Asymptotic Behavior for Textiles in von-K{\'a}rm{\'a}n regime

TL;DR

This paper investigates the combined effects of homogenization and dimension reduction on textile structures within the von-Kármán elasticity regime, deriving a limit model that captures their buckling behavior.

Contribution

It extends deformation analysis for textiles, deriving a von-Kármán plate model with linear elastic cell problems and identifying orthotropic properties for homogeneous isotropic beams.

Findings

Derived a limit von-Kármán plate model for textiles.

Showed orthotropic behavior in the limit for isotropic beams.

Analyzed buckling behavior of orthotropic textiles.

Abstract

The paper is dedicated to the investigation of simultaneous homogenization and dimension reduction of textile structures as elasticity problem with an energy in the von-K{\'a}rm{\'a}n-regime. An extension for deformations is presented allowing to use the decomposition of plate-displacements. The limit problem in terms of displacements is derived with the help of the unfolding operator and yields in the limit the von-K{\'a}rm{\'a}n plate with linear elastic cell-problems. It is shown, that for homogeneous isotropic beams in the structure, the resulting plate is orthotropic. As application of the obtained limit plate we study the buckling behavior of orthotropic textiles.