with Lo being a reference gauge length (normally 25 mm), τ the friction stress, R the fiber radius, f the fiber volume fraction, and Sg the tensile strength at length Lo. The first term is that contributed by the fibers. The second is provided by the matrix. For TMCs, σo is the matrix yield strength. For CMCs, σo ≈ 0, because of multiple matrix cracking. The main features described by this result are as follows: (i) The UTS is affected by friction. (ii) There is a weak dependence on the shape parameter, m, characterizing the individual fiber strengths. (iii) There is a strong effect of the mean reinforcement strength, Sg.

Usually, low friction is not sufficient to realize GLS conditions. Concentrated stresses persist upon fiber failure, resulting in local load sharing (LLS) [49,50]. The ensuing UTS decreases below that given by (6) and becomes scale-dependent. But the effects are not especially deleterious. In practice, the UTS scales in accordance with (3) to (5), subject to a large effective shape parameter, m* [49]. More importantly, the behavior is insensitive to the occurrence of large manufacturing flaws, because of the existence of a robust strength minimum, Smin (see Figure 7b). This attribute is understood upon examining the stress concentrations induced by large flaws. Two effects are involved: (i) One is associated with large-scale fiber bridging (LSB), governed by the index [50]

where a is the size of the flaw, E is Young's modulus, A is an anisotropy coefficient of order unity, with the subscripts f and m referring to the fibers and matrix, respectively. Based on this index, the strength contributed by the fibers through LSB is [50,51]

Note that when τ is small, η→ 0 and the flaws have no effect on the strength. (ii) The second effect is related to the notch performance enabled by large-scale inelastic strains [4,6,8,9]. These dominate at larger τ. Such strains diminish the stress concentrations around the manufacturing flaws, leading to tensile behavior independent of the flaw size, analogous to the notch insensitivity.

Inelastic Strain and Ductility

When the number density and the potency of the inelastic deformation sites increase, macroscopic inelasticity initiates at stresses below the UTS. This typically causes a marked increase in the failure strain, or ductility. The inelastic strain now allows stress redistribution. An illustration for a mechanical attachment to a CMC is presented in Figure 9 , along with an experimental validation, obtained using Moire interferometry [5,9]. The key feature is that the stress concentration factor around the hole diminishes upon increasing the load, as an inelastic zone develops. This diminished stress concentration combines with an elevation in the local UTS arising from volume scaling [50,52]. The overall effect is a component performance unaffected by the presence of the hole. These characteristics originate with the inelastic strain mechanisms, manifest in the stress-strain curves.

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