Characterization and control of the fiber-matrix interface in ceramic ...

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The multiple-matrix fracture condition set by Eq. (17) reveals that the
method is I.argely dependent on the volume of the matrix, thus on coating
thickness. If the coating is too thick, single fracture will occur. The
critical volume fraction of fiber and matrix for the transition from
single to multiple fracture are shown schematically in Figure 7.5. The
plot demonstrates that high-volume fractions of fiber are necessary far
mu1 t-iple fracture; thus, thin coatings are necessary for this technique
to be properly implemented. E’i-gure 7.6 graphically depicts the condition
for multiple-matrix fracture for a coated Nicalsn individual filament.
The inateri-als being considered are brittle ceramics I
It is assumed
that the fibers and matrix will possess very limited strain tal-erance;
thiis, fracture will occur over a small range of strain. The radial and
tangential. stresses induced by the contracting matrix as a result of
Poisson effects can he assumed to be negligible. The expressions for
stresses and strains in the matrix and fiber can be derived using the
fiber-sheath model (89-92) as shown in Figure 7.7. When a single
conti-nuous fiber embedded in a matrix is stretched in a direction
parallel to the fi.ber, tractions arise across the interEace as a result
of the difference in Poisson’s ratios of the fiber and matrix. Assuming
the fiber and matrix are circularly symmetric, the interfacial traction
is a normal stress (90) UP given by
where k is the plane-strain bulk modulus and G,
is the shear modulus of
the matrix. The stress at the interface caused by matrix contraction is