Browsing by Author "Alicandro, Roberto"

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    Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours
    (2017-01) Alicandro, Roberto; Lazzaroni, Giuliano; Palombaro, Mariapia; Mathematics
    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.
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    Interactions beyond nearest neighbours and rigidity of discrete energies: a compactness result and an application to dimension reduction
    (2016) Alicandro, Roberto; Lazzaroni, Giuliano; Palombaro, Mariapia; Mathematics
    We analyse the rigidity of discrete energies where at least nearest and nextto- 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 surface-scaled energy and we give bounds on its possible Gamma-limit. In the second part of the paper we follow the approach developed in the first part to study a discrete model for (possibly heterogeneous) nanowires. In the heterogeneous case, by applying the compactness result shown in the first part of the paper, 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.
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    Linearisation of multiwell energies
    (2017-06) Alicandro, Roberto; Dal Maso, Gianni; Lazzaroni, Giuliano; Palombaro, Mariapia; Mathematics
    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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    On the effect of interactions beyond nearest neighbours on non-convex lattice systems
    (2017-01) Alicandro, Roberto; Lazzaroni, Giuliano; Palombaro, Mariapia; Mathematics
    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.