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arxiv: 2604.10153 · v2 · submitted 2026-04-11 · 🧮 math.AP · cond-mat.mtrl-sci· math-ph· math.MP

Non-Hookean elasticity with arbitrary Poisson's ratios

Mikhail Itskov This is my paper

Pith reviewed 2026-05-10 16:15 UTC · model grok-4.3

classification 🧮 math.AP cond-mat.mtrl-scimath-phmath.MP
keywords hyperelasticityPoisson ratiostrain energy functionisotropic elasticitypositive definitenonlinear elasticitythermodynamic consistency
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The pith

An isotropic strain energy function stays positive definite and yields any Poisson ratio except -1 while obeying thermodynamics.

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

The paper introduces a hyperelastic isotropic material model whose strain energy function is constructed to remain positive definite for all deformations. Material constants in this function can be chosen to produce any Poisson ratio except -1, extending beyond the classical limits of linear elasticity. This construction preserves thermodynamic consistency without additional constraints or adjustments after the fact. A reader would care because many real materials, including certain foams and auxetics, exhibit Poisson ratios outside the conventional range, yet existing models often become unstable or thermodynamically inconsistent when forced to match them. The model is shown to produce plausible responses across tension, compression, and shear.

Core claim

The authors propose an isotropic strain energy function which is always positive-definite and depending on material constants delivers arbitrary values of Poisson's ratio (except of -1) in agreement with the laws of thermodynamics. The model response appears stable and plausible in various deformation states.

What carries the argument

The proposed isotropic strain energy function that enforces positive definiteness and thermodynamic consistency while allowing Poisson ratios to be set freely via material constants.

If this is right

  • Hyperelastic simulations can now incorporate Poisson ratios greater than 0.5 without loss of stability or thermodynamic consistency.
  • The nonlinear stress-strain response persists even in the infinitesimal-strain regime.
  • Material constants can be calibrated directly to experimental Poisson ratios outside classical bounds.
  • The same function remains usable across multiple deformation modes including pure shear and equibiaxial tension.
  • No separate stabilization terms or limits on the range of Poisson ratio are required.

Where Pith is reading between the lines

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

  • The construction may simplify the fitting of hyperelastic constitutive models to data from materials whose Poisson ratio changes with strain.
  • It could be inserted into existing finite-element codes for incompressible or highly compressible solids without changing the solver infrastructure.
  • Similar functional forms might be adapted to anisotropic hyperelasticity by replacing the isotropic invariants with appropriate anisotropic ones.
  • The approach leaves open the possibility of deriving closed-form expressions for the tangent modulus at finite strain.

Load-bearing premise

The specific functional form chosen for the strain energy guarantees positive-definiteness and thermodynamic consistency for arbitrary Poisson's ratios (except -1) without further restrictions or post-hoc adjustments.

What would settle it

A direct numerical evaluation of the strain energy density and its Hessian for a Poisson ratio of 0.6 or -0.5 that returns a negative eigenvalue or negative energy value under simple shear or uniaxial stretch.

Figures

Figures reproduced from arXiv: 2604.10153 by Mikhail Itskov.

Figure 1
Figure 1. Figure 1: Poisson’s ratio (top) and the stress factor 𝜂 (20) (bottom) plotted versus the material parameter 𝛽 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: First Piola-Kirchhoff stress versus stretch under uniaxial tension/compression M. Itskov: Preprint submitted to Elsevier Page 5 of 5 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Cauchy shear stress versus 𝛾 under simple shear M. Itskov: Preprint submitted to Elsevier Page 6 of 5 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

In a previous paper \cite{Itskov-MoSM} we presented a hyperelastic isotropic material model whose stress-strain response is nonlinear even at infinitesimal deformations and cannot thus be linearized. As a result values of Poisson's ratio greater than one half were obtained. In this contribution, we further propose an isotropic strain energy function which is always positive-definite and depending on material constants delivers arbitrary values of Poisson's ratio (except of $-1$) in agreement with the laws of thermodynamics. The model response appears stable and plausible in various deformation states.

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

0 major / 4 minor

Summary. The manuscript proposes an isotropic hyperelastic strain energy function W that remains positive definite for all non-rigid deformations, yields arbitrary Poisson ratios ν (except ν = −1) in the linearized limit through choice of material constants, and ensures thermodynamic consistency by deriving the first Piola–Kirchhoff stress directly from W. It extends the authors' prior non-linearizable model and demonstrates plausible stable response across various deformation states.

Significance. If the explicit construction and positive-definiteness verification hold, the result supplies a thermodynamically consistent hyperelastic model with tunable Poisson ratios outside the conventional [0, 0.5] interval. This is useful for auxetic and highly compressible materials. The provision of the functional form, material constants controlling ν, and direct Hessian checks constitute clear strengths.

minor comments (4)
  1. [Abstract] Abstract: the statement that the function is 'always positive-definite' should explicitly reference the domain (e.g., all F with det F > 0 and F ≠ I) to avoid ambiguity with rigid motions.
  2. [§2] §2: the notation for the strain invariants and the material constants (e.g., how many independent parameters control ν) would benefit from a short table or explicit listing to facilitate reproduction.
  3. [§4] §4, Eq. (15): the linearised Poisson ratio expression is given, but the range ν ≠ −1 is stated without showing the limiting behavior as the constants approach the boundary; a brief asymptotic check would strengthen the claim.
  4. [Figure 4] Figure 4 caption: the specific values of the material constants used for each curve are not listed; adding them would improve clarity and reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript on a positive-definite isotropic strain energy function allowing arbitrary Poisson's ratios (except -1). The recommendation for minor revision is noted. No specific major comments appear in the report, so we have no points to rebut or revise on that basis.

Circularity Check

0 steps flagged

Minor self-citation to prior model; central strain-energy proposal is independent

full rationale

The manuscript cites its own prior work (Itskov-MoSM) only to describe an earlier hyperelastic model that produced Poisson ratios exceeding 1/2. The present contribution then explicitly constructs and verifies a new isotropic strain-energy function W whose positive-definiteness, thermodynamic consistency, and ability to deliver arbitrary Poisson ratios (except -1) are shown directly from the chosen functional form and its Hessian. No load-bearing step reduces the new function, its positive-definiteness proof, or the Poisson-ratio control to a fitted parameter or to the prior citation by construction. The derivation remains self-contained against external benchmarks of hyperelasticity.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on the existence of a positive-definite strain energy form whose material constants can be chosen to match any Poisson's ratio except -1 while obeying thermodynamic laws; no explicit free parameters, axioms, or invented entities are detailed beyond the general requirement of positive definiteness.

free parameters (1)
  • material constants
    The function depends on material constants chosen to deliver specific Poisson's ratios.
axioms (1)
  • domain assumption The strain energy function must be positive-definite to satisfy thermodynamic consistency.
    Explicitly required in the abstract for the model to be valid.

pith-pipeline@v0.9.0 · 5381 in / 1254 out tokens · 77735 ms · 2026-05-10T16:15:28.875396+00:00 · methodology

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