arxiv: 2604.10623 · v1 · submitted 2026-04-12 · ⚛️ physics.comp-ph · cond-mat.other

Recognition: unknown

A unified sharp-diffusive phase-field model for bulk and interfacial cohesive fracture

Ye-Hang Qin , Ye Feng

Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:23 UTC · model grok-4.3

classification ⚛️ physics.comp-ph cond-mat.other

keywords phase-field modelingcohesive fractureinterfacial debondingsharp-diffusive modelunified cohesive lawmultiphase materialsdamage concentrationfracture simulation

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The pith

A phase-field model unifies bulk and interfacial cohesive fracture using parameters taken directly from local material properties.

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

The paper develops a phase-field approach that handles fracture in both the material interior and at its interfaces with one set of equations. It adds an analytical source term localized at interfaces and pairs it with the Ω²-model to produce sharp displacement jumps inside an otherwise diffuse damage field. This construction is meant to break the usual coupling that forces interface toughness to depend on the surrounding bulk properties. A reader would care because it removes the need for ad-hoc corrections and extreme mesh refinement when modeling layered or composite solids where interfaces often govern failure paths.

Core claim

By introducing an analytical, strongly localized interfacial source term into the phase-field formulation and leveraging the Ω²-model to manifest Dirac-like damage concentration and emergent displacement discontinuities, the proposed framework can describe the cohesive failure of both bulk and interfacial regions using a unified set of parametric equations for the cohesive law, where the model parameters are directly determined by the local material properties without the need for additional corrections.

What carries the argument

An analytical, strongly localized interfacial source term q_φ introduced into the phase-field equation and combined with the Ω²-model's capacity for Dirac-like damage concentration and emergent discontinuities.

If this is right

The model reproduces a range of prescribed interfacial cohesive laws without recalibration.
It captures the competition between debonding along interfaces and cracking through the surrounding matrix.
Interface toughness can be set independently of bulk properties.
No complex corrections or exceptionally fine local meshes near interfaces are needed.
The same equations apply to both bulk and interface regions in a single simulation.

Where Pith is reading between the lines

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

The approach could reduce setup time when modeling real composites whose interfaces have been characterized separately from the matrix.
Similar localized source terms might be adapted to other regularization problems that currently suffer from non-local coupling across material boundaries.
Direct use of measured interface properties in the parametric equations would let designers explore interface engineering without iterative fitting.
Extension to three-dimensional geometries with curved interfaces would test whether the analytical source term remains effective without further modification.

Load-bearing premise

The analytical interfacial source term together with the Ω²-model produces independent control of interface toughness and emergent discontinuities without hidden coupling or post-hoc adjustments.

What would settle it

Run a simulation of a bimaterial specimen with deliberately different prescribed toughness values at the interface and in the bulk; check whether the computed energy dissipated exactly at the interface matches the input value while the bulk response remains unaffected and no additional local mesh refinement is required.

Figures

Figures reproduced from arXiv: 2604.10623 by Ye Feng, Ye-Hang Qin.

**Figure 2.** Figure 2: The localized phase-field driving source [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: The cohesive laws with and without the source term: [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: Cohesive laws of exponential type and the [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

**Figure 5.** Figure 5: Geometry and boundary conditions for the uniaxial tension of a three-phase material (material [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

**Figure 6.** Figure 6: Convergence of the model’s cohesive law with respect to mesh size: simulations performed with a characteristic [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

**Figure 7.** Figure 7: Comparative characterisation of crack propagation in the phase-field and damage fields under uniaxial tension [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗

**Figure 8.** Figure 8: Schematic diagram of a double-cantilever beam specimen [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗

**Figure 9.** Figure 9: Force-displacement curve in the double-cantilever beam test [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗

**Figure 10.** Figure 10: Two-dimensional Double-cantilever beam test: cohesive law [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗

**Figure 11.** Figure 11: Comparative characterisation of crack propagation in the phase-field and damage fields in the double-cantilever [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗

**Figure 12.** Figure 12: Problem setup for the single-fiber reinforced composite under 2D uniaxial tension: (a) geometric configuration [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗

**Figure 13.** Figure 13: Simulated force-displacement curve for the single-fiber reinforced composite [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗

**Figure 14.** Figure 14: Comparison between the extracted numerical cohesive response at the interface and the theoretical solution [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗

**Figure 15.** Figure 15: Simulated failure process of the single-fiber composite: evolution of the phase-field [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗

**Figure 16.** Figure 16: Geometrical configuration and boundary conditions for a plate with an inclined interface. [PITH_FULL_IMAGE:figures/full_fig_p021_16.png] view at source ↗

**Figure 17.** Figure 17: Influence of interface strength on crack path selection ( [PITH_FULL_IMAGE:figures/full_fig_p023_17.png] view at source ↗

**Figure 18.** Figure 18: Influence of interface strength on crack path selection ( [PITH_FULL_IMAGE:figures/full_fig_p024_18.png] view at source ↗

read the original abstract

In traditional phase-field modeling of multiphase materials, a significant challenge arises from the non-local nature of fracture energy regularization, where interfacial toughness is inherently coupled with the properties of the surrounding bulk phases. Achieving consistency with prescribed material properties typically necessitates complex corrections and exceptionally fine local mesh refinement near the interfaces. To address this fundamental issue, we leverage the capacity of the recently proposed $\Omega^2$-model to manifest Dirac-like damage concentration and emergent displacement discontinuities, while introducing an analytical, strongly localized interfacial source term $q_{\phi}$ into the phase-field formulation. It should be emphasized that the ``sharp" nature of the proposed model manifests as a naturally emergent strong discontinuity within a continuum framework, fundamentally distinguishing it from inherently discrete approaches such as cohesive element method. This allows for the independent and precise control of interface toughness in a straightforward manner. Theoretical analysis further reveals that the proposed framework can describe the cohesive failure of both bulk and interfacial regions using a unified set of parametric equations for the cohesive law, where the model parameters are directly determined by the local material properties without the need for additional corrections. The model's versatility is numerically validated through a series of benchmarks. The results confirm that the proposed model not only accurately reproduces diverse interfacial cohesive laws but also captures the intricate competition between interfacial debonding and matrix cracking. This sharp-diffusive phase-field model may provide a robust and computationally efficient tool for predicting complex fracture trajectories in sophisticated engineering materials.

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.

Desk Editor's Note private letter to a colleague

Qin and Feng combine the Ω²-model with an analytical interfacial source term to let local material properties set both bulk and interface cohesive laws without extra corrections or mesh hacks.

read the letter

The core advance is practical: phase-field models for multiphase materials have long suffered from interface toughness bleeding into the surrounding bulk, forcing fine meshes and ad-hoc fixes. This work uses the Ω²-model's ability to produce emergent discontinuities and adds a strongly localized analytical source q_φ that decouples the interface response. The derivation shows the Euler-Lagrange equation recovers the prescribed traction-separation law directly from local G_c and δ_c values, and the numerics confirm the same parameter set reproduces both bulk cracking and debonding on one mesh across stiffness contrasts. That independence is the real contribution, and the stress-test note indicates the paper supplies the steps and consistent benchmarks rather than just asserting them. The evidence for the central claim therefore holds up better than the abstract-only read suggested. Minor gaps remain: the reported tests do not include explicit mesh-convergence tables or error bars on the recovered cohesive laws, and the behavior under large stiffness ratios or dynamic conditions is not yet shown. These are normal for a methods paper at this stage and do not undermine the unification result. The work is aimed at computational mechanics groups simulating fracture in composites or layered materials who want to avoid repeated parameter tuning. If you run phase-field codes for engineering fracture problems, the unified parametric form is worth testing. I would send it for peer review; the math is traceable, the numerics address the stated difficulty, and the practical payoff is clear enough to justify referee time.

Referee Report

2 major / 2 minor

Summary. The paper claims to introduce a unified sharp-diffusive phase-field model for cohesive fracture in both bulk and interfacial regions. By combining the recently proposed Ω²-model (which produces Dirac-like damage concentration and emergent displacement discontinuities) with a new analytical, strongly localized interfacial source term q_φ, the framework allows a single set of parametric equations for the cohesive law whose parameters are taken directly from local material properties (G_c and δ_c) with no auxiliary corrections or interface-specific mesh refinement. Theoretical analysis shows that the resulting Euler-Lagrange equation recovers the target traction-separation law on the interface, while numerical benchmarks demonstrate reproduction of diverse interfacial cohesive laws and the competition between interfacial debonding and matrix cracking on the same mesh.

Significance. If the central unification holds without hidden parameter coupling, the work is significant for computational mechanics of multiphase materials. It offers a continuum framework that naturally produces sharp discontinuities while independently prescribing interface toughness, potentially eliminating the ad-hoc corrections and extreme local refinement that plague conventional phase-field models of interfacial failure. The parameter-free character once local properties are inserted, together with the demonstrated cross-regime consistency, would make the approach attractive for engineering-scale fracture prediction.

major comments (2)

[Theoretical analysis] Theoretical analysis section: the manuscript states that the Euler-Lagrange equation recovers the prescribed traction-separation law, but the explicit steps combining the Ω²-model with the analytical q_φ must be shown to confirm that no residual coupling between bulk and interface parameters remains; this is load-bearing for the 'parameter-free' and 'unified' claims.
[Numerical validation] Numerical validation section: while benchmarks are said to confirm consistency across stiffness contrasts, the absence of reported mesh-convergence studies, quantitative error norms, and direct comparisons against established cohesive-zone or phase-field interface models leaves the accuracy of the emergent discontinuities and toughness control unsubstantiated.

minor comments (2)

[Abstract] Abstract: the phrase 'theoretical analysis further reveals' would be strengthened by a one-sentence indication of the key mathematical step (recovery of the traction-separation law).
[Model formulation] Notation: the symbol q_φ is introduced without an immediate equation reference; a parenthetical pointer to its defining expression would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation for minor revision. The comments are constructive and we address each major point below, with corresponding revisions planned for the manuscript.

read point-by-point responses

Referee: [Theoretical analysis] Theoretical analysis section: the manuscript states that the Euler-Lagrange equation recovers the prescribed traction-separation law, but the explicit steps combining the Ω²-model with the analytical q_φ must be shown to confirm that no residual coupling between bulk and interface parameters remains; this is load-bearing for the 'parameter-free' and 'unified' claims.

Authors: We agree that the explicit derivation is necessary to fully support the claims. In the revised manuscript we will expand the theoretical analysis section to present the complete step-by-step combination of the Ω²-model with the analytical interfacial source term q_φ. This will demonstrate that the Euler-Lagrange equation recovers the target traction-separation law and that no residual coupling between bulk and interface parameters remains, thereby confirming the parameter-free and unified character of the formulation. revision: yes
Referee: [Numerical validation] Numerical validation section: while benchmarks are said to confirm consistency across stiffness contrasts, the absence of reported mesh-convergence studies, quantitative error norms, and direct comparisons against established cohesive-zone or phase-field interface models leaves the accuracy of the emergent discontinuities and toughness control unsubstantiated.

Authors: We acknowledge that additional quantitative evidence will strengthen the numerical validation. In the revised manuscript we will include mesh-convergence studies under successive refinements, report quantitative error norms (such as L2 and H1 norms for displacement and phase-field fields), and provide direct comparisons against established cohesive-zone element implementations as well as other phase-field interface models. These additions will substantiate the accuracy of the emergent discontinuities and the independent toughness control across stiffness contrasts. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives a new analytical interfacial source term q_φ and demonstrates via the Euler-Lagrange equation that it recovers the target traction-separation law using parameters taken directly from local G_c and δ_c values. The Ω²-model is invoked only for its established ability to produce Dirac-like damage concentration and emergent discontinuities; the unification claim and parameter independence are obtained from the new source term and are not reduced to a fit, self-definition, or unverified self-citation. Numerical benchmarks confirm consistency across regimes without retuning, satisfying the independence condition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The model rests on the Ω²-model for Dirac-like concentration and on the assumption that an added analytical source term decouples interface toughness without introducing new fitting constants. No free parameters are explicitly introduced beyond local material properties.

axioms (1)

domain assumption The Ω²-model produces Dirac-like damage concentration and emergent displacement discontinuities within a continuum setting.
Invoked to justify the sharp-diffusive character and to enable the unified cohesive description.

invented entities (1)

analytical interfacial source term q_φ no independent evidence
purpose: To localize damage at interfaces and permit independent control of interfacial toughness.
New term introduced to solve the coupling problem; no external falsifiable prediction supplied in abstract.

pith-pipeline@v0.9.0 · 5553 in / 1354 out tokens · 60840 ms · 2026-05-10T16:23:23.815228+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 1 canonical work pages

[1]

Ordinary differential equations

Arnold, V.I., 1992. Ordinary differential equations. Springer Science & Business Media

1992
[2]

A novel phase-field based cohesive zone model for modeling interfacial failure in composites

Bian, P.L., Qing, H., Schmauder, S., 2021. A novel phase-field based cohesive zone model for modeling interfacial failure in composites. International Journal for Numerical Methods in Engineering 122, 7054–7077

2021
[3]

A unified phase-field method-based framework for modeling quasi-brittle fracture in composites with interfacial debonding

Bian, P.L., Qing, H., Schmauder, S., Yu, T., 2024. A unified phase-field method-based framework for modeling quasi-brittle fracture in composites with interfacial debonding. Composite Structures 327, 117647

2024
[4]

Phase-field modeling of anisotropic brittle fracture including several damage mechanisms

Bleyer, J., Alessi, R., 2018. Phase-field modeling of anisotropic brittle fracture including several damage mechanisms. Computer Methods in Applied Mechanics and Engineering 336, 213–236

2018
[5]

Numerical experiments in revisited brittle fracture

Bourdin, B., Francfort, G.A., Marigo, J.J., 2000. Numerical experiments in revisited brittle fracture. Journal of the Mechanics and Physics of Solids 48, 797–826

2000
[6]

A variational approach to fracture incorporating any convex strength criterion

Bourdin, B., Marigo, J.J., Maurini, C., Zolesi, C., 2025. A variational approach to fracture incorporating any convex strength criterion. arXiv preprint arXiv:2506.22558 . 26

work page arXiv 2025
[7]

Computational modelling of impact damage in brittle materials

Camacho, G.T., Ortiz, M., 1996. Computational modelling of impact damage in brittle materials. International Journal of solids and structures 33, 2899–2938

1996
[8]

Endowing explicit cohesive laws to the phase-field fracture theory

Feng, Y., Fan, J., Li, J., 2021. Endowing explicit cohesive laws to the phase-field fracture theory. Journal of the Mechanics and Physics of Solids , 104464

2021
[9]

Phase-field model for 2d cohesive-frictional shear fracture: An energetic formulation

Feng, Y., Freddi, F., Li, J., Ma, Y.E., 2024. Phase-field model for 2d cohesive-frictional shear fracture: An energetic formulation. Journal of the Mechanics and Physics of Solids 189, 105687

2024
[10]

A unified regularized variational cohesive fracture theory with directional energy decomposition

Feng, Y., Li, J., 2023. A unified regularized variational cohesive fracture theory with directional energy decomposition. International Journal of Engineering Science 182, 103773

2023
[11]

Convergence of phase-field models with emergent discontinuities to czms in multi-dimension (under review)

Feng, Y., Li, J., 2026. Convergence of phase-field models with emergent discontinuities to czms in multi-dimension (under review). Journal of the Mechanics and Physics of Solids

2026
[12]

Revisiting brittle fracture as an energy minimization problem

Francfort, G.A., Marigo, J.J., 1998. Revisiting brittle fracture as an energy minimization problem. Journal of the Mechanics and Physics of Solids 46, 1319–1342. Hansen-Dörr, A.C., de Borst, R., Hennig, P., Kästner, M., 2019. Phase-field modelling of interface failure in brittle materials. Computer Methods in Applied Mechanics and Engineering 346, 25–42

1998
[13]

Crack deflection at an interface between dissimilar elastic materials: role of residual stresses

He, M.Y., Evans, A.G., Hutchinson, J.W., 1994. Crack deflection at an interface between dissimilar elastic materials: role of residual stresses. International Journal of Solids and Structures 31, 3443– 3455

1994
[14]

Phase-field model of mode iii dynamic fracture

Karma, A., Kessler, D.A., Levine, H., 2001. Phase-field model of mode iii dynamic fracture. Physical Review Letters 87, 045501

2001
[15]

Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field fe implementations

Miehe, C., Welschinger, F., Hofacker, M., 2010. Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field fe implementations. International journal for numerical methods in engineering 83, 1273–1311

2010
[16]

A phase-field method for compu- tational modeling of interfacial damage interacting with crack propagation in realistic microstructures obtained by microtomography

Nguyen, T.T., Yvonnet, J., Zhu, Q.Z., Bornert, M., Chateau, C., 2016. A phase-field method for compu- tational modeling of interfacial damage interacting with crack propagation in realistic microstructures obtained by microtomography. Computer Methods in Applied Mechanics and Engineering 312, 567– 595. 27

2016
[17]

Paggi, M., Reinoso, J., 2017. Revisiting the problem of a crack impinging on an interface: a modeling framework for the interaction between the phase field approach for brittle fracture and the interface cohesive zone model. Computer Methods in Applied Mechanics and Engineering 321, 145–172

2017
[18]

The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models

Pham, K., Marigo, J.J., Maurini, C., 2011. The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models. Journal of the Mechanics and Physics of Solids 59, 1163–1190

2011
[19]

Adissipation-basedarc-lengthmethodforrobust simulation of brittle and ductile failure

Verhoosel, C.V., Remmers, J.J., Gutiérrez, M.A., 2009. Adissipation-basedarc-lengthmethodforrobust simulation of brittle and ductile failure. International journal for numerical methods in engineering 77, 1290–1321

2009
[20]

Phase-fieldmodelingofbrittlefractureinheterogeneous bars

Vicentini, F., Carrara, P., DeLorenzis, L., 2023. Phase-fieldmodelingofbrittlefractureinheterogeneous bars. European Journal of Mechanics-A/Solids 97, 104826

2023
[21]

Variational phase-field modeling of cohesive fracture with flexibly tunable strength surface

Vicentini, F., Heinzmann, J., Carrara, P., De Lorenzis, L., 2025. Variational phase-field modeling of cohesive fracture with flexibly tunable strength surface. Journal of the Mechanics and Physics of Solids , 106424

2025
[22]

Variational phase-field fracture modeling with interfaces

Yoshioka, K., Mollaali, M., Kolditz, O., 2021. Variational phase-field fracture modeling with interfaces. Computer Methods in Applied Mechanics and Engineering 384, 113951

2021
[23]

A triple-damage model for fibrous composite material with intra-and inter- laminar decomposition, reduced-order-homogenization and phase field method

Yue, J., Yuan, Z., 2025. A triple-damage model for fibrous composite material with intra-and inter- laminar decomposition, reduced-order-homogenization and phase field method. International Journal of Solids and Structures , 113490

2025
[24]

Crack deflection in brittle media with heterogeneous interfaces and its appli- cation in shale fracking

Zeng, X., Wei, Y., 2017. Crack deflection in brittle media with heterogeneous interfaces and its appli- cation in shale fracking. Journal of the Mechanics and Physics of Solids 101, 235–249

2017
[25]

Modelling distinct failure mechanisms in composite materials by a combined phase field method

Zhang, P., Feng, Y., Bui, T.Q., Hu, X., Yao, W., 2020. Modelling distinct failure mechanisms in composite materials by a combined phase field method. Composite Structures 232, 111551

2020
[26]

Phase field modeling of fracture in fiber reinforced composite laminate

Zhang, P., Hu, X., Bui, T.Q., Yao, W., 2019. Phase field modeling of fracture in fiber reinforced composite laminate. International Journal of Mechanical Sciences 161, 105008. 28

2019