Modelling of contact and self-contact interactions in fibre assemblies: application to the tightening of knots
Résumé
Publications in recent years have shown an interest for the modelling of contact-friction interactions. The study of the mechanical behaviour of fibrous materials such as cables, ropes or woven fabrics, for which interactions between elementary fibres can play an important role, may be a motivation for revisiting the taking into account of contacts between beams. The approach proposed in this paper has been developed in the context of the modelling of entangled media, in order to solve the mechanical equilibrium of fibre assemblies, using an implicit solution scheme. Fibres are represented by a 3D beam model describing the kinematics of each cross-section by the means of three kinematical vector fields : the position vector for the center of the cross-section, and two directors for the cross-section, unconstrained regarding both their norms and orientations. This kinematical model, with nine degrees of freedom, allows to account for planar deformation of cross-sections. Contact interactions between beams are considered through discrete contact elements that associate pairs of material particles located on the surface of beams, that are predicted to come into contact. The principle for the construction of these contact elements corresponds to a discretization of the contact problem along intermediate geometries that are assumed to approximate the actual geometry of contact in each region where two parts of beams are close enough to likely come into contact. Proximity zones, associating two parts of beams which are stated to be close enough, are first coarsely determined within the assembly of fibres. The case of self-contact is simply considered by allowing to look for proximity zones within the same beam, provided the two close intervals are seperated by an exclusion distance criterion. For each proximity zone, an intermediate geometry aimed at approximating the contact geometry is defined as the average of the two close parts of beams. Contact elements are then generated at discrete locations on this intermediate geometry, using directions related to this geometry (planes orthogonal to the intermediate geometry) to determine particles candidate to contact. Application to the tightening of knots performed on multifilamentary yarns are presented.