# Tessellated granular metamaterials with tunable elastic moduli

Nidhi Pashine  Dong Wang  Jerry Zhang  Sree Kalyan Patiballa  Sven Witthaus
Mark D. Shattuck  Corey S. O’Hern  Rebecca Kramer-Bottiglio  Department of
Mechanical Engineering & Materials Science, Yale University, New Haven,
Connecticut 06520, USA  Department of Physics, Yale University, New Haven,
Connecticut 06520, USA  Department of Applied Physics, Yale University, New
Haven, Connecticut 06520, USA  Graduate Program in Computational Biology &
Bioinformatics, Yale University, New Haven, Connecticut 06520, USA  Benjamin
Levich Institute and Physics Department, The City College of the City
University of New York, New York, New York 10031, USA

###### Abstract

Most granular packings possess shear moduli ($G$) that increase with the
applied external pressure, and bulk moduli ($B$) that increase or remain
constant with pressure. This paper presents “tessellated” granular
metamaterials for which both $G$ and the ratio $G/B$ decrease with increasing
pressure. The granular metamaterials are made from flexible tessellations
forming a ring of closed cells, each containing a small number of solid
particles. For under-constrained tessellations, the dominant contributions to
$G$ and $B$ are the particle-particle and particle-cell interactions. With
specific particle configurations in the cells, we limit the number of possible
particle rearrangements to achieve decreasing $G$ as we increase the pressure
difference between the inside and outside of the tessellation, leading to
$G/B\ll 1$ at large pressures. We further study tessellated granular
metamaterials with cells containing a single particle and many particles to
determine the variables that control the mechanical response of particle-
filled tessellations as a function of pressure.

###### keywords:

Granular metamaterials, mechanical metamaterials, atmospheric diving suit

††journal: Journal of LaTeX Templates

##  1 Introduction

Granular materials consist of collections of macroscopic particles that
interact with each other through contact forces, resulting in bulk mechanical
behaviors that range from fluid-like to solid-like [1, 2]. At low densities,
granular particles can flow past each other in a fluid-like state; at high
densities, they jam into a solid-like state. Even in their jammed state,
particles in granular media can rearrange by rolling and sliding past one
another, thus changing the contact network and the corresponding mechanical
response of the system. The mechanical properties of jammed packings, such as
the bulk $(B)$ and shear $(G)$ moduli, are dependent on these interparticle
interactions and scale as power laws with the packing fraction of the system
[3].

The ratio $G/B$ of the shear and bulk moduli quantifies the solidity of
granular materials. A liquid has $G/B=0$ whereas solids have $G/B>0$.
Moreover, $G$ and $B$ are within an order of magnitude of each other for most
materials [4, 5]. For example, the shear and bulk moduli of iron are
$77.6~{}$\mathrm{GPa}$$ and $166~{}$\mathrm{GPa}$$ [6], respectively, with a
$G/B\approx 0.47$. Further, for most metals [7, 8, 9], as well as ionic [10,
11] and non-ionic [12] crystalline solids, $G$ and $B$ increase as a function
of pressure. Most metals have a relatively constant value of $G/B$ over a wide
range of pressures [13]. In contrast, granular packings with large numbers of
particles show an increasing $G/B$ with increasing pressure above the jamming
transition, due to particle rearrangements within the packing [3, 14, 15].

This paper presents “tessellated” granular metamaterials with tunable elastic
moduli and $G/B$ values dependent on pressure. The tessellated granular
metamaterials are made of an annulus that is radially tessellated into cells,
each of which is filled with particles in identical configurations that are
close to the onset of jamming. Notably, by filling the tessellated cells with
a small number of particles, we limit the number of possible rearrangements
and thus increase our control over $B$, $G$, and $G/B$. Our results show that
the $B$ of our tessellated granular metamaterial does not always increase
monotonically with pressure, although does generally possess a positive
correlation with pressure. Due to the small number of possible local
rearrangements in the cells, we also observe a global decrease in $G$ with
increasing pressure. Combined, we find that $G/B$ for the tessellated granular
metamaterials decreases with increasing external pressure.

Figure 1: (a) Experimental setup. 3D printed tessellation with flexible joints
filled with particles. The particles as well as the tesselation are 3D printed
out of Polylactic Acid (PLA). Particles in each cell have the same
configuration, referred to as configuration ‘6 particles I’. (b) A single cell
of the tessellation showing the edges and joints. The different particle
configurations studied in this work include: (c) A single cell with particles
in configuration ‘6 particles II,’ (d) 7 particle configuration, (e) 25
particles per cell with each cell in a random particle configuration, and (f)
one particle in each cell with varying particle diameter.

##  2 Methods

### Tessellation Geometry

Our setup includes a flexible tessellation with walls that can rotate freely
about its joints to create a system with a large number of zero-energy modes.
According to Maxwell’s counting argument, a 2D tessellation with $N$ nodes and
$N_{b}$ bonds has $(2N-N_{b})$ zero energy modes, including three trivial
modes corresponding to translation and rotation [16, 17]. As shown in Fig.
1(a), our tessellation design consists of 16 trapezoid cells that are
connected together to form an annulus. Fig. 1(b) shows the structure of a
single cell. Each cell has four joints on the vertices of the trapezoid, and
two additional joints breaking up the top wall into three parts. These extra
joints add two more degrees of freedom per cell to the system. When connected
together in an annulus, the tessellation has $N$ = 64 nodes and $N_{b}$ = 80
bonds, resulting in 45 zero-energy modes in the system. When built in an
experiment, this structure is highly flexible, can undergo large deformations,
and is fully collapsible. We fill each of the cells with particles in
configurations that are derived via discrete element method (DEM) simulations.

Figure 2: All unique $6$ and $7$ particle-filled cells obtained from the DEM
simulations with the particle sizes for each configuration. Lines originating
from particle centers correspond to particle-particle or particle-wall
contacts. The configurations studied in the experiments correspond to the
largest packing fraction and are highlighted in yellow.

### Discrete Element Method Simulations

To enumerate all possible jammed packings within a cell of the tessellation,
we employ discrete element method (DEM) simulations for $N$ monodisperse,
frictionless disks in the geometry shown in Fig. 1(b). Endpoints of a cell are
held fixed while generating a jammed disk packing. Disks interact with each
other via the pairwise, purely repulsive linear spring potential energy:

$U^{pp}_{jk}=\frac{\epsilon_{pp}}{2}\left(1-\frac{r^{pp}_{jk}}{\sigma_{jk}}\right)^{2}\Theta\left(1-\frac{r^{pp}_{jk}}{\sigma_{jk}}\right),$
(1)

where $\epsilon_{pp}$ is the characteristic energy scale of the repulsive
interactions, $r^{pp}_{jk}$ is the distance between the centers of disks $j$
and $k$, $\sigma_{jk}$ is the sum of the radii of disks $j$ and $k$, and
$\Theta(\cdot)$ is the Heaviside step function. The interaction between the
$i$th side and the $j$th disk is also purely repulsive and given by:

$U^{pb}_{ji}=\frac{\epsilon_{pb}}{2}\left(1-\frac{r^{pb}_{ji}}{R_{j}}\right)^{2}\Theta\left(1-\frac{r^{pb}_{ji}}{R_{j}}\right),$
(2)

where $\epsilon_{pb}$ is the particle-wall stiffness, $r^{pb}_{ji}$ is the
shortest distance between the center of disk $j$ and the $i$th side, and
$R_{j}$ is the radius of disk $j$.

We measure the stress tensor $\Sigma_{\alpha\beta}$ in the system via the
Virial expression.
$\Sigma_{\alpha\beta}=\Sigma^{pp}_{\alpha\beta}+\Sigma^{pb}_{\alpha\beta}$
includes two terms representing the particle-particle interactions,
$\Sigma^{pp}_{\alpha\beta}$; and particle-boundary interactions,
$\Sigma^{pb}_{\alpha\beta}$.

The particle-particle stress $\Sigma^{pp}_{\alpha\beta}$ is given by

$\Sigma^{pp}_{\alpha\beta}=\frac{1}{A}\sum^{N}_{j,k}f^{pp}_{jk\alpha}r^{pp}_{jk\beta},$
(3)

where $A$ is the area of the confining boundary, $f^{pp}_{jk\alpha}$ is the
$\alpha$ component of the force on disk $j$ from disk $k$, and
$r^{pp}_{jk\beta}$ is the $\beta$ component of the separation vector from the
center of disk $k$ to the center of disk $j$.

The particle-boundary stress $\Sigma^{pb}_{\alpha\beta}$ is given by

$\Sigma^{pb}_{\alpha\beta}=\frac{1}{A}\sum^{4}_{i}\sum^{N}_{j}f^{pb}_{ji\alpha}r^{pb}_{ji\beta},$
(4)

where $f^{pb}_{ji\alpha}$ is the $\alpha$ component of the force on disk $j$
from side $i$ of the boundary, and $r^{pb}_{ji\beta}$ is the $\beta$ component
of the separation vector from the contact point between side $i$ and disk $j$
to the center of disk $j$. From the stress tensor $\Sigma_{\alpha\beta}$, we
obtain the pressure $P=(\Sigma_{xx}+\Sigma_{yy})/2$, the shear stress
$\Sigma=-\Sigma_{xy}$ in simple shear, and
$\Sigma=(\Sigma_{xx}-\Sigma_{yy})/2$ in pure shear.

To generate a jammed disk packing in the tessellation, we start with a dilute
system where disks are randomly placed in one cell and replicated in the rest
of the cells, with the packing fraction at $\phi<10^{-3}$. We then increase
the particle sizes by $\Delta\phi/\phi=2\Delta R/R=2\times 10^{-3}$, followed
by energy minimization using the fast inertia relaxation engine (FIRE)
algorithm [18]. After energy minimization, we measure the internal pressure
$P=(\Sigma^{pp}_{xx}+\Sigma^{pp}_{yy}+\Sigma^{pb}_{xx}+\Sigma^{pb}_{yy})/2$
for any given cell. Since all cells start with the same initial configuration,
the pressure, $P$ in any cell is the same. If $P$ is smaller than the target
pressure, $P_{t}=10^{-7}$, we grow the particles by $\Delta\phi$ again and
apply energy minimization. If $P>P_{t}$, we then return to the disk and
boundary configuration before the last growth step and increase $\phi$ by
$\Delta\phi/2$. We repeat this search procedure until we reach a state with
$|P-P_{t}|/P_{t}<10^{-4}$. We generate $10^{5}$ jammed packings for each $N$
with different random initial configurations and find all the unique particle-
filled cells. The unique particle configurations for $6$ and $7$ particle
systems obtained from the simulations are shown in Fig. 2.

### Experimental design

The experimental system is created by 3D printing the tessellation and
particles. From the simulated jammed configurations, we choose the particle
arrangements with the largest packing fractions for $6$ and $7$ particles per
cell, which are highlighted in Fig. 2. Even when we directly translate these
simulated configurations to experiments, there are several differences between
the simulations and the experimentally recreated packings. Instead of growing
particles inside a cell, we place them by hand, leading to differences in
particle positions and geometry of the cells. Additionally, due to the
presence of a large number of zero-energy modes, each cell can have a slightly
different geometry. These slight deviations from the designed configurations
are enough to un-jam the whole system. When we apply compressive or shear
deformations to the system, we notice that the initial response at small
strains is dominated by frictional interactions, but as we increase the
strain, the system reaches a jammed state.

The different particle configurations we study are shown in Fig. 1. We build
three designed configurations with $6$ and $7$ particles in each cell.
Additionally, we also compare these results to a system with a large number of
particles ($25$ particles) in random configurations per cell, and systems with
a single particle per cell. For the single-particle systems, we study the
effect of the variation of particle size on the mechanical response of the
tessellation. Fig. 1(a) shows the tessellation filled with a configuration of
$7$ particles in each cell. Our experimental design is 2D, and both the
tessellation walls and the disk-shaped particles have a height of
$10$\mathrm{mm}$$ in the third dimension. The tessellation and particles are
made of Polylactic Acid (PLA), 3D printed using a Prusa i3 MK3S printer and a
print infill of $20\%$. The experiments are conducted with the tessellation on
a horizontal surface of Polyethylene terephthalate (PET). The PET sheet is
covered with a thin layer of corn starch to minimize any frictional effects
between the particles and the substrate. The magnitude of frictional forces
between the experimental system and the substrate is estimated by measuring
the minimum force required to translate a filled tessellation across the
substrate and is $\mathcal{O}(1$\mathrm{N}$)$. Details on the dimensions and
geometry of the walls, as well as the particles, are included in the SI.

##  3 Results

### Bulk modulus

Figure 3: (a) Bulk compression experiments in the configuration with 7
particles per cell. A cord going around the tessellation is pulled under
tension and the resulting change in area gives the bulk modulus of the system.
When under pressure, particles and cells rearrange to create different contact
networks. (i) System before a rearrangement event—yellow lines show contacts
that disappear after rearrangement. (ii) System after a rearrangement—blue
lines show new contacts formed after the rearrangement. (b) Cord tension vs.
compressive area strain of the annulus. The slopes of the curves give the bulk
modulus. (c) Bulk modulus at different external pressures. (d) Individual
measurements and average for the $7$ particle configuration. Data highlighted
in black to show two different behaviors: particle compression (large slope)
and rearrangements (small slope). (e) Measure of how much strain change in the
$7$ particle system occurs from rearrangements. Bar chart derived from plot
(d) shows the average fraction of strain change that occurs due to particle
compression at different pressures. Bar chart is overlayed by the bulk modulus
as a function of pressure. (f-h) Same plot for the 6 and 25 particle
configurations.

In the 2D tessellation, the bulk modulus is defined as
$B=-\frac{dP}{d\gamma_{area}}$, where $P$ is the external pressure and
$\gamma_{area}$ is the area strain.
$\gamma_{area}=\frac{A^{0}-A^{\prime}}{A^{0}}$, where $A^{0}$ is the initial
area of the inner cavity of the tessellation and $A^{\prime}$ is the internal
area after applying external pressure, $P$. The experimental setup to measure
the bulk modulus is shown in Fig. 3(a). To simulate external pressure, we run
a cord around the outer edge of the tessellation and apply a known tension to
the cord. This cord tension $(T)$ exerts a normal force on the outer edge of
the tessellation, which acts as a simulated external pressure $(P)$. The
relation between $T$ and $P$ is derived in the SI and is given by
$P=\frac{Tf}{Rh}$, where $f$ is the fraction of the outer circumference that
is in contact with the cord, $R$ is the outer radius, and $h$ is the height of
the tessellation in the third dimension.

The system responds to external pressure by decreasing its internal area. As
the external pressure increases, we see intermittent particle rearrangements
in the tessellation cells. One such rearrangement is shown in Fig. 3(a)i-ii.
During a rearrangement event, certain particle contacts that existed in a
configuration are broken and new contacts are formed. These rearrangements,
which change particle-particle as well as particle-wall contacts, often modify
the shape of individual cells, which subsequently changes the internal area of
the tessellation.

To measure the pressure response of the system, we quasi-statically increase
the applied tension in the cord and measure the change in the area of the
tessellation. Fig. 3(b) shows the cord tension in the system as a function of
compressive strain. As expected, higher tension in the cord leads to higher
compressive strains. The maximum compressive strain that the system can
undergo depends on the packing fraction $(\phi)$ of the configuration. Among
the configurations studied, $\phi$ increases slightly with increasing numbers
of particles in the cells
$(\phi_{6p}=0.699,\phi_{7p}=0.747,\phi_{25p}=0.763)$. Since the tension in the
cord is proportional to external ‘pressure,’ the local slope of this curve at
any given value of pressure (tension) is a measure of the bulk modulus of the
system at that pressure. Fig. 3(c) shows bulk modulus as a function of
external pressure. Both pressure and bulk modulus are normalized by the
average Young’s modulus of the particles. Details on the estimation of Young’s
modulus are in the SI.

As seen in Fig. 3(c), the bulk modulus of each particle configuration is
different and the bulk modulus can change non-monotonically with pressure,
although it mostly increases with increasing pressure. To better understand
the relation between bulk modulus and particle rearrangements, we study the
7-particle configuration as a sample system. Fig. 3(d) overlays the averaged
data (shaded region) and individual trials (curves with points) of pressure
vs. compressive area strain. One randomly chosen curve is highlighted in
black. At very small strains ($\gamma_{area}<1\%$), almost no force is
required to strain the system because the particles are unjammed. At the onset
of jamming, each experiment presents in a staircase-like pattern, which
captures compression regimes where the particles compress against each other
(vertical lines) and undergo rearrangement events (horizontal lines). Studying
each compression and rearrangement event (Fig. 3(d) inset), we can quantify
the contributions to the strain change as coming from compression
$(\gamma_{C})$ or rearrangements $(\gamma_{R})$. Fig. 3(e) shows the strain
from compression, $\frac{<\gamma_{C}>}{<\gamma_{C}>+<\gamma_{R}>}$, as a
function of pressure in the form of a bar chart, where $<\gamma_{C}>$ and
$<\gamma_{R}>$ are averages over 10 experiments. Fig. 3(e) also shows the bulk
modulus at corresponding pressures, which is highly correlated with the
rearrangement data. This correlation persists in all the multi-particle
configurations we studied, as shown in Fig. 3(f)-(h), implying that particle
rearrangements decrease $B$ locally. However, rearrangements often lead to a
more stable packing, which leads to an overall increase in $B$ with pressure.
At higher pressures, the correlation between rearrangements and bulk modulus
goes down, as seen somewhat in the 7-particle case (Fig. 3(c)) but more
acutely in the 25-particle case (Fig. 3(h)). We suspect that at very high
pressures, there are fewer rearrangements possible and that packings with more
particles find increasingly stable configurations that contribute to $B$.

### Shear modulus

To measure the shear modulus of our tessellated granular metamaterial, we fix
the straight outer wall element of the bottom cell and apply a tangential
force on the straight outer wall of the top cell, as shown in Fig. 4(a). In a
simple shear measurement of a continuum material, the horizontal displacement
of the top wall is proportional to the applied simple shear strain. In our
system, simple shear strain is difficult to measure because a tangential force
on the top wall rotates and deforms different parts of the annulus in
different ways. Therefore, we fit an ellipse to the inner cavity of the
annulus and use its dimensions to calculate the shear strain applied to the
system. The shear strain in the system is given by
$\gamma_{shear}=\frac{a-b}{\sqrt{2}r}$, where $a$ and $b$ are the major and
minor axes of the ellipse fitted to the inner cavity and $r$ is the radius of
the initial circular cavity. The shear stress is given by
$\sigma=\frac{F}{2R\times h}$, where $F$ is the applied shear force and
$2R\times h$ is the cross-sectional area of the system. Finally, we arrive at
the shear modulus, which is given by $G=\frac{d\sigma}{d\gamma_{shear}}$. Just
as for the bulk modulus, $B$, the reported values of $G$ are normalized by
Young’s modulus of the particles. The relations for shear strain, shear
stress, and shear modulus are derived in the SI.

Figure 4: Force response to applied shear strain at different external
pressures. (a) 7-particle system, (b) 25-particle system with random
configurations in each cell, (c) 6-particle configuration I, (d) 6-particle
configuration II, and (e) 25-particles per cell in random configurations. (f)
Shear modulus at different external pressures.

The shear response of our system has two distinct regimes, as shown in Fig.
4(b-e). At small shear strains, the system is unjammed and the response is
dominated by particle-particle and particle-cell frictional interactions. Upon
shear-jamming, the tessellation requires substantially larger forces to
further shear. As expected, as the external pressure on the tessellation
increases the tessellation shear jams at smaller strains.

To estimate the shear modulus of each configuration, we measure the slope of
the curves in Fig. 4(b-e) in a strain regime where the system is shear jammed,
at force values between $2$\mathrm{N}$$ and $10$\mathrm{N}$$. The lower force
limit of $2$\mathrm{N}$$ is to ensure that the shear modulus measurement is in
the shear jammed regime and larger than the frictional forces between the
tessellation and the substrate. Fig. 4(f) shows the shear modulus for each
configuration as a function of pressure, averaged over five measurements. The
shear modulus for all of the studied configurations decreases monotonically
with pressure, which is uncharacteristic of granular materials.

Prior work has shown that, for a given jammed packing, $G$ decreases with
pressure until there is a change in the contact network [19, 20]. In our shear
experiments, the $6$-particle configurations do not undergo particle
rearrangements once the system has been shear jammed. The $7$\- and
$25$-particle systems show a few particle rearrangement events under high
shear forces, but these minor rearrangements do not change the contact network
substantially. Thus, all of the systems show a decrease in $G$ as a function
of pressure. Note that the $25$-particle system is unlike the $6$\- and
$7$-particle designed systems, as the initial particle configurations are
different in each cell and also different for each shear experiment. We
correlate this more random nature of the $25$-particle system with the larger
variation in $G$ (relative to the other systems with few particles per cell),
especially at lower pressures.

### Single particle per cell

Figure 5: One particle system. (a) Force response to bulk compression for
single particle systems with different packing fractions. Force vs shear
strain at different external pressures for increasing packing fraction: (b)
$\phi=0.730$ (particle diameter = $44$\mathrm{mm}$$) (c) $\phi=0.746$
(particle diameter = $44.5$\mathrm{mm}$$) (d) $\phi=0.763$ (particle diameter
= $45$\mathrm{mm}$$). (d) Shear modulus at different external pressures. (e)
Bulk modulus as a function of external pressure. (f) Shear modulus as a
function of pressure. Rearrangement event in $\phi=0.730$ system. Images show
configurations (g) before a rearrangement (h) after a rearrangement. Dashed
black line shows new contacts and solid blue line shows the initial geometry
of the cells.

For a better understanding of our tessellated granular material, we
investigate a simpler system with a single particle in each cell. This
configuration has minimal direction dependence and the only rearrangements
that take place involve the particle and tessellation walls. We study this
one-particle-per-cell system by varying a single parameter, the particle size.
By increasing the particle size, we increase the packing fraction in each
cell. As expected, we observe that varying the packing fraction substantially
changes the bulk and shear moduli of the tessellation. By increasing the
particle size and packing fraction, the overall flexibility of the
tessellation decreases, which then limits the maximum compression or shear the
system can undergo.

Nevertheless, even a single-particle-per-cell yields similar $B$ and $G$
trends to what we observed for the multi-particle systems. As shown in Fig.
5(a), as we increase the particle size the forces required to compress the
system increase. Looking again for a relation between $B$ and particle
rearrangements, we notice that the single-particle systems undergo
rearrangements by changing the particle-wall contacts, as shown in Fig.
5(g-h). In the single-particle setup, as we increase the packing fraction, the
initial system becomes more tightly packed, resulting in fewer possible
rearrangements, and therefore a higher bulk modulus. The bulk modulus measured
from the slopes of the force vs strain curves in Fig. 5(a) are shown in Fig.
5(e). We see that the system with the largest packing fraction, $\phi=0.763$,
which does not undergo any rearrangements, has the largest bulk modulus. The
system with intermediate packing fraction, $\phi=0.746$, does undergo
rearrangements, and the rate of increase of $B$ with pressure notably
increases after the system has rearranged.

The shear response of single-particle systems (Fig. 5(b-d)) show that
tessellations filled with larger particles shear jam at smaller strains. The
effect of external pressure on the shear response also varies with particle
size, as shown in Fig. 5(f). For the smallest particle we tested
($\phi=0.730$), the shear modulus decreases rapidly with increasing external
pressure. For the largest particle we tested ($\phi=0.763$), varying the
external pressure does not change the shear response substantially. We surmise
that, due to increased internal pressure (in the cells) from increased packing
fraction, the material system is less susceptible to an externally applied
pressure, and $G$ thus remains constant over a range of external pressures.

### Ratio of Shear to Bulk Modulus $(G/B)$

Based on the $B$ and $G$ measurements in the previous sections, we calculate
the ratio of the two moduli, $G/B$, for different configurations. $G/B$ values
for multi-particle systems are shown in Fig. 6(a). All of the multi-particle
configurations studied have $G/B=\mathcal{O}(1)$ at zero external pressure and
the value of $G/B$ decreases consistently with increasing pressure.

The relation between $G/B$ and pressure in single particle systems, shown in
Fig. 6(b), varies with packing fraction. The response of low packing fraction
systems $(\phi=0.730$ and $\phi=0.746)$ is similar to the multi-particle
systems. In the previous section, Fig. 5(e) and (f) show that for single
particle systems, $G$ decreases while $B$ increases as a function of pressure,
leading to their ratio, $G/B$ decreasing with pressure. This response is
different for the largest packing fraction, $\phi=0.763$, which starts in a
jammed state and leaves no opportunity for particle rearrangements. Increasing
the packing fraction of a given configuration effectively increases the
internal pressure in each cell. Thus, even when no external pressure is
applied, the system with $\phi=0.763$ is under pressure and therefore has a
small value of $G/B$.

Figure 6: Ratio of shear to bulk modulus $(\frac{G}{B})$ for different
particle configurations. (a) $\frac{G}{B}$ as a function of pressure for $4$
different particle configurations. All of the configurations show a monotonic
decrease in the value of $\frac{G}{B}$ as a function of pressure. (b)
$\frac{G}{B}$ as a function of pressure for a single particle per cell with
increasing size of particles (i.e., packing fraction).

## Discussion

Most prior studies of jammed granular packings have focused on systems with a
large number of particles. In these systems, it is challenging to have the
packing maintain the same interparticle contact network. As a result, studies
with a large number of particles often focus on the ensemble averages of
material properties. We show that by creating a tessellated granular system
with a small number of particles in each cell, we can control the initial
state, as well as limit the possible rearrangements in the system. Not only
has this allowed us to create a material with a $G/B$ lower than most
materials, but we have been able to reverse the typical pressure-dependent
shear response in a granular material.

We were motivated to design a material with a low $G/B$ for next-generation
atmospheric diving suits (ADS). ADS have traditionally been designed as hard-
shell submersibles, with complex anthropomorphic joints to allow articulation
while maintaining an internal pressure of 1 atm. Despite significant progress
in ADS, current designs are bulky and limit a diver’s mechanics of motion.
Ambient pressure diving suits, such as scuba gear, allow for much greater
diver maneuverability, but cannot protect the diver from extreme pressures and
from the associated physiological problems. Therefore, new materials that are
flexible but resistant to pressure gradients are needed to develop new ADS.
Such material innovation would enable ADS that allow for free bending and
twisting at body joints, while simultaneously protecting the diver from the
hydrostatic pressures at large depths. This target application further
motivated our annulus tessellation design, as it is a 2D representation of
what can later be developed into a 3D wearable sleeve or joint.

This work opens further questions regarding the design and performance of
tessellated granular metamaterials. An exciting challenge is to design and
build such a system in three dimensions. Our results suggest further studies
to understand the limits of tessellated granular metamaterial tunability,
cyclic stability, and reversibility. Herein, we focused on monodisperse
packings of circular disks with repeated configurations in each tessellated
cell, but the behavior of granular systems depends on particle properties [21,
22]. Future inquiries should include additional particle parameters such as
particle shape, stiffness, and polydispersity, which would allow for further
tuning of the mechanical response of tessellated granular metamaterials.

## Acknowledgements

This work was supported by the Office of Naval Research under Grant No.
N00014-20-1-2640.

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