**4. Discussion**

The primary transients in ufg materials are usually small because the high density of HABs limits the free path of dislocations effectively and leads to high rates of dislocation generation. They are roughly estimated [18] as d*ρ*+/d <sup>=</sup> <sup>2</sup> *<sup>M</sup>*/(*<sup>b</sup>* <sup>Λ</sup>) with <sup>Λ</sup><sup>≈</sup> *<sup>d</sup>*0<sup>≈</sup> <sup>5</sup> <sup>×</sup> <sup>10</sup>−<sup>7</sup> <sup>m</sup> and Taylor factor *M* = 3 for polycrystals. Without concurrent dynamic recovery it needs a strain interval <sup>Δ</sup><sup>+</sup><sup>≈</sup> *<sup>M</sup>*−<sup>1</sup> (*b*/*d*) (*σ*/*G*)<sup>2</sup> <sup>Δ</sup>*ρ*<sup>+</sup> to generate the full qs dislocation density *<sup>ρ</sup>*qs <sup>≈</sup> *<sup>b</sup>*−<sup>2</sup> (*σ*/*G*)<sup>2</sup> in grains of size *d* = *d*0. This means that at a stress of 150 MPa a plastic strain of Δ<sup>+</sup> = 0.011 is sufficient to generate the full qs dislocation density. This value is consistent with the primary transients extending over strain intervals > 0.01 in creep at *σ*eng = 150 MPa (Figure 10) and confirms that a qs state in the sense of Equation (1) is reached shortly after the ˙-minimum.

The aim of the present work is to learn more about the influence of the grain structure on the qs strength. As shown above, the high value of *f*HAB has a softening effect (Section 3.1) and leads to a relatively high rate sensitivity (equivalent to low stress sensitivity *n*qs) of the qs strength (Section 3.4). The qs stress exponent *n*qs ≈ 6 and a high activation energy remind one of the power laws of steady state creep that are explained by specific mechanisms of qs creep, characterized by a certain stress exponent *n*qs and rate sensitivity 1/*n*qs (see, e.g., [19–21]). For example, mechanisms 1 and 2 in Figure 13a might represent climb-controlled steady state creep and superplastic deformation, respectively. Following this approach, the power law range in Figure 12 would correspond to some mechanism 2.

**Figure 13.** Mechanisms 1 and 2 of qs deformation with (**a**) abrupt, (**b**) smooth transition.

However, there is an alternative possibility [22]. The investigated region with relatively high rate sensitivity may represent a smooth *transition* between a mechanism 1 dominating at high *σ* and a mechanism 2 dominating at low *σ* (Figure 13b). A transition region must be expected from the fact that the spacings of HABs have a wide distribution due to coexistence of small and large grains. According to the microstructural data of Figure 7, mechanism 1 would represent qs deformation with subgrain-bearing grains, while mechanism 2 would represent qs deformation with subgrain-free grains (This makes a qualitative difference to Ghosh and Raj [23] who studied the influence of a distribution of grain sizes in relation to the transition between superplastic and normal behavior, but assumed that both mechanisms of deformation are concurrently active in each grain). In the (dashed) transition region both types of grains would be present. In qualitative form this possibility has already been applied to microcrystalline Cu at 0.35 *T*<sup>m</sup> [24]. A semi-quantitative model was provided in [25] and applied to microcrystalline Cu at 0.42 *T*m [26]. To apply this model to the present case we need the qs strengths of grains with and without subgrains and the distribution of grain volumes *i* = 1, 2, ... with spacings *di* between the HABs for ufg Cu-Zr at 0.50 *T*m. This warrants assumptions based on educated guesses. Following [25,26] we make these choices:


$$
\dot{\varepsilon} \propto d^4 \sigma^8 \tag{4}
$$

from [27]; the *f*-factors were set to 0.19 (*f*-factors = 1 apply in the limiting case where all dislocations are lying at HABs, all are in dipolar configuration ready for recovery, and have unrelaxed stress fields; as this is unrealistic, *f*-values distinctly less than 1 are sensible). This choice yields the two dashed grey lines in Figure 14a for the present *Fd* and the limiting assumptions of equal stress (iso-stress) or equal strain rate (iso-rate) in all grains.

• The qs strength of crystal volumes *with* subgrains of size *<sup>w</sup>*cg qs(*σ*) is estimated by the power law ˙ ∝ *σ*<sup>15</sup> (dotted line in Figure 14a). The exponent 15 is motivated by the increase of *n*qs with

stress for *σ* > 270 MPa (Figure 12). The position of the line is supported by the result for 2Cu-Zr in Figure 2: The grain size in 2Cu-Zr is so large that all grains contain subgrains; at 150 MPa the upper bound of the initial qs strain rate of 2Cu-Zr is near 10−<sup>8</sup> s−<sup>1</sup> (Figures 2 and 3c); this is consistent with the (grey dotted) estimate for subgrain-containing grains in Figure 14a.

**Figure 14.** (**a**) Strain rate and (**b**) volume fraction of subgrain-free grains as function of stress *σ* in qs deformation; filled circles: qs data for 8/12Cu-Zr from Figure 12a,b, solid lines: model.

The solid black lines in Figure 14 show the result of the modeling. The iso-rate assumption enforces a redistribution of stress; *σ*-concentration to hard grains reduces the stresses in soft grains. This stress shielding hinders deformation of large, soft grains and so raises the flow stress level compared to the iso-stress case. The realistic situation lies between the two limits of iso-stress and iso-rate. Comparison of the model lines with the measured data shows reasonable agreement, in particular with regard to the minimal stress sensitivity *n*qs (maximal rate sensitivity 1/*n*qs) in the transition from subgrain-free to subgrain-containing grains.

We note that the difference between the two model lines is rather small in the transition region and that the choice of the solid grey straight *Fd*-line in Figure 9 has relatively little influence there. The reason may lie in a compensation effect. On the one hand, subgrain-free grains deform faster at a given stress when their grain size *d* increases (Equation (4)). On the other hand, the fraction of subgrain-free grains decreases with increasing *d* as more grains develop subgrains. This means that grain coarsening may not always have the dramatic effect expected from the *d*4-term in relation (4). At low *σ* and ˙ the situation is unclear because subgrains as well as grains coarsen during creep and the microstructural data are not precise enough for modeling. Therefore, we refrain from a detailed discussion here.
