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Theory of knitted fabrics

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-- This work is published in Physical Review X ; preprint available at arXiv.org --

Knitted materials have been around more than a thousands years and may even be older than what the archaeological findings show. The long time passed since has provided us a great experience in applying the basics of knitting technology. The success of textile industry confirms our success of knitting in application but what has been missing is that, as Samuel Poincloux from Sorbonne University and his colleagues addressed, a fundamental understanding of the physics particularly linking geometry and elasticity of knitted materials. The researchers established a first-principles theory of response using the concepts of the classical physics based on kinematics, minimization of energy and conservation of the total-length of the yarn used in knitting. This work can support innovating development in the textile industry, particularly engineering smart and composite materials.

A knitted material basically consists of stitches or more precisely a single yarn forming intertwined loops. The curved nature of stitches allows sustaining large deformations when the fabric is stretched. The yarn also slides from one stitch to the other giving extra flexibility whereas the yarn itself can only deform a small fraction of the total fabric deformation. The authors based their model on the periodic structure of the knit and the case of an unstretchable but bendable yarn reducing the complexity of the stitch pattern to simple equations with possible adaptation to different stitch patterns. When represented as a point, the stitches forms a grid or more formally a 2D-vector space. As the authors show, values of the vector components are available from stretching measurements in which a sheet of knitted nylon fabric deforms under many loading and unloading cycles of a stretching force. The measurements show that during deformation of the fabric, all the grid points follow quite straight trajectories obeying a linear relation between in- and out-plane deformation. This means that the fabric has a conserved effective area formed by the hollow stitches but it behaves as an incompressible elastic bulk material. Whenever the stitch size is larger than the diameter of the yarn, response of the individual stitch and the final shape of the fabric can be calculated using the theoretical model by inserting a couple of material specific parameters obtained from the measurements —effective stretching modulus (energy required to stretch per meter) and the effective length of the stitches (asymmetry in the deformation).

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