A Model for Predicting Set in Helices

Abstract

A model is developed to calculate the residual set in a helical fiber axis after it has been wrapped around a mandrel, set into this shape, and released. The theory used is linear viscoelasticity in conjunction with the bending and twisting equations for thin rods. The model may be applied equally well to calculate the snarling behavior of twisted yams. An example of this is presented to confirm predictions made by the model. Two cases in the setting of helices are analyzed. In the first case the released helix is not in contact with the mandrel. In this case the bending and torsional set in the helical fiber axis follows quite simply from principles of linear viscoelasticity. In the second case, the released helix is constrained to lie on the surface of the mandrel.Results of this second case are more complex, but have been computed and presented in tabular form.If wool fibers are wrapped around a mandrel into the shape of a close-coiled helix, set, and released [ 14], the shape of the released fiber depends on wool fiber type [ 14]. The differences observed have been attributed to variations in the mechanical properties of the fiber substances [ 14] or to the initial fiber crimp shape. It is the aim of this paper to enable calculations to be made of the set in the fibers in such a test, and thereby to examine whether the differences observed between wool fiber types are due to fiber substance or crimp.The fiber is assumed to have an initial helical crimp. This assumption simplifies the analysis because the shape of the fiber around the mandrel is then also helical, as is the shape of the released fiber. Fibers that have a planar crimp can be approximated by circular arcs joined together. Each of these arcs is part of a plane, close-coiled helix, and the helical solution may be easily modified to that of a planar crimped fiber by including a torsional contribution.First, for a helical fiber, setting in bending and torsion is simply related to the fiber stress relaxation in bending and torsion, respectively. Second, the equations of equilibrium for a helix forced to lie on the surface of a mandrel by a constraining couple are established by means of the principle of minimum energy [3,4]. Third, the effect of inserting twist into the helical fiber axis, prior to wrapping it around the mandrel, is included in the analysis. In this way the effect of planar fiber crimp may be deduced.A special case of inserting pretwist into a helix and then wrapping it around a mandrel is that of the snarling of a twisted yarn. In this case the pretwisted cylinder is allowed to wrap around itself, the mandrel radius corresponding to the radius of the twisted cylinder or yarn. The snarling twist predicted by the model agrees with that found experimentally [ 1,2].The theory is then used to predict the setting of a helix around a mandrel. Assuming that a crimped wool fiber can be represented by a helical shape, the apparent difference in setting behavior of straight Lincoln wool and crimped Merino wool can be largely explained by the d...

Set in Bending of Australian Cashmere and Superfine Merino

“…If the difference in the set of cashmere and superfinc wool is purely a result of fiber curvature and its relationship to mandrel curvature [ 1 ], then the permanent set values for wool on the highest mandrel curvature and cashmere on the lowest mandrel curvature would be similar, as they both have crimp to mandrel curvature ratios of 0.3-0.6. This is not the case.…”