[27,28]. These factors present important design challenges. Monolithic brittle materials subject to small-scale toughening are particularly challenging, because it is imperative that the scaling behavior be well calibrated, as well as manufacturing invariant [15]. The associated costs are often prohibitive. By introducing inelastic mechanisms that stabilize damage, through plasticity, internal friction, and so forth, progressive reduction in the scale sensitivity can be achieved. Eventually, as the inelastic mechanisms are made more effective and the number density of active inelastic strain sites is increased, macroscopic inelastic strain can be induced prior to the UTS [8,38,39]. Such deformations profoundly change the design philosophy. With sufficient inelastic deformation, the scale dependence is virtually eliminated, the UTS distribution is essentially Gaussian, and design strategies used for metals can be applied.

When size scaling of the UTS obtains, it is vividly manifest upon testing specimens that have vastly different stressed volumes, for example, tension relative to bending [27]. The goal of the testing is to characterize the population of defects that control the strength, S (Figure 7a). This population can be represented by a frequency distribution, designated g(S) dS [28], the fraction of defects per unit volume (or per unit surface area for surface flaws) that cause failure when the stresses are between S and S + dS. Then, the survival probability of a small-volume element, ∆V, subject to uniaxial stress σ is [28]

where δϕ is the failure probability. This relates to the overall failure probability Ф through [27,28],

where N is the number of volume elements in the component and Π refers to the product over all elements. G(S) for each flaw population can be fit by a power law,

where m is the shape parameter, So the scale parameter, and Su the offset parameter. However, the strength is affected by more than one defect population, and G(S) is not normally single-valued. That is, different m, So, and Su are needed to represent the defect populations over the full stress range relevant to specimen testing and design. This presents stringent testing requirements for the acquisition of accurate design data.

When inelastic deformation mechanisms operate around the defects, G(S) can be dramatically modified. These mechanisms include plastic yielding and internal friction. The most comprehensive illustration is for fiber-reinforced composites that exhibit large-scale debonding and friction, such as CMCs and TMCs. These mechanisms “blunt” and stabilize the damage emanating from failed fibers, by redistributing stresses in the surrounding material, thereby reducing the stress concentrations in the nearest fibers [40-42]. The consequence is that multiple fiber failures can occur before the UTS is reached [43], often allowing other inelastic mechanisms to be activated in the matrix. The stabilization



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