*3.2. The E*ff*ect of Strain Rate on Flow Stress*

At the usual strain state of 1 ns−1, for the strain–stress test of the model with the grain size of 18 nm, the deformation enters the plastic region after the elastic region, as shown in Figure 4a. The max stress is reached at a strain of about 4% and then the stress enters the plateau region in which the stress is called as flow stress, by following the increase of strain. From the beginning of plastic deformation (at a strain of about 3%), the dislocation density increases and becomes to be constant after entering the stable region of flow stress at a strain of about 14%.

In Figure S2, we show the strain–stress curves of grain sizes of 6 nm and 18 nm. We can find that the flow stress decreases with the decrease of strain rate, whatever the size of grains. In Figure 4b, we plot the flow stress as the function of strain rate. It is noticed that the flow stress increases rapidly with the increase of strain rate when the strain rate is more than 1 ns<sup>−</sup>1, while the flow stress decreases slowly with the strain rate decrease down to 1 ns<sup>−</sup>1. It seems there is a critical strain rate above which the mechanical properties will be affected obviously by strain rate. This may be found from the change of dislocation density with strain and strain rate. As shown in Figure 4d, the dislocation density of grain size of 6 nm changes following the strain for different strain rates. For the strain rates of 1 ns−<sup>1</sup> and 0.1 ns<sup>−</sup>1, the dislocation densities don't have an obvious difference.

**Figure 4.** (**a**) Stress and dislocation density as functions of tensile strain for nanocrystalline gold of 6 nm grain size and 18 nm grain size at a strain rate of 1 ns<sup>−</sup>1; (**b**) flow stress; (**c**) strain rate sensitivity as functions of strain rate for nanocrystalline gold; and (**d**) dislocation densities as a function of tensile strain for grain size of 6 nm under different strain rate.

There is an important parameter called as strain rate sensitivity (*m*) which can very helpful to quantify the deformation mechanism. It is defined as [14,17],

$$m = \frac{\sqrt{3}kT}{V^\*\sigma},\tag{2}$$

where *k*, *T* and σ are Boltzmann constant, absolute temperature and flow stress, respectively. In the formula, the parameter activation volume (*V*\*) can be expressed as,

$$V^\* = \sqrt{3}kT \left(\frac{\partial \ln \dot{\varepsilon}}{\partial \sigma}\right) \tag{3}$$

where . ε is the strain rate. Thus, from the relation of flow stress and strain rate, we can obtain the average value of *m* in a range of strain rate, such as 1 ns<sup>−</sup>1–0.1 ns−1.

The calculated strain rate sensitivities are shown in Figure 3c. The value of *m* decreases with the decrease of strain rate. The *m* value of 18 nm NC gold is less than that of 6 nm at the same range of strain rate. When the strain rate is below 0.1 per ns, the strain rate sensitivities of 6 nm and 18 nm NC gold are 0.063 and 0.016, respectively. This implies that the strain rate sensitivity decreases with the increase of grain size. This result is consistent with many experiments [12,14,16,22,43], the *m* of NC gold with a grain size of 30 nm is 0.01 [27]. It is reasonable to compare the strain rate sensitivity for the simulation under the strain rate lower than 0.1 per ns with the experimental value. The simulation of NC copper [28] also indicates a critical strain rate and the strain rate sensitivity decreases with the increase of grain size below the critical strain rate of 0.1 ns<sup>−</sup>1. This is consistent with the observation about strain rate sensitivity in our NC gold.

In Figure 4d, we can find that at high strain rates, the activation of dislocation occurs under relatively large strain. In Figure 5, we show the atomic structures of 6 nm NC gold under a strain of 4% with the strain rate of 0.1 ns−1and 10 ns−1. It is clear that the stacking faults are easy to form with dislocation nucleated near GBs under lower strain rates (such as 0.1 ns−1). This also results in the larger localized shear strain appearing at GBs. Thus, the deformation is easy to appear at GBs under low strain rate, and this is consistent to the observation at elastic region under small strain. At the stable flow stress region, the dislocation density is relatively high under larger strain rate (Figure 4d). This indicates that the NC gold is more prone to dislocation movement under higher strain rate (Figure 5c,g, atomic structures at the strain of 10% with strain rate of 0.1 ns−<sup>1</sup> and 10 ns<sup>−</sup>1). While at the lower strain rate, GBs are more prone to responding to the applied large strain, as the distributions of atomic shear strain in Figure 5d,h. The high dislocation density under larger strain rates is consistent to the higher stress observed in strain–stress curve.

**Figure 5.** Atomic configurations and distribution of local shear strain of grain size of 6 nm at tensile strain of 4% under the strain rate of (**a**,**b**) 0.1 ns−<sup>1</sup> and (**e**,**f**) 10 ns<sup>−</sup>1, and that at tensile strain 10% under the strain rate of (**c**,**d**) 0.1 ns−<sup>1</sup> and (**g**,**h**) 10 ns<sup>−</sup>1. In (**a**,**c**,**e**,**g**), blue, red and green represent grain interiors with fcc, stacking faults with hcp, and atoms at grain boundaries, respectively. In (**b**,**d**,**f**,**h**), the change of color from blue to red indicates the increase of atomic local shear strain.

As we know, under the assistance of GBs, the dislocations under local shear stress become easy to nucleate in NC metals. From the view of plastic flow activated thermally, the shear deformation rate for overcoming the barrier to dislocation motion is related to the activation volume by the relation [6],

$$
\dot{\gamma} \approx \exp[\left(-\Delta F + \pi\_\varepsilon^\* V^\*\right)/kT],\tag{4}
$$

where Δ*F* is the change of Helmholtz free energy and τ∗ *<sup>e</sup>* is the thermal component of total stress. The item of τ∗ *eV*<sup>∗</sup> is the contribution of thermally activated stress to reduce the energy barrier. Thus the *V*\* is related to the deformation mechanism [44–46]. Here we check the change of *V*\* by modulating the temperature and strain rate.

As shown in Figure 6a, from the strain–stress curves, the formation of structures at high temperatures (800 K) is easier in the plastic region under low strain and thus with lower flow stress, compared to the case of low temperature (300 K). In Figure 6b, the flow stress at 800 K is shown as the function of strain rate for grain sizes of 6 nm and 18 nm. It is clear that the flow stress decreases continuously following the strain rate decrease in the range of our test. It is considered that the GBs slipping are activated at 800 K for both cases of 6 nm NC and 18 nm NC.

**Figure 6.** (**a**) Stress–strain curves for nanocrystalline gold with grain sizes of 6 nm and 18 nm with a strain rate of 1 ns−<sup>1</sup> with 300 K and 800 K, (**b**) flow stress and (**c**) activation volume as functions of strain rate for grain sizes of 6 nm and 18 nm with 300 K and 800 K, and (**d**) the distribution of atomic weight as a function of atomic shear strain for 18 nm grain size with 300 K and 6 nm grain size with 600 K and 600 K at tensile strain 7.5% under the strain rates of 0.01 ns<sup>−</sup>1.

As mentioned above, the *m* below the strain rate of 1 ns−<sup>1</sup> is closely dependent on the grain size. From the relation between *m* and *V*\*, the activation volume should also be closely dependent on the grain size at the low strain rate of less than 1 ns−1. In Figure 6c, we plot the activation volume as the function of strain rate for different cases. It is noticed that in the range of 0.01 ns−1–0.1 ns−1, the activation volumes of 18 nm NC and 6 nm NC at 800 K are similar to that of 6 nm NC at 300 K and about 6.06 b3. However, the activation volume of 18 nm NC at 300 K is about 18.69 b3. The activation volumes of NC Ni and Cu from experimental strain rate tests [10,14,47] are about 10–20 b3. This is consistent to the case of 18 nm NC Au at 300 K. Thus the dislocation pile-up against GBs is the main formation mechanism for grain size 18 nm at 300 K. From the distribution of atomic weight as the function of local shear strain at the applied strain of 7.5% and strain rate of 0.01 per ns in Figure 6d, there is a second peak with large local shear strain. This is an indicator which implies GBs are important sources to nucleate the dislocations and emit the stacking faults into the grain interior. We can see in Figure 6c that temperature has little effect on the activation volume of grain size 6 nm, which implies the main deformation mechanism does not change for both temperatures (300 and 800 K). In Figure 6d, there is just one broadened peak and no second peak appears in the distribution of atomic weight for grain size of 6 nm at 800 K. Thus, it implies that the response of GBs slipping may be the main mechanism of deformation. At 300 K, though there is a second peak at large shear strain, its width is very large and implies that GBs is not only for dislocations pile-up but also the relative slipping between grains. Therefore, at high temperatures and in the case of small grain sizes, the GBs slipping are the main deformation mechanism and thus this system is with low flow stress.
