Calculus of Variations and Geometric Measure Theory
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G. Lazzaroni - M. Palombaro - A. Schlömerkemper

Rigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires

created by lazzaroni on 29 Jan 2015
modified on 20 Dec 2016


Published Paper

Inserted: 29 jan 2015
Last Updated: 20 dec 2016

Journal: Discrete Contin. Dyn. Syst. Ser. S
Volume: 10
Pages: 119-139
Year: 2017
Doi: doi:10.3934/dcdss.2017007


In the context of nanowire heterostructures we perform a discrete to continuum limit of the corresponding free energy by means of Gamma-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.


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