A general geometric framework for knitted fabric elasticity

Knitting is a process in which yarn, an essentially filament-like material, is shaped in space to form a fabric, an essentially sheet-like material, via stitching together a lattice of slip-knots. Due to fabric-level dependence on the stitch pattern, a single yarn can be used to create a large variety of fabric geometries and material responses. Moreover, the elasticity of knits remains poorly understood, as evidenced by the lackluster performance of spring-lattice models. We seek a continuum elastic model that predicts the three-dimensional shape of knitted fabric. This model should have the flexibility to be adapted to describe a wide range of stitch patterns and elasticity models. To this end, we have developed a geometric framework for relating the yarn path to the emergent surface geometry of the fabric. The generality of our approach allows for a systematic coarse-graining of yarn degrees of freedom, without a priori specification of a model of yarn elasticity. Thus, we are able to arrive at a stitch pattern-dependent, continuum elastic model of knits by assuming a simple phenomenological model of yarn, whilst allowing for the possibility of including more realistic yarn mechanics and experimental comparison.