**3. Comparison between Theoretical Results and Experimental Data**

We firstly present the *m* −*P* relationships of work-hardened mechanically polished alpha brass [49] and annealed aluminum [8], which are compared between the experimental data (black dots) and theoretical results (red lines), as illustrated in Figure 2. The creep tests are performed at room temperature with the loading force ranging from 10−<sup>4</sup> to 100 N, and the type of indenter is a Berkovich indenter [8,49]. The theoretical results are predicted by Equation (12) with *m*<sup>0</sup> and *P*<sup>∗</sup> calibrated by comparison with the converted experimental data (see the inset of Figure 2). The model parameters are listed in Table 1. An excellent agreement is observed where the SRS decreases with the increase in *P* until *m*<sup>0</sup> is reached. This decreasing tendency is ascribed to the variation of creep mechanisms, which range from the dislocation–dislocation interaction to the dislocation–solute interaction, as the former becomes dominant at a high stress level [50]. Moreover, the fitted values of *m*0, i.e., 0.016 and 0.007, for alpha brass and aluminum are, respectively, close to the literature values of 0.018 obtained by CSR tests for copper [19] and 0.01 obtained by CLH tests for aluminum [40].

**Figure 2.** Strain rate sensitivity *m*-loading force *P* relationships compared between the experimental data (black dots) and theoretical results (red lines) of (**a**) work-hardened mechanically polished alpha brass [49] and (**b**) annealed aluminum [8]. The inset figure illustrates the calibration of model parameters by comparison with experimental data.

**Table 1.** Model parameters for work-hardened alpha brass [49], annealed aluminum [8], austenitic steels [38] and annealed alpha brass [35].


The proposed *m* − *P* relation can also be expressed in a similar form as the classic Nix–Gao model [41] so that the SRS decreases with increasing indentation depth. In order to verify the *m* − *h* relation, as indicated by Equation (8), the experimental data of austenitic steel (SRJ test) [38] and annealed alpha brass (CSR test) [35] are considered. A Berkovich indenter is applied for these creep tests. In Figure 3, the comparison between theoretical results and experimental data is illustrated so that a reasonable agreement is observed for both materials. In this case, *m*<sup>0</sup> = 0.006 for austenitic steel is close to 0.0066 for 310 stainless steel [51] and *m*<sup>0</sup> = 0.0023 matches well with 0.002 obtained from [49] for annealed alpha brass. However, one may note that the experimentally measured value of *m* for the alpha brass gradually deviates from the theoretically predicted line. This is believed to originate from the effect of thermal drift during the indentation creep tests. According to the analysis of [2], the SRS without thermal drift correction *m* can be approximated as *m* ≡ *m*(1 + λ/ . *h*) 2 with λ as the thermal drift rate (of the order of <sup>±</sup>10−<sup>2</sup> nm·s−1) and . *h* as the penetration rate. With increasing indentation depth, . *h* decreases from an initial high value down to the absolute value of λ. Therefore, it is rational to observe that *m* < *m* with the increase in *h* when λ < 0, as indicated in Figure 3b.

Besides the characteristic *m* − *h* and *m* − *P* relationships as proposed above, the normalized activation volume with respect to the indentation hardness also follows a scaling law, as expressed in Equation (7). To verify this conclusion, the experimental data of copper obtained from [30,52], as well as the data of alpha brass and aluminum taken from [8], are plotted in Figure 4. Correspondingly, the scaling relation is illustrated by solid lines with the slope ∂ ln(*v*∗ /*b*3)/<sup>∂</sup> ln *<sup>H</sup>* = <sup>−</sup>2. It seems that the creep data follow this scaling law well, and similar linear relationships between ln(*v*∗ /*b*3) and ln *H* have also been noticed in the experimental data of some other FCC materials, but with the slope ranging from −1 to −3 [40]. This discrepancy may originate from ignoring the effect of strain hardening induced by the dislocation–dislocation interaction, and might also come from the influence induced by thermal drift through *v*<sup>∗</sup> = *kBT*/(*m*τ), as the simulation work of [19] has captured an obvious variation of SRS with increasing thermal drift rate.

**Figure 3.** Strain rate sensitivity *m*-indentation depth *h* relationships compared between the experimental data (black dots) and theoretical results (red lines) of (**a**) austenitic steel [38] and (**b**) annealed alpha brass [35]. The inset figure illustrates the calibration of model parameters by comparison with experimental data.

**Figure 4.** Normalized activation volume *v*∗/*b*3–hardness *H* relationships compared between experimental data (dots) and theoretical results (lines) for copper [30,52], alpha brass [8] and aluminum [8].
