Page 153 - Engineered Interfaces in Fiber Reinforced Composites

Chapter 4. Micromechanics of sfi-ess transfer 135

4.3.4. Instability of debond process

The instability condition requires that the derivative of the partial debond stress
with respect to the remaining bond length (z = L - e) is equal to or less than zero,
i.e., do$'dzdO (Kim et al., 1991). Therefore, the fiber debond process becomes
unstable if (L - C) is smaller than a critical bond length, z,,,, where the slopes of the
curves become zero in Figs. 4.23 and 4.24. At these bond lengths, the partial debond
stress, a:, corresponds to the maximum debond stress, CT;. The zmax value is
determined from Eq. (4.102) as
1 (4.104)
o(i5 - 0;)
((Ti - (Tt) + (a - 0;)
Numerical treatment of Eq. (4.104) gives z,,, values for the different composite
systems as shown in Table 4.3. It is worth emphasizing that for the Sic fiber-glass
matrix composites, z,,, values are very small relative to L,,, values, irrespective of
the fiber surface treatments and when compared to other epoxy matrix based
composites.
To show clearly how and to what extent the parameter, zmax, varies with the
properties of the interface and the composite constituents, a simple fiber pull-out
model by Karbhari and Wilkins (1990) is chosen here. This model is developed
based on the assumption of a constant friction shear stress, zfr, in the context of the
shear strength criterion for interface debonding. In this model, the partial debond
stress may be written as

(4.105)

where the frictionless debond stress, (TO, is given by

(4.106)

Eq. (4.106) is essentially similar to the solution of the debond stress derived earlier
by Takaku and Arridge (1973). The above instability condition for the partial
debond stress of Eq. (4.105) gives a rather simple equation for zmax as

(4.107)

where p4 is a complex function of o! and y, and is given by

(4.108)