Page 281 - Engineered Interfaces in Fiber Reinforced Composites

262 Engineered interfaces in fiber reinforced composites
behavior of the composite cannot be fully cxpressed by a single parameter, the
critical stress intensity factor, Klc, or the critical strain energy release rate, Grc, used
in elastic, homogeneous systems, but needs more complex functions of fracture
mechanics to describe the phenomenon.
He and Hutchinson (1989) considered a crack approaching an interface as a
continuous distribution of dislocations along a semi-infinite half space. The effect of
mismatch in elastic properties on the ratio of the strain energy release rates, GL/GT,
is related to two non-dimensional parameters, the elastic parameters of Dundurs, a
and p (Dundurs, 1968):

(6.23)

(6.24)

where p is shear modulus, v is Poisson ratio and E = E/( 1 - v2). The subscripts refer
to the cracked material 1 and the uncracked material 2, shown in Fig. 6.16.
Thcrcfore, a criterion for a crack to deflect along the interface is given by (He and
Hutchinson, 1989)

(6.25)

where GL(Y) is the fracture toughness for longitudinal splitting at a phase angle of
loading Y. c, d and e are non-dimensional complex valued functions of a and b. The
expression for the phase angle, Y, in terms of the elastic coefficient of the two media,
radius Y from the crack tip and the displacements u and u at the crack tip, Fig. 6.18,
is (Evans, 1989):

4 = tan-’(:) ,
lnr 1-p 1 1-p (6.26)
Y = 4 - --In- - tan-’ -In-
2K 1+p n 1+p

tY

Crack tip
E2 9 *2

Fig. 6.18. A crack at the bi-material interface. After Evans (1989).