e.g. volume) enable its identification from one scan to
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4
stress ratio, s1/s3
6
The baseline model is a discrete element simulation of plane strain biaxial compression under
constant confining pressure of an assembly of 5098 spherical particles [8]. A contact moment is
introduced to account for the effect of particle shape angularity, e.g. resistance to relative rotations
due to particle interlocking [23], thus providing a closer representation of the grain kinematics
encountered in Hostun sand. As stated earlier, the baseline system is not to be considered as
a model of the Hostun sample but simply a kinematical analogue. Like Hostun, the baseline
sample fails via strain localization with a single persistent shear band governing its rheological
behaviour in the post-peak stress or large strain regime. Details of this simulation such as the
contact models used, particle size, sample preparation and loading protocol are provided in the
electronic supplementary material. We have the usual discrete element output data for every
one of these particles over 299 simulation time-steps or strain states, spanning all regimes of
the loading history: strain-hardening to peak stress ratio at axial strain yy = 0.034, then strainsoftening to yy = 0.04 and, finally, a levelling off (modulo fluctuations) of the stress ratio to a
residual stress state.
..................................................
(b) Kinematics of baseline system under plane strain compression
5
the next. This process employs a backward scheme analysis of pairs of consecutive ‘previousand-current’ three-dimensional images: individual grains are identified in each image based on
information established from the previous image. The quality of the results of ID-Track depends
critically on the quality of the process of grain separation and labelling (the segmentation process).
Errors in segmentation do occur with some of these natural grain shapes, and may prevent the
tracking of those grains for the remainder of the test.
Figure 1c–e shows the displacements and rotations of all the tracked grains in a vertical slice
across four strain intervals, from the start to the end of loading history. At the first strain interval
of the test, 01–02, the macroscopic strain field is almost homogeneous. The very small deviation of
the strain field from a perfectly uniform (or affine) field with respect to the macroscopic (sample)
scale is due to a slight misalignment of the sample from the vertical. The vertical displacement in
the vertical slice (figure 1d) in 01–02 increases relatively smoothly from zero, imposed at the top
of the sample, to 0.1 mm at the bottom of the sample. This displacement field is ‘inclined’ relative
to the direction of loading (evident in the tilted gradation of the colour map) because the sample
axis is initially tilted by 1.4◦ . The key aspect to note is that the change in the displacement field
is gradual in the initial stages of loading (01–02, 02–04); as loading proceeds, particularly from
around the peak stress ratio, the displacement field portrays to an increasing extent a ‘shear band’
that is some seven to eight grains wide by the last interval 15–16. Using the benefit of hindsight,
a retrospective analysis of the displacement field from vertical and cross-sectional slices of the
sample, starting from the last interval 15–16 back to 01–02, reveals a contrasting trend: relative to
the shear band boundaries marked by the lines of distinct step change in gradient in 15–16, the
displacement field in 01–02 and 02–04 changes smoothly at a near constant rate in the direction
perpendicular to the shear band. This trend is evident for grains: along a cut normal to the shear
band (figure 1c), throughout the vertical slice (figure 1d), and in all cross-sectional slices spanning
the band from top to bottom of the sample (see the electronic supplementary material).
Up until the peak, the relatively noisy measurement of rotations appears to be randomly
distributed in space. During the intervals 08–09 and 09–10, straddling the peak stress ratio, we
observe the beginning of a localization in the rotations: see 08–09 in figure 1e (cf. the porosity maps
in figure 1b). In the stages after the peak, e.g. 15–16 in figure 1e, when the band is fully developed,
the rotations are manifestly concentrated in the shear band. The rotation and displacement fields
present the shear band in the same location, although the band appears to be wider in the rotation
field, i.e. 10–12 grains thick (figure 1e far right). Since the technique provides the axis of rotation
for each grain, the component of rotation in the plane of the cross-sectional slices can and is also
computed. From this measure, it appears that the grains inside the band do not show any more or
less spatial concentration of rotation. Thus, while grain rotations clearly concentrate in the band,
there is no evidence of bias in favour of rotations in the plane of the shear band.
We now embark on phase 2. Here our task is to construct complex networks representing the
deforming Hostun sample in state space, before examining these using appropriate network
properties. As stated earlier, the method of network construction and the network properties
depend crucially on the questions we are seeking to answer in this study. These questions are
forged around patterns of interaction between constituent grains which manifest through the
grain kinematics: i.e. collective motion of particle groups which are indicative of the evolution
of instabilities and their spatial length scales. Our overall approach is depicted in figure 2.
This approach is based on similar concepts adopted in studies of functional brain networks
which are constructed from neuronal data obtained using magnetic resonance imaging [22]. The
construction of a functional network from such data rests on a statistical analysis of the degree of
similarity of pairs of time series.
In the case of Hostun sand and the baseline sample, the temporal (strain) evolution of the
grain kinematics of each system is tracked throughout loading history. As previously discussed,
the displacement and total rotation of individual constituent grains in Hostun sand are measured
at a non-uniform sequence of 13 strain states (or twelve strain intervals). The number of sampled
strain states is too few to form a long enough time series for reliable analysis; the same can be
said of the baseline system, even though this has many more strain states than Hostun. Hence
our strategy here is to construct a complex network at each strain interval, and then analyse not
only the properties of each network but also their evolution throughout loading.
Over each strain interval, the position, displacement and rotation of each grain in the physical
space are given by ID-track. With this information, analysis in phase 2 proceeds in three steps, as
depicted in figure 2. First, we map the measured kinematical data for each grain to a state space:
this is a four-(three-) dimensional space comprising scalar components of the displacement vector
and one total rotation for the Hostun (baseline) system. Each point in the state space corresponds
to a node in the network or a grain in the physical space. Second, we assign a link that connects
two nodes in the network if the kinematics of the corresponding grains match each other: i.e. nearcoincident points in the state space with respect to Euclidean distance. This second step is crucial,
as the connections in the network determine the properties of the network. Here, we connect each
node to its k-nearest neighbours in state space, following similar concepts proposed in time-seriesbased networks [26], where k cannot be arbitrary. The arbitrary selection of too low a k-value
may result in a disconnected network with many components (subgraphs), thereby reducing
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4. Development of functional complex networks
6
Prior to the onset of force chain buckling, just before the peak stress ratio, the baseline model
deforms in an essentially affine or uniform manner—from the length scale of a small particle
cluster (i.e. a particle and its first ring of neighbours) to the macroscopic scale. This is true if we
ignore rattlers and microbands which contribute a small amount of global non-affine deformation
and dissipation in the strain hardening regime. Apart from these spatio-temporally non-persistent
structures, no localization of any material property is evident in the baseline sample, from either
the contact or the kinematical data. The essential inelastic structural rearrangements, persistent in
space and time, develop only after the first force chain buckling event. These progressively spread
but remain concentrated in the band. Once the shear band is fully developed, the sample exhibits
a pseudo steady-state behaviour. The temporal evolution of the global properties of the baseline
system fluctuates about a near constant non-zero value, with contact network properties (e.g.
average values of degree, clustering and subgraph centrality) inversely tracking the measures of
particle kinematics (e.g. average non-affine deformation) [24,25]. A burst in kinematical activity
in the shear band (unjamming) is followed by a relatively quiescient period (jamming). As the
system unjams from the collective buckling of force chains, the global kinetic energy, dissipation
and average non-affine deformation all rise to a sharp peak, while particle connectivity inside the
sample degrades. As the system jams in the ensuing interval, there is a brief and partial recovery
in the system connectivity resulting in the growth of new force chains, until the next cycle of
unjamming–jamming commences.
current configuration
a
reference configuration
a
g
g
tra c
d
k in g giv e s k in e m a
grain contacts
links decided on contacts
(ongoing study)
tic s
d
grain kinematics: functional networks
links decided on similar kinematics
(this study)
y-disp
b
d
d
b
a
g
a
g
z-disp
k-network
connect each node to k nearest
neighbours; choose smallest k to
get a single connected graph
community structure
identify communities (groups of nodes),
maximizing intraconnections within,
while minimizing interconnection
between communities
d
b
g
x-disp
d
b
g
a
closeness centrality
identify nodes with highest relative
closeness centrality: those transmitting
information to all other nodes through least
mean shorest paths
a
d
b
g
a
highest closeness
centrality
map back to physical domain of sample
Figure 2. An illustration of our three-phase strategy for characterization of Hostun sand. Analysis starts from the physical
domain of sample (phase 1). Individual grain kinematics are projected into kinematical state space (three dimensions for
grain displacement, four dimensions for grain displacement and rotation), and analysed as complex networks (phase 2).
A graph, k-network, is constructed: grains are represented by nodes, connected by links if corresponding grains exhibit similar
kinematics, i.e. nearest k-neighbours in state space with respect to Euclidean distance. The k-network is characterized using
various metrics, here centred around shortest paths. Information is then mapped back into the physical domain of the sample,
trends are interpreted, and findings tested for robustness (phase 3). (Online version in colour.)
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b
b
7
(a)
(b)
the effectiveness of global scale network information extracted by examining the shortest path
lengths. In this case, a pair of nodes from two distinct components cannot be connected through
a finite number of links, i.e. the length of the path connecting these nodes is infinite. On the
other hand, the arbitrary selection of too high a k-value may result in a very densely connected
network in which every node is linked to almost all the other nodes in the network. In this case,
results based on shortest path lengths are invariably rendered meaningless, and all other network
measures are smeared to the extent that only outliers in the data are trivially detected. Our choice
of k is not arbitrary: we solve an optimization problem to find the minimum k value (or minimum number of
links for each node) that is needed to form a single connected graph. This method of selecting a suitable
k-value strikes a balance between the two extremes of low and high k and ensures that a finite
number of non-repeating links connects any two nodes in the network [27].
In our method of connecting nodes, a different Euclidean distance threshold in the
displacement–rotation state-space applies to each network node. An alternative method would
be to use the same fixed Euclidean distance threshold for each node and connect all nodes within
this distance. As we have shown in Walker et al. [27], this leads to a different optimization problem
if one wants the resulting network to be a single connected graph; also this network would reveal
different aspects of, in response to a different set of questions on, material behaviour. For the 12
complex networks representing Hostun sand behaviour over the 12 strain intervals from the onset
of loading to failure, the minimum k value ranges from 4 to 7; those for the baseline sample range
from 3 to 6. This method of network construction and the criterion for choosing k has been recently
tested and found to capture the essential global features of the displacement fields from DIC
measurements [27]. Before proceeding to the third and final step, it is worth noting that when we
consider the extended four-dimensional coordinate space comprising the three scalar components
of the displacement vector and one total rotation, each component is normalized by the standard
deviation across all measured values for that component. In this way, the ‘dynamic range’ of
each coordinate is the same and we achieve a fairer comparison of relative Euclidean distances
in the assessment of the similarity of grain kinematics in the second step. This strategy ensures
that network connectivity is not biased in favour of a dominant difference in one coordinate: for
instance, information contained in the differences of grain displacements may be lost if the total
distance in four-dimensional coordinate space is dominated by relative rotations and vice versa.
In the third and final step of phase 2, we examine the defining properties of the functional
k-networks to reveal organizational structure and associated kinematics within the deforming
material, including: the network communities, their length scales from shortest paths and the
closeness centrality index of individual grains. Bear in mind that a useful abstract network
property may not always lend itself to an obvious physical interpretation. This makes the task of
developing satisfactory so-called correspondence rules [28] all the more challenging. Our strategy
for material characterization has been deliberately designed to allow an additional exploration of
the system in a pure mathematical context, enabling us to fully exploit developments in graph
theory and complex networks. That said, to the extent possible, we will relate the physical
..................................................
Figure 3. Representations of the functional k-networks using the spring-electrical embedding method for: (a) Hostun sand at
strain intervals 01–02, 02–04, 08–09 and 15–16, and (b) baseline at the onset of loading and at large strains when the shear band
is fully developed. In each network, the nodes (grains) are coloured by their community structure membership, later discussed
in §5. The separation of the sample into two major components mediated by the shear band is evident in the last network shown
for each system. (Online version in colour.)
nodes in the shear band
8
We now complete phase 2, before mapping the information from state space to physical space
for phase 3. Our characterization of the functional k-networks from grain kinematics, as part
of phase 2, is centred around the concept of shortest path lengths [30]. Shortest paths play
a crucial role in network representations of many phenomena, since information often travels
along the shortest and most efficient available paths, notably in communication, transport and
organizational systems [31,32]. As there are clearly multiple paths connecting any pair of distinct
nodes i and j in the networks established for the Hostun sand and baseline sample (recall figure 3),
the shortest path dij is simply that path with the minimum path length or number of links. Hence,
the range of values for dij is 1 ≤ dij ≤ ∞, where shortest path length is infinite if nodes i and j lie in
two different disconnected components (or subgraphs) of the global network. In this section, we
will motivate and establish measures at multiple resolutions of the abstract networks comprising
the mesoscale and the macroscale, before mapping this information back into the physical space
in phase 3. Recall the central question we seek to answer in this study concerns the earliest stage
in the loading history at which these measures can detect the initiation of the shear band.
(a) Community structures and length scales
Granular materials exhibit collective dynamics from the interactions among their constituent
grains. These interactions are not confined to physical, pair-wise contacts. In the lead up to and
during failure, long-range interactions (i.e. collective behaviour) become dominant, leading to
instability patterns forming at multiple spatial scales (e.g. microbands, vortices, shear bands,
buckling force chains). At the onset of shear banding, just before peak stress ratio, non-contacting
as well as contacting particles in the sample begin to move collectively, exhibiting very similar
displacements and rotations. This collective motion progressively continues until the shear band
is fully developed, at which point the two parts of the sample on either side of the band can be
observed to move in an almost rigid-body motion. We are interested in uncovering the details of
such organization in grain interactions (i.e. spatial borders and length scales), especially of groups
of grains or mesoscopic regions in the sample following a common kinematical rule—not just
when the shear band manifests post-peak stress but at all stages of loading. This organizational
structure may be determined from the so-called ‘network communities’ [30].
The communities of a complex network are groups of nodes with dense intraconnections but
relatively sparse interconnections. A method for identifying communities is by optimization of a
modularity function [33] for a candidate partition of nodes expressed as
Q=
M
i=1
(eii − a2i ),
(5.1)
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5. Rheology from network characterization
9
system and the abstract network to render transparent the physical meaning of the network
construction and analysis so as to help future formalisms of meaningful correspondence rules.
Abstract representations of the k-networks, conceived using individual grain displacement and
rotation for the baseline system and Hostun sand system, are shown in figure 3. Connections are
all that matter in these graph representations. As such, the shape of these graphs are not normally
useful, since there are an infinite number of ways to draw and visualize a graph (collection of dots
and connections) on a plane. However, a visually appealing and often quite informative method is
to position the nodes using a spring-electrical embedding algorithm [29]. As is evident in figure 3,
we find that the network shape using this particular network representation seems to reflect the
formation of the persistent shear band and is suggestive of the splitting of the sample into two
parts separated by the band. In particular, the ‘neck’ seen in the visualizations of the networks
post-peak stress corresponds to grains physically located within the shear band of the sample.
(a)
(b)
..................................................
where eii is the percentage of the number of links with both ends (nodes) being in the community,
ai is the percentage of links that start from a community [29] and the number of distinct
communities (M) is indexed by i. To initialize the process, we start by treating each node as a
community. We then successively amalgamate groups in pairs, choosing at each step that pair
which leads to the highest increase in modularity. By maximizing this function, one obtains the
‘best’ partition of the network into communities.
Figures 3 and 4 present the communities of the k-networks. To understand their meaning,
recall that the links connecting pairs of nodes are decided according to how well the kinematics
of the associated pairs of grains match. Therefore, a community is a group of grains exhibiting
very similar motions in the physical sample—to the extent that a grain in a community moves in
a manner more ‘synchronized’ with that of another grain from within its community, compared
with that of a grain from outside its community. Because grains belonging to a community exhibit
essentially synchronized motion, a community can also be thought of as a group of grains in the
sample following a common kinematical or evolution rule. (Our use of the term synchronized is
purely in the dictionary sense and should not be confused with the process of synchronization
studied in dynamical systems research.)
The communities in state space in figure 3, when mapped back to the physical sample,
are shown in figure 4. The numbers of communities for the Hostun sand sample, comprising
54 889 grains or 329 334 degrees of freedom are: 32 and 36 (44 and 30) for the initial 01–02 and
final 15–16 strain intervals of the displacement network (displacement and rotation network),
respectively. The baseline sample comprises 5098 grains or 15 294 degrees of freedom: at the
initial state, the network constructed from the grain displacements has 32 communities while
that network constructed from grain displacements and rotations has 26 communities. At the
axial strain of yy = 0.0388, when the shear band is fully developed, we observed a rise to
49 and 48 communities in the baseline sample, respectively. The increase in communities at
yy = 0.0388 relative to the initial state is mainly in the shear band and reflect the significant
non-affine grain kinematics occurring in this region during an unjamming period (sharp peaks
in figure 5; electronic supplementary material). This makes sense as the non-affine measure, i.e.
the deviation of a particle’s motion to that dictated by the local uniform strain, quantifies the
level of heterogeneity in the grain kinematics. Also the most significant unjamming event in the
loading history of this sample occurs at the strain state yy = 0.0388, where the highest peak in
the global mean non-affine curvature and strain is attained. The significant spatial variation in
grain kinematics during the unjamming stages in the baseline sample seem less pronounced in the
three-dimensional Hostun sample with many more grains, judging from the slightly less numbers
of communities developing post peak stress: for the Hostun sample, compare 34–56 communities
pre-peak stress to 29–46 communities post-peak stress.
Figure 4. Community structures from figure 3 mapped back into physical space (i.e. sample domain) show that constituent
grains form mesoscopic regions exhibiting similar kinematics. Spatial distribution of communities of the k-networks constructed
using grain displacement and rotation for representative axial strain intervals of (a) Hostun sand for 01–02, 02–04, 08–09 and
15–16, here shown as slices in the xz plane; and (b) baseline system at the beginning of loading and at the large strain regime
when the shear band is fully developed. Grains with the same colour belong to the same community. (Online version in colour.)
10
Hostun: displacement + rotation
10
4
(b)
path length
20
10
6
8
axial strain stage
10
12
0
baseline: displacement
25
20
15
10
5
communities
mean
10
0
0.02
(c)
0.04
0.06
axial strain
0.08
0.10
0
baseline: displacement + rotation
14
12
10
8
6
4
2
communities
20
mean
10
0
0.02
0.04
0.06
axial strain
0.08
0.10
0
Figure 5. Distribution of community size or average shortest path length in each community. The mean of these across
communities remains essentially constant throughout loading for the k-networks for (a) Hostun sand using displacements
and rotations, and the baseline system using (b) displacements (c) displacements and rotations. The grey colouring shows the
‘density’ of communities with a given average shortest path length at each strain state (light to dark corresponds to low to
high density). (Online version in colour.)
We next explore whether network subdivisions by communities are revealing of the evolution
of instabilities in the lead up to and during bulk failure. Such communities, or groups of grains
exhibiting common kinematics, give rise to an intermediate or mesoscopic length scale of the
topological organization of the network based only on its connectivity. To extract this length scale,
we consider each of the communities and examine their respective subnetworks involving only
the intracommunity links. We then calculate the shortest path length between all pairs of member
nodes in each community to establish the community size at each strain state: this is defined to
be the mean of the distribution of shortest path lengths in a community. The distribution of
these community sizes is informative, in particular, of prevailing length scales at each strain state
as well as throughout loading history. As shown in figure 5, these distributions are relatively
narrow and ranges from one to around 21 links for Hostun (figure 5a), for which k-networks
typically embody links in the hundreds of thousands. Also, while the community sizes lHostun in the k-networks constructed using grain displacements (displacements and rotations) ranged
from around 6–13 (5–7) links, we find that the temporal or strain evolution of the mean value of
these community sizes for the whole sample remains essentially invariant throughout loading,
i.e. around lHostun = 10 (lHostun = 6) links. A similar trend applies to the community sizes of
baseline k-networks comprising tens of thousands of links. The temporal (strain) evolution of
the mean community size maintains an essentially constant value at around lbaseline = 6–7 links
throughout loading (figure 5b, c). In general, the k-networks which take into account rotation leads
to lower community sizes. This almost invariant mean community size, derived from a relatively
narrow range of shortest path length distributions in the k-networks, is identifying and extracting
a natural mesoscopic length scale of the grain kinematics.
..................................................
mean
communities
2
path length
11
20
path length
(a)
One of the big questions in the study of localized failure in granular materials is the
‘predictability’, not of the occurrence, but of the location of the shear band. From the standpoint
of material characterization, this question on predictability translates to: is the process of strain
localization encoded in the grain kinematics and, if so, can a consummate detector of this process
be devised from the kinematical data for Hostun without need to specify a spatial scale of observation,
and how early in the loading regime can this process be detected? Recall the experimental
observations of the Hostun sample summarized briefly in §2: no distinct shear band manifest
prior to peak stress ratio. To explore this issue in the abstract state space, we turn our attention
to the graph-theoretic distance of a given node to all other nodes in the entire network, and
quantitatively characterize how efficiently information travels to and from that node. In many
real-world network systems, information travels along the shortest, i.e. most efficient, available
paths. Thus, a metric of great practical interest is that which quantifies the efficiency of information
flow through the network. One such metric is the closeness centrality of a node i which is the
reciprocal of the sum of shortest path lengths to all other nodes j in the network, i.e.
C(i) = 1
j=i dij
,
(5.2)
where dij is the shortest path length between node i and node j. To deal with disconnected
networks, it is common to multiply each C(i) by the number of nodes in the connected component
to which node i belongs. However, we have dropped this pre-factor, since our method of network
construction (i.e. choose the minimum k so that a node’s connection to its k-nearest neighbours
in state space leads to a single connected component) ensures that all nodes in the network
are reachable by design: all k-networks here have one component. Furthermore, we present our
closeness centrality findings relative to the maximum C(i), i.e. we normalize so that the maximum
value is one and the pre-factor cancels. Mathematically, equation (5.2) shows that the closeness
centrality of a node is a measure of a node’s proximity to all the other nodes in the network.
The nodes with the highest relative closeness centrality are closest to all the other nodes in the
network: hence, speed of information flow from these to the rest of the network is fastest. In a
diffusion process, a node that has high closeness centrality is likely to receive information more
quickly than others. Also, those nodes with high closeness centrality typically lie in the ‘centre’ of
the network and on the boundaries of network communities, acting as nexus or bridging points
in a network.
To some extent, we can relate the network measure of closeness centrality in state space and
the physical motion of the grains in the Hostun sample. For the k-networks constructed here,
closeness centrality can be interpreted as a measure of how close or similar a grain’s motion is to
the motion of all the other grains in the sample. In other words, the grains corresponding to those
..................................................
(b) Closeness centrality
12
In general, community length scales are a compelling attribute to examine in complex systems
because they arise as a natural outcome as opposed to a prescribed input to the analysis. In the
specific cases studied here, the significance of the persistent length scales identified above for
the physical domain warrants further scrutiny. We investigated the relation between this length
scale and its counterpart in the contact network for the baseline wherein connections represent
physical contacts. Therein communities represent true clusters of contiguous grains, and a link
is arguably of the order of the mean grain diameter. We found the mean community size of the
contact network to be similarly invariant throughout loading and is around 10 links: this is not
only of the same order as the community sizes in the k-networks, but is also consistent with the
thickness of the shear band (i.e. eight times the mean grain diameter). As we do not currently
have information on contacts for the Hostun sample, the computation for the mean community
size for its contact network will be left for future work. That said, the 6–10 links—observed from
the kinematic networks from the very first strain interval 01–02—is certainly consistent with the
shear band thickness for Hostun which is around eight grain diameters.
(a)
(b)
..................................................
nodes with the highest closeness centrality values are those which move in a manner most similar
to or representative of the motions of all the other grains in the entire sample. Keep in mind that in
state space where the network analysis is undertaken, nodes with the highest closeness centrality
values correspond to those which can access all other nodes in the network through least number of
links. Now recall that in the Hostun test, the sample was compressed from the bottom: the X-ray
micro-CT data show that the grain displacements in the initial stages of loading increase from
an imposed zero value at the top, to some finite value at the bottom of the sample. Following
the test’s reference frame (done solely for convenience as, of course, this is ultimately irrelevant
due to frame invariance), such a displacement distribution would imply that the nodes with the
highest closeness centrality are in fact symmetrically located—all along the mid-height of the
sample. To visualize this, imagine a single column of grains such that the grain displacements
increase gradually from zero for the top grain through to some value for the bottom grain. In
state space, this would lead to a linear chain of nodes: those nodes in the middle of the chain will
have the highest closeness centrality. A sample deforming under a perfectly uniform or affine
strain at the macroscopic scale would therefore exhibit a band of highest closeness centrality
grains situated symmetrically and at mid-height of the sample. We show in figures 6a and 7a,b the
resulting spatial distributions of closeness centrality when mapped back into the physical domain
of the Hostun sand sample. The most striking aspect of these patterns, similar to those observed
earlier in the community boundaries in figure 4a, is the strong bias or asymmetry from as early
as the second strain interval 02–04. Specifically, the location of the nodes (grains) with the highest
relative closeness centrality—is situated in the location of the persistent shear band that has yet
to develop. The side-on views of the sample in figure 6a, selected to match the perspective of the
slices in figure 1d, highlight the developing deformation and identify the location of those grains
whose centres correspond to the nodes with the highest closeness centrality in the k-networks.
It can be seen that such grains localize in the region of the shear band—and that such localization
is evident at the early stages of loading. Furthermore, as the horizontal slices show in figure 7a, this
‘band of grains’ propagates across the width of the sample from the bottom to the top of the
confining membrane. Observe also that while the high relative closeness centrality grains appear
to capture strain localization extremely well and follows the band as it traverses the height of
the sample, the localization at the earlier strain interval (02–04) is more heavily biased towards
the lower part of the sample (figure 7a). Indeed, the pattern in the first strain interval (01–02) in
figure 7a already shows a greater concentration of high relative closeness centrality grains from
the lower left towards the upper right corner of the sample. Given that closeness centrality is
a measure of information propagation through a network, is it possible then that this metric
is identifying the potential sites of nucleation of failure? To answer this question for sand, we
Figure 6. (a) Centres of those sand grains corresponding to nodes of highest (upper 10% percentile) relative closeness centrality
within the displacement–rotation k-networks are mapped back into the physical domain of the Hostun sand sample and shown
as red dots for four strain intervals: from left are side-on views of 01–02, 02–04, 08–09 and 15–16 from the same perspective
as those in figure 1d. The remaining grain centres are shown as blue dots. (b) Spatial distribution of maximum shear stress
from grain-scale continuum strain measurements using DIC, corresponding to the four strain intervals in (a). Notice the hint
of a possible competing band for interval 08–09 in (b), just before the peak stress at 09. In contrast, the closeness centrality
identifies only the final persistent shear band. (Online version in colour.)
13
(a)
(d)
Figure 7. Cross-sectional (xy plane) views of the sample in figure 6 for the second and final strain intervals, i.e. (a,c) 02–04 and
(b,d) 15–16, respectively. (Online version in colour.)
must wait for further developments in experimental methodology whereby force propagation
or proxies thereof within three-dimensional deforming sand samples can be observed so that the
evolution of force chains can be tracked, and force chain failure by buckling detected.
As mentioned in §2, reproducibility of results on Hostun was tested not just for baseline but
through independent calculations of continuum strain taken at high resolution, i.e. grain-scale
(measurement points being around 0.4 mm apart), using DIC [15]. The spatial distributions of the
maximum shear strain corresponding to those in figure 6a for the four selected strain intervals
discussed earlier are shown in figure 6b. Strain localization is evident, albeit faint, from as early as
02–04. Interestingly, however, the continuum strain field for 08–09 portrays a region of localization
in the lower part of the sample that is more widely spread across the sample width, than the
shear band trending upwards to the top part of the confining membrane: this may possibly be a
competing shear band emanating from the lower but opposite side of the confining membrane
inclined perpendicular to the ‘winning’ band that ultimately develops after peak stress ratio. The
cross-sectional (xy-plane) views for 08–09 of the maximum shear strain field confirms the absence
of a localization to one distinct contiguous zone in these lower regions of the sample (see the
electronic supplementary material). These trends are unlike those evident in the corresponding
closeness centrality measurements (figure 6a), which consistently show that the region of the
sample with high closeness centrality is as concentrated as in the preceding strain interval.
Closeness centrality seems able to filter out secondary factors (e.g. competing but temporary
shear bands) and are manifestly unambiguous in its identification of the shear band that will
ultimately prevail in the final stages of loading. These findings provide a direction for future
experimental design and application of our network methods. Recent simulations [34] suggest
that for samples with a large spatial variation in density, the detection of early strain localization
with DIC measures can give misleading results (i.e. the ultimate shear band forms elsewhere). The
experimental Hostun sand sample considered here may be regarded as relatively homogeneous,
thus the future testing of the generality of our methods and conclusions to more heterogeneous
sand mixtures is warranted.
Having confirmed the presence of strain localization in 02–04 for Hostun, through an
independent grain-scale strain measure using DIC, we next turned our attention to the closeness
centrality of nodes for the baseline system. Figure 8 presents the spatial distributions of the
closeness centrality when mapped back into the physical domain of the baseline sample.
The location of the nodes (grains) with the highest relative closeness centrality—manifestly
concentrates in the region of the persistent shear band—well before the nucleation of the band, as
..................................................
(c)
(b)
14
Randomization tests are statistical tests to help determine if the observed experimental data are
inconsistent with a stated hypothesis. Typically, these tests begin with the generation of new
datasets which are both consistent with the observed data and the stated hypothesis that the
observations are the result of random chance. (We refer to such datasets as surrogates.) A statistic
with the capability to distinguish whether data are inconsistent with the stated hypothesis is
calculated for both the original observed dataset and its surrogates. If the calculated value of the
statistic for the observed data is sufficiently different (e.g. to a certain significance level) to the
distribution of statistical values calculated for the surrogate data, then one can conclude with a
given uncertainty that the observed data is inconsistent with the stated hypothesis. It is vital to
remember that failure to demonstrate inconsistency with the stated hypothesis does not imply
consistency. Randomization tests and surrogate data methods are only capable of rejecting the
null hypothesis, i.e. that the observed data did not occur by chance.
An example of the use of surrogate data methods can be found in nonlinear times-series
analysis. A suite of randomization tests have been developed in an effort to detect determinism in
observed experimental data as opposed to various linear hypotheses alternatives [35]. Surrogate
datasets in complex networks can be generated by randomly rewiring the original network while
preserving certain structural properties to produce a surrogate network: for example, rewiring
to preserve node degree. Specific network quantities (e.g. shortest path length distributions) can
be calculated for the original and rewired networks to test various hypotheses about network
structure [36]. In this study, we consider randomized surrogate networks which preserve node
degree (a local property). Hence, any measure based on shortest path lengths, intended to
summarize global connectivity, should be a good test statistic to discriminate the original and
surrogate networks. The procedure we followed for random rewiring of a network to preserve
node degree is presented in detail in the electronic supplementary material.
To complete phase 3 of our material characterization, we performed the node degree
preserving randomization procedure to generate 49 surrogate networks where 10 per cent of
the links have been rewired, for each strain state of the loading programme. It is possible that
the rewired surrogate networks become disconnected and so measures based directly on shortest
path lengths will become infinite valued since, by definition, the shortest path length between two
nodes in two different disconnected components of a network is infinity. This calls for a suitable
..................................................
(c) Randomization tests
15
detected from the onset of force chain buckling. As the distributions are similar for both classes
of k-networks, we only show here results from the class of networks constructed using grain
displacement and rotation (a movie of the evolution of these distributions for many more strain
states from this class of networks and those from grain displacements only is given as an electronic
supplementary material). Again, a strong asymmetry is apparent from the onset of loading. Recall
from §3b, we observed no such persistent localization in the region of the shear band prior to
peak shear stress through any of the previous measures established for the baseline system.
Furthermore, in contrast to the kinematical k-networks developed here, the contact networks of
the baseline system at these strain states (and, for that matter, all stages) of the loading history
exhibit a high degree of symmetry—a central core encapsulating a cluster of particles having high
relative closeness centrality situated in the middle of the sample. This core is essentially invariant
throughout loading history, even though there is a clear reorganization in the communities in the
contact network (not shown).
A question that we must now address to complete phase 3 is how ‘real’ are the above results?
Is it likely that the detection of the shear band so early in the loading history, through the
k-networks—in both the Hostun sand and baseline samples—occurred simply by chance? That is,
if we constructed surrogate networks by randomly connecting points in state space regardless of
matching kinematics, would network properties of these surrogate random networks still exhibit
localization when mapped back to the material domain? Randomization tests help to answer
such questions.
(a) 3
0
–1
–2
–3
–3 –2 –1
(c) 3
0
1
2
3
0
1
2
3
2
1
0
–1
–2
–3
–3 –2 –1
(e) 3
2
1
0
–1
–2
–3
–3 –2 –1
0
1
2
3
0
0
0
1
1
1
2
2
2
3
3
3
16
..................................................
1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2
1.0 (b) 3
0.9
2
0.8
0.7
1
0.6
0
0.5
0.4
–1
0.3
0.2 –2
0.1
–3
0
–3 –2 –1
1.0 (d) 3
0.9
2
0.8
0.7
1
0.6
0
0.5
0.4
–1
0.3
0.2 –2
0.1
–3
0
–3 –2 –1
1.0 ( f ) 3
0.9
2
0.8
0.7
1
0.6
0
0.5
0.4
–1
0.3
0.2 –2
0.1
–3
0
–3 –2 –1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Figure 8. Relative closeness centrality for the k-networks constructed using displacement and rotational information for six
representative axial strain states, mapped back into the physical domain of the baseline sample: (a) at start of loading, (b)
midway through strain-hardening regime, (c) onset of force chain buckling and shear banding, (d) from peak shear stress to
strain-softening regime, (e) transition from strain-softening to full development of the shear band, (f ) a brief period of recovery
in load-carrying capacity, evident in the rise in shear stress, after the shear band is fully developed. Note that the values of
closeness centrality have been normalized to lie in the range [0,1], i.e. the maximum value of closeness centrality is scaled to
be one and the minimum is scaled to be zero with respect to the values at the considered axial strain and not across all strain
states of the loading history. Increasing values of relative closeness centrality are depicted by particle colour ordered from cool
to hot, i.e. from lowest (0) in dark blue to highest (1) in bright red. See the electronic supplementary material for a movie of this
evolution. (Online version in colour.)
shortest path length measure which remains finite even for networks with multiple components.
One such measure is the harmonic mean given by the reciprocal of the efficiency, i.e. h = 1/E where,
E=
1
1
N(N − 1)
dij
(5.3)
i=j
and dij is the shortest path length between nodes i and j and N is the total number of nodes in the
network [31]. Clearly, if dij = ∞ for two nodes in two different disconnected parts of the random
network, then its contribution to the efficiency is zero. Using the harmonic mean, we compare
Hostun: displacement + rotation
harmonic mean
min. surr.
original
50
0
1
2
3
4
5
6
7
axial strain stage
8
9
10
11
12
baseline: displacement
100
max. surr.
min. surr.
original
50
0
0.02
(c)
0.04
0.06
axial strain
0.08
0.10
baseline: displacement + rotation
100
max. surr.
min. surr.
original
50
0
0.02
0.04
0.06
axial strain
0.08
0.10
Figure 9. Randomized surrogate network tests showing the harmonic mean of the original k-network is significantly different
to node degree-preserved randomized networks The maximum and minimum values of the distribution of harmonic mean
values of the randomized surrogate networks are shown together with the value of the harmonic mean for (a) Hostun sand
k-network built using displacements and rotations, (b) baseline system k-network built using displacements, and (c) baseline
system built using displacements and rotations. The Hostun sand kinematical k-network built using solely grain displacements
exhibits similar results and is not shown. (Online version in colour.)
the k-networks of Hostun sand against their randomized surrogate networks, and demonstrate
that this shortest path length based measure (and, by implication, all other measures derived
from shortest paths) is indicative of a statistically significant, systematic pattern of connections in
Hostun sand—that is not merely a product of chance or sampling variability.
For each strain stage of the loading history, we calculate the harmonic mean of each randomly
rewired surrogate network to obtain a distribution. Next, we compare the harmonic mean value
of the kinematical network of Hostun sand for the given strain stage. If this value is ‘well into
the tails’ of the surrogate distribution, we can conclude that the connectivities in the kinematical
network of Hostun sand are sufficiently different from those in the random surrogate networks.
The results in figure 9a show that the null hypothesis is unambiguously rejected at all stages of the
loading and that any network quantity which is based on shortest paths, e.g. closeness centrality,
which localizes in the material is indeed of significance and indicative of a systematic pattern
of connections.
We gathered additional evidence of veracity of these findings by performing the same tests on
the baseline system. Again, for all strain states of the loading regime, we find that the value of the
harmonic mean calculated for the original kinematic network lies ‘far’ from the range of harmonic
mean values obtained for the rewired surrogate networks (figure 9b,c). Thus, with respect to
this node degree-preserving random rewiring method, the patterns of connections in the original
kinematical k-networks are not random—and the early and precise localization of high relative
closeness centrality in both Hostun sand and the baseline system is unlikely to be a product
of chance.
..................................................
max. surr.
(b)
harmonic mean
17
100
harmonic mean
(a)
6. Conclusion
..................................................
New experimental data on triaxial compression of Hostun sand, comprising X-ray micro-CT
measurements of displacement and rotation of individual grains over many stages of loading,
were analysed using complex networks. The central question we sought to answer was: can
measures of complex networks, suitably constructed from individual grain kinematics, detect
strain localization and, if so, do such network properties provide early detection of failure and
offer new and useful insights on material behaviour? We discovered a remarkable behaviour in
the kinematics of Hostun sand. Findings from network measures of closeness centrality suggest
that the ultimate fate of initially dense and homogeneous granular materials may be decided and
encoded in the grain kinematics in the initial stages of loading, well before peak stress ratio. The
shortest path length scales and borders of network communities corroborate this discovery.
The network property of closeness centrality, in particular, the highest relative closeness
centrality nodes, when mapped back to the material domain, correspond to the particles in the
final persistent shear band. Incredibly, this same network measure also appears to detect the shear
band proximity in the initial stages of loading—during which no localization from the observed
particle kinematics alone is evident. To eliminate the possibility that closeness centrality was not
just picking up a bias inherent in the imperfect initial conditions of the experiment, we repeated
the study using kinematical data from a DEM simulation (baseline model). Our results there
were the same as for the experimental system. Indeed no other measure from past studies of the
baseline model (and of other DEM simulations of different material properties under monotonic
compression in two and three dimensions) have detected a persistent localization at the initial
stages of loading—from an isotropic state. As a further test of reproducibility of our findings, we
examined the development of localization in the Hostun sample from independent calculations
of continuum strain taken at grain-scale resolution using DIC. A faint hint of a localization
region was detected at the same strain interval that closeness centrality first identified this
region. Interestingly, however, the region of localized strain from the DIC measurements became
somewhat dispersed just before peak stress ratio. This is unlike the manifestly unequivocal
trend in the corresponding closeness centrality measurements, which show that the region of
the sample with high closeness centrality is as concentrated as in the preceding increment.
Closeness centrality measurements are apparently not ‘distracted’ by possible secondary factors
(e.g. competing but temporary shear bands) in the grain kinematics; instead, closeness centrality
appears to have the capability of highlighting the pattern of strain localization that will ultimately
prevail in initially homogeneous samples.
We also examined the organization and length scales of the pattern of grain interactions in
Hostun sand and the baseline system using network communities, i.e. groups of grains in each
of which member grains exhibit more similar kinematics than with those from different groups.
The impetus here lies in the well-established fact that, in common with other real-world complex
systems, the complex behaviour observed in granular materials at the macroscopic level emerges
from the multi-scale interactions among its constituent units—the grains. Network communities
resolve themselves into sub-groups with an average size or shortest path length of around 6–10
links. This length scale, common to both systems, manifests in the initial stages of loading and
remains essentially invariant for the rest of loading history; also, for the baseline system where
grain contacts are known, this community size proves to be consistent with the observed thickness
of the shear band. Furthermore, the boundaries of some of the communities, when mapped back
into the physical domain, align themselves along the shear band—again from the initial stages of
loading—well before peak stress ratio.
Soil mechanics has raised the bar on grain-scale measurements. For the first time, tens of
thousands of individual grains in three-dimensional compression tests of sand (here, Hostun) can
be individually tracked in space and time. This unprecedented development raises questions on
data analysis. What is the best approach that maximizes the quality and quantity of information
one can extract from such data? How can one ‘mine’ such data to gain hidden information,
not otherwise accessible from conventional methods? What metrics can unearth deep insights,
illuminating not only what dynamical processes emerged in the deforming sample, but also
18
to him. We thank the editor and all our reviewers for constructive comments. A.T. and D.M.W. acknowledge
the support of the Australian Research Council (Discovery grant nos DP0986876 and DP120104759), the US
Army Research Office (Single Investigator grant no. W911NF1110175) and the Melbourne Energy Institute.
G.V. and E.A. acknowledge Steve Hall, Jacques Desrues and Pierre Bésuelle to whom the experimental data
on Hostun sand equally belongs. The X-ray scanner in Laboratoire 3SR was principally funded by a project of
the French research agency ANR (MicroModEx, contract no. ANR-05-BLAN-0192).
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..................................................
A.T. thanks Dr John F. Peters for many insightful discussions on strain localization and dedicates this paper
19
why? If the latter can be answered, then there is the potential to push the boundaries of
data analysis in soil mechanics—beyond the retrospective and into one capable of delivering
prospective information (i.e. identify future trends). This study demonstrates that the strategy
of abstracting kinematical information of tens of thousands of moving grains into a complex
network and then mapping information from the network analysis back to the physical domain
shows considerable promise, and appears capable of complementing and expanding on the
information obtained from more traditional data analysis, e.g. porosity maps, fabric tensors. In
particular, we draw attention to the implications of these findings in two aspects of granular
media research. For multi-scale material characterization, this study confirms that many of the
kinematical phenomena observed and extracted from two-dimensional DEM simulations exists
in reality. For continuum modelling, our findings show that the development of strain localization
is clearly not a bifurcation phenomenon, albeit bifurcation theory may still be capable of capturing
salient aspects of the fully developed ‘persistent shear localization’ region.
Experimental soil micromechanics has opened the door for research into soil characterization
and, in turn, continuum modelling to pay heed to an echo of Terzaghi’s reflections on Coulomb.
The challenge for data analysis is immense and we have demonstrated here the potential of
complex systems theory to contribute to this effort.
20
..................................................
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in sands. J. Mech. Phys. Solids 54, 22–45. (doi:10.1016/j.jmps.2005.08.009)
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Revisiting localized deformation in sand with complex systems

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