Page 44 - Fiber Fracture

MODELS OF FIBRE FRACTURE 29

INTRODUCTION
The aim of this paper is to provide a brief overview of some attempts to model fibre
fracture and is intended to complement the next chapter by Hearle devoted to the pheno-
menology of fibre fracture.
'ILvo different approaches to modelling fibre fracture are presented: the atomistic
approach, for those who use the tools of quantum mechanics, and the continuum ap-
proach that relies on continuum mechanics. In the coming years, with the advent of
nanotechnology, a hybrid approach will probably be needed to deal with nanofibres and
biological fibres.
In every approach one finds a wide range of sophistication, In the continuum
approach, the simplest (and most common) models are based on linear elastic fracture
mechanics (LEFM), a well developed discipline that requires a linear elastic behaviour
and brittle fracture, not always exhibited by fibres. Ductility and the presence of
interfaces, not to mention hierarchical structures, make modelling much more involved.
The same is true of the atomistic approach; fracture models based on bond breaking of
perfect crystals, using well established techniques of solid state physics, allow relatively
simple predictions of theoretical tensile stresses, but as soon as real crystals, with defects
and impurities, are considered, the problem becomes awkward. Nevertheless solutions
provided by these simple models - LEFM or ideal crystals - are valuable upper or
lower bounds to fibre tensile strength.
Experimental values of tensile stress of high-performance fibres are about 3 or 5 GPa,
one order of magnitude below theoretical predictions. In most cases, such discrepancies
are due to the presence of defects not properly accounted for in the models. Apart
from the difficulty of modelling such defects, what is felt in the literature is the lack
of information on the characteristics of these defects, information that is essential for
designing new models. In this respect, fibre fractography in the range of 1 pm to 1 nm
should be encouraged.

ATOMISTIC APPROACH

Prediction of the mechanical behaviour of materials beyond the elastic limit using
an atomistic approach, in particular fracture behaviour, has proved difficult. Problems
in predicting the fracture toughness, or the maximum tensile stress, from an atomistic
starting point, lie in the large length and time scales that are involved in treating
the proccss zone, associated defects and, in most cases, the plastic zone. At present,
calculations using molecular dynamics are still not capable of handling the system sizes
corresponding to the process zone around a crack tip (sometimes up to lOI3 atoms), nor
are they capable of treating the required time scales. However, simple predictions of
the theoretical tensile strength of perfect crystals based on solid state techniques can be
made and may be useful as upper limits of the attainable fibre strength.
A perfect crystal, free of defects, is expected to be the strongest form of a fibre;
the simplest 'crystal' could be a bundle of perfectly aligned linear chains of carbon
atoms covalently bonded. This oversimplified one-dimensional model may represent an