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.
|Original language||English (US)|
|Number of pages||33|
|Journal||Advances in Calculus of Variations|
|State||Published - 2021|
Bibliographical noteKAUST Repository Item: Exported on 2021-08-23
Acknowledgements: The work of Elisa Davoli has been funded by the Austrian Science Fund (FWF) project F65 “Taming complexity in partial differential systems”. Carolin Kreisbeck gratefully acknowledges the support 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. The hospitality of King Abdullah University of Science and Technology, Utrecht University, and of the University of Vienna is acknowledged. All authors are thankful to the Erwin Schr¨odinger Institute in Vienna, where part of this work was developed during the workshop “New trends in the variational modeling of failure phenomena”.