This study has been inspired by two proposed models for stable crack growth described by Sih and Kiefer (1979) and Wnuk and Sedmak (1980). Basic assumptions of the second model are reexamined and refined. Although the mathematical analyses involved in both approaches are widely different, it is shown that the physical assumptions underlying these two models are analogous. Both treatments assume a “quantum” nature of crack growth by representing extension of a stable crack as a sequence of steps each of which is executed as a finite jump in the current crack length. While Sih and Kiefer employed the numerical approach to treat the three-dimensional (3D) elastic-plastic fracture problem, here a highly idealized line-plasticity model, modified by an addition of the final stretch growth law, is used.

In earlier final stretch model the quantities essential in the formulation of the growth law, such as the growth step Δ and the opening constants δ or J, were assumed to be material constants. Now it is demonstrated that the first of these assumptions may be relaxed and replaced by a more general requirement of the constancy of the ratios COD/Δ or J/Δ. Indeed, a self-consistent theory may be constructed if only the ratio of a certain measure of the apparent material fracture toughness to the size of the growth step is maintained constant throughout the slow growth phase.

It is shown that variations in the growth step result in reduction of the true and apparent fracture toughnesses associated with the quasi-static crack, which are represented respectively by the tearing modulus δ̂/Δ and any suitable resistance parameter such as J-integral, COD, or extent of the plastic zone generated at the crack front. It is concluded that the models that are based on ideal plasticity and that ignore variations in the growth step tend to overestimate material ability to sustain a stable crack, and thus lead to non-conservative predictions regarding occurrence of terminal instability.