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arxiv: 2409.07726 · v4 · submitted 2024-09-12 · ❄️ cond-mat.soft · cond-mat.dis-nn

Mechanical hysterons with tunable interactions of general sign

Joseph D. Paulsen This is my paper

Pith reviewed 2026-05-23 21:06 UTC · model grok-4.3

classification ❄️ cond-mat.soft cond-mat.dis-nn
keywords mechanical hysterontunable interactionsrigid barslinear springsnon-reciprocal interactionshysteresiscollective behaviordesigned materials
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The pith

A bar-and-spring mechanism forms a mechanical hysteron whose interactions can be tuned to any sign and made reciprocal or non-reciprocal.

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

The paper presents a physical hysteron built from rigid bars and linear springs that exhibits controllable switching properties and fully tunable interactions between units. It supplies an explicit mapping from geometric parameters to the hysteron thresholds and interaction coefficients, including the choice of sign and reciprocity. This construction turns abstract models of interacting hysterons into a design platform where collective mechanical responses can be set by adjusting lengths and positions. A sympathetic reader would see it as a route to hardware that realizes targeted non-equilibrium behaviors under mechanical driving.

Core claim

The central claim is that a collection of rigid bars and linear springs can be arranged to function as an ideal mechanical hysteron whose individual properties and pairwise interactions are set by a derived mapping from the system parameters, allowing interactions of general sign that are either reciprocal or non-reciprocal and enabling collective behaviors to be targeted by on-the-fly geometric adjustments.

What carries the argument

The mapping from geometric parameters (bar lengths, pivot locations, spring constants) to hysteron switching thresholds and interaction strengths.

If this is right

  • Collective behaviors of multiple hysterons can be programmed by changing geometric parameters without rebuilding the devices.
  • Non-reciprocal interactions become available, producing directed or asymmetric mechanical responses.
  • The platform converts abstract hysteron models into physical designs for materials that sense and respond to mechanical inputs in specified ways.

Where Pith is reading between the lines

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

  • The same mapping could be used to create reconfigurable assemblies whose collective response is altered by external commands that shift a few pivot positions.
  • Scaling the design to three dimensions would let networks of these units perform simple mechanical computation under load.
  • Comparing the ideal mapping against measured prototypes would quantify how far real friction and compliance can be tolerated before the designed interactions deviate.

Load-bearing premise

The bar-and-spring assembly behaves exactly as an ideal hysteron, with no unmodeled effects from friction, gravity, or nonlinear deformations changing the intended properties or interactions.

What would settle it

Build a physical prototype according to the geometric parameters and measure its force-displacement response together with the interaction forces on a neighboring unit to check whether the observed thresholds and signs match the predicted mapping for both reciprocal and non-reciprocal cases.

Figures

Figures reproduced from arXiv: 2409.07726 by Joseph D. Paulsen.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: , as that rotor is bounded by larger angles. In gen￾eral, the pair γ + i,0 (±) and the pair γ − i,0 (±) shift in opposite directions, with the sign of the shift set by gij . Finally, we note that the thresholds and interactions in Eqs. 4 and 5 are independent of w. This means that one can couple together hysterons that are not adjacent, while maintaining the same design space for interactions. We also note… view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: B shows the switching thresholds as measured in our experimental realization of latching, using the pa￾rameters in the caption. The ordering of thresholds in￾deed follow Eq. 6. Plugging the experimental parameters into the kinematic model, we obtain the second set of bars, which are in good agreement with the experiment and follow the same ordering. Our mechanical system allows one to latch and release the… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
read the original abstract

Hysterons are elementary units of hysteresis that underlie many complex behaviors of non-equilibrium matter. Because models of interacting hysterons can describe disordered matter, this suggests that artificial systems could respond to mechanical inputs in precise and targeted ways. Specifying the properties of hysterons and their interactions could thus be a general method for realizing arbitrary non-equilibrium behaviors. Elastic structures including slender beams, creased sheets, and shells are clear candidates for artificial hysterons, but complete control of their interactions has seemed impractical or impossible. Here we report a mechanical hysteron composed of rigid bars and linear springs, which has controllable properties and tunable interactions of general sign that can be reciprocal or non-reciprocal. We derive a mapping from the system parameters to the hysteron properties, and we show how collective behaviors of multiple hysterons can be targeted by adjusting geometric parameters on the fly. By transforming an abstract hysteron model into a physical design platform, our work demonstrates a route toward designed materials that can sense, compute, and respond to their mechanical environment.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a mechanical hysteron built from rigid bars and linear springs. It claims that the design yields controllable hysteron properties together with tunable pairwise interactions of arbitrary sign (reciprocal or non-reciprocal). A mapping from geometric parameters to these properties is derived, and the authors assert that collective behaviors of multiple hysterons can be targeted on the fly by adjusting those parameters.

Significance. If the derived mapping is robust, the work supplies a concrete physical platform that converts abstract interacting-hysteron models into realizable mechanical units. This would be a notable advance for the design of metamaterials and responsive structures that exhibit prescribed non-equilibrium responses.

major comments (2)
  1. [Mapping derivation section] The central derivation of the parameter-to-property mapping (presented in the section following the model definition) is performed under the assumptions of perfectly rigid bars, linear springs, frictionless joints, and negligible gravity. No quantitative estimate or perturbation analysis is given for how small but realistic deviations (joint friction shifting thresholds, gravitational bias on equilibria, or finite bar compliance) would alter the interaction matrix or destroy reciprocity control. This assumption is load-bearing for the claim that geometric parameters can be adjusted on the fly to target collective behavior.
  2. [Collective behavior results] The demonstrations of collective behavior (likely in the multi-hysteron simulations or figures) rely entirely on the ideal mapping. Without either (i) an experimental realization that measures actual switching thresholds and interaction signs or (ii) a sensitivity analysis showing that the targeted behaviors survive realistic perturbations, the controllability result remains unvalidated.
minor comments (2)
  1. Notation for the interaction matrix and the sign of reciprocity is introduced without a compact table summarizing the allowed sign combinations; adding such a table would improve readability.
  2. [Abstract] The abstract states that a mapping is derived but does not indicate whether the mapping is analytic or numerical; a single sentence clarifying this would help readers assess the result at a glance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We respond point-by-point to the major comments below, clarifying the scope of our theoretical work while agreeing to strengthen certain aspects in revision.

read point-by-point responses

  1. Referee: [Mapping derivation section] The central derivation of the parameter-to-property mapping (presented in the section following the model definition) is performed under the assumptions of perfectly rigid bars, linear springs, frictionless joints, and negligible gravity. No quantitative estimate or perturbation analysis is given for how small but realistic deviations (joint friction shifting thresholds, gravitational bias on equilibria, or finite bar compliance) would alter the interaction matrix or destroy reciprocity control. This assumption is load-bearing for the claim that geometric parameters can be adjusted on the fly to target collective behavior.

    Authors: The derivation establishes an exact analytic mapping under the stated ideal conditions, which is the conventional starting point for introducing a new mechanical design platform. These assumptions isolate the geometric mechanism that enables tunable interactions of arbitrary sign. We agree that a discussion of robustness would improve the manuscript. In revision we will add a brief qualitative analysis and order-of-magnitude estimates showing how small friction or compliance shift the thresholds while preserving the ability to control interaction sign and reciprocity. revision: partial

  2. Referee: [Collective behavior results] The demonstrations of collective behavior (likely in the multi-hysteron simulations or figures) rely entirely on the ideal mapping. Without either (i) an experimental realization that measures actual switching thresholds and interaction signs or (ii) a sensitivity analysis showing that the targeted behaviors survive realistic perturbations, the controllability result remains unvalidated.

    Authors: The collective-behavior results consist of direct numerical simulations that implement the derived mapping, thereby confirming that the targeted non-equilibrium responses are achievable when the ideal mapping holds. This constitutes a theoretical validation of controllability. An experimental demonstration lies beyond the scope of the present modeling paper. We will add a sensitivity analysis in the revised manuscript to show that the principal collective behaviors remain accessible under small perturbations, thereby addressing the validation concern within the theoretical framework. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation is a forward mechanical mapping from geometry to hysteron properties

full rationale

The paper presents a physical bar-and-spring design and states that it derives a mapping from system parameters (geometry, spring constants) to hysteron switching thresholds and interaction signs/reciprocity. This is a standard forward calculation from rigid-body mechanics and linear elasticity, not a fit to data, not a self-definition, and not dependent on prior self-citations for its load-bearing steps. No equations or claims in the provided text reduce the output to the input by construction; the mapping is presented as obtained from the model assumptions rather than imposed. The central claim therefore remains independent of the inputs it maps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on the existence of a derivable mapping from geometric parameters to hysteron properties and interactions. No explicit free parameters, axioms, or invented entities are detailed.

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Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    SELClYa3DsXkFo7pRCnNbQ5925w=

    and is denoted by the arrow from −− to ++ in the second transition graph in Fig. 2B. Kinematic model.— We can use torque balance to con- nect the system parameters to the switching thresholds observed in the experiment. First, we assume the system is in a stable state S that specifies the {sj}. It is then straightforward to write down the Cartesian coordi...

    work page

  2. [2]

    J. D. Paulsen and N. C. Keim, Mechanical memories in solids, from disorder to design, Annual Review of Con- densed Matter Physics 16 (2024)

    work page 2024

  3. [3]

    Preisach, ¨Uber die magnetische nachwirkung, Zeitschrift f¨ ur physik94, 277 (1935)

    F. Preisach, ¨Uber die magnetische nachwirkung, Zeitschrift f¨ ur physik94, 277 (1935)

    work page 1935

  4. [4]

    M. L. Falk and J. Langer, Deformation and failure of amorphous, solidlike materials, Annual Review of Con- densed Matter Physics 2, 353 (2011)

    work page 2011

  5. [5]

    N. C. Keim and P. E. Arratia, Mechanical and micro- scopic properties of the reversible plastic regime in a 2d jammed material, Phys. Rev. Lett. 112, 028302 (2014)

    work page 2014

  6. [6]

    Mungan, S

    M. Mungan, S. Sastry, K. Dahmen, and I. Regev, Net- works and hierarchies: How amorphous materials learn to remember, Phys. Rev. Lett. 123, 178002 (2019)

    work page 2019

  7. [7]

    Mungan and M

    M. Mungan and M. M. Terzi, The structure of state tran- sition graphs in systems with return point memory: I. general theory, Annales Henri Poincar´ e20, 2819 (2019)

    work page 2019

  8. [8]

    M. M. Terzi and M. Mungan, State transition graph of the preisach model and the role of return-point memory, Phys. Rev. E 102, 012122 (2020)

    work page 2020

  9. [9]

    N. C. Keim, J. Hass, B. Kroger, and D. Wieker, Global memory from local hysteresis in an amorphous solid, Phys. Rev. Res. 2, 012004 (2020)

    work page 2020

  10. [10]

    N. C. Keim and D. Medina, Mechanical annealing and memories in a disordered solid, Science Advances 8, eabo1614 (2022)

    work page 2022

  11. [11]

    Shohat, Y

    D. Shohat, Y. Friedman, and Y. Lahini, Logarithmic ag- ing via instability cascades in disordered systems, Nature Physics 19, 1890 (2023)

    work page 2023

  12. [12]

    Muhaxheri and C

    G. Muhaxheri and C. D. Santangelo, Bifurcations of in- flating balloons and interacting hysterons, Phys. Rev. E 110, 024209 (2024). 10

    work page 2024

  13. [13]

    Mart´ ınez-Calvo, M

    A. Mart´ ınez-Calvo, M. D. Biviano, A. H. Christensen, E. Katifori, K. H. Jensen, and M. Ruiz-Garc´ ıa, The flu- idic memristor as a collective phenomenon in elastohy- drodynamic networks, Nature Communications 15, 3121 (2024)

    work page 2024

  14. [14]

    N. C. Keim and J. D. Paulsen, Multiperiodic orbits from interacting soft spots in cyclically sheared amorphous solids, Science Advances 7, eabg7685 (2021)

    work page 2021

  15. [15]

    C. W. Lindeman and S. R. Nagel, Multiple memory formation in glassy landscapes, Science Advances 7, eabg7133 (2021)

    work page 2021

  16. [16]

    van Hecke, Profusion of transition pathways for inter- acting hysterons, Phys

    M. van Hecke, Profusion of transition pathways for inter- acting hysterons, Phys. Rev. E 104, 054608 (2021)

    work page 2021

  17. [17]

    Szulc, M

    A. Szulc, M. Mungan, and I. Regev, Cooperative effects driving the multi-periodic dynamics of cyclically sheared amorphous solids, The Journal of Chemical Physics 156, 164506 (2022)

    work page 2022

  18. [18]

    Yasuda, P

    H. Yasuda, P. R. Buskohl, A. Gillman, T. D. Murphey, S. Stepney, R. A. Vaia, and J. R. Raney, Mechanical com- puting, Nature 598, 39 (2021)

    work page 2021

  19. [19]

    Jules, A

    T. Jules, A. Reid, K. E. Daniels, M. Mungan, and F. Lechenault, Delicate memory structure of origami switches, Phys. Rev. Res. 4, 013128 (2022)

    work page 2022

  20. [20]

    Bense and M

    H. Bense and M. van Hecke, Complex pathways and memory in compressed corrugated sheets, Proceedings of the National Academy of Sciences 118, e2111436118 (2021)

    work page 2021

  21. [21]

    Sirote-Katz, D

    C. Sirote-Katz, D. Shohat, C. Merrigan, Y. Lahini, C. Nisoli, and Y. Shokef, Emergent disorder and mechan- ical memory in periodic metamaterials, Nature Commu- nications 15, 4008 (2024)

    work page 2024

  22. [22]

    J. Liu, M. Teunisse, G. Korovin, I. R. Vermaire, L. Jin, H. Bense, and M. van Hecke, Controlled pathways and sequential information processing in serially coupled me- chanical hysterons, Proceedings of the National Academy of Sciences 121, e2308414121 (2024)

    work page 2024

  23. [23]

    Ding and M

    J. Ding and M. van Hecke, Sequential snapping and pathways in a mechanical metamaterial, The Journal of Chemical Physics 156, 204902 (2022)

    work page 2022

  24. [24]

    El Elmi and D

    A. El Elmi and D. Pasini, Tunable sequential pathways through spatial partitioning and frustration tuning in soft metamaterials, Soft Matter 20, 1186 (2024)

    work page 2024

  25. [25]

    C. W. Lindeman, T. R. Jalowiec, and N. C. Keim, Iso- lating the enhanced memory of a glassy system (2023), arXiv:2306.07177 [cond-mat.soft]

    work page arXiv 2023

  26. [26]

    L. M. Kamp, M. Zanaty, A. Zareei, B. Gorissen, R. J. Wood, and K. Bertoldi, Reprogrammable sequencing for physically intelligent under-actuated robots (2024), arXiv:2409.03737 [cs.RO]

    work page arXiv 2024

  27. [27]

    N. C. Keim, J. D. Paulsen, Z. Zeravcic, S. Sastry, and S. R. Nagel, Memory formation in matter, Rev. Mod. Phys. 91, 035002 (2019)

    work page 2019

  28. [28]

    To ensure this is an unstable equilibrium, it is necessary that y > 0, but also that the driving spring ki is suf- ficiently strong compared to any coupling springs that might disrupt this bistability

    work page

  29. [29]

    We do not presently have a precise condition on the pa- rameters that would prohibit additional stable equilibria within the interval (θ− i , θ+ i )

    work page

  30. [30]

    J. A. Barker, D. E. Schreiber, B. G. Huth, and D. H. Ev- erett, Magnetic hysteresis and minor loops: models and experiments, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 386, 251 (1983)

    work page 1983

  31. [31]

    J. P. Sethna, K. Dahmen, S. Kartha, J. A. Krumhansl, B. W. Roberts, and J. D. Shore, Hysteresis and hier- archies: Dynamics of disorder-driven first-order phase transformations, Phys. Rev. Lett. 70, 3347 (1993)

    work page 1993

  32. [32]

    N. C. Keim and J. D. Paulsen, hysteron 0.1, https://github.com/nkeim/hysteron (2020)

    work page 2020

  33. [33]

    L. J. Kwakernaak and M. van Hecke, Counting and se- quential information processing in mechanical metama- terials, Phys. Rev. Lett. 130, 268204 (2023)

    work page 2023

  34. [34]

    M. A. ten Wolde and D. Farhadi, A single-input state- switching building block harnessing internal instabilities, Mechanism and Machine Theory 196, 105626 (2024)

    work page 2024

  35. [35]

    L. P. Hyatt and R. L. Harne, Programming metastable transition sequences in digital mechanical materials, Ex- treme Mechanics Letters 59, 101975 (2023)

    work page 2023

  36. [36]

    Shohat and M

    D. Shohat and M. van Hecke, Geometric control and memory in networks of bistable elements (2024), arXiv:2409.07804 [cond-mat.soft]

    work page arXiv 2024

  37. [37]

    Stern and A

    M. Stern and A. Murugan, Learning without neurons in physical systems, Annual Review of Condensed Matter Physics 14, 417 (2023)

    work page 2023

  38. [38]

    L. E. Altman, M. Stern, A. J. Liu, and D. J. Durian, Ex- perimental demonstration of coupled learning in elastic networks, Phys. Rev. Appl. 22, 024053 (2024)

    work page 2024