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# Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang

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1864-8266
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# Homogenization in BV of a model for layered composites in finite crystal plasticity

Elisa Davoli
/ Rita Ferreira
/ Carolin Kreisbeck
Published Online: 2019-10-01 | DOI: https://doi.org/10.1515/acv-2019-0011

## Abstract

In this work, we study the effective behavior of a two-dimensional variational model within finite crystal plasticity for high-contrast bilayered composites. Precisely, we consider materials arranged into periodically alternating thin horizontal strips of an elastically rigid component and a softer one with one active slip system. The energies arising from these modeling assumptions are of integral form, featuring linear growth and non-convex differential constraints. We approach this non-standard homogenization problem via Gamma-convergence. A crucial first step in the asymptotic analysis is the characterization of rigidity properties of limits of admissible deformations in the space BV of functions of bounded variation. In particular, we prove that, under suitable assumptions, the two-dimensional body may split horizontally into finitely many pieces, each of which undergoes shear deformation and global rotation. This allows us to identify a potential candidate for the homogenized limit energy, which we show to be a lower bound on the Gamma-limit. In the framework of non-simple materials, we present a complete Gamma-convergence result, including an explicit homogenization formula, for a regularized model with an anisotropic penalization in the layer direction.

MSC 2010: 49J45; 74Q05; 74C15; 26B30

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Revised: 2019-06-23

Accepted: 2019-09-10

Published Online: 2019-10-01

Award identifier / Grant number: CZ04/2019

Funding Source: Austrian Science Fund

Award identifier / Grant number: F65

Award identifier / Grant number: I 4052-N32

Award identifier / Grant number: and V 662-N32

Funding Source: Nederlandse Organisatie voor Wetenschappelijk Onderzoek

Award identifier / Grant number: TOP2.17.012

The work of Elisa Davoli has been supported by the Austrian Science Fund (FWF) through projects F65, I 4052-N32, and V 662-N32, as well as from BMBWF through the OeAD-WTZ project CZ04/2019. Carolin Kreisbeck gratefully acknowledges the support by the Dutch Research Council (NWO) through the project TOP2.17.012 and by a Westerdijk Fellowship from Utrecht University. The research of Elisa Davoli and Carolin Kreisbeck was supported by the Mathematisches Forschungsinstitut Oberwolfach through the program “Research in Pairs” in 2017.

Citation Information: Advances in Calculus of Variations, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258,

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