The J formalism works well for any single mode among the opening mode I, the in-plane shear mode II, and the antiplane shear mode III, but even then, not when extensive crack growth precedes instability. However, except for the perfectly brittle crack, there is no detailed theoretical model for mixed-mode cracking, which is an important element in failure by shear instability. Continuum mechanical, elastic-plastic solutions are difficult to obtain for moving cracks, but such asymptotic and numerical results as are now available56 provide a promising framework for certain aspects of crack growth. Analogous results have been obtained for viscoplastic (for example, high-temperature creep) constitutive models.
Nearly all crack defect calculations are performed in two-dimensional approximations. Three-dimensional theoretical calculations are needed for cracks with curved fronts, crack interactions with curved dislocations, and crack interactions with compact second-phase particles of various shapes. Meandering and branched cracks, known to enhance toughness, represent another challenging three-dimensional problem, and work has begun in this area.57
Despite progress in treating the kinematics of rapid crack propagation,58 there remain ambiguities in understanding the problems of acceleration and inertia.
Two key relations are successful in qualitatively describing the relation of properties to microstructure on the basis of dislocation concepts. The first of these is the Hall-Petch relation between strength and grain size:
where σ is the flow stress, d is the grain diameter, and σ0 and K are material constants. The expression dates from the 1950s, but extensive work has been performed recently to verify it.59,60 The expression follows directly from dislocation pileup theory.21 The flow stress in lamellar two-phase structures such as pearlite in steel also follows Equation (1), in which case d then represents the thickness of the metal lamellae.
The second relation is the Orowan-Friedel expression for breakaway of a dislocation from pinning particles (see Figure 4).