Page 196 - Carbon Nanotube Fibres and Yarns

186 Carbon Nanotube Fibers and Yarns

α
2
E / E = cos [1 − kcosec α , ] (8.1)
y
f
where k is a “slip factor” that decreases with increasing fiber length, fine-
ness, and fiber friction, and also with increasing fiber entanglement [71].
For linear elastic fibers, the moduli in Eq. (8.1) may be replaced by their
corresponding strengths.
This equation has been used by many authors to represent the general
relationship between the tensile strength and the surface twist angle of CNT
yarns [7, 78–80]. The optimal twist angle for CNT yarns is about 20 degrees,
similar to that for conventional staple fiber spun yarns. There have also been
experimental evidences pointing to different characteristics for the strength-
twist relationship for CNT yarns. Zhao et al. [29] reported a second strength
peak at the twist angle of 27–30 degrees and ascribed the peak to structural
deformation of thin-walled CNTs, a phenomenon arising from the “nano”
dimension. To further understand this second peak, a continuum mechan-
ics model was proposed to take into account the radial pressure inside a
CNT yarn. The continuum model is based on the nanosize of CNT bundles,
which can be treated as infinitesimal (tens of nanometers). Let s(r) and P r (r)
denote respectively the cross-sectional area of and the internal radial pressure
on a CNT bundle located at radial position r. The number of CNT bundles
in a thin layer between r and r + dr is dn(r) = 2πr cosθ(r)dr/s(r). Under a tensile
strain of ξ, we have the load force on the bundle F(r) = E(r)s(r)ξ, where E(r) is
the elastic modulus of the CNT bundle. After summing up the axial compo-
nent of F(r) and dividing by the total cross-sectional area, we get the strength
when the CNT bundle with the maximum stress starts to break,

()
()
R dn rF r ()cos θ r () 2 H 2 ξ R Er rdr
y ∫
σ = = ∫ (8.2)
2
0 πR 2 R 2 0 (2 πr) + H 2
In general, as the bundle’s cross-sectional area decreases, the CNT bun-
dle is compressed more and becomes stronger, suggesting that E(r) mono-
tonically increase with s(r), i.e., E(s(r)). On the other hand, s(r) depends
on the radial pressure distribution because the CNT bundle is compress-
ible. Thus Zhao et al. [29] suggested a linear E(s(r)) and a monotonically
decreasing s(P r (r)). The main observation from this is that with decreasing
H (increasing twist angle), the maximum of P r (r) occurs at r ≈ 0.5R. This
means that the interior radial pressure could become high enough to in-
duce structural collapse for the CNTs. Zhao et al. further considered the
collapse phenomenon by setting s(r) to a small value while P r (r) is above a