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arxiv: 2605.17485 · v1 · pith:OFH2LNM7new · submitted 2026-05-17 · 🧮 math.AP · cond-mat.soft

Variational derivation of the Flamant solution for a nonlinear elastic wedge

Dominik Engl , Paul Plucinsky , Ian Tobasco This is my paper

Pith reviewed 2026-05-19 22:18 UTC · model grok-4.3

classification 🧮 math.AP cond-mat.soft
keywords nonlinear elasticityFlamant solutionelastic wedgegeometric rigidityasymptotic analysisvariational principleshyperelastic materialssingular solutions
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The pith

The Flamant solution from linear elasticity is the leading-order response of a nonlinear elastic wedge to small tip loads or displacements.

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

The paper proves that the classical Flamant solution, originally derived in linear elasticity for a concentrated force at the tip of a wedge, also gives the dominant behavior in the nonlinear setting. This holds when the wedge is slightly truncated away from the tip and the boundary data are small, for any hyperelastic energy that grows super-quadratically at large strains and even for quadratic growth when the loads remain small enough. The argument restores compactness of low-energy sequences through a logarithmic change of variables that flattens the wedge into a strip, after first establishing a geometric rigidity bound whose constant stays uniform as the truncation shrinks. The result matters because it shows that the singular linear solution survives as the precise leading term once nonlinear effects are taken into account near the tip.

Core claim

Starting from a general nonlinear hyperelastic energy, the Flamant solution characterizes the leading-order response of a slightly truncated wedge to small boundary displacements or loads. The proof proceeds by applying a logarithmic change of variables sufficiently far from the tip, which converts the problem into an asymptotic variational principle whose minimizer is precisely the Flamant displacement field; the necessary compactness follows from a uniform-in-truncation geometric rigidity inequality in L^p that is inherited from the bi-Lipschitz invariance of the Friesecke-James-Müller constant.

What carries the argument

Logarithmic change of variables together with a uniform geometric rigidity inequality on truncated wedges, which restores compactness to low-energy sequences and yields the limiting variational principle for the Flamant solution.

If this is right

  • The asymptotic result holds for every hyperelastic energy with super-quadratic growth at infinity.
  • For quadratic-growth energies the same conclusion remains valid provided the tip displacements or loads are small enough.
  • The uniform geometric rigidity inequality on truncated wedges follows directly from bi-Lipschitz invariance of the Friesecke-James-Müller constant.
  • After the logarithmic change of variables the problem reduces to an explicit asymptotic variational principle whose unique minimizer is the Flamant field.

Where Pith is reading between the lines

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

  • The same logarithmic flattening technique may adapt to other singular solutions in nonlinear elasticity, such as crack-tip fields or dislocation singularities.
  • Numerical minimization on a family of truncated wedges with successively smaller cut-offs would directly test the rate at which the computed displacement converges to the Flamant field.
  • The result indicates that nonlinear material response does not remove the leading singularity for small loads, which may simplify reduced models of stress concentration.

Load-bearing premise

The hyperelastic energy must grow at least quadratically at large strains so that low-energy maps remain close to rigid motions after the logarithmic coordinate change.

What would settle it

Construct a sequence of deformations on successively less-truncated wedges whose energy is strictly lower than that of the Flamant solution while meeting the same boundary conditions; if the energy gap does not vanish in the limit, the asymptotic claim fails.

Figures

Figures reproduced from arXiv: 2605.17485 by Dominik Engl, Ian Tobasco, Paul Plucinsky.

Figure 1.1
Figure 1.1. Figure 1.1: Basic setup of the problem. (a) The Flamant solution describes an infinite linear elastic wedge with a point force at its tip. (b) We consider a truncated nonlinear elastic wedge with prescribed forces or displacements at the truncated boundaries r = δ, 1. (c) Scaling of the boundary data versus energy for the truncated wedge. The parameters δ and ϵ control the truncation length and size of the boundary … view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: Homogenization of kirigami metamaterials. (a) The rotating squares pattern in its checkerboard configuration; (b) a mechanism deformation counter-rotates the panels and causes the pattern to dilate. (c) Fabricated sample of the rotating squares pattern made by laser cutting a rubber sheet, with panels of size ∼ ℓ and hinges of size ∼ δ. (d-e) Soft modes arising under typical loading conditions are given … view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Bi-Lipschitz equivalence of Ωδ and Ω0. (a) The map Ψ rescales and shifts the θ-coordinate to transform a wedge with angle β − α to a reference wedge with angle π/2. (b) The map Φδ acts on the δ-truncation of the reference wedge and stretches the δ-strip (in green) so that the r = δ arc deforms to match the wedge boundary (dashed). If Φδ maps Uδ bijectively onto U0, then Ψ−1 ◦ Φδ ◦ Ψ maps Ωδ bijectively o… view at source ↗
read the original abstract

Concentrated forces acting at the tip of a two-dimensional wedge give rise to the classical Flamant solution to linear elasticity, whose displacement and strain are singular at the tip of the wedge. Starting from nonlinear elasticity, we prove that the Flamant solution gives the leading order response of a slightly truncated wedge to small boundary displacements or loads. This asymptotic result holds for general hyperelastic energies with super-quadratic growth at infinity; it also holds in the borderline case of quadratic growth at infinity, so long as the tip of the wedge is subjected to small enough displacements or loads. A main point of the proof is to restore compactness to low-energy sequences. We do so by applying a logarithmic change of variables sufficiently far from the tip. To justify this change of variables, we prove a geometric rigidity inequality in $L^p$ for truncated wedge domains with a constant that is uniform in the truncation length. This follows from the bi-Lipschitz invariance of the constant in the $L^p$ Friesecke--James--M\"uller inequality. Using this change of variables, we derive an asymptotic variational principle characterizing the Flamant solution in the singular limit of an ideal wedge.

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

1 major / 1 minor

Summary. The manuscript develops a variational approach to show that the Flamant solution of linear elasticity is the leading-order asymptotic behavior for the response of a nonlinearly elastic wedge that is slightly truncated near the tip, when subjected to small boundary displacements or loads. The result applies to hyperelastic stored-energy functions with super-quadratic growth, and also to the quadratic-growth case provided the loads are sufficiently small. Compactness is restored via a logarithmic change of variables, supported by a uniform geometric rigidity estimate on the family of truncated domains.

Significance. This result, if established, would be a notable contribution to the mathematical theory of nonlinear elasticity in domains with singularities. It furnishes a variational characterization of the Flamant solution in the nonlinear setting and demonstrates how geometric rigidity estimates can be adapted to truncated wedges. The handling of the quadratic-growth borderline case under small-load assumptions is particularly interesting and extends the applicability of the method.

major comments (1)
  1. [Abstract (statement on uniform L^p geometric rigidity) and its detailed proof] The assertion that the L^p geometric rigidity inequality holds with a constant uniform in the truncation parameter ε (as stated in the abstract and used to justify the logarithmic change of variables), derived from bi-Lipschitz invariance of the Friesecke–James–Müller inequality, requires careful verification. Any bi-Lipschitz homeomorphism mapping a fixed reference domain to the truncated wedge Ω_ε must have Lipschitz constants diverging like 1/ε, because the inner boundary arc length scales as O(ε) while the reference domain has O(1) length. This would render the rigidity constant ε-dependent and potentially unbounded, preventing the uniform control needed for compactness restoration and passage to the singular limit. This point is load-bearing for the central asymptotic claim.
minor comments (1)
  1. [Abstract] The abstract is concise but could briefly indicate the precise range of p for which the L^p rigidity estimate is established.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this important technical point concerning the uniformity of the geometric rigidity constant. We agree that the original justification via bi-Lipschitz invariance requires additional verification and will revise the paper to supply a complete, self-contained argument.

read point-by-point responses

  1. Referee: The assertion that the L^p geometric rigidity inequality holds with a constant uniform in the truncation parameter ε (as stated in the abstract and used to justify the logarithmic change of variables), derived from bi-Lipschitz invariance of the Friesecke–James–Müller inequality, requires careful verification. Any bi-Lipschitz homeomorphism mapping a fixed reference domain to the truncated wedge Ω_ε must have Lipschitz constants diverging like 1/ε, because the inner boundary arc length scales as O(ε) while the reference domain has O(1) length. This would render the rigidity constant ε-dependent and potentially unbounded, preventing the uniform control needed for compactness restoration and passage to the singular limit.

    Authors: We agree that a direct transfer via a single bi-Lipschitz map from a fixed reference domain yields constants that diverge with ε, so the original appeal to invariance does not immediately guarantee uniformity. We will revise the manuscript by replacing this brief justification with a detailed proof of the uniform L^p geometric rigidity estimate. The argument proceeds by decomposing the truncated wedge into an outer annular region (where the domains are uniformly bi-Lipschitz to a fixed reference annulus) and a near-tip region (controlled by a covering argument that exploits the smallness of ε together with the super-quadratic growth of the energy). This establishes the desired ε-independent constant without relying on a globally ε-uniform bi-Lipschitz equivalence. revision: yes

Circularity Check

0 steps flagged

No circularity: asymptotic derivation from nonlinear energy is self-contained

full rationale

The paper starts from a general hyperelastic energy with super-quadratic growth and derives the Flamant solution as the leading-order limit after a logarithmic change of variables on truncated wedges. Compactness is restored via a uniform L^p geometric rigidity inequality whose constant is asserted to follow from bi-Lipschitz invariance of the external Friesecke-James-Müller theorem. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the central variational principle is obtained from independent compactness and Gamma-convergence arguments rather than renaming or re-using the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background results in nonlinear elasticity and geometric rigidity. The new element is the uniformity of the rigidity constant with respect to truncation length, obtained via bi-Lipschitz invariance.

axioms (2)
  • domain assumption The stored-energy density satisfies super-quadratic growth at infinity (or quadratic growth with small loads)
    This growth condition is explicitly required for the result to hold and enables the compactness argument after the log change of variables.
  • standard math The bi-Lipschitz invariance of the constant in the L^p Friesecke-James-Müller inequality
    Invoked to obtain a rigidity inequality uniform in the truncation length for the wedge domains.

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