pith. machine review for the scientific record. sign in

arxiv: 2604.10184 · v1 · submitted 2026-04-11 · ❄️ cond-mat.mtrl-sci · physics.geo-ph

Recognition: unknown

Brittle-to-ductile fracturing transition: A chemo-mechanical phase-field framework

Fanyu Wu , Chong Liu , Manolis Veveakis , Manman Hu
Authors on Pith no claims yet

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

classification ❄️ cond-mat.mtrl-sci physics.geo-ph
keywords chemo-mechanical couplingphase-field modelingbrittle-ductile transitionmineral dissolutionfracture propagationgeomaterialsreaction-diffusionductile fracture
0
0 comments X

The pith

A phase-field model shows that competing timescales of chemical dissolution and mechanical loading determine whether fractures in reactive materials fail in a brittle or ductile manner.

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

The paper introduces a coupled chemo-mechanical phase-field model that links mineral dissolution directly to the evolving fracture length scale and includes damage-accelerated reaction-diffusion. This framework produces an enlarged process zone where mass removal blunts the crack tip, reduces local stress concentrations, and degrades stiffness before macroscopic failure occurs. The model demonstrates that slower mechanical loading or more acidic conditions allow greater chemical interaction, resulting in gradual damage accumulation and ductile failure, while rapid loading suppresses chemistry and preserves brittle behavior. A sympathetic reader would care because the transition is controlled by a simple competition of timescales rather than separate material parameters, offering a unified explanation for observed ductilization in reactive geomaterials.

Core claim

The transition between brittle and ductile failure modes is dictated by the competing timescales of chemical degradation and mechanical deformation. Highly acidic environments enhance matrix dissolution and promote ductile fracture, whereas rapid mechanical loading limits chemical interaction and preserves brittle failure mode. The model captures an enlarged fracture process zone driven by chemical mass removal that blunts the sharp crack tip, alleviates near-tip stress concentrations, and produces a more gradual accumulation of damage with delayed onset of macroscopic failure.

What carries the argument

A phase-field framework that dynamically couples local mass removal to the fracture length scale while incorporating damage-accelerated reaction-diffusion processes.

If this is right

  • An enlarged fracture process zone develops from chemical mass removal, widening the damage region ahead of the crack.
  • Near-tip stress concentrations are alleviated, causing measurable stiffness loss before final failure.
  • Damage accumulates more gradually and macroscopic failure is delayed under conditions favoring chemical activity.
  • Ductile fracture is promoted in highly acidic environments while brittle failure is preserved under rapid mechanical loading.

Where Pith is reading between the lines

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

  • The timescale-competition view could be tested by running the same model on other reactive solids such as metals undergoing corrosion-assisted cracking.
  • Long-term predictions for underground storage or geothermal systems would need to track how local acidity evolves with fluid flow rather than assuming constant pH.
  • If the mass-removal coupling proves robust, the framework might be adapted to predict how temperature or pressure shifts the brittle-ductile boundary in reactive settings.

Load-bearing premise

The dynamic coupling of local mass removal directly to the fracture length scale together with damage-accelerated reaction-diffusion is assumed to capture the dominant physics without requiring post-hoc tuning or higher-order corrections.

What would settle it

Laboratory tests on geomaterial samples under controlled pH and strain rates that show no change in failure mode or process-zone width when acidity is varied at fixed loading speed.

Figures

Figures reproduced from arXiv: 2604.10184 by Chong Liu, Fanyu Wu, Manman Hu, Manolis Veveakis.

Figure 1
Figure 1. Figure 1: Experimental evidence of mineral dissolution from microfluidic tests. SEM-based energy-dispersive spectroscopy images of carbonate [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematics representation of the REV scale and the local scale. The figure is an idealization of the microstructure of a carbonate rock. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematics of a solid body (a-b) with Dirichlet boundary [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) Geometry and boundary conditions for the benchmark case. (b) Load–displacement curve of the single edge notched tension test, [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) Geometry and boundary conditions of the sample investigated in this study. The blue dot 2 mm ahead of the initial crack marks the [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Convergence study: (a) mesh size, (b) timestep size. [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: (a) Comparison of the evolution of phase-field damage at the front of the pre-existing crack tip in pure mechanical and chemo-mechanical [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Evolution of chemo-mechanical cracking: (a) initial stage of crack propagation, (b) during crack propagation, and (c) crack propagation [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Distribution of phase-field damage at 100 h upon exposure to acidity levels of (a) pH [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Distribution of pH at 100 h upon exposure to acidity levels of (a) pH [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Distribution of normalized mass removal at 100 h upon exposure to acidity levels of (a) pH [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Distribution of phase-field damage at failure under tensile loading rates of (a) [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Distribution of pH at failure under tensile loading rates of (a) [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Distribution of normalized mass removal at failure under tensile loading rates of (a) [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Temporal evolution of the circumferential stress at the front of the crack tip of the pre-notch sample exposed to various acidic environ [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Temporal evolution of the circumferential stress at the front of the crack tip of the pre-notch sample, considering di [PITH_FULL_IMAGE:figures/full_fig_p021_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: (a) Determination of the width of the FPZ on a di [PITH_FULL_IMAGE:figures/full_fig_p022_17.png] view at source ↗
read the original abstract

In chemically reactive environments, the mechanical integrity of geomaterials is fundamentally compromised by solid matrix dissolution. In this study, we propose a fully coupled chemo-mechanical phase-field framework to capture the dynamic interplay between mineral dissolution and fracture propagation. A key feature of the proposed model is the dynamic coupling of local mass removal to the fracture length scale, while also incorporating the damage-accelerated reaction-diffusion processes. Our results capture the development of an enlarged fracture process zone driven by chemical mass removal. This chemically induced widening blunts the sharp crack tip, alleviating the near-tip stress concentrations and causing a pronounced degradation in material stiffness before failure. Furthermore, we reveal a distinct ductilization effect, characterized by a more gradual accumulation of damage and a delayed onset of macroscopic failure. We show that the transition between brittle and ductile failure modes is dictated by the competing timescales of chemical degradation and mechanical deformation. Highly acidic environments enhance matrix dissolution and promote ductile fracture, whereas rapid mechanical loading limits chemical interaction and preserves brittle failure mode.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a chemo-mechanical phase-field framework for fracture propagation in reactive environments. It features dynamic coupling of local mass removal (dissolution) to the phase-field length scale together with damage-accelerated reaction-diffusion. Simulations illustrate an enlarged process zone, crack-tip blunting, stiffness degradation, and a transition from brittle to ductile failure modes controlled by the relative timescales of chemical degradation and mechanical loading, with acidic conditions favoring ductility and rapid loading preserving brittleness.

Significance. If the novel coupling can be shown to be thermodynamically consistent, the work would provide a mechanistic explanation for environmentally assisted ductilization in geomaterials and could guide predictions of failure under varying pH and loading rates. The emphasis on competing timescales is a clear strength and aligns with observed phenomenology; however, the absence of explicit consistency checks or quantitative experimental validation currently limits the immediate impact.

major comments (2)
  1. [Model formulation] Model formulation section: the dynamic coupling that makes the phase-field length scale l a function of local concentration c (mass removal) is introduced without derivation from a dissipation potential or free-energy functional. This leaves open whether the variational derivative still guarantees non-negative dissipation, so the reported process-zone widening and tip blunting may be regularization artifacts rather than direct consequences of timescale competition.
  2. [Results] Results section: the central claim that competing chemical and mechanical timescales dictate the brittle-to-ductile transition rests on the evolving length scale producing blunting and gradual damage accumulation. No verification of global energy balance or mesh independence for the variable length scale is supplied, which is load-bearing for the ductilization prediction.
minor comments (2)
  1. [Abstract] Abstract: the functional dependence of the length scale on concentration and the precise form of the damage-acceleration term are not stated, hindering immediate assessment of the model's novelty.
  2. [Figures] Figure captions: parameters controlling reaction rate, loading speed, and initial pH should be listed explicitly so that the timescale ratios corresponding to each panel can be reproduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The two major comments identify important points for strengthening the thermodynamic foundation and numerical robustness of the chemo-mechanical coupling. We address each below and will incorporate the suggested additions in the revised manuscript.

read point-by-point responses

  1. Referee: Model formulation section: the dynamic coupling that makes the phase-field length scale l a function of local concentration c (mass removal) is introduced without derivation from a dissipation potential or free-energy functional. This leaves open whether the variational derivative still guarantees non-negative dissipation, so the reported process-zone widening and tip blunting may be regularization artifacts rather than direct consequences of timescale competition.

    Authors: We agree that an explicit derivation from a dissipation potential would place the coupling on firmer variational ground. In the revised manuscript we will introduce the length-scale evolution directly from a dissipation potential that augments the standard phase-field fracture energy with a chemo-mechanical interaction term. We will then verify that the resulting variational derivative yields a non-negative dissipation rate for the coupled system, confirming that the observed process-zone widening and crack-tip blunting are physical consequences of the competing timescales rather than artifacts of the regularization. revision: yes

  2. Referee: Results section: the central claim that competing chemical and mechanical timescales dictate the brittle-to-ductile transition rests on the evolving length scale producing blunting and gradual damage accumulation. No verification of global energy balance or mesh independence for the variable length scale is supplied, which is load-bearing for the ductilization prediction.

    Authors: We acknowledge that explicit numerical checks are required to support the ductilization claim. In the revision we will add (i) time histories of the global energy balance (elastic, fracture, chemical, and dissipation contributions) demonstrating that the total energy remains conserved within the expected tolerance throughout the simulations, and (ii) a mesh-convergence study performed with successively refined discretizations for the variable-length-scale formulation. These verifications will be placed in the Results section and will confirm that the brittle-to-ductile transition is insensitive to mesh size and is driven by the timescale competition. revision: yes

Circularity Check

0 steps flagged

No significant circularity; transition emerges from numerical solution of coupled PDEs rather than by construction.

full rationale

The paper introduces a chemo-mechanical phase-field model with explicit coupling terms (local mass removal affecting the length scale l and damage-accelerated reaction-diffusion). The brittle-to-ductile transition is reported as an outcome of competing timescales in the solved system, not a redefinition or fit of input parameters. No self-citation chains, uniqueness theorems, or ansatzes are invoked to force the result. The derivation remains self-contained against the stated balance laws and constitutive choices.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework rests on standard phase-field fracture assumptions plus new chemo-mechanical couplings whose parameters are not enumerated in the abstract.

axioms (1)
  • standard math Phase-field regularization of sharp cracks via a length-scale parameter
    Invoked implicitly as the basis for the fracture model.
invented entities (1)
  • dynamic mass-removal-to-fracture-length-scale coupling no independent evidence
    purpose: To link chemical dissolution directly to the evolving damage zone width
    Introduced as a key feature of the proposed model; no independent evidence supplied in abstract.

pith-pipeline@v0.9.0 · 5480 in / 1212 out tokens · 68969 ms · 2026-05-10T16:16:21.495236+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

57 extracted references · 56 canonical work pages

  1. [1]

    Approximation of functional depending on jumps by elliptic functional via t-convergence

    Approximation of functional depending on jumps by elliptic functional via t-convergence. Communications on Pure and Applied Mathematics 43, 999–1036. doi:10.1002/cpa.3160430805. Amor, H., Marigo, J.J., Maurini, C.,

    work page doi:10.1002/cpa.3160430805

  2. [2]

    Journal of the Mechanics and Physics of Solids 57, 1209–1229

    Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments. Journal of the Mechanics and Physics of Solids 57, 1209–1229. doi:10.1016/j.jmps.2009.04.011. Backers, T., Stanchits, S., Dresen, G.,

    work page doi:10.1016/j.jmps.2009.04.011 2009

  3. [3]

    International Journal of Rock Mechanics and Mining Sciences 42, 1094–1101

    Tensile fracture propagation and acoustic emission activity in sandstone: The effect of loading rate. International Journal of Rock Mechanics and Mining Sciences 42, 1094–1101. doi:10.1016/j.ijrmms.2005.05.011. Barenblatt, G.I.,

    work page doi:10.1016/j.ijrmms.2005.05.011 2005

  4. [4]

    The Mathematical Theory of Equilibrium Cracks in Brittle Fracture

    The Mathematical Theory of Equilibrium Cracks in Brittle Fracture, in: Advances in Applied Mechanics. Elsevier. volume 7, pp. 55–129. doi:10.1016/S0065-2156(08)70121-2. Belytschko, T., Black, T.,

    work page doi:10.1016/s0065-2156(08)70121-2

  5. [5]

    International Journal for Numerical Methods in Engineering 45, 601–620

    Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering 45, 601–620. doi:10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S. Bilgen, C., Weinberg, K.,

    work page doi:10.1002/(sici)1097-0207(19990620)45:5

  6. [6]

    On the crack-driving force of phase-field models in linearized and finite elasticity

    On the crack-driving force of phase-field models in linearized and finite elasticity. Computer Methods in Applied Mechanics and Engineering 353, 348–372. doi:10.1016/j.cma.2019.05.009. Borden, M.J., Hughes, T.J.R., Landis, C.M., Anvari, A., Lee, I.J.,

    work page doi:10.1016/j.cma.2019.05.009 2019

  7. [7]

    Computer Methods in Applied Mechanics and Engineering 312, 130–166

    A phase-field formulation for fracture in ductile materials: Finite deformation balance law derivation, plastic degradation, and stress triaxiality effects. Computer Methods in Applied Mechanics and Engineering 312, 130–166. doi:10.1016/j.cma.2016.09.005. Borja, R.I., Chen, W., Odufisan, A.R.,

    work page doi:10.1016/j.cma.2016.09.005 2016

  8. [8]

    Journal of the Mechanics and Physics of Solids 173, 105198

    A constitutive framework for rocks undergoing solid dissolution. Journal of the Mechanics and Physics of Solids 173, 105198. doi:10.1016/j.jmps.2023.105198. Bourdin, B., Francfort, G.A., Marigo, J.J.,

    work page doi:10.1016/j.jmps.2023.105198 2023

  9. [9]

    Journal of the Mechanics and Physics of Solids 48, 797–826

    Numerical experiments in revisited brittle fracture. Journal of the Mechanics and Physics of Solids 48, 797–826. doi:10.1016/S0022-5096(99)00028-9. Bourdin, B., Francfort, G.A., Marigo, J.J.,

    work page doi:10.1016/s0022-5096(99)00028-9

  10. [10]

    Journal of Geotechnical and Geoenvironmental Engineering 130, 728–739

    Oedometric Tests on Artificially Weathered Carbonatic Soft Rocks. Journal of Geotechnical and Geoenvironmental Engineering 130, 728–739. doi:10.1061/(ASCE)1090-0241(2004)130:7(728). Chen, J., Hu, M.,

    work page doi:10.1061/(asce)1090-0241(2004)130:7(728 2004

  11. [11]

    Geomechanics for Energy and the Environment 37, 100526

    Single fluid-driven crack propagation in analogue rock assisted by chemical environment. Geomechanics for Energy and the Environment 37, 100526. doi:10.1016/j.gete.2023.100526. Chen, J., Hu, M.,

    work page doi:10.1016/j.gete.2023.100526 2023

  12. [12]

    URL:https: //hal.science/hal-05343935

    Chemically enhanced fluid-driven crack propagation in analogue rock, in: 3rd International Conference on Energy Geotechnics 2025, Navier Laboratory, Ecole Nationale des Ponts et Chauss´ees, Universit´e Gustave Eiffel, CNRS, Paris, France. URL:https: //hal.science/hal-05343935. Chukwudozie, C., Bourdin, B., Yoshioka, K.,

    2025

  13. [13]

    Computer Methods in Applied Mechanics and Engineering 347, 957–982

    A variational phase-field model for hydraulic fracturing in porous media. Computer Methods in Applied Mechanics and Engineering 347, 957–982. doi:10.1016/j.cma.2018.12.037. Ciantia, M.O., Castellanza, R., Crosta, G.B., Hueckel, T.,

    work page doi:10.1016/j.cma.2018.12.037 2018

  14. [14]

    Engineering Geology 184, 1–18

    Effects of mineral suspension and dissolution on strength and compressibility of soft carbonate rocks. Engineering Geology 184, 1–18. doi:10.1016/j.enggeo.2014.10.024. Ciantia, M.O., Hueckel, T.,

    work page doi:10.1016/j.enggeo.2014.10.024 2014

  15. [15]

    G ´eotechnique 63, 768–785

    Weathering of submerged stressed calcarenites: chemo-mechanical coupling mechanisms. G ´eotechnique 63, 768–785. doi:10.1680/geot.SIP13.P.024. Cui, C., Ma, R., Mart´ınez-Pa˜neda, E.,

    work page doi:10.1680/geot.sip13.p.024

  16. [16]

    Journal of the Mechanics and Physics of Solids 147, 104254

    A phase field formulation for dissolution-driven stress corrosion cracking. Journal of the Mechanics and Physics of Solids 147, 104254. doi:10.1016/j.jmps.2020.104254. Dugdale, D.S.,

    work page doi:10.1016/j.jmps.2020.104254 2020

  17. [17]

    Journal of the Mechanics and Physics of Solids 212, 106585

    A variational phase-field model for anisotropic fracture accounting for multiple cohesive lengths. Journal of the Mechanics and Physics of Solids 212, 106585. doi:10.1016/j.jmps.2026.106585. Fei, F., Choo, J., Liu, C., White, J.A.,

    work page doi:10.1016/j.jmps.2026.106585 2026

  18. [18]

    International Journal for Numerical and Analytical Methods in Geomechanics 46, 841–868

    Phase-field modeling of rock fractures with roughness. International Journal for Numerical and Analytical Methods in Geomechanics 46, 841–868. doi:10.1002/nag.3317. Fei, F., Costa, A., Dolbow, J.E., Settgast, R.R., Cusini, M.,

    work page doi:10.1002/nag.3317

  19. [19]

    International Journal for Numerical and Analytical Methods in Geomechanics 47, 3065–3089

    A phase-field model for hydraulic fracture nucleation and propagation in porous media. International Journal for Numerical and Analytical Methods in Geomechanics 47, 3065–3089. doi:10.1002/nag.3612. Francfort, G.A., Marigo, J.J.,

    work page doi:10.1002/nag.3612

  20. [20]

    Journal of the Mechanics and Physics of Solids , author =

    Revisiting brittle fracture as an energy minimization problem. Journal of the Mechanics and Physics of Solids 46, 1319–1342. doi:10.1016/S0022-5096(98)00034-9. Gaston, D., Newman, C., Hansen, G., Lebrun-Grandi´e, D.,

    work page doi:10.1016/s0022-5096(98)00034-9

  21. [21]

    Nuclear Engineering and Design 239, 1768–1778

    MOOSE: A parallel computational framework for coupled systems of nonlinear equations. Nuclear Engineering and Design 239, 1768–1778. doi:10.1016/j.nucengdes.2009.05.021. Guo, Y ., Na, S.,

    work page doi:10.1016/j.nucengdes.2009.05.021 2009

  22. [22]

    Computer Methods in Applied Mechanics and Engineering 419, 116645

    A reactive-transport phase-field modelling approach of chemo-assisted cracking in saturated sandstone. Computer Methods in Applied Mechanics and Engineering 419, 116645. doi:10.1016/j.cma.2023.116645. Gu´evel, A., Meng, Y ., Peco, C., Juanes, R., Dolbow, J.E.,

    work page doi:10.1016/j.cma.2023.116645 2023

  23. [23]

    Journal of the Mechanics and Physics of Solids 181, 105427

    A Darcy–Cahn–Hilliard model of multiphase fluid-driven fracture. Journal of the Mechanics and Physics of Solids 181, 105427. doi:10.1016/j.jmps.2023.105427. Horne, R., Genter, A., McClure, M., Ellsworth, W., Norbeck, J., Schill, E.,

    work page doi:10.1016/j.jmps.2023.105427 2023

  24. [24]

    Nature Reviews Clean Technology 1, 148–160

    Enhanced geothermal systems for clean firm energy generation. Nature Reviews Clean Technology 1, 148–160. doi:10.1038/s44359-024-00019-9. Hu, M., Hueckel, T.,

    work page doi:10.1038/s44359-024-00019-9

  25. [25]

    G ´eotechnique 63, 313–321

    Environmentally enhanced crack propagation in a chemically degrading isotropic shale. G ´eotechnique 63, 313–321. doi:10.1680/geot.SIP13.P.020. Hu, M., Hueckel, T.,

    work page doi:10.1680/geot.sip13.p.020

  26. [26]

    Geomechanics for Energy and the Environment 19, 100114

    Modeling of subcritical cracking in acidized carbonate rocks via coupled chemo-elasticity. Geomechanics for Energy and the Environment 19, 100114. doi:10.1016/j.gete.2019.01.003. Hu, T.,

    work page doi:10.1016/j.gete.2019.01.003 2019

  27. [27]

    Computers and Geotechnics 36, 934–943

    Feedback mechanisms in chemo-mechanical multi-scale modeling of soil and sediment compaction. Computers and Geotechnics 36, 934–943. doi:10.1016/j.compgeo.2009.02.005. 22 Ilgen, A.G., Newell, P., Hueckel, T., Espinoza, D.N., Hu, M.,

    work page doi:10.1016/j.compgeo.2009.02.005 2009

  28. [28]

    (Eds.), Science of Carbon Storage in Deep Saline Formations

    Coupled Chemical-Mechanical Processes Associated With the Injection of CO2 into Subsurface, in: Newell, P., Ilgen, A.G. (Eds.), Science of Carbon Storage in Deep Saline Formations. Elsevier, pp. 337–359. doi:10.1016/B978-0-12-812752-0.00015-0. Ip, S.C.Y ., Borja, R.I.,

    work page doi:10.1016/b978-0-12-812752-0.00015-0

  29. [29]

    Journal of the Mechanics and Physics of Solids 181, 105441

    Modeling heterogeneity and permeability evolution in a compaction band using a phase-field approach. Journal of the Mechanics and Physics of Solids 181, 105441. doi:10.1016/j.jmps.2023.105441. Jiang, W., Hu, T., Aagesen, L.K., Biswas, S., Gamble, K.A.,

    work page doi:10.1016/j.jmps.2023.105441 2023

  30. [30]

    Theoretical and Applied Fracture Mechanics 119, 103348

    A phase-field model of quasi-brittle fracture for pressurized cracks: Application to UO2 high-burnup microstructure fragmentation. Theoretical and Applied Fracture Mechanics 119, 103348. doi:10.1016/j.tafmec.2022. 103348. Kristensen, P.K., Niordson, C.F., Mart´ınez-Pa˜neda, E.,

    work page doi:10.1016/j.tafmec.2022 2022

  31. [31]

    Journal of the Mechanics and Physics of Solids 143, 104093

    A phase field model for elastic-gradient-plastic solids undergoing hydrogen embrit- tlement. Journal of the Mechanics and Physics of Solids 143, 104093. doi:10.1016/j.jmps.2020.104093. Kristensen, P.K., Niordson, C.F., Mart´ınez-Pa˜neda, E.,

    work page doi:10.1016/j.jmps.2020.104093 2020

  32. [32]

    Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 379, 20210021

    An assessment of phase field fracture: crack initiation and growth. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 379, 20210021. doi:10.1098/rsta.2021.0021. Ling, B., Sodwatana, M., Kohli, A., Ross, C.M., Jew, A., Kovscek, A.R., Battiato, I.,

    work page doi:10.1098/rsta.2021.0021 2021

  33. [33]

    Proceedings of the National Academy of Sciences 119, e2122520119

    Probing multiscale dissolution dynamics in natural rocks through microfluidics and compositional analysis. Proceedings of the National Academy of Sciences 119, e2122520119. doi:10.1073/ pnas.2122520119. Liu, C., Kang, X., Calo, V .M., Hu, M., Regenauer-Lieb, K., 2026a. Automatically adaptive finite element method via residual minimization onto dual norms ...

    work page doi:10.1016/j.cma.2025.118581 2025

  34. [34]

    Liu, S., Wang, Y ., Geng, X., Wu, W., 2026b

    doi:10.3390/ma11101921. Liu, S., Wang, Y ., Geng, X., Wu, W., 2026b. A generalized higher-order phase-field model for brittle fracture in anisotropic rocks. Journal of the Mechanics and Physics of Solids 210, 106526. doi:10.1016/j.jmps.2026.106526. Mart´ınez-Pa˜neda, E., Golahmar, A., Niordson, C.F.,

    work page doi:10.3390/ma11101921 2026

  35. [35]

    Computer Methods in Applied Mechanics and Engineering 342, 742–761

    A phase field formulation for hydrogen assisted cracking. Computer Methods in Applied Mechanics and Engineering 342, 742–761. doi:10.1016/j.cma.2018.07.021. Miehe, C., Dal, H., Sch¨anzel, L.M., Raina, A.,

    work page doi:10.1016/j.cma.2018.07.021 2018

  36. [36]

    International Journal for Numerical Methods in Engineering 106, 683–711

    A phase-field model for chemo-mechanical induced fracture in lithium-ion battery electrode particles. International Journal for Numerical Methods in Engineering 106, 683–711. doi:10.1002/nme.5133. Miehe, C., Hofacker, M., Welschinger, F., 2010a. A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator...

    work page doi:10.1002/nme.5133 2010

  37. [37]

    Miehe, F

    Phase field modeling of fracture in multi-physics problems. Part III. Crack driving forces in hydro-poro-elasticity and hydraulic fracturing of fluid-saturated porous media. Computer Methods in Applied Mechanics and Engineering 304, 619–655. doi:10.1016/ j.cma.2015.09.021. Miehe, C., Welschinger, F., Hofacker, M., 2010b. Thermodynamically consistent phase...

    work page doi:10.1002/nme.2861 2015

  38. [38]

    Cement and Concrete Research 199, 108012

    Fracture in concrete: X-ray tomography with in-situ testing, digital volume correlation and phase-field modeling. Cement and Concrete Research 199, 108012. doi:10.1016/j.cemconres.2025.108012. Mollaali, M., Yoshioka, K., Lu, R., Montoya, V ., Vilarrasa, V ., Kolditz, O.,

    work page doi:10.1016/j.cemconres.2025.108012 2025

  39. [39]

    International Journal for Numerical Methods in Engineering 126, e7621

    Variational Phase-Field Fracture Approach in Reactive Porous Media. International Journal for Numerical Methods in Engineering 126, e7621. doi:10.1002/nme.7621. Mo¨es, N., Dolbow, J., Belytschko, T.,

    work page doi:10.1002/nme.7621

  40. [40]

    International Journal for Numerical Methods in Engineering , author =

    A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 46, 131–150. doi:10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J. Nguyen, T.T., Yvonnet, J., Bornert, M., Chateau, C.,

    work page doi:10.1002/(sici)1097-0207(19990910)46:1

  41. [41]

    Journal of the Mechanics and Physics of Solids 95, 320–350

    Initiation and propagation of complex 3D networks of cracks in heterogeneous quasi- brittle materials: Direct comparison betweenin situtesting-microCT experiments and phase field simulations. Journal of the Mechanics and Physics of Solids 95, 320–350. doi:10.1016/j.jmps.2016.06.004. Peerlings, R.H.J., De Borst, R., Brekelmans, W.a.M., De Vree, J.H.P.,

    work page doi:10.1016/j.jmps.2016.06.004 2016

  42. [42]

    International Journal for Numerical Methods in Engineering 39, 3391–3403

    Gradient Enhanced Damage for Quasi-Brittle Materials. International Journal for Numerical Methods in Engineering 39, 3391–3403. doi:10.1002/(SICI)1097-0207(19961015)39:19<3391::AID-NME7>3. 0.CO;2-D. Peerlings, R.H.J., Geers, M.G.D., de Borst, R., Brekelmans, W.A.M.,

    work page doi:10.1002/(sici)1097-0207(19961015)39:19

  43. [43]

    International Journal of Solids and Structures 38, 7723–7746

    A critical comparison of nonlocal and gradient-enhanced softening continua. International Journal of Solids and Structures 38, 7723–7746. doi:10.1016/S0020-7683(01)00087-7. Permann, C.J., Gaston, D.R., Andr ˇs, D., Carlsen, R.W., Kong, F., Lindsay, A.D., Miller, J.M., Peterson, J.W., Slaughter, A.E., Stogner, R.H., Martineau, R.C.,

    work page doi:10.1016/s0020-7683(01)00087-7

  44. [44]

    SoftwareX 11, 100430

    MOOSE: Enabling massively parallel multiphysics simulation. SoftwareX 11, 100430. doi:10.1016/j.softx.2020. 100430. Ruan, H., Rezaei, S., Yang, Y ., Gross, D., Xu, B.X.,

    work page doi:10.1016/j.softx.2020 2020

  45. [45]

    A thermo-mechanical phase-field fracture model: Application to hot cracking simulations in additive manufacturing.Journal of the Mechanics and Physics of Solids, 172: 105169, 2023

    A thermo-mechanical phase-field fracture model: Application to hot cracking simulations in additive manufacturing. Journal of the Mechanics and Physics of Solids 172, 105169. doi:10.1016/j.jmps.2022.105169. Rutqvist, J., Chijimatsu, M., Jing, L., Millard, A., Nguyen, T.S., Rejeb, A., Sugita, Y ., Tsang, C.F.,

    work page doi:10.1016/j.jmps.2022.105169 2022

  46. [46]

    Part 3: Effects of THM coupling in sparsely fractured rocks

    A numerical study of THM effects on the near-field safety of a hypothetical nuclear waste repository—BMT1 of the DECOV ALEX III project. Part 3: Effects of THM coupling in sparsely fractured rocks. International Journal of Rock Mechanics and Mining Sciences 42, 745–755. doi:10.1016/j.ijrmms.2005.03.012. Schuler, L., Ilgen, A.G., Newell, P.,

    work page doi:10.1016/j.ijrmms.2005.03.012 2005

  47. [47]

    Computer Methods in Applied Mechanics and Engineering 362, 112838

    Chemo-mechanical phase-field modeling of dissolution-assisted fracture. Computer Methods in Applied Mechanics and Engineering 362, 112838. doi:10.1016/j.cma.2020.112838. Stefanou, I., Sulem, J.,

    work page doi:10.1016/j.cma.2020.112838 2020

  48. [48]

    Journal of Geophysical Research: Solid Earth 119, 880–899

    Chemically induced compaction bands: Triggering conditions and band thickness. Journal of Geophysical Research: Solid Earth 119, 880–899. doi:10.1002/2013JB010342. Tang, X., Hu, M.,

    work page doi:10.1002/2013jb010342

  49. [49]

    Rock Mechanics and Rock Engineering 56, 515–533

    A Reactive-Chemo-Mechanical Model for Weak Acid-Assisted Cavity Expansion in Carbonate Rocks. Rock Mechanics and Rock Engineering 56, 515–533. doi:10.1007/s00603-022-03077-2. Tang, X., Hu, M.,

    work page doi:10.1007/s00603-022-03077-2

  50. [50]

    Rock Mechanics and Rock Engineering doi:10.1007/s00603-025-04485-w

    The Effect of Chemical Environment on Crack Propagation in Pressure-Sensitive Rocks. Rock Mechanics and Rock Engineering doi:10.1007/s00603-025-04485-w. 23 Vafaie, A., Cama, J., Soler, J.M., Kivi, I.R., Vilarrasa, V .,

    work page doi:10.1007/s00603-025-04485-w

  51. [51]

    Renewable and Sustainable Energy Reviews 178, 113270

    Chemo-hydro-mechanical effects of CO2 injection on reservoir and seal rocks: A review on laboratory experiments. Renewable and Sustainable Energy Reviews 178, 113270. doi:10.1016/j.rser.2023.113270. Wu, F., Hu, M.,

    work page doi:10.1016/j.rser.2023.113270 2023

  52. [52]

    Journal of Geophysical Research: Solid Earth 130, e2024JB030896

    Influence of Carbon Dioxide on Micro-Cracking in Calcite: An Atomistic Scale Investigation. Journal of Geophysical Research: Solid Earth 130, e2024JB030896. doi:10.1029/2024JB030896. Wu, F., Sac-Morane, A., Rattez, H., Veveakis, M., Hu, M.,

    work page doi:10.1029/2024jb030896

  53. [53]

    International Journal of Rock Mechanics and Mining Sciences 196, 106319

    Onset of reactive brittle cracking in sandstones: DEM-informed phase-field modeling. International Journal of Rock Mechanics and Mining Sciences 196, 106319. doi:10.1016/j.ijrmms.2025.106319. Wu, J.Y .,

    work page doi:10.1016/j.ijrmms.2025.106319 2025

  54. [54]

    Journal of the Mechanics and Physics of Solids 103, 72–99

    A unified phase-field theory for the mechanics of damage and quasi-brittle failure. Journal of the Mechanics and Physics of Solids 103, 72–99. doi:10.1016/j.jmps.2017.03.015. Wu, J.Y ., Hong, Y .F.,

    work page doi:10.1016/j.jmps.2017.03.015 2017

  55. [55]

    Journal of the Mechanics and Physics of Solids 172, 105168

    Crack nucleation and propagation of electromagneto-thermo-mechanical fracture in bulk superconductors during magnetization. Journal of the Mechanics and Physics of Solids 172, 105168. doi:10.1016/j.jmps.2022.105168. Wu, T., De Lorenzis, L.,

    work page doi:10.1016/j.jmps.2022.105168 2022

  56. [56]

    Computer Methods in Applied Mechanics and Engineering 312, 196–223

    A phase-field approach to fracture coupled with diffusion. Computer Methods in Applied Mechanics and Engineering 312, 196–223. doi:10.1016/j.cma.2016.05.024. Zietlow, W.K., Labuz, J.F.,

    work page doi:10.1016/j.cma.2016.05.024 2016

  57. [57]

    International Journal of Rock Mechanics and Mining Sciences 35, 291–299

    Measurement of the intrinsic process zone in rock using acoustic emission. International Journal of Rock Mechanics and Mining Sciences 35, 291–299. doi:10.1016/S0148-9062(97)00323-9. 24

    work page doi:10.1016/s0148-9062(97)00323-9