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

## TL;DR

This paper studies the mathematical properties of functionals on piecewise rigid functions, focusing on integral representation and $ ext{Gamma}$-convergence, with implications for material modeling and variational problems involving rigid behaviors.

## Contribution

It extends the theory of $ ext{Gamma}$-convergence and integral representation to functionals on piecewise rigid functions, a setting relevant for modeling rigid materials.

## Key findings

- Established integral representation results for functionals on piecewise rigid functions.
- Proved $ ext{Gamma}$-convergence properties under general assumptions.
- Laid groundwork for future analysis of lower semicontinuity and relaxation in this context.

## Abstract

We analyze 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 where the derivative in each component is constant and lies in a set without rank-one connections. Such functionals account for interfacial energies in the variational modeling of materials which locally show a rigid behavior. Our results are based on localization techniques for $\Gamma$-convergence and a careful adaption of the global method for relaxation (Bouchitt\'e et 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.