A simple model for the forces acting on a single fiber as it is withdrawn from a tangled fiber assembly is proposed. Particular emphasis is placed on understanding the dynamics of the reptating fiber with respect to the entanglement of fibers within the tuft. The resulting two-parameter model captures the qualitative features of experimental simulation. The model is extended to describe the breakup of a tuft. The results show good agreement with experiment and indicate where a tuft is most likely to fracture based on the density of fiber endpoints. DOI: 10.1103/PhysRevE.74.041806 PACS number͑s͒: 36.20.Ϫr, 62.40.ϩi, 46.55.ϩd, 46.70.Ϫp
I. MOTIVATIONThere has been considerable interest in understanding the physical properties of media that are partly composed of fibers or nematic objects; examples are fiber suspensions ͓1-3͔, fiber reinforced composites ͓4,5͔, polymers ͓6͔, and liquid crystals ͓7͔. For textile fibers within dense tufts, their relative orientations within an assembly and the number of frictional contact points govern the internal forces of the tuft. In compression and for small deformations in extension the bending of individual fibers allows the whole assembly to distort without significant slippage between fibers and an anisotropic elastic model can be used ͓8͔. However under higher extensional loads, the fibers must slip over each other causing an irreversible deformation and it is this type of distortion that we are concerned with here. Modelling these interactions, in particular aggregate or bulk behavior, is not well understood but there has been some recent work. For example, Refs. ͓9,10͔ present models for extensional deformation that are based on planar constrained fiber mechanics and Refs. ͓11,12͔ consider fiber withdrawal from an ad hoc point of view. More recently Wilkins ͓13͔ studied the dynamics of fiber slippage under small cyclic loads. A different approach was taken in Ref. ͓14͔, which derives a threedimensional ͑3D͒ phenomenological continuum model that includes density, alignment, order, and entanglement as dependent variables.The original aim of this work was to understand and quantify the forces acting inside a tuft of fibers as it is processed by the carding machine, but the theory developed here applies to any mass of long tangled fibers which are subject to extensional forces. As a tuft goes through the carding machine it is subjected to a series of forces aimed at teasing, separating and aligning the fibers so that an ordered lap is obtained. Typically a tuft is torn apart either under tension or under shearing forces, and the forces resisting the deformation of the tuft will be those acting between neighboring fibers in the tuft; see Ref. ͓15͔ for a recent experimental study.In order to aid the creation of a model for the deforming tuft a series of experiments were set up in order to elucidate the dependence of the forces on the parameters of the problem. The experimental work was conducted by Mahmoudi at the Centre for Technical Textiles, University of Leeds, and some of ...
