Squish Jamming
Samuel Poincloux1, ∗ and Kazumasa A. Takeuchi1, 2, †arXiv:2404.09773v1 [cond-mat.soft] 15 Apr 2024

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1
Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
Institute for Physics of Intelligence, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan(Dated: April 16, 2024)

A wide range of disordered materials, from biological to geological assemblies, feature discreteelements undergoing large shape changes. How significant geometrical variations at the microscopic
scale affect the response of the assembly, in particular rigidity transitions, is an ongoing challenge
in soft matter physics. However, the lack of a model granular-like experimental system featuring
large and versatile particle deformability impedes advances. Here, we explore the oscillatory shear
response of a sponge-like granular assembly composed of highly compressible elastic rings. We
highlight a progressive rigidity transition, switching from a fluid-like to a solid-like response byincreasing density or decreasing shear amplitude. The rearranging fluid state consists of crystal
clusters separated by melted regions; in contrast, the solid state remains amorphous and absorbs all
imposed shear elastically. We rationalise this transition by uncovering an effective, attractive shear
force between rings that emerges from a friction-geometry interplay. If friction is sufficiently highcompared to shear, the extent of the contacts between rings, captured analytically by elementarygeometry, controls the rigidity transition.

Particulate assemblies, such as granular media, canswitch their mechanical behaviour between a solid anda fluid-like response depending on their density, appliedstress or interactions. This solid-fluid transition is oftremendous importance across length scales and disciplines, with the elemental particles varying in size, shape,and stiffness. Such jamming or yielding transitions are,for instance, underlying the onset of geological flows [1],the transport and clogging in industrial food processing[2], and are at the heart of numerous biological processes[3].
For rigid particles without internal degrees of freedom,
the rigidity transition is mainly controlled by the average
number of constraints imposed by the number of neighbours [4], including for non-spherical particles [5]. Even
when particles are softer and can moderately deform theirshape upon contact, such as emulsions or foams, the number of contacts remains an essential factor [6–8]. However, particle deformations may reach up to the order ofthe particle size for a broader class of materials, from biological tissues to porous soils. By adapting their shape
to external constraints, such squishy particles [9] acquire
additional degrees of freedom, potentially affecting thefluid-solid transition ingredients.
Recent works highlight the role of shape and deformability in the jamming transition of highly deformable particles, both in thermal [10, 11] and athermal [12–14] assemblies. Dedicated tools have also been developed toquantify the large shape changes that cells undergo whileembedded in a deforming tissue [15]. The shape of thecells in epithelial tissues is one of the essential parameters governing a rigidity transition [16–18]. In geological flows, the shape of constitutive particles undergoes
significant changes primarily via breaking [19], which inturn have a critical effect on the rheology [20] and jamming transition [1, 21]. With very soft grains that do notbreak, compaction far above jamming highlights that not
only the coordination number increases with compaction,
but also the size of the contacts [22, 23].
Despite these recent advances, a clear picture of how
extreme deformability affects the fluid-solid transition isstill lacking. It is an ongoing challenge in soft matterphysics and related fields [24]. Although advanced numerical methods, such as the vertex model [18], the material point method [25] and phase field approach [26, 27],
have been used, numerical investigations remain incredibly challenging owing to the non-linear couplings between geometry and interactions at the local scale, anda large number of elements. As such, a simple experimental model featuring large deformation at the grain
scale would be essential to explore this question. One
may leverage the recent advances in granular metamaterials [28] and metafluids [29], whose properties are governed by geometrical non-linearities at the particle scale,to control grains compressibility.
Here, we introduce assemblies of compressible elastic
rings as an experimental model to explore the role oflarge grain deformations in the rheology of disordered
media. Via oscillatory shear experiments, we highlighta solid-fluid transition controlled by the shear amplitude
and the density. While the transition is characterisedby the absence or presence of structural rearrangements,
compression-induced large geometrical deformations help
the assembly absorb shear without rearranging in thesolid phase. Finally, we argue that this transition ismainly driven by the size of the frictional contacts between the compressed rings via effective adhesive interactions under shear, which stem from the interplay between
geometrical deformations and friction.

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Fluid-solid transition in a sheared assembly ofcompressible rings

We consider a two-dimensional assembly of frictional,
elastic, and ring-shaped grains (see Fig. 1a and Supplementary Video 1). The assembly is compressed into an
imposed area and sheared by oscillatory displacementof the top boundary as shown in the setup schematicin Fig. 1a. The rings are all identical with a thickness
t = 1.5 mm, radius r = 3.3 mm and height h = 10 mm.
Rings are fabricated via moulding of polymerising liquidsilicone (see Methods and Supplementary Fig. 1), thencoated with talc powder to avoid adhesion and ensure
dry frictional interactions. The compression response of
the rings to a loading-unloading cycle (Fig. 1b) attests toa perfect elasticity with no remaining permanent deformations and high reproducibility, and is well recoveredby the finite element method (FEM) (see Methods fordetails). In addition, the onset of the compression response features a small stiffness decrease (Fig. 1c). Deformed rings being slightly softer, a compressed assembly of such rings develops heterogeneous deformations
as shown in the top view displayed in Fig. 1d (see also
Supplementary Video 1). This heterogeneous deformation upon compression is reminiscent of the mechanics
of cellular materials [30–32], including solid foams [33],with the critical difference that here the unit cells do notadhere each other and can therefore rearrange.
The assembly’s response to shear is explored by systematically varying the shear amplitude γ0 and the density of rings Φ. The oscillatory shear γ is imposed ata fixed maximum shear rate of γ˙0 = 1.33 × 10−2 s−1 ,low enough to ensure a quasi-static regime where inertial
or vibrational effects are negligible. The density is adjusted by varying the number of rings in the shear celland quantified by Φ = N πrh2 /A with rh = r + t/2 theexternal radius of the rings, N the number of rings and A
the area of the shear cell. This is the density that would
correspond to the undeformed area of the rings, and we
focus on the regime above the hexagonal packing densityπ≈ 0.91. Initialisation is made by dispersingΦh = 2√3
the rings in the cell area and then manually lowering andfixing the top arm at the desired height. The ring assembly is, therefore, initially in a mechanically quenchedstate. An acrylic plate on top of the assembly holds therings in 2D, preventing out-of-plane buckling.
The mechanical response to shear is probed locally via
image analysis and globally via a force sensor. Picturesfrom an overhanging camera are taken regularly during the cycles, such that the position, shape and neighbours of each ring are identified and tracked all along theshear cycles (Fig. 1e; see also Methods and Supplementary Video 2). A force sensor, placed between the drivingscrew and the top arm, records the force FX necessary
to shear the assembly. The shear force FX for Φ = 1.0

and all investigated shear amplitudes is shown in Fig. 1f.
Unless stated otherwise, the presented measures represent the system in the steady state (see Supplementary
Table 1 for the cycles used in the analysis). The highreproducibility of the shear response in Fig. 1f testifies
that the assembly indeed reached a steady state.
We then assess whether the ring assembly responds
more in a solid-like (jammed) or fluid-like (yielded) fashion. To assess this response, a classic measure for particulate systems under oscillatory loading is the irreversible
displacement of the particles between two consecutive cycles [34–39]. We define ∆u(i, c) as the norm of the irreversible displacement of ring i occurring between thestart and end of cycle c. In Fig. 1d, rings are coloured bytheir value of log (∆u/r) and the associated irreversible
displacement is drawn with a red arrow (magnified by afactor 4). Even at such a high density, irreversible displacements occur but are highly heterogeneous, with values spanning orders of magnitude within the assembly.
To explore how the irreversibility depends on the density and shear amplitude, for each parameter pair (Φ, γ0 )considered, we compute ⟨∆u⟩, the averaged irreversible
displacement over all tracked rings in the steady state.
The resulting map is shown in Fig. 1g. Two zones can be
identified: a fluid-like regime with significant irreversibility, which smoothly transitions to a solid-like zone withbarely any irreversible displacement, by increasing density or decreasing shear (see also non-affine trajectories
in Supplementary Video 3).
The transition between the fluid-like and solid-like features is observed not only in the rings’ local movement
but also via the global mechanical response. From theshear response FX (γ) of the whole assembly, we compute for each cycle the first harmonic of the storage andloss moduli, G′ and G′′ , respectively [40] (see separated
map of G′′ and G′ in Extended Data Fig. 1). Identifying when the ratio G′′ /G′ is greater than 1 is a standard method to evaluate if the assembly yields or remains essentially solid [41]. The map of log(⟨G′′ ⟩ / ⟨G′ ⟩)shown in Fig. 1h, with ⟨G′′ ⟩, ⟨G′ ⟩ the moduli averaged
over the cycles in the steady state, recovers qualitatively
the solid- and fluid-like regions identified above from theirreversible displacements (Fig. 1g). Indeed, the region
with high irreversible displacements coincides with a loss
modulus significantly larger than the storage one and inversely for the dominantly reversible region.
Exploring the mechanical response to shear at very
high density and shear amplitude, we identify a progressive jamming transition of the ring assembly by either
reducing the shear amplitude or increasing the density.
To understand the driving factors of this transition, we
first quantify the structural properties of the assembly.

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Fig. 1 | Fluid-solid transition under oscillatory shear of an assembly of compressible rings. a, Schematic of the experimental setup. A ring assembly is confined within a 2D cell with a fixed area. Oscillatory shear γ = (γ0 /2) sin 2γ˙ 0 t/γ0
is applied to the assembly via lateral motion of the top arm while the bottom one is fixed. Both top and bottom arms aredecorated with half-solid disks of the same dimension as the rings to transmit shear. A transparent acrylic plate placed on top
keeps the rings in the plane. b, Experimental and simulated (FEM) indentation response fy (δy ), and c, associated stiffness
dfy
of a single ring compressed between two flat plates. The experimental curve is the averaged response over five differentdδyrings from the assembly; upon loading and unloading, the shaded area shows the minimal and maximal values reached by therings during the compression cycle. The corresponding FEM response is made via a 3D simulation of the neo-Hookean
material model with Young’s modulus E = 1 MPa and Poisson’s ratio ν = 0.45, reproducing the ring and indentationgeometry. The experiments and FEM, which agree quantitatively, feature a slight softening upon indentation, leading to solid
foam-like deformation heterogeneities upon compression of the ring assembly. d, Top view of the ring assembly (Φ = 1.04 andγ0 = 0.53) and representative of the pictures used for image analysis. Here, each tracked ring is coloured over its detectedarea by the value of its irreversible displacement log(∆u/r). The corresponding displacement is shown by a red arrow(up-scaled by a factor 4). e, For each ring, we measure via image analysis its position (geometric centre), shape, neighbours,
and extent of each contact (length of the blue segments). We define two neighbourhood types: Voronoi neighbours share acommon edge in the Voronoi tessellation, and touching neighbours are in contact with a finite contact extent ℓc . The thinorange silicone layer is applied in the inner part of the ring to help with detection and further imaging processes. See also
Supplementary Video 2. f, Shear response FX (γ) of the assembly for Φ = 1.00 and for shear amplitude γ0 ranging from 0.07
to 0.7. Six cycles in the steady state are displayed for each γ0 . g, Intensity of the averaged irreversible displacementlog(⟨∆u⟩ /r) for varying density Φ and shear amplitude γ0 . The average is calculated over all tracked rings and in the steadystate. h, Map of the ratio between the loss and storage moduli of the assembly, computed from the shear response Fx (γ) andaveraged over the cycles in the steady state. The corresponding Fx (γ) for Φ = 1.00 are shown in f, each coloured by its
corresponding value of log(⟨G′′ ⟩/⟨G′ ⟩). In g,h, and subsequent maps, the white dotted line is a geometric prediction
introduced thereafter (Eq. (1)), but now serves as a reference to help compare the maps and identify solid and fluid regions.

A partially crystallised fluid and an amorphous solid

The presence or absence of irreversible displacements
upon cyclic shear in the steady state highlights that ring
rearrangements and associated structural deformations
play a crucial role in the mechanical response of the assembly [42, 43]. We then measure the structural differences between the solid and fluid regions and how they
evolve from their initial quenched state. With all rings
having the same size, the configuration with the highest density and minimal shape deformation (and hence
minimum elastic energy) is the hexagonal packing. We
thus use the hexagonal bond orientational order Ψ6 as
the structural indicator of the assembly [44]. For eachring j, the hexagonal order is computed by Ψ6 (j) =Pnj 6iαjk(1/nj )| k=1e| with nj the number of Voronoi neighbours and αjk the angle between the line passing throughthe centre of the two rings and the x axis (the directionof the shear). The value of Ψ6 is 1 for a ring in a perfecthexagonal lattice and continuously decreases to 0 as the
structure deviates from the hexagonal one.
As the assembly accumulates shear cycles, we compare
in Fig. 2 (see also Supplementary Video 4) the structural
evolution of configurations for two cases located near theborder between the jammed and yielded regions, witheach case positioned on the opposite side of the border.
After mechanical quenching, both configurations start inan amorphous state with a flat probability distribution
P (Ψ6 ), but their evolution with cycling strongly differ.
In the yielded region, for (γ0 , Φ) = (0.4, 1.00) shown in4number of shear cycle

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0Fig. 2 | Crystallising fluid and amorphous solid. a,Left, probability density profiles of the hexagonal bondorientational order Ψ6 for an assembly at(γ0 , Φ) = (0.4, 1.00) in the yielded region. The profiles areshown for the assembly before shear and after 20, 40 and 60
shear cycles. Right, corresponding snapshots of the assembly
after cycling, where each ring is coloured by its Ψ6 value.
The assembly shows a progressive growth of crystalline
regions separated by melted ones. b, Same quantities as in a
but for an assembly with higher density and same shear
amplitude, (γ0 , Φ) = (0.4, 1.04), now located in the jammed
region. Here, no significant structural changes are observed,
and despite high shear, the assembly remains amorphous.
See also Supplementary Video 4. c, Map of the proportion ofrings belonging to crystal clusters (with Ψ6 > 0.75) in thesteady state. The corresponding snapshots of the assemblies
in their steady states, with the rings coloured by their Ψ6values, are shown on the right for γ0 = 0.25 and all
investigated densities. See also Supplementary Video 5 forthe time evolution of Ψ6 .

Fig. 2a, cycling induces drastic structural changes withthe progressive growth of distinct crystal clusters. The
probability density increases close to Ψ6 = 1 while itdecreases for low values. In contrast, for larger densities within the jammed region (with (γ0 , Φ) = (0.4, 1.06)
shown in Fig. 2b), no significant structural changes occur

upon cycling, and the assembly remains in an amorphous
phase. We then measure the crystallised portion of theassembly, or proportion of rings having Ψ6 > 0.75, in thesteady state over the parameter space (Fig. 2c). All theassemblies in the solid region feature almost no structuralchanges and remain amorphous, while the proportion of
the crystal clusters decreases with increasing density inthe fluid region (Fig. 2c; see also Supplementary Video 5).
Rearranging rings within a given surface area tend to
crystallise to reduce their overall elastic energy, as thehexagonal packing is the most compact structure without shape deformations. However, the growth of crystal
clusters is limited by the presence of rings deformed bycompression, which necessarily exist for high densities
and prevent crystallisation.
The structure of the ring assembly in its steady state
differs drastically between the yielded and jammed regions. The fluid-like region is characterised by hexagonal
crystal clusters separated by a melted, disordered phase.
This partial crystallisation is typical of granular media
under sufficient shear where reorganisations lead to structural changes [45–48]. Even if essentially crystallised,
the yielded phase still displays a fluid-like mechanicalresponse with irreversible displacements (Fig. 1g,h). In
contrast, no crystallisation occurs in the jammed region,
where the assembly remains amorphous almost without
rearrangements. Even under high shear, the rings stick
together to keep the same structure. In the following,
we delve into the dynamics during shear cycles to investigate the interplay between structural and geometricaldeformations of the rings.

Large geometrical deformations absorb shear andhinder rearrangements

We now focus on the deformation of rings during shear
cycles to understand how yielded and jammed regions
respond to shear. The geometry, of the rings√is quantified by the shape index q = (ring perimeter)/ ring area,
a classic quantity used in highly deformable assemblies
like epithelial tissues [16, 17] to assess how far a convex
shape√is from a circle. The shape index takes the valueq = 2 π ≈ 3.54 for a perfect circle and increases when
the ring gets deformed. As the rings are slender objects,
they deform mostly by flexion rather than stretching.
Thus, q is mainly increased by reducing the area while
the perimeter is almost unchanged. This contrasts withemulsions or tissues where most deformations originate
from a change in perimeter and not the area [7, 27]. The
perimeter and area of the rings are obtained via image
analysis, and a Fourier decomposition is used to smooththe pixelated shape and reduce the shape to a few descriptors [27]. The resulting measure of q during a shear
cycle is shown in Fig. 3a for one ring within an assemblywith parameters (γ0 , Φ) = (0.4, 1.00). For each cycle c

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density,Fig. 3 | Competition between geometrical andstructural deformations. a, The geometrical deformationof the rings is quantified by the shape index q. The
evolution of q within a cycle is shown for one ring in a
(γ0 , Φ) = (0.4, 1.00) assembly. Each marker goes with anassociated ring snapshot. We extract q̃ and ∆q, the average
and amplitude variation from the shape trajectory. b,c,
Average shape of the assembly q̃ and shape variation ∆q as
a function of shear amplitude for each density. The values
are averaged over all tracked rings in the steady state. d,e,
Map of the geometrical Sc and topological Tc strains of theassembly. The total strains here are the cumulative sums ofthe norm of the frame-to-frame strains estimated over thewhole assembly and in the steady state.

and tracked ring, we define two quantities from the shapetrajectory: the average deformation q̃ = ⟨q⟩cycle and thegeometrical fluctuation ∆q = max(q)cycle − min(q)cycle .

In the steady state, the mean shape index and fluctuation are then averaged over all tracked rings, andtheir dependence on shear is compared for each density
Fig. 3b,c. As expected, the average geometrical deformation q̃ increases steadily with the density Fig. 3b. For
all densities, increasing shear amplitude drives significant
shape fluctuations during shear cycles, and the higher thedensity, the more pronounced the geometrical changes(Fig. 3c). Thus, a higher density contributes, in addition
to naturally increasing the average geometrical deformation, to promoting shape changes for a given imposed
shear. Ring assemblies, therefore, use their compressibility and shape-changing abilities to absorb imposed exter-

nal shear, and they do so more efficiently as the density
increases.
To grasp the combined role of geometrical changes andstructural rearrangements leading to jammed or yielded
assemblies, we quantify these two aspects by dedicatedstrains derived for epithelial tissue deformations [15]. For
each ring and between each time frame, we track thebond vectors between touching neighbours (Fig. 1e andSupplementary Video 6). The geometrical strain S ∗ isderived from the change in bond length and orientationand quantifies changes in size and shape within the assembly. In contrast, the topological strain T ∗ , based on
the appearance or creation of bonds due to rearrangements, captures the change in the topology of the contact
network and then quantifies the structural deformationsof the assembly. More details on these quantities are
given in the Methods, and their step-by-step derivation
is in the Appendix of Ref.[15]. The strains S ∗ and T ∗ arecomputed between each time frame and for the whole assembly. Their norm is then summed over all theRcycles inthe steadyR states to obtain the quantities Sc = t ∥S ∗ ∥dt
and Tc = t ∥T ∗ ∥dt shown in Fig. 3d,e. From the map oflog Sc over (γ0 , Φ) in Fig. 3d, we observe that geometricalchanges increase significantly with both shear amplitude
and density as observed in Fig. 3c. By contrast, the mapof log Tc in Fig. 3e shows that structural deformations,while increasing with shear amplitude, are diminished byhigher densities. As a result, the region where changes
in the topology of the contact network are significant coincides with the fluid region identified in Fig. 1g,h. Yet,the geometrical deformations vary regardless of the fluid
or solid regions.
In contrast to confluent bi-dimensional active cell models where more significant deformations lead to yielding[18], here, deforming the assembly by increasing the density fosters jamming by absorbing shear and hindering
structural rearrangements. We then investigate the ringring interactions to understand the mechanisms underwhich large shape deformations impede structural ones.

Geometry and friction concur to generate effective
shear adhesion between rings.

We introduce a microscopic ersatz comprising two halfrings sequentially compressed and sheared to measure theinteraction between rings. Then, we will link the sign ofthe shear force, which would promote or prevent structural rearrangements, to the assembly’s fluid-solid transition.
We measure, via FEM and experiments, the shear response of two clamped rings compressed by an amountδy (Fig. 4a and Supplementary Video 7). With FEM, we
systematically vary the frictional interaction via the friction coefficient µ of the sheared interfaces (Fig. 4b) andthe compression δy between the rings (Fig. 4c). Experi-

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1.1-1Fig. 4 | Friction and geometrical deformations produce an effective shear adhesion between the rings. a,
Ring-ring interaction experiments and FEM simulations. Two facing half-rings are compressed over a distance δy and thensheared laterally until they lose contact. Displayed are two FEM snapshots showing the contact pressure between the rings
after compression, before (top) and after (bottom) a lateral shear displacement of x = r. See also Supplementary Video 7.
b,c, The resulting shear interaction force fx is measured as a function of the lateral displacement x. Negative (positive)
values indicate an attractive (repulsive) force between the rings. Experimental responses are averaged over 3 distinct pairs,
and shaded areas represent max and min values obtained from the different pairs. Panel b shows the shear response fx (x) forδy = 950 µm and the friction coefficient µ = 0.1, 0.3, 0.6 varied in the FEM. The experimental response matches that of FEM
for µ = 0.6. Panel c shows the shear response fx (x) for µ = 0.6, for increasing confinement δy = 200, 600, 950 µm. In this highfriction regime, fx remains negative until the separation of the rings. We measure xr (δx , µ), the displacement needed toswitch from attractive (fx < 0) to repulsive (fx > 0) shear interactions. d, Experimental measure of the attractive-repulsive
threshold shear γr (Φ), obtained by translating local quantities (δy , xr ) to global ones (Φ, γr ), together with results from FEM(dashed lines) and the geometrical predictions (dotted lines) for µ = 0.1, 0.3, 0.6. This curve γr (Φ) for µ = 0.6 is the one
drawn in the phase diagrams shown so far (e.g., Fig. 1g,h), marking the fluid-solid transition of the assembly (fluid-like for
γ0 > γr and solid-like otherwise). e, (Φ, γ0 ) map of the difference between the imposed shear displacement x0 and themeasured maximal contact size ⟨ℓmax⟩, averaged over all tracked rings in the steady state. The geometrical prediction andclocal measurements of the attractive-repulsive threshold γr correspond to the region where y0 ≈ ⟨ℓmax⟩. In this high friction
cand large geometrical deformations limit, the extent of the contacts ℓc mainly controls the jamming transition.

mentally, we test rings taken from the assembly by applying the same compression values as in FEM. We find thebest agreement between the experiments and the FEM
for µ = 0.6, constituting a measure of the experimental friction coefficient. In the absence of friction, theshear force fx is instantly positive (repulsive) as elastic
deformations push the rings away from each other. With
friction, however, fx starts negative (attractive) and a finite lateral displacement xr (δy , µ) is required to make itrepulsive (fx > 0). The synergy of geometrical deformation and friction thus produces an attractive interaction,or effective adhesion, between the rings until sufficient
shear is applied. The role of shape change to generate
an effective adhesion under shear is reminiscent of therheology of cohesive granular media [49], where the low
stiffness of the grains also participates in producing ahigher effective adhesion. We argue next that the presence or absence of this effective adhesion under shear isat the root of the assembly’s fluid or solid macroscopic
response.
We now give a geometrical prediction of the finite lateral displacement xr necessary to switch the shear interactions force from attractive to repulsive, i.e., fx (xr ) = 0.
This distance is a function of the initial compression δyand the friction between the rings, µ. Two cases emerge
in the way xr (δy , µ) is determined. In the first case, therings remain in contact when fx switches from negative
to positive (e.g., Fig. 4b for µ = 0.1 and 0.3). In thiscase, assuming Amontons-Coulomb frictional properties
and reducing the contacting interfaces to a point, one candirectly link the sign of the shear force fx to the lateraldisplacement and obtain xr (δy , µ) = µ(2rh −δy ). By con-7
trast, in the second case where fx remains negative until
the rings are fully separated (e.g., Fig. 4c), xr is givenby the extent ofp the initial overlap between the rings:
xr (δy , µ) = 2rh 1 − (1 − δy /(2rh ))2 . Now that we havea prediction for the microscopic ring-ring interactions, wemust translate the compression δy and local lateral displacement x to the global variable density Φ and shear
γ. Assuming an hexagonal structure of density Φh , anisotropic local overlap δy directly translates into a densityby Φ(δy ) = Φh /(1 − δy /(2rh ))2 . For the shear γ, too, byusing its local counterpart, i.e., the local shear imposed
between the two half rings, we obtain γ = x/(2rh − δy ).
Schematics of the underlying geometrical predictions areshown in Extended Data Fig. 2. By combining the expressions of xr (δy , µ) with the expressions of Φ(δy ) andγ(y, δy ), we can finally express the repulsive shear γr at
which the interactions become repulsive as a function of
global variables (γ, Φ), as follows:(qΦΦh − 1 if γr < µ(1)
γr (Φ, µ) =µif γr ≥ µ
In Fig. 4d, we compare γr from the prediction against
FEM and experimental measurements and find a perfectmatch. In the high friction case (γr < µ), the thresholdγr (Φ) is p
simply set by the extent of contacts, yielding
γr (Φ) = Φ/Φh − 1. Experimentally, the explored density range and the estimated friction coefficient µ ≈ 0.6
are such that we remain in this high friction limit. In soft
spheres without shape changes, the jamming transition
also features a scaling between yielding and density [50]but occurs at a much lower density and with a different
scaling.
With repulsive interactions between rings, we expect
structural rearrangements to be promoted and, inversely,
hindered if interactions are attractive and act as an effective adhesive force. The shear γr can then be viewed as ayield shear, where rearrangements leading to fluid-like response are expected for γ0 > γr , and a solid-like jammed
assembly for γ0 < γr . Indeed, the line drawn by the geometric prediction of Eq. (1) qualitatively separates thefluid and solid-like regions in the (Φ, γ0 ) parameter space(Fig. 1g,h), which also accompany significant changes of
the contact networkp
topology (Fig. 3e). This geometricboundary γr (Φ) = Φ/Φh − 1 is set by the extent of
the contact between rings, which increases with density.
To verify that the contact extent is a major factor controlling the fluid-solid transition, we measure ℓmax
, theclargest contact length among all touching neighbours, foreach ring and at each time step. We then compute thedifference between the equivalent local lateralp displacement amplitude x0 = γ0 (2rh − δy ) = γ0 2rh Φh /Φ and⟨ℓmax⟩, averaged in the steady state (Fig. 4e). The geoc
metrical prediction of γr matches with the region wherex0 ≈ ⟨ℓc max ⟩. The fluid-solid regions are then defined bywhether the shear is large enough to overcome the effec-

tive adhesion and move the rings by a distance greaterthan the maximal contact length. Therefore, in contrast
with rigid granular media where the number of contacts
controls the jamming transition [4], the transition withsquishy grains in a high friction limit seems to be strongly
controlled by the extent of the contacts (Extended Data
Fig. 3).

Discussion

We established that large shape deformation may radically modify the mechanisms by which an assembly
yields. At high density, far into the jammed state, thenumber of contacts becomes irrelevant, and their extent, coupled with frictional interactions, becomes crucial. The contact size is intrinsically linked to the geometrical deformations of the elements, so the shape also
has a crucial role in the rigidity transition. Geometry
and friction concur to generate an attractive force between rings under shear, and sufficient shear must be
applied to overcome this effective adhesion. In the highfriction limit, the shear must be enough to separate therings completely to overcome the adhesion; hence, a simple geometric formula (Eq. (1)) sets the transition between repulsive and attractive interactions. Below thistransition (when the interactions are attractive), structural rearrangements are strongly hindered, leading to a
jammed assembly.
However, these results are valid only within significant friction and shear limits. Our geometrical model
implies that a frictionless assembly yields at an infinitesimal shear as γr = µ. By considering only the interactions between two rings, we neglect collective effects,
such as the caging [51], that would require a finite shear
to yield even without friction. How caging is affected bysignificant shape changes is still an open question.
Another striking property of the ring assembly is thedrastic structure change between the fluid and solid state.
Counterintuitively, the fluid phase features highly crystallised portions, while the solid remains in an amorphous
structure. Similarly to the link between active matterand sheared granular media [52], one may also draw aparallel between this fluid-like crystal under shear andactive crystals [53, 54].
From an engineering perspective, assemblies of highly
deformable elements combine frictional interactions andstructural rearrangement to dissipate energy while keeping a significant elastic component [55]. Ring assemblies
then have great potential as a tunable shock absorber,with the density serving as a knob to tune dissipation
and response to a shock, keeping even a high dissipative
potential in the solid and reversible state. To develop
this axis, it would be essential to explore the 3D version
of such assemblies and their response to inertial forces.

8∗

poincloux@phys.aoyama.ac.jp; Current affiliation: Department of Physical Sciences, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara, Kanagawa 252-5258,
Japan†
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Methods

Ring fabrication
Rings are fabricated from silicone elastomer (Elite Double 22, Zhermack) in a urethane mould (Clear Flex 50,
Smooth-On). The ring’s material is chosen to remain
elastic without permanent deformation, even under highand prolonged deformations. The mould material has
been selected for its deformability, which eases demoulding without tearing the rings, and its very low affinity
with silicone, ensuring no adhesion with the rings andreusability. The rings have a thickness of t = 1.5 mm, a
midline radius r = 3.3 mm and height h = 10 mm. To
help with later image analysis, a thin layer (1 mm) ofdyed (Silc Pig Electric Orange, Smooth-On) silicone isapplied on the inner portion of the ring surface (fromr − t/2 to r). This extra layer does not affect the mechanics of the ring (the additional layer is not reproduced

10in FEM but still compares well with the experiments,
Fig. 1b,c). The rings are slightly coated with talc powder to avoid adhesion and ensure reproducible frictional
properties. See Supplementary Fig. 1 for an illustrationof the protocol.
Shear setup
The shear cell comprises a fixed bottom arm and a movable top arm with movements constrained parallel tothe bottom. The two sides of the cell freely rotate respectively to the top and bottom arms via joints of radius 40 mm, and freely slide through the top arm via
roller bearings to adjust their length under large oscillatory shear. The distance between the two side arms is
W = 300 mm, and the one between the top and bottom
arm is T = 75 mm. Both bottom and top arms are decorated by rigid (3D-printed in PLA and then glued) halfdisks of radius rh = r +t/2 = 4.05 mm (external radius of
the rings). The whole setup rests on a fixed acrylic plate,
and the ring assembly is confined vertically by a 3 mm
thick acrylic plate blocked in the vertical direction but
free to move laterally. The top arm imposes shear via a
driving screw controlled by a stepper motor (AZM 46ACTS10R, Oriental Motor). The motor
imposes a cyclic
horizontal motion X = (X0 /2) sin 2Ẋ0 t/X0 , with X0

the displacement amplitude and Ẋ0 = 1 mm/s the maximum shear speed, kept constant for all the amplitudes
considered. The resulting oscillatory shear is γ = X/T .
The density of the rings is calculated by Φ = N πrh 2 /A,
where A = 16500 mm2 is the area of the shear cell, evaluated by taking into account the size of the joints andthe top and bottom granular boundary conditions. With
the highest shear considered, this area varies between
16500 mm2 and 17500 mm2 within a shear cycle (∼ 6% ofvariations). Consequently, densities are defined with uncertainties of ∼ ±0.02. The motor, force sensor, and camera are controlled and synchronised via a custom Labview
program.
Force measurements
All forces reported here are measured with a flat six-axis(forces and moments along the three directions) force sensor (USX10-H10-500N-C, TecGihan) and transferred to
a computer via 2 data acquisition cards (USB-6001, National Instrument). To measure the shear response of
the assembly (Fig. 1f ), the sensor is inserted between
the top arm and the driving screw. The top arm isslightly lifted from the bottom acrylic plate to avoid
measuring the frictional response of the setup. All six
axes of the sensor are recorded at a rate of 20 Hz. The
first harmonic of the loss
R 2πand storage moduli are computed by G′ = (2/πγ0 ) 0 Fy (θ) sin θdθ/W H and G′′ =R 2π(2/πγ0 ) 0 Fy (θ) cos θdθ/W H with θ = (t − tc )2γ˙0 /γ0the normalised time since the start of cycle c [38]. To
measure the compression response of the rings (Fig. 1b,c)and their interactions (Fig. 4b,c), a distinct setup is used
where positions are imposed manually via a micrometricstage and 100 mm long arm levers, and forces are measured via the moments output of the sensor. For the compression response, one free ring is confined between two
rigid walls, while for the interaction experiments, a PLA
3D-printed part holding half of the rings into a groove
clamps the pair of rings. For one interaction experiment,
the six axes of the sensor are recorded continuously at
100 Hz while compression and shear are imposed via manually moving the micrometric stage. The displacement ismade by step of 50 µm every 10 s. For every step, theposition is changed between 0 and 5 s, and the reported
value for this new position is average over the time interval between 7 and 9 s. The arm lever’s high but finite
stiffness is measured in compression and shear and used
to correct the displacement imposed on the rings.
Finite Element Method (FEM)
The mechanical tests using full 3D FEM were performed
with the software COMSOL Multiphysics 6.1. We use
symmetries to model only the necessary portions of thering for the compression (Fig. 1b,c) and shear (Fig. 4)
simulations. The rings are cylinders of height H along
the z axis and of external radius rh in the xy plane. Rings
are compressed and sheared in the xy plane, and we assume stress-free surfaces at z = 0 and z = H. Fromsymmetries, we model only half of the rings, from z = 0
to z = H/2, with a free surface at z = 0 and a mirror
symmetry (no outward displacement) at z = H/2. The
tetrahedralmesh size is fixed and set to a minimum of√
rt/5, chosen to optimise convergence. The material is
the Neo-Hookean model with Young modulus E = 1 MPa
and Poisson’s ratio ν = 0.45. The mechanical tests are
performed via a stationary approach and by imposing
displacements. For the compression tests (Fig. 1b,c), thering is compressed between two flat and infinitely rigid
blocks, and contacts are solved via the Nitsche method.
For the interaction tests (Fig. 4), two facing rings are cut
again along the x direction to form semi-circles. To clamp
the rings, no displacement in the xz plane is imposed on
the cut surfaces of the top ring. The position of the bottom ring, first along y for compression and then x forshearing, is also controlled by imposing the displacementof its cut surfaces. The contacts are also solved using the
Nitsche method, but this time adding a Coulomb frictional interaction term with a unique friction coefficientµ. The resulting shear force fx is then obtained by integrating the σyx component of the stress on the surfaces at
y = 0 of the bottom ring, where the shear displacement
is imposed.
Image acquisition and analysis
Pictures are acquired via a camera (EMVC-CB1400C3
with an EMVL-MP814 lens, Misumi) placed on topof the assembly.
Indirect lighting is obtained byshining a light-emitting diode panel on a whiteboard. Irrespective of shear amplitude, 15 pictures
per shear cycle are taken (30 pictures for (γ0 , Φ) =(0.400, 0.94), (0.533, 0.94), (0.533, 1.00) where tracking is11more challenging), and a 5 s pause between the cycles
is implemented to ensure that each cycle’s first andlast pictures correspond to γ = 0. The pictures consist of 4608 × 1300 pixels, resulting in a resolution of
13.05 pixel/mm. Then, a custom-made Matlab script
processes all the pictures. Using the orange-coloured inner part of the rings, they are detected and separated
via colour and saturation thresholding. A group of pixels within its contour is attributed to each detected ring
and, for instance, used to represent ring-specific quantities via colour coding, as in Fig. 1d. The geometrical centre of the pixels defines then the position of thering, while its shape, or contour, is measured via a circle Fourier decomposition [27]. The polar coordinates
R(θ) of the outermost pixels of the contour are measured with the geometrical centre of the ring as the origin. Using the fast Fourier transform, the Fourier coefficients (ak , bk ) of the contour are measured such that
M
P
R(θ) = rh(ak cos θ + bk sin θ) up to M = 5. In addik=0

tion to the position and shape of the ring, we also detect
their neighbours via two methods. First, we find topological neighbours via their position by performing theDelaunay tessellation. Based on the resulting Voronoi
diagram, we defined Voronoi neighbours as rings sharing a common edge and no farther than 2.5r. From
this Voronoi neighbourhood, quantities such as the orientational bond order parameter Ψ6 are computed (seeFig. 2). The other neighbourhood detection method aims
to detect the rings in contact. A given ring’s contour isdilated by 8 pixels, and its touching neighbours are those
whose contour overlaps. Based on these touching rings,
we then compute the geometrical and topological strains(Fig. 3d,e, detailed below) and the contact extents forFig. 4e. To measure the extent of the contacts ℓc , wecompute the long axis of the overlapping contours. Indeed, each overlap has an approximate ellipsoidal shapewith its long axis along the contacting surfaces (see ringoverlaps in Extended Data Fig. 2b). This measure overestimates the actual contact length because of the dilatation of the central ring, and we found that multiplying
by 0.7 gives a visually convincing measure.
Geometrical and topological strains
To distinguish quantitatively the geometrical deformations (change of shape) from the structural or topologicalones (change in neighbours), we follow the method introduced in Ref. [15]. Initially developed for epithelial tissue
development, this strain quantification can be directly
applied to our ring assembly. The method is describedin great detail in the Appendix section of Ref. [15], sohere, we only give the general methodology and highlight the specificities of our ring assemblies. Strain evaluation is based on the variations of the texture tensor
of the assembly computed from the links between eachring. The texture tensor of the assembly is defined as

M =

P1 P⃗
lik ⊗ ⃗lik , with ⃗lik the vector linking the
2
i

k∼i

position of ring i to its neighbour k, i spans all tracked
ring of the assembly and k all neighbours of ring i and⃗lik ⊗ ⃗lik = ⃗lik⃗lT is the tensor product of ⃗lik by itself. In
ikcontrast with epithelial tissues, ring assemblies are notconfluent, so we must specify what neighbours we consider here. As rings interact only via contact, we consider only touching rings as neighbours to compute thetexture tensor. Between two consecutive time frames,
the strains are computed from the difference of texture
tensor M between the two frames. The difference withinlinks that are preserved between the two frames will be
the base to compute the geometrical strain S∗ , while theappearing and disappearing links due to rearrangements
only (no fusion or division, unlike in epithelial tissues)
will contribute to the topological strain T∗ . To make thedifference in texture tensor dimensionless and endowedwith strain properties, such as insensitivity to rigid motion, a rigorous normalisation detailed in Ref. [15] is necessary. Between two time frames, we can compute twostrain tensors, S∗ and T∗ , rigorously quantifying the geometrical and topological deformation of the ring assembly. To compare how the assemblies deform during shear
cycles as a function of the control parameters (Φ, γ0 ),we take the norm of these strain tensors and sum themover the 6 shear cycles in the steady state. The norm
is the square root of the sum of the squared elementsofR∗
the matrices.
The
resultingquantities,
S
=
∥S
∥dt
c
t
R
and Tc = t ∥T∗ ∥dt, are the scalar quantities shown in
Fig. 3d,e.
Acknowledgments.
This work benefited from numerous enriching discussions
with, but not limited to, A. Ikeda, N. Saito, S. Ishihara, Y. Lou, H. Ikeda, F. Lechenault, C. Scalliet, M.
Henot, I. Cantat, B. Chakraborty, and B. Andreotti.
We thank R. Tosaka for the construction of a prototype of the shear device. This work is supported in partby KAKENHI from the Japan Society for the Promotion of Science (Grant Nos.JP22KF0084, JP20H00128,
JP19H05800, JP24K00593). S.P. acknowledges financial
support from the Japanese Society for the Promotion of
Science as a JSPS International Research Fellow.
Authors contribution.
S.P. and K.A.T. designed the experiments. S.P. performed the experiments, the numerical simulations, thedata analysis, and conceived the model. All authors analysed and discussed the results and contributed to writing
the paper.
Code and data availability
The Zenodo repository 10.5281/zenodo.10961344, contains the experimental data for the force and from image
analysis for ring assemblies and single ring experiments.
The raw images of the first five cycles for γ0 = 0.40 andΦ = 1.00 and the MATLAB code to process them. It also
contains the FEM data and associated COMSOL files.

12

a

-3

b
0.6

-4
0.4
-5

0.2

0

0.95

1 1.05density,

1.1-6

shear amplitude,

shear amplitude,

0.6

-3

-4
0.4
-5

0.2

0

0.95

1 1.05density,

1.1-6

Extended Data Fig. 1 | Loss and storage moduli across the parameter space. The first harmonic of the loss andstorage moduli are shown in a and b, respectively. The moduli are normalised by the Young modulus of the rings, E = 1 MPa.

13
a

b

c

d

Extended Data Fig. 2 | Schematic for the geometrical predictions. a, First case to estimate the lateral displacement
xr which for which the lateral force fx becomes repulsive (fx > 0). The contact is reduced to a single point with anorientation α given by the line linking the two intersection points of the half-circles. Assuming frictional Coulomb interactions,the amplitude of tangential Tt and normal Tn contact forces are linked by the friction coefficient µ. The lateral force fx isthe resultant of the contact forces projected on the lateral axis: fx = (Tn + Tt ) · ex . The expression for xr is then found bysolving the equation fx (x = xr ) = 0. This prediction is valid if the two rings are still in contact at x = xr = µ(2rr − δy ). b, If
the rings are not in contact at x = xr = µ(2rr − δy ), we have to consider the second case where fx remains attractive until
complete separation of the rings. In that case, xr is the no-overlap distance given by the initial contact extent ℓc after
compression. c, To link the global assembly density Φ and the local compression δy , we assume an isotropic and homogeneouscompression of an assembly initially in a hexagonal packing. The density is estimated by considering a central ring and thehexagon with corners at the centre of its 6 neighbours and by taking the ratio between the area of the ring inside the hexagon(1 full ring + 6 third of rings) and the area of the hexagon. In the hexagonal lattice, the hexagon edge is 2rh and reduces to
2rh − δy under compression, while the area of the rings is constant (Φ is estimated from the area of undeformed rings). d,
Shear is a scaleless quantity defined as the ratio between the lateral displacement and the height of the system. The shear
defined for the assembly as γ= X/T is then identical to the one in the local interaction experiments, γ = x/(2rh − δy ).

14

a

5.5

b
0.6

0.4

5

0.2

0

0.95

1 1.05density,

1.1

cshear amplitude,

shear amplitude,

0.6

4.5

1.5

0.4

0

0.2

-0.5

0.95

1 1.05density,

1.1

d

-1

1.5
0.6

1.0
0.4
0.5

0.2

0.95

1 1.05density,

1.1

0

shear amplitude,

shear amplitude,

0.5

0

0.6

0

1

1.0
0.4
0.5

0.2

0

0.95

1 1.05density,

1.10Extended Data Fig. 3 | The average coordination number is irrelevant for highly compressed assemblies. a,
Average of the coordination number Zr of the assemblies in the steady state. The coordination number of a ring is defined as
the number of touching neighbours. No clear tendency with γ0 , Φ, or the fluid/solid regions is noticeable. b, Reproduction ofFig. 4e, showing that the relative value between the shear distance x0 and the averaged maximal contact length ⟨ℓc max ⟩ marks
a clear separation between fluid and solid-like regions. c, local average lateral displacementpx0 imposed to the assembly overthe parameter space (γ0 , Φ). It is a deterministic quantity computed following x0 = γ0 2rh Φh /Φ. d, For each tracked ring inthe steady state, the contact length ℓc with each of its touching neighbours is measured (see Methods). At each time-step, thehighest ℓc of each ring is averaged over all rings, and then all time-steps to give ⟨ℓc max ⟩. While x0 mainly increases with theshear γ0 ,p⟨ℓc max ⟩ is increasing with density Φ. The two values coincide around the fluid-solid separation line predicted byγr (Φ) = Φ/Φh − 1.

Supplementary Material for “Squish Jamming”
Samuel Poincloux∗
Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

Kazumasa A. Takeuchi†

arXiv:2404.09773v1 [cond-mat.soft] 15 Apr 2024

Department of Physics, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan andInstitute for Physics of Intelligence, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan(Dated: April 16, 2024)of shear cycles. The video shows, for all the experiments, snapshots of the assembly at the startof each cycle with each tracked ring coloured byits Ψ6 value. The assembly is darkened when it
reaches its maximal cycling number.

VIDEO CAPTIONS

Supplementary Video 1: Video of the first ten shear
cycles for the assemblies with (γ0 , Φ) = (0.4, 0.94)(top) and (γ0 , Φ) = (0.4, 1.16) (bottom).
Supplementary Video 2: Illustration of the measurements made on the rings, here shown for a single
ring and parameters (γ0 , Φ) = (0.4, 1.0). The shapeis highlighted in red, and the Voronoi diagram is infine white dotted lines. Each touching ring and its
corresponding bond vector is white, while the contact width is blue.
Supplementary Video 3: Non-affine trajectories of
the rings in the steady state. The affine displacement due to the instantaneous shear strain is retrieved for each ring trajectory and time frame
along shear cycles.

Supplementary Video 6: Bond vector dynamics ofthe assemblies. The video shows, for all the experiments, snapshots of the assembly upon cyclingin the steady state. For each picture, the links thatare conserved are coloured in white, the ones disappearing at the next frame in blue, and the ones
appearing from the previous frame in red.
Supplementary Video 7: Video showing the FEM
simulation of ring-ring experiments, with snapshotsof the rings on the left, and the corresponding
shear force fx (x) in the right. The results forµ = 0.3 and µ = 0.6 are shown for a compressionof δy = 950 µm.
SUPPLEMENTARY TABLES AND FIGURES

Supplementary Video 4: An animated version of
Fig.2a,b showing the probability distribution of Ψ6with the number of shear cycles.
Supplementary Video 5: Stroboscopic evolution of
the structure of the assemblies with the number∗†

poincloux@phys.aoyama.ac.jp
kat@kaztake.org

2
a

b

c

d

e

f

Negative mold
(PLA, D printed)

Positive mold(Silicone, Smooth-On)

Negative mold
(Uretane, Smooth-On)

Molded rings(Silicone, Zhermack)

Ring holder and cover(PLA, 3D printed)

Finished rings(Silicone + dye, Zhermack)

Supplementary Fig. 1 | Fabrication process of the rings. A negative mould 3D printed with PLA (a) is used tofabricate a positive mould made of silicone (b). The silicone mould itself is used to fabricate a urethane negative mould (c),in which the rings are made (d). Then, the rings are placed in a 3D printed holder with a cover (e), leaving only the internal
half of the ring top surface visible. A thin layer of dyed silicone is then coated on the surface to give the final rings (f ). Before
experiments, the rings are placed in a sealed plastic bag with talc powder and thoroughly shaken to ensure a homogeneous
coating of the rings. The multiple moulds were used to satisfy the following constraints: the mould needs to be flexible toensure demoulding without breaking the rings, silicone sticks to silicone moulds, but not at all with urethane (also a softpolymer), urethane sticks to everything except silicone. Only the choice of the silicone for the rings and its coloured top is
crucial (Elite Double 22, Zhermack) to ensure no plastic deformation of the rings, even under prolonged and highdeformations. The red scale bar in the pictures measures 10 mm.

γ0 = 0.667(X0 = 50 mm)

Nc = 100
Rtracked = 84.8%

Nc = 15
Rtracked = 91.2%

Nc = 15
Rtracked = 95.9%

Nc = 15
Rtracked = 96.7%

γ0 = 0.533(X0 = 40 mm)

Nc = 100
Rtracked = 92.9%

Nc = 50
Rtracked = 95.9%

Nc = 40
Rtracked = 97.4%

Nc = 45
Rtracked = 97.5%

γ0 = 0.400(X0 = 30 mm)

Nc = 100
Rtracked = 98.3%

Nc = 100
Rtracked = 98.7%

Nc = 60
Rtracked = 99.1%

Nc = 75
Rtracked = 95.3%

γ0 = 0.267(X0 = 20 mm)

Nc = 100
Rtracked = 99.3%

Nc = 100
Rtracked = 98.4%

Nc = 100
Rtracked = 100%

Nc = 100
Rtracked = 99.7%

γ0 = 0.133(X0 = 10 mm)

Nc = 100
Rtracked = 100%

Nc = 100
Rtracked = 100%

Nc = 100
Rtracked = 99.4%

Nc = 100
Rtracked = 99.2%

γ0 = 0.067(X0 = 5 mm)

Nc = 100
Rtracked = 99.7%

Nc = 100
Rtracked = 99.7%

Nc = 100
Rtracked = 100%

Nc = 100
Rtracked = 100%

Φ = 0.94(N = 296)

Φ = 1.00(N = 320)

Φ = 1.06(N = 340)

Φ = 1.12(N = 360)

Supplementary Table 1 | Experimental parameters for the shear experiments. The shear amplitude γ0 is set by
the lateral displacement amplitude of the top arm, X0 , while the density Φ is controlled by the number of rings N . For each
set of the parameters (γ0 , Φ), the total number of cycles Nc (including the transient and the steady state) and the proportionof rings successfully tracked Rtracked are indicated. The last 6 available cycles (for instance, from cycle 45 to 50 if Nc = 50)
are considered to be in the steady state and are used for the analyses presented in this work. For instance, for the parameter(γ0 , Φ) = (0.400, 1.06), the measures are made on the cycle [55, 56, 57, 58, 59, 60]. Progressive and out-of-plane extrusion ofsome rings limits the number of valid cycles for high shear and density. Due to the extreme deformability of the rings, upon
cycling, some rings manage to escape the plane of the assembly despite the cover plate by out-of-plane deformations. In
Supplementary Fig. 2, we verify that the results and following conclusions are not dependent on which cycles we consider as
the steady state and make the measurements.

3
b
-1

0.2

0

-4

0.95

1 1.05density,

1.1

-5

0

0.2

0.95

1 1.05density,

1.1

-1

e

d
1

0.6
shear amplitude,

0.4

0

0

0.4

-1
0.2

0.95

1 1.05density,

1.1

75%
0.4
50%

0.2

0

0.95

1 1.05density,

1.1

f
1

0.6

0

0.4

-1
0.2

-2
0

100%
0.6

shear amplitude,-3

shear amplitude,

-2

0.4

c

0.6

shear amplitude,

shear amplitude,

0.6

1

25%

1
0.6

shear amplitude,

a

0.5
0.4

0

0.2

-0.5

-2
0

0.95

1 1.05density,

1.1

0

0.95

1 1.05density,

1.1-1Supplementary Fig. 2 | The quantities in the (γ0 , Φ) parameter space are not sensitive to the definition of thesteady state. The parameters map presented in the main figures are reproduced here, for the cycle numbers fixed to be 10
to 15 for all the parameters, the highest numbers common to all parameters. a, Irreversible displacement. b, Ratio between
the viscous and elastic moduli. c, Proportion of rings in a crystal cluster. d, Geometrical strain. e, Topological strain. f,
Difference between the imposed lateral displacement and maximum contact length. While some quantitative differences canbe observed in the proportion of crystal (c) and shear moduli (b), conveying that the steady state may not be fully settled forthe cycle numbers chosen here, the tendency and evolution remain essentially the same. The choice of the cycles to take into
account is then not a determining factor in analysing the response of ring assemblies.