Homogenisation of the undulations of a single yarn. Applications to the modelling of the traction

Considering the yarn as the elementary constituent of fabric at the microscale, the mechanics of single-yarn motion and behaviour is investigated, whereby the impact of the thread undulations on the deformation mechanisms of the yarn are analysed. Considering that the yarn can be modelled as an undulated beam, a perturbation method is involved in order to set up the limit model of a straight yarn. Assuming the undulations to be quasi-periodical at a mesoscopic scale, a zoom is made on a period of undulation, so that the equilibrium problem of the yarn is embedded in a one undulation parameter (the ratio of the curvilinear length of a period to the projected length) series of problem. The limit behaviour for a vanishing undulation parameter is obtained, using the multiscale asymptotic expansion method. Equivalent mechanical rigidity in traction and flexion is further evaluated, the expressions of which reflect the actual undulation of the yarn, via the average over one period of trigonometric functions of the rotation field along the mean line of the beam. Traction curves for a single yarn in a woven structure are simulated, considering different initial shapes of the yarn, that do reflect the essential trends of the observed J-shaped curve behaviour of real fabric samples.