00906nas a2200121   4500008004100000245009600041210006900137520045600206100002300662700002400685700002400709856005100733       2017                        en  d00aDerivation of a rod theory from lattice systems with interactions beyond nearest neighbours0 aDerivation of a rod theory from lattice systems with interaction3 aWe 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.1 aAlicandro, Roberto1 aLazzaroni, Giuliano1 aPalombaro, Mariapia  uhttp://urania.sissa.it/xmlui/handle/1963/3526901090nas a2200121   4500008004100000245009000041210006900131520064600200100002300846700002400869700002400893856005100917       2017                        en  d00aOn the effect of interactions beyond nearest neighbours on non-convex lattice systems0 aeffect of interactions beyond nearest neighbours on nonconvex la3 aWe 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.1 aAlicandro, Roberto1 aLazzaroni, Giuliano1 aPalombaro, Mariapia  uhttp://urania.sissa.it/xmlui/handle/1963/3526801406nas a2200133   4500008004100000245004000041210004000081520101100121100002301132700002101155700002401176700002401200856004801224       2017                        en  d00aLinearisation of multiwell energies0 aLinearisation of multiwell energies3 aLinear 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.1 aAlicandro, Roberto1 aDal Maso, Gianni1 aLazzaroni, Giuliano1 aPalombaro, Mariapia  uhttp://preprints.sissa.it/handle/1963/3528801420nas a2200133   4500008004100000245009400041210006900135260001000204520094900214100002401163700002401187700002501211856005001236       2015                        en  d00aRigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires0 aRigidity of threedimensional lattices and dimension reduction in  bSISSA3 aIn 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 large1 aLazzaroni, Giuliano1 aPalombaro, Mariapia1 aSchlomerkemper, Anja  uhttp://urania.sissa.it/xmlui/handle/1963/7494