**4. Crystal Dislocation Pile-Ups at Small Dimensions**

The reduction in numbers of dislocations in pile-ups within ultrafine crystal or grain size materials raises an issue of analogous pile-up lengths and crack sizes associated with brittle fracturing of conventional microstructures. Petch determined an inverse square root of grain size dependence for the cleavage fracture strength of a combination of α-iron and related steel materials with grain sizes larger than ten microns and also provided the now widely accepted explanation for the dependence in terms of the dislocation pile-up model [28]. Just afterwards, Stroh provided a follow-up calculation of the condition whereby the concentrated stress at the tip of the pile-up was the same as for a Griffith-type cleavage crack [29]. The similar relationship of crystal grain size and crack size dependencies for describing fracture strengths has been reviewed [30].

Figure 3 provides a comparison of the stress concentrations at a distance, *r*, ahead of two pile-ups and cracks for 5 and 49 dislocations in the pile-ups [12]. In the model figure description, τ*<sup>23</sup>* and τ<sup>∞</sup> are the local shear stress and applied shear stress; *Ai* is dislocation position within the pile-up divided by the characteristic length, *a'* = *Gb*/*2*πτ∞, in which *G* is the shear modulus and *b* is dislocation the Burgers vector. The *n* = 5 and *n* = 49 open circles point to the level of the stress ratios at the pile-up tips. The Griffith stress concentration follows an *r*−1/<sup>2</sup> dependence as expected and is approximately followed at small (*r*/*a'*) by the forward pile-up stress concentration. At large (*r*/*a'*), the pile-up stress concentration is seen to follow an *r*−<sup>1</sup> dependence as is sometimes employed to represent the pile-up as a multiple Burgers vector dislocation of strength, *nb*.

**Figure 3.** Comparison of the concentrated stresses ahead of dislocation pile-ups consisting of 5 or 49 dislocations and Griffith cracks of equal lengths [12].

The notable result shown in Figure 3 is that the smaller dislocation pile-up does not achieve a sufficient stress concentration to be modeled as a Griffith crack. On such theoretical basis, therefore, one might expect that cleavage would be more difficult to initiate within an ultrafine grain size material. In fact, experimental evidence for the greater difficulty of initiating cleavage in ultrafine grain size α-iron material has been reported by Hohenwarter and Pippan [31]. The result adds to the well-known experimental observation of lowering the ductile-to-brittle transition behavior of steel and related bcc metals through refinement of the polycrystal grain size, particularly via measurements obtained in Charpy impact testing [30].
