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

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Fiber length becomes important after matrix failure. When a brittle
matrix fractures, the fiber and matrix debond and the ultimate fracture
strength of the composite will then depend on the forces to overcome the
friction between the fibers and matrix during fiber pull-out and to
fracture the reinforcing fibers. Once debonding has occurred, the
frictional resistance to the fiber pulling out of the matrix is
determined by the normal compressive forces exerted on the fiber by the
matrix (141-143). A semiempirical expression for the tensile stress on a
fiber during pull-out in a glass or carbon-fiber-reinforced cement is
given as
where Be is the embedded length of the fiber, Ti is the interfacial shear
stress, df is the fiber diameter, and k is a fitting constant. As shown
earlier, there is a minimum length over which efficient load transfer can
occur. The argument can be extended to determine the influence of
interfacial frictional stress on the ultimate strength of a composite by
assuming that the interfacial frictional stress decreases linearly with
stress on the fiber (see Figure 7.4).
The postcracking strength of a
fiber-reinforced composite is then
where Vf is the volume fraction of fiber and the variables are as in the
previous equation and
where E is the modulus, Y is the Poisson's ratio, and p is the
coefficient of friction between the fiber and matrix.