1. Introduction


For the last several years, based on the topologicalconservation for rods undergoing large deformations, Van der Heijden and hiscoworkers have published several papers on the mechanical behavior of twisted rods(Van der Heijden et al., 2003). Recently, Tran et al. (2006a, b) have appliedthis 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, afeature of interest for fiber interactions in yarn structures, will beconsidered. This self-contact force via the inter-strand pressures isdetermined at the balance situation across a range of the equilibriumconfigurations of a series of multiply twisted yarns.


2. Mathematical Formulation


Consider a yarn made from n strands of radius r and length Lwhose centerlines are wound on a cylinder of radius R in a right handed helix(Fig.1). Let ψ, θ, φ (threeEulerian angles) be the angular rotation of the single yarn around the cylinderaxis (X3), the helical ply angle of the strand and the twist angle of a strand,respectively. Each strand is considered as an elastic inextensible unshearablecircular single yarn.


2.1 Geometrical Constraints of a Multi-ply Yarn usingTopological Conservation Conditions


Topological studies on the behavior of closed rodsundergoing arbitrary deformations have defined the concepts of link, twist andwrithe (Fuller, 1978), where the link can be thought of as the number of turnsput into a rod before gluing the ends of the rod together to form the closedrod. According to the topological conservation law (Fuller, 1978), during thedeformation of the closed rod the link, Lk (spinning twist in our case) isinvariant and is expressed as follows

L k = T w+ W r , (1)


where Tw is the total internal twist in the yarnafter the closed yarn is allowed to deform under the action of the torque inthe yarn, and Wr, called the writhe, is a measure for the out-of-planedeformation. Since the yarns cross-section is assumed circular, the twist andwrithe 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 isequal to either 1 or 0 corresponding to an odd or even number of strands in themulti-ply yarn. Based on these equations, we find the relation between thespinning 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 theposition of a curve in space ri(s), where s is the arc length along the centralaxis of the yarn. The force and moment balance equations for a single strand inthe multiply structure are

 

where F and M are the internal force and moment, respectively, and p is the pressure (contact force per unit length) on the strand exerted by the neighbor strands. Taking into account the geometry of the strands in the multi-ply we can write for the pressure:



where pj+1,j and pj-1,j are the pressures exerted on strand (j) by strands (j+1) and (j-1), respectively. After using the topological constraint, one finds the expression for the total pressure on a strand as follows


where B, ρ and n are the bending stiffness of a strand, the ratio of the cylinder radius and strand radius and the number of strands, respectively. The end force F0 and end moment M0 are both applied axially to the overall yarn.


3. Pressure on the Strands of a Multi-ply Yarn


Some single yarns of different counts were modelled to investigate the inter-strand and total pressures at the balanced point of multi-ply yarns comprising 2, 4 and 6 strands. Figure 2 describes the inter-strand pressure and radial total pressure on strands of the 2, 4 and 6-ply yarns plotted against the spinning twist of the original single yarns at the balance point for strand count of 40 Tex, after removing external forces. These results identify some interesting behavior for the multi-ply yarns. While the differences in the total pressure are negligible at low spinning twist (for example, less than 300tpm for 40 Tex), a clear difference is observed for the inter-strand pressure. Generally, the effect of the number of strands on these two kinds of pressure shows two contrasting effects. As the number of strands increases, the total pressure decreases but the interstrand pressure increases. Furthermore, the total pressure tends to reach a stable value with increasing spinning twist. This stable total pressure is particularly evident for the 6-ply yarns from 40 Tex single yarns. The effect of inter-strand pressures may have very practical consequences in relation to fabric performance. The inter-strand pressures may influence the lateral compression of strands and influence the bulk in the strand and multi-ply yarn as well as the movement of fibers in the yarn structure. The latter effect has potential to affect the shedding of fibers and the formation of pills in textile products.


In order to further evaluate the influence of structural properties of a multiply yarn on the inter-strand and total pressures, three multi-ply yarns of the same resultant count (80 Tex) prepared from 2x40tex, 4x20tex and 6x13.3tex were considered. Figure 3 depicts the total pressures and interstrand pressure with respect to spinning twist for the three multi-ply yarns. The results in Figure 3 confirm the influence of the number of strands on the total pressure and inter-strand pressure on strands. Due to the limitation of this extended abstract further results will be presented at the conference.



 

4. Conclusion


The self-contact via inter-strand and axial total pressures in multi-ply yarns has been considered and provides a basis for studying related yarn and fabric structural properties, e.g., yarn bulk, fiber migration and pilling.


References


1)     Fuller, F.B., 1978. Decomposition of the Linking of a Closed Ribbon: A Problem from Molecular Biology, Proc. Nat. Acad. Sci. USA 75, 3557- 3561.

2)    Neukirch, S. and van der Heijden, G.H.M., (2002), Geometry and mechanics of uniform n-plies: from engineering ropes to biological filaments, J. Elasticity 69, 41-72.

3)    Tran, C.D., Phillips, D.G. and van der Heijden G.H.D., (2006a), Issues in Processing Unstable Twisted Fibre Assemblies, in the 4th International Simulation Conference, pp. 325-329. The European Technology Institute and The European Simulation Society, Palermo, Italia.

4)    Tran, C.D., van der Heijden, G.H.M. and Phillips D.G., (2006b), Application of Topological Conservation to model Key Features of Zero-torque Multi-ply Yarns, submitted, J. Text. Inst.

5)    Van der Heijden, G.H.M., Neukirch, S., Goss, V.G.A. and Thompson, J.M.T., (2003), Instability and self-contact phenomena in the writhing of clamped rods, Int. J. Mech. Sci. 45, 161-196.


About the Authors:


The authors are associated with CSIRO Textile and Fibre Technology, Australia and Centre for Nonlinear Dynamics, Civil Engineering Building, UCL, London, UK.



To read more articles on Textile, Industry, Technical Textile, Dyes & Chemicals, Machinery, Fashion, Apparel, Technology, Retail, Leather, Footwear & Jewellery,  Software and General please visit https://articles.fibre2fashion.com


To promote your company, product and services via promotional article, follow this link: https://www.fibre2fashion.com/services/article-writing-service/content-promotion-services.asp