On the anomalous elasticity in the mechanical response of amorphous solids

Gilles Tarjus; Giulio Biroli; Misaki Ozawa

arxiv: 2602.19299 · v2 · submitted 2026-02-22 · ❄️ cond-mat.soft · cond-mat.stat-mech

On the anomalous elasticity in the mechanical response of amorphous solids

Gilles Tarjus , Misaki Ozawa , Giulio Biroli This is my paper

Pith reviewed 2026-05-15 20:33 UTC · model grok-4.3

classification ❄️ cond-mat.soft cond-mat.stat-mech

keywords amorphous solidsanomalous elasticityquadrupolar defectselasto-plastic modelEshelby singularitiesplastic rearrangementsmechanical perturbation

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The pith

Mechanical perturbations in amorphous solids create quadrupolar defects only within a region set by the perturbation size, producing anomalous elasticity locally and conventional elasticity beyond.

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

The paper examines the mechanical response of amorphous solids, which combines elastic deformation with plastic events driven by localized irreversible rearrangements. These rearrangements correspond to Eshelby-like quadrupolar singularities in the displacement field. The authors show that for non-macroscopic perturbations the density of such quadrupoles is zero in the thermodynamic limit. The defects instead form only inside a region whose size scales with the spatial extent of the perturbation. This produces anomalous elastic behavior near the perturbation while restoring standard elasticity at larger distances. Elasto-plastic model simulations confirm the localized quadrupoles and the associated modulus renormalization but do not recover the dipole-screening signatures reported elsewhere.

Core claim

The response of amorphous solids to a mechanical perturbation consists of elastic and plastic deformation, with the latter mediated by localized irreversible rearrangements associated with Eshelby-like quadrupolar singularities. For non-macroscopic perturbations the density of these quadrupolar defects vanishes in the thermodynamic limit. Plastically active quadrupoles therefore emerge only inside a region whose size scales as the perturbation extent ℓ. This mechanism yields anomalous elasticity on scale ℓ close to the perturbation and conventional elasticity beyond.

What carries the argument

The elasto-plastic model that treats localized irreversible rearrangements as Eshelby-like quadrupolar singularities and tracks their contribution to the displacement field under an imposed perturbation.

If this is right

Anomalous elasticity is confined to distances comparable to the perturbation size ℓ.
The effective shear modulus is renormalized only inside the region set by ℓ.
Conventional long-wavelength elasticity is recovered at scales larger than ℓ.
The elasto-plastic model reproduces localized plastic quadrupoles but shows no dipole-screening signatures.

Where Pith is reading between the lines

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

Observed anomalous screening in atomistic studies may arise only for specific macroscopic or long-range perturbations not covered by the present analysis.
Mesoscopic modeling of amorphous solids should incorporate a scale-dependent crossover from anomalous to classical elasticity rather than uniform screening.
Adding explicit long-range interactions to the elasto-plastic model could be tested to see whether it generates the missing dipole-screening effects.

Load-bearing premise

The density of quadrupolar events created by an imposed non-macroscopic perturbation vanishes in the thermodynamic limit.

What would settle it

A simulation or experiment that measures a non-vanishing density of quadrupolar defects persisting throughout the system in the thermodynamic limit for any localized perturbation would falsify the claim.

read the original abstract

The response of amorphous solids to a mechanical perturbation consists in an elastic and a plastic deformation. The latter is mediated by localized irreversible rearrangements associated with Eshelby-like quadrupolar singularities in the displacement field. It has recently been argued that a density of such singularities leads to an anomalous elastic behavior taking the form of screening effects, which goes beyond classical elastic predictions. Here, we reexamine this scenario using general theoretical arguments and a description in terms of an elasto-plastic model, which we compare with atomistic simulations of the canonical Eshelby inclusion geometry. We discuss the conditions under which a finite, i.e., nonvanishing, density of quadrupolar events is created by an imposed perturbation. We argue that, except when the perturbation is macroscopic, there are many situations in which the density of quadrupolar defects is zero in the thermodynamic limit. In these cases, we find that plastically active quadrupoles emerge in a region whose size generically scales as the spatial extent $\ell$ of the mechanical perturbation. This mechanism leads to anomalous elasticity on a scale $\ell$ close to the perturbation and to conventional elasticity beyond. The simulations of the elasto-plastic model reproduce the emergence of plastic quadrupoles in a region set by $\ell$ and the associated renormalization of the effective shear modulus, but they do not exhibit the dipole-screening signatures reported in atomistic and experimental studies. Our analysis delineates the scale-dependent breakdown of long-wavelength elasticity in amorphous materials and suggests directions for incorporating anomalous screening into mesoscopic modeling frameworks.

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

The paper argues that quadrupolar defect density vanishes in the thermodynamic limit for non-macroscopic perturbations, producing local anomalous elasticity but no dipole screening in their elasto-plastic model.

read the letter

The main thing to know is that the authors claim the density of Eshelby quadrupoles created by a localized perturbation goes to zero in the thermodynamic limit except when the perturbation is macroscopic. This produces anomalous elasticity only near the perturbation scale ℓ and recovers conventional elasticity farther away. They support this with general thermodynamic arguments plus simulations of an elasto-plastic model in the canonical Eshelby inclusion setup. The simulations show plastic activity confined to a region whose size scales with ℓ and recover the expected local renormalization of the shear modulus. That part is clean and directly comparable to atomistic geometry. The model does not, however, produce the dipole-screening signatures reported in atomistic and experimental studies, which the authors note explicitly. The soft spot is that the elasto-plastic model omits the long-range elastic couplings that nucleate extra quadrupoles at distances much larger than ℓ in real systems. Because the zero-density result relies on this model choice, the thermodynamic-limit claim is not fully general and depends on whether those couplings are essential. The paper is useful for anyone working on mesoscopic descriptions of amorphous solids and scale-dependent elasticity. It qualifies recent claims without overreaching and deserves a serious referee to check the model implementation and the precise conditions under which the density really vanishes.

Referee Report

2 major / 2 minor

Summary. The paper argues that the mechanical response of amorphous solids to localized perturbations involves Eshelby quadrupolar defects whose average density vanishes in the thermodynamic limit except for macroscopic driving. Using general thermodynamic-limit considerations and simulations of an elasto-plastic model, it shows that plastic activity remains confined to a region whose size scales with the perturbation extent ℓ. This produces anomalous (screened) elasticity on scales ~ℓ near the perturbation and recovers conventional elasticity at larger distances. The model reproduces the local renormalization of the shear modulus but lacks the dipole-screening signatures seen in atomistic simulations and experiments.

Significance. If the central claim holds, the work clarifies the scale-dependent breakdown of long-wavelength elasticity in amorphous solids by linking it to the spatial confinement of plastic events rather than a finite defect density. The combination of thermodynamic-limit reasoning with direct comparison of elasto-plastic simulations to the canonical Eshelby inclusion geometry is a strength, offering a mesoscopic framework that could guide incorporation of anomalous effects into larger-scale models. However, the result remains tied to the specific model assumptions.

major comments (2)

[§3 and model description] The claim that quadrupolar density vanishes in the L→∞ limit for non-macroscopic perturbations (abstract and §3) rests on the elasto-plastic model whose simulations confine activity to ~ℓ. Yet the model is stated to omit the long-range elastic couplings responsible for dipole screening in atomistic work; if those couplings are restored, they could nucleate a finite density at distances ≫ℓ, undermining the zero-density premise even for localized driving.
[§4] Table or figure comparing model to atomistic Eshelby geometry (likely §4): the reported absence of dipole-screening signatures is presented as consistent with zero density, but without a quantitative test of how screening length scales with system size L, it is unclear whether the thermodynamic-limit conclusion is robust or an artifact of the truncated interactions.

minor comments (2)

[Introduction] Notation for the perturbation size ℓ and the screening length should be introduced earlier and used consistently to avoid confusion between local anomalous and bulk conventional regimes.
[§3] The discussion of conditions for finite versus vanishing quadrupolar density would benefit from an explicit list or flowchart summarizing the macroscopic versus localized cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments. We address each major comment point by point below, providing clarifications based on the general thermodynamic arguments and simulation results presented in the paper. Where appropriate, we indicate revisions that will be incorporated in the revised version.

read point-by-point responses

Referee: [§3 and model description] The claim that quadrupolar density vanishes in the L→∞ limit for non-macroscopic perturbations (abstract and §3) rests on the elasto-plastic model whose simulations confine activity to ~ℓ. Yet the model is stated to omit the long-range elastic couplings responsible for dipole screening in atomistic work; if those couplings are restored, they could nucleate a finite density at distances ≫ℓ, undermining the zero-density premise even for localized driving.

Authors: We respectfully disagree that restoring long-range couplings would undermine the zero-density result for localized (non-macroscopic) perturbations. The thermodynamic-limit argument in §3 is independent of interaction range: it follows from the fact that the energy cost of creating a finite density of quadrupoles at distances much larger than ℓ exceeds any gain from the localized driving when L→∞. The elasto-plastic model is introduced precisely to isolate the local plastic response without long-range terms, and its confinement of activity to scale ℓ is consistent with this general reasoning. We will add a short paragraph in the revised §3 clarifying that the zero-density conclusion is not an artifact of truncated interactions but follows from energy minimization for ℓ ≪ L. revision: partial
Referee: [§4] Table or figure comparing model to atomistic Eshelby geometry (likely §4): the reported absence of dipole-screening signatures is presented as consistent with zero density, but without a quantitative test of how screening length scales with system size L, it is unclear whether the thermodynamic-limit conclusion is robust or an artifact of the truncated interactions.

Authors: We agree that an explicit scaling analysis would strengthen the presentation. In the simulations reported in §4, the system size L is chosen to be several times larger than the perturbation scale ℓ, and no dipole-screening signatures appear at any scale. Because the model omits the long-range couplings that produce screening in atomistic studies, the absence of screening is expected and supports the zero-density premise rather than being an artifact. We will add a brief discussion and, if feasible within the model, a supplementary plot showing the absence of any emergent screening length that grows with L. revision: partial

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper's central claim—that the density of quadrupolar defects vanishes in the thermodynamic limit except for macroscopic perturbations, producing scale-dependent anomalous elasticity—is derived from general theoretical arguments combined with simulations of an elasto-plastic model. The model shows plastic activity confined to a region of size ~ℓ with associated renormalization of the shear modulus, without any reduction of the claimed outcome to a fitted parameter defined by the result itself or to a self-citation chain. No load-bearing step equates a prediction to its input by construction, and the derivation remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on treating plastic events as independent Eshelby quadrupoles whose density is controlled by perturbation scale, with the elasto-plastic model as the computational framework.

axioms (2)

domain assumption Plastic rearrangements can be represented as localized Eshelby-like quadrupolar singularities in the displacement field
Standard modeling choice in amorphous plasticity literature invoked throughout the abstract.
domain assumption The density of such quadrupoles in the thermodynamic limit is zero unless the perturbation is macroscopic
Central theoretical step used to derive the scale-dependent behavior.

pith-pipeline@v0.9.0 · 5581 in / 1250 out tokens · 26366 ms · 2026-05-15T20:33:36.922860+00:00 · methodology

discussion (0)

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Relation between the paper passage and the cited Recognition theorem.

elasto-plastic models ... kinetic rules ... GEshelby_ij ... no dipole-screening signatures

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Topological Defects in Amorphous Solids
cond-mat.mtrl-sci 2026-04 unverdicted novelty 2.0

Topological defects can be identified in glasses and may provide a first-principles framework for their mechanical response and spatiotemporal dynamics.