3—the Mechanics of Staple-Fibre Yarns Part Ii: Analysis and Results

The Tangent Compliance of Staple-fibre Bundles in Tension

“…The number of fibres aiding the withdrawal of the fibre at the location of the section will be all those that will ultimately move to the left but with shorter tails than L, namely: L Hence [7,8], since * Grosberg [4] has shown that ^i' is given by The value of P can be obtained by locating the maximum normal G tress on any plane parallel to the z-axls. Let a and a be the transverse stresses on the previous increment in two orthogonal directions x and y lying in a plane perpendicular to the sliver axis z.…”

“…The tail-length distribution and packing fraction can now be corrected for the mass transfer that has taken place during the increment by using Equation (8). The process can be repeated for further strain Increments.…”

“…Hearle [6] has provided a derivation for C for the case of a yarn composed of fibres all of the same length, and van Luijk et al [7,8] have carried out an analysis that applies to specific migration paths in spun yarns, again on the assumption that all the fibres have an Identical length. The following analysis allows for the more realistic case where the fibres are of various lengths as defined by the length-biassed fibre-length distribution, N^(L).…”

“…The maximum force, WF, that can be exerted on a shorter slipping tail of lengtii L will be given [7][8][9] where P = transverse pressure on the silver, WF -withdrawal force per unit length when P " 0, )i' = propor tlonall ty f ac tor (with dlmens ions of length )*, and C -proportionality factor accounting for the fact that, whereas some fibres In contact with the tail will oppose its withdrawal, others will aid it.…”

“…If we are given the stresses and strains from the previous increment, the critical length can be calculated by using the principles of single-fibre withdrawal theory previously used by Hearle [6] and van Luijk et al [7,8].…”