%0 Report
%D 2017
%T Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours
%A Roberto Alicandro
%A Giuliano Lazzaroni
%A Mariapia Palombaro
%X We study continuum limits of discrete models for (possibly heterogeneous) nanowires. The lattice energy includes at least nearest and next-to-nearest neighbour interactions: the latter have the role of penalising changes of orientation. In the heterogeneous case, we obtain an estimate on the minimal energy spent to match different equilibria. This gives insight into the nucleation of dislocations in epitaxially grown heterostructured nanowires.
%G en
%U http://urania.sissa.it/xmlui/handle/1963/35269
%1 35575
%2 Mathematics
%4 1
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-01-16T12:46:44Z
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%0 Report
%D 2017
%T On the effect of interactions beyond nearest neighbours on non-convex lattice systems
%A Roberto Alicandro
%A Giuliano Lazzaroni
%A Mariapia Palombaro
%X We analyse the rigidity of non-convex discrete energies where at least nearest and next-to-nearest neighbour interactions are taken into account. Our purpose is to show that interactions beyond nearest neighbours have the role of penalising changes of orientation and, to some extent, they may replace the positive-determinant constraint that is usually required when only nearest neighbours are accounted for. In a discrete to continuum setting, we prove a compactness result for a family of surface-scaled energies and we give bounds on its possible Gamma-limit in terms of interfacial energies that penalise changes of orientation.
%G en
%U http://urania.sissa.it/xmlui/handle/1963/35268
%1 35574
%2 Mathematics
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%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-01-16T12:38:33Z
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%0 Report
%D 2017
%T Linearisation of multiwell energies
%A Roberto Alicandro
%A Gianni Dal Maso
%A Giuliano Lazzaroni
%A Mariapia Palombaro
%X 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.
%G en
%U http://preprints.sissa.it/handle/1963/35288
%1 35594
%2 Mathematics
%4 1
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-06-22T09:07:10Z
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%0 Report
%D 2015
%T Rigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires
%A Giuliano Lazzaroni
%A Mariapia Palombaro
%A Anja Schlomerkemper
%X In the context of nanowire heterostructures we perform a discrete to continuum limit of the corresponding free energy by means of Γ-convergence techniques. Nearest neighbours are identified by employing the notions of Voronoi diagrams and Delaunay triangulations. The scaling of the nanowire is done in such a way that we perform not only a continuum limit but a dimension reduction simultaneously. The main part of the proof is a discrete geometric rigidity result that we announced in an earlier work and show here in detail for a variety of three-dimensional lattices. We perform the passage from discrete to continuum twice: once for a system that compensates a lattice mismatch between two parts of the heterogeneous nanowire without defects and once for a system that creates dislocations. It turns out that we can verify the experimentally observed fact that the nanowires show dislocations when the radius of the specimen is large
%I SISSA
%G en
%U http://urania.sissa.it/xmlui/handle/1963/7494
%1 7623
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2015-01-30T07:54:33Z
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