Page 341 - Fiber Fracture

FRACTURE OF NATURAL POLYMERIC FIBRES 323

Fig. 10. The geometry referred to in the derivation of Eq. 6, pertaining to cylindrical fibres.

lc''r lX'" (5)
this distance:
$
do
=
&
Explicit integration of Eq. 5, followed by rearrangement, leads to:

So xuf is an increasing function of r: an increased fibre thickness will increase the
capacity of the interfacial surface area to transfer load to the fibre (in proportion to r),
but will also (and even more effectively) increase the capacity of the fibre cross-section
to carry that load (in proportion to r2).
It follows that much of the material in the fibre is wasted, in that the tensile strength
is not being properly exploited along almost the whole length of the fibre! This problem
is exacerbated in thicker fibres. The load near the ends of the fibres could be carried
adequately by a thinner fibre cross-section, compared to the load near the middle of
fibres. A less wasteful use of material, and a more efficient exploitation of the fibre
properties, would therefore be achieved if the fibres were to taper from the middle
towards the ends. Also, regions of the fibre having a smaller cross-section would then be
able to undergo a larger elastic deformation, thus matching more closely the deformation
of the matrix; therefore, the shear stress concentration in the matrix near the fibre ends
would be reduced. These gains have been recognised and discussed qualitatively in the
context of sea cucumber and sea urchin collagen fibres, which are appropriately tapered
(Trotter and Koob, 1989; Trottcr ct a]., 1994,2000b).
It is instructive to consider how the force balance in Eq. 3 and the critical half-length
(transfer distance) of the fibres in Eq. 6 are affected by allowing the fibres to taper
towards their ends. Reference should be made to Fig. 11, which again considers the
shear stress acting on a small length dx near the end of a fibre. The tensile stress in the
fibre again increases by do over this distance, and the fibre radius increases by dr. From
the geometry of Fig. 1 1,
dr
-dx = tan& i.e. r = xtane (7)