Extreme Mechanics Letters 63 (2023) 102055

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Extreme Mechanics Lettersjournal homepage: www.elsevier.com/locate/eml

Tessellated granular metamaterials with tunable elastic moduli
Nidhi Pashine a , Dong Wang a , Jerry Zhang a , Sree Kalyan Patiballa a,f , Sven Witthaus a,b ,∗Mark D. Shattuck e , Corey S. O’Hern a,b,c,d , Rebecca Kramer-Bottiglio a ,aDepartment of Mechanical Engineering & Materials Science, Yale University, New Haven, CT 06520, USA
Department of Physics, Yale University, New Haven, CT 06520, USA
Department of Applied Physics, Yale University, New Haven, CT 06520, USAdGraduate Program in Computational Biology & Bioinformatics, Yale University, New Haven, CT, 06520, USAe
Benjamin Levich Institute and Physics Department, The City College of the City University of New York, NY, New York 10031, USAf
Department of Mechanical Engineering, The University of Alabama, Tuscaloosa, AL 35487, USAbcarticleinfo

Article history:
Received 20 March 2023
Received in revised form 7 June 2023
Accepted 10 July 2023
Available online 17 July 2023
Keywords:
Granular metamaterials
Mechanical metamaterials
Atmospheric diving suita b s t r a c t
Most granular packings possess shear moduli (G) that increase with the applied external pressure, andbulk 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 granularmetamaterials are made from flexible tessellations forming a ring of closed cells, each containing asmall 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 thecells, 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 ≪ 1 at largepressures. 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.
© 2023 Elsevier Ltd. All rights reserved.

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 solidlike [1,2]. At low densities, granular particles can flow past eachother in a fluid-like state; at high densities, they jam into asolid-like state. Even in their jammed state, particles in granular
media can rearrange by rolling and sliding past one another, thuschanging the contact network and the corresponding mechanicalresponse 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 lawswith the packing fraction of the system [3].
The ratio G/B of the shear and bulk moduli quantifies thesolidity of granular materials. A liquid has G/B = 0 whereas
solids have G/B > 0. Moreover, G and B are within an order ofmagnitude of each other for most materials [4,5]. For example,
the shear and bulk moduli of iron are 77.6 GPa and 166 GPa [6],respectively, with a G/B ≈ 0.47. Further, for most metals [7–9],∗ Corresponding author.
E-mail address: rebecca.kramer@yale.edu (R. Kramer-Bottiglio).
https://doi.org/10.1016/j.eml.2023.102055
2352-4316/© 2023 Elsevier Ltd. All rights reserved.

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[3,14–17].
This paper presents ‘‘tessellated’’ granular metamaterials withtunable elastic moduli and G/B values dependent on pressure. The
tessellated granular metamaterials are made of an annulus that isradially 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 ofparticles, we limit the number of possible rearrangements andthus increase our control over B, G, and G/B. Our results show thatthe 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.

N. Pashine, D. Wang, J. Zhang et al.

Extreme Mechanics Letters 63 (2023) 102055

the system. This structure, when built in experiments, displays
flexibility (and collapsibility) at low energy cost because it isunder-coordinated with many unconstrained degrees of freedom.
We fill each of the cells with particles in configurations that arederived via discrete element method (DEM) simulations of 2D
disks bounded by 1D walls.
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 othervia the pairwise, purely repulsive linear spring potential energy:
pp
Ujk

=

ϵpp

pp

(
1−

2

rjk

)2

(
Θ 1−

σjk

pp

rjk

)
,

σjk

(1)

where ϵpp is the characteristic energy scale of the repulsive interpp
actions, rjk is the distance between the centers of disks j and k, σjk
is the sum of the radii of disks j and k, and Θ (·) is the Heaviside
step function. The interaction between the ith side and the jth
disk is also purely repulsive and given by:
pb
Uji

=

ϵpb

pb

(
1−

2

rji

)2

(
Θ 1−

Rj

pb

rji

Rj

)
,

(2)
pb

where ϵpb is the particle–wall stiffness, rji is the shortest distance
between the center of disk j and the ith side, and Rj is the radiusof disk j.
We measure the stress tensor Σαβ in the system via the Virial
pppb
expression. Σαβ = Σαβ + Σαβ includes two terms representpp
ing the particle–particle interactions, Σαβ ; and particle-boundary

Fig. 1. (a) Experimental setup. 3D printed tessellation with flexible joints filledwith particles. The particles as well as the tessellation 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) 7particle 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.

pb

interactions, Σαβ .
pp
The particle–particle stress Σαβ is given by
N

pp

Σαβ =

1 ∑ pp pp
fjkα rjkβ ,
A

(3)

j,k

pp

where A is the area of the confining boundary, fjkα is the αpp

2. Methods

component of the force on disk j from disk k, and rjkβ is the βTessellation geometrycomponent of the separation vector from the center of disk k to
the center of disk j.
pb
The particle-boundary stress Σαβ is given by

Our setup includes a flexible tessellation with walls that canrotate freely about its joints to create a system with a large
number of zero-energy modes. The tessellation design depends
on several factors including the size of the particles, number ofparticles per cell, and packing fraction. We empirically determined the cell size to fit fixed ranges for the above parameters,
which resulted in a table-top tessellation with 10−20 cells. Related work [18] has shown that the properties of tessellated
granular metamaterials do not have a strong dependence on thenumber of cells. As shown in Fig. 1(a), our tessellation 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. According to
Maxwell’s counting argument, a 2D tessellation with N nodes andNb bonds has (2N − Nb ) zero energy modes, including three trivial
modes corresponding to translation and rotation [19,20]. Whenconnected together in an annulus, the tessellation has N = 64
nodes and Nb = 80 bonds, resulting in 45 zero-energy modes in

pb

Σαβ =

4

N

i

j

1 ∑ ∑ pb pb
fjiα rjiβ ,
A

(4)

pb

where fjiα is the α component of the force on disk j from side i ofpb

the boundary, and rjiβ is the β component of the separation vector
from the contact point between side i and disk j to the centerof disk j. From the stress tensor Σαβ , we obtain the pressure
P = (Σxx + Σyy )/2, the shear stress Σ = −Σxy in simple shear,and Σ = (Σxx − Σyy )/2 in pure shear.
To generate a jammed disk packing in the tessellation, we startwith a dilute system where disks are randomly placed in one celland replicated in the rest of the cells, with the packing fraction
at φ < 10−3 . We then increase the particle sizes uniformly by∆φ/φ = 2∆R/R = 2 × 10−3 , followed by energy minimization
using the fast inertia relaxation engine (FIRE) algorithm [21]. Afterenergy minimization, we measure the internal pressure P =
pp
pp
pb
pb
(Σxx + Σyy + Σxx + Σyy )/2 for any given cell. Since all cells startwith the same initial configuration, the pressure, P in any cell is2

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Extreme Mechanics Letters 63 (2023) 102055

Fig. 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 centerscorrespond to particle–particle or particle–wall contacts. The configurations studied in the experiments correspond to the largest packing fraction and are highlighted
in yellow. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

the same. If P is smaller than the target pressure, Pt = 10−7 , wegrow the particles by ∆φ again and apply energy minimization.
If P > Pt , we then return to the disk and boundary configuration
before the last growth step and increase φ by ∆φ/2. We repeat
this search procedure until we reach a state with |P − Pt |/Pt <10−4 . For a given set of boundary conditions (e.g., a trapezoidalcell) and number of disks (N), there are a finite number (Ns ) of
jammed disk packings. Each initial random configuration of disks
maps to one of these Ns packings. There are Ns = 19 distinct
jammed packings for N = 6, and Ns = 16 distinct packings for
N = 7. We carried out simulations for 105 sets of random initial
disk configurations for all N. The frequency with which eachdistinct jammed packing occurs depends on the details of thejammed packing generation protocol (e.g., the packing fraction
increment and rate of energy relaxation) [22]. Since the target
pressure (Pt ) is low, the jammed packings do not depend strongly
on the functional form of the interparticle forces. All the distinct
packings for N = 6 and N = 7 are shown in Fig. 2.

random configurations per cell, and systems with a single particle
per cell. For the single-particle systems, we study the effect ofthe variation of particle size on the mechanical response of thetessellation. 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 havea height of 10 mm in the third dimension. The tessellation andparticles 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 witha 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 isestimated by measuring the minimum force required to translatea filled tessellation across the substrate and is O(1 N). Details on
the dimensions and geometry of the walls, as well as the particles,
are included in the SI.

Experimental design

3. Results

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 inFig. 2. All the particles in the chosen configurations have at least
three points of contact, ensuring that all the particles are experiencing nonzero interparticle forces, are in force balance, and there
are no ‘rattlers’. Even directly translating these simulated configurations to experiments, there are several differences between thesimulations and the experimentally recreated packings. Instead ofgrowing particles inside a cell, we place them by hand, leading todifferences 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. Theseslight 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 responseat 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

Bulk modulus
In the 2D tessellation, the bulk modulus is defined as B =, where P is the external pressure and γarea is the area
− dγdParea
strain. γarea =A0 −A′, where A0
A0

is the initial area of the innercavity of the tessellation and A′ 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 andapply a known tension to the cord. This cord tension (T ) exerts anormal force on the outer edge of the tessellation, which acts as asimulated external pressure (P). The relation between T and P is
Tf
derived in the SI and is given by P = Rh , where f is the fraction of
the outer circumference that is in contact with the cord, R is theouter 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 suchrearrangement 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,3

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Extreme Mechanics Letters 63 (2023) 102055

Fig. 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 arearrangement 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 externalpressures. (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 fromplot (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 asa function of pressure. (f–h) Same plot for the 6 and 25 particle configurations. (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

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(γarea < 1%), almost no force is required to strain the systembecause 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 thestrain change as coming from compression (γC ) or rearrange⟨γC ⟩ments (γR ). Fig. 3(e) shows the strain from compression, ⟨γ ⟩+⟨γ,⟩

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 quasistatically increase the applied tension in the cord and measure
the change in the area of the tessellation. Fig. 3(b) shows thecord 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 canundergo depends on the packing fraction (φ ) of the configuration. Among the configurations studied, φ increases slightly withincreasing numbers of particles in the cells (φ6p = 0.699, φ7p =0.747, φ25p = 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 afunction 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 nonmonotonically with pressure, although it mostly increases withincreasing pressure. To better understand the relation between
bulk modulus and particle rearrangements, we study the 7particle configuration as a sample system. Fig. 3(d) overlays

CR

as a function of pressure in the form of a bar chart, where ⟨γC ⟩and ⟨γR ⟩ 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 stablepacking, which leads to an overall increase in B with pressure.
At higher pressures, the correlation between rearrangements andbulk 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 findincreasingly stable configurations that contribute to B.
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Extreme Mechanics Letters 63 (2023) 102055

Fig. 4. Force response to applied shear strain at different external pressures. (a) Shear measurement experiment with particles in the 6-particle configuration I ineach cell. (b) Shear response of 6-particle configuration I under different external pressures. The dotted red line is a guide to show the slope in the unjammed
regime and the red dashed line refers to the response in the jammed regime at no external pressure. (c) 6-particle configuration II, (d) 7-particle configuration, and(e) 25-particles per cell in random configurations. (f) Shear modulus at different external pressures.

Shear modulus

modulus for all of the studied configurations decreases monotonically with pressure, which is uncharacteristic of granularmaterials.
Prior work has shown that, for a given jammed packing, G
decreases with pressure until there is a change in the contact
network [23,24]. 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, butthese minor rearrangements do not change the contact network
substantially. Thus, all of the systems show a decrease in G as afunction 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 eachshear experiment. We correlate this more random nature of the
25-particle system with the larger variation in G (relative to theother systems with few particles per cell), especially at lower
pressures.

To measure the shear modulus of our tessellated granularmetamaterial, 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). We measure the shear modulusof the granular metamaterial over a range of external pressures,
which are generated by applying tension to a cord that is wrappedaround the outer edge of the tessellation. In a simple shear measurement of a continuum material, the horizontal displacementof 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
a−b
in the system is given by γshear = √, where a and b are the
2r
major and minor axes of the ellipse fitted to the inner cavity
and r is the radius of the initial circular cavity. The shear stressis given by σ = 2RF×h , where F is the applied shear force and2R × h is the cross-sectional area of the system. Finally, we arriveat the shear modulus, which is given by G = dγdσ . Just as forshear
the bulk modulus, B, the reported values of G are normalized byYoung’s modulus of the particles. The relations for shear strain,
shear stress, and shear modulus are derived in the SI.
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, thetessellation 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 N and
10 N. The lower force limit of 2 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 afunction of pressure, averaged over five measurements. The shear

Single particle per cell
For a better understanding of our tessellated granular material,
we investigate a simpler system with a single particle in eachcell. This configuration has minimal direction dependence andthe only rearrangements that take place involve the particle andtessellation walls. We study this one-particle-per-cell system byvarying a single parameter, the particle size. By increasing theparticle 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 flexibilityof the tessellation decreases, which implies that we need larger
forces to deform the tessellation to the same strain.
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 thatthe single-particle systems undergo rearrangements by changing
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Extreme Mechanics Letters 63 (2023) 102055

Fig. 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) φ = 0.730 (particle diameter = 44 mm) (c) φ = 0.746 (particle diameter = 44.5 mm) (d) φ = 0.763 (particle
diameter = 45 mm). (d) Shear modulus at different external pressures. (e) Bulk modulus as a function of external pressure. (f) Shear modulus as a function ofpressure. Rearrangement event in φ = 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 interpretation of the references to colour in this figure legend, the reader is referredto the web version of this article.)

G/B = O(1) at zero external pressure and the value of G/Bdecreases 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 (φ = 0.730 and φ =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 isdifferent for the largest packing fraction, φ = 0.763, whichstarts 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
φ = 0.763 is under pressure and therefore has a small value ofG/B.

the particle–wall contacts, as shown in Fig. 5(g–h). In the singleparticle setup, as we increase the packing fraction, the initialsystem 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 withthe largest packing fraction, φ = 0.763, which does not undergo
any rearrangements, has the largest bulk modulus. The systemwith intermediate packing fraction, φ = 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 (φ = 0.730), the shear modulus decreases rapidly with increasing external pressure. For thelargest particle we tested (φ = 0.763), varying the externalpressure does not change the shear response substantially. We
surmise that, due to increased internal pressure (in the cells) fromincreased packing fraction, the material system is less susceptible
to an externally applied pressure, and G thus remains constantover a range of external pressures.

Discussion
Most prior studies of jammed granular packings have focusedon systems with a large number of particles. In these systems, it ischallenging 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 asmall number of particles in each cell, we can control the initialstate, 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 typicalpressure-dependent shear response in a granular material.

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 have6

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Extreme Mechanics Letters 63 (2023) 102055

between ∼ 0.5 (for G/B ≈ 1) and ∼ 1 (for G/B ≈ 0.01) withν increasing with pressure. At high pressures, ν ∼ 1, which implies that the material is nearly incompressible. This work opens
further questions regarding the design and performance of tessellated granular metamaterials. Recent related work [18] has shownthat mechanical properties of tessellated granular materials canbe further tuned by changing the particle–wall interactions. An
exciting challenge is to design and build such a system in three
dimensions. Our results suggest further studies to understandthe limits of tessellated granular metamaterial tunability, cyclicstability, 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 [25,26]. Future inquiries should include
additional particle parameters such as particle shape, stiffness,and polydispersity, which would allow for further tuning of themechanical response of tessellated granular metamaterials.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared
to influence the work reported in this paper.

Data availability
Data will be made available on request.
Acknowledgments
This work was supported by the Office of Naval Research, USA
under Grant No. N00014-20-1-2640.
Fig. 6. Ratio of shear to bulk modulus ( GB ) for different particle configurations.
(a)

G
B

Appendix A. Supplementary data

as a function of pressure for 4 different particle configurations. All of

the configurations show a monotonic decrease in the value ofGB

as a function

Supplementary material related to this article can be foundonline at https://doi.org/10.1016/j.eml.2023.102055.

of pressure. (b) GB as a function of pressure for a single particle per cell withincreasing size of particles (i.e., packing fraction).

References
We were motivated to design a material with a low G/B fornext-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 formuch greater diver maneuverability, but cannot protect the diverfrom extreme pressures and from the associated physiologicalproblems. Therefore, new materials that are flexible but resistantto pressure gradients are needed to develop new ADS. Suchmaterial innovation would enable ADS that allow for free bending
and twisting at body joints, while simultaneously protecting thediver 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. We further note that the resultspresented herein are normalized by the constituent material’s
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