The aim of this joint DFG project with the department of Applied Mechanics at FAU Erlangen is to create simulations to optimize the structure of:
- woven or knitted fabrics
- 3D knitwear
- fiber materials with heterogeneous microstructures
A special focus will be given on the contacts between individual threads or fibers. The consideration of contact leads to a non-linear problem and the different geometric length scales make direct numerical simulation extremely complex. This is why a multi-scale approach has been chosen to facilitate dimension reduction of the problem.
The Problem has Two Small Parameters:
- the relationship between periodic or representative patterns in the fabrics and the overall fabric dimensions
- the relationship between fiber (or yarn) diameter and its length
Mathematically asymptotic methods are named homogenization or dimension reduction relative to the first or second small parameter. In homogenization, the aim is a scale separation on the basis of auxiliary problems using a periodicity or representative structural unit cell to facilitate the derivation of the overall effective material behavior of the fabric.
A dimension reduction (asymptotic in terms of the fiber thickness) reduces the fabric to a beam network, where the total displacement can be computed as a superposition of the stretch, bending, and torsion of one-dimensional beams. The mechanical contact between the beams must be explicitly taken into account.