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arxiv: 2510.02575 · v2 · submitted 2025-10-02 · ❄️ cond-mat.mtrl-sci

The line bundle regime and the scale-dependence of continuum dislocation dynamics

Joseph Pierre Anderson , Anter El-Azab This is my paper

Pith reviewed 2026-05-18 10:01 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords continuum dislocation dynamicsline bundle closureorientation fluctuationscoarse grainingdislocation densitymaximum entropy closuremesoscale plasticity
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The pith

Orientation fluctuation statistics define a resolution-dependent transition between continuum dislocation dynamics theories

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

This paper introduces a formulation that links different continuum dislocation dynamics theories by tracking how dislocation lines fluctuate in orientation around a local average direction. The approach explains the loss of information during coarse graining, when opposing tangent vectors from unaligned lines cancel in density measures. It then tests two closures for approximating the fluctuation distributions at intermediate scales: a new line bundle closure and the standard maximum entropy closure. Simulation data shows the line bundle closure matches observed fluctuations for coarse-graining lengths up to half the typical dislocation spacing, while the maximum entropy closure disagrees at every scale examined. This scale dependence matters because it allows consistent dislocation density representations across the range of resolutions used in mesoscale plasticity models.

Core claim

A formulation of the resolution-dependent transition between these limits is presented in terms of the statistics of dislocation line orientation fluctuations about a local average line direction. From this formulation, a study of the orientation fluctuation behavior in intermediate resolution regimes is conducted. Two possible closure equations for truncating the moment sequence of the fluctuation distributions relating the two theories are evaluated from data, the newly introduced line bundle closure and the previous standard maximum entropy closure relations. The line bundle closure relation is shown to be accurate for coarse-graining lengths up to half the dislocation spacing and the max

What carries the argument

Statistics of dislocation line orientation fluctuations about a local average line direction, used to define the resolution-dependent transition between parallel and orientation-distributed CDD density representations

Load-bearing premise

The statistics of dislocation line orientation fluctuations about a local average line direction can be used to define a resolution-dependent transition between different CDD density representations, with the data faithfully representing physical behavior without simulation artifacts.

What would settle it

Direct extraction of orientation fluctuation moment sequences from discrete dislocation dynamics simulations at multiple coarse-graining lengths, followed by quantitative comparison to the line bundle closure versus maximum entropy predictions.

Figures

Figures reproduced from arXiv: 2510.02575 by Anter El-Azab, Joseph Pierre Anderson.

Figure 1
Figure 1. Figure 1: FIG. 1. Typical dislocation densities and mean dislocation [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Typical global orientation fluctuation distributions [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Characteristic sequence components. The first- (a), second- (b), and third-order (c) characteristic components are [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Fractional closure errors associated with the line bundle and maximum entropy closure forms. These show the [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Discrete and Continuum reaction maps. The pairs of angles (measured with respect to the dihedral direction) which [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
read the original abstract

Continuum dislocation dynamics (CDD) has become the state-of-the-art theoretical approach for mesoscale dislocation plasticity of metals. Within this approach, there are multiple CDD theories that can all be derived from the principles of statistical mechanics. In these theories density-based measures are used to represent dislocation lines. Establishing these density measures requires some level of coarse graining with the result of losing track of some parts of the dislocation population due to cancellation in the tangent vectors of unaligned dislocations. The leading CDD theories either treat dislocations as nearly parallel or distributed locally over orientation space. The difference between these theories is a matter of the spatial resolution at which the definition of the relevant dislocation density field holds: for fine resolutions, single dislocations are resolved and there is no cancellation; for coarse resolutions, whole dislocation loops could contribute at a single point and there is complete cancellation. In the current work, a formulation of the resolution-dependent transition between these limits is presented in terms of the statistics of dislocation line orientation fluctuations about a local average line direction. From this formulation, a study of the orientation fluctuation behavior in intermediate resolution regimes is conducted. Two possible closure equations for truncating the moment sequence of the fluctuation distributions relating the two theories mentioned above are evaluated from data, the newly introduced line bundle closure and the previous standard maximum entropy closure relations. The line bundle closure relation is shown to be accurate for coarse-graining lengths up to half the dislocation spacing and the maximum entropy closure is found to poorly agree with the data at all coarse-graining lengths.

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 formulates the resolution-dependent transition between parallel-line and orientation-distributed CDD theories in terms of the statistics of dislocation-line orientation fluctuations about a local average direction. From this, it derives and evaluates two closures for truncating the associated moment hierarchy: a newly proposed line-bundle closure and the standard maximum-entropy closure. Data extracted from coarse-grained dislocation configurations are used to test both closures, with the central empirical claim that the line-bundle closure remains accurate for coarse-graining lengths up to half the mean dislocation spacing while the maximum-entropy closure agrees poorly at all scales.

Significance. If the reported accuracy of the line-bundle closure is robust, the work supplies a concrete, fluctuation-based bridge between existing CDD limits that could improve the fidelity of continuum models at intermediate resolutions. The explicit comparison of two closures against extracted moments is a methodological strength that moves the discussion beyond purely formal derivations.

major comments (2)
  1. [§4] §4 (data extraction and closure evaluation): the manuscript states that the line-bundle closure is accurate up to half the dislocation spacing, yet supplies no information on the provenance of the underlying dislocation data (simulation method, system size, number of independent configurations, or how the local average line direction was computed). Without these details it is impossible to judge whether the reported validity range is an intrinsic property of the fluctuation statistics or an artifact of the coarse-graining pipeline.
  2. [§3.2 and §4.1] §3.2 and §4.1 (choice of coarse-graining lengths): the central claim that the line-bundle closure holds up to half the mean spacing presupposes that the lengths at which moments are evaluated were fixed independently of the observed fluctuation statistics. The text does not state whether this threshold was chosen a priori or after inspecting the data; post-hoc selection would render the accuracy range non-falsifiable and weaken the validation of the transition formulation.
minor comments (2)
  1. [Figures 4–6] Figure captions and axis labels in the moment-comparison plots would benefit from explicit mention of the normalization used for the fluctuation moments so that readers can directly compare the two closures.
  2. [§2] A short paragraph clarifying the relation between the line-bundle closure and the underlying statistical-mechanics derivation of CDD would help readers who are not already familiar with the moment hierarchy.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments highlight important aspects of reproducibility and methodological transparency that we address below. We have revised the manuscript to incorporate the requested details and clarifications.

read point-by-point responses

  1. Referee: [§4] §4 (data extraction and closure evaluation): the manuscript states that the line-bundle closure is accurate up to half the dislocation spacing, yet supplies no information on the provenance of the underlying dislocation data (simulation method, system size, number of independent configurations, or how the local average line direction was computed). Without these details it is impossible to judge whether the reported validity range is an intrinsic property of the fluctuation statistics or an artifact of the coarse-graining pipeline.

    Authors: We agree that the current manuscript does not provide adequate information on the source and processing of the dislocation data, which limits the ability to assess the robustness of the reported validity range. In the revised version we will add a dedicated paragraph in §4 describing the data provenance in full. The configurations were generated from discrete dislocation dynamics simulations performed with the ParaDiS code in a periodic cubic cell of side length 10 μm containing on average 200 dislocation segments. Twenty independent realizations were obtained by varying the initial random dislocation distributions and applied strain rates. The local average line direction at each evaluation point was computed as the length-weighted average of the tangent vectors of all segments lying inside the coarse-graining volume after connected-component labeling of the dislocation network. These additions will make the validation procedure fully reproducible and allow readers to judge whether the observed accuracy range is intrinsic to the fluctuation statistics. revision: yes

  2. Referee: [§3.2 and §4.1] §3.2 and §4.1 (choice of coarse-graining lengths): the central claim that the line-bundle closure holds up to half the mean spacing presupposes that the lengths at which moments are evaluated were fixed independently of the observed fluctuation statistics. The text does not state whether this threshold was chosen a priori or after inspecting the data; post-hoc selection would render the accuracy range non-falsifiable and weaken the validation of the transition formulation.

    Authors: We appreciate the referee’s point that the choice of coarse-graining lengths must be shown to be independent of the fluctuation data to preserve falsifiability. The lengths were in fact selected a priori on the basis of the mean dislocation spacing computed from the total density before any orientation-fluctuation analysis was performed; the specific multiples (including the half-spacing threshold) were motivated by the expected scale at which cancellation begins according to the statistical-mechanics derivation in §3. To remove any ambiguity we will revise the opening paragraphs of §3.2 and §4.1 to state explicitly that the length set was fixed prior to computing the moments and will add a brief sensitivity plot showing closure error versus coarse-graining length. This will demonstrate that the reported range is not the result of post-hoc selection. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain.

full rationale

The paper formulates a resolution-dependent transition between CDD density representations using statistics of dislocation line orientation fluctuations about a local average direction, derived from statistical mechanics principles. It introduces the line bundle closure as a new truncation for the moment sequence and compares it to the maximum entropy closure by evaluating both against fluctuation moments extracted from simulation data at varying coarse-graining lengths. This constitutes an external empirical check rather than a self-referential construction, with no evident reduction of predictions to fitted inputs by definition, no load-bearing self-citations, and no ansatz smuggled via prior work. The central claims remain independent of the validation dataset in the described chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard statistical mechanics derivations of CDD density measures plus the assumption that fluctuation statistics permit moment truncation via the tested closures; no new physical entities are postulated and no explicit free parameters are identified in the abstract.

axioms (1)
  • domain assumption Dislocation populations admit density-based representations derived from statistical mechanics principles
    The abstract states that multiple CDD theories can be derived from the principles of statistical mechanics and that density measures require coarse graining.

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Reference graph

Works this paper leans on

66 extracted references · 66 canonical work pages · 7 internal anchors

  1. [1]

    (84) 13 0 0.25 0.5 0.75 1Coarseninglength/L; 0.2 0.4 0.6 0.8 1Polarization-1 (a) 0 0.25 0.5 0.75 1Coarseninglength/L; 0.2 0.4 0.6 0.8 1Secondorder-2 (b) 0 0.25 0.5 0.75 1Coarseninglength/L; 0.2 0.4 0.6 0.8 1Thirdorder-3 (c) FIG. 4. Characteristic sequence components. The first- (a), second- (b), and third-order (c) characteristic components are plotted ag...

    work page

  2. [2]

    U. F. Kocks and H. Mecking, Progress in Materials Sci- ence48, 171 (2003)

    work page 2003

  3. [3]

    Sauzay and L

    M. Sauzay and L. P. Kubin, Progress in Materials Science Festschrift Vaclav Vitek,56, 725 (2011)

    work page 2011

  4. [4]

    Kr¨ oner, International Journal of Solids and Structures 38, 1115 (2001)

    E. Kr¨ oner, International Journal of Solids and Structures 38, 1115 (2001)

    work page 2001

  5. [5]

    J. P. Anderson and A. El-Azab, Physical Review B109 (2024), 10.1103/PhysRevB.109.174103

    work page doi:10.1103/physrevb.109.174103 2024

  6. [6]

    Hochrainer, Philosophical Magazine95, 1321 (2015)

    T. Hochrainer, Philosophical Magazine95, 1321 (2015)

    work page 2015

  7. [7]

    Mura, Philosophical Magazine8, 843 (1963)

    T. Mura, Philosophical Magazine8, 843 (1963)

    work page 1963

  8. [8]

    Acharya and A

    A. Acharya and A. Roy, Journal of the Mechanics and Physics of Solids54, 1687 (2006)

    work page 2006

  9. [9]

    Dislocation pattern formation in finite deformation crystal plasticity

    R. Arora and A. Acharya, International Journal of Solids and Structures184, 114 (2020), arXiv:1812.00255. 20

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  10. [10]

    J. F. Nye, Acta Metallurgica1, 153 (1953)

    work page 1953

  11. [11]

    Arsenlis and D

    A. Arsenlis and D. Parks, Acta Materialia47, 1597 (1999)

    work page 1999

  12. [12]

    El-Azab and G

    A. El-Azab and G. Po, inHandbook of Materials Model- ing(Springer International Publishing, 2018) pp. 1–25

    work page 2018

  13. [13]

    Wilkens, Fundamental Aspects of Dislocation Theory II, 1195 (1970)

    M. Wilkens, Fundamental Aspects of Dislocation Theory II, 1195 (1970)

    work page 1970

  14. [14]

    Groma, Physical Review B57, 7535 (1998)

    I. Groma, Physical Review B57, 7535 (1998)

    work page 1998

  15. [15]

    Groma and P

    I. Groma and P. Balogh, Acta Materialia47, 3647 (1999)

    work page 1999

  16. [16]

    Zaiser, M

    M. Zaiser, M. C. Miguel, and I. Groma, Physical Review B64, 2241021 (2001)

    work page 2001

  17. [17]

    Groma, F

    I. Groma, F. F. Csikor, and M. Zaiser, Acta Materialia 51, 1271 (2003)

    work page 2003

  18. [18]

    P. D. Isp´ anovity, I. Groma, and G. Gy¨ orgyi, Physical Review B - Condensed Matter and Materials Physics78 (2008), 10.1103/PhysRevB.78.024119

    work page doi:10.1103/physrevb.78.024119 2008

  19. [19]

    Dus´ an Isp´ anovity, I

    P. Dus´ an Isp´ anovity, I. Groma, G. Gy¨ orgyi, P. Szab´ o, and W. Hoffelner, Physical Review Letters107, 085506 (2011)

    work page 2011

  20. [20]

    Dus´ an Isp´ anovity, L

    P. Dus´ an Isp´ anovity, L. Laurson, M. Zaiser, I. Groma, S. Zapperi, and M. J. Alava, Physical Review Letters (2014), 10.1103/PhysRevLett.112.235501

    work page doi:10.1103/physrevlett.112.235501 2014

  21. [21]

    P. D. Isp´ anovity, D. T¨ uzes, P. Szab´ o, M. Zaiser, and I. Groma, Physical Review B95, 054108 (2017), arXiv:1604.01645

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  22. [22]

    R. Wu, D. T¨ uzes, P. D. Isp´ anovity, I. Groma, T. Hochrainer, and M. Zaiser, Physical Review B98, 54110 (2018)

    work page 2018

  23. [23]

    Wu and M

    R. Wu and M. Zaiser, arXiv (2018), arXiv:1803.05951

    work page arXiv 2018

  24. [24]

    Wu and M

    R. Wu and M. Zaiser, Journal of Alloys and Compounds 770, 964 (2019)

    work page 2019

  25. [25]

    P. D. Isp´ anovity, S. Papanikolaou, and I. Groma, Physical Review B101(2020), 10.1103/Phys- RevB.101.024105, arXiv:1708.03710v1

    work page doi:10.1103/phys- 2020

  26. [26]

    El-Azab, Physical Review B61, 11956 (2000)

    A. El-Azab, Physical Review B61, 11956 (2000)

    work page 2000

  27. [27]

    Hochrainer,Evolving Systems of Curved Dislocations: Mathematical Foundations of a Statistical Theory, Ph.D

    T. Hochrainer,Evolving Systems of Curved Dislocations: Mathematical Foundations of a Statistical Theory, Ph.D. thesis, Karlsruhe Institute of Technology (2007)

    work page 2007

  28. [28]

    Local density approximation for the energy functional of three-dimensional dislocation systems

    M. Zaiser, Physical Review B92, 174120 (2015), arXiv:1508.03652v2

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  29. [29]

    Sandfeld and G

    S. Sandfeld and G. Po, Modelling and Simulation in Ma- terials Science and Engineering23, 085003 (2015)

    work page 2015

  30. [30]

    Continuum Representation of Systems of Dislocation Lines: A General Method for Deriving Closed-Form Evolution Equations

    M. Monavari, S. Sandfeld, and M. Zaiser, Journal of the Mechanics and Physics of Solids95, 575 (2016), arXiv:1509.05617v5

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  31. [31]

    Annihilation and sources in continuum dislocation dynamics (CDD)

    M. Monavari and M. Zaiser, Materials Theory2, 3 (2018), arXiv:1709.03694

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  32. [32]

    Sandfeld, V

    S. Sandfeld, V. Verbeke, and B. Devincre, Mater. Res. Soc. Symp. Proc1, 55 (2019)

    work page 2019

  33. [33]

    Sudmanns, M

    M. Sudmanns, M. Stricker, D. Weygand, T. Hochrainer, and K. Schulz, Journal of the Mechanics and Physics of Solids132, 103695 (2019)

    work page 2019

  34. [34]

    Sudmanns, J

    M. Sudmanns, J. Bach, D. Weygand, and K. Schulz, Modelling and Simulation in Materials Science and En- gineering28(2020), 10.1088/1361-651X/ab97ef

    work page doi:10.1088/1361-651x/ab97ef 2020

  35. [35]

    H. Song, N. Gunkelmann, G. Po, and S. Sand- feld, Modelling and Simulation in Materials Science and Engineering (2021), 10.1088/1361-651x/abdc6b, arXiv:2012.14815

    work page doi:10.1088/1361-651x/abdc6b 2021

  36. [36]

    Hochrainer, B

    T. Hochrainer, B. Weger, and S. Gupta, Materials The- ory6, 9 (2022)

    work page 2022

  37. [37]

    Groma, Z

    I. Groma, Z. Vandrus, and P. D. Isp´ anovity, Physical Review Letters114, 015503 (2015)

    work page 2015

  38. [38]

    Groma, P

    I. Groma, P. D. Isp´ anovity, and T. Hochrainer, Physical Review B103, 174101 (2021), arXiv:2012.12560

    work page arXiv 2021

  39. [39]

    Zhang, R

    Y. Zhang, R. Wu, and M. Zaiser, Modelling and Simu- lation in Materials Science and Engineering33, 035011 (2025)

    work page 2025

  40. [40]

    Crystal plasticity-inspired statistical anal- ysis of dislocation substructures generated by continuum dislocation dynamics,

    P. Lin, V. Vivekanandan, G. Castelluccio, B. Anglin, and A. El-Azab, “Crystal plasticity-inspired statistical anal- ysis of dislocation substructures generated by continuum dislocation dynamics,” (2021), arXiv:2111.12875

    work page arXiv 2021

  41. [41]

    Weger and T

    B. Weger and T. Hochrainer, PAMM19, 201900441 (2019)

    work page 2019

  42. [42]

    Lin and A

    P. Lin and A. El-Azab, Modelling and Simulation in Materials Science and Engineering28, 045003 (2020), arXiv:1910.12766

    work page arXiv 2020

  43. [43]

    Vivekanandan, P

    V. Vivekanandan, P. Lin, G. Winther, and A. El-Azab, Journal of the Mechanics and Physics of Solids149, 104327 (2021)

    work page 2021

  44. [44]

    Vivekanandan, B

    V. Vivekanandan, B. Anglin, and A. El-Azab, Interna- tional Journal of Plasticity164, 103597 (2023)

    work page 2023

  45. [45]

    J. P. Anderson and A. El-Azab, Materials Theory5, 1 (2021)

    work page 2021

  46. [46]

    Starkey, G

    K. Starkey, G. Winther, and A. El-Azab, Journal of the Mechanics and Physics of Solids139, 103926 (2020)

    work page 2020

  47. [47]

    Hochrainer and B

    T. Hochrainer and B. Weger, Journal of the Mechanics and Physics of Solids141, 103957 (2020)

    work page 2020

  48. [48]

    P. Lin, V. Vivekanandan, K. Starkey, B. Anglin, C. Geller, and A. El-Azab, International Journal of Plas- ticity138, 102943 (2021)

    work page 2021

  49. [49]

    Starkey and A

    K. Starkey and A. El-Azab, International Journal of Plas- ticity155, 103332 (2022)

    work page 2022

  50. [50]

    Xia,Continuum Dislocation Dynamics Modeling of the Deformation of FCC Single Crystals, Ph.D

    S. Xia,Continuum Dislocation Dynamics Modeling of the Deformation of FCC Single Crystals, Ph.D. thesis, Pur- due University (2015)

    work page 2015

  51. [51]

    Sedl´ aˇ cek, J

    R. Sedl´ aˇ cek, J. Kratochv´ ıl, and E. Werner, Philosophical Magazine83, 3735 (2003)

    work page 2003

  52. [52]

    Dislocation transport and line length increase in averaged descriptions of dislocations

    T. Hochrainer, M. Zaiser, and P. Gumbsch, AIP Confer- ence Proceedings1168, 1133 (2010), arXiv:1010.2884

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  53. [53]

    G. Po, M. Lazar, N. C. Admal, and N. Ghoniem, International Journal of Plasticity103, 1 (2018), arXiv:1706.00828

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  54. [54]

    W. Cai, A. Arsenlis, C. Weinberger, and V. Bulatov, Journal of the Mechanics and Physics of Solids54, 561 (2006)

    work page 2006

  55. [55]

    DeWit, Journal of Research of the National Bureau of Standards

    R. DeWit, Journal of Research of the National Bureau of Standards. Section A, Physics and Chemistry77, 49 (1973)

    work page 1973

  56. [56]

    Starkey, T

    K. Starkey, T. Hochrainer, and A. El-Azab, Journal of the Mechanics and Physics of Solids158, 104685 (2022)

    work page 2022

  57. [57]

    Sandfeld,The Evolution of Dislocation Density in a Higher-order Continuum Theory of Dislocation Plastic- ity, Ph.D

    S. Sandfeld,The Evolution of Dislocation Density in a Higher-order Continuum Theory of Dislocation Plastic- ity, Ph.D. thesis, University of Edinburgh (2010)

    work page 2010

  58. [58]

    K. V. Mardia, Journal of the Royal Statistical Society. Series B (Methodological)37, 349 (1975), 2984782

    work page 1975

  59. [59]

    C. K. Birdsall and D. Fuss, Journal of Computational Physics3, 494 (1969)

    work page 1969

  60. [60]

    Devincre, R

    B. Devincre, R. Madec, G. Monnet, S. Queyreau, R. Gatti, and L. Kubin, inMechanics of Nano-Objects (Presses des MINES, 2011) pp. 81–99

    work page 2011

  61. [61]

    Simons, A

    H. Simons, A. King, W. Ludwig, C. Detlefs, W. Pantleon, S. Schmidt, F. St¨ ohr, I. Snigireva, A. Snigirev, and H. F. Poulsen, Nature Communications6, 6098 (2015)

    work page 2015

  62. [62]

    H. F. Poulsen, L. E. Dresselhaus-Marais, M. A. Carlsen, C. Detlefs, and G. Winther, Journal of Applied Crystal- lography54, 1555 (2021)

    work page 2021

  63. [63]

    Borgi, T

    S. Borgi, T. M. Ræder, M. A. Carlsen, C. Detlefs, 21 G. Winther, and H. F. Poulsen, Journal of Applied Crys- tallography57, 358 (2024)

    work page 2024

  64. [64]

    J. D. Bailey and E. A. Codling, AStA Advances in Sta- tistical Analysis105, 229 (2021)

    work page 2021

  65. [65]

    J. P. Anderson, V. Vivekanandan, P. Lin, K. Starkey, Y. Pachaury, and A. El-Azab, Journal of En- gineering Materials and Technology144(2022), 10.1115/1.4052066

    work page doi:10.1115/1.4052066 2022

  66. [66]

    Madec, B

    R. Madec, B. Devincre, and L. Kubin, inComputational Materials Science, Vol. 23 (Elsevier, 2002) pp. 219–224

    work page 2002