Page 178 - Engineered Interfaces in Fiber Reinforced Composites

160 Engineered interfaces in fiber reinforced composites

1
8m(a,z) = -{2aokvfo - yo( 1 - 2kv,)[l - exp(-h)](~ - o)} . (4.147)
Em
Hence, for the fiber pull-out, the residual strains after complete unloading when the
external stress is zero now become:
1
&(a,z) = --o(l - 2kvf)[l - exp(-h)]a , (4.148)
Ef
1
.
8m-res(~,~) - 2kv,)[l - exp(-h)]~ (4.149)
--yyo(l
=
Em
Consequently, the residual relative displacement in fiber pull-out is obtained by
combining Eqs. (4.141), (4.148) and (4.149):
1 1 - exp(-Ae) loo .
-
6, = OlEf [a + y - 2k(avf + yvm)] (4.150)
Similarly, the expressions for 6 and 6, for the fiber push-out are derived as:

1
[E"~(u,z) - E~,(u,z)]~z = -(1 - 2vfk)b
Ef
0
1
- - + y - 2k(avf + yv,,,)]
[M
MEf

Eqs. (4.140) and (4.150)-(4.152) are used to evaluate the response of the model
composites in cyclic loading and the displacements 6 and 6, can be expressed as a
function of the alternating stress, Ao, and the number of cycles, N. In experiments,
degradation of the interface properties, e.g., the coefficient of friction, p or
A(= 2@/u), can also be expressed in terms of the cyclic loading parameters, Au and
N. In practice 6 and 6, can be measured using optical methods (with a microscope)
or by means of more complicated instruments (see for example Naaman et al.
(1992)) in fiber pull-out. Alternatively, they can be directly determined from the load
and load-point displacement records in the case of fiber push-out.
It is envisaged that the degradation of the frictional interface properties and the
corresponding increase in the relative displacements eventually lead to debond crack
growth once the debond criterion is satisfied. The debond criterion based on the
energy balance theory given by Eq. (4.35) under monotonic loading can be rewritten as

G = G(~o,ct;,e) 2 Gic > (4.153)

where pois the original value of the coefficient of friction before degradation and o:
represents the applied fiber stress corresponding to debond length, e. Under cyclic
loading the debond criterion becomes