# Compacting an assembly of soft balls far beyond the jammed state: insights
from 3D imaging

Jonathan Barés<EMAIL_ADDRESS>LMGC, Université de
Montpellier, CNRS, Montpellier, France  Manuel Cárdenas-Barrantes
<EMAIL_ADDRESS>CFTC, Universidade de Lisboa, Lisbonne, Portugal  Gustavo
Pinzón<EMAIL_ADDRESS>Université Grenoble Alpes, Grenoble
INP, CNRS, 3SR, F-38000 Grenoble, France  Edward Andò<EMAIL_ADDRESS>EPFL Center for Imaging, École Polytechnique Fédérale de Lausanne (EPFL)
Mathieu Renouf<EMAIL_ADDRESS>LMGC, Université de
Montpellier, CNRS, Montpellier, France  Gioacchino Viggiani
<EMAIL_ADDRESS>Université Grenoble Alpes, Grenoble INP, CNRS,
3SR, F-38000 Grenoble, France  Emilien Azéma<EMAIL_ADDRESS>LMGC, Université de Montpellier, CNRS, Montpellier, France  Institut
Universitaire de France (IUF), Paris, France

###### Abstract

Very soft grain assemblies have unique shape-changing capabilities that allow
them to be compressed far beyond the rigid jammed state by filling void spaces
more effectively. However, accurately following the formation of these systems
by monitoring the creation of new contacts, the changes in grain shape, and
measuring grain-scale stresses is challenging. We developed an experimental
method overtaking these challenges and connecting their microscale behavior to
their macroscopic response. By tracking the local strain energy during
compression, we reveal a transition from granular-like to continuous-like
material. Mean contact geometry is shown to vary linearly with the packing
fraction, which is supported by a mean field approximation. We also validate a
theoretical framework which describes the compaction from a local view. Our
experimental framework provides insights into the granular micro-mechanisms
and opens new perspectives for rheological analysis of highly deformable grain
assemblies in various fields ranging from biology to engineering.

##  I Introduction

Compressing a disordered assembly of grains is a seemingly simple process that
has been carried out since the dawn of time by humanity to store, transport or
transform matter. The objectives are quite basic: either to improve the
strength of the resulting material, or to optimize the space by reducing the
volume taken by the grains. In the most common case of hard grains (i.e.,
grains with high stiffness relatively to the applied pressure, $P$), the
compression generally ends once the grains cannot rearrange any more, meaning
a mechanically stable state is reached unless grains break. This so called
jammed state [37, 8, 7], mainly depends on the grain morphologies (size [46]
and shape [26, 35]), the friction between them [26], and the loading process
[8]. The packing fraction $\phi$ cannot exceed that of the Random Close
Packing (RCP), and the coordination number $Z$ (i.e., the average number of
contacts per grain) cannot be higher than $6$ for spherical ($12$ for non-
spherical) grain assemblies. It is now well documented that the applied
stresses are transmitted by an inhomogeneous network of forces where only a
small proportion of grains bear the strongest forces [50, 49]. The elastic
properties of the assembly, around the jammed state, cannot be directly
deduced from grains elasticity, but result from a subtle intertwining between
the values of the packing fraction, the coordination number and contact
properties [39, 3, 33]. The same is true for the coordination number, which
evolves as a power law of the packing fraction [22, 38, 32].

But, what if we consider soft and even squishy grains [5] (i.e., grains having
a low stiffness relatively to the applied pressure) while continuing to
compress the packing? The grains would deform; changing their shapes
elastically or in a more complex manner, without breaking to accommodate for
the internal mechanical and geometrical constraints. As a result, the solid
fraction would exceed that of the RCP, and it would even be possible to
entirely fill the space. A new world, yet poorly understood, is opening up
with these materials made of soft, squishy, deformable grains. Such materials
play a major role in various fields of natural sciences (foams, colloidal
suspensions, biological cells, blood clogging in a vein, etc) and in many
industrial processes (emulsions, metallic powders, sintered material, rubber
mixtures, etc). In soft granular assemblies, the macro- and micro- scale
behaviors are no longer uniquely ruled by the steric exclusions and the
topological disorder of contacts, but also by the bulk behavior of each
constitutive grain. Including the change in grain shape in addition to the
inherently multi-contact nature of a granular system in a realistic 3D
modeling is a vast emerging topic, with challenging experimental, numerical,
and theoretical issues [5, 11].

From a numerical point of view, these last years have seen the advent of new
strategies in the form of Discrete Element Methods (DEM) coupled with
different continuum approaches [5]. Although technical challenges still
remain, especially in terms of computational time, existing studies, mainly in
2D and few in 3D [25, 34, 2], have allowed to glimpse new microscopic
mechanisms that take place from contacts scale until the inside of the grains.
For example, the power law dependence of the coordination number with the
packing fraction is still valid far beyond the jammed state [63, 45, 12, 16,
13], even for assemblies of initially non-circular grains [15]. Local grain
rearrangements, sometimes dramatic, can still occur even after the jamming
point [27, 20, 9, 6], but at the same time the stress distribution becomes
more homogeneous as the packing fraction increases. Friction also contributes
to a better homogenization of the contact force network [15]. By implementing
some of these new microscopic descriptors into a rigorous micro-mechanical
framework, Cantor et al. (in 2D [12]), and Cardenas-Barrantes et al., in 2D
[14, 13] and 3D [16], have stated compaction equations fully determined by the
evolution of the microstructure, resolving thus a long lasting issue verified
through the very large number of ad hoc models that have been proposed over
the last few decades [64, 17, 53, 43, 48, 45].

Still, there is no experimental validation, especially in 3D, of all these
advances. Even more important, no clear strategies exist to explore highly
deformable granular systems far beyond the jammed state, from the particle
scale.

More generally, whether it is in 2D or 3D, the measurement of contacts
(position and orientation) and forces or stresses in the grains, as well as
the description of their evolution, are mostly based on inverse problem
methods. Close to the jammed state it is always assumed that the contacts obey
a Hertz type law [36]. It is then possible to detect the contacts and to
extrapolate the resulting interaction forces using different local field
measurements such the deformation of the grain [29, 10], a photoelastic [21,
68] or thermoelastic [30] signal, or raw measurements by digital image
correlation (DIC) [41]. These inverse methods have also been successfully
extended to 2D non-circular grains assemblies [65, 23] and also to 3D analysis
using X-ray tomography with different granular materials [51]. Careful image
analysis work also allows some inroads into the problem [66].

Far from the jammed point, at least two challenging issues appear: ($i$)
Hertz’s law is no longer valid, and ($ii$) it is difficult to follow the shape
of the particles while detecting new contacts. This last point is particularly
exacerbated in 3D and at large packing fraction since it requires high image
resolution. Moreover, it is not always possible to use a direct optical
approach to measure local particle properties, and non trivial tomographic
reconstructions may be necessary. Most of the 3D experimental studies based on
highly deformable, squishy, grains are macroscopic and focus on general
properties such as stress release [28, 59], seismic isolation [56, 58], or
foundation damping [42]. To the best of our knowledge, only the work of S.
Mukhopadhyay and J. Peixinho in 2011 [44], using fluorescent hydrogel spheres,
associated with a tomographic reconstruction, was able to follow the local
evolution of a compressed packing up to a density of $0.85$. In particular,
they described the evolution of particle connectivity, but unlike what is
obtained numerically, they do not report power-law dependencies between the
coordination number and packing fraction.

In this work, we present a new 3D experimental technique that measures the
displacement field directly at the grain scale (within grains), without any
material or geometrical assumptions. This is performed by combining advanced
Digital Volume Correlation (DVC) methods together with accurate x-ray
tomography imaging. This method is applied to the paradigmatic case of axial
compression with prevented radial strain: a cylindrical sample containing $60$
bidisperse silicone beads of diameter $3$ and $4$ mm is compressed vertically.
Micrometer scale glass beads are trapped in the particle material to form a 3D
homogeneous random pattern to facilitate DVC. The granular medium is
quasistatically compressed in an x-ray scanner to obtain density images of the
evolving system, halting loading during scans. Experiments have been repeated
four times. The major outcome of this work is the simultaneous measurement of
the topological (contacts and shapes) and bulk properties (internal strain
fields) of each grain throughout the whole compression process up to very high
packing fraction. This permits the inferrence of the grain-scale stress
tensors from the loading history and the material rheology. In a context where
3D simulations are still rare and highly demanding in terms of time, our
method allows the description of the microstructural and micromechanical
properties of any squishy granular system deep in the jammed state. This
allows the validation and extension to uniaxial compression constitutive laws
recently introduced numerically and in 2D configurations [12, 16], in view of
theoretical modeling.

Figure 1: Raw and post-processed data: (a): Middle slice of the x-ray scanner
density matrix for different levels of compression. The system is compressed
vertically from the bottom. (b): Corresponding 3D reconstruction of the
particle surfaces from the DVC results obtained from the x-ray scans. (c):
Evolution of the shape and contact surfaces (in red) of a given particle when
compressed.

##  II Results

###  II.1 Compaction and connectivity evolution laws

Figure 1a and b shows tomographic slices and corresponding 3D reconstructions
of one of our samples for different levels of compression: from an initially
jammed state to a highly compact one. During the compression process the voids
within the assembly are progressively filled due to the deformation of the
grains. Meanwhile, the packing fraction $\phi$ increases from $\phi_{0}\approx
0.53$ (jammed state) to $\phi_{max}\approx 0.9$ for the most compact state
obtained.

In Fig. 2, deep in the jammed state, we present the evolution of the global
vertical strain $\varepsilon=-\ln(H/H_{0})$, with $H_{0}$ the initial height
of the cylinder (a, blue curves), the coordination number $Z$ (a, black
curves), and the applied vertical stress $P$ (b), as a function of the packing
fraction $\phi$. Basically, $\varepsilon$ decreases as $\phi$ increases
following, with a good approximation, a logarithmic law:
$\varepsilon\sim-\ln(\phi/\phi_{0})$. The evolution of $Z$ is such that it
increases in a non-linear way with $\phi$ from $Z_{0}\approx 4.5$ to
$Z_{max}\approx 8.5$, as $\phi$ goes from $\phi_{0}$ to $\phi_{max}$. As shown
in the previous 2D numerical/experimental works [22, 38, 32, 63, 12] and
recently in 3D [16], this increase follows a power law:

$Z-Z_{0}=\xi\sqrt{\phi-\phi_{0}},$ (1)

where $\xi\approx 6$ is fitted from experimental data. The same power-law is
observed in our 3D experiments (Fig.2(a), black solid line). The small
variations from the prediction are due to the experimental uncertainty in the
contact detection limited by the image resolution. At large packing fraction
we note that the system most likely enters a different regime. Thus, and to
the best of our knowledge, our experimental results confirm for the first time
the validity of the phenomenological law between $Z$ and $\phi$ up to a
certain density. This law has been often verified numerically or in 2D model
experiments only [22, 38, 32, 63, 12, 16].

Along with the evolution of $Z$, the compaction of the assembly is also
characterised by the $P$ vs. $\phi$ curves, shown in Fig. 2b. We find that $P$
slowly increases with $\phi$ at low density, and then diverges as the packing
fraction tends to $\phi^{*}_{max}\approx 1$, the “theoretical” maximum packing
fraction. This divergence is expected since the assembly of soft grains begins
to behave as a rigid body when the grains almost fill all the voids. By
combining the micromechanical formulation of the granular stress tensor [4],
its limit to small deformations and the evolution of the particle
connectivity, we deduce the following equation for the compaction curve $P$
vs. $\phi-\phi_{0}$ (see details in the Methods section):

$P=E^{*}\frac{2(1+2\mu_{M})}{3\pi\Gamma^{3/2}}\frac{\phi^{*}_{max}-\phi_{0}}{\phi_{0}^{3/2}}Z\phi(\phi-\phi_{0})^{1/2}\ln\left(\frac{\phi^{*}_{max}-\phi_{0}}{\phi^{*}_{max}-\phi}\right)$
(2)

Where $E^{*}=E/2(1-\nu^{2})$ is the reduced Young modulus, $\mu_{M}=0.25$ is
related to the macroscopic friction of an assembly of spherical grains,
$\Gamma=3.89$ is a structural parameter linearly relating the macroscopic
deformation and the mean contact deflection, and $Z$ is the coordination
number given by Eq.1. The power of the proposed compaction law is that it is
based only on physical parameters that are easy to measure experimentally. As
we can see, our data are very well fitted by Eq.2. This validates and extends
to the case of 3D uniaxial compression a recently introduced micromechanical
framework allowing to design compaction laws for different loading conditions
and material properties [12].

Figure 2: Macroscopic evolution. (a): Evolution of the global uniaxial strain,
$\varepsilon$, (in blue) and of the coordination number increase from the
jamming point, $Z-Z_{0}$, (in black) as a function of the relative packing
fraction, $\phi-\phi_{0}$. Points are experimental data (different symbols
stand for different experiments) while plain lines are analytical models. The
line for global strain is a scaling of $-\ln\left({\phi}/{\phi_{0}}\right)$,
while the black line is a plot of Eq.1. (b) Compaction curve: applied vertical
stress, $P$ as a function of $\phi-\phi_{0}$. Points are experimental data
while the plain line is the analytical model given by Eq.2.

###  II.2 Quantitative description of grain shapes evolution

Figure 3: Grain shape evolution: (a) Schematic view in 2D of the major and
minor directions of a particle and their orientation with respect to the
vertical direction. $\theta$ is the elevation angle of the minor direction.
(b) Evolution of the mean grain aspect ratio, $\bar{\lambda}$ (in blue), of
the average of the sinus of the minor direction elevation,
$\bar{\epsilon}_{p}$ (in black) and of the average grain asphericity,
$\bar{\alpha}$ (green) as a function of the packing fraction relative to the
jamming point, $\phi-\phi_{0}$. Points are experimental data (different
symbols stand for different experiments). Blue and black plain lines are
averaged curves and the green one is the power-law fit:
$\bar{\alpha}=1+(\phi-\phi_{0})^{3/2}$. (c) Evolution of the average shape of
the grains as a function of $\phi-\phi_{0}$. Curves are circumferentially
averaged over each grain of the packing.

The strength of our experimental setup and data post-processing is to obtain
precise information at the grain scale, such as the evolution of their shape
as well as the evolution of the contact geometries, when the whole assembly is
compacted. For example, Fig.1c shows the evolution of the shape of one of the
grains located in the center of the system, which progressively shifts from
spherical to flattened polyhedral shape, with increasing contact areas.

We define the asphericity parameter by
$\alpha=\frac{a_{p}^{3/2}}{3\sqrt{4\pi}}$, where $a_{p}$ and $v_{p}$ are the
surface area and volume, respectively, of a given particle $p$. By
construction, $\alpha$ equals one for a sphere and is higher for any other
geometry. $\bar{\alpha}$ is the average of this quantity over all the grains
of the packing. Figure 3(b - green curves) displays the evolution of
$\bar{\alpha}$ as a function of the excess of packing fraction
$\phi-\phi_{0}$. We find that the asphericity increases as a power law:
$\bar{\alpha}=1+(\phi-\phi_{0})^{\beta}$, with $\beta=3/2$ and $1$
corresponding to the initial asphericity. Interestingly, a similar trend has
recently been observed in 2D/3D numerical simulations of compaction of soft
disks/spheres assemblies [13, 16]. Thus, our experiments confirm this
seemingly universal geometric feature of soft particles compaction.

Above the loss of sphericity it is important to analyse how particles deform.
To do so, for each grain, we consider the evolution of its aspect ratio
$\lambda$ and its average orientation $\epsilon$. $\lambda$ is defined as the
ratio of the shortest over the longest straight lines joining opposite
boundaries of a given grain and crossing its center of mass. $\epsilon$ is the
sinus of $\theta$, the elevation angle of the shortest direction (see in
Fig.3a). As observed in Fig.3b, the average particle aspect ratio
$\bar{\lambda}$ decreases with $\phi$ from $1$ (spherical shape) to values
close to $0.6$. Meanwhile, their average orientation $\bar{\epsilon}$
increases from $0.5$ (anisotropic) to $1$ (horizontally expanded) for
$\phi\gtrapprox 0.72$. From these observations, a schematic picture emerges to
describe the compaction from a grain-scale perspective. The grains elongate in
the surrounding voids mainly by increasing the largest distance, contributing
to the creation of new contacts aligned along the compression direction. In
the other directions, the mean grain aspect ratio also increases by shrinking
their shortest length. This is also well evidenced in Fig.3(c), which shows
the average circular profile of the grains. At first this profile is perfectly
semi-circular and it then gets progressively larger in width and thinner in
height.

###  II.3 Linking contact geometry evolution to packing fraction

Along with the grain shape, the evolution of the contact surface geometries
between grains is also probed during the compression. As an illustration,
Fig.4a displays the evolution of a typical contact surface within the sample.
Qualitatively, close to $\phi_{0}$ the shape of a contact can be considered as
circular. However, as the compression goes deeper into the jammed state, the
contact surface gets larger and exhibits a non-trivial concave shape. As for
the grain shape, for each contact, we define the mean contact surface
acircularity $\gamma=\frac{p_{c}^{2}}{4\pi s_{c}}$, where $q_{c}$ and $s_{c}$
are the perimeter and the surface of this contact, respectively.
$\bar{\gamma}$ is the average of this quantity over all the contacts of the
packing. In Fig.4b the blue curve shows the evolution of $\bar{\gamma}$, as a
function of the excess packing fraction $\phi-\phi_{0}$. We find that, on
average, the contact surfaces can be considered as circular up to fairly high
packing fraction, close to $0.83$. Before this density $\bar{\gamma}\approx
1$, but beyond, $\bar{\gamma}$ increases suddenly to values close to $1.2$.
This highlights the strong deviation of the contact surfaces from a circular
shape. The shape of this curve is reminiscent of the divergence of $P$
observed in Fig.2b. Along with $\gamma$, the contact geometry is also
characterised by the mean curvature of the contact surfaces $l$. This latter
is defined by the contact non-planarity, the mean distance between the real
contact surface and its average plane. This quantity is then normalised by the
mean particle radius $R_{p}$. The evolution of its average value $\bar{l}$ as
a function of the excess of packing fraction $\phi-\phi_{0}$ is shown in
Fig.4c. We observe that $\bar{l}$ slightly increases with $\phi$ and remains
overall less than $1\%$ of the mean grain radius until, as for $\bar{\gamma}$,
a cross-over is observed at $\phi\approx 0.83$. Then, a rapid increase is
observed and $\bar{l}$ exceeds more than $1\%$ of the mean grain radius. In
other words, from a certain level of compression, the grains indent each other
and the contact surfaces are no longer symmetrical. This leads to concave-
shaped contacts as observed in Fig.4a.

To quantify the evolution of the size of the contact, we define the relative
effective contact radius $\kappa=\frac{\sqrt{s_{c}/\pi}}{R_{p}}$. Using a mean
field approach that combines the geometrical properties of an Hertzian contact
together with the $Z$ vs. $\phi$ relation (Eq.1), we obtain (See Methods
Section):

$\bar{\kappa}\propto\phi-\phi_{0}.$ (3)

We see in Fig.4b that the linear dependency predicted by Eq.3 is well
satisfied up to large $\phi$ values deep in the jammed state. However, from
$\phi\approx 0.83$, the data slightly deviate from this prediction since the
circularity and flatness assumptions of the contact surface are not valid
anymore. Going further in the contact analysis, in Fig.4b we show the
evolution of the mean contact direction
$\bar{\omega}=\langle\sin\theta_{c}\rangle_{c}$, where $\theta_{c}$ is the
elevation angle of the normal angle to the contacts. We find that
$\bar{\omega}$ increases from $0.5$ (anisotropy) at $\phi_{0}$ to $\sim 0.75$
(mainly vertical) for $\phi\rightarrow\phi_{max}^{*}$. This indicates that the
contact surfaces are preferentially oriented perpendicular to the direction of
compression as the packing fraction is increased. This is consistent with the
preferential direction of grain elongation discussed in the previous section.

Figure 4: Contact geometry evolution. (a): Evolution of the shape of a given
contact when the system is compressed. Projections of the contact contours on
three orthogonal directions are also shown. (b,c): Evolution of the average
contact acircularity, $\bar{\lambda}$ (a-blue), the average effective contact
radius, $\bar{\kappa}$ (a-black), the average contact non-planarity, $\bar{l}$
(b-blue) and the average contact direction elevation, $\bar{\omega}$ (b-black)
as a function of the distance of the packing fraction from the jamming point,
$\phi-\phi_{0}$. Points are experimental data (different symbols stand for
different experiments) while plain lines are averaged curves except for
$\bar{\kappa}$ where it is a linear fit with slope $2.94\pm 0.03$.

###  II.4 Deeper within the grains: strain and energy density fields

The presented method not only allows the measurement of the 3D evolution of
grain boundaries, but also provides the local displacement field inside each
grain, even deep in the jammed state, for highly deformed grains. The right
Cauchy-Green deformation [54] tensor and its von Mises norms are deduced from
this displacement field [61]. Also, assuming an hyperelastic behavior of the
silicone used to make the beads [61, 63, 62], we compute the energy density
stored in the material [54, 63], $\rho$. Figure 5(a) shows the evolution of
the mean von Mises strain when the granular system is loaded. We find that the
mean von Mises strain increases first linearly with the packing fraction. On
average, this implies a proportionality relation between the local and the
global strain applied to the system. Still for higher packing fraction, from
$\phi\approx 0.83$ the local strain enters a regime where it increases more
rapidly. In fig.4(b), we show the evolution of the average energy density when
the system is compressed. This latter quantity increases quadratically with
the packing fraction. This is a direct consequence of the linear relation
between the strain and the packing fraction. It implies that even for quite
high packing fractions, from a coarse-grained point of view, the system
behaves as a solid. Still, when looking at the statistical distribution of the
energy field, presented in fig.5(c), this analogy is no longer true. Indeed,
the probability density function (PDF) of $\rho$ is Gaussian-like for gently
squeezed system but for more compressed systems, it displays an exponential
tail reminiscent of what is observed for the PDF of the interaction forces
between rigid grains [50].

Figure 5: Local mechanical evolution. Evolution of the average von Mises
strain (a) and of the average energy density, $\bar{\rho}$ (b) as a function
of the distance of the packing fraction from the jamming point,
$\phi-\phi_{0}$. Points are experimental data (different symbols stand for
different experiments) while in (a) the line show a linear fit with slope
$3\pm 0.1$ and in (b) plain curve is the quadratic fit $5\times
10^{5}(\phi-\phi_{0})^{2}$. (c) Evolution of the probability density function
(PDF) of the energy density, $P(\rho)$ when a system is compressed.

##  III Discussion

We have overcome one of the main experimental challenges recently identified
as such by the granular matter community [5, 52, 40]. The novel experimental
framework we present in this paper permits to travel from macro to micro
observables –and conversely– of a highly squeezed granular system deep in the
jammed state up to very large packing fraction. The strain tensor in the
material of each particle of the whole system is recorded during the
compression process permitting to compute the evolution of the corresponding
stress tensor and energy density. The change in shape of the particles and
their contacts are deduced and the classical –strain vs. stress– measurements
are also monitored. No classical mechanics observables of the system are kept
unknown during the whole loading process opening the way to a thorough
analysis of this complex multi-scale mechanical process.

This novel set of information permitted first to check that the
phenomenological exponential law between the system coordination number, $Z$,
and packing fraction, $\phi$, still holds in 3D. So far this had been only
observed for 2D experiments [22, 32, 38, 63]. This point extends the domain of
validity of this relation and suggests that more contacts are created than
what was initially expected. The compaction law linking the pressure applied
to the system and its packing fraction, built for 2D systems and in 3D for
isotropic compression have been extended to the case of uniaxially loaded 3D
systems. The strength of this theoretical framework, derived from the micro-
mechanical expression of the granular stress tensor, is to be based only on
physical parameters that are easy to measure experimentally. Our experimental
results outstandingly validate this law even deep in the jammed state. This
would permit to predict with great precision the behaviour of a loaded system
just by knowing the exact nature of the grains.

Moreover, the results presented in these series of 3D experiments shows that
the mean contact geometry varies linearly with the packing fraction. Just like
for the compaction law, this relation is supported by a mean field
approximation developed in the small deformation framework. It is remarkable
that even deep in the jammed state, where grains are extremely deformed
through multiple contacts and that the Hertz contact hypothesis are highly
violated [36, 62], the mean-field analytical framework still holds and gives
greatly reliable results. This point is good omen to find a unified
rheological law describing the mechanical behaviour of a system from the
jammed state up to bulk material whatever the loading process.

Still by tracking the local strain energy during compression, we reveal that,
for very high packing fraction, a transition from granular-like to continuous-
like material exists. For a packing fraction close to $0.83$, the average
energy density exits a quadratic law with the packing fraction and the average
strain exits a linear law. The description of these quantities remains complex
to catch analytically in these high density regimes. They correspond to
regimes where particles are highly distorted to fill the last remaining
porosities. This is is agreement with the fact that above $\phi\approx 0.83$,
the average particle asphericity also exits a well defined power-law behaviour
with the packing fraction. Below this threshold the material remains
remarkably well described by the material rheology taking granular structure
into account; on the contrary, above, it enters a more extreme regime where
observables vary abruptly. This point, that has been also observed in 2D [61,
13], clearly evidences a sharp crossover in the material behaviour.

Our experimental framework evidences heuristic and analytical laws linking
different observables of densely packed granular systems. This provides
insights into the granular micro-mechanisms and how they are responsible for a
specific macroscopic behavior. On top of permitting to predict the evolution
of a densely packed system, our findings permit to deduce the global state of
a system through local observations –and conversely. This opens new
perspectives for rheological analysis of highly deformable grain assemblies in
various fields ranging from biology [19, 67, 40], engineering [18] and
geophysics [55, 57], to name just a few.

##  IV Methods

Figure 6: (a): picture of the custom-made silicone particles. The large
particles are $4$ mm wide, while the small ones have a diameter of $3$ mm.
(b): schematic view of the in-situ uniaxial compression set-up. (c) picture of
the experimental set-up.

###  IV.1 Sample preparation and in-situ compression

The granular samples are made of $60$ silicone beads, $24$ of diameter $4$ mm
and $36$ of diameter $3$ mm (see fig.6-a). These particles are molded with
SortaClear 18 silicone by SmoothOn company [1], mixed with $35$% in mass of
glass beads of diameter between $40$ and $70$ $\mu$m. The silicone is viscous
enough for the glass beads not to sediment during polymerisation. Just before
molding, the silicone is degased to remove air bubbles. Each grain is coated
with talc powder to avoid the particles to stick together. They end up having
a friction coefficient of $0.59\pm 0.15$. The Young modulus of the material is
$E=0.663$ MPa and its Poisson ratio is $\nu=0.289$.

To account for the variability of the initial state, $4$ independent
experiments are performed. Each specimen is prepared by placing particles
individually and layer by layer inside an oedometer of
$15\text{\,}\mathrm{mm}$ in diameter. To avoid segregation and crystallisation
effects, each “layer” consists of $2$ large and $3$ small particles. The
particles are deposited until the height of the assembly reaches
$15\text{\,}\mathrm{mm}$, yielding a specimen aspect ratio of 1. The oedometer
is then closed and mounted on top of the rotation stage of the x-ray
tomography cabin, built by RX-Solutions [60, 31]. The loading system is
located below the setup, thus the loading is applied from the lower plate of
the oedometer, as shown in Fig.6-c. The oedometric compression is performed
“in-situ”, i.e., imaging the specimen while performing the test step-by-step:
the specimen is compressed at a loading speed of
$4\text{\,}\mathrm{\SIUnitSymbolMicro m}$, using displacement steps of
$250\text{\,}\mathrm{\SIUnitSymbolMicro m}$, after which, the loading system
is paused and the specimen is imaged. To perform a scan, $3$ x-ray projections
are averaged for each of the 1120 different radial positions (see Fig.6-a),
and a 3D field is reconstructed using the filtered back-projection algorithm
proposed by [24], as implemented within the XAct software provided by RX-
Solutions, saved on a 16-bit data range. The x-ray tomography scans are
performed using a pixel size of
$15\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}$,
resulting in more than $200$ pixels per particle diameter. The loading force
is measured externally, and each specimen is compressed until a packing
fraction of $\phi\approx 1$ is reached.

Figure 7: (a)-(b): evolution of 3d contours of the particles as obtained from
DVC. The contact areas are shown in red. (c)-(d) corresponding packing showing
stored energy density fields. (e): 2D view of a x-ray slice zoomed on the
contact between two grains. Green dots are potential contact points where the
average x-ray density is probed inside the green rectangles. The average value
of this density in these subzones constitutes a contact detection threshold.
(f) evolution of the number of contacts in the whole system by varying the
threshold on the contact criterion.

###  IV.2 Numerical method for post-processing

From each experiments we obtain $\sim 25$ $16$ bit density matrices of
dimension $1200\times 1200\times 1800$. We first convert the matrices into $8$
bit. Then we isolate the large cluster of connected grains from the rest of
the system on the initial undeformed image by means of thresholding. Eroding
sequentially this cluster, we isolate the centers of the grains. To segment
grains in the initial image we detect voxels on the borders of this large
cluster containing all the grains. Considering these voxels one by one we
check if the lines going from it to the two closest particle centers stay
included in the cluster. If so, that means the voxel belongs to a contact
border. These voxels forming penny shaped clusters give the border of the
contact, where we have to separate particles. We look for the plane in which
each of these clusters are included and cut the big cluster of grains along
these planes. As shown in the bottom left part of fig.1 we obtain the clusters
of voxels covering each particle.

From the initial particle clustering, regular cubic meshes are built inside
each grain. They form a network of correlation cell centers that will be
tracked from one image to the next all along the compression. The distance
between the nodes of this network is $12$ vx. From one image to the next, the
solid rigid motion of each grain is first computed: on the first image, around
the closest correlation cell center from the particle center of mass, a cubic
sub-matrix of side $36$ vx is extracted. By means of Fourier transform it is
convolved with the submatrix at the same location in the next image. The
position of the maximum of convolution gives the particle translation with a
vx accuracy. To improve this displacement measurement, we minimize the squared
difference between the first submatrix and the second one deformed by a first
order shape function. Once the solid rigid motion of each particle is
obtained, by applying the same process as the one described for the particle
center of mass to each correlation cell center we obtain the displacement
field inside each particle. In this latter computation step, the correlation
cell size is $18$ vx and if the correlation criterion is low enough, then the
shape function is just a translation. The process is sequentially repeated for
each consecutive pair of images.

From the displacement field obtained from DVC, the displacement of the
particle borders is deduced and the evolution of the particle shapes is
tracked along the compression process. To measure contact surfaces, we first
discretize the border into a network of points separated by $18$ vx. for each
of these points, if they are close to another point belonging to another
particle we extract a parallelepipedic submatrix centered around this point
and aligned perpendicularly from the boundary as shown in fig.7-e. We compute
the average value of this submatrix. The higher this density value, the higher
the quantity of matter in the submatrix, so the higher the probability of
contact. In fig.7-f we have plotted the evolution of the number of contacts
for different thresholds on this density value to consider the contact as
real. We observe that the density of curves increases for a threshold of
$120$, which is close to the mean density ($2^{8}/2$). This is the threshold
we picked.

###  IV.3 Derivation of the compaction equation for 3D uniaxial geometry

We briefly recall the main ingredients of the theoretical framework presented
in [16] and we extend it to fit uniaxial 3D geometry. In this case, $P$ is
related to the granular stress tensor through its $zz$ component by
$P=\sigma_{zz}$. The granular stress tensor is given by
$\bm{\sigma}=n_{c}\langle\bm{f^{c}}\otimes\bm{\ell^{c}}\rangle_{c}$ [4], where
$\bm{\ell^{c}}$ is the vector between the centers of the grains and
$\bm{f^{c}$}theforcevectoratacontact$c.n_c=N_c/V$isthedensityofcontact,with$N_c$thetotalnumberofcontactsinthevolume$V$.Consideringasmallparticlesizedistributionaroundthemeandiameter$d$and$V_p
= (π/6)d^3$thevolumeofasphere$p$,weobtain$n_c
≃3Zϕ/(πd^3)$,with$Z=2N_c/N_p$thecoordinationnumber.Fromthedefinitionof$σ_zz$viatheprincipalvaluesof$σ$,weget:\begin{equation}P\simeq\alpha\frac{\phi
Z}{\pi}\sigma_{\ell},\end{equation}with$σ_ℓ = ⟨f^c ⋅ℓ^c
⟩_c/d^3$,ameasureofthemeancontactstressand$α$aconstantrelatedtovariousmicroscopicparameterspertainingtothecontactandforcenetworkspecificofuniaxialcompression,buthardlyaccessiblebymeansofexperiments.However,theoreticaldevelopments\cite[cite]{[\@@bibref{Number}{Ouadfel2001}{}{}]}haveshownthatthesemicroscopicparameterscanbeaddedtogethertobuildtheshearstrengthparameter$μ_M$ofthematerialallowingtowrite$α∼1+2μ_M$.Foranassemblyoffrictionalspheres$μ_M$iscloseto$0.25$.Notethat,Eq.\ref{eq:Pzz_sigma}ismostlyknowninitssimplerform\cite[cite]{[\@@bibref{Number}{Agnolin2007c,Andreotti2013}{}{}]},where$μ_M$isfixedto$0$whenconsideringthemeanstressfromthetraceof$σ$.\par
Closetothejammedstate({\it
i.e.},inthecaseofsmalldeformations),assumingthatthecontactforcesfollowaHertzlawandverifyingthat(asdiscussedinSec.\ref{Sec_Energy})themacroscopicdeformation$ε$islinearlyrelatedtothemeancontactsdeformation$⟨δ^c/R_p⟩_c$,with$δ^c$thereductionofradiusatacontact$c$(viaaparameter$Γ∼3.8$measuredinourexperiments),wecanrewriteEq.\ref{eq:Pzz_sigma}as$P_SD
=
-α2E*3πΓ3/2Z_0ϕln^(3/2)(ϕϕ0)$.Now,deepinthejammedstate,werelyonthemacroscopichypothesisformulatedbyCarrollandKim\cite[cite]{[\@@bibref{Number}{Carroll1984}{}{}]}.Usingananalogybetweenthecompactionprocessandthecollapseofacavitywithinanelasticmediumtheyshownthat,necessarily,$P
∝ln[(ϕ^*_max-ϕ)/(ϕ^*_max-
ϕ_0)]$.Bycombiningthislargedeformationapproach,togetherwiththesmalloneandEq.\ref{eq:Pzz_sigma}thecompactionequationbecomestheonepresentedinSec.\ref{Sec_epsilon_Z_Phi_P}.\par$

###  IV.4 Predicting the linearity between $\bar{l}$ and $\phi$

First, let’s remark that $\bar{l}$ can be rewritten as a mean over all grains
and contacts: $\bar{l}\sim Z.r_{c}/R_{p}$. Then, since we have verified (see
Sec.II.1) that $(\phi-\phi_{0})/\phi_{0}\sim\varepsilon$ and also that
$\varepsilon\sim\langle\delta^{c}/R_{p}\rangle_{c}$ (see Sec.II.4), we get the
following proportionality between $Z$ and $\delta$:
$Z\sim\sqrt{\delta/R_{p}}$. Finally, from the Hertz theory we get that the
radius of the contact surface between two spheres touching at a contact $c$ is
given by $r_{c}\sim\sqrt{R^{*}\delta^{c}}$, with $R^{*}\equiv R_{p}$ the
effective radius and $\delta^{c}$ the reduction of radius at the contact.
Injecting the two above proportionalities within the definition of $\bar{l}$
leads to $\bar{l}\sim\sqrt{\delta}\sqrt{\delta}\sim\phi-\phi_{0}$.

###  IV.5 Data availability

The data that support the findings of this study are available from the
corresponding author.

###  IV.6 Code availability

The codes that support the findings of this study are available from the
corresponding author.

###  IV.7 Author contributions

E.An., J.B., E.Az. and M.R. conceived the experiment; J.B. made the samples;
G.P. and M.C.-B. performed the experiments; J.B. and G.P. post-processed the
data; J.B., M.C.-B., G.P., E.An., E.Az. and M.R. analyzed the data; M.C.-B.,
E. Az. and M.R. set-up the analytical model; E.Az., J.B., G.P., M.C.-B.
drafted the manuscript; all authors reviewed the manuscript.

###  IV.8 Acknowledgements

We are grateful to G. Camp and S. Devic for helping with the sample
preparation.

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