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\begin{document}

\title{Asymptotic behavior for textiles with loose contact}



\author[1]{Riccardo Falconi}%
\author[2]{Georges GRISO}%
\author[3]{Julia Orlik}%
\author[1]{Stephan Wackerle}%
\affil[1]{Fraunhofer-Institut fur Techno und Wirtschaftsmathematik ITWM}%
\affil[2]{UPMC Paris 6}%
\affil[3]{Fraunhofer ITWM}%


\vspace{-1em}



  
  \date{May 20, 2024}


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\begin{abstract}
The paper is dedicated to the modeling and asymptotic investigation of a
linear elasticity problem, in the form of variational inequality, for a
textile structure. The textile is made of long and thin fibers crossing
each others, forming a periodic squared domain. The domain is clamped
only partially and an in plane sliding between the fibers is bounded by
a contact function, which is chosen to be loose. We also assume a
non-penetration condition for the fibers. Both partial clamp and loose
contact arise a domain split, leading to different behaviors in each of
the four parts. The homogenization is made via unfolding method, with an
additional dimension reduction to further simplify the problem. The four
cell problems are inequalities heavily coupled by the outer plane
macro-micro constraints, while the macroscopic limit problem results to
be an inequality of Leray-Lions type with only macro in plane
constraints. On both scales, no uniqueness is expected.%
\end{abstract}%



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