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Polyconvexity for incompressible inversion-symmetric energies of Valanis–Landel type

This paper establishes sufficient conditions for the polyconvexity of incompressible, inversion-symmetric Valanis–Landel type energies by deriving an explicit polyconvex extension and demonstrating that while certain exponential Hencky-type energies are rank-one convex, the standard quadratic Hencky energy is not.

Original authors: Ionel-Dumitrel Ghiba, Maximilian P. Wollner, Patrizio Neff

Published 2026-09-28

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CC BY 4.0

Original authors: Ionel-Dumitrel Ghiba, Maximilian P. Wollner, Patrizio Neff

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where materials, from the rubber in a tire to the soft tissue in a human body, could be predicted with absolute certainty as they stretch, twist, and compress. This is the realm of nonlinear elasticity, a branch of physics and mathematics dedicated to understanding how solid objects change shape under force. For engineers and scientists, the goal is to find a mathematical recipe that describes a material's energy—how much effort it takes to deform it—without running into impossible scenarios where the material would spontaneously tear apart or behave in a physically nonsensical way. A key requirement for this stability is a property called rank-one convexity, a technical condition that ensures the material responds smoothly and predictably when pulled in different directions. Without it, the mathematical models used to design everything from airplane wings to medical implants could fail, leading to catastrophic errors in simulation.

For decades, researchers have relied on a class of energy formulas known as Valanis–Landel type, which break down a material's total energy into simple, separate parts based on how much it stretches in three different directions. These formulas are popular because they are intuitive and easy to work with, often resembling a sum of individual stretch effects. However, a lingering question has been whether these simple, separate parts are enough to guarantee the material's stability. Specifically, scientists wondered if a material that looks stable when you look at its stretching in isolation would remain stable when those stretches interact in complex, three-dimensional ways. The answer, it turns out, is not as straightforward as it seems.

In a recent study, a team of mathematicians from Romania, Austria, and Germany tackled this problem by focusing on a specific, symmetric type of energy where the material behaves the same way whether it is stretched or compressed by the same amount. They investigated a family of energy functions that depend on the logarithm of the stretch, a measure that treats large and small deformations with a balanced perspective. The researchers discovered that for these materials to be truly stable, the simple act of adding up the energy of each stretch is not enough. Instead, they found a precise mathematical condition that the energy function must satisfy to ensure stability. They proved that if the energy function follows a specific rule regarding how it curves as the stretch increases, it can be extended into a larger, more complex mathematical structure that guarantees stability. This structure, known as a polyconvex extension, acts as a safety net, ensuring that the material will never behave erratically under any deformation.

The team's work provides a clear, one-dimensional test that engineers can use to check if a proposed energy formula is safe. If the formula passes this test, it is guaranteed to be stable. However, the study also delivers a crucial warning. The researchers demonstrated that a very famous and widely used energy model, known as the quadratic Hencky energy, which simply squares the logarithmic stretch, fails this stability test. Even though this model looks perfectly reasonable and symmetric on the surface, it is not stable when the material undergoes a specific type of shearing motion, like sliding layers of a deck of cards past one another. This finding overturns the assumption that simple convexity in the stretch variable is sufficient for stability. It shows that a formula can look smooth and well-behaved in isolation but still hide dangerous instabilities when the material is subjected to real-world forces.

By establishing these rigorous conditions, the researchers have provided a new tool for designing better materials. Their work clarifies exactly which mathematical forms are safe to use and which are not, preventing the use of models that might look good on paper but fail in practice. They showed that while some complex, exponential versions of these energy formulas are indeed stable and can be trusted, the simpler, classic versions are not. This distinction is vital for anyone relying on computer simulations to predict how materials will behave, ensuring that the virtual models used to build the physical world are grounded in mathematical reality. The study confirms that in the complex dance of deformation, simplicity is not always a virtue, and a deeper, more careful look at the underlying mathematics is required to ensure safety and reliability.

Technical Summary: Polyconvexity for Incompressible Inversion-Symmetric Energies of Valanis–Landel Type

Problem Statement
The paper addresses the challenge of verifying material stability for incompressible isotropic elastic energies of the Valanis–Landel type. These energies are defined as a separable sum of a scalar function of the principal stretches (singular values of the deformation gradient $F$ ). While Legendre–Hadamard ellipticity (rank-one convexity) is a necessary condition for material stability, its direct analytical verification is often burdensome. A stronger, more tractable condition is polyconvexity, which implies rank-one convexity. The authors focus on the specific subclass of energies that satisfy term-wise inversion symmetry, i.e., $w(\lambda) = w(\lambda^{-1})$ , which can be expressed as $W(F) = \sum \psi(|\log \lambda_i|)$ . The central problem is to determine tractable conditions on the scalar function $\psi$ that ensure the energy is polyconvex (and thus rank-one convex) on the special linear group $SL(3)$ .

Methodology
The authors employ a combination of spectral analysis, convexity theorems, and direct verification along rank-one lines.

Spectral Identity: The core of the methodology relies on the identity valid on $SL(3)$ : $\nu_i(F + \text{Cof } F) = \lambda_i + \lambda_i^{-1} = 2\cosh(|\log \lambda_i|)$ , where $\nu_i$ denotes the singular values. This identity links the logarithmic strain measure to the singular values of the sum of the deformation gradient and its cofactor.
Polyconvex Extension: Using Ball's convexity theorem for functions of singular values, the authors construct an explicit polyconvex extension for the energy. They define a transformation $g(s) = \psi(\text{arcosh}(s/2))$ for $s \ge 2$ . If $g$ admits a convex and non-decreasing extension $\bar{g}$ to $[0, \infty)$ , the energy $W_\psi$ is shown to be the restriction of the polyconvex function $P(X, Y, d) = \sum \bar{g}(\nu_i(X+Y))$ to $SL(3)$ .
Differential Conditions: The paper derives a one-dimensional differential sufficient condition for the existence of such an extension $\bar{g}$ . This involves analyzing the derivatives of $\psi$ to ensure $g$ is convex and non-decreasing on $[2, \infty)$ and can be extended smoothly to $[0, 2]$ .
Direct Verification: A direct proof of rank-one convexity is provided by analyzing the energy along admissible rank-one lines $F_t = F + t a \otimes b$ contained in $SL(3)$ , utilizing the Sherman–Morrison formula to track the evolution of singular values.

Key Contributions and Results

Explicit Polyconvex Extension: The paper establishes that if the transformed function $g(s) = \psi(\text{arcosh}(s/2))$ has a convex, non-decreasing extension $\bar{g}$ to $[0, \infty)$ , then the Valanis–Landel energy $W_\psi(F) = \sum \psi(|\log \lambda_i|)$ is the restriction of an explicit polyconvex function to $SL(3)$ .
One-Dimensional Sufficient Condition: The authors derive a specific differential inequality: $\psi''(x) \ge \psi'(x) \coth x$ for $x > 0$ (along with $\psi'(0)=0$ and $\psi' \ge 0$ ). Satisfying this condition guarantees the existence of the required extension $\bar{g}$ , thereby ensuring polyconvexity.
Exponential Hencky-Type Energy: As a primary application, the paper proves that the energy $W_{\text{exp}, k}(F) = \mu \sum \exp(k \log^2 \lambda_i)$ is rank-one convex on $SL(3)$ for $k \ge 1/6$ . This energy admits a polyconvex extension defined by a piecewise function involving $\exp(k \text{arcosh}^2(s/2))$ .
Counterexample (Quadratic Hencky Energy): The paper demonstrates that scalar convexity of the function in terms of $\log \lambda$ alone is insufficient for rank-one convexity. Specifically, the quadratic Hencky energy $W_H(F) = \mu \sum \log^2 \lambda_i$ (where $\psi(x) = x^2$ ) fails to be rank-one convex on $SL(3)$ . A simple-shear computation shows that the second derivative of the energy along a rank-one line becomes negative, violating convexity.

Significance and Claims
The paper claims to provide a "tractable" framework for verifying the stability of a broad class of incompressible, inversion-symmetric energies. By reducing the complex multi-dimensional problem of rank-one convexity on $SL(3)$ to a one-dimensional differential condition on the scalar generator $\psi$ , the authors offer a practical tool for constructing stable material models.

The work highlights that while the Valanis–Landel form is intuitive and separable, simple convexity of the scalar function with respect to the stretch variable is not sufficient for stability. The derived criterion successfully identifies a generalization of the Hencky energy (the exponential form) that is stable, while explicitly ruling out the standard quadratic Hencky energy in the incompressible 3D setting. The authors note that their condition is sufficient for the existence of the specific polyconvex extension derived but do not claim it is necessary for rank-one convexity in general. The results are framed as a contribution to the theoretical understanding of constitutive requirements for material stability, particularly in the context of logarithmic strain energies.

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