{"id":"f44f73f1-b8ed-4a5c-adf9-099c4358f0b3","arxiv_id":"2508.05638","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"A neural-network constitutive model with built-in thermodynamic consistency, convexity, and normality represents anisotropic damage via a second-order damage tensor and structural tensors.","lead":"The authors present a machine-learning constitutive model in which thermodynamic consistency and convexity are enforced by construction, enabling it to represent direction-dependent damage in soft materials. The model tracks damage with a tensorial internal variable and fits multiaxial hydrogel data better than an isotropic version.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Damage-evolution equations (3.15) and (4.4) ignore the psi_stress derivatives that the final conjugate forces (5.4) include, so the implemented model is not the thermodynamically consistent one claimed.","rationale":"Good-faith reading: the paper contributes a data-driven anisotropic damage framework with ICNN-based polyconvex energies, convex decreasing attenuation functions, normality corrections, and a staggered training scheme. The reader's chosen weakest assumption, the fixed co-axial damage axes in (3.2), is explicitly acknowledged and confined to the implemented case, so it limits generality but does not contradict the paper's stated scope. The unacknowledged derivation gap between (3.10)/(3.15) and (5.4) is more damaging: it touches the central theoretical guarantee, and it is not resolved by the numerical dissipation plots because those plots cannot show consistency unless the evolution used is the derivative of the energy actually plotted. Independent support such as code or data is absent, so the gap cannot be checked post hoc. The coaxial restriction and the in-sample experimental comparison remain secondary concerns. Overall the reader's CONDITIONAL verdict is appropriate: the framework is promising and likely repairable, but the implemented model is not currently proven to be the thermodynamically consistent model described in the conclusions.","tokens_in":26022,"tokens_out":9679,"duration_ms":106624,"concrete_test":"Re-derive the evolution law for the total energy (5.12)-(5.13), and in the incompressible case (6.3) with (6.7), by differentiating y_i = -partial psi^e/partial alpha_i including partial psi_stress/partial alpha_i. Implement both the published update (4.4)/(3.15) and the corrected update on the double-network hydrogel dataset, and compare the alpha_i trajectories, predicted stress histories, and cumulative dissipation y_i dot-alpha_i. If they differ at any training strain, the paper's numerical results were produced by a model different from the stated one; the thermodynamic-consistency claim would then require a corrected derivation and re-trained fits before acceptance.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.2 derives the damage-evolution laws starting from (3.10), where the conjugate forces are y_i = -(d p_i/d alpha_i) bar-psi_i and y_0 analogously; (3.12)-(3.15) and the damage-induced-anisotropy counterpart (4.4) follow by differentiating only that expression. Section 5.3 then changes the model: the final energy (5.12)-(5.13) contains the normality-correction terms psi_stress, and (5.4) defines the actual conjugate forces as y_i = -(d p_i/d alpha_i) hat-psi_i^ICNN - d psi_stress/d alpha_i, and similarly for y_0. Because psi_stress depends on alpha_0, alpha_i through p_0, p_i, the extra terms d psi_stress/d alpha_i are generally nonzero away from C = I. No corrected version of (3.15) or (4.4) is supplied. Consequently the alpha_i-update implemented for the fits is not the derivative of the stated free-energy function, and the dissipation inequality (2.6)_2 is not established for the model actually fitted to the hydrogel data. This is not a cosmetic discrepancy: alpha_i controls the stiffness degradation, and the R-regularization in (7.1) only shrinks the residual coefficients; it does not remove the missing derivatives. This undercuts the central 'thermodynamically consistent, as designed' claim even in the coaxial setting that the paper explicitly restricts itself to.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a data-driven constitutive framework for anisotropic damage in solids, built around a second-order damage tensor, an ICNN-parameterized strain energy with polyconvex invariant inputs, convex decreasing attenuation functions, a damage potential with KKT conditions, and additional normality-correction terms. Damage evolution is formulated through conjugate thermodynamic forces, and a decoupled training scheme is introduced to separate unloading-branch energy fitting from damage-evolution fitting. The framework is demonstrated on synthetic incompressible isotropic, transversely isotropic, and compressible orthotropic benchmarks, and on experimental double-network hydrogel data under multiaxial loading, where the anisotropic model is reported to outperform an isotropic damage model.","tokens_in":26255,"tokens_out":6186,"duration_ms":72187,"significance":"If the theoretical claims are fully established, this would be a useful contribution: it combines several nontrivial constraints (objectivity, polyconvexity, normality, and thermodynamic consistency) in a trainable constitutive model and demonstrates predictive capability on multiaxial damage data. The decoupled training scheme and the explicit treatment of damage-induced anisotropy are practically valuable. The paper also honestly acknowledges that non-negativity of the energy is verified numerically on the training range rather than proven globally. The main theoretical concern, detailed below, concerns the consistency between the damage evolution equations used in the implementation and the conjugate forces associated with the final energy, which is load-bearing for the central 'thermodynamically consistent as designed' claim.","major_comments":[{"comment":"The damage evolution equations (3.15) and (4.4) are derived from the conjugate forces in Eq. (3.10), namely y_i = -p'_i(alpha_i) bar-psi_i and y_0 = -p'_0(alpha_0) bar-psi_0. However, the actual model defined in Section 5.3 includes the normality-correction term psi_stress, and the final conjugate forces in Eq. (5.4) are y_i = -p'_i(alpha_i) hat-psi_i^ICNN - d psi_stress/d alpha_i, with an analogous expression for y_0. Because psi_stress depends on alpha_i and alpha_0 through p_i and p_0, the extra derivatives d psi_stress/d alpha_i are generally nonzero. No corrected version of (3.15) or (4.4) is supplied, and the experimental validation in Section 8 explicitly states that the anisotropic model is trained using (3.15)_1 and (4.4). The alpha_i-update actually implemented is therefore not the derivative of the stated Helmholtz free energy. The dissipation inequality (2.6)_2 is established for the energy and evolution equations only when the conjugate forces are those of Eq. (3.10); it is not established for the model that includes psi_stress. The numerical dissipation plots in Sections 8 cannot close this gap unless the plotted dissipation is computed with the full conjugate forces and the corrected evolution equations. This is a load-bearing inconsistency for the central claim that the formulation is thermodynamically consistent by construction.","section":"Sections 3.2, 4, 5.3, and 8, Eqs. (3.15), (4.4), (5.4), (5.12)-(5.13)"},{"comment":"The damage tensor is assumed to remain coaxial with the virgin material symmetry axes and to have fixed principal directions throughout the damage process. The invariant basis in Eq. (3.7) is built from the fixed structural tensors L_i, so the model cannot represent a damage state whose principal directions rotate relative to the material axes, for example under non-proportional shear with evolving crack orientation. This is an explicit assumption, and Remark 2 sketches a generalization, but the abstract and concluding remarks present the model as a general anisotropic damage model for orthotropic materials without this qualification. Since the experimental validation is limited to biaxial and planar tests where the loading directions coincide with the symmetry axes, the scope of the model should be stated explicitly in the abstract and conclusions, or the generalized formulation of Remark 2 should be implemented and tested to support the unqualified claim.","section":"Section 3.1, Eq. (3.2), and Section 9"}],"minor_comments":[{"comment":"The paper correctly notes that non-negativity of the trained energy is verified only on the training range. The concluding sentence that 'numerical results across all examples demonstrate that the resulting strain energy density functions are non-negative' should be qualified as a data-range verification, not a global property, to avoid overstating the physics guarantee.","section":"Section 2.7 and Section 9"},{"comment":"The function 'softplus' is used without definition. Although it is standard in machine learning, a brief definition or reference would improve readability for the mechanics audience.","section":"Section 5.3, Eq. (5.7)"},{"comment":"The text refers to 'Mullin's effect' in one place; this should be 'Mullins effect' for consistency with the literature and with the rest of the manuscript.","section":"Section 8, experimental example"},{"comment":"The comparison between the isotropic and anisotropic models would be clearer if a quantitative measure of fit error (for example, normalized RMS error per loading mode) were reported, since the qualitative improvement of the anisotropic model is otherwise assessed only visually from Figs. 5 and 6.","section":"Section 8, experimental example"}],"recommendation":"major_revision","confidential_remarks":"The main blocker is the mismatch between the damage evolution equations and the conjugate forces of the final energy. This is fixable but requires re-deriving the evolution laws with the psi_stress contributions and retraining or re-validating the examples. The coaxial fixed-direction assumption is a scope limitation rather than an internal error, but it should be more prominently qualified. The paper otherwise contains a substantial and well-motivated framework with promising numerical demonstrations."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper builds a plausible data-driven anisotropic damage model using second-order damage tensors, ICNN polyconvex potentials, convex decreasing attenuation functions, and a staggered training scheme. The combination is genuinely new: prior work was either isotropic or analytical, and the synthetic benchmarks are clean. The experimental example on double-network hydrogels shows the anisotropic model fits the multiaxial data much better than the isotropic version. That part is worth reading.\n\nThe main problem is a load-bearing inconsistency in the thermodynamics. Section 3.2 derives the damage evolution laws from conjugate forces y_i = -dp_i/dalpha_i * psi_i_bar, before the normality-correction terms are added. In Section 5.3 the final energy includes those corrections via psi_stress, and the conjugate forces in Eq. (5.4) correctly include d psi_stress/d alpha_i. But no corrected version of Eqs. (3.15) or (4.4) is given, and the experimental example explicitly uses Eq. (4.4). Since psi_stress depends on alpha_i through p_i(alpha_i), the missing derivatives are nonzero. The implemented alpha-update is therefore not the derivative of the stated free energy, and the dissipation inequality (2.6)_2 is not established for the model actually fitted. The R-regularization in Eq. (7.1) does not fix this; it only shrinks the residual coefficients, not the missing derivative terms.\n\nTwo other soft spots, in proportion. The co-axiality assumption—damage principal directions fixed to material symmetry axes—is stated in Remark 2, so it is an acknowledged simplification, but it limits the model to non-rotating damage fabrics. And the experimental validation is in-sample: all loading modes are used for training, so claims of prediction are overstated. No code or data are provided, which hurts reproducibility.\n\nIf the evolution equations are corrected and the dissipation inequality is demonstrated for the actual implemented model, the framework has real value. As it stands, the paper deserves a serious referee, but the referee should insist on seeing the corrected evolution laws and ideally a check of the dissipation inequality on the fitted model. I would not desk-reject; I would send it to review expecting major revision.","headline":"A genuinely new anisotropic damage architecture, undermined by a mismatch between the derived evolution laws and the final energy—worth review but not acceptance as is.","tokens_in":26917,"tokens_out":7012,"would_cite":false,"duration_ms":69366,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A physics-augmented neural network learns anisotropic damage from stress–strain data, reproducing direction-dependent softening in hydrogels.","keywords":["anisotropic damage","continuum damage mechanics","damage-induced anisotropy","input convex neural networks","polyconvexity","thermodynamic consistency","Mullins effect","double-network hydrogels"],"falsifier":"Train the model on a multiaxial protocol and then subject the material to non-proportional loading, such as uniaxial pre-stretch followed by shear at 45 degrees; if measurements of damage orientation (for example from X-ray scattering or birefringence) show the damage axes rotating away from the original symmetry axes, the co-axial assumption fails and the predicted stresses will diverge from the measured ones.","tokens_in":25653,"feed_emoji":"⚙️","tokens_out":10017,"duration_ms":103075,"temperature":0.7,"pith_summary":"Soft materials lose stiffness unevenly: cracks and broken polymer bonds leave a material softer in some directions than others, a family of behaviors known as damage-induced anisotropy. This paper proposes a machine-learning constitutive model that learns such direction-dependent degradation directly from stress–strain measurements. The central claim is that representing damage as a second-order tensor and writing the strain-energy density as a convex neural network of deformation and damage invariants yields a thermodynamically consistent, polyconvex model for compressible and incompressible orthotropic materials. On multiaxial experiments in double-network hydrogels, the anisotropic model reproduces equal-biaxial, unequal-biaxial, and planar responses where an isotropic damage model misses important stress components. If the claim holds, physics-safe learned constitutive equations of this type could replace hand-chosen degradation functions in simulations of soft-tissue and polymer failure.","feed_headline":"Neural damage model learns directional softening from stretch data","feed_subtitle":"Thermodynamically consistent network captures direction-dependent softening that isotropic damage models miss.","key_machinery":"The load-bearing object is the free-energy representation $\\bar\\psi_e(C,L_i,\\alpha_i,\\alpha_0)$: an isotropic function of the right Cauchy–Green tensor $C$, structural tensors $L_i$ for material symmetry, and scalar amplitudes $\\alpha_0,\\alpha_i$ from the second-order damage tensor $D_{\\rm aniso}=\\sum_i \\alpha_i L_i$. Polyconvex invariants $I_C=C:I$, $II_C=\\mathrm{cof}(C):I$, $III_C=\\det C$, $\\tilde I^{(i)}=C:\\tilde L^{(i)}$, $\\tilde J^{(i)}=\\mathrm{cof}(C):\\tilde L^{(i)}$ feed non-decreasing input-convex neural networks, whose composition preserves convexity. The attenuation envelope $p(\\alpha)$ is a trainable weighted sum of convex decreasing functions $(1-\\alpha/a)^{q_j}$, and damage evolves through potentials $g_i=G_i(y_i)-G_i(r_i)$ with consistency and loading/unloading conditions. A two-stage training procedure first fits the unloading branches to learn the elastic energy, then fits damage evolution on the full cyclic data.","core_discovery":"On the paper's own terms, the discovery is a general data-driven anisotropic damage formulation. The Helmholtz free energy is written as an isotropic function of deformation invariants and structural tensors, with damage encoded by a second-order damage tensor whose principal directions are assumed to remain co-axial with the virgin material's symmetry axes. Non-decreasing input-convex neural networks parameterize the strain-energy potentials, so polyconvexity holds by construction; attenuation of energy under damage is a trainable weighted sum of convex decreasing functions $(1-\\alpha/a)^{q_j}$, generalizing the classical $(1-d)$ degradation; and damage evolution is governed by conjugate thermodynamic forces through damage potentials with thresholds and KKT consistency conditions. Additional terms are included so that energy and stress vanish at the undeformed state, satisfying the normality condition. The authors demonstrate accurate recovery of synthetic isotropic, transversely isotropic, and compressible orthotropic responses, and show that on double-network hydrogel data the anisotropic model captures the dominant multiaxial behavior where an isotropic damage model falls short.","pith_inferences":["An implicit extension: adding independent structural tensors for damage axes would let the same architecture track rotating damage fabrics under shear or non-proportional loading, relaxing the co-axial assumption.","Because the authors note that multiple internal states can fit the same stress–strain data, pairing the model with damage-orientation measurements could disambiguate the hidden state and make predictions more physical.","The inclusion of $III_C$ in the potentials means volumetric damage is representable, pointing toward applications in rocks and concrete, though the reported examples are dominated by deviatoric loading.","A natural test of extrapolation is to train on one loading mode and predict another, such as training on uniaxial and planar data and testing on pure shear; the paper's experiments do not isolate this."],"forward_implications":["The same formulation covers incompressible and compressible orthotropic materials, so one architecture can model gels, elastomers, and soft tissue without switching constitutive forms.","Because polyconvexity and thermodynamic consistency are enforced by construction, the learned strain-energy function can be used directly in finite-element simulations of failure.","The nonlinear convex decreasing attenuation functions replace the classical $(1-d)$ degradation, so the model can represent richer damage evolution while keeping energy non-negative in the demonstrated range.","The decoupled training strategy learns elastic response from unloading branches and damage evolution from full cycles, reducing the cost of fitting path-dependent models and improving scalability.","For the double-network hydrogel validation, the anisotropic model reproduces the dominant multiaxial response in equal-biaxial, unequal-biaxial, and planar tests, whereas the isotropic model misses key stress components."],"supporting_citations":[{"why":"the isotropic data-driven damage framework that the present anisotropic formulation generalizes","marker":"[1]"},{"why":"supplies the experimental multiaxial double-network hydrogel data used to validate damage-induced anisotropy","marker":"[12]"},{"why":"introduces the strain-based damage threshold formulation that the damage evolution here generalizes","marker":"[22]"},{"why":"establishes the representation of anisotropic scalar functions as isotropic functions with structural tensors","marker":"[40]"},{"why":"provides the polyconvex invariant family and generalized structural tensor construction used as network inputs","marker":"[43]"},{"why":"shows how structural tensors handle anisotropic inelastic behavior when damage and material axes are not aligned, the route to relaxing the co-axial assumption","marker":"[49]"},{"why":"supplies the consistent extra terms that enforce the normality (zero energy and stress at the undeformed state) condition","marker":"[60]"},{"why":"provides the input-convex neural network architecture that guarantees polyconvexity by construction","marker":"[63]"},{"why":"introduces the damage potential and conjugate-force evolution that the damage equations are built on","marker":"[67]"}],"fun_headline_variants":["Neural net learns anisotropic damage with built-in thermodynamics","Polyconvex NN models damage directionality in solids","Data-driven damage model respects thermodynamics and anisotropy","Anisotropic damage from convex neural networks","Physics-augmented ML captures direction-dependent softening"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise, stated in Section 3.1, is that damage principal directions stay fixed and aligned with the material's original symmetry axes throughout loading; the paper also notes in Sections 2.6 and 2.7 that growth conditions are omitted and energy non-negativity is verified numerically rather than by construction.","fun_headline_variants_meta":{"raw":{"variants":["Neural net learns anisotropic damage with built-in thermodynamics","Polyconvex NN models damage directionality in solids","Data-driven damage model respects thermodynamics and anisotropy","Anisotropic damage from convex neural networks","Physics-augmented ML captures direction-dependent softening"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000174,"raw_usage":{"total_tokens":1310,"prompt_tokens":999,"completion_tokens":311,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":615,"completion_tokens_details":{"reasoning_tokens":240}},"tokens_in":615,"tokens_out":311,"duration_ms":3604,"temperature":1.0,"reasoning_tokens":240,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T14:55:23.228477+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train the model on a multiaxial protocol and then subject the material to non-proportional loading, such as uniaxial pre-stretch followed by shear at 45 degrees; if measurements of damage orientation (for example from X-ray scattering or birefringence) show the damage axes rotating away from the original symmetry axes, the co-axial assumption fails and the predicted stresses will diverge from the measured ones.","supporting_citations":[{"cited_title":"Strain-and stress-based continuum damage models—i. formulation,","cited_arxiv_id":null,"evidence_quote":"introduces the strain-based damage threshold formulation that the damage evolution here generalizes"},{"cited_title":"Data-driven continuum damage mechanics with built-in physics,","cited_arxiv_id":null,"evidence_quote":"the isotropic data-driven damage framework that the present anisotropic formulation generalizes"},{"cited_title":"Distinctive characteristics of internal fracture in tough double network hydrogels revealed by various modes of stretching,","cited_arxiv_id":null,"evidence_quote":"supplies the experimental multiaxial double-network hydrogel data used to validate damage-induced anisotropy"},{"cited_title":"Structural tensors for anisotropic solids,","cited_arxiv_id":null,"evidence_quote":"establishes the representation of anisotropic scalar functions as isotropic functions with structural tensors"},{"cited_title":"A class of orthotropic and transversely isotropic hyperelastic constitutive models based on a polyconvex strain energy function,","cited_arxiv_id":null,"evidence_quote":"provides the polyconvex invariant family and generalized structural tensor construction used as network inputs"},{"cited_title":"Using structural tensors for inelastic material modeling in the finite strain regime–a novel approach to anisotropic damage,","cited_arxiv_id":null,"evidence_quote":"shows how structural tensors handle anisotropic inelastic behavior when damage and material axes are not aligned, the route to relaxing the co-axial assumption"},{"cited_title":"Neural networks meet hyperelasticity: A guide to enforcing physics,","cited_arxiv_id":null,"evidence_quote":"supplies the consistent extra terms that enforce the normality (zero energy and stress at the undeformed state) condition"},{"cited_title":"Input convex neural networks,","cited_arxiv_id":null,"evidence_quote":"provides the input-convex neural network architecture that guarantees polyconvexity by construction"},{"cited_title":"On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects,","cited_arxiv_id":null,"evidence_quote":"introduces the damage potential and conjugate-force evolution that the damage equations are built on"}],"review_version":1}