*2.3. Crack Initiation at Pre-Existing Stress Concentrations*

Because cracks are high energy defects, it is difficult for them to be initiated in otherwise defect-free crystals. Most fracture mechanics tests using the American Society of Testing and Materials (ASTM) criteria [21] are conducted using notched specimens and, sometimes, notched and pre-cracked specimens. The stress field of a notch is characterized by the elastic stress concentration factor Kt and the notch tip radius, ρ. Figure 3a shows an incipient crack initiated at the notch tip. The stress at the notch tip corresponds to Kt σ but decreases with distance depending on the notch tip radius, ρ, approaching the remote stress, σ. For sharp notches, the stress gradient is sharp, while for blunt notches, the rate of decrease is slower. We have analyzed the growth of a short crack at the notch tip using elastic-plastic fracture mechanics [16]. The results are shown schematically in Figure 3c. The stress intensity factor for the short crack increases sharply from zero, decreases to some minimum, and then increases slowly with a further increase in the crack length. When the short crack length is zero, K for the short crack is also zero. The sharp increase is due to the very high notch tip stresses. Hence the initial sharp increase can be considered as within the process zone or from the point of dislocations within the core region of the notch. The decrease of K as the short crack grows is due to the gradient in the notch tip stress field. Further increase in the K value arises as the crack grows due to the remote applied stress since K increases with the crack length for a given stress. Hence, the depicted behavior of Ksc is expected due to the notch tip stress gradient. It may be noted that for just purely elastic calculation, Ksc monotonically increases and does not show the observed minimum [22].

**Figure 3.** Crack initiation at a notch tip. (**a**) The stress field ahead of the notch tip. (**b**) Crack initiation at grain boundaries ahead of the notch tip due to hydrostatic stresses. (**c**) The variation of the stress intensity factor Ksc of the incipient crack growing nearing a notch tip.

For the continuous growth of the initiated crack at the notch tip, the minimum must exceed the threshold stress intensity factor, Kth for crack growth. Otherwise, the incipient crack that is growing in the high-stress field of the notch is arrested when Ksc drops below Kth. The minimum of the Ksc value is related to the internal stress (notch tip stress) magnitude and its gradient. For very sharp notches (ρ ~ 0), the stress gradient can be sharp, leading to arrest of the growing short crack leading to non-propagating cracks at sharp notches. This is observed, particularly under fatigue, leading to fatigue stress concentration factor, KFC, differing from the elastic stress concentration factor, Kt. The magnitude of the stress at the notch tip also depends on the applied stress, σapl. We have shown [16] that the minimum applied stress needed for the continuous growth of incipient crack near the stress concentration can be expressed as:

$$
\sigma\_{apl} = \frac{2\mathcal{K}\_{th}}{(\mathcal{K}\_t)^{1.3} \times \sqrt{\rho}} \tag{1}
$$

where Kth corresponds to the threshold for crack growth. It can be a threshold for any subcritical crack growth (thresholds for fatigue, stress corrosion, corrosion-fatigue, sustained load, or even for a fracture, such as K1C). Kt and ρ are elastic stress concentration factor and notch-tip root radius, respectively. The equation has been successfully applied to the extensive notch-fatigue data available in the open literature. Recently, the equation has been applied to determine the pit to crack transition under corrosion fatigue [23].

Figure 3b also shows the crack initiation ahead of the notch tip. This can occur at grain boundary junctions (accentuated by the presence of carbide particles or inclusions) due to high hydrostatic stresses present. Several models have been developed assuming such nucleation [24–26]. We show later a similar problem was analyzed using discrete dislocations where short crack nucleation ahead of a blunt crack is considered. Figure 4 shows typical results of short crack initiation and growth near notches with different Kt values but for a fixed ρ, showing how minimums in Ksc values become sharper with increasing Kt value.

**Figure 4.** Typical results of actual calculations of Ksc (called Kpl due to elastic-plastic stress fields) for different Kt values but for a fixed ρ, showing the minimum in Ksc.
