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arxiv: 2604.03063 · v1 · submitted 2026-04-03 · ⚛️ physics.app-ph

Nonlinear interface effects in multilayered structures: vibro-acoustic modeling and experimental analysis

Antoine Demiquel , Kerem Ege , Emmanuel Gourdon This is my paper

Pith reviewed 2026-05-13 18:28 UTC · model grok-4.3

classification ⚛️ physics.app-ph
keywords nonlinear interfacesmultilayer beamsZig-Zag formulationequivalent bending stiffnessvibro-acoustic modelinglaser vibrometryimperfect interfacesglass-epoxy-glass
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The pith

The equivalent bending stiffness of multilayer beams with imperfect interfaces depends on excitation level due to nonlinear interface behavior.

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

The paper models multilayer beams using a Zig-Zag formulation that links adjacent layers through stress continuity and displacement discontinuity at imperfect interfaces. This reduces the number of unknowns while yielding an equivalent Kirchhoff-Love plate model whose frequency-dependent bending stiffness captures the overall dynamic response. Laser vibrometry experiments on a glass-epoxy-glass beam, analyzed with the Corrected Force Analysis Technique, show that this stiffness changes measurably with excitation amplitude. A sympathetic reader would care because the result indicates that real-world vibration and acoustic performance of layered composites cannot be predicted accurately by linear models when loads vary in strength.

Core claim

The paper establishes that the dynamic response of a three-layer beam with imperfect interfaces depends on the excitation level, with observed variations in the equivalent bending stiffness revealing the nonlinear nature of the interfacial behavior. The Zig-Zag formulation with coupling conditions reduces kinematic unknowns without losing accuracy, and the derived equivalent plate model is validated experimentally on a glass-epoxy-glass beam under various excitation levels.

What carries the argument

Zig-Zag formulation with interfacial coupling conditions of stress continuity and displacement discontinuity, from which an equivalent frequency-dependent bending stiffness is derived in a Kirchhoff-Love plate model.

If this is right

  • Nonlinear interface effects must be included when predicting the vibro-acoustic response of multilayer beams at varying load levels.
  • The frequency-dependent equivalent stiffness offers a reduced-order way to characterize global behavior without full layer-by-layer kinematics.
  • The approach applies directly to glass-epoxy-glass beams and indicates similar nonlinear signatures in other imperfectly bonded composites.
  • Experimental extraction via the Corrected Force Analysis Technique provides a practical method to quantify interface nonlinearity in situ.

Where Pith is reading between the lines

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

  • Amplitude dependence could shift resonance frequencies or alter damping in composite panels used in vehicles or aircraft.
  • Extending the model from beams to plates would allow prediction of two-dimensional wave propagation and sound radiation influenced by the same nonlinearity.
  • Design of interfaces with controlled nonlinearity might enable structures whose acoustic properties change with operating intensity.

Load-bearing premise

The Zig-Zag formulation with interfacial coupling conditions accurately captures the nonlinear interfacial behavior while the equivalent Kirchhoff-Love plate formulation represents the global structural response.

What would settle it

If the equivalent bending stiffness extracted from vibrometry measurements remained constant across increasing excitation levels, or if model predictions failed to match the measured stiffness changes.

Figures

Figures reproduced from arXiv: 2604.03063 by Antoine Demiquel, Emmanuel Gourdon, Kerem Ege.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Outline of the considered three-layered structure composed of stiff skin layers (light [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Example of interface behavior characterized by a nonlinear interface constitutive [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Equivalent bending stiffness and (b) equivalent wavenumbers obtained from the pro [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Lateral, bottom, and top views of the experimental glass/epoxy/glass three-layer beam [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Procedure to isolate the longitudinal bending wavenumber from the measured displacement [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Influence of the interface parameter [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Panels (a, b, c) compare the dynamically equivalent Young’s modulus obtained from exper [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
read the original abstract

This paper presents an experimental and theoretical study of the nonlinear behavior of imperfect interfaces in multilayer structures using an equivalent vibro-acoustic approach. The multilayer system is modeled through a Zig-Zag formulation, in which interfacial coupling conditions, stress continuity and displacement discontinuity, relate the kinematics of adjacent layers while preserving an independent description of each layer. This framework significantly reduces the number of kinematic unknowns without compromising the model accuracy. An equivalent Kirchhoff-Love plate formulation is then introduced to derive a frequency-dependent bending stiffness representative of the global structural response. Experimental measurements of the transverse displacement field are performed using laser vibrometry and processed via the Corrected Force Analysis Technique (CFAT).The results demonstrate that the dynamic response of a three-layer beam with imperfect interfaces depends on the excitation level. In particular, variations in the equivalent bending stiffness are observed, revealing the nonlinear nature of the interfacial behavior. The proposed methodology is applied to a glass-epoxy-glass multilayer beam under various excitation levels.

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 paper develops a Zig-Zag kinematic model for multilayer beams incorporating interfacial stress continuity and displacement discontinuity, which is reduced to an equivalent frequency-dependent bending stiffness via a Kirchhoff-Love plate formulation. Laser-vibrometry measurements on a glass-epoxy-glass beam are inverted with the Corrected Force Analysis Technique (CFAT) to show that the recovered equivalent bending stiffness varies with excitation amplitude, which is interpreted as evidence of nonlinear interfacial behavior.

Significance. If the experimental inversion is shown to be free of linear-operator artifacts, the work supplies a reduced-order modeling route and an experimental diagnostic for amplitude-dependent interface effects in layered structures, with relevance to composite design and vibro-acoustic prediction.

major comments (2)
  1. [Experimental analysis and CFAT processing] The CFAT inversion (described in the experimental section) applies the linear Kirchhoff-Love operator to reconstruct equivalent forces and stiffness from measured displacement fields. When the true interface response is nonlinear, the displacement field does not satisfy the linear PDE pointwise; the linear inverse can therefore return an amplitude-dependent “equivalent” stiffness even in the absence of interfacial nonlinearity. No section provides a forward nonlinear simulation, bias quantification, or exclusion of alternative linear sources (material damping, boundary compliance) to demonstrate that CFAT remains unbiased at the observed excitation levels.
  2. [Equivalent plate derivation] The reduction from the Zig-Zag model to a single equivalent Kirchhoff-Love plate (Section on equivalent formulation) assumes that a linear, frequency-dependent bending stiffness fully captures the global response across excitation amplitudes. For genuinely nonlinear interfaces this equivalence is not guaranteed to be unique or amplitude-independent; the manuscript does not derive or test the conditions under which the effective stiffness remains a valid descriptor.
minor comments (2)
  1. [Abstract] The abstract states that “variations in the equivalent bending stiffness are observed” but supplies neither the magnitude of the variation, the number of excitation levels tested, nor uncertainty estimates, which weakens the reader’s ability to judge the strength of the central claim.
  2. [Modeling section] Notation for the interfacial coupling conditions and the definition of the frequency-dependent bending stiffness should be collected in a single table or appendix for clarity.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the detailed and constructive comments on our manuscript. We address each major point below and have revised the manuscript to incorporate clarifications and additional discussion where feasible.

read point-by-point responses

  1. Referee: [Experimental analysis and CFAT processing] The CFAT inversion (described in the experimental section) applies the linear Kirchhoff-Love operator to reconstruct equivalent forces and stiffness from measured displacement fields. When the true interface response is nonlinear, the displacement field does not satisfy the linear PDE pointwise; the linear inverse can therefore return an amplitude-dependent “equivalent” stiffness even in the absence of interfacial nonlinearity. No section provides a forward nonlinear simulation, bias quantification, or exclusion of alternative linear sources (material damping, boundary compliance) to demonstrate that CFAT remains unbiased at the observed excitation levels.

    Authors: We acknowledge that CFAT relies on the linear Kirchhoff-Love assumptions, so nonlinear interface behavior can in principle introduce reconstruction artifacts. However, the amplitude dependence we recover is systematic, repeatable across multiple modes and frequencies, and vanishes at the lowest excitation levels where the structure behaves linearly. To address the concern, we have added a dedicated paragraph in the experimental section that (i) establishes the linear baseline from low-amplitude data, (ii) shows that conventional linear mechanisms (viscoelastic damping, boundary compliance) cannot reproduce the observed magnitude of stiffness variation, and (iii) discusses the expected bias order for the displacement amplitudes used. A complete forward nonlinear simulation to quantify residual bias lies outside the present scope and is noted as future work. revision: yes

  2. Referee: [Equivalent plate derivation] The reduction from the Zig-Zag model to a single equivalent Kirchhoff-Love plate (Section on equivalent formulation) assumes that a linear, frequency-dependent bending stiffness fully captures the global response across excitation amplitudes. For genuinely nonlinear interfaces this equivalence is not guaranteed to be unique or amplitude-independent; the manuscript does not derive or test the conditions under which the effective stiffness remains a valid descriptor.

    Authors: The equivalent-plate reduction is presented as an approximate reduced-order model valid when interfacial nonlinearity remains moderate and the response can still be represented by a single frequency-dependent stiffness parameter. We have revised the derivation section to state these assumptions explicitly, to sketch the steps by which the Zig-Zag kinematics collapse to an effective bending stiffness, and to note that the mapping is not claimed to be unique for strong nonlinearity. Within the experimental range reported, the effective stiffness provides a compact descriptor that matches the measured global response; we have added a short numerical check confirming consistency in the linear limit. revision: yes

standing simulated objections not resolved
  • A full forward nonlinear time-domain simulation to quantify CFAT bias exactly, which would require a separate modeling framework beyond the scope of the current study.

Circularity Check

0 steps flagged

No significant circularity; standard formulations and independent experimental processing

full rationale

The paper's modeling chain starts from the established Zig-Zag kinematic description with interfacial stress continuity and displacement discontinuity, then reduces to an equivalent Kirchhoff-Love plate to obtain a frequency-dependent bending stiffness. This reduction follows directly from the layer-wise equilibrium and continuity conditions without redefining any output as an input. The experimental claim relies on laser vibrometry data inverted via the standard CFAT procedure (linear operator applied to measured fields). No parameter is fitted to a data subset and then relabeled as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and the observed amplitude dependence of stiffness is reported as a direct experimental outcome rather than a constructed result. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are identifiable. The framework appears to rest on standard assumptions from plate theory and interface modeling without new postulated entities.

pith-pipeline@v0.9.0 · 5471 in / 1187 out tokens · 64336 ms · 2026-05-13T18:28:39.273700+00:00 · methodology

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