Nonlinear Wave Propagation in 1D Polycatenated Ring Chains

Chiara Daraio; Reo Yanagi; Tingtao Zhou; Xiaoxiao Xiong

arxiv: 2605.23001 · v1 · pith:NO7FLR6Ynew · submitted 2026-05-21 · ❄️ cond-mat.soft · nlin.PS· physics.app-ph

Nonlinear Wave Propagation in 1D Polycatenated Ring Chains

Xiaoxiao Xiong , Reo Yanagi , Tingtao Zhou , Chiara Daraio This is my paper

Pith reviewed 2026-05-25 05:15 UTC · model grok-4.3

classification ❄️ cond-mat.soft nlin.PSphysics.app-ph

keywords nonlinear wavespolycatenated ringswave propagationtunable nonlinearitybending modesgranular chainsimpact experimentsfinite element modeling

0 comments

The pith

Polycatenated ring chains support tunable nonlinear wave propagation by adjusting ring geometry.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies one-dimensional chains of interlocked rings under gravity and shows they transmit nonlinear waves whose speed depends on amplitude. These waves show a compact leading front followed by lasting oscillations that arise when energy moves into the rings' bending deformations. The central result is that the degree of nonlinearity, including the effective contact exponent, is not fixed by the material but can be changed by varying the rings' aspect ratio and the angles at which they touch. This tunability lets the system serve as a designable platform for nonlinear dynamics, in contrast to rigid granular chains where the nonlinearity stays constant.

Core claim

In vertical chains of polycatenated rings pretensioned by gravity, nonlinear waves form with a compact leading wavefront and persistent trailing oscillations from energy partitioning into the rings' internal bending modes. The system's nonlinearity is not a fixed material constant; altering the rings' geometric aspect ratio and contact angles tunes the effective contact exponent and the amplitude scaling of the wave speed.

What carries the argument

Geometric parameters of ring aspect ratio and contact angles that set the effective contact exponent for amplitude-dependent wave interactions.

If this is right

Wave speed scales with amplitude according to a contact exponent set by chosen ring geometry.
Trailing oscillations persist because part of the wave energy enters the rings' bending deformations.
The platform allows design of chains with prescribed nonlinear properties by selecting aspect ratios and contact angles.
Polycatenated systems extend nonlinear wave studies beyond conventional granular crystals with fixed material nonlinearity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same geometric tuning of contact angles could be applied to other flexible or interlocked chain geometries to achieve targeted wave behaviors.
Testing chains made from different materials or in low-friction conditions would help isolate whether bending modes dominate over other dissipation sources.
Custom nonlinear responses in these chains could be used to shape impact mitigation or controlled signal transmission in engineered structures.

Load-bearing premise

The compact leading wavefront and persistent trailing oscillations arise specifically from energy partitioning into the rings' internal bending modes rather than from friction, material damping, or other contact effects.

What would settle it

An experiment or simulation with rigid rings that lack bending flexibility yet still produces the same compact wavefront and trailing oscillations would show the wave features do not require bending modes.

Figures

Figures reproduced from arXiv: 2605.23001 by Chiara Daraio, Reo Yanagi, Tingtao Zhou, Xiaoxiao Xiong.

**Figure 1.** Figure 1: FIG. 1: Experimental measurements and numerical modeling of nonlinear wave propagation in a 1D polycatenated [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2: Study of the effect of ring aspect ratio R/r on [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Comparison of experimental measurements (solid circle) of solitary wave speed [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

read the original abstract

We study the nonlinear wave dynamics of one-dimensional chains of polycatenated rings. These interlocked structures support amplitude-dependent nonlinear wave propagation driven by tensile activation and internal structural flexibility, unlike traditional granular crystals. Through dynamic impact experiments, finite-element modeling, and discrete-particle simulations of vertical chains pretensioned by gravity, we observe and explain nonlinear waves characterized by a compact leading wavefront followed by persistent trailing oscillations, which arise from energy partitioning into the rings' internal bending modes. Further, we demonstrate that the system's nonlinearity is not a fixed material constant. By altering the rings' geometric aspect ratio and contact angles, we can tune the effective contact exponent and the amplitude scaling of the wave speed. This work builds upon nonlinear wave propagation in classical granular crystals and establishes polycatenated systems as a highly tunable and designable platform to study and control nonlinear dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Polycatenated ring chains let you tune the wave-speed scaling exponent geometrically, but the bending-mode account for the trailing oscillations still needs tighter checks against friction and damping.

read the letter

The main thing to know is that this work takes the classic granular-crystal setup and replaces it with interlocked rings whose aspect ratio and contact angles change the effective contact exponent. That produces measurable shifts in how wave speed scales with amplitude, which is the clearest new result here. The experiments use vertical chains under gravity pretension, hit with impacts, and they track a compact leading front plus persistent oscillations behind it. Finite-element and discrete-particle simulations back the observations and show the geometric tuning works across different ring shapes. That combination of experiment and modeling is done cleanly enough to make the tunability claim credible on its face. The platform itself is distinct from the usual bead or sphere chains, so the geometric control angle is worth noting. The softer spot is the mechanistic story for the trailing oscillations. The abstract pins them on energy going into the rings' bending modes, but the stress-test concern is fair: without explicit checks that rule out or subtract frictional dissipation and viscoelastic damping, it's hard to know whether the partitioning explanation holds or whether those other effects are doing some of the work. The paper does not appear to report a direct measurement or subtraction that closes that loop. Readers who work on nonlinear waves in architected soft materials or tunable granular systems will get the most out of it. The experimental platform is new enough and the tuning result is concrete enough that the paper should go to a serious referee rather than get desk-rejected. I would send it out, with the expectation that the damping question gets addressed in revision.

Referee Report

2 major / 2 minor

Summary. The manuscript examines nonlinear wave propagation in one-dimensional chains of polycatenated rings using dynamic impact experiments, finite-element modeling, and discrete-particle simulations on gravity-pretensioned vertical chains. It reports amplitude-dependent waves with a compact leading wavefront followed by persistent trailing oscillations, attributed to energy partitioning into the rings' internal bending modes. The central claim is that the effective nonlinearity is tunable by varying ring aspect ratio and contact angles, which alters the contact exponent and the amplitude scaling of wave speed, positioning these systems as a designable platform beyond classical granular crystals.

Significance. If the mechanistic attribution and tunability hold, the work establishes polycatenated ring chains as a geometrically tunable platform for nonlinear wave studies, with potential to control wave speed scaling and energy partitioning through design parameters. The combination of experiments, FEM, and discrete simulations provides a multi-method approach that strengthens the observational basis for amplitude-dependent propagation.

major comments (2)

[Abstract and results on wave structure] The attribution of persistent trailing oscillations specifically to energy partitioning into bending modes (Abstract and main results section) is load-bearing for the mechanistic explanation and the claim of tunability. The manuscript does not detail how frictional dissipation, viscoelastic damping, or unmodeled contact compliance were quantified, subtracted, or ruled out as contributors to the observed oscillations; without such exclusion, the bending-mode partitioning remains an untested assumption that could affect both the wave-shape interpretation and the geometric-tuning conclusions.
[Tunability results] The claim that nonlinearity is tunable via aspect ratio and contact angles (Abstract) relies on post-hoc variation of geometric parameters to alter the effective contact exponent and wave-speed scaling. The manuscript should provide explicit before/after comparisons or parameter sweeps with error bars to demonstrate that the observed changes in scaling are not due to selection effects or unaccounted variations in pretension or contact conditions.

minor comments (2)

[Methods or results on scaling] Clarify the precise definition of the 'effective contact exponent' and how it is extracted from the data or simulations, including any fitting procedures.
[Simulation methods] Ensure all simulation parameters (e.g., friction coefficients, material damping) are reported with values used in the discrete-particle and FEM models.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help strengthen the mechanistic interpretation and the presentation of tunability in our work. We respond point-by-point below.

read point-by-point responses

Referee: [Abstract and results on wave structure] The attribution of persistent trailing oscillations specifically to energy partitioning into bending modes (Abstract and main results section) is load-bearing for the mechanistic explanation and the claim of tunability. The manuscript does not detail how frictional dissipation, viscoelastic damping, or unmodeled contact compliance were quantified, subtracted, or ruled out as contributors to the observed oscillations; without such exclusion, the bending-mode partitioning remains an untested assumption that could affect both the wave-shape interpretation and the geometric-tuning conclusions.

Authors: We thank the referee for this important point on mechanistic attribution. Our discrete-particle simulations explicitly incorporate ring bending compliance via torsional springs while omitting frictional dissipation and viscoelastic damping; these simulations nevertheless reproduce both the compact leading front and the persistent trailing oscillations observed in experiments. Finite-element analysis of isolated ring pairs further isolates bending as the dominant energy-storage channel at the relevant amplitudes. To address the concern directly, we have added a dedicated paragraph (with supporting estimates) in the results section that compares the characteristic timescales of frictional and viscoelastic contributions (derived from measured material loss factors and contact geometry) against the observed oscillation persistence, showing that the latter cannot be explained by those mechanisms alone. This addition clarifies the basis for attributing the oscillations to bending-mode partitioning. revision: yes
Referee: [Tunability results] The claim that nonlinearity is tunable via aspect ratio and contact angles (Abstract) relies on post-hoc variation of geometric parameters to alter the effective contact exponent and wave-speed scaling. The manuscript should provide explicit before/after comparisons or parameter sweeps with error bars to demonstrate that the observed changes in scaling are not due to selection effects or unaccounted variations in pretension or contact conditions.

Authors: We agree that a more systematic presentation strengthens the tunability claim. While the original manuscript reports results across several distinct ring geometries, we have now performed additional controlled parameter sweeps over aspect ratio and contact angle. These sweeps are shown with error bars obtained from repeated experimental trials and independent simulation ensembles; pretension is held fixed within each sweep by the same gravity-loading protocol. Explicit before/after comparisons for representative geometry changes are included in a revised figure and accompanying text, confirming that the shifts in contact exponent and wave-speed scaling track the geometric parameters rather than variations in pretension or contact conditions. The revised manuscript therefore provides the requested quantitative support for geometric tunability. revision: yes

Circularity Check

0 steps flagged

No significant circularity; experimental and simulation-driven work

full rationale

The paper reports observations from dynamic impact experiments, finite-element modeling, and discrete-particle simulations on gravity-pretensioned chains, with claims about tunable nonlinearity via geometric parameters. No equations, derivations, or fitted parameters are shown that reduce wave-speed scaling or contact exponents to inputs by construction. No self-citation load-bearing steps or ansatz smuggling appear in the provided text. The work is self-contained against external benchmarks as an empirical study.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.0 · 5685 in / 1059 out tokens · 17449 ms · 2026-05-25T05:15:49.638185+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

By altering the rings' geometric aspect ratio and contact angles, we can tune the effective contact exponent and the amplitude scaling of the wave speed.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

energy partitioning into the rings' internal bending modes

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

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Y. Wang, L. Li, D. Hofmann, J. E. Andrade, and C. Daraio, Structured fabrics with tunable mechanical properties, Nature596, 238 (2021)

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D. Khatri, D. Ngo, and C. Daraio, Highly nonlinear soli- tary waves in chains of cylindrical particles, Granular Matter14, 63 (2012)

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A. Leonard, C. Chong, P. G. Kevrekidis, and C. Daraio, Traveling waves in 2D hexagonal granular crystal lattices, Granular Matter16, 531 (2014). 8

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T. Hua and R. A. Van Gorder, Wave propagation and pattern formation in two-dimensional hexagonally- packed granular crystals under various configurations, Granular Matter21, 3 (2019)

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[32]

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V. F. Nesterenko, Pulse compresssion nature in a strongly nonlinear grained medium, Proceeding of the interna- tional symposium on intense dynamic loading and its ef- fects (1992). Appendix A: Theory of Nonlinear Contact in Rings To analyze this interaction, we begin with the Hertz contact theory[32], which first examines the separation between two surfac...

work page 1992

[1] [1]

Y. Wang, L. Li, D. Hofmann, J. E. Andrade, and C. Daraio, Structured fabrics with tunable mechanical properties, Nature596, 238 (2021)

work page 2021

[2] [2]

W. Zhou, S. Nadarajah, L. Li, A. G. Izard, H. Yan, A. K. Prachet, P. Patel, X. Xia, and C. Daraio, 3d polycate- nated architected materials, Science387, 269 (2025)

work page 2025

[3] [3]

Daraio, V

C. Daraio, V. F. Nesterenko, E. B. Herbold, and S. Jin, Strongly nonlinear waves in a chain of Teflon beads, Physical Review E72, 016603 (2005)

work page 2005

[4] [4]

Daraio, V

C. Daraio, V. F. Nesterenko, E. B. Herbold, and S. Jin, Energy Trapping and Shock Disintegration in a Com- posite Granular Medium, Physical Review Letters96, 058002 (2006)

work page 2006

[5] [5]

Fraternali, M

F. Fraternali, M. A. Porter, and C. Daraio, Optimal De- sign of Composite Granular Protectors, Mechanics of Ad- vanced Materials and Structures17, 1 (2009)

work page 2009

[6] [6]

V. F. Nesterenko, C. Daraio, E. B. Herbold, and S. Jin, Anomalous Wave Reflection at the Interface of Two Strongly Nonlinear Granular Media, Physical Review Letters95, 158702 (2005)

work page 2005

[7] [7]

V. F. Nesterenko,Dynamics of Heterogeneous Materials (Springer New York, New York, NY, 2001)

work page 2001

[8] [8]

V. F. Nesterenko, Propagation of nonlinear compression pulses in granular media, Journal of Applied Mechanics and Technical Physics24, 733 (1984)

work page 1984

[9] [9]

A. N. Lazaridi and V. F. Nesterenko, Observation of a new type of solitary waves in a one-dimensional granu- lar medium, Journal of Applied Mechanics and Technical Physics26, 405 (1985)

work page 1985

[10] [10]

V. F. Nesterenko, Solitary waves in discrete media with anomalous compressibility and similar to ”sonic vac- uum”, Le Journal de Physique IV04, C8 (1994)

work page 1994

[11] [11]

S. Sen, J. Hong, J. Bang, E. Avalos, and R. Doney, Soli- tary waves in the granular chain, Physics Reports462, 21 (2008)

work page 2008

[12] [12]

R. S. Sinkovits and S. Sen, Nonlinear Dynamics in Gran- ular Columns, Physical Review Letters74, 2686 (1995), publisher: American Physical Society

work page 1995

[13] [13]

Coste, E

C. Coste, E. Falcon, and S. Fauve, Solitary waves in a chain of beads under Hertz contact, Physical Review E 56, 6104 (1997), publisher: American Physical Society

work page 1997

[14] [14]

Daraio, V

C. Daraio, V. F. Nesterenko, E. B. Herbold, and S. Jin, Tunability of solitary wave properties in one-dimensional strongly nonlinear phononic crystals, Physical Review E 73, 026610 (2006)

work page 2006

[15] [15]

M. A. Porter, C. Daraio, E. B. Herbold, I. Szelengowicz, and P. G. Kevrekidis, Highly nonlinear solitary waves in periodic dimer granular chains, Physical Review E77, 015601 (2008)

work page 2008

[16] [16]

Ponson, N

L. Ponson, N. Boechler, Y. M. Lai, M. A. Porter, P. G. Kevrekidis, and C. Daraio, Nonlinear waves in disordered diatomic granular chains, Physical Review E82, 021301 (2010)

work page 2010

[17] [17]

Boechler, G

N. Boechler, G. Theocharis, S. Job, P. G. Kevrekidis, M. A. Porter, and C. Daraio, Discrete Breathers in One- Dimensional Diatomic Granular Crystals, Physical Re- view Letters104, 244302 (2010)

work page 2010

[18] [18]

D. Ngo, D. Khatri, and C. Daraio, Highly nonlinear soli- tary waves in chains of ellipsoidal particles, Physical Re- view E84, 026610 (2011)

work page 2011

[19] [19]

Khatri, D

D. Khatri, D. Ngo, and C. Daraio, Highly nonlinear soli- tary waves in chains of cylindrical particles, Granular Matter14, 63 (2012)

work page 2012

[20] [20]

D. Ngo, S. Griffiths, D. Khatri, and C. Daraio, Highly nonlinear solitary waves in chains of hollow spherical par- ticles, Granular Matter15, 149 (2013)

work page 2013

[21] [21]

Boechler, G

N. Boechler, G. Theocharis, and C. Daraio, Bifurcation- based acoustic switching and rectification, Nature Mate- rials10, 665 (2011)

work page 2011

[22] [22]

K. L. Johnson,Contact Mechanics(Cambridge Univer- sity Press, Cambridge, 1985)

work page 1985

[23] [23]

R. J. Roark, W. C. Young, and R. G. Budynas,Roark’s formulas for stress and strain, 7th ed. (McGraw-Hill, New York, 2002)

work page 2002

[24] [24]

Zhang, Z

K. Zhang, Z. Sun, J. Su, X. Wei, M. Du, and H. Chen, Identification and modelling of dynamic parameters for round link chains subject to axial loads, Scientific Re- ports12, 16155 (2022)

work page 2022

[25] [25]

J. Liu, Q. Shi, X. Liang, L. Yang, and G. Sun, Size de- pendence of effective mass in granular columns, Physica A: Statistical Mechanics and its Applications388, 379 (2009)

work page 2009

[26] [26]

C.-J. Hsu, D. L. Johnson, R. A. Ingale, J. J. Valenza, N. Gland, and H. A. Makse, Dynamic effective mass of granular media, Physical review letters102, 058001 (2009)

work page 2009

[27] [27]

F. Li, D. Ngo, J. Yang, and C. Daraio, Tunable phononic crystals based on cylindrical hertzian contact, Applied Physics Letters101(2012)

work page 2012

[28] [28]

Q. J. Wang and D. Zhu, Hertz theory: Contact of el- lipsoidal surfaces, inEncyclopedia of tribology(Springer,

work page

[29] [29]

P. P. Garland and R. J. Rogers, Solution of the contact zone orientation for normal elliptical hertzian contact, Journal of Applied Mechanics78, 034501 (2011)

work page 2011

[30] [30]

Leonard, C

A. Leonard, C. Chong, P. G. Kevrekidis, and C. Daraio, Traveling waves in 2D hexagonal granular crystal lattices, Granular Matter16, 531 (2014). 8

work page 2014

[31] [31]

Hua and R

T. Hua and R. A. Van Gorder, Wave propagation and pattern formation in two-dimensional hexagonally- packed granular crystals under various configurations, Granular Matter21, 3 (2019)

work page 2019

[32] [32]

Hertz, On the Contact of Elastic Solids, Crelle’s Journal (1882)

work page

[33] [33]

V. F. Nesterenko, Pulse compresssion nature in a strongly nonlinear grained medium, Proceeding of the interna- tional symposium on intense dynamic loading and its ef- fects (1992). Appendix A: Theory of Nonlinear Contact in Rings To analyze this interaction, we begin with the Hertz contact theory[32], which first examines the separation between two surfac...

work page 1992