Page 54 - Fiber Fracture

MODELS OF FIBRE FRACTURE 39

it is possible, in principle, to calculate the variability of the fracture strength. Without
it, one has to resort to statistical methods. The statistical variation of strength of brittle
solids is usually described using a Weibull distribution. Coleman (1958) gave reasons
why such a distribution of strength is plausible for fibres whose strength is independent
of the rate of loading, and he computed the mean fibre strength as:

where L is the fibre length, m the Weibull modulus, 00 is the characteristic strength,
defined as the stress at which the failure probability of a fibre of length Lo is 0.63 and r
the gamma function.
The Weibull approach, although useful when comparing fibres, does not take account
of the failure mechanism. On the other hand, the deterministic theory of LEFM
overcomes some of the limitations of the Weibull method but is applicable only when
the initial crack geometry is known. There are intermediate cases where the crack
geometry is known but not its precise location, as, for example, inner circular cracks
located randomly. In these circumstances, LEFM can provide answers to questions
such as: what is the maximum rupture stress for a postulated maximum defect size of
unknown location?
Some of these questions have been answered in a recent paper (Guinea et a]., 2002b)
for a very simple family of defects: inner circular cracks and surface circular cracks,
both in planes normal to the fibre axis. The main result is plotted in Fig. 5, showing
the dependence of minimum rupture stress uc on crack radius. This plot may be used
to set boundaries for some problems in brittle fibre fracture analysis, Le.: to evaluate
the maximum circular defect size when the rupture stress is known or, conversely, to
estimate the rupture stress when defects, of unknown location, can be modelled as
circular cracks.
The way of modelling brittle fracture of fibres is paved by LEFM, and good agree-
ment between experimental results has been found. Future refinements will improve the
accuracy of the predictions, in particular for anisotropic fibres where cracks propagate
in mixed mode. More inputs from fractography and physical aspects of fracture will be
helpful in modelling defects able to initiate cracks.

Ductile Behaviour

Modelling fibre ductile fracture is an involved problem, even for homogeneous and
macro-defect-free fibres. Under tensile loading, fibres eventually reach an instability
point, where strain hardening cannot keep pace with loss in cross-sectional area, and a
necked region forms beyond the maximum load. A central crack is nucleated, spreads
radially and finally, when approaching the fibre surface, propagates along localised shear
planes, at roughly 45" to the axis, to form the 'cone' part of the fracture. This is the
typical cup-and-cone fracture of a ductile failure after a tensile test.
Exceptions to this form of ductile fracture appear in microstructurally clean high-
purity metal fibres, which sometimes neck down to a point. On the contrary, in ductile
fibres containing large defects, or deep surface notches, necking is absent and a brittle