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REVIEW 2 major objections 6 minor 39 references

Energetic L2 coupling of field dislocation mechanics to the phase-field crystal only matches compatible distortions, is blind to dislocation topology, and cannot stop unnatural core spreading.

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2026-07-12 09:21 UTC pith:7UCEBIRK

load-bearing objection Solid negative result: the Acharya–Viñals L2 (and p-norm) energetic penalty only forces compatible mismatch, so it cannot reconcile FDM cores with PFC crystallography. the 2 major comments →

arxiv 2607.01284 v2 pith:7UCEBIRK submitted 2026-07-01 cond-mat.mtrl-sci

On the limits of the energetic coupling between field dislocation mechanics and phase field crystal

Aymane Graini , Jorge Vi\~nals , Manas V. Upadhyay This is my paper
classification cond-mat.mtrl-sci
keywords dislocation mechanicsphase field crystalSwift-Hohenbergplasticitycrystal defectsfield dislocation mechanicsenergetic coupling
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tests a proposed continuum marriage of two complementary defect models: field dislocation mechanics, which correctly solves the mechanical initial-boundary-value problem for continuously distributed dislocations, and the phase-field crystal model, which encodes lattice structure and naturally keeps dislocation cores compact. The marriage is an energy penalty that punishes the L2 mismatch between the elastic distortion of the first model and the configurational distortion of the second. Variational analysis shows that the penalty forces only the divergences of the two fields to agree, so it can match only their compatible, curl-free parts; the incompatible, divergence-free parts that carry all Burgers-vector topology remain invisible. Boundary loads therefore reach the phase field by slow phase diffusion rather than elastic relaxation, and numerical simulations confirm that core spreading in field dislocation mechanics is not suppressed. The same structural limits hold for a general p-norm energetic penalty, so pure energy matching of the two distortions cannot fully reconcile dislocation mechanics with crystallography.

Core claim

The L2 energetic penalty between elastic distortion from field dislocation mechanics and configurational distortion from the phase-field crystal produces only a divergence-driven forcing in the phase-field evolution. That forcing matches solely the compatible (curl-free) parts of the two fields and is insensitive to the incompatible (divergence-free) elastic distortion that carries dislocation topology. Mechanical boundary conditions are transmitted by phase diffusion rather than elasticity, the coupling cannot prevent unnatural core spreading, and the same drawbacks persist for any p-norm energetic coupling.

What carries the argument

The functional derivative of the L2 penalty with respect to the phase-field order parameter, which reduces to a forcing proportional to the divergence of the difference between configurational and elastic distortions and therefore annihilates all incompatible content.

Load-bearing premise

The claim that only the divergence of the distortion mismatch enters the phase-field forcing rests on representing configurational distortion through slowly varying complex amplitudes obtained by Gaussian demodulation of the order parameter.

What would settle it

Deliberately mismatch the incompatible parts of the two distortion fields while keeping their divergences equal, then evolve under the penalty alone: if the phase field still equalizes the incompatible parts or suppresses field-dislocation-mechanics core spreading, the central claim fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Any pure energetic mismatch of the two distortion fields leaves the two dislocation densities free to evolve independently.
  • Boundary loads reach the phase-field crystal only by phase diffusion, never by instantaneous elastic relaxation.
  • Core localization cannot be imported from the phase-field crystal into field dislocation mechanics by an energetic penalty of this type alone.
  • A phase-field-crystal-driven plastic flux can keep a single dislocation density but does not guarantee non-negative plastic dissipation.
  • Integrating continuum dislocation mechanics with crystallography requires couplings beyond pure energy penalties on distortion mismatch.

Where Pith is reading between the lines

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

  • Kinematic constraints that directly identify the two dislocation densities, rather than energetic penalties on their parent distortions, may be needed to transfer topology between the models.
  • The same divergence-only limitation would appear in any continuum scheme that couples two distortion fields solely through an L2 or p-norm energy of their difference when one field is reconstructed from demodulated complex amplitudes.
  • The empty window between the minimum coupling needed for fast penalty relaxation and the maximum that avoids pinning suggests parameter tuning alone cannot rescue the present construction.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The manuscript analyzes the energetic L2 penalty coupling between Field Dislocation Mechanics (FDM) elastic distortion Ue and Phase Field Crystal (PFC) configurational distortion Q proposed by Acharya and Viñals (Phys. Rev. B 102, 064109, 2020). Through a variational calculation of δFpen/δψ (Appendix B, Eqs. 31–37 and 52–67), it shows that the bulk forcing depends only on ∇·(Q−Ue) and therefore matches only the compatible parts of the two distortions, remaining blind to the incompatible, dislocation-carrying parts. The same structure implies that mechanical boundary data are transmitted by phase diffusion of the complex amplitudes rather than by elastic relaxation (Eqs. 39–41), and that a residual incompatible mismatch can persist at equilibrium (Stokes–Helmholtz argument, Eq. 38). The analysis is extended to a family of p-norm penalties (Eqs. 44–46), which retain the same divergence-driven character. Numerical experiments on an immobile dipole, free-core evolution, and dipole annihilation under both FDM-driven and PFC-driven plasticity confirm dual dislocation densities, residual core spreading, an empty operating window for the coupling strength cpen, and (in the PFC-driven case) non-positive plastic dissipation.

Significance. The paper supplies a clear, structural negative result on a coupling that was proposed as a thermodynamically consistent route to combine crystallography with continuum dislocation mechanics. The central variational reduction (forcing depends only on ∇·(Q−Ue)) is clean, independent of the particular inverse map from amplitudes to ψ, and is corroborated by independent numerical tests that exhibit dual densities, core spreading, and dissipation issues. The empty cpen window (Section 6.1) and the regularity/log-divergence observation (Section 4.3) are useful quantitative diagnostics for anyone attempting related couplings. If the conclusions hold, they usefully delimit the domain of applicability of pure energetic mismatch penalties between Ue and Q and redirect the community toward alternative (e.g., kinematic or mixed) couplings. The work is therefore of genuine interest to the dislocation-mechanics and phase-field communities.

major comments (2)
  1. Abstract and final sentence of the conclusion state that “even in the most general case, an energetic coupling suffers from the same drawbacks.” Section 4.4 only treats the one-parameter family fpen = (1/(2p))[(Ue−Q):A:(Ue−Q)]p and shows that the functional derivative retains a divergence structure (Eq. 45). That is not the most general energetic coupling (for example, penalties involving curls of the distortions, or direct penalties on α−eα, are not covered). The claim should be narrowed to “general p-norm penalties of the distortion mismatch” so that the abstract matches the derivation that is actually given.
  2. Section 6.2 and Eq. (30): the demonstration that plastic dissipation ∫σ:Jψ can be negative is numerical and configuration-specific (dipole annihilation). The structural argument that no consistent evolution for ψ̇ can guarantee positivity when Jψ is taken from the amplitudes alone is stated but not proved for general loadings. A short analytic counter-example or a clearer statement of the scope of the negativity claim would strengthen the load-bearing conclusion that the PFC-driven route is thermodynamically flawed.
minor comments (6)
  1. Figure 1 caption: panels are listed as (a)–(e) but the text refers to (b) dislocation density and (d) contour of the penalty forcing; the ordering of (c) and (e) is easy to misread. A single sentence clarifying the layout would help.
  2. Figure 7 caption: “v ,d” contains a stray space/comma; should be vd.
  3. Eq. (48) and the fit for Bref: the reference trajectory is obtained from uncoupled PFC (cpen=0). It would be useful to state explicitly in the main text (not only Appendix D) that the same Bref is then used for all coupled FDM-driven runs, so that the comparison is controlled.
  4. Reference [36] (Volterra) contains the typo “cirps” for “corps”; also several author names and titles could be checked for consistency with the published versions.
  5. Notation: both α and eα appear for the two dislocation densities; a brief reminder at the start of Section 5 that they are allowed to evolve independently under FDM-driven plasticity would reduce reader confusion.
  6. Section 3: the neglect of cross terms between ψ̇ and elastic stress is stated without citation to the original PFCFDM paper or a short justification; one sentence would suffice.

Circularity Check

0 steps flagged

No significant circularity: the paper critically analyzes a prior coupling via independent variational structure and numerics; self-citations supply the model under test, not the negative conclusions.

full rationale

This is a critical analysis of the Acharya–Viñals 2020 energetic coupling, not a derivation that re-packages its own inputs as predictions. The central negative claims (forcing depends only on ∇·(Q−Ue), blindness to incompatible/dislocation-carrying parts, phase-diffusive rather than elastic transmission of BCs, inability to prevent FDM core spreading, same structure for general p-norms, and non-positive plastic dissipation under PFC-driven plasticity) follow from the variational derivative of the published free-energy penalty (Eqs. 31–37, 45) together with independent numerical experiments (immobile dipole, free-core evolution, FDM- vs PFC-driven annihilation). Self-citations to Acharya–Viñals 2020 and Upadhyay–Viñals 2024 define the model being tested and supply background on the mismatch between Q and Ue; they are not used as uniqueness theorems or load-bearing external facts that force the paper’s conclusions. The amplitude demodulation and small-deformation Q formula are standard PFC machinery, and the paper notes that the divergence-only structure is independent of the particular inverse map. No fitted parameter is renamed a prediction, no ansatz is smuggled in via self-citation, and no known empirical pattern is merely renamed. Score 1 reflects only the ordinary presence of self-citations that are not load-bearing for the negative result.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 0 invented entities

The central negative claim rests on standard continuum and PFC constructions plus the specific L2 (and p-norm) penalty form from Acharya–Viñals 2020. Free parameters (cpen, csh, B, r, ψ̄, numerical mollifiers) control numerics and the empty operating window but are not fitted to force the variational conclusion. No new physical entity is postulated; the paper analyzes an existing coupling. The load-bearing modeling axioms are small-strain FDM kinematics, Stokes–Helmholtz decomposition, nonconserved PFC dynamics with amplitude demodulation, and the definition of Q from complex amplitudes.

free parameters (5)
  • c_pen (penalty coupling strength)
    Hand-chosen energetic coefficient varied across simulations (0–8 range); enters both stress and phase-field forcing and defines the empty operating window cmin vs cmax.
  • c_sh (Swift–Hohenberg energy weight)
    Hand-chosen relative weight of crystallographic energy vs penalty; set to 1, 10, or 100 in different sections to slow or speed PFC dynamics.
  • B (dislocation drag / inverse mobility)
    Constitutive mobility in FDM-driven vd = (1/B) f_PK; Bref≈0.095 fitted to uncoupled PFC annihilation trajectory, then scanned as Bref and Bref/95.
  • r (PFC quench depth) and average ψ̄
    PFC phase-diagram parameters fixed at r=−1.2, ψ̄=−0.5 to select hexagonal lattice and isotropic Lamé coefficients; not derived from the coupling analysis.
  • Gaussian mollifier widths (a0 for demodulation; a0/4 for Q smoothing)
    Numerical regularization choices that keep Fpen finite near cores and define amplitude envelopes; affect core regularity and eα mollification.
axioms (6)
  • domain assumption Small-strain FDM kinematics: ∇u = Ue + Up, α = ∇×Ue = −∇×Up, Stokes–Helmholtz split of Up, and mechanical equilibrium ∇·σ=0.
    Section 2.1; standard FDM setup on which the coupling is built.
  • domain assumption PFC free energy is the Swift–Hohenberg functional and ψ evolves by nonconserved gradient flow (with fixed average), not conserved H−1 dynamics.
    Section 2.2 and Eq. 12; modeling choice from Acharya–Viñals treating mass density and lattice distortion as independent.
  • domain assumption Configurational distortion Q is defined from complex amplitudes via Q = −(d/N) Σ qn ⊗ Im(∇An/An), with An obtained by Gaussian demodulation of ψ.
    Eqs. 14–16; standard amplitude representation of PFC used to define the penalty argument.
  • ad hoc to paper The Helmholtz free energy is Fel[Ue] + csh Fsh[ψ] + cpen (1/2)∥Ue−Q∥²_L2, and cross forces/currents between ψ and elastic stress are neglected.
    Section 3, Eqs. 21–26; the specific coupling under test, inherited from Acharya–Viñals 2020.
  • standard math Stokes–Helmholtz compatible and incompatible parts are L2-orthogonal, so matching divergences does not force matching curls.
    Used in Eq. 38 to conclude residual Fpen > 0 when only ∇·(Q−Ue)=0.
  • standard math For p-norm generalizations, the variational derivative still involves a divergence of a nonlinear function of (Ue−Q), hence the same phase-diffusion structure.
    Section 4.4, Eqs. 44–46; extends the negative result beyond pure L2.

pith-pipeline@v1.1.0-grok45 · 29038 in / 3870 out tokens · 39154 ms · 2026-07-12T09:21:34.082991+00:00 · methodology

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read the original abstract

This paper investigates the energetic coupling between Field Dislocation Mechanics (FDM) and the Phase Field Crystal (PFC) model proposed in Phys. Rev. B 102, 064109, 2020. While FDM correctly solves the initial boundary value problem of a continuum body with dislocation fields, PFC captures the underlying crystallographic structure. The coupling, which penalizes the $L^2$ distance between elastic distortion from FDM and configurational distortion from PFC in the $L^2$ sense, had been proposed to reconcile dislocation mechanics with crystallography in a single continuum framework. Variational analysis reveals that the coupling term acts as a divergence-driven forcing in the phase-field evolution that matches only the compatible (curl-free) parts of the distortion fields. Consequently, its contributions are insensitive to the incompatible (divergence-free) elastic distortion carrying all the information on dislocation topology. Furthermore, the nature of the configurational distortion causes mechanical boundary conditions to be transmitted diffusively from FDM to PFC rather than elastically. Numerical simulations demonstrate that this coupling cannot prevent the unnatural core spreading in FDM. Finally, it is shown that even in the most general case, an energetic coupling suffers from the same drawbacks, which limits its ability to integrate dislocation mechanics with crystallography.

Figures

Figures reproduced from arXiv: 2607.01284 by Aymane Graini, Jorge Vi\~nals, Manas V. Upadhyay.

Figure 1
Figure 1. Figure 1: (a) Order parameter field in the presence of a single edge dislocation at the center of a 2D hexagonal lattice. (b) Corresponding dislocation density αxz. (d) Contour plot of the penalty-induced forcing for this configuration. In both, the gray circle shows the extent of the core. (c) A horizontal slice along y = 0 of the penalty forcing whereas (e) show the behavior of the xx of Q and U e . The vertical s… view at source ↗
Figure 2
Figure 2. Figure 2: Immobile edge dislocation dipole in Climb configuration. (a): Order parameter ψ at t = 0 and the dislocation density tensor α. (b) relaxation of ∥U e − Q∥ with increasing cpen. (c), corresponding evolution of ∥∇ · (U e − Q)∥. At t = 0, in the absence of coupling (cpen = 0), the elastic distortion U e and the configurational lattice distortion Q exhibit different spatial distributions( [PITH_FULL_IMAGE:fig… view at source ↗
Figure 3
Figure 3. Figure 3: a [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Fpen at t = 0 and t = ∞ for different cpen (a) t = 0 (b) t = tf (c) t = 0 (d) t = tf [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Vertical profiles at x = L/2 of the distortion fields for cpen = 2, compared with the uncoupled elastic solution for the same defect configuration: (a) xx component at t = 0; and at t = tf in (b). (c) yy component at t = 0; and at t = tf in (d). 5.2 Evolving dislocation core in a periodic domain without macroscopic loading Before investigating dislocation motion under an applied loading, we first examine t… view at source ↗
Figure 7
Figure 7. Figure 7: d [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: Horizontal profiles at y = 3H/8 of the distortion fields for cpen = 2, compared with the uncoupled elastic solution for the same defect configuration: (a) xy component at t = 0; and at t = tf in (b). (c) yx component at t = 0; and at t = tf in (d). As we activate the coupling with cpen = 2, the behavior of the system remains qualitatively similar (Figures 7.b and 7.e). The spreading of the core is still no… view at source ↗
Figure 8
Figure 8. Figure 8: a [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: Evolution of the dislocation density fields [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: a [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: Initial configuration of the dislocation annihilation simulation. [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: In (a) and (b),Time evolution of the dislocation core positions for two values of B.Dashed lines track the PFC density αe and solid lines the FDM density α. (c) shows the snapshot of FDM cores for different values of B and cpen, (d) shows the PFC cores for the same parameters. spatially uniform correction U corr(x, t) = U (x, t) + dt Seff : Σ − U (t) + U p (t) + cpen Seff : Qsym(t) [PITH_FULL_IMAGE:figure… view at source ↗
Figure 10
Figure 10. Figure 10: a [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: i [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: a [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 11
Figure 11. Figure 11: Evolution of a dislocation dipole during PFC-driven glide and annihilation: [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: (a): Evolution of the coordinate of the cores for different values of cpen. (b) to (d) same evolution for fixed values of cpen and different applied macroscopic stresses τ . Reference PFC uncoupled motion shown in solid line. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Plastic and PFC dissipation in the uncoupled case (a) and coupled case (b) for both PFC and FDM driven cases. Shaded part and dashed black line marks the annihilation start (cores are less than a0 distance). 21 [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Decay of Fpen in penalty only case for for an immobile dipole in climb. For numerical validation. C.2 Dislocation Transport: Edge Dislocation in 2D In the framework of Field Dislocation Mechanics (FDM), the evolution of the dislocation density tensor α is governed by the transport equation: α˙ + ∇ × α × v d  = 0. (76) We consider a 2D setting in the (x, y) plane, assuming invariance along the z-axis and … view at source ↗
Figure 15
Figure 15. Figure 15: Time evolution of αe and the Peach-Koehler (pk) fit for B in eq. 48 References [1] A. Acharya. A model of crystal plasticity based on the theory of continuously distributed dislo￾cations. Journal of the Mechanics and Physics of Solids, 49(4):761–784, Apr. 2001. [2] A. Acharya. Microcanonical Entropy and Mesoscale Dislocation Mechanics and Plasticity. Journal of Elasticity, 104(1-2):23–44, Aug. 2011. [3] A… view at source ↗

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Works this paper leans on

39 extracted references

  1. [1]

    A. Acharya. A model of crystal plasticity based on the theory of continuously distributed dislo- cations. Journal of the Mechanics and Physics of Solids, 49(4):761–784, Apr. 2001

    2001

  2. [2]

    A. Acharya. Microcanonical Entropy and Mesoscale Dislocation Mechanics and Plasticity.Journal of Elasticity, 104(1-2):23–44, Aug. 2011

    2011

  3. [3]

    Acharya, L

    A. Acharya, L. Angheluta, and J. Viñals. Elasticity versus phase field driven motion in the phase field crystal model.Modelling and Simulationin Materials Science and Engineering, 30(6):064005, Sept. 2022

    2022

  4. [4]

    Acharya and A

    A. Acharya and A. Roy. Size effects and idealized dislocation microstructure at small scales: Predictions of a Phenomenological model of Mesoscopic Field Dislocation Mechanics: Part I. Journal of the Mechanics and Physics of Solids, 54(8):1687–1710, Aug. 2006

    2006

  5. [5]

    Acharya and J

    A. Acharya and J. Viñals. Field dislocation mechanics and phase field crystal models.Physical Review B, 102(6):064109, Aug. 2020

    2020

  6. [6]

    P. M. Anderson, J. P. Hirth, and J. Lothe.Theory of Dislocations. Cambridge University Press, Jan. 2017

    2017

  7. [7]

    Arora, R

    A. Arora, R. Arora, and A. Acharya. Mechanics of micropillar confined thin film plasticity.Acta Materialia, 238:118192, Oct. 2022

    2022

  8. [8]

    R. Arora. Computational Approximation of Mesoscale Field Dislocation Mechanics at Finite Deformation. Thesis, Carnegie Mellon University, Mar. 2019

    2019

  9. [9]

    Arora, X

    R. Arora, X. Zhang, and A. Acharya. Finite element approximation of finite deformation dislo- cation mechanics. Computer Methods in Applied Mechanics and Engineering, 367:113076, Aug. 2020

    2020

  10. [10]

    Berry, M

    J. Berry, M. Grant, and K. R. Elder. Diffusive atomistic dynamics of edge dislocations in two dimensions. Physical Review E, 73(3):031609, Mar. 2006. 31

    2006

  11. [11]

    Bradbury, R

    J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, and Q. Zhang. JAX: Composable transformations of Python+NumPy programs, 2018

    2018

  12. [12]

    P. Y. Chan, G. Tsekenis, J. Dantzig, K. A. Dahmen, and N. Goldenfeld. Plasticity and Dislocation Dynamics in a Phase Field Crystal Model.Physical Review Letters, 105(1):015502, June 2010

    2010

  13. [13]

    S. M. Cox and P. C. Matthews. Exponential Time Differencing for Stiff Systems.Journal of Computational Physics, 176(2):430–455, Mar. 2002

    2002

  14. [14]

    M. C. Cross and P. C. Hohenberg. Pattern formation outside of equilibrium.Reviews of modern physics, 65(3):851, 1993

    1993

  15. [15]

    Devincre, V

    B. Devincre, V. Pontikis, Y. Brechet, G. Canova, M. Condat, and L. Kubin. Three-Dimensional Simulations of Plastic Flow in Crystals. In M. Mareschal and B. L. Holian, editors,Microscopic Simulations of Complex Hydrodynamic Phenomena, pages 413–423. Springer US, Boston, MA, 1992

    1992

  16. [16]

    K. R. Elder and M. Grant. Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals.Physical Review E, 70(5):051605, Nov. 2004

    2004

  17. [17]

    K. R. Elder, M. Katakowski, M. Haataja, and M. Grant. Modeling Elasticity in Crystal Growth. Physical Review Letters, 88(24):245701, June 2002

    2002

  18. [18]

    K. R. Elder, N. Provatas, J. Berry, P. Stefanovic, and M. Grant. Phase-field crystal modeling and classical density functional theory of freezing.Physical Review B, 75(6):064107, Feb. 2007

    2007

  19. [19]

    Eymard, T

    R. Eymard, T. Gallouët, and R. Herbin. Finite volume methods. In Solution of Equation in R (Part 3), Techniques of Scientific Computing (Part 3), volume 7 ofHandbook of Numerical Analysis, pages 713–1018. Elsevier, Jan. 2000

    2000

  20. [20]

    StrongStabilityPreservingRunge-KuttaandMultistep Time Discretizations

    S.Gottlieb, D.Ketcheson, andC.-W.Shu. StrongStabilityPreservingRunge-KuttaandMultistep Time Discretizations. WORLD SCIENTIFIC, 2011

    2011

  21. [21]

    Halperin

    B. Halperin. Physics of defects. In Physics of Defects, number XXXV in Proceedings of the Les Houches Summer School, pages 814–857. North Holland Publishing Company, r. balian, m. kleman, and j.-p. poirier edition, 1981

    1981

  22. [22]

    Heinonen, C

    V. Heinonen, C. V. Achim, J. M. Kosterlitz, S.-C. Ying, J. Lowengrub, and T. Ala-Nissila. Con- sistent Hydrodynamics for Phase Field Crystals. Physical Review Letters, 116(2):024303, Jan. 2016

    2016

  23. [23]

    DYNAMICALTHEORYOFDISLOCATIONS

    A.M.Kosevich. DYNAMICALTHEORYOFDISLOCATIONS. SovietPhysicsUspekhi, 7(6):837, June 1965

    1965

  24. [24]

    G. F. Mazenko. Vortex Velocities in the $\mathit{O}(\mathit{n})$ Symmetric Time-Dependent Ginzburg-Landau Model. Physical Review Letters, 78(3):401–404, Jan. 1997

    1997

  25. [25]

    T. Mura. Continuous distribution of moving dislocations.Philosophical Magazine, 8(89):843–857, May 1963

    1963

  26. [26]

    A. Roy. Finite element approximation of field dislocation mechanics.Journal of the Mechanics and Physics of Solids, 53(1):143–170, Jan. 2005

    2005

  27. [27]

    Skaugen, L

    A. Skaugen, L. Angheluta, and J. Viñals. Dislocation dynamics and crystal plasticity in the phase-field crystal model.Physical Review B, 97(5):054113, Feb. 2018. 32

    2018

  28. [28]

    Skaugen, L

    A. Skaugen, L. Angheluta, and J. Viñals. Separation of Elastic and Plastic Timescales in a Phase Field Crystal Model.Physical Review Letters, 121(25):255501, Dec. 2018

    2018

  29. [29]

    Skogvoll, L

    V. Skogvoll, L. Angheluta, A. Skaugen, M. Salvalaglio, and J. Viñals. A phase field crystal theory of the kinematics of dislocation lines.Journal of the Mechanics and Physics of Solids, 166:104932, Sept. 2022

    2022

  30. [30]

    Skogvoll, J

    V. Skogvoll, J. Rønning, M. Salvalaglio, and L. Angheluta. A unified field theory of topological defects and non-linear local excitations.npj Computational Materials, 9(1):1–13, July 2023

    2023

  31. [31]

    Skogvoll, M

    V. Skogvoll, M. Salvalaglio, and L. Angheluta. Hydrodynamic phase field crystal approach to interfaces, dislocations, and multi-grain networks, Sept. 2022

    2022

  32. [32]

    Stefanovic, M

    P. Stefanovic, M. Haataja, and N. Provatas. Phase-Field Crystals with Elastic Interactions. Physical Review Letters, 96(22):225504, June 2006

    2006

  33. [33]

    M. V. Upadhyay and J. Viñals. Coupling Phase Field Crystal and Field Dislocation Mechanics for a consistent description of dislocation structure and elasticity.European Journal of Mechanics - A/Solids, 108:105419, Nov. 2024

    2024

  34. [34]

    van Leer

    B. van Leer. Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method.Journal of Computational Physics, 32(1):101–136, July 1979

    1979

  35. [35]

    S. N. Varadhan, A. J. Beaudoin, A. Acharya, and C. Fressengeas. Dislocation transport using an explicit Galerkin/least-squares formulation. Modelling and Simulation in Materials Science and Engineering, 14(7):1245–1270, Oct. 2006

    2006

  36. [36]

    Sur l’equilibre des cirps elastiques multiplement connexes.Ann

    Voltera. Sur l’equilibre des cirps elastiques multiplement connexes.Ann. Ecole Norm. Super, 24:401–517, 1907

    1907

  37. [37]

    Wijnen, R

    J. Wijnen, R. H. J. Peerlings, J. P. M. Hoefnagels, and M. G. D. Geers. A discrete slip plane model for simulating heterogeneous plastic deformation in single crystals.International Journal of Solids and Structures, 228:111094, Oct. 2021

    2021

  38. [38]

    Zhang, A

    X. Zhang, A. Acharya, N. J. Walkington, and J. Bielak. A single theory for some quasi-static, supersonic, atomic, and tectonic scale applications of dislocations.Journal of the Mechanics and Physics of Solids, 84:145–195, Nov. 2015

    2015

  39. [39]

    S. J. Zhou, D. L. Preston, P. S. Lomdahl, and D. M. Beazley. Large-scale molecular dynamics simulations of dislocation intersection in copper.Science (New York,N.Y.), 279(5356):1525–1527, Mar. 1998. 33

    1998