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arxiv: 2602.10677 · v2 · submitted 2026-02-11 · ❄️ cond-mat.stat-mech

Analytic Nonlinear Theory of Shear Banding in Amorphous Solids

Avanish Kumar , Itamar Procaccia This is my paper

Pith reviewed 2026-05-16 03:39 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords shear bandingamorphous solidsnonlinear elasticitydisplacement fieldplastic dipolesenergy functionalinstabilityelastic moduli
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The pith

A derived energy functional has a vanishing Hessian eigenvalue that predicts the critical stress for shear band formation in amorphous solids.

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

This paper develops an analytic nonlinear theory for the shear banding instability in athermal quasi-statically sheared amorphous solids. It derives nonlinear equations for the displacement field that incorporate the effects of plastic events, modeled as distributed dipoles causing stress screening and new length scales. Solving the weakly nonlinear amplitude equation yields analytic expressions for displacement fields in shear bands and shows how elastic moduli control band width from ductile to brittle responses. The central result is an energy functional whose Hessian develops an eigenvalue that reaches zero at the instability, giving a prediction for the critical accumulated stress. A sympathetic reader would care because this offers a first-principles analytic route to the onset of failure in amorphous materials without full numerical simulation.

Core claim

We derive nonlinear equations for the displacement field, including the consequences of plastic deformation on the mechanical response of amorphous solids. The plastic events collectively induce distributed dipoles that are responsible for screening effects and the creation of typical length-scales that are absent in classical elasticity theory. The nonlinear theory exposes an instability that results in the creation of shear bands. By solving the weakly nonlinear amplitude equation we present analytic expressions for the displacement fields associated with shear bands, explaining the role of the elastic moduli that determine the width of a shear band from ductile to brittle characteristics.

What carries the argument

The energy functional derived from the nonlinear displacement equations closed by dipole screening, whose Hessian eigenvalue vanishes at the shear-banding instability.

If this is right

  • Analytic displacement fields become available for shear bands and depend on elastic moduli.
  • Shear band width varies continuously between ductile and brittle limits set by material properties.
  • The critical accumulated stress is given by the vanishing of the Hessian eigenvalue.
  • Instability arises specifically from screening and length scales introduced by plastic dipoles.

Where Pith is reading between the lines

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

  • The framework could be extended to describe band propagation after onset.
  • Material properties might be tuned to select desired ductility by shifting the critical stress.
  • Dipole screening ideas may apply to instabilities in other disordered systems.
  • Direct comparison to molecular dynamics of sheared glasses would test the predicted critical stress.

Load-bearing premise

Collective plastic events are modeled as distributed dipoles that induce screening and new length scales to close the nonlinear equations for the displacement field.

What would settle it

Measurement of the accumulated stress at shear band onset in a sheared amorphous solid, checked against the value where the Hessian eigenvalue of the derived energy functional reaches zero.

Figures

Figures reproduced from arXiv: 2602.10677 by Avanish Kumar, Itamar Procaccia.

Figure 1
Figure 1. Figure 1: FIG. 1. Examples of shear-band profiles in the ductile case. [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The shear associated with the solution plotted in [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Examples of shear-band profiles in the brittle case. [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Examples of shear profiles in the brittle case. The [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
read the original abstract

The aim of this paper is to offer an analytic theory of the shear banding instability in amorphous solids that are subjected to athermal quasi-static shear. To this aim we derive nonlinear equations for the displacement field, including the consequences of plastic deformation on the mechanical response of amorphous solids. The plastic events collectively induce distributed dipoles that are responsible for screening effects and the creation of typical length-scales that are absent in classical elasticity theory. The nonlinear theory exposes an instability that results in the creation of shear bands. By solving the weakly nonlinear amplitude equation we present analytic expressions for the displacement fields that is associated with shear bands, explaining the role of the elastic moduli that determine the width of a shear band from ductile to brittle characteristics. We derive an energy functional whose Hessian possesses an eigenvalue that goes to zero at the shear-banding instability, providing a prediction for the critical value of the accumulated stress that results in an instability.

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 / 1 minor

Summary. The paper derives nonlinear equations for the displacement field in athermal quasi-static shear of amorphous solids, closing them by modeling collective plastic events as distributed dipoles that generate screening and new length scales absent from classical elasticity. An energy functional is constructed whose Hessian possesses an eigenvalue that vanishes at the shear-banding instability, yielding an analytic prediction for the critical accumulated stress; the weakly nonlinear amplitude equation is solved to obtain explicit displacement fields inside shear bands and to relate elastic moduli to band width and the ductile-to-brittle crossover.

Significance. If the central construction holds, the work supplies an analytic route to the onset and morphology of shear bands in amorphous solids, with explicit dependence on elastic moduli that could unify ductile and brittle responses. This would be a notable advance for the field provided the dipole closure can be placed on firmer footing.

major comments (2)
  1. [Abstract] Abstract: the prediction that the Hessian eigenvalue reaches zero at a specific critical stress is presented as following from the closed nonlinear system, yet the dipole representation used to introduce screening lengths is introduced by postulate rather than derived; because this closure directly determines the functional whose eigenvalue supplies the instability threshold, the critical-stress claim rests on an unverified modeling assumption whose validity in the nonlinear regime is not independently checked.
  2. [Abstract] Abstract (and the derivation of the energy functional): it is unclear whether the dipole strengths or the resulting screening length are fixed by independent microscopic considerations or are effectively calibrated to the same instability data used to test the zero-eigenvalue condition; if the latter, the reported critical stress is not a genuine prediction but a consistency check within the same closure.
minor comments (1)
  1. [Abstract] Abstract: the phrase 'weakly nonlinear amplitude equation' is used without stating its explicit form or the order of the expansion employed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and detailed report. We address each major comment below. The dipole closure is a modeling assumption motivated by the Eshelby picture, but we agree that its status and the independence of parameters require clearer exposition; we will revise accordingly while maintaining that the critical stress follows directly from the closed equations.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the prediction that the Hessian eigenvalue reaches zero at a specific critical stress is presented as following from the closed nonlinear system, yet the dipole representation used to introduce screening lengths is introduced by postulate rather than derived; because this closure directly determines the functional whose eigenvalue supplies the instability threshold, the critical-stress claim rests on an unverified modeling assumption whose validity in the nonlinear regime is not independently checked.

    Authors: We acknowledge that the distributed-dipole representation is introduced as a closure to capture collective screening by plastic events rather than being derived from the microscopic dynamics within this manuscript. This choice is standard in the amorphous-plasticity literature because each localized rearrangement is mechanically equivalent to an Eshelby inclusion (force dipole). The resulting screening length then enters the energy functional, so the vanishing of the Hessian eigenvalue is indeed a property of the closed model. We will revise the abstract and the derivation section to state the modeling assumption explicitly and to note that its validity in the fully nonlinear regime remains to be verified by direct comparison with particle simulations; no such independent check is provided in the present work. revision: partial

  2. Referee: [Abstract] Abstract (and the derivation of the energy functional): it is unclear whether the dipole strengths or the resulting screening length are fixed by independent microscopic considerations or are effectively calibrated to the same instability data used to test the zero-eigenvalue condition; if the latter, the reported critical stress is not a genuine prediction but a consistency check within the same closure.

    Authors: The dipole strength and the emergent screening length are fixed by independent microscopic considerations: they are expressed in terms of the typical plastic-event size and the local yield strain, both taken from prior athermal-quasistatic simulations that characterize individual rearrangements (references to be added). These parameters are not adjusted to reproduce the instability threshold; once inserted, the nonlinear equations yield the critical accumulated stress as the value at which the lowest Hessian eigenvalue crosses zero. We will add an explicit paragraph in the derivation of the energy functional that traces the screening length back to these microscopic inputs, thereby clarifying that the critical stress is a prediction of the closed theory rather than a consistency check. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from explicit modeling assumptions to eigenvalue prediction.

full rationale

The paper postulates collective plastic events as distributed dipoles to close the nonlinear displacement equations and introduce screening lengths, then derives the energy functional from that closed system. The zero-eigenvalue condition on the Hessian is obtained directly from the resulting functional and supplies the critical-stress prediction. No step reduces by construction to a fitted parameter renamed as prediction, no self-citation chain bears the load of the central claim, and the dipole closure is introduced as an ansatz rather than smuggled via prior self-citation. The derivation is therefore self-contained relative to its stated premises and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on treating plastic events as distributed dipoles (an invented modeling device) and on standard continuum assumptions for the displacement field; no free parameters are explicitly named in the abstract, but elastic moduli appear as inputs that control band width.

free parameters (1)
  • elastic moduli
    Enter the analytic solution for band width and control the ductile-to-brittle transition.
axioms (1)
  • domain assumption Plastic events collectively induce distributed dipoles responsible for screening and new length scales
    Invoked to close the nonlinear equations and to generate the instability absent in classical elasticity.
invented entities (1)
  • distributed dipoles from plastic events no independent evidence
    purpose: To produce screening effects and characteristic length scales in the displacement field
    Postulated without independent falsifiable evidence in the abstract; the entity is introduced to explain the instability.

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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.

  1. 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.

Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages · cited by 1 Pith paper

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    Analysis of the LHS Let us label the first four terms on the LHS of Eq. (35) as Z(0) k , Z(1) k , Z(2) k , Z(3) k , and the final long cubic group as Z(4) k : LHSk = Σ αβ dk,αβ   Z(0) k + ˜Aαkγδ dγ,δα   Z(1) k + 1 2 ˜Aαkγδ ( dµ,γα dµ,δ +dµ,γ dµ,δα )    Z(2) k (43) + ˜Aαβγδ ( dγ,δα dk,β +dγ,δ dk,βα )    Z(3) k + 1 2 ˜Aαβγδ ( dµ,γα dµ,δ dk,β +...

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    (55) Using this in the definitions (48), (50), (52) and (54) we find µ (1) =µ, µ (2) =µ (3) = 0, µ (4) =λ + 2µ

    Simplifying for homogeneous isotropic systems Presently we are ready to simplify the result of the previous subsection, taking into account the explicit form of the elastic tensor in homogeneous isotropic systems: ˜Aαβγδ =λ δαβ δγδ +µ (δαγ δβδ +δαδδβγ ). (55) Using this in the definitions (48), (50), (52) and (54) we find µ (1) =µ, µ (2) =µ (3) = 0, µ (4) =...

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    To this aim we choose the background stress in the form Σαβ = ( 0 Σ Σ 0 )

    Simplifying for simple shear Next we will specialize our equations for the case of simple shear. To this aim we choose the background stress in the form Σαβ = ( 0 Σ Σ 0 ) . (57) At present this tensor is off diagonal in an unspecified co- ordinate frame that we must make explicit. To this aim we introduce a pair n, t of orthonormal unit vectors as- sociated...

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    In that case we can estimate α 4 (f ′)4 ≈ B 4 (f 2 − f 2 0 )2

    Inner solution In the core of the shear band we assume that the quar- tic term in f ′ is dominant over the quadratic term, i.e ǫ2(f ′)2/ 2 ≫ ǫ1. In that case we can estimate α 4 (f ′)4 ≈ B 4 (f 2 − f 2 0 )2. (104) 11 Since shear band solution f (ξ) satisfied |f (ξ)| ≤ f0, we can take twice a square root of Eq. (104) to find f ′ ≈ ( B ǫ2 ) 1/ 4 √ f 2 0 − f 2...

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    The unprojected energy functional U [d] We note that the governing equilibrium equation is an Euler-Lagrange equation so it should be derived form the variation of anenergy functional U [d], that is, δU[d] = 0 for all admissible variations η satisfying η|∂ Ω = 0, such that after integration by parts we get a PDE of the form (61). Guided by Eq. 61, we writ...

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    Second variation: Full unprojected Hessian in bilinear form a. Prestress-gradient term From (B15): 16 δU∞ [d;η] = ∫ Σαβ dk,β ηk,α . We vary this with respect to d in direction ζ: dk,β ↦→ dk,β +tζk,β . Thus δ2U∞ [d;η,ζ ] = ∫ Ω Σαβ ζk,β ηk,α d2x. (B20) This is already symmetric in ( η,ζ ) if Σ αβ is symmetric (interchanging dummy indices α ↔ β ). b. Local t...

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    Full unprojected Hessian in bilinear form Collect (B20), (B23), and (B28): δ2U [d;η,ζ ] = ∫ Ω Σαβ ηk,α ζk,β d2x +δ2Uel[d;η,ζ ] + ∫ Ω ηkMkζ (d)ζζd2x, (B29) with Mkζ (d) = Lkζ − 3Tkζqrdqdr and δ2Uel given by (B28). This is the full unprojected Hessian. 17

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