Uniform distribution of dislocations in Peierls-Nabarro models for semi-coherent interfaces

Marcello Ponsiglione; Riccardo Scala; Silvio Fanzon

Uniform distribution of dislocations in Peierls-Nabarro models for semi-coherent interfaces

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read Edge dislocations at a semi-coherent interface prefer uniform spacing, and the rescaled energy Γ-converges to a constant plus the H^{1/2} seminorm.

desk verdict Solid variational-materials paper whose central claims hold up; the reader's alleged algebraic flaw in Theorem 4.2 does not survive re-derivation. read the letter →

arxiv 1908.04222 v2 pith:CEGJCAUV submitted 2019-08-12 math.AP

Silvio Fanzon , Marcello Ponsiglione , Riccardo Scala This is my paper · ORCID

classification math.AP MSC 74N0574N1549J45

keywords Peierls-Nabarromodeledgedislocationssemi-coherentinterfacesGamma-convergenceH^{1/2}seminormuniformdislocationdistributionvariationalmethodsdensity

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 paper proves a variational answer to a classical question: when two crystals with slightly different lattice spacings meet, where do the dislocations that accommodate the mismatch prefer to sit? In a simplified Peierls–Nabarro model whose energy is the $H^{1/2}$ seminorm of a scalar interfacial displacement, the answer is asymptotic uniformity. As the interface length tends to infinity, the rescaled energy $\Gamma$-converges to a limit made of a positive constant $c_\infty$, the minimal energy per unit length of the dislocation array, plus the $H^{1/2}$ seminorm of any extra macroscopic displacement. Minimizers' dislocation densities converge weakly-* to the constant $\lambda/(\delta(\lambda+\Lambda))$, and in a periodic one-dimensional circle model with a core cutoff every minimizer is exactly evenly spaced. The interest is that the paper reaches this pattern without assuming periodicity in advance.

What carries the argument

The load-bearing object is the $H^{1/2}$ seminorm of the interfacial displacement, restricted by admissibility: the derivative $u'$ takes only the two values $\lambda$ (elastic matching) and $-\Lambda$ (dislocation core), and the cores have fixed length $\delta$. The proof machinery first rescales the interface length by $l$ and the amplitude by $\sqrt{l}$, so that the energy per unit length stays finite; then the double integral is split into short-range diagonal blocks, whose limit contributes the constant $c_\infty$, and off-diagonal terms, which pass to the limiting $H^{1/2}$ seminorm. In the circle model the energy is recast as a sum of convex functions $f(y)=-\log|y|+2|y|$ of the pairwise spacings, and Jensen's inequality forces equal spacings.

What would settle it

Run a two-dimensional linear-elastic relaxation of a finite semi-coherent interface with the standard symmetrized-gradient energy, record the dislocation positions for growing interface length, and check whether their density approaches $\lambda/(\delta(\lambda+\Lambda))$. If it does not, the $H^{1/2}$-seminorm simplification is the point of failure; within the paper's own model, replacing the kernel $|x-y|^{-2}$ by $|x-y|^{-2s}$ with $s\ne 1/2$ would test whether even spacing survives the change of nonlocality.

Watch

Extended reading notes

Core claim

The central discovery is that uniform spacing of dislocations is a consequence of the model, not an assumption. Theorem 3.1 states that the rescaled functionals $F_l(w)=\int_0^1\int_0^1 |w(x)-w(y)|^2/|x-y|^2\,dx\,dy$ $\Gamma$-converge, as $l\to\infty$, to $F_\infty(w)=c_\infty+\int_0^1\int_0^1 |w(x)-w(y)|^2/|x-y|^2\,dx\,dy$ for $w\in H^{1/2}(0,1)$, and $+\infty$ otherwise, where $c_\infty>0$ is the limit of the minimal energies per unit length. Theorem 2.6 identifies the asymptotic dislocation density: for minimizers, $\mu_l=\frac{1}{l}\sum_{i=1}^{N_l} \delta_{x_i/l}$ converges weakly-* to $\lambda/(\delta(\lambda+\Lambda))$. In the simplified periodic setting of Section 4, Theorem 4.2 shows that among $N$ dislocations on the circle with minimal separation $\rho$, every minimizer is evenly spaced at distance $1/N$. Together these results give a rigorous derivation of the periodic-uniform dislocation arrays that are routinely assumed in the physical literature.

Load-bearing premise

The argument stands on identifying the elastic energy of the crystal with the $H^{1/2}$ seminorm of a scalar interfacial displacement; if the true energy depends on the symmetrized gradient of a two-dimensional strain and couples shear components, the uniform-array conclusion need not follow.

Editorial extensions

If this is right

For a long interface, the minimum energy per unit length has a well-defined limit $c_\infty$, so bulk boundary conditions do not affect the cost of the dislocation array.
Any minimizer's dislocation density becomes uniform in the limit, with the value $\lambda/(\delta(\lambda+\Lambda))$ fixed by the lattice mismatch and the core length.
The limiting energy separates into the constant $c_\infty$ plus the $H^{1/2}$ seminorm of the macroscopic displacement, so any further dislocations or strain beyond the uniform array are penalized by exactly that seminorm.
In the periodic-circle submodel with $N$ dislocations and a core cutoff, every minimizer is evenly spaced at mutual distance $1/N$.

Reading between the lines

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

If the same rescaling is applied to fractional kernels $|x-y|^{-2s}$, the uniform-density phenomenon may persist, but the exact even-spacing theorem on the circle relies on convexity of $-\log|y|+2|y|$ and is likely special to $s=1/2$; testing $s\ne 1/2$ would delineate the mechanism.
The limiting decomposition suggests a practical two-scale computational recipe not spelled out in the paper: compute $c_\infty$ once from a periodic cell problem, then solve a continuum $H^{1/2}$ variational problem for the far-field displacement.
Theorem 2.6 establishes only weak-* convergence of the dislocation density; proving genuine periodicity of the limiting array would require controlling the boundary layers near $0$ and $l$ that the authors leave open, for instance by a stronger compactness argument for minimizers.
The paper's comparison with phase-separation energies suggests a broader conjecture: fractional-order nonlocal repulsion at a critical exponent may generically select periodic patterns, and the same method might extend to two-dimensional interfaces where dislocations form networks rather than lines.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

What you should know: this is a careful, honest paper, and the one substantive objection in the reader's report—an algebraic error in Theorem 4.2—does not hold up. I re-derived equations (69), (70), and (71) by hand. Starting from the first variation in (69), substituting the definition of Δ(xi) and collecting terms gives exactly the expression in (70): for right neighbors you get 4(1−2r)/r, for left neighbors 4(−1−2l)/|l|. Differentiating (71) with respect to xi gives the same, because each ordered pair contributes twice and the outer factor 2 turns the per-pair derivative into the same 4(1/d−2) or 4(−1/d+2). The computation is correct as written.

What is actually new: the paper gives a rigorous Gamma-convergence analysis of a Peierls-Nabarro-type model for semi-coherent interfaces, showing that the rescaled energy converges to a limit consisting of a positive constant c∞ plus the H^{1/2} seminorm of the macroscopic displacement. It also proves that minimizers have uniformly distributed dislocation densities in the weak-* sense, and, in the simplified periodic circle model, that evenly spaced dislocations are the unique minimizers. No free parameters are fitted; c∞ is defined as a limit of minimal energies, not used as an adjustable input. That is a real, nontrivial step.

The soft spots are real but proportionate. The stored elastic energy is replaced by the H^{1/2} seminorm of the interfacial displacement, and the authors state plainly that this is a mathematical simplification. So the uniform-array conclusion is rigorously tied to the model energy, not to a full symmetrized-gradient elastic energy. The Gamma-limsup construction is intricate and several admissibility checks are left to the reader; they look routine, but a referee should ask for them to be written out. The paper also honestly notes that true periodicity of dislocations in the original model remains open—the equi-spacing result lives in the simplified S^1 model with a cutoff. None of this undermines the main theorems.

Who this is for: anyone working in variational models of dislocations, Gamma-convergence, or pattern formation in semi-coherent interfaces. It deserves a serious referee. I would not desk-reject it; I would send it to review, and I would expect it to be accepted after the deferred checks are filled in.

Referee Report

0 major / 5 minor

Summary. The paper introduces a variational Peierls-Nabarro-type model for edge dislocations at semi-coherent interfaces, in which the interfacial displacement u satisfies u' in {lambda, -Lambda} and dislocation cores have a fixed length delta. The main results are: existence of the asymptotic minimal energy c_infinity (Theorem 2.3); weak-* convergence of the normalized dislocation densities of minimizers to the uniform density Lambda/(delta(Lambda+lambda)) (Theorem 2.6); Gamma-convergence of the rescaled H^{1/2} energies F_l to F_infinity = c_infinity + H^{1/2} seminorm (Theorem 3.1); and, in a periodic circle model with a core cutoff rho and Lambda tending to infinity, the fact that every minimizer is evenly spaced on the circle (Theorem 4.2). The authors state explicitly that the H^{1/2} energy is a mathematical simplification of linearized elasticity and that proving true periodicity in the original interval model remains open.

Significance. If the results hold, they provide a rigorous Gamma-convergence derivation of uniform dislocation spacing in a simplified Peierls-Nabarro model, connecting the asymptotic energy constant c_infinity to uniformly distributed dislocation arrays and, in the periodic setting, to even spacing on S^1. The proof is largely self-contained and the model limitations are honestly declared, which is a strength. I checked the stress-test concern about the first-variation computation in Theorem 4.2: the reader's alleged algebraic error does not survive re-derivation, since (70) is algebraically equivalent to (69) and matches the derivative of (71). Within the explicitly stated scope of the model energy, I found no load-bearing mathematical error. The physical scope is limited by the H^{1/2} scalar simplification, but this is a modeling limitation declared by the authors rather than an internal inconsistency.

minor comments (5)

[Section 3.2, after (32)] The admissibility of the recovery sequence g_l is asserted with the words 'the check is left to the reader.' Since this check is part of the proof of Theorem 3.1, a few sentences should be added verifying that g_l satisfies the derivative constraint in (9) and that the inserted points N_wl are separated from 1/l X_wl in the sense of (8). The claim is fillable from properties i)-iii), but it should not be left implicit.
[Section 4.2, Step 2] The statement that the quantity in (70) 'coincides with the partial derivative' of (71) is sufficient only if one also explains why the two functionals differ by a constant on the relevant connected components of the configuration space. The authors should state that on each chamber of configurations with a fixed cyclic order the open set {d(x_i,x_j)>rho} is connected and that the equality of first variations extends by continuity to the closure, so that minimality of E^N_rho follows from minimality of tilde E^N_rho.
[Section 3.2, first paragraph] The reduction to piecewise-affine w with alpha_i in R\{0} should be justified explicitly, because the construction of the points N_wl in property ii) only covers intervals with nonzero slope. A standard density argument in H^{1/2} can handle zero-slope intervals, but it is not written.
[Global] There are several typographical errors that should be corrected: 'atsemi-coherent' in the abstract, 'whithin' in Section 1.3, and 'enstablishes' in Section 3.2.
[Section 4.2, Step 1] The sentence 'the other case is similar and will yield the same result' for epsilon<0 is acceptable, but the symmetry used in passing from I_rho to 2 integral over (rho,1/2) in (60) should be spelled out, since the sign conventions in this first-variation computation are delicate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: c_infinity is recovered from the same functional by Gamma-convergence, not fitted; uniform spacing is derived, not assumed.

full rationale

The paper's central result is a Gamma-convergence theorem for the rescaled H^{1/2} seminorm functionals F_l(w), and the constant c_infinity in the limit functional is not a fitted or externally imposed input: it is defined in Theorem 2.3 as the limit of the minimal energies c_l = min_u (1/l) E_l(u), and the Gamma-limit proof recovers this same constant from the diagonal energy blocks, so no parameter is adjusted to force the conclusion. The uniform dislocation density lambda/(delta(lambda+Lambda)) in Theorem 2.6 is a kinematic consequence of the derivative constraint u' = lambda - (lambda+Lambda) chi and the weak convergence of minimizers to zero, not an ansatz. Theorem 4.2 is established by computing the first variation of E_N^rho and matching it with the derivative of the explicitly written functional ~E_N^rho in equations (69)-(71), then minimizing each convex term G_k by Jensen; the algebra is internally consistent, because each neighbor pair appears twice and the outer factor 2 yields 4(1/d-2) for right neighbors and 4(-1/d+2) for left neighbors. The only self-citations, [6] and [7], are contextual references to related prior work and are not load-bearing: the model, admissible classes, and convergence proofs are self-contained. The paper explicitly declares the H^{1/2} simplification a 'mere mathematical simplification' and states that proving true periodicity in the original model remains open; these are honest modeling limitations and scope caveats, not circular steps.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

All parameters lambda, Lambda, delta are prescribed by the lattice geometry, not fitted. The central claim depends on modeling choices: the scalar H^{1/2} energy in place of symmetrized linear elasticity, the two-slope admissible displacement class with fixed core length delta, and in Section 4 the sharp-interface cutoff rho and the relation delta=lambda/(N(lambda+Lambda)). No new physical entities are introduced. These assumptions are stated explicitly in the paper, but they limit the physical scope.

assumptions (5)

ad hoc to paper The physical elastic energy of a crystal can be replaced by the H^{1/2} seminorm of the scalar interfacial displacement u, rather than an energy depending on the symmetrized gradient of a two-dimensional displacement field.
Invoked in the Introduction and Section 1.2 as the stored elastic energy; the authors call it a mere mathematical simplification. All results concern this model energy.
domain assumption Admissible displacements have u' in {lambda, -Lambda} on intervals, with dislocation cores, where u'=-Lambda, being disjoint intervals of fixed length delta, while no minimal length is imposed on elastic intervals where u'=lambda.
Definitions (8), (9), and (16) in Section 2.1 encode the lattice mismatch and core-radius structure. This two-slope constraint and fixed core size are essential to the scaling and to c_infinity>0.
ad hoc to paper In the periodic circle model, the number of dislocations N is linked to the core size by delta=lambda/(N(lambda+Lambda)), and the limit Lambda to infinity with N fixed describes the semi-coherent to coherent transition.
Equation (45) in Section 4.1 enforces periodicity of the displacement and a fixed dislocation count. It is specific to the simplified S1 setting.
domain assumption In the sharp-interface circle model, the limiting energy is computed with a fixed cutoff rho>0 around each dislocation core, and the step function h has jumps -lambda/N at each dislocation.
Equations (51) through (54). The cutoff avoids the infinite H^{1/2} energy of a step function, and the minimizer statement is for the energy E^N_rho.
standard math Standard functional analysis facts hold: compactness and lower semicontinuity of H^{1/2}, Poincare-Wirtinger, BV compactness, Jensen's inequality, and strict convexity of f(y)=max(-log y, -log(1-y)) + 2 min(y,1-y) on (0,1).
Used throughout Sections 2 through 4, for example in Proposition 2.5, Theorem 2.6, and Step 3 of Theorem 4.2.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Uniform distribution of dislocations in Peierls-Nabarro models for semi-coherent interfaces." pith.science (2026). https://pith.science/paper/CEGJCAUV

@misc{pith2026190804222,
  author       = {Pith},
  title        = {Pith review of: Uniform distribution of dislocations in Peierls-Nabarro models for semi-coherent interfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CEGJCAUV}},
  note         = {Machine review of arXiv:1908.04222}
}

abstract

In this paper we introduce Peierls-Nabarro type models for edge dislocations at semi-coherent interfaces between two heterogeneous crystals, and prove the optimality of uniformly distributed edge dislocations. Specifically, we show that the elastic energy $\Gamma$-converges to a limit functional comprised of two contributions: one is given by a constant $c_\infty>0$ gauging the minimal energy induced by dislocations at the interface, and corresponding to a uniform distribution of edge dislocations; the other one accounts for the far field elastic energy induced by the presence of further, possibly not uniformly distributed, dislocations. After assuming periodic boundary conditions and formally considering the limit from semi-coherent to coherent interfaces, we show that $c_\infty$ is reached when dislocations are evenly-spaced on the one dimensional circle.

Figures

Figures reproduced from arXiv: 1908.04222 by the authors.

**Figure 1.** Left: reference configuration. Top Right: purely elastic deformation, with relative displacement (∆ − δ)/2. Bottom Right: deformation leading to an edge dislocation, with relative displacement ∆ 4 − δ 2 . Here ∆ 2 and δ 2 are the lattice spacing of C − and C +, respectively, and the convenience of the prefactor 1 2 will be commented later on ( [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

28 extracted references · 27 canonical work pages

[1]

Alberti, G

G. Alberti, G. Bouchitté, and P. Seppecher. Phase Transition with the Line-Tension Eﬀect.Arch. Rational Mech. Anal., 144:1–46, 1998

work page 1998
[2]

Alicandro, G

R. Alicandro, G. Lazzaroni, and M. Palombaro. On the eﬀect of interactions beyond nearest neigh- bours on non-convex lattice systems.Calc. Var. Partial Diﬀerential Equations, 56(42), 2017

work page 2017
[3]

Alicandro, G

R. Alicandro, G. Lazzaroni, and M. Palombaro. Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours.Netw. Heterog. Media, 13:1–26, 2018

work page 2018
[4]

Ariza and M

M. Ariza and M. Ortiz. Discrete crystal elasticity and discrete dislocations in crystals.Arch. Rational Mech. Anal., 178(2):149–226, 2005

work page 2005
[5]

Dipierro, G

S. Dipierro, G. Palatucci, and E. Valdinoci. Dislocation dynamics in crystals: a macroscopic theory in a fractional laplace setting.Comm. Math. Phys., 333(1061-1105), 2015

work page 2015
[6]

Fanzon, M

S. Fanzon, M. Palombaro, and M. Ponsiglione. A Variational Model for Dislocations at Semi-coherent Interfaces. Journal of Nonlinear Science, 27(5):1436–1461, 2017

work page 2017
[7]

Fanzon, M

S. Fanzon, M. Palombaro, and M. Ponsiglione. Derivation of Linearised Polycrystals from a Two- Dimensional System of Edge Dislocations.SIAM J. Math. Anal., 51(5):3956–3981, 2019

work page 2019
[8]

Focardi and A

M. Focardi and A. Garroni. A1D macroscopic phase ﬁeld model for dislocations and a second order Γ-limit. SIAM Multiscale Model. Simul., 6(4):1098–1124, 2007. UNIFORM DISTRIBUTION OF DISLOCATIONS AT SEMI-COHERENT INTERF ACES 27

work page 2007

Show all 28 references

[9]

Fonseca, N

I. Fonseca, N. Fusco, G. Leoni, and M. Morini. A model for dislocations in epitaxially strained elastic ﬁlms.J. Math. Pures Appl., 111(9):126–160, 2018

2018
[10]

M.FriedrichandU.Stefanelli.Crystallizationinaone-dimensionalperiodiclandscape. J. Stat. Phys., 2019 (To appear)

2019
[11]

Garroni and S

A. Garroni and S. Müller.Γ-limit of a Phase-Field Model of Dislocations.SIAM Journal on Mathe- matical Analysis, 36(6):1943–1964, 2005

1943
[12]

Garroni and S

A. Garroni and S. Müller. A model for dislocations in epitaxially strained elastic ﬁlms.Arch. Rational Mech. Anal., 181(3):535–578, 2006

2006
[13]

Giuliani and F

A. Giuliani and F. Theil. Long range order in atomistic models for solids. arXiv e-prints, arXiv:1907.07923, 2019

1907 arXiv
[14]

González and R

M. González and R. Monneau. Slow motion of particle systems as a limit of a reaction-diﬀusion equation with half-laplacian in dimension one.Discrete Contin. Dyn. Syst., 32:1255–1286, 2012

2012
[15]

P. Hirsch. Nucleation and propagation of misﬁts dislocations in strained epitaxial layer systems. In Proceedings of the Second International Conference Schwäbisch Hall, Fed. Rep. of Germany, July 30-August 3 1990

1990
[16]

Kreutz and P

L. Kreutz and P. Piovano. Microscopic validation of a variational model of epitaxially strained crystalline ﬁlm.arXiv e-prints, arXiv:1902.06561, 2019

1902 arXiv
[17]

Lauteri and S

G. Lauteri and S. Luckhaus. An Energy Estimate for Dislocation Conﬁgurations and the Emergence of Cosserat-Type Structures in Metal Plasticity.arXiv e-prints, arXiv:1608.06155, 2016

2016 arXiv
[18]

Lazzaroni, M

G. Lazzaroni, M. Palombaro, and A. Schlömerkemper. Rigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires.Discrete Contin. Dyn. Syst. Ser. S, 10:119–139, 2017

2017
[19]

Mironescu and A

P. Mironescu and A. Pisante. A variational problem with lack of compactness forH 1/2(S1; S1) maps of prescribed degree.J. Funct. Anal., 217:249–279, 2004

2004
[20]

S. Müller. Singular perturbations as a selection criterion for periodic minimizing sequences.Calc. Var. Partial Diﬀerential Equations, 1:169–204, 1993

1993
[21]

Müller and M

S. Müller and M. Palombaro. Derivation of a rod theory for biphase materials with dislocations at the interface.Calculus of Variations and Partial Diﬀerential Equations, 48(3-4):315–335, 2013

2013
[22]

F. R. N. Nabarro. Dislocations in a simple cubic lattice.Proc. Phys. Soc., 59:256–272, 1947

1947
[23]

F. R. N. Nabarro.Theory of Crystal Dislocations. Clarendon Press, Oxford, 1967

1967
[24]

Ohta and K

T. Ohta and K. Kawasaki. Equilibrium morphology of block copolymer melts. Macromolecules, 19:2621–2632, 1986

1986
[25]

Patrizi and V

S. Patrizi and V. E. Crystal dislocations with diﬀerent orientations and collisions.Arch. Rational Mech. Anal., 217:231–261, 2015

2015
[26]

R. Peierls. The size of a dislocation.Proc. Phys. Soc., 52:34–37, 1940

1940
[27]

W. T. Read and W. Shockley. Dislocation models of crystal grain boundaries.Phys. Rev., 78:275–289, May 1950

1950
[28]

G. Castelnuovo

J. Van Der Merwe. On the stresses and energies associated with inter-crystalline boundaries.Proc. Phys. Soc. A, 63:616–637, 1950. (Silvio Fanzon)(Corresponding author) University of Graz, Institute of Mathematics and Scientific Computing, Heinrichstraße 36, 8010 Graz, Austria ...

1950

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Alberti, G

G. Alberti, G. Bouchitté, and P. Seppecher. Phase Transition with the Line-Tension Eﬀect.Arch. Rational Mech. Anal., 144:1–46, 1998

work page 1998

[2] [2]

Alicandro, G

R. Alicandro, G. Lazzaroni, and M. Palombaro. On the eﬀect of interactions beyond nearest neigh- bours on non-convex lattice systems.Calc. Var. Partial Diﬀerential Equations, 56(42), 2017

work page 2017

[3] [3]

Alicandro, G

R. Alicandro, G. Lazzaroni, and M. Palombaro. Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours.Netw. Heterog. Media, 13:1–26, 2018

work page 2018

[4] [4]

Ariza and M

M. Ariza and M. Ortiz. Discrete crystal elasticity and discrete dislocations in crystals.Arch. Rational Mech. Anal., 178(2):149–226, 2005

work page 2005

[5] [5]

Dipierro, G

S. Dipierro, G. Palatucci, and E. Valdinoci. Dislocation dynamics in crystals: a macroscopic theory in a fractional laplace setting.Comm. Math. Phys., 333(1061-1105), 2015

work page 2015

[6] [6]

Fanzon, M

S. Fanzon, M. Palombaro, and M. Ponsiglione. A Variational Model for Dislocations at Semi-coherent Interfaces. Journal of Nonlinear Science, 27(5):1436–1461, 2017

work page 2017

[7] [7]

Fanzon, M

S. Fanzon, M. Palombaro, and M. Ponsiglione. Derivation of Linearised Polycrystals from a Two- Dimensional System of Edge Dislocations.SIAM J. Math. Anal., 51(5):3956–3981, 2019

work page 2019

[8] [8]

Focardi and A

M. Focardi and A. Garroni. A1D macroscopic phase ﬁeld model for dislocations and a second order Γ-limit. SIAM Multiscale Model. Simul., 6(4):1098–1124, 2007. UNIFORM DISTRIBUTION OF DISLOCATIONS AT SEMI-COHERENT INTERF ACES 27

work page 2007