6432 Threshold Stress Intensity Factor

In ductile monolithic materials the transition of a short crack or cracklike defect into a catastrophic fatigue crack is related to a boundary condition known as: (a) fatigue limit or (b) threshold stress intensity factor, Kth, [111, 112]. In brief, the Kth boundary represents crack tip strain conditions able to create plasticity damage at some certain distance ahead of the crack tip.

In uMMCs, the evaluation of a similar boundary condition is a puzzling task due to the number of parameters involved in the fatigue damage process. During the early stages of research, many workers supported the idea that crack arrest in

MMCs can be defined in a way similar to crack arrest in monolithic material [54, 58]. In detail, they assumed that when the crack growth rate is approximately 10-8 mm/cycle and no crack progression is detected for at least 107 cycles, then conditions of crack arrest prevail. Undoubtedly, such empirical approach is not able to provide numerical solutions and consequently information for design. In 1996, de los Rios et al. [96] published a work where crack arrest (threshold) is achieved when the crack strain conditions cannot overcome the FCE and therefore cannot propagate plasticity. In other words, if the crack cannot develop the required shear stresses at the interface to initiate debonding, crack arrest conditions should be assumed. These hypothetical conditions for crack arrest are shown in Fig. 6.25.

In terms of mathematical modeling, crack arrest is achieved when two boundary conditions are met: (a) the crack contacts the fiber (negligible plasticity) and (b) the shear stress at the interface is still lower than the interfacial shear strength. According to Fig. 6.17, the above boundary conditions can be written as:

ni = U2 ~ 1, which proclaims that the crack tip plasticity is minimum, and

FIGURE 6.25 Conditions for crack arrest. (a) FCE starts instance plastic zone contacts the fiber. (b) and (c) Further propagation of crack against fiber results into plastic zone condensation and higher shear stresses at interface. (d) Crack arrest.

FIGURE 6.25 Conditions for crack arrest. (a) FCE starts instance plastic zone contacts the fiber. (b) and (c) Further propagation of crack against fiber results into plastic zone condensation and higher shear stresses at interface. (d) Crack arrest.

which proclaims that the developed shear stress at the interface is lower than the interfacial shear strength. Since n\ = n2 ~ 1, then sin-1«! ~ sin-1^ ~ 1 and cos-1 n2 a + d + am). Using these approximations into Eq. 6.18, the maximum allowed applied stress, which would still lead to crack arrest of a particular crack length, a, yields

n Va + d + ain where o3d is given by Eq. 6.19 and a^ represents defect or notch size.

Equation 6.31, shown in Fig. 6.26, represents a theoretical Kitagawa-Takahashi (KT) curve for the uMMCs. However, in contrast to the monolithic materials where the true fatigue limit is the highest stress level, which is unable to transform a fatigue flaw into a fatigue crack, in uMMCs the true fatigue limit corresponds to the inability of an already established fatigue crack to propagate beyond one or more fiber rows. The accuracy of Eq. 6.31, considering a1 = 0, can be seen in Fig. 6.27.

CRACK LENGTH (m)

FIGURE 6.26 Crack arrest curve for 32% SCS-6/Ti-15-3 uMMC. For calculations values of o3d = 1173 MPa, an = 140 ^m and d = 140 ^m were used. Negligible bridging stress was considered. Such simplification is reinforced from fact that small cracks are expected to be arrested by first or second fiber row. Consequently, contribution of bridging stress is expected to be negligible or minimum.

CRACK LENGTH (m)

FIGURE 6.26 Crack arrest curve for 32% SCS-6/Ti-15-3 uMMC. For calculations values of o3d = 1173 MPa, an = 140 ^m and d = 140 ^m were used. Negligible bridging stress was considered. Such simplification is reinforced from fact that small cracks are expected to be arrested by first or second fiber row. Consequently, contribution of bridging stress is expected to be negligible or minimum.

■ Vf = 35% from Ref.[54] • Vf = 36% from Ref.[58]

■ Vf = 35% from Ref.[54] • Vf = 36% from Ref.[58]

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