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Polyconvexity does not imply true-stress-true-strain monotonicity in the incompressible three-dimensional case

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Pith's one-line read Polyconvexity alone does not force true Cauchy stress to rise with true Hencky strain for incompressible three-dimensional hyperelasticity.

desk verdict Clean, explicit counterexample that finally separates polyconvexity from TSTS-M (and rank-one convexity from TSTS-M) in incompressible 3-D hyperelasticity. read the letter →

arxiv 2607.06568 v1 pith:P7Y2H6M5 submitted 2026-06-10 physics.class-ph cond-mat.mtrl-scimath-phmath.MP

Dominik K. Klein , Maximilian P. Wollner , Patrizio Neff This is my paper · ORCID
classification physics.class-phcond-mat.mtrl-scimath-phmath.MP
keywords hyperelasticityincompressibilitypolyconvexitytrue-stress-true-strainmonotonicityHill’sinequalityLegendre-Hadamardellipticityrank-oneconvexityconstitutiveinequalities
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks which constitutive inequalities are needed for idealized isotropic hyperelasticity when the material is constrained to be incompressible in three dimensions. Polyconvexity is known to give existence theorems and real wave speeds, while true-stress-true-strain monotonicity (TSTS-M) is the multi-axial requirement that the Cauchy stress must increase with the Hencky strain. Earlier work had shown that Ball’s sufficient conditions for polyconvexity already imply TSTS-M; the open question was whether every polyconvex energy does the same. The authors settle the question negatively: they construct an explicit isotropic energy built from softplus functions of the signed singular values that is polyconvex for every real choice of coefficients, yet produces a non-monotone uniaxial Cauchy stress. Consequently polyconvexity (and the weaker rank-one convexity it implies) is not by itself enough to guarantee a physically reasonable stress-strain response.

What carries the argument

The isotropic softplus potential (2.1) written as a sum of four softplus functions of linear forms in the signed singular values and their products; its polyconvexity follows at once from the necessary-and-sufficient characterization of Wiedemann & Peter, while its loss of TSTS-M is read off from the closed-form uniaxial stress formula (2.7).

What would settle it

Direct numerical evaluation of the uniaxial Cauchy stress (2.7) at the two stretches 1.5 and 2.5 for the stated parameters: if the stress at 1.5 is not strictly larger than the stress at 2.5, or if an independent check shows that the energy fails to be polyconvex, the claim collapses.

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Extended reading notes

Core claim

In the incompressible three-dimensional setting, polyconvexity does not imply true-stress-true-strain monotonicity. An explicit isotropic potential that is polyconvex for all real parameters nevertheless yields a Cauchy stress that decreases over an interval of uniaxial stretch, violating TSTS-M (and therefore also Hill’s inequality).

Load-bearing premise

The proof that the softplus construction is polyconvex rests entirely on the recent necessary-and-sufficient characterization of isotropic polyconvexity in terms of signed singular values; if that characterization has a gap for this particular family, the counter-example fails.

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

Referee Report

0 major / 4 minor

Summary. The manuscript constructs an explicit isotropic incompressible hyperelastic potential (2.1) based on the softplus function of a linear combination of signed singular values and their products. Theorem 2.1 asserts that this potential is polyconvex for every real triple (a,b,c) by verifying the three necessary-and-sufficient conditions of Wiedemann & Peter (2026) / Geuken et al. (2026): convexity in the six signed-singular-value arguments, Π_{3}-invariance by construction, and lower semi-continuity. For the concrete coefficients (a,b,c)=(−5,−14,−22) the closed-form uniaxial Cauchy stress (2.7) is shown to be non-monotone (numerical values σ(1.5)≈14.5>σ(2.5)≈12.3), so TSTS-M fails (Theorem 2.2). The same potential remains rank-one convex (hence LH-elliptic) and produces a monotone true shear stress in simple shear. The two corollaries therefore establish that neither polyconvexity nor rank-one convexity implies TSTS-M in the three-dimensional incompressible setting, completing the diagram of constitutive implications summarized in Figure 1.

Significance. The result closes a previously open logical gap in the hierarchy of constitutive inequalities for idealized isotropic incompressible elasticity. Earlier work had shown that Ball’s sufficient conditions for polyconvexity already imply TSTS-M, and that the implication fails in the compressible three-dimensional case; the present counter-example demonstrates that the implication also fails once the full (necessary-and-sufficient) characterization of polyconvexity is admitted. The construction is elementary once the cited characterization is granted, supplies closed-form stress expressions, and is immediately usable as a test case for numerical schemes that rely solely on polyconvexity. The paper therefore supplies a clean, falsifiable negative answer that will be of lasting reference value.

minor comments (4)
  1. In the sentence preceding Corollary 2.3 the word “satisfsy” is misspelled; correct to “satisfy”.
  2. Figure 3 caption and the surrounding text refer to “ρ” in places where the uniaxial Cauchy stress is denoted σ; unify the notation.
  3. The asymptotic value τ o28 of the shear stress is stated after (2.5); a one-line derivation of this limit (using the known limits of the sigmoid) would make the claim self-contained.
  4. References to the authors’ own concurrent preprints (Wollner et al. 2026a,b; Klein et al. 2026a,b) are numerous; a short clarifying sentence that the present counter-example is independent of those works would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: explicit counterexample verifies polyconvexity and TSTS-M violation independently by direct construction and closed-form computation.

full rationale

The central claim (Corollary 2.3) is established by an explicit isotropic potential (2.1) built from the softplus of a linear form in the signed singular values. Theorem 2.1 verifies polyconvexity for arbitrary real coefficients by checking the three necessary-and-sufficient conditions of the external characterization (Wiedemann–Peter 2026 / Geuken et al. 2026 Cor. 2): convexity follows because softplus is convex and θ is linear (hence each summand is convex), Π₃-invariance holds by the four-term construction, and lower semi-continuity is immediate from continuity. Theorem 2.2 then supplies the closed-form uniaxial Cauchy stress (2.7) and exhibits a concrete numerical decrease (σ(1.5)≈14.5 > σ(2.5)≈12.3). Neither step reduces to a fitted quantity, a self-definition, or a load-bearing self-citation; prior papers by the same authors supply only background relations (Ball’s sufficient conditions imply TSTS-M, the overall diagram of implications). The argument is therefore self-contained against the external characterization it invokes and contains no circular reduction.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim is a pure mathematical separation result. It rests on standard differential calculus, the classical definitions of polyconvexity/rank-one convexity/TSTS-M, and one external characterization theorem for isotropic polyconvexity. Three real coefficients are chosen by hand to exhibit the violation; they are free parameters of the counterexample, not fitted quantities. No new physical entities are postulated.

free parameters (1)
  • a,b,c (material coefficients of the softplus potential) = (-5, -14, -22)
    Chosen by hand as (-5,-14,-22) solely to produce a visible non-monotonic uniaxial response while preserving polyconvexity; any other triple that yields a decrease would serve equally well.
assumptions (4)
  • domain assumption Wiedemann-Peter (2026) necessary-and-sufficient characterization of isotropic polyconvexity via signed singular values (convexity of h, Pi(3)-invariance, lower semi-continuity)
    Invoked as Theorem 1.5 and used as the sole criterion for proving polyconvexity of the constructed potential (Theorem 2.1).
  • standard math Polyconvexity implies rank-one convexity (Ciarlet 1988)
    Used to obtain the secondary corollary that rank-one convexity also fails to imply TSTS-M.
  • domain assumption TSTS-M is equivalent to strict convexity of the reduced energy in logarithmic principal stretches for incompressible isotropic materials
    Cited from Baaser (2026) and used to confirm the loss of TSTS-M via the reduced potential plot.
  • standard math Standard definitions of Cauchy stress, Hencky strain, softplus and sigmoid functions, and the special linear group SL(3)
    Background continuum-mechanics and calculus facts used throughout Sections 1-2.

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Pith. "Pith review of Polyconvexity does not imply true-stress-true-strain monotonicity in the incompressible three-dimensional case." pith.science (2026). https://pith.science/paper/P7Y2H6M5

@misc{pith2026260706568,
  author       = {Pith},
  title        = {Pith review of: Polyconvexity does not imply true-stress-true-strain monotonicity in the incompressible three-dimensional case},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P7Y2H6M5}},
  note         = {Machine review of arXiv:2607.06568}
}
read the original abstract

We study constitutive conditions of hyperelastic potentials for incompressible material behavior in three dimensions. By means of a counterexample, we show that polyconvexity does not imply true-stress-true-strain monotonicity. Thus, polyconvexity alone is not strong enough to guarantee a physically reasonable response for idealized elasticity.

Figures

Figures reproduced from arXiv: 2607.06568 by the authors.

Figure 1
Figure 1. Overview of various constitutive constraints and their relation in isotropic incompressible hyperelasticity. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Visualization of the softplus function (𝑥) employed for the potential in (2.1). The stress response of the potential, cf. (2.5) and (2.7), depends on the sigmoid function (𝑥) = D𝑥(𝑥), which is the first derivative of the softplus function. Polyconvexity is fulfilled if and only if (i) ℎ is convex, (ii) ̃ℎ is Π3 -invariant, which includes the six permutations introduced in (1.12) and the four symmetries ̃ℎ(𝜈1 ,… view at source ↗
Figure 3
Figure 3. Evaluation of the potential in (2.1) with parameters [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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    math.AP 2026-08 accept novelty 5.0 of 10

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