Calculus of Variations and Geometric Measure Theory
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R. Alicandro - M. Cicalese - L. De Luca

Screw dislocations in periodic media: variational coarse graining of the discrete elastic energy

created by cicalese on 01 Nov 2021
modified on 07 May 2022

[BibTeX]

Accepted Paper

Inserted: 1 nov 2021
Last Updated: 7 may 2022

Journal: Nonlinear Analysis
Year: 2021
Doi: https://doi.org/10.1016/j.na.2022.112871

Abstract:

We study the asymptotic behavior, as the lattice spacing ε tends to zero, of the discrete elastic energy induced by topological singularities in an inhomogeneous $\varepsilon$ periodic medium within a two-dimensional model for screw dislocations in the square lattice. We focus on the $\mid\log\varepsilon\mid$ regime which, as $\varepsilon\to 0$, allows the emergence of a finite number of limiting topological singularities. We prove that the $\Gamma$-limit of the $\mid\log\varepsilon\mid$ scaled functionals as $\varepsilon\to 0$ equals the total variation of the so-called “limiting vorticity measure” times a factor depending on the homogenized energy density of the unscaled functionals.


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