Materials for engineering, 3rd Edition - (Malestrom)

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Materials for engineering
A fibre of length l c will carry an average stress ( σ f ) of only σ * f /2 and a
general expression for the average fibre stress for a fibre of length l, is thus
σ
= σ (1 – l /2 l)
f f * c
and the composite strength, for l > l c , calculated from the mixture rule on this
basis is thus:
σ = σ f (1 – l /2 l) + σ (1 – f )
[6.10]
c * f * f c m f
Equations [6.8] and [6.10] compare the strengths of continuous and short
fibre composites, and it emerges that 95% of the strength of a continuous
fibre composite can be achieved in a short-fibre composite with fibres only
ten times as long as the critical length. On the other hand, it must also be
remembered that, in a panel made of a chopped fibre composite, the fibres
may be randomly oriented in the plane of the sheet. Only a fraction of the
fibres will be aligned to permit a tensile force to be transferred to them, so
their contribution to the strength is correspondingly reduced.
6.4.3 Fracture behaviour of composites
Work of fracture in axial tensile parallel to the fibres
If a rod of composite material containing a transverse notch or crack in the
matrix (Fig. 6.19) is subjected to an axial force, the volume of material at the
minimum cross-section will experience the greatest stress and so it will start
to extend. This region will also suffer a lateral Poisson contraction, however,
giving rise to transverse tensile stresses parallel to the plane of the notch. If
a crack starts to run in a direction perpendicular to the fibres, the transverse
tensile stress will cause the fibre/matrix interface to pull apart, or ‘debond’,
as shown in Fig. 6.19(b). By this process the crack is deflected along the
weak interface in the same way as when a notched stick of bamboo wood is
bent. It does not snap into two pieces: it may splinter, but no cracks run into
the material since its structure consists of strong bundles of cellulose fibres
separated by lignin-based material at relatively weak-yielding interfaces.
Returning to Fig. 6.19, if the specimen is progressively further extended,
the fibres across the crack face will bear the load and will eventually fracture.
The component will eventually separate into two pieces by a process of fibre
pull-out (Fig. 6.19(d)).
The total work of fracture of a fibre composite is thus composed of
several terms, namely the work of debonding the fibre/matrix interface, the
work of deformation of the matrix as the crack opens and the work of fibre
pull-out. In fibre composites with a brittle matrix, the latter term often dominates
the toughness and its magnitude may be estimated as follows for a composite
containing a volume fraction f f of fibres of length l (> l c ).