1. Introduction
For the last several years, based on the topological
conservation for rods undergoing large deformations, Van der Heijden and his
coworkers have published several papers on the mechanical behavior of twisted rods
(Van der Heijden et al., 2003). Recently, Tran et al. (2006a, b) have applied
this method to consider the mechanical properties of a balanced multi-ply yarn.
In the present work, the self-contact force on strands of a multi-ply yarn, a
feature of interest for fiber interactions in yarn structures, will be
considered. This self-contact force via the inter-strand pressures is
determined at the balance situation across a range of the equilibrium
configurations of a series of multiply twisted yarns.
2. Mathematical Formulation
Consider a yarn made from n strands of radius r and length L
whose centerlines are wound on a cylinder of radius R in a right handed helix
(Fig.1). Let ψ, θ, φ (three
Eulerian angles) be the angular rotation of the single yarn around the cylinder
axis (X3), the helical ply angle of the strand and the twist angle of a strand,
respectively. Each strand is considered as an elastic inextensible unshearable
circular single yarn.
2.1 Geometrical Constraints of a Multi-ply Yarn using
Topological Conservation Conditions
Topological studies on the behavior of closed rods
undergoing arbitrary deformations have defined the concepts of link, twist and
writhe (Fuller, 1978), where the link can be thought of as the number of turns
put into a rod before gluing the ends of the rod together to form the closed
rod. According to the topological conservation law (Fuller, 1978), during the
deformation of the closed rod the link, Lk (spinning twist in our case) is
invariant and is expressed as follows
L k = T w
+ W r , (1)
where Tw is the total internal twist in the yarn
after the closed yarn is allowed to deform under the action of the torque in
the yarn, and Wr, called the writhe, is a measure for the out-of-plane
deformation. Since the yarns cross-section is assumed circular, the twist and
writhe Wr are determined for a multi-ply yarn as follows, respectively
(Neukirch and van der Heijden, 2002)
where k3 is the local twist of strands, σ is
equal to either 1 or 0 corresponding to an odd or even number of strands in the
multi-ply yarn. Based on these equations, we find the relation between the
spinning twist per unit length of yarn, τ, and k3 as follows
2.2 Kinetic Governing Equations
The configuration of a strand (i) is specified by the
position of a curve in space ri(s), where s is the arc length along the central
axis of the yarn. The force and moment balance equations for a single strand in
the multiply structure are
Free Download
