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arxiv: 2605.16837 · v1 · pith:PNHIVQOVnew · submitted 2026-05-16 · ❄️ cond-mat.mtrl-sci

Learning inelastic constitutive models from stress-strain data under hard thermodynamic constraints

Filippo Masi This is my paper

Pith reviewed 2026-05-19 21:09 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords constitutive modelinginelastic materialsthermodynamic constraintsmachine learninggranular mediastress-strain dataneural networkshistory-dependent behavior
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The pith

Embedding non-equilibrium thermodynamics, objectivity and stability as hard constraints inside a neural network lets it learn consistent inelastic constitutive models from limited stress-strain data alone.

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

The paper builds a learning framework that bakes the laws of non-equilibrium thermodynamics, frame indifference and material stability directly into the architecture as unbreakable constraints rather than soft penalties. This lets the model extract constitutive relations for inelastic solids and granular materials from ordinary macroscopic loading curves without needing full internal state measurements. A reader should care because history-dependent materials still defeat most data-driven methods when training data are scarce, yet this approach claims to produce models that respect energy dissipation and generalize to loading histories never seen in training. The work shows the same architecture recovers interpretable internal variables on its own and correctly predicts new phenomena such as hysteresis loops that were absent from the data.

Core claim

The thermodynamics-constrained learning framework embeds the principles of non-equilibrium thermodynamics, objectivity and stability as hard, scalable constraints to learn constitutive models from standard macroscopic data. Analytical benchmarks demonstrate that the method learns thermodynamically consistent and robust constitutive models for a range of inelastic materials of increasing complexity. At inference the resulting models generalise to more demanding, unobserved paths and autonomously recover interpretable internal variables that capture path-dependent evolution. When applied to granular media trained on discrete-element simulations, the model discovers the underlying constitutive,

What carries the argument

Neural architecture that treats thermodynamic consistency, objectivity and stability as hard architectural constraints enforced at every step of training and inference.

If this is right

  • Models trained on limited paths still predict responses under cyclic loading, including hysteresis loops absent from the training set.
  • The framework recovers interpretable internal variables that track path-dependent evolution without being given them explicitly.
  • The same architecture works across materials of increasing complexity from simple analytical benchmarks to heterogeneous granular media.
  • Constitutive equations can be discovered using only macroscopic stress-strain histories obtained from simulated or real experiments.

Where Pith is reading between the lines

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

  • Similar hard-constraint designs could be tested on other history-dependent systems such as viscoelastic polymers or fatigue in metals where internal variables are also hidden.
  • If the approach scales, it would let engineers replace expensive full-field simulations with fast, thermodynamically safe surrogate models trained on sparse experimental curves.
  • The recovered internal variables might be compared directly to measurable quantities like fabric tensors in granular experiments to validate their physical meaning.

Load-bearing premise

That making thermodynamic and objectivity rules into hard, scalable constraints inside the network is sufficient to guarantee physical consistency and robust generalization when the only available data are limited macroscopic stress-strain curves for complex, heterogeneous materials.

What would settle it

Train the model on the reported stress-strain paths for a granular specimen, then test it on a cyclic loading sequence that produces clear hysteresis; if the predicted dissipation or loop area deviates significantly from independent discrete-element or laboratory measurements, the central claim fails.

Figures

Figures reproduced from arXiv: 2605.16837 by Filippo Masi.

Figure 1
Figure 1. Figure 1: Discovering constitutive models from stress–strain data with hard-constrained non-equilibrium thermodynamics. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Loading protocol and stress data for the one-dimensional benchmarks used for training. From left to right: (a) [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Predictions at inference of the response of the elasto-plastic (EP) material under unobserved loading protocols: [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Predictions at inference of the response of the hypo-plastic (HP) material under unobserved loading protocols: [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Predictions at inference of the response of the elasto-visco-plastic (EVP) material under velocity stepping and [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Predictions at inference of the response of the elasto-plastic material with isotropic hardening (EPH) under an [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Predictions at inference of the response of the elasto-plastic material with isotropic hardening (EPH) in terms [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Loading protocols for the training process in terms of (a) deviatoric versus volumetric stress ( [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Predictions at inference for an unobserved protocol (simple shear, i.e., [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Ground-truth and learned free energy function (top) and dissipation rate (bottom). From left to right: (a) [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Sample generation: (i) creation of random particle clouds with different seeds (horizontal axis) and (ii) compaction with different values of the friction coefficient 𝜇𝑐 to control the degree of densification (vertical axis). Five randomly sampled particle clouds are considered. Each cloud is subjected to isotropic consolidation (up to 𝑝 = 100 kPa) considering four different values of 𝜇𝑐, resulting in 20 … view at source ↗
Figure 12
Figure 12. Figure 12: Isotropic extension loading protocols for the training process in terms of (a) solid fraction versus isotropic [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Drained triaxial protocols for the training process in terms of (top) deviatoric stress versus strain ( [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Selection of the number of internal variables. (a) Training loss versus epochs for [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Training predictions for monotonic drained triaxial compression tests at [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Training predictions for monotonic drained triaxial compression tests at [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Inference predictions for unobserved drained triaxial compression tests with three loading–unloading sequences [PITH_FULL_IMAGE:figures/full_fig_p023_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Inference predictions for unobserved drained triaxial compression tests with eight loading–unloading sequences [PITH_FULL_IMAGE:figures/full_fig_p024_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Inference predictions for unobserved cyclic drained triaxial compression tests at [PITH_FULL_IMAGE:figures/full_fig_p025_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Inference predictions for an unobserved simple-shear test with unloading at [PITH_FULL_IMAGE:figures/full_fig_p026_20.png] view at source ↗
read the original abstract

Machine learning approaches informed by physics have offered new insights into the discovery of constitutive models from data, helping overcome some limitations of traditional constitutive modelling while reducing the cost of otherwise computationally intensive simulations. Yet, most existing approaches only partially enforce the requirements of physics and thermodynamics, leaving open questions about their consistency across a broad range of material behaviours and their ability to generalise robustly to unseen loading paths when only limited measurements are available. This work establishes a thermodynamics-constrained learning framework whose architecture embeds the principles of non-equilibrium thermodynamics, objectivity and stability as hard, scalable constraints to learn constitutive models from standard macroscopic data. Analytical benchmarks involving stress-strain loading paths demonstrate that the method learns thermodynamically consistent and robust constitutive models for a range of inelastic materials of increasing complexity. At inference, the resulting models generalise to more demanding, unobserved paths and can autonomously recover interpretable internal variables that capture path-dependent evolution. The framework is then applied to granular media, prototypical heterogeneous and history-dependent materials for which constitutive modelling remains challenging. Trained on numerically simulated experiments based on the discrete element method, the model discovers the underlying constitutive equations and predicts responses under cyclic loading, including the emergence of hysteresis absent from the training data, relying solely on macroscopic stress-strain histories. The findings indicate that enforcing non-equilibrium thermodynamics through hard constraints represents a principled route to robust, consistent, and scalable data-driven discovery of constitutive models.

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

2 major / 2 minor

Summary. The manuscript proposes a neural-network framework that embeds non-equilibrium thermodynamics, objectivity, and stability as hard architectural constraints to learn inelastic constitutive models directly from macroscopic stress-strain histories. Analytical benchmarks on materials of increasing complexity are used to verify thermodynamic consistency and generalization; the method is then applied to DEM-simulated granular media, where it recovers internal variables and predicts cyclic responses including emergent hysteresis absent from the training data.

Significance. If the central claims hold, the work supplies a scalable route to thermodynamically consistent constitutive discovery from limited macroscopic measurements, with the hard-constraint design and autonomous recovery of internal variables as particular strengths. Successful extrapolation to unobserved paths for heterogeneous, history-dependent materials would be a meaningful advance over soft-penalty or post-hoc correction approaches.

major comments (2)
  1. [Granular media application and results] The central claim that hard embedding of thermodynamic principles suffices for robust generalization and unique recovery of interpretable internal variables is load-bearing for the granular-media results. Macroscopic stress-strain data alone are known to be insufficient to uniquely identify micromechanical state evolution; the manuscript should demonstrate (e.g., via additional state-space probes or comparison with known DEM internal quantities) that the learned variables remain physically meaningful outside the training distribution rather than merely satisfying the inequalities on seen paths.
  2. [Numerical experiments on granular media] The generalization experiments report successful prediction of hysteresis on cyclic paths not present in training. However, the paper must clarify whether the enforced dissipation structure or potential form can produce non-physical responses when loading paths enter regions of state space that are only weakly constrained by the limited macroscopic data; a concrete counter-example or sensitivity test would address this risk.
minor comments (2)
  1. Notation for the internal variables and the precise form of the hard constraints (e.g., how objectivity is imposed) should be introduced earlier and used consistently across equations and figures.
  2. Figure captions for the benchmark and granular results should explicitly state the training versus test loading paths to make the generalization claims immediately verifiable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and insightful comments on our manuscript. We address each major comment in detail below, outlining our responses and the revisions we plan to incorporate to strengthen the presentation of the granular-media results.

read point-by-point responses

  1. Referee: [Granular media application and results] The central claim that hard embedding of thermodynamic principles suffices for robust generalization and unique recovery of interpretable internal variables is load-bearing for the granular-media results. Macroscopic stress-strain data alone are known to be insufficient to uniquely identify micromechanical state evolution; the manuscript should demonstrate (e.g., via additional state-space probes or comparison with known DEM internal quantities) that the learned variables remain physically meaningful outside the training distribution rather than merely satisfying the inequalities on seen paths.

    Authors: We agree that stress-strain data alone cannot guarantee unique recovery of micromechanical states, and our manuscript does not claim uniqueness. Instead, the hard thermodynamic constraints restrict admissible evolutions, enabling generalization even when internal variables are not directly observed. In the analytical benchmarks, the autonomously recovered variables match known physical quantities (e.g., equivalent plastic strain or hardening variables). For the DEM granular application, physical meaningfulness is evidenced by the model's ability to predict emergent hysteresis on cyclic paths absent from training. To address the request for further demonstration, we will add state-space probes in a new subsection of the revised manuscript. These will include (i) evolution plots of the learned internal variables along extrapolated paths and (ii) comparison of their trends against expected micromechanical signatures from the underlying DEM simulations, confirming consistency beyond the training distribution. revision: yes

  2. Referee: [Numerical experiments on granular media] The generalization experiments report successful prediction of hysteresis on cyclic paths not present in training. However, the paper must clarify whether the enforced dissipation structure or potential form can produce non-physical responses when loading paths enter regions of state space that are only weakly constrained by the limited macroscopic data; a concrete counter-example or sensitivity test would address this risk.

    Authors: The architecture embeds a convex free-energy potential and a non-negative dissipation function as hard constraints, which by construction guarantees thermodynamic consistency (non-negative dissipation, frame-indifference, and material stability) for any input state. This prevents many classes of non-physical behavior even in extrapolation. We acknowledge that explicit verification in sparsely sampled regions strengthens the claim. In the revised manuscript we will add a sensitivity analysis subsection that evaluates the trained model on deliberately chosen loading paths entering weakly constrained regions of state space. We will report the resulting stress responses and internal-variable trajectories, confirming they remain physically admissible. No counter-examples of non-physical behavior appeared in our existing tests, but the new analysis will provide the requested clarification and quantify robustness. revision: yes

Circularity Check

0 steps flagged

No significant circularity: external thermodynamic constraints applied to data-driven learning

full rationale

The paper's core derivation embeds non-equilibrium thermodynamics, objectivity and stability as hard architectural constraints rather than deriving them from or fitting them to the stress-strain data. Training occurs on macroscopic measurements, with generalization and internal-variable recovery presented as emergent outcomes of the constrained architecture. No self-definitional steps, fitted-input predictions, or load-bearing self-citations appear in the abstract or described framework. The approach is self-contained against external thermodynamic principles and does not reduce its claimed predictions to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard thermodynamic principles treated as domain assumptions and on the modeling choice that hard architectural constraints suffice for generalization from limited data; no explicit free parameters or invented entities are described in the abstract.

axioms (1)
  • domain assumption Principles of non-equilibrium thermodynamics, objectivity and stability
    Invoked as hard constraints embedded in the learning architecture to ensure consistency.

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