True-Stress–True-Strain Monotonicity

• True-stress–true-strain monotonicity is a constitutive requirement ensuring that true stress, derived from the Kirchhoff or Cauchy stress, increases with Hencky logarithmic strain.
• It encompasses tensorial, spectral, and rate-based formulations, clarifying its relationships with Hill’s inequality, polyconvexity, and Legendre–Hadamard ellipticity.
• The concept is pivotal in both theoretical modeling and experimental validation, influencing numerical simulations, material stability analyses, and neural-network constitutive models.

True-stress–true-strain monotonicity (TSTS-M) is a constitutive monotonicity requirement formulated for the pairing of true stress—usually the Cauchy stress $\sigma$, or the Kirchhoff stress $\tau = J\sigma$—with true strain taken as the Hencky logarithmic strain, either in left form $h=\log V$ or right form $E=\log U$. In isotropic hyperelasticity it expresses, in a multiaxial sense, that stress should increase with logarithmic strain; recent work has clarified its tensorial, spectral, and rate-based formulations, its reduction to Hill’s inequality in incompressibility, and its nontrivial relation to polyconvexity and Legendre–Hadamard ellipticity (Wollner et al., 10 Sep 2025, Wollner et al., 19 May 2026).

1. Constitutive setting and stress–strain pairing

The standard kinematic objects are the deformation gradient $F$, the left and right Cauchy–Green tensors

$B=FF^\top,\qquad C=F^\top F,$

and the associated stretch tensors

$V=\sqrt{B},\qquad U=\sqrt{C}.$

For isotropic response, the principal stretches $\lambda_i>0$ are the singular values of $F$, and the principal logarithmic strains are $\log \lambda_i$. The cited works use the Hencky strain in either left or right form; for isotropy, $\tau = J\sigma$0 and $\tau = J\sigma$1 have the same principal stretches and hence the same principal logarithmic stretches (Wollner et al., 10 Sep 2025).

In the hyperelastic setting, the stored energy may be written in logarithmic variables, for example as $\tau = J\sigma$2. The Kirchhoff stress is then work-conjugate to logarithmic strain:

$\tau = J\sigma$3

while the Cauchy stress is

$\tau = J\sigma$4

In principal directions this becomes

$\tau = J\sigma$5

For incompressible response, $\tau = J\sigma$6, so $\tau = J\sigma$7 and the true-stress–true-strain pairing becomes particularly direct (Wollner et al., 10 Sep 2025, Wollner et al., 19 May 2026).

This constitutive pairing is not merely terminological. In the cited literature, the logarithmic strain is singled out because it is conjugate to the Kirchhoff stress and because monotonicity in $\tau = J\sigma$8 admits both finite-difference and differential characterizations that are compatible with isotropy, spectral calculus, and corotational objective rates (Wollner et al., 10 Sep 2025).

2. Formal definitions and equivalent formulations

The local, strong form is commonly denoted TSTS-M$\tau = J\sigma$9 and requires positive definiteness of the Jacobian of the stress map with respect to logarithmic strain:

$h=\log V$0

A global finite-difference form, denoted TSTS-M$h=\log V$1 in the cited work, is

$h=\log V$2

Under convexity of the admissible logarithmic-strain set, TSTS-M$h=\log V$3 implies TSTS-M$h=\log V$4 (Wollner et al., 10 Sep 2025).

For isotropic hyperelasticity, the spectral form is especially useful. TSTS-M$h=\log V$5 is equivalent to positive definiteness of the matrix

$h=\log V$6

In other words, the principal Cauchy stresses must be a strictly monotone function of the principal logarithmic strains. This yields a concrete operational test in principal-value space (Wollner et al., 10 Sep 2025).

A complementary formulation uses objective stress rates. For an isotropic Cauchy-elastic law $h=\log V$7, the corotational stability postulate

$h=\log V$8

with the Zaremba–Jaumann rate, is equivalent to positive definiteness of the symmetric part of $h=\log V$9; the same exact equivalence holds for the corotational logarithmic rate, and a corresponding Green–Naghdi statement is formulated as a conjecture (Neff et al., 2024).

In incompressible elasticity, the admissible logarithmic strains are trace-free and $E=\log U$0. The monotonicity condition then reduces to Hill’s inequality on the isochoric manifold,

$E=\log U$1

which the recent literature refers to as the weak Hill inequality because it is imposed only on the trace-free subspace rather than on all symmetric tensors (Ghiba et al., 17 Jun 2026).

At a more abstract level, if a constitutive mapping is a primary matrix function,

$E=\log U$2

then TSTS-M is equivalent to scalar monotonicity of the generator $E=\log U$3: for $E=\log U$4 spectral laws, Hilbert–Schmidt monotonicity on $E=\log U$5 is equivalent to $E=\log U$6 on the relevant interval (Martin et al., 2014).

3. Relation to polyconvexity, ellipticity, and Hill’s inequality

Polyconvexity is the requirement that

$E=\log U$7

with $E=\log U$8 convex in its arguments. It implies quasiconvexity and rank-one convexity, hence the Legendre–Hadamard condition. In differentiable form, Legendre–Hadamard ellipticity reads

$E=\log U$9

These conditions have a clear mathematical stability role, but the cited work emphasizes that they do not by themselves encode the same mechanical monotonicity content as TSTS-M$F$0 (Wollner et al., 10 Sep 2025).

The mechanical scope of the conditions is different. Rank-one convexity, and therefore Legendre–Hadamard ellipticity, enforces monotone true shear stress in simple shear. TSTS-M$F$1, by contrast, enforces monotone axial true stress in unconstrained uniaxial extension or compression. In compressible isotropic elasticity the two requirements are therefore independent: polyconvexity does not imply monotone axial true stress, and TSTS-M$F$2 does not imply monotone shear response or Legendre–Hadamard ellipticity (Wollner et al., 10 Sep 2025).

The incompressible setting is more structured but still dimension-dependent. In objective isotropic elasticity on $F$3, rank-one convexity and polyconvexity are equivalent, and both imply the weak Hill inequality. The argument proceeds through reduction to a one-parameter logarithmic variable on the deviatoric subspace and convexity of the reduced energy in that variable (Ghiba et al., 17 Jun 2026).

In three-dimensional incompressible isotropic hyperelasticity, the implication is not valid in full generality, but recent work proves it for a large and practically relevant subclass. If the energy satisfies Ball’s sufficient conditions for isotropic polyconvexity in terms of principal stretches and cofactors, then strict convexity in logarithmic strain on the trace-free subspace follows, which yields Hill’s inequality and hence global TSTS monotonicity for incompressible response (Wollner et al., 19 May 2026).

4. Model classes, examples, and counterexamples

Several families furnish positive examples in the incompressible setting. Ogden-type energies

$F$4

are polyconvex and satisfy Hill’s inequality and TSTS-M for $F$5. More generally, if

$F$6

with $F$7 convex and non-decreasing, strictly so in $F$8 or $F$9, then the energy is polyconvex and satisfies Hill’s inequality. The same work also gives a counterexample showing that TSTS-M does not imply polyconvexity:

$B=FF^\top,\qquad C=F^\top F,$0

satisfies Hill’s inequality but is neither rank-one convex nor polyconvex (Wollner et al., 19 May 2026).

In compressible isotropic elasticity, explicit counterexamples separate the major constitutive conditions. One polyconvex energy,

$B=FF^\top,\qquad C=F^\top F,$1

has a proper linearization but fails monotonicity of the axial true stress in unconstrained uniaxial extension. Conversely,

$B=FF^\top,\qquad C=F^\top F,$2

satisfies the sufficient invariant-based TSTS-M$B=FF^\top,\qquad C=F^\top F,$3 conditions used in that paper but has non-monotone true shear stress in simple shear, so it cannot be rank-one convex (Wollner et al., 10 Sep 2025).

A separate obstruction concerns energies built solely from scalar logarithmic strain measures. For $B=FF^\top,\qquad C=F^\top F,$4, there is no strictly monotone function $B=FF^\top,\qquad C=F^\top F,$5 such that either

$B=FF^\top,\qquad C=F^\top F,$6

or

$B=FF^\top,\qquad C=F^\top F,$7

is rank-one convex on $B=FF^\top,\qquad C=F^\top F,$8. Moreover, in dimension $B=FF^\top,\qquad C=F^\top F,$9 a volumetric–isochoric split of the form

$V=\sqrt{B},\qquad U=\sqrt{C}.$0

cannot be rank-one convex if $V=\sqrt{B},\qquad U=\sqrt{C}.$1 is strictly monotone. By contrast, the planar case admits a sharp positive characterization, and $V=\sqrt{B},\qquad U=\sqrt{C}.$2 is polyconvex for $V=\sqrt{B},\qquad U=\sqrt{C}.$3 (Martin et al., 2017).

Hencky-type models illustrate the distinction between monotonicity of $V=\sqrt{B},\qquad U=\sqrt{C}.$4 and monotonicity of $V=\sqrt{B},\qquad U=\sqrt{C}.$5. The quadratic Hencky energy is globally convex in $V=\sqrt{B},\qquad U=\sqrt{C}.$6, so Hill’s inequality holds for $V=\sqrt{B},\qquad U=\sqrt{C}.$7; however, because

$V=\sqrt{B},\qquad U=\sqrt{C}.$8

the corresponding $V=\sqrt{B},\qquad U=\sqrt{C}.$9–$\lambda_i>0$0 map generally fails TSTS monotonicity in compression. Earlier work on the exponentiated Hencky family proved global monotonicity of $\lambda_i>0$1 and TSTS-M$\lambda_i>0$2 for $\lambda_i>0$3 on bounded-distortion domains, together with global invertibility of the true-stress–true-strain relation in dimensions two and three (Neff et al., 2014). A later numerical survey reports analytical positive results for preferred exponentiated Hencky split forms under $\lambda_i>0$4, $\lambda_i>0$5, $\lambda_i>0$6, and $\lambda_i>0$7, while also documenting that many classical compressible models do not satisfy TSTS-M$\lambda_i>0$8 (Ghiba et al., 29 Aug 2025).

5. Incompressible isotropy and neural-network constitutive models

A recent development is the explicit concurrent enforcement of polyconvexity and TSTS monotonicity in incompressible isotropic hyperelasticity. The key representation is

$\lambda_i>0$9

with $F$0 convex, permutation-invariant in the first three and last three variables separately, and monotonically increasing in all six arguments. Under incompressibility,

$F$1

so strict monotonicity of $F$2 in either the first or last three arguments, or strict convexity of $F$3, yields strict convexity of $F$4 on trace-free directions. This is equivalent to Hill’s inequality, and integration then gives global TSTS monotonicity for incompressible response (Wollner et al., 19 May 2026).

The same paper shows that, in incompressible isotropic elasticity, Hill’s inequality is equivalent to strict convexity of the reduced potential

$F$5

because admissible variations satisfy $F$6. This reduced-convexity statement is important because full convexity in $F$7 may fail even when the incompressible monotonicity condition holds.

These constitutive results were then embedded into four physics-augmented neural-network architectures. PANN–I uses the invariants $F$8, PANN–$F$9 uses $\log \lambda_i$0, PANN–$\log \lambda_i$1 uses principal stretches and their cofactors with permutation averaging, and PANN–$\log \lambda_i$2 uses signed singular values with $\log \lambda_i$3-invariance. The first three architectures guarantee polyconvexity and TSTS-M a priori; PANN–$\log \lambda_i$4 guarantees polyconvexity via signed-singular-value criteria, while TSTS-M is not automatic from Ball’s route. The models were calibrated to Treloar rubber, EPDM, and DLP-50 data under UX, BX, and PS loading using mean-squared error in first Piola–Kirchhoff stress $\log \lambda_i$5, Adam, batch size $\log \lambda_i$6, learning rate $\log \lambda_i$7, $\log \lambda_i$8 steps, and five runs. Although the architectures satisfy the same constitutive constraints a priori, they display different approximation power and pronounced differences in extrapolation (Wollner et al., 19 May 2026).

6. Numerical evidence and usage beyond ideal hyperelasticity

A broad numerical survey on $\log \lambda_i$9 confirms that TSTS-M$\tau = J\sigma$00 is selective. In that survey, Biot, Saint-Venant–Kirchhoff, Mooney–Rivlin, generalized Neo-Hooke, and Simo–Pister models are reported as negative for $\tau = J\sigma$01–$\tau = J\sigma$02 monotonicity; Ciarlet–Geymonat appears as inconclusive or weak positive numerics; quadratic Hencky satisfies Hill’s inequality for $\tau = J\sigma$03–$\tau = J\sigma$04 but generally fails $\tau = J\sigma$05–$\tau = J\sigma$06 monotonicity in compression; exponentiated Hencky, a limited-compressibility variant, Benam, and tension–compression symmetric energies show positive analytical or numerical evidence. The same survey also emphasizes that monotonicity depends strongly on the stress–strain pair: Biot stress versus $\tau = J\sigma$07 may be monotone even when Cauchy stress versus $\tau = J\sigma$08 is not, whereas $\tau = J\sigma$09–$\tau = J\sigma$10 monotonicity follows directly from convexity in logarithmic strain (Ghiba et al., 29 Aug 2025).

Separate experimental literatures use the phrase in a descriptive sense for measured true-stress–true-strain curves. In axisymmetric tensile tests on 303 stainless steel, contour-based extraction of local diameter and curvature, followed by a Bridgman-type triaxiality correction, recovers equivalent uniaxial true stress–true strain curves that remain monotonic and collapse across smooth and notched specimens; manufactured finite-element validations reported stress errors below $\tau = J\sigma$11 for smooth and below $\tau = J\sigma$12 for notched geometries (Ganzenmüller et al., 2022). In HDPE, local Cauchy true stress–true strain measured in the neck by 3D DIC was monotone increasing under both constant strain-rate and constant grip-rate protocols, with a quasi-plateau at the lowest rates but no negative-slope segment (André et al., 2020). At the molecular scale, MEAM simulations of single $\tau = J\sigma$13-alkane molecules and a generalized single polyethylene chain likewise produced monotone increasing true Cauchy and true virial stress–true strain curves up to bond rupture under all three area and volume definitions considered (Nouranian et al., 2016).

Taken together, the cited works show that true-stress–true-strain monotonicity has two closely related but distinct roles. In constitutive theory it is a precise operator monotonicity condition in logarithmic strain, with local, global, spectral, and corotational-rate formulations. In computational and experimental practice it is also used to assess whether a model or measured response preserves the physically expected increase of true stress with increasing true strain. The central contemporary result is not that TSTS-M supersedes polyconvexity or ellipticity, but that it captures a different aspect of physically reasonable response; recent progress therefore focuses on enforcing it concurrently with convexity-based stability conditions rather than treating any one of them as sufficient on its own (Wollner et al., 10 Sep 2025).