give values for metals74 and ceramics,75 respectively, of at least 10 and 3 times the surface energy; thus, a dissipation process other than the creation of surface is still active. Progress has been made in understanding the DBTT for the critical mode I stress intensity KIc(MPa m1/2) that represents the resistance of a material to crack propagation. The intensified strain field of a crack causes a crack nucleation event that could be related to grain size at a critical distance ahead of the crack.74 The event could correspond to the cracking or decohering of a second-phase particle such that the crack then propagates in an unstable manner into the matrix. The problem then becomes statistical and involves the spacing, position, and crack nucleation probability of particles and the position-dependent stress field of the crack;75 some fractographic support exists for such a model.76 Near the DBTT, particle cracking may not be the critical event. Instead the crack may propagate across a single grain and then be arrested. Nonpropagating microcracks spanning grains have been observed in heat-treated steel, with a maximum density at the DBTT.77 The critical event would then be statistical as before but now involves unstable propagation of a favorably situated and oriented microcrack.
Dislocation pileups at the crystal plasticity level of description of plastic deformation, or shear instabilities at a more macroscopic level, could be important in providing stress concentrations to enhance crack nucleation. Even when a crack tip is stressed at the Griffith level, theoretical calculations show that there is usually enough of a stress concentration at the tip to move several dislocations, a tendency that is more likely the lower the strength of the material.
In ceramic materials where the matrix is normally completely resistant to dislocation motion, toughening can still be achieved by screening of the crack tip in a manner analogous to dislocation screening. In this case, the screening is provided by metastable dispersed particles (e.g., ZrO2 in Al2O3) that undergo a phase transformation in the stress field of the crack tip and produce transformation stresses that screen the tip in the sense of decreasing the local stress that would tend to extend the crack.78–80 Analogous toughening effects can be achieved if nonpropagating microcracks form in the region of the crack tip.80,81
The concepts of flow and fracture in alloys are sufficiently sound and tested to provide at least qualitative guidelines for alloy design, a situation that did not exist in 1960. An ideal alloy should be strengthened primarily by hard dispersed second-phase particles, because these give large work hardening, resistance to failure by plastic instability, and thus some damage tolerance. The particles should be as fine as possible, but greater than the bypass size of about 1 nm, to minimize crack nucleation or decohesion. The