## Od 620

d where l is the embedded fiber length, d is the fiber diameter, and t is the interfacial shear strength. Published results in the literature indicate values for interfacial shear strength of SCS-6/Ti-15-3 and SCS-6/Ti-6-4 in the range of 124-148 MPa and 138-156 MPa [31, 94]. Furthermore, experiments have shown that the interfacial shear strength increases for longer embedded fiber length or thicker specimens and gradually approaches a constant value for fiber lengths approximately 4-5 times the fiber diameter (d = 140 ^m for the Textron SCS-6 [95]) [46]. Nevertheless, Eq. 6.19 is limited by the composite ultimate tensile strength for debonding to be achieved before fiber failure. Such a premise can be considered as the distinction between a strong and a weak interface bond.

To determine the flow response of the composite is a difficult task. This is due to uncertainties raised by the presence and contribution of the fiber within the plastic zone [69]. At short crack lengths, close to the interfiber spacing, it is rational to assume that the crack tip plastic zone is fiber free and hence the flow response of the uMMC is matrix flow dependent. However, as the crack length increases, the possibility of having a number of fibers within the plastic zone is strong and rational. A better understanding of the way fibers are entering the plastic zone is provided by the following steps, recognized by the TZMM. The condition for crack propagation is achieved as follows: with the crack tip positioned between two fibers as shown in Fig. 6.17, the level of the stress a3 is given by Eq. 6.18. On further crack growth the level of a3 increases due to the increase in n\ until a3 attains the value for debonding, derived by Eq. 6.19. This is happening at a critical crack length defined by ni1c. The value of ni1c and, therefore, the crack tip position at the critical point, is obtained by substituting Eq. 6.19 into Eq. 6.18 and solving for ni1. At this point the crack tip plasticity constraining effect of the fiber is overcome since the plasticity is now allowed to pass around the fiber and become constrained once again by the next fiber. This behavior, which is in direct agreement with work published by Schulte and Minoshima [7], justifies that the major factor controlling the fatigue resistance of the material, especially at short crack lengths, is the plasticity constraining effect (PCE). The above are schematically shown in Fig. 6.18.

Undoubtedly, an accurate application of Eq. 6.18 requires a sound determination of the composite flow stress. If we considered that the crack is long

FIGURE 6.18 Conditions for crack propagation. At (a) the crack tip plastic zone contacts the high stiffness fiber. As the crack propagates further, the plastic zone is squeezed between the crack tip and the intact fiber (b). As a result a tensile stress is starting to build up around the fiber (constrain effect). When constrain effect has become sufficient to initiate debonding (c), the plastic flow propagates round the fiber and the constrain effect relaxes (d).

FIGURE 6.18 Conditions for crack propagation. At (a) the crack tip plastic zone contacts the high stiffness fiber. As the crack propagates further, the plastic zone is squeezed between the crack tip and the intact fiber (b). As a result a tensile stress is starting to build up around the fiber (constrain effect). When constrain effect has become sufficient to initiate debonding (c), the plastic flow propagates round the fiber and the constrain effect relaxes (d).

enough to contain fibers, then the flow response ahead of the crack tip would be controlled by the matrix yield stress and by the high stiffness phase. Such collaboration is justifiable by examining stress-strain curves of the constituent materials, (Fig. 6.19).

The fact that similar behavior to that shown in Fig. 6.19 has been observed in most uMMCs convinced many workers to accept that an isostrain condition between the fiber and the matrix within the plastic zone is somehow justifiable [48, 96, 97]. Based on the above, the flow response of the uMMC can be written as [96]:

Assuming that debonding is always achieved before fiber failure, the crack propagates through the matrix without breaking the fibers. Subsequently, intact fibers located behind the crack tip slide in relation to the matrix, producing friction (bridging) stresses a1, which reduce the crack driving force and improve the fatigue resistance of the MMC. The friction stress at each bridged fiber row

in the crack zone is calculated as follows [48]:

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