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Deflection of the main crack in the direction of the interface was reported, particularly for short crack lengths in the SCS-6/Ti-15-3 composite [64, 68].

Matrix Crack Shielding by Fibers Several analyses [8, 85] have revealed that when a crack approaches a fiber or an elastic boundary stiffer than the medium where the crack propagates, the crack tip opening displacement (at a given stress intensity) decreases as the crack tip comes closer to a fiber manifested by a larger stress transferred from the matrix to the fiber (FCE). This is due to the constraint effect provided by the fibers on crack tip plasticity while the fiber-matrix interface is unbroken. Detailed analysis is provided later.

Crack Bridging by Unbroken Fibers In fiber bridging modeling there are two basic groups of models. Those based on a direct connection between FB and crack propagation energy and those that are treating FB as a distinct phenomenon.

The first work to consider the steady-state cracking of matrix cracks in a ceramic matrix composite during monotonic loading was published by Aveston et al. in 1971 [86]. The model, also known as the ACK model, was based on the determination of the matrix stress intensity factor, Km, through a strain energy balance before and after cracking. Using shear lag assumptions, and full crack bridging, they determined that for conditions of steady-state cracking, the stress intensity factor is independent of the crack length and is controlled only by the transition zone between the crack tip and the onset of steady-state cracking. In terms of the model the matrix is considered load free since the fibers support full load. Additionally, in cases where the crack is partially bridged, the contribution of the unbridged crack portion to the stress intensity factor is negligible.

In 1986 Budiansky et al. [87] suggested a new energy balance approach, similar to the ACK analysis, to describe crack growth. In terms of the model, the fibers are initially bonded, while debonding may be achieved by the passage of the crack. Both the ACK and the Budiansky model are referred to as steady-state fiber bridging (SSFB) models [88].

Another class of models was introduced in 1985 and 1987 by Marshall et al. [81] and McCartney [89], respectively. These models combine continuum fracture mechanics principles and micromechanics analysis to determine stress intensity factor solutions for an arbitrary size matrix crack, subjected to monotonic loading. According to these models (also known as generalized fiber-bridging models, GFB [88]) the friction stresses developed by the intact fibers within the matrix crack wake are idealized by an unknown uniform closure pressure. The evaluation of the closure pressure in the GFB models is obtained by combining crack opening displacement solutions from continuum fracture mechanics and from micromechanics analysis. Even though the models differ from each other in the methodology followed to relate those two issues, identical steady-state solutions (as derived from the SSFB models [86, 87]) are used as boundaries to characterize Km. The formulation of the GFB models as applied to fatigue loading was developed by McMeeking and Evans [90].

According to the GFB models, the restraining effect of the fiber causes a reduction in both the crack surface displacements and the crack tip stresses. Based on the Marshall, Cox, and Evans analysis (also known as MCE) [81], the composite stress intensity factor is defined by superposition of the normal stress intensity factor due to the remote stress on an unbridged crack and that due to the friction stresses due to fiber bridging. Using micromechanics analysis, the friction stresses were idealized as continuous, but with varying distributed crack flank pressure. The friction or bridging stress in relation to the fiber stress is given by p(x) = Of(x)Vf, (6.12)

where p(x) is the crack flank pressure, Of(x) is the fiber stress at a given distance x from the crack mouth, and Vf is the fiber volume fraction. Equation 6.12 could be considered valid only in cases where at least one fiber is positioned within the crack.

The friction stress can be related to the stress intensity factor, Ktip, through a modified Sneddon-Lowengrub equation [91]. The Sneddon-Lowengrub equation describes in a convenient form cracks in infinite bodies, loaded by arbitrary crack flank pressure distributions. In the case of a straight crack embedded in an infinite medium, Ktip is written as [88]:

where a represents the crack length, p(X) is the friction stress at X, X is the normalized distance along the crack length defined as X = x/a, and is the remote stress.

To evaluate Eq. 6.13, friction stresses are related to the crack opening displacement (COD) p(x) <x Vu (x). Such direct relation is based on the assumption that there is strain compatibility between the fiber and the matrix in the slip regime while outside the slip regime the effect of the shear stress is negligible (Fig. 6.15).

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