Page 45 - Fiber Fracture

30 M. Elices and J. Llorca

idealised polymer$bre. The next step, a two-dimensional model, could be a rolled-up
crystalline sheet of atoms; this cylinder can be used as a model for a nanotubeJibre.
Finally, a three-dimensional perfect array of atoms may idealise a crystal whisker. In the
next sections these three topics are considered more closely.
A crude estimate of the tensile strength of a fibre that has the advantage of being
applicable to all fibres, whatever the detailed nature of their interatomic forces, is
the model due to Polanyi (1921) and Orowan (1949). This model, summarised in the
classical book of Kelly and Macmillan (1986), gives the maximum tensile stress a,,
as:

where E is the Young modulus, y is the surface energy per unit area, and the
equilibrium separation of atomic planes. In spite of its simplicity, this model gives, in
many instances, the correct order of magnitude. More recently, accurate calculations
have been made using approximate potential functions describing the interactions
between atoms in the crystal lattice, and some results are quoted in subsequent
paragraphs.
To check the accuracy of theoretical predictions, or to have an estimate of the strength
of perfect fibres, tensile tests have to be performed. To this end, fibres free from flaws
and defects must be produced and tested. Whiskers are nearly the ideal samples for
three-dimensional models and the strongest one should be a whisker of a covalently
bonded crystal; to the authors’ knowledge, no diamond whiskers have yet been tested,
but results obtained for Si and Ge whiskers are of the same order of magnitude
as theoretical predictions. Brenner’s seminal papers on tensile strength of whiskers,
particularly Brenner (1956) on whiskers of iron, copper and silver, are still a good
reference. Two-dimensional models have been checked by testing carbon nanotubes;
experimental results by Yu et al. (2000) agree quite well with atomistic predictions by
Bcmholc (last chapter in this volume). Simplified one-dimensional models for polymer
fibres give values one order of magnitude above experimental ones; atomistic failure
models for polymer fibres need to be improved by considering the presence of defects,
the degree of crystallinity and the role of secondary bonds in cross-linking chains.
Tensile stresses in commercial fibres are, in general, one order of magnitude below
theoretical values. As already emphasised, this discrepancy is due to the presence of
defects, a fact supported by the increase of the rupture stress as the fibre diameter, or the
gauge length, decreases. Extrapolation of these stresses to very small diameters gives,
in some cases, values near the theoretical ones. Although high-performance fibres have
small diameters, about 10 pm, there is still plenty of room for defects, some of them
of nanometer size. Realistic atomistic modelling should consider such defects and the
subsequent triggered cracks or plastic deformation. This has been done with success for
‘simple’ nanotubes (see paper by Bemholc et al. in this volume), but more information
is needed - with the help of TEM or AFM - to foster the scientists’ ingenuity in
modelling fibres in a realistic way. The next section is a brief summary of results of
simple computations for perfect fibres, together with some experimental measurements
to give a flavour of atomistic modelling.