Page 177 - Engineered Interfaces in Fiber Reinforced Composites

Chapter 4. Micromechanics of stress transfer 159
1 1

6 = [E;(a,z) - E;,(a,z)]dz = -(1 - 2Vfk)CG
Ef
0
1
+
-_ [. - 2k(avr + l/Vm)] [‘““‘);I - 1 -e o(a-a) (4.140)
1
y
.Ef
in fiber pull-out. The residual relative displacement, a,, is the sum of the residual
strains in the fiber, &(a, z), and matrix, GpreS(a,z), after complete unloading over
the debonded region
P
(4.141)

The solution for Eq. (4.141) requires a knowledge of the stress (and strain)
distributions after unloading, which can be obtained in a procedure similar to that
for loading with minor modifications. The sign of the IFSS during unloading has to
be altered while other conditions of equilibrium remain the same for both fiber pull-
out and fiber push-out. In particular, the equilibrium condition between the external
and internal stresses given by Eq. (4.87) is still valid during unloading. Accordingly,
the condition for the stress transfer from the fiber to the matrix during unloading is
governed by Eq. (4.128) for fiber pull-out. Therefore, solving these equations with
other equilibria and boundary conditions given in Sections 4.3 and 4.4 yields the
following solutions for the FAS and MAS during unloading in fiber pull-out:

.f(~) = a + w(a - a)[l - exp(-k)] (4.142)
dm(z) = -yw(o - o)[l - exp(-k)] . (4.143)

Also, from the general relations between strains and stresses given by Eqs. (4.8) and
(4.9), and the additional radial stress q1 (a,z) of Eq. (4.18), the strains in the fiber and
matrix at the interface for fiber pull-out are obtained as:

Therefore, from Eqs. (4.142)-(4.145):
1
€;(a,.) = -{w(1 - 2kvf)[l - exp(-h)](8 - G)
Ef
+ (1 - 2kvr + wk)o} , (4.146)