Mechanics of a Model Frictional Knitted Fabric
We present a study of the shapes and mechanical properties of a knitted loop made of elastic and frictional yarns. First, for non-frictional yarns, we investigate how the elastic energy of a loop varies with its shape. This allows us to quantify the energetic contributions of stretching and bending, and to identify energetically favorable loop configurations. We then consider the case of thin frictional yarns. In this case, the stability of a loop can be reduced, even in the presence of friction, to a simple planar \textit{elastica} problem. Friction gives rise to a multiplicity of stable loop shapes in a knitted fabric at rest, \textit{i.e.}, in the absence of externally applied forces. We show that, in the limit of thin yarns, a 2D \textit{elastica} model can predict these rest shapes. Furthermore, the force-deformation curves under uniaxial stretching can be obtained straightforwardly within this framework. Measurements of loop dimensions and forces during uniaxial stretching of nylon knitted fabrics allow us to assess the qualitative and quantitative validity of the planar \textit{elastica} model. Finally, we examine the limitations of this model for dense knitted fabrics.