Calculus of Variations and Geometric Measure Theory
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M. Friedrich - F. Solombrino

Functionals defined on piecewise rigid functions: Integral representation and $\Gamma$-convergence

created by solombrin on 12 Apr 2019
modified on 15 Apr 2019

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Inserted: 12 apr 2019
Last Updated: 15 apr 2019

Year: 2019

Abstract:

We analyse integral representation and $\Gamma$-convergence properties of functionals defined on \emph{piecewise rigid functions}, i.e., functions which are piecewise affine on a Caccioppoli partition whose derivative in each component is constant and lies in a set without rank-one connections. Such functionals are customary in the variational modeling of materials which locally show a rigid behavior, and account for interfacial energies, e.g., for polycrystals or in fracture mechanics. Our results are based on localization techniques for $\Gamma$-convergence and a careful adaption of the global method for relaxation (Bouchitt\'eet al. 1998, 2001) to this new setting, under rather general assumptions. They constitute a first step towards the investigation of lower semicontinuity, relaxation, and homogenization for free-discontinuity problems in spaces of (generalized) functions of bounded deformation.


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