Driven Odd Elasticity in Passive Mechanical Metamaterials

Harold S Park; Mohamad Rahimi

REVIEW 3 major objections 4 minor 31 references

Driven Odd Elasticity in Passive Mechanical Metamaterials

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Periodically driven chiral gears make a square-lattice metamaterial respond as an odd elastic solid, with a one-way shear modulus of 3125 GPa that enables non-conservative work and boundary-localized skin modes.

desk verdict A genuinely new chiral-gear ratchet mechanism for driven odd elasticity, plausibly demonstrated in FEM — but A is measured from a single monotonic protocol, the main text admits gear-phase dependence, and no reverse-loading test establishes that A is a true material constant. read the letter →

arxiv 2607.13997 v1 pith:377YJLHA submitted 2026-07-15 cond-mat.soft

Mohamad Rahimi , Harold S Park This is my paper · ORCID

classification cond-mat.soft

keywords oddelasticitymechanicalmetamaterialchiralgearsnon-reciprocitynon-Hermitianskineffectnon-conservativeworkpassivecomponentsdrivensolids

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that a square-lattice metamaterial fitted with chirally toothed gears, driven by a periodic rotation at its boundaries, behaves as an odd elastic solid. The key result is an odd shear modulus A = 3125 GPa that connects normal strain to shear stress while the converse connection is absent, making the effective elasticity tensor asymmetric. Because of this asymmetry, a closed cycle of normal and shear strain produces net non-conservative work, and the lattice's equations of motion become non-Hermitian, yielding boundary-localized skin modes with one-way amplification. The structure is entirely passive: no internal energy source, electronic feedback, or robotic control is required, the driving motion itself supplying the energy. If the claim holds, odd elasticity moves from active matter and feedback-controlled devices to ordinary mechanical metamaterials.

What carries the argument

The central object is the chiral gear attached to the lattice boundary, whose asymmetric tooth geometry (contact lengths l1 ≠ l2) and clearance δ produce a one-way mechanical ratchet. During a normal strain step, the gear momentarily separates from the wall and re-engages at the next tooth, locking a new shear displacement into the lattice; repeated strain steps accumulate shear in a single direction regardless of gear phase. The driven periodic rotation provides the energy input, and the time-averaged response over a driving period defines the effective elastic constants. The gear tooth length l sets the shear increment per strain step, giving a design handle on A.

What would settle it

Measure Δσ_yx/Δϵ_yy under two different gear initial phases or two driving frequencies; if the ratio changes, A is protocol-dependent and the non-conservative-work and skin-effect predictions do not follow. Alternatively, run a closed normal-shear strain cycle and measure the net work; if the net work vanishes once frictional dissipation is subtracted, the ratcheting response is dissipative, not non-conservative elastic work.

Watch

Extended reading notes

Core claim

The central discovery is that a ratcheting gear mechanism converts each increment of normal strain into a locked-in increment of shear strain, so that the time-averaged shear stress grows linearly with normal strain. The authors express the response as C = C_e + C_o, where C_e is the isotropic symmetric part (bulk modulus B = 214.3 GPa, shear modulus G = 115.4 GPa) and C_o contains a single non-zero odd modulus A = Δσ_yx/Δϵ_yy = 3125 GPa. This non-reciprocal term means extension produces shear while shear produces no extension, breaking Maxwell-Betti reciprocity. The mechanism relies on the chiral gear teeth's unequal contact lengths (l1 and l2) to bias the shear direction when the gear dise

Load-bearing premise

The odd shear modulus A is a genuine time-averaged material constant of the driven metamaterial, independent of the gear's initial phase, the driving frequency and amplitude, the friction coefficient, the clearance, and the size of the normal strain increments.

Editorial extensions

If this is right

Odd elasticity can be realized with passive components, eliminating the need for embedded energy sources, feedback loops, or robotic controllers.
The large odd modulus (A = 3125 GPa, about 27 times the shear modulus) means non-conservative work per strain cycle is substantial and should be measurable.
A normal-strain input is converted into one-way shear output, so the structure acts as a mechanical rectifier or strain-direction valve.
The predicted non-Hermitian skin effect makes the lattice a passive directional amplifier/attenuator for elastic waves at open boundaries.
Because A depends on gear tooth geometry, the odd response can be tuned by design rather than by external control parameters.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The main-text evidence for A rests on a single driving protocol; whether A stays constant under changes of driving frequency, amplitude, friction coefficient, or strain-step size is deferred to the SI, so the 'material constant' status of A is the point to scrutinize.
A direct test of reciprocity would be to apply shear strain first and measure normal stress; if any normal stress appears, the tensor is not simply the claimed C_o.
The non-conservative work calculation treats the strain cycle as closed in strain space, but the gear re-engagement involves friction and dissipation; separating genuine odd work from frictional loss would clarify the energetics.
The same chiral-ratchet concept could be ported to other lattices or to three dimensions, potentially producing odd bulk responses or odd terms in other stress components; a topology search over gear tooth shapes could optimize A.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Punchline: the chiral-gear ratchet is a genuinely new mechanism, the FEM demonstration is internally consistent, and the paper is honest — but the odd modulus A = 3125 GPa is not yet established as a material constant, and the statics and dynamics claims ride entirely on that. I agree with your conditional verdict, with two shifts in emphasis.

What's new and good: the mechanism directly targets the open challenge from Huang et al. [27] — driven odd elasticity without electronics or feedback. Physically it makes sense: each normal-strain step detaches the gears, they re-contact at the next tooth, and the lattice picks up a shear increment. The paper also states in the main text, not in the SI, that depending on gear phase a strain step can produce no jump at all or shear in only one direction. That admission is the key to the soft spot.

The load-bearing question is whether A is a modulus or a ratchet artifact. A comes from monotonic positive strain steps only; there is no reverse-loading test, no phase sweep, no step-size or frequency dependence shown. If the shear offset is a one-way configurational change, Eq. (6)'s non-conservative work is the ordinary work of a hysteretic plastic material, and the difference between that and odd elasticity is exactly whether reversing the cycle reverses the work sign. The stress-test note is right on the facts; I'd only soften 'unsupported' to 'not established.' The consequences do follow if A is a true effective modulus — the missing piece is evidence that it is.

The circularity point is real but should be stated precisely. Sections C and D don't validate A; they compute what follows from inserting it into the tensor. That is a standard effective-medium move, not a logical circle. The bigger gap is dynamics: the time-averaged modulus is assumed to govern the instantaneous equations of motion, and no dynamic simulation of the actual metamaterial confirms the skin effect. The NHSE is a prediction of the effective continuum model, not a demonstrated property of the mechanism.

Minor: 'passive' in the title overstates — the components are passive, the system is externally driven. No mesh convergence or UQ in the main text; robustness is deferred to an SI that is not in the version we have.

Bottom line: a serious, honest, citable proposal of a new mechanism whose central premise needs one decisive test — reverse loading at controlled gear phase. Send it to peer review; a good referee will ask exactly the right questions.

Referee Report

3 major / 4 minor

Summary. The authors propose a metamaterial made of a square lattice with chiral gears attached at top and bottom, driven by continuous oscillatory gear rotation. Under stepwise normal strain, the gears ratchet to new tooth positions, producing an increment of time-averaged shear stress. From this they extract an odd shear modulus A = Δσyx/Δεyy = 3125 GPa (Eq. 3), giving an asymmetric elasticity tensor (Eq. 2). The paper then argues analytically that this A yields non-conservative work in a closed normal/shear strain cycle (Section C) and, through the resulting non-Hermitian dynamical matrix, the non-Hermitian skin effect (Section D). The central reported result is a single FEM demonstration, with robustness, parameter sensitivity, and mesh-convergence details deferred to an SI that is not included in the manuscript.

Significance. If the measured A were established as a genuine, protocol-independent effective modulus, this would be a notable advance: it would show driven odd elasticity in a passive mechanical structure without electronic feedback, with concrete static and dynamic consequences. The proposed gear mechanism is physically plausible, and Sections C and D correctly work out the consequences of an elasticity tensor of the form of Eq. (2). However, the paper's own text in Section B admits a phase dependence that undermines the material-constant interpretation of A, and the later sections use that same A rather than providing independent validation. The significance therefore depends entirely on whether the missing SI and additional controls establish A as a well-defined material property.

major comments (3)

[Section B, Eq. (3)] A is measured from a single monotonic sequence of positive normal-strain increments. The main text itself states that 'depending on the phase of the gear with respect to the wall tied to the metamaterial, application of normal strain can only result in either no jump (and no odd shear strain), or odd shear occurring in only one direction.' This directly implies that the measured Δσyx/Δεyy is phase- and history-dependent. No reversed loading, no phase sweep, no frequency/amplitude sweep, and no step-size convergence study are reported in the main text. Until such tests show that A is independent of these control parameters, the time-averaged constitutive tensor in Eqs. (1)-(2) is not established, and all subsequent claims that rely on A are unsupported.
[Sections C and D, Eqs. (6) and (8)] The non-conservative work and the non-Hermitian skin effect are not independent tests of the mechanism; they are analytic consequences of substituting the fitted A into Eq. (2). For example, Eq. (6) gives a nonzero closed-cycle work for any constitutive relation of the form Eq. (4), including a purely plastic or ratcheting material with an asymmetric incremental response. The paper does not report a direct FEM simulation of the full closed strain cycle on the gear-lattice system, nor a direct dynamic simulation of the finite metamaterial with open boundaries. Without such simulations, the paper has not shown that the metamaterial actually produces non-conservative work or the NHSE; it has only shown that if A is a material constant, these effects follow.
[General (Section B, Fig. 2)] The robustness claims are repeatedly deferred to an SI that is not included in the submitted manuscript. The main text does not provide the mesh density, element type, time-step size, averaging window, or convergence criteria for the ABAQUS simulations, and no uncertainty is attached to the reported A = 3125 GPa. Given that this value is more than an order of magnitude larger than the Young's modulus E = 300 GPa, and that the mechanism is discrete (one tooth jump per strain increment), the step-size dependence and averaging procedure are load-bearing. The authors should either include the SI or summarize the required convergence/sensitivity data in the main text.

minor comments (4)

[Equations (1)-(2)] The notation uses both σyx/σxy and εyx/εxy as independent components. Please clarify whether εxy and εyx are engineering shear strains or tensor shear strains, and how they are defined in terms of displacement gradients. As written, the repeated G entries in C_e are ambiguous and affect the derivation of Eq. (8).
[Section B, Fig. 2(c)] The text says the time-averaged shear stress 'converges to a constant after a few periods.' Please specify the averaging window, the number of periods, and the convergence tolerance used to define the equilibrium value.
[Section C, Fig. 3] It is unclear whether the strain loop in Fig. 3 is a schematic of Eq. (4) or an actual simulated path. Please state explicitly which, and if it is analytic, note that the FEM model was not used to verify the work integral.
[References] References [13] and [26] appear to refer to the same paper; please check and consolidate. Also, a brief comparison with the driven odd elasticity model of Huang et al. [27] in the main text (not just the introduction) would help the reader understand what is new.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the odd modulus A is measured, and the statics/dynamics consequences are deductive rather than used to fit A.

full rationale

The derivation chain is A = Δσyx/Δϵyy (Eq. 3), measured in ABAQUS from the time-averaged shear stress under normal-strain increments. This measured value is then inserted into the constitutive tensor (Eq. 2) and used to compute non-conservative work (Eq. 6) and the non-Hermitian skin effect (Eq. 8). These downstream results are mathematical consequences of having a nonzero off-diagonal modulus; they are not used to define or fit A, and the paper does not claim them as independent empirical predictions of the simulation. No load-bearing step is an equation that equals its own input. The only self-citation, ref. [4], is cited alongside textbooks for Maxwell-Betti reciprocity and is not load-bearing. The text's admission that the gear-jump depends on the gear phase ('application of normal strain can only result in either no jump... or odd shear occurring in only one direction, depending on the phase') is a genuine limitation for whether A is a robust material constant, but that is a correctness/validity threat, not circularity: even if the mechanism is ratchet-like, the statics/dynamics are not used as evidence for A. Therefore no circular step is exhibited.

Assumptions & free parameters 8 free parameters · 4 assumptions · 0 invented entities

The paper's novelty rests on many hand-chosen design and protocol parameters plus one extracted modulus A. No independent prediction or external validation is provided; all structural conclusions follow from the measured A.

free parameters (8)

Gear clearance δ = 0.15 mm
Chosen to allow gear detachment and re-latching during normal-strain steps; the ratchet mechanism requires this gap.
Friction coefficient μ = 0.1
Contact friction enables the non-reciprocal gear jump; no sensitivity analysis is shown in the main text.
Driving frequency ω_d and amplitude Θ = 12π rad/s, 0.01 rad
Continuous external drive is the energy input; the effective response may depend on these values.
Normal strain increment Δϵyy = 0.04%
Defines the quasi-static steps and is the denominator of the odd modulus A.
Asymmetric contact lengths l1, l2 = not specified
The asymmetry (chirality) is the source of one-directional shear bias, but numerical values are not given.
Gear tooth geometry (l, h) = l=3.5 mm, h=1 mm
Tooth size sets the shear increment per jump and thus the magnitude of odd shear.
Bottom wall foundation stiffness k = 5×10^6 N/m
Chosen wall compliance affects the equilibrium shear after each gear jump.
Odd modulus A = 3125 GPa
Extracted from a single simulation via Eq. (3); used to compute non-conservative work and NHSE, so all downstream results rest on this fitted value.

assumptions (4)

domain assumption The time-averaged response under continuous periodic driving is described by a linear elastic constitutive tensor with a constant odd modulus A (Eqs. 1-3).
The entire odd-elasticity characterization and all downstream results rely on this effective quasi-static linear representation; if the response is rate-dependent or protocol-dependent, A is not a material constant.
domain assumption The ABAQUS frictional contact model faithfully simulates the discrete gear-tooth jump and lock-in mechanism.
The central mechanism depends entirely on contact/friction behavior; no experimental validation or analytical benchmark is provided.
domain assumption Odd-elasticity theory from Scheibner et al. [5], including the non-conservative work expression and the NHSE condition, applies unchanged to the effective driven metamaterial.
The statics and dynamics sections take this theory as given and insert the measured A; no independent derivation or experimental test is provided.
domain assumption The quasi-static strain increments equilibrate to a steady time-averaged state before each new increment.
The authors state the time-scale separation and show some convergence in Fig. 2(c), but no quantitative convergence criterion or error bars are given.

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Cite this review

Pith. "Pith review of Driven Odd Elasticity in Passive Mechanical Metamaterials." pith.science (2026). https://pith.science/paper/377YJLHA

@misc{pith2026260713997,
  author       = {Pith},
  title        = {Pith review of: Driven Odd Elasticity in Passive Mechanical Metamaterials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/377YJLHA}},
  note         = {Machine review of arXiv:2607.13997}
}

read the original abstract

We present a mechanical mechanism leveraging passive mechanical components, i.e. chiral gears and a square lattice metamaterial, to demonstrate driven odd elasticity in a mechanical metamaterial. The mechanism couples tension and shear in a non-reciprocal way, resulting in an odd shear modulus. The emergence of this odd shear modulus enables non-conservative work in a standard quasistatic strain cycle, and further enables the non-Hermitian skin effect in dynamics. Our results demonstrate that odd elasticity can be achieved in mechanical structures using passive elements without electronic components coupled with feedback or robotic control systems.

Figures

Figures reproduced from arXiv: 2607.13997 by the authors.

**Figure 2.** FIG. 2. (a) Illustration of periodic (driven) oscillations applied to each gear. (b) Normal strain increments [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Illustration of non-conservative work enabled by the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. (a) Spectrum for odd elastic metamaterial for both [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [27]

Huang, R

R. Huang, R. Mandal, C. Scheibner, and V. Vitelli, Odd elasticity in driven granular matter, Arxiv , 2311.18720 (2023)

arXiv 2023

[2] [1]

Fruchart, C

M. Fruchart, C. Scheibner, and V. Vitelli, Odd viscosity and odd elasticity, Annual Review of Condensed Matter Physics14, 471 (2023)

2023

[3] [2]

Coulais, D

C. Coulais, D. Sounas, and A. Alu, Static non-reciprocity in mechanical metamaterials, Nature542, 461 (2017)

2017

[4] [3]

J. D. Achenbach,Reciprocity in elastodynamics(Cam- bridge University Press, 2003)

2003

[5] [4]

Montazeri, M

A. Montazeri, M. Rahimi, and H. S. Park, Non- reciprocity and asymmetric elasticity in twisting chiral metamaterials, International Journal of Mechanical Sci- ences287, 109990 (2025)

2025

[6] [5]

Scheibner, A

C. Scheibner, A. Souslov, D. Banerjee, P. Surowka, W. T. M. Irvine, and V. Vitelli, Odd elasticity, Nature Physics16, 475 (2020)

2020

[7] [6]

X. N. Liu, G. L. Huang, and G. K. Hu, Chiral effect in plane isotropic micropolar elasticity and its application to chiral lattices, Journal of the Mechanics and Physics of Solids60, 1907 (2012)

1907

[8] [7]

Fernandez-Corbaton, C

I. Fernandez-Corbaton, C. Rockstuhl, P. Ziemke, P. Gumbsch, A. Albiez, R. Schwaiger, T. Frenzel, M. Kadic, and M. Wegener, New twists of 3d chiral meta- materials, Advanced Materials31, 1807742 (2019)

2019