Pith. sign in

REVIEW 3 major objections 5 minor 63 references

Defect-free and defective adaptations of crystalline sheets to stretching deformation

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper claims that a crystalline sheet under large uniaxial stretching adapts not by smooth flow but by intermittent plastic shear events, and that the final fracture mode is selected mainly by the sheet's width.

desk verdict A systematic simulation study that plausibly identifies two fracture modes in stretched 2D crystals; the qualitative picture is likely sound, but the quantitative phase boundary is protocol-sensitive and should be treated as provisional. read the letter →

arxiv 2506.05784 v1 pith:2PYJ3D3A submitted 2025-06-06 cond-mat.soft cond-mat.mtrl-sciphysics.class-phphysics.comp-ph

Ranzhi Sun , Zhenwei Yao This is my paper
classification cond-mat.softcond-mat.mtrl-sciphysics.class-phphysics.comp-ph
keywords crystallinesheetplasticdeformationlatticetiltingdislocationfractureLennard-Jonestriangularintermittentshear
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

The paper studies how a two-dimensional crystalline sheet, modeled as Lennard-Jones particles on a triangular lattice and wrapped around a gradually expanding cylinder, responds to large uniaxial stretching. It claims that the sheet does not deform smoothly: it undergoes a series of discrete, irreversible plastic shear instabilities, each marked by a sudden lattice tilt and an abrupt drop in energy. Ultimately the sheet fractures through one of two distinct routes: a defect-free route in which the entire lattice tilts in quantized steps, described by a geometric formula, or a defective route in which dislocations become anchored and open into elongated vacancy cracks. The choice between these routes is governed mainly by the sheet width, with wider sheets favoring the defective route. If correct, the result implies that the plastic response of 2D crystals is a sequence of history-dependent instabilities rather than a gradual yield, and that fracture geometry can be controlled by sample shape.

What carries the argument

The load-bearing object is the geometric step-count formula sin θ = (√3/4π)(ℓ(Γ)/R(Γ)) N_s, which connects the macroscopic tilt angle of the entire lattice to the total number of step-like boundary defects produced by plastic shear. The argument also uses the Peach–Koehler force to explain why dislocation pairs glide in anti-parallel directions along shear bands, and Griffith-type crack analysis to describe how anchored dislocations extend into elongated vacancies. These elements together turn the observed intermittent events into a predictive picture of how sheets choose between defect-free tilting and defect-based fracture.

What would settle it

Run the identical expansion protocol with a different relaxation scheme (for example finite-temperature molecular dynamics at a small but nonzero temperature, or a conjugate-gradient minimizer) on sheets of the same geometries, and compare the recorded sequences of critical Γ values and the defect-free/defective boundary as a function of width; if the intermittent steps smear out or the phase boundary shifts substantially, the zero-temperature protocol is not representative of the physical sheet.

Watch

Extended reading notes

Core claim

The central discovery is that the adaptation of a crystalline sheet to large uniaxial stretching is intermittent and can be classified into two fracture categories. In the defect-free category, plastic shear occurs along lattice-aligned shear bands, producing step-like changes in the tilt angle of the entire lattice; the tilt follows the quantitative relation sin θ = (√3/4π)(ℓ(Γ)/R(Γ)) N_s, where N_s is the total number of boundary steps, ℓ(Γ) is the bond length, and R(Γ) is the cylinder radius. In the defective category, which occurs in wider sheets, isolated dislocations remain in the relaxed states and are anchored in space, serving as seeds that grow into elongated vacancies and eventually interior or boundary fractures. The paper further shows that the first plastic event occurs at a nearly universal expansion factor Γ1 = 0.126 ± 0.008, while the fracture point Γf varies widely, and that the defect-free to defective transition shifts with noise and with initial lattice orientation.

Load-bearing premise

The load-bearing premise is that deterministic, zero-temperature steepest-descent relaxation after each 0.7 percent expansion step produces the physically relevant plastic response of a 2D crystal, so that thermal activation, strain rate, and dynamic effects would not change the intermittent event sequences, the critical expansion values, or the width-based fracture classification.

Editorial extensions

If this is right

  • If the claim is correct, the plastic response of a 2D crystalline sheet under uniaxial stretching is a sequence of discrete, history-dependent shear instabilities rather than a smooth flow.
  • The step-count formula provides a direct geometric rule: the tilt angle is set by the number of boundary steps, so measuring step heights on a stretched sheet gives the tilt angle.
  • Wide sheets fail by defect proliferation (anchored dislocations opening into cracks), while narrow sheets fail by defect-free shear along lattice-aligned bands; this gives a geometric control knob for fracture mode.
  • The first plastic instability occurs at a nearly universal expansion of about 12.6 percent, independent of sheet size and aspect ratio, while the complete fracture point is highly variable, indicating that early yield is reproducible but final failure is sensitive.
  • Noise and initial lattice orientation shift the defect-free/defective boundary and can change whether the 30-degree tilted lattice switches locally or globally to the 60-degree configuration.

Reading between the lines

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

  • A testable extension is to run the same expansion protocol with finite-temperature molecular dynamics or with a different relaxation algorithm; if the intermittent event sequences and the width-based phase boundary persist, the zero-temperature picture is robust, and if not, the reported critical values are protocol-dependent.
  • The defect-free/defective transition with width resembles a brittle-to-ductile crossover in 2D materials, suggesting that the same geometric criterion might help predict fracture modes in monolayer crystals or colloidal sheets.
  • The noise-induced shift of the transition boundary implies that thermal fluctuations could favor the defect-free glide route in real experiments, which would make the defective route more prominent at low temperature or high strain rate.
  • The reconnection events observed after apparent fracture suggest that a stretched sheet can heal if the two patches rotate into registry; this could be exploited in designing self-healing particulate packings.
Share X Bluesky LinkedIn Reddit HN

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 manuscript studies the plastic response of a triangular Lennard-Jones crystalline sheet wrapped on a slowly expanding cylinder. The authors report that under large uniaxial stretching the sheet adapts through intermittent, irreversible plastic shear events, visible as step-like increases of the lattice tilt angle, drops in energy, and zig-zag behavior of the mean bond length. They propose a geometric model, Eq. (10), relating the tilt angle to the number of boundary steps, the bond length, and the cylinder radius. Fracture is classified into defect-free processes, where dislocations appear only transiently and the sheet slides along shear bands, and defective processes, where anchored dislocations grow into elongated vacancies that ultimately disconnect the sheet; Table I summarizes the resulting width-dependent phase diagram. The paper also discusses noise effects, which shift the defect-free/defective boundary, and the abrupt 30-to-60 degree tilting transition for differently oriented lattices.

Significance. If the central claims hold, the paper would extend the study of 2D crystalline sheets beyond elastic wrinkling into the plastic regime, proposing that the athermal plastic response is a sequence of discrete, history-dependent shear instabilities and that the fracture mode can be selected by sheet geometry. The elastic check in Fig. 1(b), the geometric consistency check in Fig. 2(a), and the explicit tests of step-size and noise sensitivity are commendable features. However, the quantitative phase boundary and the reported fracture strains are not yet established: the phase diagram entries are single deterministic steepest-descent trajectories, the fracture strain varies strongly with step size, and Eq. (10) is asserted without derivation. These are load-bearing issues for the paper's quantitative claims, though the qualitative picture of intermittent plastic shear and the defect-free/defective distinction appears defensible.

major comments (3)
  1. [Sec. III B 4 and Table I] The central quantitative claims—the fracture strain Γf and the width-controlled defect-free/defective boundary—rest on single deterministic steepest-descent runs. The manuscript itself reports Γf = 0.749 ± 0.228 when the step size is varied (Sec. III B 4) and reports that adding noise at c0 = 0.1s shifts W0′ and even produces a nonmonotonic case at L0 = 15 (Sec. III D). Since each cell of Table I is one trajectory with no ensemble statistics, the phase diagram cannot currently be distinguished from the solver's path dependence. The authors should provide ensemble statistics over noise realizations or initial perturbations, report the distribution of fracture modes per geometry, and show that the defect-free/defective classification and Γf are stable under these variations.
  2. [Sec. III B 1, Eq. (10)] Eq. (10) is introduced with 'geometric arguments show' but no derivation is given. The quantities Ns and ℓ(Γ) are measured from the same simulations that produce θ, so the agreement between the black and red curves in Fig. 2(a) is a consistency check rather than an independent validation. To make the geometric model load-bearing, the authors should derive Eq. (10) from the step and lattice geometry, state the assumptions explicitly, and ideally test it on configurations not used to extract Ns and ℓ.
  3. [Sec. II and Sec. III D] The premise that a 0.7% incremental expansion followed by steepest-descent relaxation produces the physically relevant plastic response is not adequately validated. The paper shows that Γ1 is robust (0.129 ± 0.008) but Γf is highly sensitive to step size, and the transition threshold W0′ shifts with a small noise amplitude. This does not refute the qualitative intermittency, but it means the quantitative phase boundary and Γf are not robust predictions. The authors should either use a more controlled sampling of the energy landscape (e.g., multiple random initial perturbations, conjugate-gradient minimization from several seeds, or low-temperature Langevin dynamics) or explicitly frame the results as protocol-specific observations rather than material properties.
minor comments (5)
  1. [Sec. III B 4] The 'statistical analysis' that yields Γ1 = 0.126 ± 0.008 appears to be the scatter across a small set of geometries; the number of samples and the nature of the distribution should be stated.
  2. [Table I and Fig. 3(a)] The defect-free and defect-based categories are distinguished by blue and black font colors; if the paper is printed in grayscale, this distinction will be lost, so an additional symbol or label should be used.
  3. [Sec. III B 1] The fracture criterion (neck width less than about two lattice spacings) is practical but arbitrary; the sensitivity of Γf to this cutoff should be discussed, since a different threshold will change the reported fracture strains.
  4. [Sec. III B 3] The statement that the system 'breaks the originally Ck symmetry of the system' is not fully defined in the text or the figure caption; the integer k and the meaning of the symmetry should be clarified.
  5. [Sec. III C] The difference between the defect states Sd and Sd′ is clear only from the figures; a brief textual definition or a small table of state symbols would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central results are empirical simulations plus a consistency-check geometric relation, and the self-citations are not load-bearing.

full rationale

After walking the derivation chain, I find no circular step that reduces a prediction to its inputs. The central claims—intermittent plastic shear events, the defect-free versus defective fracture classification, and the width-controlled transition—are based on direct simulation observations and are not produced by fitting a parameter to the target quantity. The geometric model in Eq. (10) is the closest item to a 'prediction': the paper states that 'at each value of Γ, we calculate the mean bond length of the stretched lattice in mechanical equilibrium for the value of the quantity ℓ(Γ), and count the total number of steps Ns,' and then compares the resulting sinθ to the measured tilt angle. This is a consistency check of a geometric identity, not an independent prediction, and it involves no fitted parameters; the step count Ns and bond length ℓ are measured from the same configurations, so the agreement validates the geometric bookkeeping rather than deriving a new physical result from independent inputs. The classification into defect-free and defective categories is definitional, but the substantive width-dependent transition is an empirical Table I observation, not a tautology. The self-citations (Refs. 24 and 25) supply the Poisson ratio and a comparison vortex phenomenon; both are accompanied by independent references (Refs. 48-49) or are not load-bearing for the claimed results. The sensitivity of Γf and W0′ to step size and noise is a protocol-robustness concern, which falls under correctness risk rather than circularity. Therefore the circularity score is 0.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central results rest on a standard LJ model, a zero-temperature relaxation protocol, and continuum concepts such as linear elasticity, Peach-Koehler force, and Griffith cracks imported to the discrete lattice. The only hand-chosen numerical parameters are the expansion increment, relaxation step size, fracture criterion, and noise amplitude; no parameter is fitted to experimental or external data. No new physical entities are postulated.

free parameters (4)
  • Expansion increment p = 0.7% per step
    Chosen to approximate quasi-static expansion; it sets the strain discretization and affects which energy minima are reached. Not fitted to data.
  • Steepest descent step size s = 5×10^-4
    Chosen for energy relaxation; Sec. IIIB4 shows changing it from 5×10^-4 to 10^-3 changes Γf, so the numerical protocol influences fracture strain.
  • Fracture criterion = neck width < 2 lattice spacings
    Hand-chosen operational definition of complete fracture; changing it would shift Γf values.
  • Noise amplitude c0 = 0.1s
    Ad hoc noise level in Sec. IIID; affects the defect-free to defective crossover width W0'.
assumptions (6)
  • domain assumption Triangular LJ lattice can be treated as an isotropic 2D elastic sheet with Poisson ratio σ=1/3.
    Used in Sec. IIIA for the strain field and in Eq. (11) for bond dispersion; the value is cited from Refs. [24,48,49], not measured here.
  • standard math Cylindrical surface has zero Gaussian curvature, so it maps isometrically to the plane and planar elasticity applies.
    Invoked in Sec. IIIA to justify Cartesian coordinates on the unfolded sheet; standard differential geometry.
  • domain assumption Steepest descent relaxation after each 0.7% expansion finds the mechanically relevant minimum energy state at zero temperature.
    Defines the quasistatic protocol in Sec. II; it excludes thermal activation and rate effects, which is the paper's main modeling assumption.
  • domain assumption Peach-Koehler force and Griffith crack propagation criteria, derived for continua, apply to the discrete LJ lattice.
    Used in Sec. IIIB3 for dislocation glide and in Sec. IIIC for vacancy growth; a transfer of continuum theory to a small lattice.
  • standard math Delaunay triangulation of relaxed configurations correctly identifies coordination and disclinations.
    Standard topological analysis described in Sec. II; used to define defect states and Burgers vectors.
  • domain assumption Lennard-Jones potential represents covalently bonded particulate packings.
    Model choice stated in Sec. I; affects relevance to real materials but not the internal mechanics of the simulation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Defect-free and defective adaptations of crystalline sheets to stretching deformation." pith.science (2026). https://pith.science/paper/2PYJ3D3A

@misc{pith2026250605784,
  author       = {Pith},
  title        = {Pith review of: Defect-free and defective adaptations of crystalline sheets to stretching deformation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2PYJ3D3A}},
  note         = {Machine review of arXiv:2506.05784}
}
read the original abstract

The elastic response of the crystalline sheet to the stretching deformation in the form of wrinkles has been extensively investigated. In this work, we extend this fundamental scientific question to the plastic regime by exploring the adaptations of crystalline sheets to the large uniaxial mechanical stretching. We reveal the intermittent plastic shear deformations leading to the complete fracture of the sheets wrapping the cylinder. Specifically, systematic investigations of crystalline sheets of varying geometry show that the fracture processes can be classified into defect-free and defective categories depending on the emergence of topological defects. We highlight the characteristic mechanical and geometric patterns in response to the large stretching deformation, including the shear-driven intermittent lattice tilting, the vortex structure in the displacement field, and the emergence of mobile and anchored dislocations as plastic excitations. The effects of noise and initial lattice orientation on the plastic deformation of the stretched crystalline sheet are also discussed. These results advance our understanding of the atomic level on the irreversible plastic instabilities of 2D crystals under large uniaxial stretching and may have potential practical implications in the precise engineering of structural instabilities in packings of covalently bonded particulate systems.

Figures

Figures reproduced from arXiv: 2506.05784 by the authors.

Figure 1
Figure 1. FIG. 1: The schematic plot of the model system and the pre [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The adaptation of the crystalline sheet to the gradual expansion of the cylinder leads to the intermittent plastic shear [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Characterization of the intermittent plastic deforma [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Characteristic defect structures define the stable states in the fracture process of the crystalline sheet. (a)-(c) The [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Effect of noise on the fracture of the crystalline sheet. [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The abrupt tilting transition of the crystalline lattices initially at the tilt angle of 30 degrees. The table shows the [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

63 extracted references · 59 canonical work pages

  1. [1]

    Shear-driven lattice tilting To explore the response of the crystalline sheet upon the gradual expansion of the cylinder, we first analyze a typical example case presented in Fig. 2. An important observation is the tilting of the strongly stretched crys- talline lattice. The tilting phenomenon is quantified in the plot of the tilt angle θ against the expa...

    work page

  2. [2]

    Geometric analysis of bond length and displacement field We proceed to analyze the variation of the bond length in the plastic shear deformation of the case in Fig. 2(a). In the upper panel in Fig. 2(b), the mean bond length ⟨ℓ⟩ is plotted against the expansion factor Γ. The standard deviation of the bond length distribution is indicated by the error bars...

    work page

  3. [3]

    Glide of dislocations in intermediate states In the mechanical relaxation process for the case pre- sented in Fig. 2, dislocations are observed within the shear band in the intermediate states; the intermediate states refer to the particle configurations in the mechan- ical relaxation process of the system at a given value of the expansion factor Γ. The r...

    work page

  4. [4]

    Characterization of the intermittency of plastic deformations We systematically investigate the plastic deformations in the expansion of crystalline sheets of varying geometry. The value of Γ is recorded for each plastic event in the expansion process, including the formation of steps on the boundary of the sheet and the relocation of particles in the int...

    work page

  5. [5]

    Cerda and L

    E. Cerda and L. Mahadevan, Phys. Rev. Lett. 90, 074302 (2003)

    work page 2003

  6. [6]

    Marder, R

    M. Marder, R. D. Deegan, and E. Sharon, Physics Today 60, 33 (2007)

    work page 2007

  7. [7]

    Audoly and Y

    B. Audoly and Y. Pomeau, Elasticity and Geometry (Ox- ford University Press, Oxford, 2010)

    work page 2010

  8. [8]

    Chopin, V

    J. Chopin, V. D´ emery, and B. Davidovitch, J. Elast.119, 137 (2015)

    work page 2015

Show all 63 references
  1. [9]

    Yu and L

    T. Yu and L. Zhang, Int. J. Mech. Sci. 28, 729 (1986)

    1986

  2. [10]

    G´ eminard, R

    J.-C. G´ eminard, R. Bernal, and F. Melo, Eur. Phys. J. E 15, 117 (2004)

    2004

  3. [11]

    A. K. Harris, P. Wild, and D. Stopak, Science 208, 177 (1980)

    1980

  4. [12]

    Cerda, K

    E. Cerda, K. Ravi-Chandar, and L. Mahadevan, Nature 419, 579 (2002)

    2002

  5. [13]

    S. P. Lacour, S. Wagner, Z. Huang, and Z. Suo, Appl. Phys. Lett. 82, 2404 (2003)

    2003

  6. [14]

    T. Wang, Y. Yang, and F. Xu, Proc. R. Soc. London, Ser. A 478, 20220149 (2022)

    2022

  7. [15]

    Sharon, B

    E. Sharon, B. Roman, and H. L. Swinney, Phys. Rev. E 75, 046211 (2007)

    2007

  8. [16]

    Li, Y.-P

    B. Li, Y.-P. Cao, X.-Q. Feng, and H. Gao, Soft Matter 8, 5728 (2012)

    2012

  9. [17]

    Friedl, F

    N. Friedl, F. G. Rammerstorfer, and F. D. Fischer, Com- put. Struct. 78, 185 (2000)

    2000

  10. [18]

    Lecieux and R

    Y. Lecieux and R. Bouzidi, Adv. Eng. Softw. 50, 57 (2012)

    2012

  11. [19]

    C. Wang, L. Lan, and H. Tan, Phys. Chem. Chem. Phys. 15, 2764 (2013)

    2013

  12. [20]

    Puntel, L

    E. Puntel, L. Deseri, and E. Fried, J. Elast. 105, 137 (2011)

    2011

  13. [21]

    G. M. Grason and B. Davidovitch, Proc. Natl. Acad. Sci. U.S.A. 110, 12893 (2013)

    2013

  14. [22]

    Chopin and A

    J. Chopin and A. Kudrolli, Soft Matter 12, 4457 (2016)

    2016

  15. [23]

    Jules, F

    T. Jules, F. Lechenault, and M. Adda-Bedia, Phys. Rev. E 102, 033005 (2020)

    2020

  16. [24]

    Miguel, A

    M.-C. Miguel, A. Mughal, and S. Zapperi, Phys. Rev. Lett. 106, 245501 (2011)

    2011

  17. [25]

    Negri, A

    C. Negri, A. L. Sellerio, S. Zapperi, and M.-C. Miguel, Proc. Natl. Acad. Sci. 112, 14545 (2015)

    2015

  18. [26]

    N. P. Mitchell, V. Koning, V. Vitelli, and W. Irvine, Nat. Mater. 16, 89 (2017)

    2017

  19. [27]

    M. Liu, S. Yu, L. He, and Y. Ni, Soft Matter 18, 5906 (2022)

    2022

  20. [28]

    Chen and Z

    J. Chen and Z. Yao, Soft Matter 18, 5323 (2022)

    2022

  21. [29]

    Sun and Z

    H. Sun and Z. Yao, Phys. Rev. E 109, 044802 (2024)

    2024

  22. [30]

    Politi, G

    P. Politi, G. Grenet, A. Marty, A. Ponchet, and J. Villain, Phys. Rep. 324, 271 (2000)

    2000

  23. [31]

    G. Meng, J. Paulose, D. R. Nelson, and V. N. Manoha- ran, Science 343, 634 (2014)

    2014

  24. [32]

    K¨ ohler, R

    C. K¨ ohler, R. Backofen, and A. Voigt, Europhys. Lett. 111, 48006 (2015)

    2015

  25. [33]

    L. Ma, X. Liu, A.-k. Soh, L. He, C. Wu, and Y. Ni, Soft Matter 15, 4391 (2019)

    2019

  26. [34]

    Dinsmore, M

    A. Dinsmore, M. F. Hsu, M. Nikolaides, M. Marquez, A. Bausch, and D. Weitz, Science 298, 1006 (2002)

    2002

  27. [35]

    Palacci, S

    J. Palacci, S. Sacanna, A. P. Steinberg, D. J. Pine, and P. M. Chaikin, Science 339, 936 (2013)

    2013

  28. [36]

    Bowick and P

    M. Bowick and P. Chaikin, Nat. Mater. 15, 1151 (2016)

    2016

  29. [37]

    D. L. Caspar and A. Klug, in Cold Spring Harbor Sym- posia on Quantitative Biology (Cold Spring Harbor Lab- oratory Press, 1962), vol. 27, pp. 1–24

    1962

  30. [38]

    Lidmar, L

    J. Lidmar, L. Mirny, and D. R. Nelson, Phys. Rev. E 68, 051910 (2003)

    2003

  31. [39]

    W. Roos, R. Bruinsma, and G. Wuite, Nat. Phys. 6, 733 (2010)

    2010

  32. [40]

    J. E. Jones, Proc. R. Soc. London, Ser. A. 106, 463 (1924)

    1924

  33. [41]

    J. N. Israelachvili, Intermolecular and Surface Forces (Academic Press, New York, 2011), 3rd ed

    2011

  34. [42]

    L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 3rd ed. (Butterworth-Heinemann, Oxford, UK, 1986)

    1986

  35. [43]

    Mughal, H

    A. Mughal, H. Chan, D. Weaire, and S. Hutzler, Phys. Rev. E 85, 051305 (2012)

    2012

  36. [44]

    Amir and D

    A. Amir and D. R. Nelson, Proc. Natl. Acad. Sci. U.S.A. 109, 9833 (2012)

    2012

  37. [45]

    D. R. Nelson, Annu. Rev. Biophys. 41, 371 (2012)

    2012

  38. [46]

    A. Amir, J. Paulose, and D. R. Nelson, Phys. Rev. E 87, 042314 (2013)

    2013

  39. [47]

    Mughal and D

    A. Mughal and D. Weaire, Phys. Rev. E 89, 042307 (2014)

    2014

  40. [48]

    D. A. Beller and D. R. Nelson, Phys. Rev. E 94, 033004 (2016)

    2016

  41. [49]

    J. A. Snyman and D. N. Wilke, Practical Mathematical Optimization (Springer, New York, 2005)

    2005

  42. [50]

    D. R. Nelson, Defects and Geometry in Condensed Matter Physics (Cambridge University Press, Cambridge, 2002)

    2002

  43. [51]

    Struik, Lectures on Classical Differential Geometry (Dover Publications, New York, 1988), 2nd ed

    D. Struik, Lectures on Classical Differential Geometry (Dover Publications, New York, 1988), 2nd ed

    1988

  44. [52]

    H. S. Seung and D. R. Nelson, Phys. Rev. A 38, 1005 (1988)

    1988

  45. [53]

    Berinskii and H

    I. Berinskii and H. Altenbach, Acta Mech. 228, 683 (2017)

    2017

  46. [54]

    Asaro and W

    R. Asaro and W. Tiller, Metall. Mater. Trans. B. 3, 1789 (1972)

    1972

  47. [55]

    Grinfeld, in Dok

    M. Grinfeld, in Dok. Akad. Nauk SSSR 290, 1358 (1986) [Sov. Phys. Dokl. 31, 831 (1986)]

    1986

  48. [56]

    D. J. Srolovitz, Acta Metall. 37, 621 (1989)

    1989

  49. [57]

    J. P. Hirth and J. Lothe, Theory of Dislocations (John Wiley & Sons, 1982)

    1982

  50. [58]

    Peach and J

    M. Peach and J. Koehler, Phys. Rev. 80, 436 (1950)

    1950

  51. [59]

    A. A. Griffith, Philos. Trans. R. Soc. London A 221, 163 (1921)

    1921

  52. [60]

    Ippolito, A

    M. Ippolito, A. Mattoni, L. Colombo, and N. Pugno, Phys. Rev. B 73, 104111 (2006)

    2006

  53. [61]

    K. S. Cheung and S. Yip, Phys. Rev. Lett. 65, 2804 (1990)

    1990

  54. [62]

    Mattoni, L

    A. Mattoni, L. Colombo, and F. Cleri, Phys. Rev. Lett. 95, 115501 (2005)

    2005

  55. [63]

    Timoshenko and J

    S. Timoshenko and J. Goodier, Theory of Elasticity 14 (McGraw-Hill Book Company, 1951)

    1951

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.