For strong dispersoids ϕ approaches zero and dislocations can completely bow out and bypass particles, leaving behind loops of dislocation that encircle the particles. The loops can assume various configurations, readily understandable on the basis of Ashby’s concept of geometrically necessary dislocations.64 These dislocations give rise to rapid work hardening in dispersion-strengthened alloys, of great benefit in inhibiting plastic instability and in giving rise to long-range back stresses.65

Ductile Fracture

In recent years ductile tensile fracture has been classified in three types.66 The first is necking to a point or chisel point, as might occur for a pure fcc metal. The second is deformation or necking terminated by a shear instability leading to a mixed-mode crack following the shear trace, as might occur for a nominally pure bcc metal. The third involves necking or deformation leading to void formation at inclusions or second-phase particles and crack propagation by void linking through either local necking of ligaments or shear localization. The latter process is most pertinent to complex engineering alloys. Figure 5 shows the crack propagation process. Particles crack or decohere under the influence of the crack strain field and thereby nucleate a void. The void grows and limits the plastic flow to a region whose extent is of the order of the void spacing.67 Thus, the smaller the void spacing, the less the plastic flow, the lower the energy release rate, the lower Jc, and the less the toughness. Smaller void spacings are associated with weak interfacial cohesion, brittle particles, large particles, and small spacing of particles. The particle size enters because the nucleation of a crack or a decohesion becomes less probable as the particle size decreases. A rough estimate for spherical particles indicates that the critical local stress for decohesion is proportional to the inverse square root of the particle size and that decohesion should not occur below a critical size of about 20 nm. For very fine particles, of approximately 1 nm, the particles become ineffective as obstacles. These numbers would change somewhat for other particle shapes, particularly those with sharp salient features. Hence, a “window” of sizes exists for optimum dispersion strengthening and toughening.

Theoretical calculations, with some experimental support, also indicate that voids, once formed, increase the susceptibility of a material to failure by macroscopic shear instability.68 The susceptibility to shear instability is much greater under plane-strain conditions and when work hardening is low.69,70 Surface instability in the form of surface rumpling is also a precursor to bulk shear instability and is amenable to experimental study.70

The presence of a metastable phase that transforms in the presence of a local stress or strain concentration provides another means of improving toughness. In transformation-induced plasticity, when a material necks or

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