**4. Discussion**

The microstructure study demonstrated that the v1w1 regime of the HPTE deformation, carried out at 100 ◦C, resulted in the formation of a gradient microstructure in CP copper. This is not surprising since HPTE combines both extrusion and shear deformation modes. The shearing mode gradually increases from the sample´s center to the edge (Equation (1)). Consequently, the central area of the HPTE-processed rod is characterized by the presence of a fine-grained structure, whereas in the mid-radius and edge areas are characterized by an ultrafine-grained structure. Apparently, large grains were gradually refined with an increasing strain, resulting in an increased fraction of the fine grains, as shown in Figure 4a. Even in the central area of the HPTE rod the majority of grains (about 80–85%) are the newly refined grains with the sizes in the range of ~2–5 μm (Figure 3a). In this area, new dislocation boundaries were actively formed and the difference between D15 and D3 is significant (Table 1).

At a larger distance from the center, the applied strain increased, and the formation rate of new GBs became slower since both grain sizes, D3 and D15, were practically the same at the mid-radius and at the edge, indicative of no further grain refinement. This indicates that the saturation of the grain size refinement was reached, which is typical for other SPD methods [32]. Following grain refinement

saturation, the misorientations at the already-formed interfaces were growing resulting in an increase of the HABs fraction (Figure 5). Note that this increase occurred at different rates at different grain facets and the difference between D15 and D3 still persisted up to the highest strain reached in our experiment. GB distribution maps like one shown in Figure 6d clearly demonstrate that in all sample areas, the microstructure is complex, i.e., the LABs and HABs represent an interpenetrated network. In other words, a crystallite (grain or subgrain) can be simultaneously bounded by high angle and low angle boundaries. Interestingly, the density of SSD showed an enhanced increase at the edge of the HPTE-processed sample in comparison with that of GND (Figure 6c). This fact suggests that the multiplication rate of SSD was higher than their recovery/annihilation rate at the edge (also in the mid-radius) area, compared to the kinetics of the same processes in the central area. The difference in density for both kinds of dislocations (GND and SSD) is consistent with the published data [33], showing that: (i) in well-annealed metallic materials the GND density is 20%–30% lower than the total dislocation density; and (ii) that both values are strongly correlated.

The microstructure resulting from HPTE processing has similar features with the gradient microstructure of CP Cu after free-end torsion [31,34]. Nevertheless, several distinctions should be mentioned here. HPTE provided a more efficient grain refinement compared to that achievable with free-end torsion. Furthermore, instead of a coarse-grained structure in the center and a fine-grained structure at the periphery, we observed an FG structure at the center and a UFG structure at the periphery after HPTE. After HPTE, we observed a more homogeneous grain structure in the entire sample with no grain elongation, as it was observed after the free-end torsion.

CP Cu samples deformed via the HPTE under the v1w1 regime demonstrated a yield strength and an ultimate strength close to the corresponding values obtained after the free-end torsion with strain 1.3 [31]. The elongation to failure of Cu was reduced for more than 1.5 times compared to the elongation after free end torsion. By contrast, the ductility after HPTE remained at the level of the annealed state.

The microhardness saturation value of 125 HV is very close to the microhardness of oxygen-free (OF) copper deformed by HPTE under the v1w1 regime at room temperature [12], and also close to OF copper microhardness after HPT [8,15] (Table 2). The strength of the HPTE-processed copper is comparable with that of copper after ECAP, and it is not as high as that after HPT [8].

The change of the microhardness along the radius of the Cu rod cross-section allows to analyze the contribution of various hardening mechanisms after HPTE in different sample areas (Figure 8a). In fact, after HPTE treatment the variation of hardness is determined by the distribution of strain in the sample affecting the dislocation density values and the degree of grain refinement which gives rise to Hall–Petch hardening [35]. The inhomogeneity of the strain distribution in the HPTE-processed rod (Equation (1)) leads to the formation of the gradient structure. The tensile sample geometry (Figure 2b) made it possible to conduct mechanical testing on the sample with gradient structure.

We assume that experimentally obtained strength in CP copper samples after HPTE results from the following hardening mechanisms: accumulation of dislocations (σρ), Hall–Petch strengthening and subgrains (σH-P), solid solution strengthening (σSSS), and second-phase particle strengthening (σSPPS).

The contribution of stored dislocations to the yield strength can be estimated using the Taylor Equation:

$$
\sigma\_{\rm P} = \alpha \text{MGb} \sqrt{\rho},
\tag{3}
$$

where M is Taylor factor, α is a constant, G is the shear modulus, b is the absolute value of the average Burgers vector, and ρ is the total dislocation density. We can estimate strengthening due to the grain refinement using the Hall–Petch relation:

$$
\sigma\_{\rm H\cdot P} = \sigma\_0 + \text{kd}^{-1/2},\tag{4}
$$

where σ<sup>0</sup> is the friction stress, k is a constant, and d is the mean grain size in the corresponding sample area. The sample contained material from central and mid-radius areas. The microstructure in both areas is complex because the grains are bounded by both HABs and LABs. Therefore, it is not clear which GB misorientation threshold and grain size should be used in Equation (3). Prior in-situ TEM study [36] had shown that in FCC metals with low and middle stacking fault energy, the LABs with Θ > 3◦ and special GBs provide effective barriers for the dislocation slip. Therefore, we can assume that the effective grain size, d, in the Hall–Petch equation is equal to D3. In Hall–Petch strengthening it is also necessary to account for the input from nanotwins. Twin boundaries are the obstacles for dislocations slip. The volume fraction of twin boundaries was 15.4% and the effective grain size d for the Hall–Petch strengthening calculation is D3 with twins = 0.25 μm, estimated from several ACOM TEM maps. This value is smaller than D3 obtained from EBSD maps (0.4 μm, Table 1) due to higher angular resolution of ACOM TEM technique as previously discussed.

Solid solution strengthening is complex since the solute atoms can interact with dislocations by the elastic, electrical, short-range and long-range interactions [37]. For the Cu-Al substitutional solid solution, the elastic interaction is most important. This elastic interaction can be associated with local short-range mean normal stress. In the linear approximation, the elastic interaction is assessed as Paierls-Nabarro forces taking into account the atomic radius misfits and alloying element concentrations [37]. As a result, solid solution strengthening leads to an increase in the friction stress σ0. Solid solution strengthening for the Cu–Al single crystal was measured experimentally and, depending on the concentration of Al in solid solution, a linear dependence of the σsss was obtained [38]:

$$
\sigma\_{\rm SSS} = M \mathbf{4} \mathbf{c}^{\frac{2}{3}},\tag{5}
$$

where c is the concentration of Al in the Cu–Al solid solution.

Second-phase particle strengthening was provided by the large (about 1 micrometer in diameter) particles and it can be calculated using the Orovan's formula for the non-coherent particles:

$$
\sigma\_{\rm SPSS} = \frac{2\alpha \text{MG} b}{d\_p \sqrt{\pi/6} f},\tag{6}
$$

The parameters (σ0, k, α, G, b) used for computing the various strengthening mechanisms described above were taken from [22,39,40], whereas the structure parameters D3, ρ, *dp*, *f* are given in Tables 1 and 3.


**Table 3.** Measured parameters used for the strengthening calculations.

As can be seen in Table 4, the calculated contributions from the solid solution and the precipitation strengthening are small and they provide less than 10% to the YScalc value, while the dislocation strengthening is 3–4 times higher, which provides ~50% to the YScalc in the initial state and only 20% of the YScalc in HPTE-processed copper. The low contribution of dislocation strengthening to YScalc in the HPTE-processed copper is likely related to the elevated temperature of the HPTE processing of 100 ◦C, which promoted recovery of dislocation density. Therefore, the main contribution (~60%) to the strength of HPTE-processed Cu is due to the GB strengthening via the Hall–Petch mechanism.


**Table 4.** Contributions (all are given in MPa) of various strengthening mechanisms in CP Cu before and after HPTE. The testing sample contained central and mid-radius areas of the rod.

\* YScalc is an arithmetic sum of the contributions from different strengthening mechanisms. \*\* YSM is the yield strength calculated by the mixture rule.\*\*\* YSexp is the experimental strength after HPTE.

In order to compare the calculated (YScalc) and the experimentally measured (YSM) yield strengths of the entire HPTE-processed sample we can use the rule of mixture:

$$Y\text{S}\_M = Y\text{S}\_\text{C}V\_\text{C} + Y\text{S}\_{MR}V\_{MR} \tag{7}$$

where YSC and YSMR are the calculated values of the yield strength of the material from the center and mid-radius area, respectively (Table 4), and VC and VMR are the volume fractions in these same areas. According to the microhardness distribution along the radius of the HPTE-processed sample (Figure 8a), the hardness is low in a narrow area with the radius of 1.2 mm around the rod axis. Therefore, the volume fraction of this part of the tensile specimen is only 0.12. Consequently, the majority of the volume fraction is contributed by mid-radius area. The calculated value of YSM = 374.6 MPa is in a good agreement with experimentally measured yield strength of 370 MPa. A part of the HPTE-processed rod with high hardness and strength values was removed upon machining of the tensile specimen (Figure 8b). A rapid strengthening and a short uniform elongation range characterized the stress–elongation curve of copper after HPTE. A decrease of uniform elongation from ~16% in the annealed state to ~2% after the HPTE deformation is a typical behavior for the materials produced by SPD methods. However, as the deformation is localized, the elongation to fracture reaches 30%, which is two times higher than post-necking elongation of the annealed copper (Figure 8b). Such a behavior is typical for materials with ultrafine-grained microstructure [41–43] and most likely related to the increased strain rate sensitivity of such materials. Otherwise, the strength of the entire specimen would be even higher. Therefore, HPTE processing of CP copper provides large elongation to failure and high strengthening mostly due to the Hall–Petch mechanism. The uniform elongation (~2%) had not been improved despite the presence of gradient microstructure as proposed in [7]. Most likely, this approach could not be realized in the HPTE-processed copper because both central and outer layers were sufficiently strengthened, and the material demonstrated tensile behavior similar to that of the homogeneous UFG materials.
