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FIGURE 6.15 Schematic representation of slip regime. Note that length of slip regime is not constant, but it increases as crack propagates further and bridging fibers remain intact.

FIGURE 6.15 Schematic representation of slip regime. Note that length of slip regime is not constant, but it increases as crack propagates further and bridging fibers remain intact.

According to MCE the closure pressure is given by:

where u(x) is the COD at x, t is the shear resistance of the interface, R is the fiber radius, and Ef,Ec,Em represent the elastic moduli of the fiber, composite and matrix, respectively.

As an improved solution to the MCE closure pressure, the shear lag model was further modified by McCartney [89] in order to make the model energetically consistent. The MCE closure pressure was calculated as:

Moreover, later numerical comparisons between the two models have revealed that the crack opening displacement profile obtained by Eq. 6.15 is identical to that obtained by Eq. 6.14 only if the shear resistance is reduced by a factor of 3.2 [79]. Additionally, the lack of a standard method for obtaining the correct shear resistance (it was mentioned before that different methods could produce great variations) urged Kantzos [92] to suggest an alternative solution [known as the fiber pressure model (FPM)] for the determination of the closure pressure. The closure pressure in FPM is assumed to be equal to the stress carried by the fibers in the bridged region averaged out over the total bridged area (a — a0). The problem of the shear stress parameter was overcome by suggesting w 6 wa0[0.5(w — a0) — (x — a0)] 1 w — a0

where w is the specimen width, a0 and a are the initial notch length and total crack length, and x is the distance to the bridged area measured from the free surface.

Even though most of the models described above do capture the essential features of fatigue damage, that is, matrix cracking and fiber-matrix debonding, there are several limitations to the ability of the models to predict the crack driving force: (a) the friction stresses are idealized as a continuum closure pressure; (b) the one-dimensional micromechanical analysis used to relate COD and closure pressure is based on the assumption that the fiber in the wake of the matrix crack is far from the crack tip, and so any crack tip effect is negligible [88]; (c) the complex and time-dependent conditions ahead of the crack tip are not modeled (i.e., fiber failure, extensive debonding, and the corresponding differences in composite fatigue resistance); (d) the models claimed to be applicable to situations in which cracks are long compared to the distance between two successive fibers (interfiber spacing) and partially bridged [81]; and finally (e) high crack tip plasticity coupled with crack tip-fiber interactions are difficult to reconcile with linear elastic fracture mechanics (LEFM), since matrix small-scale yielding and homogeneous continuum mechanics principles are violated in composites [88].

To overcome similar problems, de los Rios et al. [4] suggested that the fatigue crack growth in uMMCs should be addressed in terms of the crack tip opening displacement (CTOD). The model, originally developed by Navarro and de los Rios [93], implements the representation of the crack and its plastic zone by means of dislocations subjected to an applied stress a, as first developed by Bilby et al. [47] in 1963 for monolithic materials. In its original form, the model considers infinitesimal dislocations distributed within two regions or zones, one for the crack itself and the other for the plastic zone. In 1995 [4], the model was extended further by considering a third zone to represent cases where the plastic zone is blocked by grain boundaries (this is common for short cracks). Such an approach was argued to be more realistic in physical terms since in the two-zone system an infinite stress level is sustained by the grain boundary. In terms of the three-zone system, also known as three-zone micromechanical model (TZMM), the plastic zone size (slip band ahead of the crack tip) is blocked by the grain boundary and remains blocked until the stress in the third zone, that is, the grain boundary, attains the required critical level for dislocations to cross this zone. Adapting the same system for a uMMC, the three zones of the crack system are: (a) the crack, (b) the plastic zone, and (c) the plastic constrained zone at the fiber

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