**5. Crystal Size-Dependent Strain Rate Sensitivity**

A dislocation mechanics description of loading rate dependence based on thermally-activated dislocation motion has provided a valuable additional avenue for the investigation of crystal size effects at micro- to nano-scale sizes. An early model description of the behavior for conventional grain size hcp cadmium involved separation of the strain rate sensitivity property into grain volume and grain boundary components in the relation [32]:

$$(1/\upsilon^\*) = (1/\upsilon\_0 \, ^\*) + (k\_\varepsilon/2 \mathbf{m}\_\mathsf{T} \tau\_\mathsf{C} \mathbf{v}\_\mathsf{C} ^\*) \ell^{-1/2} \tag{1}$$

The parameters in Equation (1) are an activation volume, *v\** = *A\*b* = *kBT[{*Δ*ln(d*γ/*dt)}*/Δτ*}]T*, in which *A\** is an activation area; *kB* is Boltzmann's constant; *T* is temperature; Δ*ln(d*γ/*dt)* is the imposed change in the plastic shear strain rate; Δτ is the accompanying change in shear stress; *k*<sup>ε</sup> is the H–P stress intensity, *mT* is the Taylor orientation factor; τ*<sup>C</sup>* is the local shear stress at the grain boundary; and is the grain size. The first term, (*1*/*v0\**), measures the rate-dependent contribution of the dislocation motion within the grain volume and the second term, involving the factor (*1*/*vC\**), is a measure of the thermally-activated resistance to transmission of plastic flow across the grain boundary [33]. The product (τ*CvC\**) is constant, and thus Equation (1) follows an H–P type dependence.

The current Figure 4 is an added-to version of a previous description of copper and nickel measurements in which a comparison was made with the Hall–Petch type grain size prediction given in Equation (1) [34]. The upper curve in the figure is for T = 195 K and the lower curve for 300 K. Transition from grain size strengthening to weakening is shown to occur in the smaller grain size regime at the "jump" in the filled-triangle values of (*1*/*v\**). The unmistakable jump by an approximate order-of-magnitude leads to *v\** values of near atomic dimensions that are generally determined for higher temperature (creep) deformations. However, additional open circle points shown in the figure, as obtained from measurements reported by Chen et al. [35], are seen to follow the H–P grain size strengthening prediction. The positions of these points near to the H–P dependence are significant in terms of the grain boundary structures being stable and continuing to resist slip transmission from within the grains. These same authors provided confirmation of H–P grain size strengthening via separately reported hardness measurements. In related research on nano-grained copper and nickel materials, Zhou et al. have given emphasis to the importance of establishing thermal stability of the grain boundary structures in such materials [36]. Otherwise, Figure 4 shows *v\** to be an effective monitor for detecting the onset of grain size weakening, for example, in line with the determination of material creep behavior as described in a previous report by Armstrong et al. [37].

**Figure 4.** Strain rate sensitivity-based reciprocal activation volume measurements for copper and nickel materials spanning conventional and micro- to nano-scale grain sizes [34,35].
