On the effect of interactions beyond nearest neighbours on non-convex lattice systems

Alicandro, Roberto; Lazzaroni, Giuliano; Palombaro, Mariapia

doi:10.1007/s00526-017-1129-5

Cited by 10 publications

(10 citation statements)

References 25 publications

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“…This is the subject of the last section of the paper. Our aim is to show that the role of the singular term in the continuum model is played in this setting by interactions beyond nearest neighbours, which prevent too many jumps from one well to another (see also [4]). We focus on the simple case of a two-dimensional discrete system governed by pairwise harmonic interactions between nearest and next-to-nearest neighbours, corresponding to a total interaction energy with two wells.…”

Section: Contentsmentioning

confidence: 99%

Derivation of a Linearised Elasticity Model from Singularly Perturbed Multiwell Energy Functionals

Alicandro

Maso

Lazzaroni

et al. 2018

Arch Rational Mech Anal

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Mariapia (2018) Derivation of a linearised elasticity model from singularly perturbed multiwell energy functionals.Abstract. Linear elasticity can be rigorously derived from finite elasticity under the assumption of small loadings in terms of Gamma-convergence. This was first done in the case of one-well energies with super-quadratic growth and later generalised to different settings, in particular to the case of multi-well energies where the distance between the wells is very small (comparable to the size of the load). In this paper we study the case when the distance between the wells is independent of the size of the load. In this context linear elasticity can be derived by adding to the multi-well energy a singular higher order term which penalises jumps from one well to another. The size of the singular term has to satisfy certain scaling assumptions whose optimality is shown in most of the cases. Finally, the derivation of linear elasticty from a two-well discrete model is provided, showing that the role of the singular perturbation term is played in this setting by interactions beyond nearest neighbours.

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Section: Contentsmentioning

confidence: 99%

Derivation of a Linearised Elasticity Model from Singularly Perturbed Multiwell Energy Functionals

Alicandro

Maso

Lazzaroni

et al. 2018

Arch Rational Mech Anal

Self Cite

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“…for some positive constants γ f and γ s depending on the film and substrate material, respectively. Notice that it would be equivalent to introduce a dependence on discrete deformations y also in the definition of E S ε by considering the sum in (2) as extended over the elements of the deformed lattice y(L ε (Ω h )), if we restrict to small deformations y, i.e., deformations y that do not change the topology of the lattice or, in other words, such that #N ε (y(x)) = #N ε (x) for every x ∈ L ε (Ω h ).…”

Section: Introductionmentioning

confidence: 99%

Microscopic Validation of a Variational Model of Epitaxially Strained Crystalline Films

Kreutz¹,

Piovano²

2021

SIAM J. Math. Anal.

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A discrete-to-continuum analysis for free-boundary problems related to crystalline films deposited on substrates is performed by Γ-convergence. The discrete model here introduced is characterized by an energy with two contributions, the surface and the elastic-bulk energy, and it is formally justified starting from atomistic interactions. The surface energy counts missing bonds at the film and substrate boundaries, while the elastic energy models the fact that for film atoms there is a preferred interatomic distance different from the preferred interatomic distance for substrate atoms. In the regime of small mismatches between the film and the substrate optimal lattices, a discrete rigidity estimate is established by regrouping the elastic energy in triangular-cell energies and by locally applying rigidity estimates from the literature. This is crucial to establish pre-compactness for sequences with equibounded energy and to prove that the limiting deformation is one single rigid motion. By properly matching the convergence scaling of the different terms in the discrete energy, both surface and elastic contributions appear also in the resulting continuum limit in agreement (and in a form consistent) with literature models. Thus, the analysis performed here is a microscopical justifications of such models.

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“…This is why much effort has been spent in computing the elastic energy induced by periodic distribution of dislocations; we refer to the monograph [23, Sec 3.3] for a comprehensive overview. A relevant theoretical question is to understand optimal configurations of dislocations without assuming their periodicity: rigorous proofs that dislocations are favorable with respect to purely elastic deformations for large interfaces, as well as energy scaling properties have been recently faced in a variety of physical systems related to grain boundaries and epitaxial growth, starting from discrete or semi-discrete models of dislocations [2,3,6,7,9,10,13,[15][16][17][18]21]. The goal of this paper is to analyze the simplified version of the Peierls-Nabarro model, based on the minimization of the energy in (1), without assuming any periodicity on u.…”

Section: Introductionmentioning

confidence: 99%

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

Fanzon

Ponsiglione

Scala³

2020

Calc. Var.

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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-converges to a limit functional comprised of two contributions: one is given by a constant c ∞ > 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 ∞ is reached when dislocations are evenly-spaced on the one dimensional circle.

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On the effect of interactions beyond nearest neighbours on non-convex lattice systems

Cited by 10 publications

References 25 publications

Derivation of a Linearised Elasticity Model from Singularly Perturbed Multiwell Energy Functionals

Derivation of a Linearised Elasticity Model from Singularly Perturbed Multiwell Energy Functionals

Microscopic Validation of a Variational Model of Epitaxially Strained Crystalline Films

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

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