Page 342 - Fiber Fracture

324 C. Viney

,

o o+do
Fig. 11. The geometry refemed to in the derivation of Eq. 14, pertaining to tapered fibres.

and the interface area accommodated in the distance dr is

Area = 27rrdx + 2indrdx = 2nrdx+ndrdr % 27rrdx (8)
provided that 6 is small so that the second-order term containing the product of two
differentials can be ignored. 6 is indeed found to be small for the collagen fibres under
consideration here: their length-to-width ratio is of the order of 2000 (Trotter et al.,
2000b).
A simple force balance now gives
t(2nrdx)cosQ % r(2nrdr) = da(nr2) (9)
Substitution of Eq. 7 into Eq. 9, followed by simplification, leads to
2t dx
do=--
tan0 x
If the stress in a fibre increases from a very small value qf (at a point xif that is
arbitrarily close to the end of the fibre) to the failure stress a,f (at a point x,f that ideally
is at the midpoint of the fibre), Eq. 10 can be integrated between the corresponding
limits to obtain an expression in which xuf again represents the transfer distance
l: do = - 1, (1 1)
x.f dx
2t
tan6
Explicit integration of Eq. 1 1, followed by rearrangement, then leads to:

Therefore
2t
a,f 25 - lnxuf - C (13)
tan 6
where C is a constant that subsumes events at the end of the fibre. The adoption of a small,
non-zero lower limit when performing the integration is a device that circumvents the need
to consider the logarithm of zero in the calculation. A similar approach is conventionally