## 6431 Initial Cracks

Effect of Residual Stresses One of the main factors affecting the performance of uMMCs is the development of residual stresses during manufacturing. These stresses, which arise from the difference in the thermal expansion coefficients (CTE) between the metallic matrix and the fiber, may be responsible for the presence of radial cracks observed in the as-received condition [104, 105] (Fig. 6.22).

In most uMMCs, the matrix CTE, am, exceeds that of the fiber, af, thus, after cooling from the processing temperature, the matrix is subjected to an axial tensile stress am. Large matrix tensile stresses may induce premature crack initiation, especially in the presence of external loading [68]. The residual thermal stresses in the matrix can be evaluated as [104]:

Ec where AT is the effective temperature range. It should be noted that residual stresses developed during the fabrication process at absolute temperatures greater than half of the melting point of the matrix, are not considered as a result of the matrix stress relaxation due to creep or viscoplastic flow [62]. Furthermore, when the matrix contracts more than the fiber Aa > 0, as in the case of a single long fiber embedded in an infinite matrix, the matrix is placed in hoop tension, which can be simulated by Lame distribution [106]:

where R is the fiber radius and x is the distance from the fiber center. Equation 6.28 proclaims that the maximum stress due to the relaxation of the residual stresses is situated at the broken fiber radius, x = R. Furthermore, Eq. 6.28 can be expressed in terms of stress intensity factor by including the geometric features of broken fibers at the edge and the projected length of an initial crack:

Generally, maximum matrix cracking as a result of residual stress relaxation is expected in the case of broken fibers (due to machining). To encounter the effect of such geometric discontinuities, elastic stress concentration formulas are usually used. Considering a semicircular corner crack configuration to represent the geometry of a broken fiber, Eq. 6.28 can be written as [107]:

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