*2.5. Role of Internal Stresses and the Modified Kitagawa-Takahashi Diagram*

From Griffith's equation, we note that the stress for crack growth increases with decreasing crack length according to the relation:

$$
\sigma\_{\text{apl}} = \frac{\sqrt{(2E\gamma)}}{\sqrt{\pi a (1 - \nu)}} \tag{2}
$$

Where the required applied stress increases with decreasing size of the crack approaching infinity when the crack length goes to zero. Figure 6 shows this in a log–log plot. For ductile materials, with the increase in the applied stress, the material yields before a crack is initiated. From Figure 6, we can define the minimum internal stresses, and the gradients required for crack initiation can be defined by what we call an internal stress triangle. In some sense, we are generating the local internal stresses, by way of dislocation pile-ups, to augment the applied stress for the crack initiation and growth, until the remote applied stress is sufficient to sustain further growth of the initiated crack. The internal stress triangle thus defines both the magnitude and gradient required for incipient crack initiation and its growth. Based on this figure, the following points can be made: (a) In addition to the magnitude of the stress, its gradient is also involved in sustaining the growth of the incipient crack. If the gradient is too sharp, as in the case of sharp notches, the initiated crack may be arrested if the stress falls below Griffith's line. (b) Furthermore, it is difficult to separate the initiation vs. growth as both are simultaneously involved since Griffith's condition corresponds to the maxima in the total energy (unstable equilibrium). It may be possible that the initiated crack can be stabilized due to oxidation of the mating crack surfaces, but this is a separate issue.

**Figure 6.** (**a**) Griffith crack representation with yield stress defining the required minimum internal stress magnitude and gradient. (**b**) Parallel representation of the modified Kitagawa–Takahashi diagram for subcritical crack growth.

Figure 6b shows a similar behavior that can be extracted using the modified Kitagawa–Takahashi diagram, which was initially developed for fatigue failure. The crack-growth threshold stress-intensity factor replaces Griffith's criterion, and the endurance limit replaces the yield stress. Our modification involves extending the threshold line beyond the endurance limit, thereby defining the internal stress triangle. At the endurance of a smooth specimen, close to 107 cycles are needed for the crack to initiate and grow. These cycles are required for the development of the needed internal stresses and their gradients for an incipient crack to form and grow. The initiation and growth of the short crack in the endurance have been accounted for by the fracture mechanics community by invoking the similitude break down and proposing that the short crack threshold is different from that of long crack thresholds due to crack closure. We have shown using the dislocation theory that the crack-closure concept is inherently faulty in the plane strain regim e, and no similitude break down is needed to account for the short crack growth behavior. The short crack grows due to the presence of both applied and in situ generated internal stresses arising from inhomogeneous deformations in the polycrystalline materials. The thresholds do not depend on the crack size, and one has to properly account for the local build-up of internal stresses and their gradients due to dislocation pile-ups. A detailed review of short crack growth was provided recently [30].

Internal stresses are difficult to determine. One can compute them based on some physical models such as dislocation pile-ups. The Kitagawa diagram provides some way to estimate the required minimum internal stresses and their gradients for the incipient crack to grow at the threshold condition. The stresses are higher than those causing acceleration of the crack growth while the stresses are lower than those causing crack arrest. Overloads and underloads, for example, can change local internal stresses (sometimes referred to as residual stresses, which are only a subset of the internal stresses) and contribute to changes in the crack growth kinetics. From equilibrium consideration, the internal stresses are self-equilibrating. That is, there will always be a plus/minus type with the net result of maintaining the specimen in equilibrium. The fact is internal stresses resulting from inhomogeneous deformations are involved in the nucleation and propagation of the cracks in specimens, even though they are difficult to measure.

## *2.6. Role of Chemical Forces*

Figure 7 provides a simple case where chemical forces manifest in terms of reduction in the surface energy of the crack surfaces, thereby reducing the required applied stress needed for a Griffith crack, for example, to be initiated and grow (Equation (2)). The total energy as a function of the crack length is reduced (Figure 7a), and, correspondingly, the applied stress needed for a given crack length is reduced (Figure 7b). The micromechanisms involved in the reduction of crack tip driving force can be

complex. Nonetheless, the net result from the point of engineering considerations is that there is a reduction in the required applied stress to contribute to the same crack length or crack growth rate. Hence, we can formally define the mechanical equivalent of chemical forces based on the reduction in the required stress to cause the same crack growth rate. This is shown in Figure 7b. To compute the chemical stresses involved, we need, therefore, crack growth in the inert medium as a reference state. For corrosion fatigue, fatigue in an inert medium can be used as a reference state. However, care should be exercised since fatigue is a two-load parametric problem requiring σmax and Δσ for Stress vs. number of cycles for failure (S-N fatigue) fatigue or two stress intensity factors (Kmax, ΔK) for fatigue crack growth [30]. However, for stress corrosion or sustained load crack growth, there is no subcritical crack growth in an inert environment for a reference state. Only the fracture toughness value in an inert medium provides the reference.

**Figure 7.** Role of chemical forces and their quantification. (**a**) Total energy as a function of crack length and (**b**) log(stress) vs. log(crack length) plot.
