Linked loops on a lattice: Formulating knitted fabric elasticity with coarse-grained curve mechanics

Knitted fabric is a 2D mechanical metamaterial constructed from loops of quasi-1D yarn, topologically entangled in 3D. Different fabrics are designed by combining knitted stitches, the basic unit of linked yarn, in a variety of patterns. One stitch can be transformed into another merely by swapping the order in which segments of yarn cross over each other. It is therefore remarkable that different stitch patterns give rise to dramatically different macroscopic elastic responses. We explore this relationship between microstructure and macroscopic response by coarse-graining the yarn-level mechanics. This leads to a reduced lattice model that depends only on the configurations and spatial arrangements of the cross-over regions. In the low-stress regime, the fabric elasticity is governed by collective deformations of the cross-over regions and thus depends on stitch pattern. In the high-stress regime, the yarn mechanics within individual cross-overs is dominant, leading to nonlinear strain-stiffening that is observed in experiments. To conclude, we show how this coarse-grained model additionally provides a microscopic basis for the mechanics of curved fabric.