*2.4. Crack Initiation at the In Situ Generated Stress-Concentrations*

Since stress concentrations are essential for the initiation of cracks in the material, if there are no pre-existing cracks or stress concentrations, then the inhomogeneous deformations produce localized stress concentrations via the formation of dislocation pile-ups, as discussed by several authors, beginning with Eshelby [27]. The typical internal stress profile ahead of a dislocation pile-up deduced by Eshelby is shown in Figure 5, which was experimentally confirmed by X-ray analysis by Gao et al. [28]. The internal stress profile is not much different from that of a notch tip stress field. In fact, it was shown that most of the stress fields at stress concentrations follow a similar pattern [29]. The implication is that incipient cracks that form at stress concentrations will also have typical profiles for their stress intensity factor, as shown in Figures 3 and 4, with minima that must exceed the governing threshold for continuous crack growth. Otherwise, initiated growing short cracks are arrested. Experimentally, one finds many short cracks are initiated near the surface under, for example, corrosion, but only one or two grow and thus contribute to specimen failure.

**Figure 5.** Stress field in the next grain due to dislocation pile-up, from Eshelby, and X-ray data from Gao et al. (2014).
