**3. Crystal (Grain) Size-Dependent Strengths**

Brenner [6] found an inverse dependence of strength on the specimen wire diameter for α-iron and copper whiskers. The same dependence has been found for integrated circuit-connected measurements on gold micro-wires [21] as well as for thin copper wires [22] and other nickel nano-pillar [23] materials, also with consideration taken into account of the nano-polycrystal grain structures. The status of nano-polycrystal micro-pillar strength properties have been reviewed by Shahbeyk et al. [24]. Kiener et al. [25] have provided an analysis of the strength properties on the basis of the small volumes of the materials that were tested.

Figure 2 illustrates on an expanded abscissa scale a further connection of the size dependence described previously for a compilation of strength properties obtained on conventional and ultrafine crystal (grain) size of α-iron and steel materials [26]. In the figure, the continuous slightly curved line is an extrapolation of the Hall–Petch (H–P) inverse square root of grain size measured at ambient temperature for a number of steel materials. The filled-square points connected by a dashed line just below the extrapolated H–P dependence are pioneering measurements reported for patented eutectoid steel wire materials. The filled triangle points were also obtained more recently for eutectoid steel wire materials but at smaller effective grain sizes as is indicated as well for the filled circle points that were obtained for ball-milled α-iron material. The topmost filled diamond point is a latest measurement reported by Li et al. [7] for severely drawn eutectoid material.

Of particular importance in Figure 2 is a dot-dashed dislocation pile-up model description of the Hall–Petch dependence fitted to the reported H–P parameters at larger grain sizes and which dependence is shown to transition at ultrafine grain size from an -<sup>−</sup>1/<sup>2</sup> to an -<sup>−</sup><sup>1</sup> dependence. Such size-dependent transition has been predicted for the variation in H–P measurements assessed earlier for copper materials [27]. The transition occurs when the dislocation pile-up length is sufficiently reduced such that only one dislocation loop, *n* = 1.0, is able to be produced within a restricted slip plane length to overcome the nano-polycrystal grain boundary resistance. On such a restricted dislocation number basis, the dislocation model description for an H–P dependence connects with the described strength dependence of whiskers, nano-wires, and micro-pillars on their specimen diameters.

**Figure 2.** Strength of α-iron and steel materials on the basis of an inverse square root of grain size dependence that transitions to an inverse size dependence depending on the level of the dislocation pile-up stress intensity; an expanded version of compiled results is reported in reference [26].
