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Polyconvexity and rank-one convexity imply the weak Hill inequality for incompressible isotropic materials in two dimensions.

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load-bearing objection The paper gives several direct proofs that polyconvexity and rank-one convexity imply weak Hill's inequality for isotropic materials with F in SL(2).

arxiv 2606.18879 v1 pith:W7UZRSHB submitted 2026-06-17 math.AP math-phmath.MP

Polyconvexity implies Hill's inequality in {rm SL}(2)

Ionel-Dumitrel Ghiba , Patrizio Neff , Maximilian P. Wollner This is my paper
classification math.AP math-phmath.MP
keywords polyconvexityHill inequalityrank-one convexityincompressible elasticitytwo-dimensional elasticityisotropic hyperelasticityLegendre-Hadamard ellipticity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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The paper examines relations between constitutive conditions for nonlinear isotropic elasticity. It proves that when the deformation gradient satisfies det F equals 1, both Legendre-Hadamard ellipticity and polyconvexity imply the weak Hill inequality. Hill's inequality is the monotonicity of the Kirchhoff stress tensor with respect to the logarithmic stretch tensor. This linkage arises specifically under the two-dimensional incompressible constraint even though the conditions remain independent for compressible materials. The connections are shown via several alternative proofs.

Core claim

For isotropic hyperelastic materials with deformation gradient in SL(2), both LH-ellipticity (rank-one convexity) and polyconvexity imply the weak Hill inequality, which is the monotonicity of the Cauchy stress with respect to the logarithmic stretch tensor.

What carries the argument

The SL(2) restriction together with isotropy, which converts the true stress-true strain monotonicity condition into Hill's inequality.

Load-bearing premise

The deformation gradient must satisfy det F exactly equal to one and the material response must be isotropic.

What would settle it

An explicit isotropic polyconvex stored-energy function on SL(2) for which the associated Kirchhoff stress fails to be monotone in the logarithmic stretch would disprove the implication.

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Editorial analysis

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Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that, for isotropic hyperelastic materials with deformation gradient restricted to SL(2), both Legendre-Hadamard ellipticity (rank-one convexity) and polyconvexity imply the weak form of Hill's inequality (monotonicity of the Kirchhoff stress with respect to the logarithmic stretch). Multiple alternative proofs are supplied that work directly with the principal logarithmic stretches and the isotropic representation of the stored-energy function, using the det F = 1 condition to convert the relevant monotonicity statements.

Significance. If the implication holds, the work clarifies the relations among a priori independent constitutive conditions in the incompressible two-dimensional setting, where TSTS-M+ reduces to Hill's inequality. The provision of several direct, explicit proofs is a strength that increases confidence in the result. This contributes to the constitutive theory of nonlinear elasticity by identifying when rank-one convexity, polyconvexity, and Hill monotonicity are simultaneously satisfied under the stated restrictions.

minor comments (2)
  1. The transition from the compressible TSTS-M+ condition to Hill's inequality under det F = 1 is invoked repeatedly; a single consolidated statement of the precise weak inequality (including the role of the hydrostatic pressure) in §1 or §2 would improve readability.
  2. Notation for the principal logarithmic stretches and the isotropic stored-energy function is introduced in §3; ensuring consistent use of the same symbols across all alternative proofs would reduce the chance of reader confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation to accept the manuscript. No major comments were provided in the report.

Circularity Check

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Direct implication proof; no circularity detected

full rationale

The manuscript supplies explicit mathematical proofs that LH-ellipticity and polyconvexity each imply the weak form of Hill's inequality for isotropic hyperelastic materials with F restricted to SL(2). The restriction to det F = 1 and isotropy is stated at the outset as the setting in which the implication is shown, not derived from the target statement. No fitted parameters, self-definitional relations, or load-bearing self-citations appear in the derivation chain; the work consists of direct algebraic and analytic manipulations on the principal logarithmic stretches and the isotropic stored-energy function. The central claim therefore remains independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted.

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read the original abstract

For compressible nonlinear isotropic elasticity it is well known that rank-one convexity, polyconvexity and the monotonicity of the Cauchy stress tensor with respect to the logarithmic stretch tensor (the true stress-true strain monotonicity, TSTS-M$^+$) are independent constitutive conditions which should, however, all together be satisfied for a physically meaningful description of idealized elastic materials. In the incompressible case, TSTS-M$^+$ turns into Hill's inequality since the Cauchy stress $\sigma$ reduces to the Kirchhoff stress $\tau$. Hill's inequality requires then monotonicity of the Kirchhoff stress in terms of the logarithmic stretch tensor evaluated for incompressible response. In this paper we clarify how the a priori independent notions of Legendre-Hadamard ellipticity (LH), polyconvexity and Hill's inequality are nevertheless intimately connected. More precisely, by providing several alternative proofs, we show that both LH-ellipticity (rank-one convexity) and polyconvexity imply the weak Hill inequality in the incompressible two-dimensional case.

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