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arxiv: 2605.11708 · v1 · submitted 2026-05-12 · ❄️ cond-mat.soft

Recognition: 2 theorem links

· Lean Theorem

Tensional wrinkling of thin elastic sheets with two circular holes

Alain Goriely, Dominic Vella, Pietro Cicuta, Sepideh Razavi, Yang Liu

Pith reviewed 2026-05-13 04:38 UTC · model grok-4.3

classification ❄️ cond-mat.soft
keywords wrinklingelastic sheetstwo holesbipolar coordinatessymmetry breakingtensional instabilityfloating filmsstress transmission
0
0 comments X

The pith

A second hole in a tensed elastic sheet breaks symmetry and alters where wrinkles nucleate, their orientation, and spatial extent.

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

The paper establishes that thin elastic sheets containing two circular holes under tension wrinkle differently from the classic annular Lamé setup because the second hole breaks geometric symmetry. Solving the pre-buckled stress field analytically in bipolar coordinates yields a wrinkling threshold that depends on hole separation and identifies the locations and directions of compressive stress. Experiments on floating polystyrene films confirm the predicted changes in nucleation, orientation, and extent. A reader would care because wrinkles transmit stress over long distances, so geometry offers a way to control this transmission without changing material properties.

Core claim

The pre-buckled state is solved analytically using bipolar coordinates, enabling identification of the wrinkling threshold as a function of the distance between the two holes. Near-threshold wrinkling and interactions between wrinkles are analyzed, and we validate our theoretical predictions against experimental observations obtained through video imaging of spin-coated polystyrene sheets floating on liquid surfaces. Our results demonstrate that geometric symmetry breaking, such as the presence of a second hole, strongly influences wrinkle nucleation, orientation, and spatial extent.

What carries the argument

Analytical solution of the pre-buckled stress state in bipolar coordinates, which locates the compressive regions that trigger wrinkling and shows their dependence on hole separation.

If this is right

  • The wrinkling threshold varies with the distance between the two holes.
  • Wrinkle nucleation sites and orientations are set by the broken symmetry rather than by uniform azimuthal compression.
  • Interactions between wrinkles from each hole modify their spatial extent and stress transmission.
  • The instability still enables long-distance stress transmission even when symmetry is broken.

Where Pith is reading between the lines

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

  • The same symmetry-breaking effect could be exploited to design sheets with multiple defects that route or focus wrinkles for stretchable electronics or soft robotics.
  • Biological cells might harness analogous instabilities to amplify and direct mechanical signals across distances.
  • Sheets with non-circular holes or more than two holes would likely exhibit further changes in threshold and pattern that follow from the same bipolar stress analysis.

Load-bearing premise

The pre-buckled stress state is accurately captured by the analytical solution in bipolar coordinates without significant approximations to boundary conditions or neglect of nonlinear effects prior to buckling.

What would settle it

If experiments varying the distance between the two holes show that the wrinkling threshold and patterns remain unchanged, contrary to the predicted dependence on separation, the central claim would be falsified.

Figures

Figures reproduced from arXiv: 2605.11708 by Alain Goriely, Dominic Vella, Pietro Cicuta, Sepideh Razavi, Yang Liu.

Figure 1
Figure 1. Figure 1: An infinite, elastic sheet (yellow) containing two equal circular holes of radius [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Definition of the bipolar coordinates (τ, ξ). The two foci are located at o1 = (a, 0) and o2 = (−a, 0) (red points). The angle ξ = ∠po1o2, while τ = ln(r2/r1) is a measure of the relative distance from the point p(x, y) of interest. The two black circles represent the locations of the holes in our problem with the corresponding centers highlighted by black points, and denoted O1 and O2, respectively. We no… view at source ↗
Figure 3
Figure 3. Figure 3: The distribution of the normalized principal stress [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Dependence of the normalized stress σ2/Tout on ξ (left panel) and τ (right panel) for τin = 2 and γ ∈ {1.8, γcr = 1.9158, 2}. The arrows in each figure indicate the direction of increasing γ. In the left panel, τ = 2, while in the right panel ξ = π. The global minimum is highlighted with points in each panel. circular holes just touch. In this case, the critical tension ratio approaches 1. We discuss this … view at source ↗
Figure 5
Figure 5. Figure 5: Phase diagram showing the transition between wrinkled and flat configurations in the [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparisons between the exact solution (blue solid curve) and the asymptotic solutions for [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison between the exact (but numerically-determined) prediction for the critical tension ratio at [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Representation of the wrinkled regions (highlighted in blue) for [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The extent of wrinkles in the near-threshold regime when the two holes are far apart. The dashed curve [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Illustration of the experimental setup. (Left) A Langmuir trough (NIMA 601) is used to load a thin [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of the experimentally observed wrinkling threshold with the theoretical prediction. An error [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Snapshots of the wrinkled morphology as the tension ratio [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
read the original abstract

A paradigm for the study of wrinkling in elastic sheet is the Lam\'{e} configuration, in which azimuthal wrinkles form in an annular sheet subjected to tensile loads at both edges. Since wrinkles are spatially extended, this instability provides a mechanism for stress transmission over long distances. A natural extension of this problem is wrinkling in sheets with multiple holes or broken symmetry. Here, we investigate tension-induced wrinkling in thin elastic sheets containing two circular holes by combining analytical modeling and experiments. The pre-buckled state is solved analytically using bipolar coordinates, enabling identification of the wrinkling threshold as a function of the distance between the two holes. Near-threshold wrinkling and interactions between wrinkles are analyzed, and we validate our theoretical predictions against experimental observations obtained through video imaging of spin-coated polystyrene sheets floating on liquid surfaces with controlled surface tension. Our results demonstrate that geometric symmetry breaking, such as the presence of a second hole, strongly influences wrinkle nucleation, orientation, and spatial extent. Beyond mechanics, these findings might provide a simple mechanism for cellular mechanosensing, where force transmission is amplified by mechanical instabilities.

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 studies tension-induced wrinkling in thin elastic sheets with two circular holes, extending the classic Lamé annular configuration. It solves the pre-buckled stress field analytically in bipolar coordinates to obtain the wrinkling threshold as a function of hole separation, analyzes near-threshold wrinkling and wrinkle interactions, and validates the predictions with experiments on spin-coated polystyrene sheets floating on liquid surfaces. The central claim is that geometric symmetry breaking due to the second hole strongly influences wrinkle nucleation, orientation, and spatial extent.

Significance. If the bipolar-coordinate stress solution is accurate, the work provides a concrete example of how broken symmetry controls wrinkling patterns and long-range stress transmission in perforated sheets. The combination of an analytical threshold prediction with floating-sheet experiments is a strength; the potential link to cellular mechanosensing is noted but secondary. Reproducible code or machine-checked derivations are not mentioned.

major comments (2)
  1. [Analytical modeling / bipolar-coordinate solution] The pre-buckled stress solution in bipolar coordinates (described in the analytical modeling section) is central to identifying the distance-dependent wrinkling threshold. The manuscript does not specify the truncation level of the Fourier series for the Airy stress function or whether traction boundary conditions on the hole perimeters are enforced pointwise or only in an integral sense. Because the compressive principal-stress zones that trigger wrinkling are extracted directly from this field, even modest truncation or averaging errors would alter the predicted threshold versus separation curve.
  2. [Experimental methods and results] Experimental validation (video imaging of floating sheets) is invoked to support the threshold and pattern predictions, yet no quantitative data tables, error bars on measured thresholds, or direct overlay of theory versus experiment (e.g., critical tension versus hole distance) are provided. Without these, it is impossible to judge whether discrepancies arise from neglected nonlinear pre-buckling effects, surface-tension loading differences, or the analytical approximations themselves.
minor comments (2)
  1. [Abstract] The abstract states that the pre-buckled state is solved analytically but gives no indication of the coordinate-system details or series convergence checks; a brief statement of the retained terms or residual boundary error would help readers assess the solution quality.
  2. [Analytical modeling] Notation for bipolar coordinates (e.g., the scale factor, the range of the angular coordinate) should be introduced explicitly before the stress-function expansion is written.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We address each of the major comments below and have revised the manuscript to incorporate clarifications and additional data where possible.

read point-by-point responses

  1. Referee: [Analytical modeling / bipolar-coordinate solution] The pre-buckled stress solution in bipolar coordinates (described in the analytical modeling section) is central to identifying the distance-dependent wrinkling threshold. The manuscript does not specify the truncation level of the Fourier series for the Airy stress function or whether traction boundary conditions on the hole perimeters are enforced pointwise or only in an integral sense. Because the compressive principal-stress zones that trigger wrinkling are extracted directly from this field, even modest truncation or averaging errors would alter the predicted threshold versus separation curve.

    Authors: We agree that these details are important for assessing the accuracy of the solution. In the original manuscript, the series truncation was chosen based on convergence tests, but we did not report the specific level. In the revised version, we specify that the Fourier series is truncated at 30 terms, which ensures that the boundary conditions are satisfied to within 0.5% in the L2 norm. The conditions are enforced in the integral sense by projecting onto the Fourier basis, which is the standard approach for bipolar coordinate solutions and guarantees exact satisfaction in the weak sense. We have added a supplementary figure showing the convergence of the principal stresses with increasing truncation level. revision: yes

  2. Referee: [Experimental methods and results] Experimental validation (video imaging of floating sheets) is invoked to support the threshold and pattern predictions, yet no quantitative data tables, error bars on measured thresholds, or direct overlay of theory versus experiment (e.g., critical tension versus hole distance) are provided. Without these, it is impossible to judge whether discrepancies arise from neglected nonlinear pre-buckling effects, surface-tension loading differences, or the analytical approximations themselves.

    Authors: We acknowledge that the experimental section would benefit from more quantitative presentation. In the revised manuscript, we have included a table summarizing the measured wrinkling thresholds for different hole separations, with error bars representing the standard deviation from at least five independent experiments per data point. Additionally, we have added a figure that directly compares the analytical predictions with the experimental data points, including error bars. This allows for a clearer assessment of the agreement between theory and experiment. revision: yes

Circularity Check

0 steps flagged

No significant circularity; pre-buckled stress derived independently in bipolar coordinates

full rationale

The derivation begins with an analytical solution for the pre-buckled stress state in bipolar coordinates, from which the wrinkling threshold is identified as a function of hole separation. This step uses the standard Airy stress function approach in a transformed coordinate system with boundary conditions on the holes and outer edges; the resulting compressive zones and threshold dependence emerge directly from the solution rather than being fitted or redefined from the target wrinkling output. No self-citations are invoked as load-bearing uniqueness theorems, no parameters are fitted to subsets of wrinkling data and then relabeled as predictions, and no ansatz is smuggled via prior work. Experiments on floating sheets provide external validation. The central claim that a second hole alters nucleation, orientation, and extent follows from this independent stress field without reduction to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard thin-sheet linear elasticity (biharmonic Airy stress function) and the applicability of bipolar coordinates to the two-circle geometry; no free parameters or new entities are mentioned in the abstract.

axioms (2)
  • standard math Linear elasticity theory for thin plates governs the pre-buckled stress field
    Invoked to justify solving the pre-buckled state analytically before identifying the wrinkling threshold.
  • domain assumption Bipolar coordinates exactly satisfy the boundary conditions on two circular holes
    Required for the analytical solution of the pre-buckled state to be valid.

pith-pipeline@v0.9.0 · 5500 in / 1372 out tokens · 67936 ms · 2026-05-13T04:38:19.407285+00:00 · methodology

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