Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours

Roberto Alicandro; Giuliano Lazzaroni; Mariapia Palombaro

doi:10.3934/nhm.2018001

Article Contents

2018, Volume 13, Issue 1: 1-26. Doi: 10.3934/nhm.2018001

This issue Previous Article Next Article Stochastic homogenization of maximal monotone relations and applications

Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours

1.
Università di Cassino e del Lazio meridionale, Dipartimento di Ingegneria Elettrica e dell’Informazione, via Di Biasio 43, Cassino (FR), 03043, Italy
2.
Università di Napoli Federico Ⅱ, Dipartimento di Matematica e Applicazioni, via Cintia, Monte S. Angelo, Napoli, 80126, Italy
3.
University of Sussex, Department of Mathematics, Pevensey 2 Building, Falmer Campus, Brighton, BN1 9QH, United Kingdom

Received: February 2017

Revised: November 2017

Published: March 2018

RA and GL received funding from the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). When this research was carried out, GL was affiliated to SISSA and supported by the European Research Council under the Grant No. 290888 "Quasistatic and Dynamic Evolution Problems in Plasticity and Fracture"; he was then affiliated to the University of Vienna and supported by the Austrian Science Fund (FWF) project P27052. MP was partially supported by the EU Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie project ModCompShock agreement No. 642768

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Abstract

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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Mathematics Subject Classification: Primary: 74B20; Secondary: 74K10, 74E15, 74G65, 49J45.

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Figure 1. The six tetrahedra in the Kuhn decomposition of a three-dimensional cube

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Figure 2. Two possible recovery sequences for the profile at the centre of the figure. Here we picture only a part of the wire containing just one species of atoms, therefore the transition at the interface is not represented. A kink in the profile may be reconstructed by folding the strip, i.e., mixing rotations and rotoreflections (left); or by a gradual transition involving only rotations or only rotoreflections (right). In the limit, the former recovery sequence gives a positive cost, while the latter gives no contribution. If the stronger topology is chosen, the appropriate recovery sequence will depend on the value of the internal variable, which defines the orientation of the wire

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Figure 3. Lattices with dislocations: choice of the interfacial nearest neighbours in $\mathcal{L}_\varepsilon(\rho, k)$ and $H \mathcal{L}_\varepsilon(\rho, k)$ for a Delaunay triangulation

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