Materials for engineering, 3rd Edition - (Malestrom)

Composite materials 209
(a) (b) (c) (d)
6.19 Propagating transverse crack in an element of fibre composite.
When fracture occurs by the breaking of fibres, then all those fibres which
end within a distance l c /2 of the cross-section which crosses the break will
pull out of the matrix. The fraction pulling out is thus = l c /l. Assuming the
fibre/matrix interfacial shear strength τ is maintained constant, the work to
pull out one fibre of length x is
work = πr
2
∫
0
∫
x
σ dx
2
= πr (2 τx/ r)dx
= πrτx 2
0
x
For a volume fraction of fibres f f , the number of fibres per unit area =
f f /πr 2 . In order to pull out, the fibres can only have a length up to l c /2 above
the plane of fracture: if this distance from the fibre end is x, then the work
of pullout is exactly πrτx 2 . (N.B. if the distance is (x + dx), the work is still
the same, since dx is vanishingly small).
Considering all the possible positions along the fibre where the plane of
fracture might be, the probability of it intersecting exactly at x is zero, but the
probability of it intersecting over the length dx (i.e. of pulling out a length
between x and x + dx must be:
d x/total length = d x/ 1 l
2 c
So the total work of pullout is,
l0
/2
2
2
W = ( f / πr ) l / l πrτx d x/( l /2)
f
ffτlc 3
= 12 rl
c
∫
0
c