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REVIEW 3 major objections 5 minor 100 references

Matrix-free phase-field modeling of fracture in micromechanical testing simulations of inelastic materials

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A new phase-field fracture model treats damage and inelastic deformation as rheologically separated, so crack growth is driven purely by elastic strain energy while yield stress and viscosity remain intact.

desk verdict Serious computational-mechanics paper with a genuinely new constitutive assembly and a strong HPC implementation, but the physical validation is qualitative and the perfect-plastic branch harbors a positive-feedback issue that needs a definitive answer. read the letter →

arxiv 2607.21150 v1 pith:4227HOWI submitted 2026-07-23 physics.comp-ph cond-mat.mtrl-sciphysics.app-ph

Fabio Di Gioacchino , Rezgar Shakeri , Zachary Atkins , Karen Stengel , Layla Ghaffari , Jeremy Thompson , Jed Brown This is my paper · ORCID
classification physics.comp-phcond-mat.mtrl-sciphysics.app-ph
keywords phase-fieldfractureviscoplasticityrheologicalelementmatrix-freefiniteelementsp-multigridGPUcomputingmicrostructure-resolvedsimulationductile
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This work claims that phase-field fracture can be coupled to visco-elastoplasticity by inserting a fracture element in series with the inelastic rheological block, so that damage and inelasticity interact only through homogenization-length-scale effects. The authors implement this in a matrix-free, GPU-portable finite-element framework and demonstrate that it reproduces characteristic inelastic responses and crack patterns in synthetic microstructures: X-shaped cracking in compression and cup-cone fracture in tension. If the claim holds, it offers a computationally tractable route to simulating fracture in 3D microstructural specimens without coupling damage to plastic strain. The central innovation is constitutive, not just computational: damage deactivates load transfer but never degrades the inelastic properties themselves.

What carries the argument

The central object is the rheological fracture element (F) placed in series with the Perić–Dettmer constitutive block. It acts like a switch: as the phase-field damage variable grows, the elastic stiffness of the block is degraded through a quadratic degradation function, but the inelastic branches (yield stress, viscosity, hardening) are unaffected. Damage is driven solely by the tensile component of the elastic strain energy density. This serial assembly encodes the microstructural-scale hypothesis that fully damaged zones can no longer accumulate inelastic deformation, while inelastic deformation elsewhere does not directly promote damage.

What would settle it

A concrete test: perform an in-situ tensile test on a particle-reinforced alloy while tracking plastic strain and crack growth. If new plastic strain is observed to accumulate inside or immediately ahead of a propagating crack within the damaged region, the model's assumption of mutual exclusion would be violated. Alternatively, compare simulated force–displacement curves at multiple stress triaxialities with experimentally measured ductile fracture data; the model would fail if it cannot reproduce the strong triaxiality dependence of ductility characteristic of void-growth-driven fracture.

Watch

Extended reading notes

Core claim

The paper introduces a rheological fracture element assembled in series with the Perić–Dettmer block (parallel Hooke, Maxwell, and Prandtl branches). In this assembly, damage accumulation deactivates the block's ability to transmit stress, while the inelastic parameters — yield stress, viscosity, hardening — remain undegraded. Only the elastic strain energy, specifically its tensile part, drives damage evolution. The authors show that this framework, implemented with high-order matrix-free finite elements and p-multigrid preconditioning on GPUs, can reproduce experimentally documented crack patterns such as the X-shaped failure in compressed granular composites and cup-cone fracture in tensi

Load-bearing premise

The load-bearing assumption is that, at the resolved microstructural scale, plastic strain does not directly promote damage and damage does not degrade the yield surface — inelastic and fracture properties interact only at homogenization length scales.

Editorial extensions

If this is right

  • If this framework is correct, phase-field fracture models can be extended to inelastic materials without introducing plastic strain terms into the damage driving force, simplifying constitutive calibration.
  • The model should reproduce cup-cone fracture and X-shaped compression crack patterns in 3D microstructural simulations, providing a testable link between microstructure and macroscopic failure morphology.
  • The matrix-free, GPU-portable implementation makes high-resolution 3D fracture simulations of realistic microstructures (about 20M degrees of freedom) feasible on moderate clusters, enabling systematic parameter studies.
  • The decoupling suggests that damage and plasticity can be modeled independently at the resolved scale, with their interaction emerging only after homogenization — a potentially useful simplification for multiscale methods.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The model effectively assumes that plastic strain and damage are spatially exclusive: where damage forms, further plastic accumulation ceases. This sharp prediction could be tested with in-situ X-ray tomography or digital image correlation during ductile fracture.
  • If inelastic deformation does promote damage nucleation in real materials (as in Gurson-type void growth), the model would fail in regimes where plastic strain precedes fracture. Direct comparison with quantitative ductile-fracture data at different stress triaxialities would reveal this.
  • The serial-assembly idea naturally extends to anisotropic plasticity (e.g., crystal plasticity): damage could be driven by volumetric strain energy only, which might explain void nucleation at grain boundaries and particles without coupling to slip activity.
  • A homogenized version of this model might produce an effective damage–plasticity coupling at the macroscale without explicitly degrading the yield surface, offering a new perspective on continuum damage mechanics.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper presents a phase-field fracture framework coupled to a Perić–Dettmer finite-strain visco-elasto-plastic constitutive model, with a newly introduced 'rheological fracture element' assembled in series with the inelastic rheological block. The model is implemented in the open-source Ratel library using matrix-free high-order finite elements with p-multigrid preconditioning on GPUs. Three three-dimensional proof-of-concept simulations are reported: a sharp-notched plate in shear, a particle-reinforced viscoelastic composite in compression, and a particle-reinforced visco-elasto-plastic composite in tension. The central claim is that the framework reproduces characteristic inelastic responses and crack propagation patterns at the microstructural scale, with damage driven solely by tensile elastic strain energy and without degrading the underlying inelastic properties.

Significance. The computational contribution is genuine: this appears to be the first matrix-free phase-field fracture study that includes large-strain inelasticity and GPU execution, with a detailed appendix of consistent tangents, reproducible microstructure-generation scripts, and a demonstration at roughly 20 million degrees of freedom on a few GPU nodes. The series-assembly idea is conceptually attractive for microstructure-resolved simulations because it separates fracture from inelastic constitutive degradation. However, the load-bearing constitutive hypothesis has a serious self-amplification issue in the perfect-plastic branch, and the claimed validation against experimental crack patterns is only qualitative. If the constitutive issue is resolved, the paper would be a valuable contribution to high-throughput micromechanical test simulation.

major comments (3)
  1. [§3, Eqs. (51)–(52), (A8), Appendix C (C24)] In the active perfect-plastic regime, the transmitted deviatoric stress is damage-independent, but the stored deviatoric energy used to drive damage is ψ_d = μ||ε_e_d||² = σ0²/(6μg(ϕ)²), which grows as 1/g² as damage accumulates. Since H = max(H_n, ψ+) in Eq. (62), damage raises its own driving force even at fixed deformation. This positive feedback contradicts the series-element picture of Fig. 2 and the §5.3 statement that damage does not promote additional plastic flow. It also means crack morphology in Example 3 may be controlled by ζ and η rather than by Gc. Please provide a numerical probe, e.g., a uniformly stressed perfect-plastic bar under fixed displacement, showing whether damage grows spontaneously, or reformulate the damage driving force for inelastic branches.
  2. [§5, Examples 2–3, Figs. 9–14] The central validation claim rests on qualitative visual agreement with experimental patterns. No quantitative metric is reported: crack path angle, load-drop slope, localization width, or comparison with a reference ductile-fracture phase-field model. Because the damage viscosity ζ and residual stiffness η are chosen primarily for numerical stability (§5.1), and because of the positive feedback in the perfect-plastic branch, the reported cup-cone and X-shaped patterns could be artifacts of these regularization parameters. I request a sensitivity study over ζ and η (at least an order of magnitude variation) and at least one quantitative comparison with experimental data or an established ductile-fracture model.
  3. [§2.3–§3, Eqs. (49)–(51)] The local Maxwell/Prandtl updates use μ_degr = g(ϕ)μ inside the return mapping, while the history variable H is computed from ψ+ evaluated with the resulting ε_e_d. Since ε_e_d then depends on ϕ, the stress is not the variational derivative of a single total potential ∫(gψ+ + ψ−) plus fracture energy. This breaks the variational structure that underlies the phase-field formulation and makes the ϕ-dependence of H nonphysical. Please clarify how thermodynamic consistency is maintained, or derive the system from a well-defined incremental potential.
minor comments (5)
  1. [§2.3 heading] Typo: 'Parallel Assembly fof' should be 'of'.
  2. [Abstract vs §5] The abstract says simulations were run on an 'El Capitan high performance computing prototype,' while §5 says all simulations were run on 'Tioga.' If Tioga is the El Capitan prototype, please state this explicitly.
  3. [§5.2] The viscous damping coefficient for the lubricated platen condition is not given a numerical value. Since it directly affects the boundary condition and the reported force–displacement response, please report the value used.
  4. [Fig. 2 caption] The fracture element is drawn as a solid containing a sharp crack, while the phase-field regularizes cracks over a finite length l0. The schematic and the diffuse-interface representation should be reconciled for clarity.
  5. [§5.3] The statement that 'damage does not induce plastic softening through yield surface contraction' is asserted without discussing the growth of ψ_d in the perfect-plastic branch. This issue should be addressed directly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the constitutive assumption is independent of the simulation outputs; self-citations are infrastructural only.

full rationale

The paper's central derivation—a serial rheological fracture element that leaves inelastic properties undegraded and drives damage only from elastic strain energy—is a constitutive assumption introduced by the authors and implemented directly in Eqs. (44)–(54), not a quantity fitted to the outputs it later 'predicts.' Material parameters (Table 2) are prescribed inputs, not calibrated to the reported force-displacement curves or crack patterns. The numerical examples are checked qualitatively against external references (notched-plate shearing against the phase-field literature; PBX compression X-shaped patterns against Manner et al. and Mehrdad et al.; cup-cone fracture against Tvergaard & Needleman). The self-citations (Ratel [60], Shakeri et al. [70] for log1p numerics, Brown et al. [54] for p-multigrid, and the group's UQ paper [93]) are infrastructural or contextual and do not carry the constitutive argument. The internal question raised by Appendix C—that in the perfect-plastic Prandtl regime the stored deviatoric energy grows as 1/g(phi)^2 (Eq. C30) and the transmitted stress is damage-independent (Eq. 52/C24)—is a physical consistency concern about the proposed model, not a circular reduction of a prediction to its inputs. No equation in the paper reduces to a fitted constant or to the paper's own prior result. Therefore no significant circularity is present.

Assumptions & free parameters 3 free parameters · 6 assumptions · 1 invented entities

The central derivations rest on established phase-field and Perić–Dettmer machinery, but the paper introduces one key assumption (serial fracture element with no direct damage-inelasticity coupling) and several hand-tuned numerical parameters (η, ζ, damping coefficient) that affect the qualitative outcomes. The synthetic material parameters are assumed inputs, not fitted. No new physical entities with independent evidence are proposed.

free parameters (3)
  • residual stiffness factor η = 1e-3 (Examples 1 and 3); 1e-2 (Example 2, lubricated case)
    Chosen to prevent a singular Jacobian at full damage; the value is increased in Example 2 specifically to maintain numerical stability with contact, and it affects post-peak crack opening and the force-drop tail.
  • damage viscosity ζ = 1e-2 MPa·s (Examples 1 and 3; Example 2 value not clearly stated)
    Viscous regularization parameter for the monolithic phase-field solver; the paper acknowledges that higher ζ promotes more damage nucleation sites and crack branching, altering the predicted fracture pattern toward 'better agreement' with experiments.
  • viscous damping coefficient for lubricated platens = Not specified numerically
    Chosen 'sufficiently large to suppress convergence issues associated with near-null space modes'; it modifies lateral resistance at the contact surfaces and thereby influences whether X-shaped or single-crack patterns emerge.
assumptions (6)
  • standard math Standard phase-field fracture variational formulation (Griffith energy, AT1/AT2 crack density, Amor tension-compression split, history variable for irreversibility)
    Adopted from the literature (Section 3, Eqs. 43–47, 62) without modification; the paper relies on these as established tools.
  • standard math Perić–Dettmer multiplicative finite-strain inelasticity with exponential map integrator and logarithmic strains
    Used as the underlying inelastic constitutive framework (Section 2), originally from Perić & Dettmer [59]; the paper specializes it to Hencky materials and deviatoric inelasticity.
  • ad hoc to paper Damage is driven solely by tensile elastic strain energy ψ+ and does not degrade inelastic properties; fracture element is assembled in series
    This is the core constitutive hypothesis introduced by the paper (Section 3, Eqs. 48–52). It is not derived from lower-scale physics and is not quantitatively validated against ductile-fracture experiments.
  • domain assumption At the resolved microstructure scale, inelastic and fracture interactions are captured purely by microstructure morphology and homogenized fields, not by material-point coupling
    The paper's stated motivation (Figs. 1–2); the separation of scales is assumed, not proven, for the two target materials (PBX, particle-reinforced alloy).
  • domain assumption Mesh resolution with h ≤ l0 using P2 elements is sufficient to resolve steep damage gradients without spurious oscillations
    Based on Jodlbauer et al. [52]; no mesh-convergence study is performed in this paper for the new coupled problem.
  • domain assumption The synthetic particle-matrix microstructures and the material parameters in Table 2 are representative of PBX and particle-reinforced alloys
    The parameters are chosen by hand or from literature for the particle and matrix phases; no calibration to specific experimental batches is reported, so the claim of 'reproducing' material behavior is only qualitative.
invented entities (1)
  • Rheological fracture element (F) assembled in series with the Perić–Dettmer block
    purpose: Introduces damage as a serial rheological element that degrades only elastic strain energy while leaving inelastic properties (yield, viscosity, hardening) unmodified; couples inelastic and fracture behavior only at homogenization length scales.
    This is a modeling construct rather than a new physical entity. It has no falsifiable handle outside the model; its validity rests on the paper's constitutive hypothesis and the qualitative match to known crack patterns, not on independent experiments or predictions.

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Cite this review

Pith. "Pith review of Matrix-free phase-field modeling of fracture in micromechanical testing simulations of inelastic materials." pith.science (2026). https://pith.science/paper/4227HOWI

@misc{pith2026260721150,
  author       = {Pith},
  title        = {Pith review of: Matrix-free phase-field modeling of fracture in micromechanical testing simulations of inelastic materials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4227HOWI}},
  note         = {Machine review of arXiv:2607.21150}
}
read the original abstract

Resolving steep damage gradients across diffuse cracks in the phase-field modeling of fracture favors the use of high-order finite elements, for which matrix-free methods can provide superior performance and scalability. Here, we implement the Peric & Dettmer constitutive framework for visco-elastoplastic materials in an open source solid mechanics library supporting matrix-free operators for high-order finite elements with p-multigrid preconditioning on GPUs. We introduce a rheological fracture element assembled in series so that inelastic and fracture properties can appear to affect each other only at homogenization length scales. Numerical simulations of tensile and compressive tests are conducted for synthetic particle-matrix microstructures on an El Capitan high performance computing prototype. Results are shown to reproduce characteristic inelastic responses and crack propagation patterns.

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Reference graph

Works this paper leans on

100 extracted references · 39 canonical work pages

  1. [1]

    A review of extended/generalized finite element methods for material modeling.Modelling and Simulation in Materials Science and Engineering.2009;17(4):043001

    Belytschko T, Gracie R, Ventura G. A review of extended/generalized finite element methods for material modeling.Modelling and Simulation in Materials Science and Engineering.2009;17(4):043001. doi: 10.1088/0965-0393/17/4/043001

    work page doi:10.1088/0965-0393/17/4/043001 2009

  2. [2]

    The variational approach to fracture.Journal of Elasticity.2008;91(1-3):5–148

    Francfort GA, Bourdin B, Marigo JJ. The variational approach to fracture.Journal of Elasticity.2008;91(1-3):5–148. doi: 10.1007/s10659-007- 9107-3

    doi:10.1007/s10659-007- 2008

  3. [3]

    Phase-field material point method for brittle fracture.International Journal for Numerical Methods in Engineering

    Kakouris EG, Triantafyllou SP. Phase-field material point method for brittle fracture.International Journal for Numerical Methods in Engineering. 2017;112(12):1750–1776. _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.5580doi: 10.1002/nme.5580

    work page doi:10.1002/nme.5580doi: 2017

  4. [4]

    Phase-field modeling of brittle fracture using an efficient virtual element scheme.Computer Methods in Applied Mechanics and Engineering.2018;341:443–466

    Aldakheel F, Hudobivnik B, Hussein A, Wriggers P. Phase-field modeling of brittle fracture using an efficient virtual element scheme.Computer Methods in Applied Mechanics and Engineering.2018;341:443–466. doi: 10.1016/j.cma.2018.07.008

    work page doi:10.1016/j.cma.2018.07.008 2018

  5. [5]

    Adaptive phase-field modeling of brittle fracture using the scaled boundary finite element method.Computer Methods in Applied Mechanics and Engineering.2019;355:284–307

    Hirshikesh , Pramod A, Annabattula R, Ooi E, Song C, Natarajan S. Adaptive phase-field modeling of brittle fracture using the scaled boundary finite element method.Computer Methods in Applied Mechanics and Engineering.2019;355:284–307. doi: 10.1016/j.cma.2019.06.002

    work page doi:10.1016/j.cma.2019.06.002 2019

  6. [6]

    Extension of high-fidelity time-domain spectral element formulation for phase-field modeling of fracture: A static analysis.Engineering Fracture Mechanics.2025;317:110908

    Li F, Su Y , Chen Y , Sun X. Extension of high-fidelity time-domain spectral element formulation for phase-field modeling of fracture: A static analysis.Engineering Fracture Mechanics.2025;317:110908. doi: 10.1016/j.engfracmech.2025.110908

    arXiv 2025

  7. [7]

    A review on phase-field models of brittle fracture and a new fast hybrid formulation.Computational Mechanics.2015;55(2):383–405

    Ambati M, Gerasimov T, De Lorenzis L. A review on phase-field models of brittle fracture and a new fast hybrid formulation.Computational Mechanics.2015;55(2):383–405. doi: 10.1007/s00466-014-1109-y

    doi:10.1007/s00466-014-1109-y 2015

  8. [8]

    Gradient damagevsphase-field approaches for fracture: Similarities and differences.Computer Methods in Applied Mechanics and Engineering.2016;312:78–94

    Borst dR, Verhoosel CV . Gradient damagevsphase-field approaches for fracture: Similarities and differences.Computer Methods in Applied Mechanics and Engineering.2016;312:78–94. doi: 10.1016/j.cma.2016.05.015

    work page doi:10.1016/j.cma.2016.05.015 2016

Show all 100 references
  1. [9]

    Crack nucleation in variational phase-field models of brittle fracture.Journal of the Mechanics and Physics of Solids.2018;110:80–99

    Tanné E, Li T, Bourdin B, Marigo JJ, Maurini C. Crack nucleation in variational phase-field models of brittle fracture.Journal of the Mechanics and Physics of Solids.2018;110:80–99. doi: 10.1016/j.jmps.2017.09.006

    2018 doi

  2. [10]

    A FEniCS implementation of the phase field method for quasi-static brittle fracture.Frontiers of Structural and Civil Engineering.2019;13(2):380–396

    Hirshikesh , Natarajan S, Annabattula RK. A FEniCS implementation of the phase field method for quasi-static brittle fracture.Frontiers of Structural and Civil Engineering.2019;13(2):380–396. doi: 10.1007/s11709-018-0471-9

    2019 doi

  3. [11]

    GPFniCS: A generalised phase field method to model fracture.SoftwareX.2023;24:101594

    Kumar M, Alessi R, Salvati E. GPFniCS: A generalised phase field method to model fracture.SoftwareX.2023;24:101594. doi: 10.1016/j.softx.2023.101594

    2023

  4. [12]

    An open-source moose implementation of phase-field modeling of fracture in functionally graded materials.Procedia Structural Integrity.2024;58:115–121

    Sidharth PC, Rao BN. An open-source moose implementation of phase-field modeling of fracture in functionally graded materials.Procedia Structural Integrity.2024;58:115–121. doi: 10.1016/j.prostr.2024.05.019

    2024 doi

  5. [13]

    PhaseFieldX: An Open-Source Framework for Advanced Phase-Field Simulations.Journal of Open Source Software

    Castillón M. PhaseFieldX: An Open-Source Framework for Advanced Phase-Field Simulations.Journal of Open Source Software. 2025;10(108):7307. doi: 10.21105/joss.07307

    2025 doi

  6. [14]

    PhaFiDyn: An explicit dynamic phase field damage model implementation.SoftwareX

    Barki A, Zghal J, Gallimard L, Bruant I, Davenne L. PhaFiDyn: An explicit dynamic phase field damage model implementation.SoftwareX. 2025;31:102227. doi: 10.1016/j.softx.2025.102227

    2025

  7. [15]

    A detailed guide to an open-source implementation of the hybrid phase field method for 3D fracture modeling in deal.II.Computer Physics Communications.2026;319:109901

    Munshi WN, Fehling M, Bangerth W, Annavarapu C. A detailed guide to an open-source implementation of the hybrid phase field method for 3D fracture modeling in deal.II.Computer Physics Communications.2026;319:109901. doi: 10.1016/j.cpc.2025.109901

    2026

  8. [16]

    A phase-field description of dynamic brittle fracture.Computer Methods in Applied Mechanics and Engineering.2012;217-220:77–95

    Borden MJ, Verhoosel CV , Scott MA, Hughes TJR, Landis CM. A phase-field description of dynamic brittle fracture.Computer Methods in Applied Mechanics and Engineering.2012;217-220:77–95. doi: 10.1016/j.cma.2012.01.008

    2012 doi

  9. [17]

    An explicit phase field method for brittle dynamic fracture.Computers & Structures.2019;217:45–56

    Ren HL, Zhuang XY , Anitescu C, Rabczuk T. An explicit phase field method for brittle dynamic fracture.Computers & Structures.2019;217:45–56. doi: 10.1016/j.compstruc.2019.03.005

    2019 doi

  10. [18]

    Evaluation of variational phase-field models for dynamic brittle fracture.Engineering Fracture Mechanics

    Mandal TK, Nguyen VP, Wu JY . Evaluation of variational phase-field models for dynamic brittle fracture.Engineering Fracture Mechanics. 2020;235:107169. doi: 10.1016/j.engfracmech.2020.107169

    2020

  11. [19]

    Phase-field cohesive fracture theory: A unified framework for dissipative systems based on variational inequality of virtual works

    Feng Y , Li J. Phase-field cohesive fracture theory: A unified framework for dissipative systems based on variational inequality of virtual works. Journal of the Mechanics and Physics of Solids.2022;159:104737. doi: 10.1016/j.jmps.2021.104737

    2022

  12. [20]

    Multi-phase-field modeling of anisotropic crack propagation for polycrystalline materials

    Nguyen TT, Réthoré J, Yvonnet J, Baietto MC. Multi-phase-field modeling of anisotropic crack propagation for polycrystalline materials. Computational Mechanics.2017;60(2):289–314. doi: 10.1007/s00466-017-1409-0

    2017 doi

  13. [21]

    Phase-field modeling of anisotropic brittle fracture including several damage mechanisms.Computer Methods in Applied Mechanics and Engineering.2018;336:213–236

    Bleyer J, Alessi R. Phase-field modeling of anisotropic brittle fracture including several damage mechanisms.Computer Methods in Applied Mechanics and Engineering.2018;336:213–236. doi: 10.1016/J.CMA.2018.03.012

    2018 doi

  14. [22]

    Crack kinking in a variational phase-field model of brittle fracture with strongly anisotropic surface energy.Journal of the Mechanics and Physics of Solids.2019;125:502–522

    Li B, Maurini C. Crack kinking in a variational phase-field model of brittle fracture with strongly anisotropic surface energy.Journal of the Mechanics and Physics of Solids.2019;125:502–522. doi: 10.1016/j.jmps.2019.01.010

    2019 doi

  15. [23]

    Phase field modeling of anisotropic silicon crystalline cracking in 3D thin-walled photovoltaic laminates

    Liu Z, Lenarda P, Reinoso J, Paggi M. Phase field modeling of anisotropic silicon crystalline cracking in 3D thin-walled photovoltaic laminates. International Journal of Fracture.2025;249(1):19. doi: 10.1007/s10704-024-00821-3 24 DI GIOACCHINOET AL

    2025 doi

  16. [24]

    Phase-field modeling of anisotropic fracture of additively manufactured parts.Engineering Fracture Mechanics

    Ruan H, Peng XL. Phase-field modeling of anisotropic fracture of additively manufactured parts.Engineering Fracture Mechanics. 2025;323:111160. doi: 10.1016/j.engfracmech.2025.111160

    2025

  17. [25]

    A phase-field model for fatigue crack growth.Journal of the Mechanics and Physics of Solids

    Lo YS, Borden MJ, Ravi-Chandar K, Landis CM. A phase-field model for fatigue crack growth.Journal of the Mechanics and Physics of Solids. 2019;132:103684. doi: 10.1016/j.jmps.2019.103684

    2019

  18. [26]

    A framework to model the fatigue behavior of brittle materials based on a variational phase-field approach.Computer Methods in Applied Mechanics and Engineering.2020;361:112731

    Carrara P, Ambati M, Alessi R, De Lorenzis L. A framework to model the fatigue behavior of brittle materials based on a variational phase-field approach.Computer Methods in Applied Mechanics and Engineering.2020;361:112731. doi: 10.1016/j.cma.2019.112731

    2020

  19. [27]

    An acceleration scheme for the phase field fatigue fracture simulation with a concurrent temporal homogenization method

    Yang S, Shen Y . An acceleration scheme for the phase field fatigue fracture simulation with a concurrent temporal homogenization method. Computer Methods in Applied Mechanics and Engineering.2023;416:116294. doi: 10.1016/j.cma.2023.116294

    2023

  20. [28]

    A review on phase field models for fracture and fatigue.Engineering Fracture Mechanics.2023;289:109419

    Li P, Li W, Li B, et al. A review on phase field models for fracture and fatigue.Engineering Fracture Mechanics.2023;289:109419. doi: 10.1016/j.engfracmech.2023.109419

    2023

  21. [29]

    A Crystal Plasticity Phase-Field Study on the Effects of Grain Boundary Degradation on the Fatigue Behavior of a Nickel-Based Superalloy.Materials.2025;18(14):3309

    Liu P, Chen Z, Zhao X, Dong J, Jiang H. A Crystal Plasticity Phase-Field Study on the Effects of Grain Boundary Degradation on the Fatigue Behavior of a Nickel-Based Superalloy.Materials.2025;18(14):3309. doi: 10.3390/ma18143309

    2025 doi

  22. [30]

    A generalised, multi-phase-field theory for dissolution-driven stress corrosion cracking and hydrogen embrittlement.Journal of the Mechanics and Physics of Solids.2022;166:104951

    Cui C, Ma R, Martínez-Pañeda E. A generalised, multi-phase-field theory for dissolution-driven stress corrosion cracking and hydrogen embrittlement.Journal of the Mechanics and Physics of Solids.2022;166:104951. doi: 10.1016/j.jmps.2022.104951

    2022

  23. [31]

    Numerical Simulation for Hydrogen-Assisted Cracking: An Explicit Phase-Field Formulation.Materials.2023;16(4):1708

    Wang D, Ma F, Chen H. Numerical Simulation for Hydrogen-Assisted Cracking: An Explicit Phase-Field Formulation.Materials.2023;16(4):1708. doi: 10.3390/ma16041708

    2023 doi

  24. [32]

    Zhang D, Ma S, Nachtsheim J, Zhang S, Markert B. A variational phase-field framework for multiphysics modelling of degradation and stress corrosion cracking in biodegradable magnesium alloys.Journal of the Mechanics and Physics of Solids.2024;190:105694. doi: 10.1016/j.jmps.20...

    2024

  25. [33]

    A Continuous Damage Mechanics Model for Ductile Fracture.Journal of Engineering Materials and Technology.1985;107(1):83–89

    Lemaitre J. A Continuous Damage Mechanics Model for Ductile Fracture.Journal of Engineering Materials and Technology.1985;107(1):83–89. doi: 10.1115/1.3225775

    1985 doi

  26. [34]

    Gurson AL. Continuum Theory of Ductile Rupture by V oid Nucleation and Growth: Part I—Yield Criteria and Flow Rules for Porous Ductile Media.Journal of Engineering Materials and Technology.1977;99(1):2–15. doi: 10.1115/1.3443401

    1977 doi

  27. [35]

    Analysis of the cup-cone fracture in a round tensile bar.Acta Metallurgica.1984;32(1):157–169

    Tvergaard V , Needleman A. Analysis of the cup-cone fracture in a round tensile bar.Acta Metallurgica.1984;32(1):157–169. doi: 10.1016/0001- 6160(84)90213-X

    1984 doi

  28. [36]

    An Experimental Study of the Polycrystalline Plasticity of Austenitic Stainless Steel.International Journal of Plasticity.2015;74

    Di Gioacchino F, Quinta Da Fonseca J. An Experimental Study of the Polycrystalline Plasticity of Austenitic Stainless Steel.International Journal of Plasticity.2015;74. doi: 10.1016/j.ijplas.2015.05.012

    2015 doi

  29. [37]

    An experimental study of the polycrystalline plasticity of lamellar titanium aluminide.International Journal of Plasticity.2019;118

    Edwards T, Di Gioacchino F, Clegg W. An experimental study of the polycrystalline plasticity of lamellar titanium aluminide.International Journal of Plasticity.2019;118. doi: 10.1016/j.ijplas.2019.02.013

    2019 doi

  30. [38]

    Reduced partitioning of plastic strain for strong and yet ductile precipitate-strengthened alloys.Scientific Reports.2018;8(1):8698

    Jones RD, Di Gioacchino F, Lim H, et al. Reduced partitioning of plastic strain for strong and yet ductile precipitate-strengthened alloys.Scientific Reports.2018;8(1):8698. doi: 10.1038/s41598-018-26917-0

    2018 doi

  31. [39]

    Phase field modeling of ductile fracture at finite strains: A variational gradient-extended plasticity-damage theory

    Miehe C, Aldakheel F, Raina A. Phase field modeling of ductile fracture at finite strains: A variational gradient-extended plasticity-damage theory. International Journal of Plasticity.2016;84:1–32. doi: 10.1016/j.ijplas.2016.04.011

    2016 doi

  32. [40]

    Phase-field modeling of ductile fracture.Computational Mechanics.2015;55:1013–1040

    Ambati M, Gerasimov T, De Lorenzis L. Phase-field modeling of ductile fracture.Computational Mechanics.2015;55:1013–1040. doi: 10.1007/s00466-015-1151-4

    2015 doi

  33. [41]

    A phase-field/gradient damage model for brittle fracture in elastic–plastic solids.International Journal of Plasticity.2015;65:269–296

    Duda FP, Ciarbonetti A, Sánchez PJ, Huespe AE. A phase-field/gradient damage model for brittle fracture in elastic–plastic solids.International Journal of Plasticity.2015;65:269–296. doi: 10.1016/j.ijplas.2014.09.005

    2015 doi

  34. [42]

    Crack phase-field model equipped with plastic driving force and degrading fracture toughness for ductile fracture simulation.Computational Mechanics.2022;69(1):151–175

    Han J, Matsubara S, Moriguchi S, Kaliske M, Terada K. Crack phase-field model equipped with plastic driving force and degrading fracture toughness for ductile fracture simulation.Computational Mechanics.2022;69(1):151–175. doi: 10.1007/s00466-021-02087-1

    2022 doi

  35. [43]

    Attaining regularization length insensitivity in phase-field models of ductile failure.Computer Methods in Applied Mechanics and Engineering.2021;384:113936

    Talamini B, Tupek MR, Stershic AJ, et al. Attaining regularization length insensitivity in phase-field models of ductile failure.Computer Methods in Applied Mechanics and Engineering.2021;384:113936. doi: 10.1016/j.cma.2021.113936

    2021

  36. [44]

    Svolos L, Tran QT, Boureima ID, Anghel V , Garikipati K, Mourad HM. A phase-field fracture formulation for generalized standard ma- terials: The interplay between thermomechanics and damage.Journal of the Mechanics and Physics of Solids.2025;201:106154. doi: 10.1016/j.jmps.2025.106154

    2025

  37. [45]

    A Micropolar Phase Field Fracture Model for Elastoplastic Solids Applied to Concrete Failure.International Journal for Numerical Methods in Engineering.2025;126(19):e70140

    Abrari Vajari S, Neuner M, Arunachala PK, Linder C. A Micropolar Phase Field Fracture Model for Elastoplastic Solids Applied to Concrete Failure.International Journal for Numerical Methods in Engineering.2025;126(19):e70140. _eprint: https://onlinelibrary.wiley.com/doi/pdf/10....

    2025 doi

  38. [46]

    Yin B, Kaliske M. A ductile phase-field model based on degrading the fracture toughness: Theory and implementation at small strain.Computer Methods in Applied Mechanics and Engineering.2020;366:113068. doi: 10.1016/j.cma.2020.113068 Matrix-free phase-field modeling of fracture...

    2020

  39. [47]

    Khalil Z, Elghazouli AY , Martínez-Pañeda E. A generalised phase field model for fatigue crack growth in elastic–plastic solids with an efficient monolithic solver.Computer Methods in Applied Mechanics and Engineering.2022;388:114286. doi: 10.1016/j.cma.2021.114286

    2022

  40. [48]

    3D micromechanical progressive failure simulation for fiber-reinforced composites.Composite Structures

    Zhang P, Yao W, Hu X, Bui TQ. 3D micromechanical progressive failure simulation for fiber-reinforced composites.Composite Structures. 2020;249:112534. doi: 10.1016/j.compstruct.2020.112534

    2020

  41. [49]

    Cheng J, Tu X, Ghosh S. Wavelet-enriched adaptive hierarchical FE model for coupled crystal plasticity-phase field modeling of crack propagation in polycrystalline microstructures.Computer Methods in Applied Mechanics and Engineering.2020;361:112757. doi: 10.1016/j.cma.2019.112757

    2020

  42. [50]

    dD, Lim H

    Rezwan AA, Aragon NK, Oca Zapiain M. dD, Lim H. Coupled phase field damage and crystal plasticity analysis of intragranular fracture: The role of crystallographic orientation and voids.International Journal of Plasticity.2025;195:104372. doi: 10.1016/j.ijplas.2025.104372

    2025

  43. [51]

    Xiao Y , Peng X, Wang D, et al. Phase-field simulation study on the mechanical behavior and fracture mechanism of SiC-p/Al composites considering microstructural effects under uniaxial tension.Composites Communications.2025;58:102540. doi: 10.1016/j.coco.2025.102540

    2025

  44. [52]

    Matrix-free multigrid solvers for phase-field fracture problems.Computer Methods in Applied Mechanics and Engineering.2020;372

    Jodlbauer D, Langer U, Wick T. Matrix-free multigrid solvers for phase-field fracture problems.Computer Methods in Applied Mechanics and Engineering.2020;372. arXiv: 1902.08112 Publisher: Elsevier B.V .doi: 10.1016/j.cma.2020.113431

    2020 arXiv

  45. [53]

    A matrix-free approach for finite-strain hyperelastic problems using geometric multigrid.International Journal for Numerical Methods in Engineering.2020;121(6):1224–1248

    Davydov D, Pelteret JP, Steinmann P. A matrix-free approach for finite-strain hyperelastic problems using geometric multigrid.International Journal for Numerical Methods in Engineering.2020;121(6):1224–1248. doi: 10.1002/nme.6336

    2020 doi

  46. [54]

    Performance Portable Solid Mechanics via Matrix-Free $p$-Multigrid

    Brown J, Barra V , Beams N, et al. Performance Portable Solid Mechanics via Matrix-Free $p$-Multigrid. arXiv:2204.01722 [cs]; 2022

    2022 arXiv

  47. [55]

    Matrix-free higher-order finite element methods for hyperelasticity.Computer Methods in Applied Mechanics and Engineering.2025;435:117600

    Schussnig R, Fehn N, Munch P, Kronbichler M. Matrix-free higher-order finite element methods for hyperelasticity.Computer Methods in Applied Mechanics and Engineering.2025;435:117600. doi: 10.1016/j.cma.2024.117600

    2025

  48. [56]

    Parallel Matrix-Free Higher-Order Finite Element Solvers for Phase-Field Fracture Problems.Mathematical and Computational Applications.2020;25(3):40

    Jodlbauer D, Langer U, Wick T. Parallel Matrix-Free Higher-Order Finite Element Solvers for Phase-Field Fracture Problems.Mathematical and Computational Applications.2020;25(3):40. arXiv: 2005.00331 Publisher: MDPI AGdoi: 10.3390/mca25030040

    2020 arXiv

  49. [57]

    Liu X, Réthoré J, Lubrecht AA. An efficient matrix-free preconditioned conjugate gradient based multigrid method for phase field modeling of fracture in heterogeneous materials from 3D images.Computer Methods in Applied Mechanics and Engineering.2022;388:114266. doi: 10.1016/j...

    2022

  50. [58]

    Matrix-Free Geometric Multigrid Preconditioning of Combined Newton-GMRES for Solving Phase-Field Fracture with Local Mesh Refinement

    Kolditz LM, Wick T. Matrix-Free Geometric Multigrid Preconditioning of Combined Newton-GMRES for Solving Phase-Field Fracture with Local Mesh Refinement. In: Sequeira A, Silvestre A, Valtchev SS, Janela J., eds.Numerical Mathematics and Advanced Applications ENUMATH 2023, Volu...

    2023

  51. [59]

    A computational model for generalized inelastic materials at finite strains combining elastic, viscoelastic and plastic material behaviour.Engineering Computations.2003

    Peric´ D, Dettmer W. A computational model for generalized inelastic materials at finite strains combining elastic, viscoelastic and plastic material behaviour.Engineering Computations.2003. doi: 10.1108/02644400310488862

    2003 doi

  52. [60]

    Ratel: Performance portable solid mechanics with libCEED and PETSc.Journal of Open Source Software.2026;11(118):8388

    Atkins ZR, Brown J, Gioacchino FD, et al. Ratel: Performance portable solid mechanics with libCEED and PETSc.Journal of Open Source Software.2026;11(118):8388. doi: 10.21105/joss.08388

    2026 doi

  53. [61]

    libCEED: Fast algebra for high-order element-based discretizations.Journal of Open Source Software

    Brown J, Abdelfattah A, Barra V , et al. libCEED: Fast algebra for high-order element-based discretizations.Journal of Open Source Software. 2021;6(63):2945. doi: 10.21105/joss.02945

    2021 doi

  54. [62]

    PETSc Users Manual

    Balay S, Abhyankar S, Adams MF, et al. PETSc Users Manual. Tech. Rep. ANL-95/11 - Revision 3.6, Argonne National Laboratory; : 2015

    2015

  55. [63]

    A model for finite strain elasto-plasticity based on logarithmic strains: Computational issues.Computer Methods in Applied Mechanics and Engineering.1992;94(1):35–61

    Peri´c D, Owen DRJ, Honnor ME. A model for finite strain elasto-plasticity based on logarithmic strains: Computational issues.Computer Methods in Applied Mechanics and Engineering.1992;94(1):35–61. doi: 10.1016/0045-7825(92)90156-E

    1992 doi

  56. [64]

    Elastic-Plastic Deformation at Finite Strains.Journal of Applied Mechanics.1969;36(1):1–6

    Lee EH. Elastic-Plastic Deformation at Finite Strains.Journal of Applied Mechanics.1969;36(1):1–6. doi: 10.1115/1.3564580

    1969 doi

  57. [65]

    Simo J, Hughes T.Computational Inelasticity. 7. Springer, 1998

    1998

  58. [66]

    A theory of finite viscoelasticity and numerical aspects.International Journal of Solids and Structures.1998;35(26– 27):3455–3482

    Reese S, Govindjee S. A theory of finite viscoelasticity and numerical aspects.International Journal of Solids and Structures.1998;35(26– 27):3455–3482. doi: 10.1016/S0020-7683(97)00217-5

    1998 doi

  59. [67]

    Eterovic AL, Bathe KJ. A hyperelastic-based large strain elasto-plastic constitutive formulation with combined isotropic-kinematic hardening using the logarithmic stress and strain measures.International Journal for Numerical Methods in Engineering.1990;30(6):1099–1114. doi: 1...

    1990 doi

  60. [68]

    Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic-viscoplastic solids

    Weber G, Anand L. Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic-viscoplastic solids. Computer Methods in Applied Mechanics and Engineering.1990;79(2):173–202. doi: 10.1016/0045-7825(90)90131-5

    1990 doi

  61. [69]

    Borden MJ, Hughes TJ, Landis CM, Anvari A, Lee IJ. 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.2016;312:130–166...

    2016 doi

  62. [70]

    Stable numerics for finite-strain elasticity.International Journal for Numerical Methods in Engineering.2024:e7563

    Shakeri R, Ghaffari L, Thompson JL, Brown J. Stable numerics for finite-strain elasticity.International Journal for Numerical Methods in Engineering.2024:e7563. doi: 10.1002/nme.7563 26 DI GIOACCHINOET AL

    2024 doi

  63. [71]

    A variational formulation of constitutive models and updates in non-linear finite viscoelasticity.International Journal for Numerical Methods in Engineering.2006;65(11):1831–1864

    Fancello E, Ponthot JP, Stainier L. A variational formulation of constitutive models and updates in non-linear finite viscoelasticity.International Journal for Numerical Methods in Engineering.2006;65(11):1831–1864. doi: 10.1002/nme.1525

    2006 doi

  64. [72]

    Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments.Journal of the Mechanics and Physics of Solids.2009;57(8):1209–1229

    Amor H, Marigo JJ, Maurini C. Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments.Journal of the Mechanics and Physics of Solids.2009;57(8):1209–1229. doi: 10.1016/j.jmps.2009.04.011

    2009 doi

  65. [73]

    On the ductile enlargement of voids in triaxial stress fields*.Journal of the Mechanics and Physics of Solids.1969;17(3):201–

    Rice JR, Tracey DM. On the ductile enlargement of voids in triaxial stress fields*.Journal of the Mechanics and Physics of Solids.1969;17(3):201–

    1969

  66. [74]

    On degradation functions in phase field fracture models.Computational Materials Science.2015;108:374–384

    Kuhn C, Schlüter A, Müller R. On degradation functions in phase field fracture models.Computational Materials Science.2015;108:374–384. doi: 10.1016/j.commatsci.2015.05.034

    2015 doi

  67. [75]

    Svolos L, Plohr JN, Manzini G, Mourad HM. On the convexity of phase-field fracture formulations: Analytical study and comparison of various degradation functions.International Journal of Non-Linear Mechanics.2023;150:104359. doi: 10.1016/j.ijnonlinmec.2023.104359

    2023

  68. [76]

    A fracture-controlled path-following technique for phase-field modeling of brittle fracture

    Singh N, Verhoosel C, De Borst R, Van Brummelen E. A fracture-controlled path-following technique for phase-field modeling of brittle fracture. Finite Elements in Analysis and Design.2016;113:14–29. doi: 10.1016/j.finel.2015.12.005

    2016 doi

  69. [77]

    Bharali R, Goswami S, Anitescu C, Rabczuk T. A robust monolithic solver for phase-field fracture integrated with fracture energy based arc-length method and under-relaxation.Computer Methods in Applied Mechanics and Engineering.2022;394:114927. doi: 10.1016/j.cma.2022.114927

    2022

  70. [78]

    How regularization concepts interfere with (quasi-)brittle damage: a comparison based on a unified variational framework.Continuum Mechanics and Thermodynamics.2022;34(6):1517–1544

    Langenfeld K, Kurzeja P, Mosler J. How regularization concepts interfere with (quasi-)brittle damage: a comparison based on a unified variational framework.Continuum Mechanics and Thermodynamics.2022;34(6):1517–1544. doi: 10.1007/s00161-022-01143-2

    2022 doi

  71. [79]

    Miehe C, Hofacker M, Welschinger F. A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits.Computer Methods in Applied Mechanics and Engineering.2010;199(45-48):2765–2778. arXiv: 1011.1669v3 ISBN: 0045-7825doi: 10...

    2010 arXiv

  72. [80]

    A line search assisted monolithic approach for phase-field computing of brittle fracture.Computer Methods in Applied Mechanics and Engineering.2016;312:276–303

    Gerasimov T, De Lorenzis L. A line search assisted monolithic approach for phase-field computing of brittle fracture.Computer Methods in Applied Mechanics and Engineering.2016;312:276–303. doi: 10.1016/j.cma.2015.12.017

    2016 doi

  73. [81]

    Phase field fracture modelling using quasi-Newton methods and a new adaptive step scheme.Theoretical and Applied Fracture Mechanics.2020;107:102446

    Kristensen PK, Martínez-Pañeda E. Phase field fracture modelling using quasi-Newton methods and a new adaptive step scheme.Theoretical and Applied Fracture Mechanics.2020;107:102446. doi: 10.1016/j.tafmec.2019.102446

    2020

  74. [82]

    In Situ Imaging during Compression of Plastic Bonded Explosives for Damage Modeling.Materials

    Manner VW, Yeager JD, Patterson BM, et al. In Situ Imaging during Compression of Plastic Bonded Explosives for Damage Modeling.Materials. 2017;10(6):638. doi: 10.3390/ma10060638

    2017 doi

  75. [83]

    In-Situ Internal Deformation Measurement of an IDOX Particulate Estane Binder Composite via Backward Digital V olume Correlation.Experimental Mechanics.2026;66(1):103–119

    Mehrdad E, Javadzadeh PB, Ren Y , et al. In-Situ Internal Deformation Measurement of an IDOX Particulate Estane Binder Composite via Backward Digital V olume Correlation.Experimental Mechanics.2026;66(1):103–119. doi: 10.1007/s11340-025-01253-3

    2026 doi

  76. [84]

    MicroStructPy: A statistical microstructure mesh generator in Python.SoftwareX.2020;12:100595

    Hart KA, Rimoli JJ. MicroStructPy: A statistical microstructure mesh generator in Python.SoftwareX.2020;12:100595. doi: 10.1016/j.softx.2020.100595

    2020

  77. [85]

    Dawson-Haggerty M. Trimesh. https://github.com/mikedh/trimesh; 2019

    2019

  78. [86]

    Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities.International Journal for Numerical Methods in Engineering.2009;79(11):1309–1331

    Geuzaine C, Remacle JF. Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities.International Journal for Numerical Methods in Engineering.2009;79(11):1309–1331. doi: 10.1002/nme.2579

    2009 doi

  79. [87]

    NETGEN—An advancing front 2D/3D-mesh generator based on abstract rules.Computing and Visualization in Science.1997;1(1):41–

    Schöberl J. NETGEN—An advancing front 2D/3D-mesh generator based on abstract rules.Computing and Visualization in Science.1997;1(1):41–

    1997

  80. [88]

    micromorph / STLs postprocessing for 3Dmeshing of composites · GitLab

    Di Gioacchino F. micromorph / STLs postprocessing for 3Dmeshing of composites · GitLab. https://gitlab.com/micromorph/stls-postprocessing; 2024

    2024

  81. [89]

    doi: 10.1007/s007910050004

    doi

  82. [90]

    hypre: A Library of High Performance Preconditioners

    Falgout RD, Yang UM. hypre: A Library of High Performance Preconditioners. In: Sloot PMA, Hoekstra AG, Tan CJK, Dongarra JJ., eds. Computational Science — ICCS 2002Springer 2002; Berlin, Heidelberg:632–641

    2002

  83. [91]

    micromorph / Synthetic composite · GitLab

    Di Gioacchino F. micromorph / Synthetic composite · GitLab. https://gitlab.com/micromorph/synthetic-composite; 2025

    2025

  84. [92]

    Kokkos 3: Programming Model Extensions for the Exascale Era.IEEE Transactions on Parallel and Distributed Systems.2022;33(4):805–817

    Trott CR, Lebrun-Grandié D, Arndt D, et al. Kokkos 3: Programming Model Extensions for the Exascale Era.IEEE Transactions on Parallel and Distributed Systems.2022;33(4):805–817. doi: 10.1109/TPDS.2021.3097283

    2022

  85. [93]

    PETSc/TAO developments for GPU-based early exascale systems.The International Journal of High Performance Computing Applications.2025;39(2):306–325

    Mills RT, Adams MF, Balay S, et al. PETSc/TAO developments for GPU-based early exascale systems.The International Journal of High Performance Computing Applications.2025;39(2):306–325. doi: 10.1177/10943420241303710

    2025 doi

  86. [94]

    A new mechanism of strain transfer in polycrystals.Scientific Reports.2020;10(1)

    Di Gioacchino F, Edwards T, Wells G, Clegg W. A new mechanism of strain transfer in polycrystals.Scientific Reports.2020;10(1). doi: 10.1038/s41598-020-66569-7

    2020 doi

  87. [95]

    Calibrating a finite-strain phase-field model of fracture for bonded granular materials with uncertainty quantification.Journal of Applied Physics.2025;138(23):234903

    Schmid AC, Jensen E, Di Gioacchino F, et al. Calibrating a finite-strain phase-field model of fracture for bonded granular materials with uncertainty quantification.Journal of Applied Physics.2025;138(23):234903. doi: 10.1063/5.0300527

    2025 doi

  88. [96]

    Strain gradient crystal plasticity: size-dependentdeformation of bicrystals.Journal of the Mechanics and Physics of Solids

    Shu JY , Fleck NA. Strain gradient crystal plasticity: size-dependentdeformation of bicrystals.Journal of the Mechanics and Physics of Solids. 1999;47(2):297–324. doi: 10.1016/S0022-5096(98)00081-7

    1999 doi

  89. [97]

    Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density.Acta Materialia

    Arsenlis A, Parks D. Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density.Acta Materialia. 1999;47(5):1597–1611. doi: 10.1016/S1359-6454(99)00020-8 Matrix-free phase-field modeling of fracture for micromechanical testing simulations ...

    1999 doi

  90. [98]

    Ratel iMPM: A Performance-Portable, Matrix-Free Implicit Material Point Method for Exascale Systems

    Atkins ZR, Brown J, Di Gioacchino F. Ratel iMPM: A Performance-Portable, Matrix-Free Implicit Material Point Method for Exascale Systems. Manuscript submitted to SC26: The International Conference for High Performance Computing, Networking, Storage and Analysis; 2026. How to c...

    2026

  91. [99]

    Lattice incompatibility and a gradient theory of crystal plasticity.Journal of the Mechanics and Physics of Solids.2000;48(8):1565–1595

    Acharya A. Lattice incompatibility and a gradient theory of crystal plasticity.Journal of the Mechanics and Physics of Solids.2000;48(8):1565–1595. doi: 10.1016/S0022-5096(99)00075-7

    doi

  92. [217]

    doi: 10.1016/0022-5096(69)90033-7

    doi

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.