Page 283 - Carbon Nanotube Fibres and Yarns

272 Carbon Nanotube Fibers and Yarns

11.2 Yarn transformation due to insertion
of extremely high twist

Although the torsional properties discussed in this section are based on
yarns made from textile fibers, the conclusions are also applicable to the
torsional behaviors of twisted CNT yarns, but in some cases with certain
caveats.

11.2.1 Twist-induced longitudinal contraction
The most commonly used geometrical model of twisted yarns consists of a
series of coaxial helices. According to this model, all the fibers follow perfect
helices with a common axis which coincides with the yarn axis. The helix
angle (θ) of fibers on the yarn surface can be related to the twist (T, turns
per unit length, the length of yarn with one turn of twist is h=1/T) and the
radius r of the yarn by
π
tanθ = 2 rT (11.1)
This means that the fiber helix angle will increase as the yarn radius
increases while the yarn twist T maintains a constant.
Obviously, all the fiber helix paths are longer than the corresponding length
of the yarn axis. In average, the ratio between fiber length and yarn length is
R = sec 2 ( / 2θ ) = /2 (1 + cosθ ) (11.2)
t
This ratio represents the shortening of the yarn when a twist T is in-
serted to an initially zero twist yarn, and is known as twist retraction of the
yarn [22]. Note that R t is calculated on the basis of average fiber length in
a twisted yarn length, instead of the length of fibers on the yarn surface,
although the twist retraction of a yarn in Eq. (11.2) is related only to the
helical angle of surface fibers on the yarn, which is referred to as the twist
angle of the yarn.

11.2.2 Torque in a twisted yarn
The torque developed in a twisted yarn depends on the mechanical state
of the constituent fibers in tension, torsion, and bending [23]. Because the
fiber diameter is one order of magnitude smaller than the yarn diameter for
traditional textile fibers and the moment of inertia for the fiber cross section
is proportional to the fourth power of the fiber diameter, the contributions
of the fiber torsion and bending to the total yarn torque are quite small.
Postle et al. [24] showed that the yarn torque due to fiber tensile stresses