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arxiv: 1906.08087 · v1 · pith:J6S72A3Enew · submitted 2019-06-19 · ❄️ cond-mat.soft · physics.comp-ph

Generalization of the Zabolotskaya equation to all incompressible isotropic elastic solids

Michel Destrade , Edvige Pucci , Giuseppe Saccomandi This is my paper

Pith reviewed 2026-05-25 20:05 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.comp-ph
keywords Zabolotskaya equationnonlinear elasticityincompressible solidsshear wavesmultiple scaleswave polarizationelastic wavessoft solids
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The pith

Linear polarization is impossible for general nonlinear two-dimensional shear waves in isotropic incompressible elastic solids because anti-plane and in-plane motions remain coupled.

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

The paper derives a system of coupled nonlinear equations for small but finite amplitude elastic shear waves that include both anti-plane displacement and general in-plane motion. This system applies to every isotropic incompressible nonlinear elastic solid and reduces to the scalar Zabolotskaya equation only in the compressible case. Because the coupling terms cannot be removed for a generic strain-energy function, a pure anti-plane wave necessarily generates in-plane components. The same equations show that a Gaussian beam driven by a purely polarized source produces only odd harmonics while even a small in-plane perturbation at the source creates a second harmonic of comparable size to the fifth harmonic.

Core claim

Using a multiple scales expansion, the authors obtain an asymptotic system of coupled nonlinear equations that govern the propagation of combined anti-plane and in-plane shear motions in any isotropic incompressible hyperelastic solid. For a general strain-energy function the coupling between the two classes of motion cannot be eliminated, so that linear polarization is impossible for generic nonlinear two-dimensional shear waves. The equations recover the known Zabolotskaya equation when the material is compressible and also admit special exact solutions in which linear polarization is preserved.

What carries the argument

The multiple-scales-derived system of two coupled nonlinear partial differential equations for the anti-plane displacement and the in-plane displacement potential.

If this is right

  • A pure linearly polarized shear-beam source generates only odd harmonics.
  • Adding slight in-plane noise to the source produces a second harmonic whose amplitude equals that of the fifth harmonic.
  • Special exact shear motions with preserved linear polarization exist inside the general theory.
  • The coupled system replaces the scalar Zabolotskaya equation for every incompressible isotropic solid.

Where Pith is reading between the lines

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

  • The derived equations supply a practical model for predicting harmonic generation in soft tissues or rubber under finite shear.
  • Numerical integration of the coupled system could test how sensitive polarization purity is to small perturbations in other geometries.
  • The impossibility of linear polarization may extend to other two-dimensional nonlinear wave problems once incompressibility is imposed.

Load-bearing premise

The assumed ordering of small but finite amplitude together with slow transverse variation remains valid for every incompressible isotropic strain-energy function.

What would settle it

An experiment that launches a purely anti-plane finite-amplitude shear wave into a general isotropic incompressible solid and measures no in-plane displacement component at any distance would falsify the claim that the motions remain coupled.

Figures

Figures reproduced from arXiv: 1906.08087 by Edvige Pucci, Giuseppe Saccomandi, Michel Destrade.

Figure 1
Figure 1. Figure 1: Spectrum generated by a large shear excitation in gelatine: experimental results of Esp´ındola et al. [6] (shared by G. Pinton and D. Esp´ındola). We consider a regular perturbative solution of the GZ system (17) via a new small parameter ε, w = εw1 + ε 2w2 + ε 3w3 + ε 4w4 + ε 5w5 + . . . , ψ = εψ1 + ε 2ψ2 + ε 3ψ3 + ε 4ψ4 + ε 5ψ5 + . . . , where wi , ψi are functions to be determined at each order. For a g… view at source ↗
Figure 2
Figure 2. Figure 2: Sketches of the results in Section 3: (a) A pure anti-plane shear Gaussian beam generates only odd harmonics, at 100, 300, 500 Hz, while (b) An initially noisy Gaussian beam (with a small in-plane component) additionally generates a harmonic at 200 Hz. identity (28) holds and the first equation of the GZ system (17) gives the following equation for ψ2, ψ2,ηηττ + 2ψ2,τττχ = 0, (38) as in the previous sectio… view at source ↗
read the original abstract

We study elastic shear waves of small but finite amplitude, composed of an anti-plane shear motion and a general in-plane motion. We use a multiple scales expansion to derive an asymptotic system of coupled nonlinear equations describing their propagation in all isotropic incompressible non-linear elastic solids, generalizing the scalar Zabolotskaya equation of compressible nonlinear elasticity. We show that for a general isotropic incompressible solid, the coupling between anti-plane and in-plane motions cannot be undone and thus conclude that linear polarization is impossible for general nonlinear two-dimensional shear waves. We then use the equations to study the evolution of a nonlinear Gaussian beam in a soft solid: we show that a pure (linearly polarised) shear beam source generates only odd harmonics, but that introducing a slight in-plane noise in the source signal leads to a second harmonic, of the same magnitude as the fifth harmonic, a phenomenon recently observed experimentally. Finally, we present examples of some special shear motions with linear polarisation.

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 uses a multiple-scales expansion to derive a system of coupled nonlinear PDEs that generalizes the Zabolotskaya equation to small-but-finite-amplitude two-dimensional shear waves (anti-plane plus in-plane) propagating in arbitrary incompressible isotropic hyperelastic solids. It concludes that the coupling between anti-plane and in-plane components cannot be removed for a general strain-energy function W, implying that linearly polarized waves are impossible except in special cases. The reduced system is then applied to the evolution of a nonlinear Gaussian beam, where a pure anti-plane source produces only odd harmonics while a small in-plane perturbation generates a second harmonic comparable in size to the fifth; examples of special linearly polarized motions are also exhibited.

Significance. If the non-degeneracy of the coupling coefficients is established, the work supplies a parameter-free asymptotic model applicable to the entire class of incompressible isotropic solids, extending the compressible Zabolotskaya theory and furnishing a direct link to recent experiments on harmonic generation in soft solids. The derivation from the general constitutive law via standard multiple-scale ordering, without ad-hoc restrictions on W beyond isotropy and incompressibility, is a clear methodological strength.

major comments (2)
  1. [Asymptotic reduction and statement of the reduced system] Derivation of the coupled system: the central claim that coupling between anti-plane and in-plane motions is irreducible for general W requires that the leading-order nonlinear coefficients (generated by the first and second derivatives of W evaluated at the identity) remain O(1) under the chosen amplitude/transverse scaling. The manuscript supplies neither the explicit algebraic combinations of these derivatives that must be nonzero nor a demonstration that they fail to vanish on a dense open set within the space of admissible strain-energy functions. Without this, the assertion that linear polarization is impossible for general nonlinear two-dimensional shear waves rests on an unverified non-degeneracy assumption.
  2. [Gaussian beam evolution] Application to the Gaussian beam (the section containing the numerical illustrations): the reported generation of a second harmonic of magnitude comparable to the fifth when a small in-plane component is added to the source is presented as a generic consequence of the coupled system. This conclusion is load-bearing for the experimental relevance claim, yet the text does not verify that the relevant coefficient ratios remain bounded away from zero for the family of W used in the simulations.
minor comments (2)
  1. [Abstract] The abstract states the result for “all incompressible isotropic elastic solids” while the body later exhibits special cases with linear polarization; a brief clarifying sentence on the precise meaning of “general” versus “special” would improve readability.
  2. [Constitutive assumptions] Notation for the strain-energy function and its derivatives at the identity should be introduced once, with a single consistent symbol, rather than re-defined in each section where the coefficients appear.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We address each of the major comments below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Asymptotic reduction and statement of the reduced system] Derivation of the coupled system: the central claim that coupling between anti-plane and in-plane motions is irreducible for general W requires that the leading-order nonlinear coefficients (generated by the first and second derivatives of W evaluated at the identity) remain O(1) under the chosen amplitude/transverse scaling. The manuscript supplies neither the explicit algebraic combinations of these derivatives that must be nonzero nor a demonstration that they fail to vanish on a dense open set within the space of admissible strain-energy functions. Without this, the assertion that linear polarization is impossible for general nonlinear two-dimensional shear waves rests on an unverified non-degeneracy assumption.

    Authors: We agree that making the non-degeneracy explicit would strengthen the presentation. The multiple-scales derivation in the manuscript yields the coefficients as specific combinations of the derivatives of W at the identity. We will include these explicit algebraic expressions in a revised Section 2 and add a paragraph demonstrating that the vanishing set has positive codimension in the function space of admissible strain-energy functions, hence the generic case has irreducible coupling. This addresses the concern directly. revision: yes

  2. Referee: [Gaussian beam evolution] Application to the Gaussian beam (the section containing the numerical illustrations): the reported generation of a second harmonic of magnitude comparable to the fifth when a small in-plane component is added to the source is presented as a generic consequence of the coupled system. This conclusion is load-bearing for the experimental relevance claim, yet the text does not verify that the relevant coefficient ratios remain bounded away from zero for the family of W used in the simulations.

    Authors: The simulations in the manuscript employ the incompressible neo-Hookean strain-energy function (and mention of Mooney-Rivlin in the general discussion), for which the nonlinear coefficients are known explicitly and the relevant ratios are nonzero and O(1). We will add a brief calculation in the revised manuscript verifying that these ratios are bounded away from zero for the specific W used, confirming the generic behavior observed in the numerics. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is an independent asymptotic reduction

full rationale

The paper derives the generalized system via multiple-scales expansion applied to the standard nonlinear elasticity equations for incompressible isotropic solids. The resulting coupled PDEs are obtained directly from the strain-energy function W without any fitted parameters, self-referential definitions, or load-bearing self-citations that reduce the central claim to prior inputs. The statement that coupling cannot be undone follows from the explicit form of the nonlinear coefficients generated by the expansion, which remain general for arbitrary admissible W. No predictions are made that collapse to the input data by construction, and the derivation chain is self-contained against the underlying balance laws.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation rests on the standard constitutive assumptions of nonlinear incompressible isotropic elasticity and the validity of the multiple-scales ansatz; no new free parameters, ad-hoc entities, or invented quantities are introduced.

axioms (2)
  • domain assumption The solid is isotropic and incompressible
    Invoked to restrict the strain-energy function and to enforce the constraint on the deformation gradient.
  • domain assumption Multiple scales expansion with slow transverse variation is applicable to small but finite amplitude waves
    Used to obtain the asymptotic system from the full equations of motion.

pith-pipeline@v0.9.0 · 5700 in / 1299 out tokens · 30134 ms · 2026-05-25T20:05:54.653709+00:00 · methodology

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Reference graph

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