Page 119 - Engineered Interfaces in Fiber Reinforced Composites

102 Engineered interfaces in fiber reinforced composites

(4.1 1)
do'(z) 2
f-
dz - --zj(a,z) , (4.12)
a
(4.13)

where y = a2/(b2 - a2) is the volume ratio of the fiber to the matrix. It is assumed
here that the plane normal to the z-direction remains plane in plane strain
deformation of the matrix. The average MAS is thus defined by

b2
dm(z) = - ]&(r,z)rdr (4.14)
-a2
a
In the bonded region (- (L - e) <z Q (L - e)), the applied stress is transferred from
the matrix to the fiber through the IFSS, q(a,z), such that the equilibrium condition
can be obtained by combining Eqs. (4.1 1) and (4.12) as

(4.15)

Since the matrix shear stress, C(r,z), has to be compatible with IFSS, zi(a,z), and
the matrix cylindrical surface is stress free (Zhou et al. 1993)
Y(b2 - $1
zz(r,z) = qyz) . (4.16)
ar
Also, the axial displacement is continuous at the bonded interface (i.e.
Urm(a,z) = q(a,z)). Combining Eqs. (4.8)-(4.10) and (4.16), and differentiating with
respect to z gives

(4.17)

An additional radial stress, ql(a,z), acts at the interface that arises from the
differential Poisson contraction between the fiber and the matrix when the matrix is
subjected to an axial tension at remote ends. 41 (a, z) is obtained from the continuity
of tangential strain at the interface (Le. efe(a,z) = eL(a,z)) (Gao et al., 1988)

(4.18)

Therefore, combining Eqs. (4. IO), (4.1 I), (4.17) and (4.18) yields a second-order
differential equation for the FAS