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A two-stage physics-informed Gaussian-process model learns intact elasticity and damage evolution of hyperelastic materials from uniaxial tension data, then predicts compression and shear failure with physically plausible behavior.

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2026-08-03 00:08 UTC pith:4HX6SO4J

load-bearing objection Clever two-stage GPR setup for damage, but the claimed out-of-sample generalization is undermined by a rank-deficient inversion in Stage I that leaves the response functions non-unique. the 4 major comments →

arxiv 2602.11414 v1 pith:4HX6SO4J submitted 2026-02-11 cs.CE cond-mat.mtrl-scicond-mat.soft

A physics-informed data-driven framework for modeling hyperelastic materials with progressive damage and failure

Kshitiz Upadhyay This is my paper
classification cs.CE cond-mat.mtrl-scicond-mat.soft
keywords Data-driven constitutive modelsHyperelasticityDamage and failurePhysics-informed machine learningGaussian process regressionSoft materialsBrain tissue mechanics
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper argues that progressive damage and failure in hyperelastic soft materials can be learned as a two-stage inference problem: first recover the intact (undamaged) elastic response from the early part of a single uniaxial tension test, then learn a stress-reduction factor that depends on the intact strain energy. The central test is whether the resulting model generalizes to deformation modes it was never trained on. On synthetic data it does, reproducing compression and shear failure trends; on brain-tissue data it extracts a critical failure energy from limited experimental measurements. If this holds, it offers a data-efficient and interpretable path to constitutive modeling of soft-material failure without prescribing an analytical damage law.

Core claim

The paper claims that the total stress response of an energy-limited hyperelastic material can be decomposed into an intact elastic part—expressed through volumetric and isochoric response functions in a fixed tensor basis—multiplied by a scalar stress-reduction factor χ(W) that depends solely on the intact strain energy density W. This structure is learned directly from data: Stage I uses Gaussian process regression to learn the intact response functions from the undamaged portion of a uniaxial test, and Stage II uses a constrained Gaussian process to learn χ(W) from the full loading curve, enforcing monotonicity, non-negativity, and complete failure at large W. The paper shows that the res

What carries the argument

The central object is the generalized stress representation S = χ(W)[ζ(J)G1 + J^{-2/3}Γ1(Ī1,Ī2)G2 + J^{-2/3}Γ2(Ī1,Ī2)G3], with tensor basis G1 = C^{-1}, G2 = Dev(I), G3 = Dev(Ĉ). This decomposition separates the constitutive law into intact elasticity (learned by two GPR models in Stage I) and damage (learned by a third constrained GPR model in Stage II). The Stage II model is the key new mechanism: it maps intact strain energy density W to the stress-reduction factor χ, with penalties that enforce χ ≥ 0, dχ/dW ≤ 0, and χ → 0 via augmented failure points, converting ordinary hyperelastic learning into energy-limited failure modeling.

Load-bearing premise

The method hinges on the assumption that the early, user-selected stretch range of a uniaxial test is purely intact (χ = 1) and that the volumetric and isochoric response functions can be uniquely recovered from the diagonal components of stress and deformation available in that test.

What would settle it

Train the model on synthetic uniaxial tension data generated from an energy-limited constitutive law with distinct failure energies for tension and shear (i.e., a Lode-dependent χ). If the learned χ is a single scalar function of W, the model will predict the same saturation energy across modes and mismatch the ground truth, showing that the scalar-energy assumption fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A single uniaxial tension test may provide enough information to predict compression and shear response, reducing the experimental burden for soft-material characterization.
  • The inferred stress-reduction factor and saturating energy give a physically interpretable internal damage variable that can be used in finite element simulations; the paper derives a consistent tangent stiffness tensor.
  • The framework enables direct estimation of the critical failure energy ψf from mechanical test data, allowing quantitative toughness comparisons such as gray matter versus white matter.
  • Enforcing monotonicity and non-negativity of χ prevents non-physical healing and stress recovery in extrapolation, a failure mode observed in unconstrained baselines.
  • The method is data-efficient—51 uniaxial tension points sufficed—making it applicable in settings where large multiaxial datasets are unavailable.

Where Pith is reading between the lines

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

  • A natural extension, not pursued in the paper, would automate the intact-regime cutoff (currently user-selected) by detecting the onset of damage from the stress-strain response itself, removing the main remaining human input.
  • The framework assumes damage is governed by a single scalar energy measure; for materials whose failure energy depends on loading mode (e.g., different tension and shear toughness), this scalar assumption would likely break down, as the paper itself notes in its outlook.
  • Since damage is learned from monotonic loading only, the model as presented does not handle unloading-reloading hysteresis; adding a history-dependent switch for irreversibility is mentioned as a straightforward but untested extension.
  • The two-stage decomposition could be adapted to anisotropic damage by enriching the tensor basis with structural tensors, though identifiability of additional response functions would need careful re-examination.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper proposes a two-stage Gaussian-process-regression (GPR) constitutive framework for energy-limited hyperelastic materials with progressive damage. Stage I learns the intact (undamaged) volumetric and isochoric response functions ζ(J), Γ1(Ī1,Ī2), Γ2(Ī1,Ī2) by inverting Eq. (21) on an intact subset of uniaxial tension data. Stage II learns a stress-reduction factor χ(W) as a function of the intact strain energy density W, with non-negativity, monotonicity, and complete-failure constraints enforced through penalty terms and artificial χ=0 data augmentation. The model is validated on synthetic Mooney–Rivlin-type data with out-of-sample compression and shear tests, and applied to experimental brain tissue data under an incompressibility assumption. The central claim is that the two-stage physics-informed GPR model simultaneously reproduces correct stress trends and energy-limited failure behavior while generalizing to deformation modes not seen in training.

Significance. The two-stage structure is well motivated, and the paper makes a genuine effort to test out-of-sample behavior rather than only in-distribution interpolation. The synthetic compression and shear experiments are meaningful generalization tests, and the use of exact-inference GPR to enforce a stress-free reference state is a useful ingredient. The penalty-based enforcement of monotonic, non-negative damage evolution is also a reasonable design choice. If the identifiability and thermodynamic-consistency gaps identified below are addressed, the framework would be a useful contribution to data-driven constitutive modeling of soft-tissue failure. However, as it stands, the central generalization claim rests on an underdetermined Stage I inversion and on a path-energy that is not guaranteed to be well defined, so the current evidence is not sufficient for acceptance.

major comments (4)
  1. [§3.2.1, Eq. (21); §5.1.3] The Stage I inversion is rank-deficient for the uniaxial training data. Under Eq. (46), C, G1=C^-1, G2=Dev(I), and G3=Dev(C̄) are all diagonal with equal 22- and 33-components. Thus each column of A in Eq. (21) lies in the same two-dimensional subspace of symmetric tensors with S22=S33, and b lies in that subspace. The 9×3 least-squares problem has a one-dimensional null space. The paper does not state how the solution is selected (minimum-norm least squares is the usual default) or demonstrate that the resulting ζ, Γ1, Γ2 are insensitive to that choice. Since the out-of-sample shear stress in Eq. (18) depends on Γ1 and Γ2 separately, the accurate shear predictions in Fig. 6 are not guaranteed by the uniaxial training data; they reflect the particular null-space component produced by the linear solver and by GPR smoothing. This is load-bearing for the Section 5.1.3 claim that the model i
  2. [§3.2–3.3, Eq. (28)] Stage I does not enforce hyperelastic compatibility. The representation in Eq. (20) with independently learned ζ, Γ1, Γ2 does not ensure that the predicted intact stress S_intact is the gradient 2∂W/∂C of a scalar potential W. In three dimensions, the response functions must satisfy cross-derivative/compatibility conditions for S_intact to be conservative. No such constraints are imposed. Consequently, the line integral W=∫ S_intact : (1/2)dC in Eq. (28) is path-dependent in general. Because Stage II defines χ(W) as a function of W and then uses it under arbitrary deformation modes, the damage prediction is not well posed unless path-independence is established. The thermodynamic-consistency claim in §3.4 is therefore stronger than what is actually enforced. The authors should either impose compatibility (for example, by learning a scalar potential or adding compatibility constraints) or
  3. [§4.2, Eqs. (55)–(58)] The finite-kinematics formulas used for the brain-tissue application appear inconsistent. For the incompressible uniaxial deformation F=diag(λ,λ^{-1/2},λ^{-1/2}), the exact relations are S11=P11/λ for the first Piola–Kirchhoff stress P, and dE11=λ dλ, so W=∫_1^λ S11 λ dλ (equivalently ∫ S11 dE11). The paper instead uses S11=P11/(1+λ) in the Fig. 2 caption and W=∫ (S_intact)_11 dλ in Eq. (58). These are not equivalent to Eq. (28) and appear to be small-strain/engineering-strain expressions. Because W is the input to the Stage II damage model, this error propagates into χ(W) and into the reported critical failure energies in Table 2. The authors should correct the stress conversion and the energy integral, or provide an explicit justification that the retained notation means something else.
  4. [§5.2, Fig. 9, Table 2] The predicted critical failure energy ψf is not an independent material property inferred from the experimental data. Its asymptotic saturation is imposed by the artificial χ=0 points augmented over W ∈ [1.3Wpeak, 2.6Wpeak], and the monotonicity/non-negativity constraints are concentrated on W ∈ [0.8Wpeak, 1.3Wpeak] (§5.2). The Stage I intact cutoffs are also chosen individually by visual inspection. The values in Table 2 (e.g., GM = 4.22 and 4.28 kPa) could be artifacts of these modeling choices rather than tissue properties. A systematic sensitivity analysis with respect to a, b, Nc, Λnn, Λmono, and the intact cutoff is required before ψf is reported as a physical quantity.
minor comments (4)
  1. [Title/Abstract] There is a typo in the running title and first line: 'hyp erelastic' should be 'hyperelastic'.
  2. [§3.3.2, Eq. (23)] The nugget term is typeset as αδ′_{zz}; it should be αδ_{zz′} (Kronecker delta on the two inputs). The notation should be cleaned up.
  3. [§3.4] The objectivity statement 'S(C)=R^T S(RCR^T)R' is confusing and appears to be the inverse transformation. The standard statement is S(QCQ^T)=Q S(C) Q^T for orthogonal Q. Please correct.
  4. [General] No data or code availability statement is included. Given the many hyperparameter choices (nuggets, cutoffs, penalty weights, augmentation ranges), release of the code and synthetic-data generator would substantially improve reproducibility.

Circularity Check

1 steps flagged

Critical failure energy ψf is imposed by construction, not independently predicted; core out-of-sample generalization remains non-circular.

specific steps
  1. self definitional [Section 3.3.2 (Complete failure at very large strains, around Eq. (40)) and Section 5.2 (Prediction of strain energy density saturation and critical failure energy, Fig. 9 / Table 2)]
    "In all cases, ψ(λ) increases initially and then asymptotically approaches a constant value, i.e., ψf , at large stretches, signifying complete failure and energy saturation. This asymptotic behavior arises directly from the constrained GPR learning of the stress-reduction factor χ(W ) in Stage II, which enforces monotonicity, non-negativity, and complete failure."

    Stage II defines the damage model to satisfy lim_{W→∞} χ(W)=0, implemented by prescribing artificial χ=0 points at W∈[a·Wpeak,b·Wpeak]. Since ψ = ∫ S dλ = ∫ χ(W)S_intact dλ, the saturation of ψ and therefore the value of ψf are mathematical consequences of the imposed χ→0 constraint and the user-chosen a,b, not independent empirical discoveries. The paper itself states that the asymptotic behavior 'arises directly' from these constraints, so the 'predicted critical failure energy' in Table 2 is substantially determined by model construction rather than by the data.

full rationale

The main out-of-sample generalization claim is not circular: compression and shear predictions are obtained by evaluating the Stage I GPR response functions and the Stage II damage function χ(W) at deformation states and energy values not used in training. The good shear/compression behavior is therefore genuinely extrapolated, even though Stage I's least-squares inversion (Eq. (21)) is rank-deficient under uniaxial loading and leaves Γ1, Γ2 non-unique; that is an identifiability limitation, not a reduction of predictions to inputs. The self-citations ([8], [51]) are contextual/future-work and not load-bearing. However, the paper's inference of the critical failure energy ψf does reduce to construction: the complete-failure constraint forces χ→0 at artificial points and thereby forces ψ to saturate, so the asymptotic ψf is a built-in consequence of the model rather than a data-driven prediction. This partial circularity in a claimed central output yields a score of 6; the independent out-of-sample content prevents a higher score.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 0 invented entities

The pipeline rests on several hand-tuned knobs: intact cutoff, nuggets, penalty weights, constraint points, and artificial failure augmentation. The last one is the most consequential because the reported psi_f values are strongly shaped by where and how chi=0 is imposed beyond the data.

free parameters (5)
  • intact stretch cutoff = lambda=1.25 (synthetic); lambda=1.89, 1.38, 1.37, 1.80 (brain datasets)
    User-defined boundary separating assumed-undamaged from damaged data; selected 'based on visual inspection' for brain data (Section 5.2). Wrong cutoff corrupts Stage I response functions.
  • GPR nugget alpha = Mvol: 1e-5/1e-2; Miso: 1e0 (synthetic), 0.5 (brain); Mdam: 1e-4 (synthetic), 2.5e-3 (brain)
    Chosen by hand to balance fitting accuracy and numerical stability; Supplementary S1 shows predictions vary strongly with alpha.
  • penalty weights Lambda_nn, Lambda_mono = 1e3
    User-defined; non-negativity and monotonicity are soft constraints, not hard guarantees (Eq. 41).
  • constraint points D_cons = Nc=30 over W in [0.8 Wpeak, 1.7 Wpeak] (synthetic) or [6,15] kPa (brain)
    Locations and range chosen manually; constraints are only enforced at these points, not globally.
  • failure augmentation range and values = W in [1.3 Wpeak, 2.6 Wpeak], chi=0
    Artificial data points force chi to zero and control the asymptotic failure energy; this directly shapes the inferred psi_f values in Table 2.
axioms (6)
  • domain assumption Material isotropy: stress depends only on J, Ibar1, Ibar2
    Eq. (6) and Section 3.4; applied to brain tissue despite known white-matter anisotropy.
  • domain assumption Incompressibility of brain tissue (J=1)
    Section 4.2; necessary because no volumetric data are available.
  • ad hoc to paper The intact response functions are compatible with a single scalar potential W
    Eqs. (27)-(28) integrate predicted intact stress to define W; independent GPR outputs Gamma1,Gamma2 are not required to satisfy the hyperelastic compatibility condition, so path-independence of W is not guaranteed.
  • domain assumption Damage is governed solely by scalar intact energy W (energy limiter)
    Stage II Mdam: W->chi; inherited from Volokh energy limiters; Section 3.3.1.
  • domain assumption Monotonic loading without unloading/reloading; no hysteresis
    Section 6: irreversibility under unloading is not considered in this work.
  • ad hoc to paper GPR exact inference with nugget alpha > 0
    Section 3.2.2 claims the stress-free reference state is enforced exactly because alpha->0, but nonzero nuggets are used at most training points, so exact interpolation does not hold.

pith-pipeline@v1.3.0-alltime-deepseek · 35374 in / 15888 out tokens · 154274 ms · 2026-08-03T00:08:18.475717+00:00 · methodology

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read the original abstract

This work presents a two-stage physics-informed, data-driven constitutive modeling framework for hyperelastic soft materials undergoing progressive damage and failure. The framework is grounded in the concept of hyperelasticity with energy limiters and employs Gaussian Process Regression (GPR) to separately learn the intact (undamaged) elastic response and damage evolution directly from data. In Stage I, GPR models learn the intact hyperelastic response through volumetric and isochoric response functions (or only the isochoric response under incompressibility), ensuring energetic consistency of the intact response and satisfaction of fundamental principles such as material frame indifference and balance of angular momentum. In Stage II, damage is modeled via a separate GPR model that learns the mapping between the intact strain energy density predicted by Stage I models and a stress-reduction factor governing damage and failure, with monotonicity, non-negativity, and complete-failure constraints enforced through penalty-based optimization to ensure thermodynamic admissibility. Validation on synthetic datasets, including benchmarking against analytical constitutive models and competing data-driven approaches, demonstrates high in-distribution accuracy under uniaxial tension and robust generalization from limited training data to compression and shear modes not used during training. Application to experimental brain tissue data demonstrates the practical applicability of the framework and enables inference of damage evolution and critical failure energy. Overall, the proposed framework combines the physical consistency, interpretability, and generalizability of analytical models with the flexibility, predictive accuracy, and automation of machine learning, offering a powerful approach for modeling failure in soft materials under limited experimental data.

Figures

Figures reproduced from arXiv: 2602.11414 by Kshitiz Upadhyay.

Figure 1
Figure 1. Figure 1: Schematic of the proposed physics-informed data- [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Second Piola–Kirchhoff stress components [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: In-distribution model performance under uniaxia [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Process-level interpretation of the two-stage ph [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Out-of-sample generalization under uniaxial com [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Out-of-sample generalization under simple shear [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Role of penalty-based constrained optimization d [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Application of the proposed two-stage physics-in [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Predicted total strain energy density ψ versus stretch λ for brain tissue under uniaxial tension, shown for two gray matter (GM) datasets and two white matter substructures (WM-o and WM-p). Predictions are obtained by extrapolating the trained two-stage physics-informed GPR-based model beyond the experimental range. In all cases, ψ(λ) asymptotically approaches a saturation value, identifying the critical f… view at source ↗

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Reference graph

Works this paper leans on

101 extracted references · 36 canonical work pages

  1. [1]

    Puglisi, G

    G. Puglisi, G. Saccomandi, Multi-scale modelling of rubber-lik e materials and soft tissues: an appraisal, Proceedings of the Royal Society A: Mathematical, Physical an d Engineering Sciences 472 (2187) (2016) 20160060. doi:10.1098/rspa.2016.0060

    arXiv 2016

  2. [2]

    Volokh, Mechanics of soft materials, Springer Singapore, 2016

    K. Volokh, Mechanics of soft materials, Springer Singapore, 2016. doi:https://doi.org/10. 1007/978-981-13-8371-7 . 28

    2016

  3. [3]

    Avril, Hyperelasticity of Soft Tissues and Related Inverse Prob lems, in: CISM International Centre for Mechanical Sciences, Courses and Lectures, Vol

    S. Avril, Hyperelasticity of Soft Tissues and Related Inverse Prob lems, in: CISM International Centre for Mechanical Sciences, Courses and Lectures, Vol. 573, Springe r International Publishing, 2017, pp. 37–66. doi:10.1007/978-3-319-45071-1_2

    work page doi:10.1007/978-3-319-45071-1_2 2017

  4. [4]

    Xiang, D

    Y. Xiang, D. Zhong, S. Rudykh, H. Zhou, S. Qu, W. Yang, A Review of Physical ly Based and Thermodynamically Based Constitutive Models for Soft Materials, Jou rnal of Applied Mechanics, Transactions ASME 87 (11) (2020) 1–21. doi:10.1115/1.4047776

    work page doi:10.1115/1.4047776 2020

  5. [5]

    Volokh, On modeling failure of rubber-like materials, Mechani cs Research Communications 37 (8) (2010) 684–689

    K. Volokh, On modeling failure of rubber-like materials, Mechani cs Research Communications 37 (8) (2010) 684–689. doi:10.1016/j.mechrescom.2010.10.006

    work page doi:10.1016/j.mechrescom.2010.10.006 2010

  6. [6]

    Konale, V

    A. Konale, V. Srivastava, On modeling fracture of soft polymers, Mech anics of Materials 206 (April) (2025) 105346. doi:10.1016/j.mechmat.2025.105346

    arXiv 2025

  7. [7]

    S. N. S. H. Chittajallu, A. Richhariya, K. M. Tse, V. Chinthapenta, A R eview on Dam- age and Rupture Modelling for Soft Tissues, Bioengineering 9 (1) (2022) 26. doi:10.3390/ bioengineering9010026

    2022

  8. [8]

    Y. C. Chandrashekar, K. Upadhyay, A constitutive framework for tensi on-compression failure asym- metry in soft materials (2025). arXiv:2512.12614. URL https://arxiv.org/abs/2512.12614

    Pith/arXiv arXiv 2025

  9. [9]

    L. R. G. Treloar, The elasticity of a network of long-chain molecules—II , Transactions of the Faraday Society 39 (1943) 241–246. doi:10.1039/TF9433900241

    work page doi:10.1039/tf9433900241 1943

  10. [10]

    Rivlin, Large elastic deformations of isotropic materials

    R. Rivlin, Large elastic deformations of isotropic materials. I. Fundam ental concepts, Philosoph- ical Transactions of the Royal Society of London. Series A, Mathematical and P hysical Sciences 240 (822) (1948) 459–490. doi:10.1098/rsta.1948.0002

    arXiv 1948

  11. [11]

    Mooney, A Theory of Large Elastic Deformation, Journal of Applied Physi cs 11 (9) (1940) 582–592

    M. Mooney, A Theory of Large Elastic Deformation, Journal of Applied Physi cs 11 (9) (1940) 582–592. doi:10.1063/1.1712836

    doi:10.1063/1.1712836 1940

  12. [12]

    Rivlin, Large elastic deformations of isotropic materials IV

    R. Rivlin, Large elastic deformations of isotropic materials IV. furthe r developments of the general theory, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 241 (835) (1948) 379–397. doi:10.1098/rsta.1948.0024

    arXiv 1948

  13. [13]

    R. W. Ogden, Large deformation isotropic elasticity – on the correlation of theory and experiment for incompressible rubberlike solids, Proceedings of the Royal So ciety of London. A. Mathematical and Physical Sciences 326 (1567) (1972) 565–584. doi:10.1098/rspa.1972.0026

    arXiv 1972

  14. [14]

    A. N. Gent, A New Constitutive Relation for Rubber, Rubber Chemist ry and Technology 69 (1) (1996) 59–61. doi:10.5254/1.3538357

    doi:10.5254/1.3538357 1996

  15. [15]

    G. A. Holzapfel, Nonlinear Solid Mechanics: A Continuum Approach for Engin eering, John Wiley & Sons, Ltd, Chichester, England, 2000

    2000

  16. [16]

    Upadhyay, G

    K. Upadhyay, G. Subhash, D. Spearot, Thermodynamics-based stabil ity criteria for constitutive equations of isotropic hyperelastic solids, Journal of the Mechanics and Physics of Solids 124 (2019) 115–142. doi:10.1016/j.jmps.2018.09.038

    work page doi:10.1016/j.jmps.2018.09.038 2019

  17. [17]

    Destrade, G

    M. Destrade, G. Saccomandi, I. Sgura, Methodical fitting for mathemat ical models of rubber-like materials, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473 (2198) (2017) 20160811. doi:10.1098/rspa.2016.0811

    arXiv 2017

  18. [18]

    Upadhyay, G

    K. Upadhyay, G. Subhash, D. Spearot, Hyperelastic constitutive mo deling of hydrogels based on primary deformation modes and validation under 3D stress states, Int ernational Journal of Engineering Science 154 (2020) 103314. doi:10.1016/j.ijengsci.2020.103314

    arXiv 2020

  19. [19]

    Beda, An approach for hyperelastic model-building and parameters estimation a review of constitutive models, European Polymer Journal 50 (1) (2014) 97–108

    T. Beda, An approach for hyperelastic model-building and parameters estimation a review of constitutive models, European Polymer Journal 50 (1) (2014) 97–108. doi:10.1016/j.eurpolymj. 2013.10.006

    work page doi:10.1016/j.eurpolymj 2014

  20. [20]

    C. Wex, S. Arndt, A. Stoll, C. Bruns, Y. Kupriyanova, Isotropic incom pressible hyperelastic models for modelling the mechanical behaviour of biological tissues: a revie w, Biomedical Engineering / Biomedizinische Technik 60 (6) (2015) 577–592. doi:10.1515/bmt-2014-0146. 29

    work page doi:10.1515/bmt-2014-0146 2015

  21. [21]

    H. B. Khaniki, M. H. Ghayesh, R. Chin, M. Amabili, Hyperelastic str uctures: A review on the mechanics and biomechanics, International Journal of Non-Linear Mechanics 148 (September 2022) (2023) 104275. doi:10.1016/j.ijnonlinmec.2022.104275

    arXiv 2022

  22. [22]

    J. N. Fuhg, G. Anantha Padmanabha, N. Bouklas, B. Bahmani, W. Sun, N. N. Vlassis , M. Flaschel, P. Carrara, L. De Lorenzis, A Review on Data-Driven Constitu tive Laws for Solids, Archives of Computational Methods in Engineering 32 (3) (2025) 1841–1883. doi:10.1007/ s11831-024-10196-2

    2025

  23. [23]

    Kumar, D

    S. Kumar, D. M. Kochmann, What Machine Learning Can Do for Computational Solid Mechan- ics, in: Current Trends and Open Problems in Computational Mechanic s, Springer International Publishing, Cham, 2022, pp. 275–285. doi:10.1007/978-3-030-87312-7_27

    doi:10.1007/978-3-030-87312-7_27 2022

  24. [24]

    Chen, Recurrent neural networks (RNNs) learn the constitutiv e law of viscoelasticity, Compu- tational Mechanics 67 (3) (2021) 1009–1019

    G. Chen, Recurrent neural networks (RNNs) learn the constitutiv e law of viscoelasticity, Compu- tational Mechanics 67 (3) (2021) 1009–1019. doi:10.1007/s00466-021-01981-y

    doi:10.1007/s00466-021-01981-y 2021

  25. [25]

    S. Tang, Y. Li, H. Qiu, H. Yang, S. Saha, S. Mojumder, W. K. Liu, X. Guo, MAP123-EP : A mechanistic-based data-driven approach for numerical elastoplastic an alysis, Computer Methods in Applied Mechanics and Engineering 364 (2020) 112955. doi:10.1016/j.cma.2020.112955

    arXiv 2020

  26. [26]

    S. Tang, G. Zhang, H. Yang, Y. Li, W. K. Liu, X. Guo, MAP123: A data-driven approach to use 1D data for 3D nonlinear elastic materials modeling, Computer Methods in Applied Mechanics and Engineering 357 (2019) 112587. doi:10.1016/j.cma.2019.112587

    arXiv 2019

  27. [27]

    L. Lu, M. Dao, P. Kumar, U. Ramamurty, G. E. Karniadakis, S. Suresh, Extr action of mechanical properties of materials through deep learning from instrumented ind entation, Proceedings of the National Academy of Sciences of the United States of America 117 (13) (2020) 7052–7062. doi: 10.1073/pnas.1922210117

    doi:10.1073/pnas.1922210117 2020

  28. [28]

    As’ad, P

    F. As’ad, P. Avery, C. Farhat, A mechanics-informed artificial neural network approach in data- driven constitutive modeling, International Journal for Numerical Me thods in Engineering 123 (12) (2022) 2738–2759. doi:10.1002/nme.6957

    doi:10.1002/nme.6957 2022

  29. [29]

    Ostadrahimi, A

    A. Ostadrahimi, A. Teimouri, K. Upadhyay, G. Li, Physics-informed d ata-driven constitutive mod- eling of compressible, nonlinear soft materials under multiaxial cyc lic loading, International Journal of Mechanical Sciences 312 (April 2025) (2026) 111250. doi:10.1016/j.ijmecsci.2026.111250

    arXiv 2025

  30. [30]

    Peirlinck, J

    M. Peirlinck, J. A. Hurtado, M. K. Rausch, A. B. Tepole, E. Kuhl, A un iversal material model subroutine for soft matter systems, Engineering with Computers (0123456789) (2024). doi:10. 1007/s00366-024-02031-w

    2024

  31. [31]

    Q. He, D. W. Laurence, C. H. Lee, J. S. Chen, Manifold learning based dat a-driven modeling for soft biological tissues, Journal of Biomechanics 117 (2021) 110124. doi:10.1016/j.jbiomech. 2020.110124

    arXiv 2021

  32. [32]

    Linka, M

    K. Linka, M. Hillg¨ artner, K. P. Abdolazizi, R. C. Aydin, M. Itskov, C . J. Cyron, Constitu- tive artificial neural networks: A fast and general approach to predictiv e data-driven consti- tutive modeling by deep learning, Journal of Computational Physics 429 ( 2021) 110010. doi: 10.1016/j.jcp.2020.110010

    arXiv 2021

  33. [33]

    K. A. Kalina, L. Linden, J. Brummund, P. Metsch, M. K¨ astner, Automat ed constitutive modeling of isotropic hyperelasticity based on artificial neural networks, Compu tational Mechanics 69 (1) (2022) 213–232. doi:10.1007/s00466-021-02090-6

    doi:10.1007/s00466-021-02090-6 2022

  34. [34]

    F. Masi, I. Stefanou, Evolution TANN and the identification of internal v ariables and evolution equations in solid mechanics, Journal of the Mechanics and Physics of Sol ids 174 (September 2022) (2023) 105245. doi:10.1016/j.jmps.2023.105245

    arXiv 2022

  35. [35]

    Flaschel, S

    M. Flaschel, S. Kumar, L. De Lorenzis, Unsupervised discovery of i nterpretable hyperelastic con- stitutive laws, Computer Methods in Applied Mechanics and Engine ering 381 (2021) 113852. doi:10.1016/j.cma.2021.113852

    arXiv 2021

  36. [36]

    Flaschel, S

    M. Flaschel, S. Kumar, L. De Lorenzis, Discovering plasticity mo dels without stress data, npj Computational Materials 8 (1) (2022) 1–10. doi:10.1038/s41524-022-00752-4 . 30

    doi:10.1038/s41524-022-00752-4 2022

  37. [37]

    H. Hu, L. Qi, X. Chao, Physics-informed Neural Networks (PINN) for computati onal solid me- chanics: Numerical frameworks and applications, Thin-Walled Structu res 205 (2025) 112495. doi:https://doi.org/10.1016/j.tws.2024.112495

    arXiv 2025

  38. [38]

    Jeong, M

    I. Jeong, M. Cho, H. Chung, D.-N. Kim, Data-driven nonparametric ident ification of material behavior based on physics-informed neural network with full-field data, Computer Methods in Applied Mechanics and Engineering 418 (PB) (2024) 116569. doi:10.1016/j.cma.2023.116569

    arXiv 2024

  39. [39]

    Eggersmann, T

    R. Eggersmann, T. Kirchdoerfer, S. Reese, L. Stainier, M. Ortiz , Model-Free Data-Driven in- elasticity, Computer Methods in Applied Mechanics and Engineerin g 350 (2019) 81–99. doi: 10.1016/j.cma.2019.02.016

    work page doi:10.1016/j.cma.2019.02.016 2019

  40. [40]

    Kirchdoerfer, M

    T. Kirchdoerfer, M. Ortiz, Data-driven computational mechanics , Computer Methods in Applied Mechanics and Engineering 304 (2016) 81–101. doi:10.1016/j.cma.2016.02.001

    doi:10.1016/j.cma.2016.02.001 2016

  41. [41]

    Z. Yang, C. H. Yu, M. J. Buehler, Deep learning model to predict c omplex stress and strain fields in hierarchical composites, Science Advances 7 (15) (2021). doi:10.1126/SCIADV.ABD7416

    work page doi:10.1126/sciadv.abd7416 2021

  42. [42]

    D. K. Klein, M. Fern´ andez, R. J. Martin, P. Neff, O. Weeger, Polyconvex anisotropic hyperelasticity with neural networks, Journal of the Mechanics and Physics of Solids 159 ( June 2021) (2022) 104703. doi:10.1016/j.jmps.2021.104703

    arXiv 2021

  43. [43]

    Thakolkaran, Y

    P. Thakolkaran, Y. Guo, S. Saini, M. Peirlinck, B. Alheit, S. Kumar, C an KAN CANs? Input- convex Kolmogorov-Arnold Networks (KANs) as hyperelastic constitutive art ificial neural networks (CANs), Computer Methods in Applied Mechanics and Engineering 443 (2025). doi:10.1016/j. cma.2025.118089

    arXiv 2025

  44. [44]

    J. Wang, T. Li, F. Cui, C. Y. Hui, J. Yeo, A. T. Zehnder, Metamodeling of constitutive model using Gaussian process machine learning, Journal of the Mechanics and P hysics of Solids 154 (June) (2021) 104532. doi:10.1016/j.jmps.2021.104532

    arXiv 2021

  45. [45]

    J. N. Fuhg, M. Marino, N. Bouklas, Local approximate Gaussian process regr ession for data-driven constitutive models: development and comparison with neural networ ks, Computer Methods in Applied Mechanics and Engineering 388 (2022) 114217. doi:10.1016/j.cma.2021.114217

    arXiv 2022

  46. [46]

    J. N. Fuhg, N. Bouklas, On physics-informed data-driven isotropic and anisotropic constitutive models through probabilistic machine learning and space-filling samp ling, Computer Methods in Applied Mechanics and Engineering 394 (2022) 114915. doi:10.1016/j.cma.2022.114915

    arXiv 2022

  47. [47]

    A. L. Frankel, R. E. Jones, L. P. Swiler, Tensor Basis Gaussian Proces s Models of Hyperelastic Materials, Journal of Machine Learning for Modeling and Computing 1 (1) (2020) 1–17. doi: 10.1615/jmachlearnmodelcomput.2020033325

    work page doi:10.1615/jmachlearnmodelcomput.2020033325 2020

  48. [48]

    M. Liu, L. Liang, W. Sun, A generic physics-informed neural network- based constitutive model for soft biological tissues, Computer Methods in Applied Mechanics and Engineering 372 (2020) 113402. doi:10.1016/j.cma.2020.113402

    arXiv 2020

  49. [49]

    Konale, V

    A. Konale, V. Srivastava, A physics-informed neural network for mode ling fracture without gradient damage: Formulation, application, and assessment, Journal of the Mechanics and Physics of Solids 206 (PA) (2026) 106395. doi:10.1016/j.jmps.2025.106395

    arXiv 2026

  50. [50]

    Ostadrahimi, A

    A. Ostadrahimi, A. Teimouri, K. Upadhyay, G. Li, Stress softening dam age in strongly nonlinear viscoelastic soft materials: A physics-informed data-dri ven constitutive model with time–temperature coupling, International Journal of Plasticity 197 (Dec ember 2025) (2026) 1–31. doi:10.1016/j.ijplas.2025.104582

    arXiv 2025

  51. [51]

    Upadhyay, J

    K. Upadhyay, J. N. Fuhg, N. Bouklas, K. T. Ramesh, Physics-informed d ata-driven discovery of constitutive models with application to strain-rate-sensitive s oft materials, Computational Me- chanics (jun 2024). doi:10.1007/s00466-024-02497-x

    doi:10.1007/s00466-024-02497-x 2024

  52. [52]

    K. Y. Volokh, Hyperelasticity with softening for modeling material s failure, Journal of the Me- chanics and Physics of Solids 55 (10) (2007) 2237–2264. doi:10.1016/j.jmps.2007.02.012. 31

    work page doi:10.1016/j.jmps.2007.02.012 2007

  53. [53]

    Volokh, Loss of ellipticity in elasticity with energy limiters , European Journal of Mechanics - A/Solids 63 (2017) 36–42

    K. Volokh, Loss of ellipticity in elasticity with energy limiters , European Journal of Mechanics - A/Solids 63 (2017) 36–42. doi:10.1016/j.euromechsol.2016.10.003

    work page doi:10.1016/j.euromechsol.2016.10.003 2017

  54. [54]

    Abu-Qbeitah, M

    S. Abu-Qbeitah, M. Jabareen, K. Y. Volokh, Quasi-static crack propagation in soft materials using the material-sink theory, International Journal of Mechanical Scienc es 248 (December 2022) (2023) 108160. doi:10.1016/j.ijmecsci.2023.108160

    arXiv 2022

  55. [55]

    A. Faye, Y. Lev, K. Y. Volokh, The effect of local inertia around the crack -tip in dynamic fracture of soft materials, Mechanics of Soft Materials 1 (1) (2019). doi:10.1007/s42558-019-0004-2

    work page doi:10.1007/s42558-019-0004-2 2019

  56. [56]

    Y. Lev, A. Faye, K. Volokh, Thermoelastic deformation and failure of rub berlike materials, Journal of the Mechanics and Physics of Solids 122 (2019) 538–554. doi:10.1016/j.jmps.2018.09.033

    work page doi:10.1016/j.jmps.2018.09.033 2019

  57. [57]

    V. Tac, E. Kuhl, A. B. Tepole, Data-driven continuum damage mechanics w ith built-in physics, Extreme Mechanics Letters 71 (August) (2024) 102220. doi:10.1016/j.eml.2024.102220

    arXiv 2024

  58. [58]

    Gaetano, F

    D. Gaetano, F. Greco, L. Leonetti, A. Pascuzzo, Investigation of failure i n anisotropic composite structures via an efficient data-driven multiscale strategy, Meccan ica 60 (10-11) (2025) 3235–3256. doi:10.1007/s11012-025-02008-z

    work page doi:10.1007/s11012-025-02008-z 2025

  59. [59]

    D. Liu, X. Shao, X. Fu, C. Chen, K. Elkhodary, S. Tang, Data-driven nonlo cal damage mechanics and fracture of shells, Engineering Fracture Mechanics 316 (November 2024) (2025) 110864. doi: 10.1016/j.engfracmech.2025.110864

    arXiv 2024

  60. [60]

    C. J. Rojas, J. L. Boldrini, M. L. Bittencourt, Parameter identific ation for a damage phase field model using a physics-informed neural network, Theoretical and App lied Mechanics Letters 13 (3) (2023) 100450. doi:10.1016/j.taml.2023.100450

    arXiv 2023

  61. [61]

    Abdusalamov, J

    R. Abdusalamov, J. Weise, M. Itskov, Rediscovering the Mulli ns effect with deep symbolic regres- sion, International Journal of Plasticity 179 (June) (2024) 104037. doi:10.1016/j.ijplas.2024. 104037

    work page doi:10.1016/j.ijplas.2024 2024

  62. [62]

    A. Ani, R. Nakka, G. Subhash, J.-F. Molinari, S. A. Ponnusami, Machine le arning for computa- tional fracture and damage mechanics— Status and perspectives, Enginee ring Fracture Mechanics 332 (September 2025) (2026) 111778. doi:10.1016/j.engfracmech.2025.111778

    arXiv 2025

  63. [63]

    K. Y. Volokh, Review of the energy limiters approach to modeling fail ure of rubber, Rubber Chemistry and Technology 86 (3) (2013) 470–487. doi:10.5254/rct.13.87948

    work page doi:10.5254/rct.13.87948 2013

  64. [64]

    E. A. Kearsley, Note: Strain Invariants Expressed as Average Stretche s, Journal of Rheology 33 (5) (1989) 757–760. doi:10.1122/1.550063

    doi:10.1122/1.550063 1989

  65. [65]

    H. G. Kilian, An interpretation of the strain-invariants in largely st rained networks, Colloid and Polymer Science 263 (1) (1985) 30–34. doi:10.1007/BF01411245

    work page doi:10.1007/bf01411245 1985

  66. [66]

    P. K. Currie, The Attainable Region of strain-invariant space for el astic materials, International Journal of Non-Linear Mechanics 39 (5) (2004) 833–842. doi:10.1016/S0020-7462(03)00059-3

    work page doi:10.1016/s0020-7462(03)00059-3 2004

  67. [67]

    Xing-gui, X

    F. Xing-gui, X. Jin-sheng, C. Xiong, L. Hui, T. Bing-dong, Hyperelastic Mod el with Energy Limiter for EPDM and Its Numerical Application, Journal of Physics: Conference S eries 2235 (1) (2022) 012088. doi:10.1088/1742-6596/2235/1/012088

    work page doi:10.1088/1742-6596/2235/1/012088 2022

  68. [68]

    Mythravaruni, K

    P. Mythravaruni, K. Y. Volokh, Failure of Rubber Bearings Under Combi ned Shear and Compres- sion, Journal of Applied Mechanics 85 (7) (2018) 1–4. doi:10.1115/1.4040018

    work page doi:10.1115/1.4040018 2018

  69. [69]

    Mythravaruni, K

    P. Mythravaruni, K. Y. Volokh, On Incompressibility Constraint an d Crack Direction in Soft Solids, Journal of Applied Mechanics 86 (10) (oct 2019). doi:10.1115/1.4044089

    work page doi:10.1115/1.4044089 2019

  70. [70]

    Pissarenko, M

    A. Pissarenko, M. A. Meyers, The materials science of skin: Analysis , characterization, and mod- eling, Progress in Materials Science 110 (2020) 100634. doi:10.1016/j.pmatsci.2019.100634

    arXiv 2020

  71. [71]

    W. Li, X. Y. Luo, An Invariant-Based Damage Model for Human and Animal Skins, Annal s of Biomedical Engineering 44 (10) (2016) 3109–3122. doi:10.1007/s10439-016-1603-9 . 32

    work page doi:10.1007/s10439-016-1603-9 2016

  72. [72]

    Volokh, Prediction of arterial failure based on a microstructural b i-layer fiber–matrix model with softening, Journal of Biomechanics 41 (2) (2008) 447–453

    K. Volokh, Prediction of arterial failure based on a microstructural b i-layer fiber–matrix model with softening, Journal of Biomechanics 41 (2) (2008) 447–453. doi:10.1016/j.jbiomech.2007. 08.001

    work page doi:10.1016/j.jbiomech.2007 2008

  73. [73]

    Volokh, Modeling failure of soft anisotropic materials with applic ation to arteries, Journal of the Mechanical Behavior of Biomedical Materials 4 (8) (2011) 1582–1594

    K. Volokh, Modeling failure of soft anisotropic materials with applic ation to arteries, Journal of the Mechanical Behavior of Biomedical Materials 4 (8) (2011) 1582–1594. doi:10.1016/j.jmbbm. 2011.01.002

    work page doi:10.1016/j.jmbbm 2011

  74. [74]

    Williams, C

    C. Williams, C. Rasmussen, Gaussian processes for regression, in : D. Touretzky, M. Mozer, M. Hasselmo (Eds.), Advances in Neural Information Processing System s, Vol. 8, MIT Press, 1995. URL https://proceedings.neurips.cc/paper_files/paper/1995/file/ 7cce53cf90577442771720a370c3c723-Paper.pdf

    1995

  75. [75]

    M. L. Stein, Interpolation of Spatial Data, Springer Series in Statist ics, Springer New York, New York, NY, 1999. doi:10.1007/978-1-4612-1494-6 . URL http://link.springer.com/10.1007/978-1-4612-1494-6

    doi:10.1007/978-1-4612-1494-6 1999

  76. [76]

    Mat´ ern, Spatial Variation, Vol

    B. Mat´ ern, Spatial Variation, Vol. 36 of Lecture Notes in Statistics, Sp ringer New York, New York, NY, 1986. doi:10.1007/978-1-4615-7892-5 . URL http://link.springer.com/10.1007/978-1-4615-7892-5

    doi:10.1007/978-1-4615-7892-5 1986

  77. [77]

    Laurent, R

    L. Laurent, R. Le Riche, B. Soulier, P.-A. Boucard, An Overview of Gradi ent-Enhanced Metamod- els with Applications, Archives of Computational Methods in Engineeri ng 26 (1) (2019) 61–106. doi:10.1007/s11831-017-9226-3

    doi:10.1007/s11831-017-9226-3 2019

  78. [78]

    Y. M. Thant, T. Wakamiya, M. Nukunudompanich, K. Kameda, M. Ihara, S. Manz hos, Kernel regression methods for prediction of materials properties: Recent d evelopments, Chemical Physics Reviews 6 (1) (mar 2025). doi:10.1063/5.0242118

    work page doi:10.1063/5.0242118 2025

  79. [79]

    P. S. Palar, L. Parussini, L. Bregant, K. Shimoyama, L. R. Zuhal, On kern el functions for bi- fidelity Gaussian process regressions, Structural and Multidisci plinary Optimization 66 (2) (2023)

    2023

  80. [80]

    doi:10.1007/s00158-023-03487-y

    work page doi:10.1007/s00158-023-03487-y

Showing first 80 references.