Page 120 - Engineered Interfaces in Fiber Reinforced Composites

Chapter 4. Micromechanics of stress transfer 103

(4.19)

where the coefficients AI and A2 are complex functions of the elastic properties and
~~
geometric factors of the constituents, and are given by:
2[a( I - 2hf) + y( 1 - 2h~rn)]
A1 = , (4.20)
(1 + vm)[2yb2 ln(b/u) - u2]
A2 =- (1 - 2kvm)
-
A1 a(l -2kvf)+y(l-2kvm) > (4.21)

k = (mvf + YVm)/[E(I - vf) + 1 + Vm + 2~1. The solution of FAS is subjected to the
following boundary conditions

o'(L-4 =o'(-(L-e)) =gt=w(~j+o)[l -exp(-~)] (4.22)

where w1 = v,/(avf + yv,). A is the reciprocal length giving the effective frictional
shear stress transfer and i7 is the asymptotic debond stress for long embedded length.
These parameters are related to the interfacial properties in the debonded region,
namely the coefficient of friction, p, and the residual fiber clamping stress, 40, as:

(4.23)

(4.24)

where wo = avf/(avf + yvrn). In Eq. (4.22) is defined as the crack tip debond stress
at the boundary between the bonded and debonded regions at z = f(L - e) where
the FAS given in Eq. (4.25) must be continuous. Therefore, from the consideration
of cylindrical, elastic fiber and matrix, the solution of FAS and the corresponding
MAS and IFSS are obtained for the bonded region (-(L - e) <z< (L - e)):

azf(z) = --d - gcr - @) cosh fiz (4.25)
A2
A1 coshU/;T;(L-l)

(4.26)

(4.27)

(4.28)

In the debonded regions (-L<z< - (L - e) and (L - f2) <z<L), frictional slip
occurs between the fiber and matrix and the stress transfer is governed by the
Coulomb friction law for a constant coefficient of friction, p