*2.1. Size E*ff*ect of the Activation Volume*

Considering that the total length of GNDs is λ = π*ha*/*b* [41] and the number of dislocation loops equals *N* = *h*/*b*, the average length of the dislocation segment within the creep region yields

$$d\_G^\* = \frac{\lambda}{N} = \pi a = \pi h \cot \theta \tag{1}$$

where *a* and *h* are, respectively, the contact radius and indentation depth. θ is the angle between the indenter and sample. In addition, the activation distance between each dislocation loop becomes

$$d\_G^\* = s = \frac{ba}{h} \tag{2}$$

*Crystals* **2020**, *10*, 898

When one further considers the obstacle-determined dislocation plasticity [24,42], the activation volume determined by GNDs can then be expressed as

$$w\_G^\* = b l\_G^\* d\_G^\* = \pi b^2 \cot^2 \theta h = \frac{3\pi b}{2} \frac{1}{\rho\_G} \tag{3}$$

where ρ*<sup>G</sup>* = 3 tan<sup>2</sup> θ/(2*bh*) is the depth-dependent GND density, as defined by Nix and Gao [41]. Equation (3) indicates that *v*∗ *<sup>G</sup>* increases proportionally with *h* but varies inversely with ρ*G*. A similar evolution tendency has already been observed for swaged and annealed copper where the activation volume increases almost linearly with *h* [18]. Given that the activation distance is a constant, as indicated by Equation (2), the variation of *v*∗ *<sup>G</sup>* is then realized to be determined by the evolution of *l* ∗ *G*.

As the creep process goes on, the density of GNDs gradually decreases due to the expansion of the creep region with increasing indentation depth. Although the transition of pre-existing statistically stored dislocations (SSDs) to GNDs might be possible by the cross-slip mechanism [43], its influence on the decrease in GND density may not be obvious, as the dominant type of dislocation is the edge dislocation for FCC materials considered in this work. Then, the thermal activation of SSDs tends to dominate the creep deformation. For crystalline materials with large grain size, the average segment length and activation distance of SSDs can be, respectively, estimated as *l* ∗ <sup>S</sup> <sup>∼</sup> *<sup>k</sup>*/ <sup>√</sup>ρ*<sup>S</sup>* and *<sup>d</sup>*<sup>∗</sup> <sup>S</sup> <sup>∼</sup> 1/ <sup>√</sup>ρ*<sup>S</sup>* with ρ*<sup>S</sup>* being the density of SSDs [24]. Then, to be consistent with Equation (3), the activation volume related to SSDs could be taken as

$$v\_S^\* = b l\_S^\* d\_S^\* = \frac{3\pi b}{2} \frac{1}{\rho\_S} \tag{4}$$

where ρ*<sup>S</sup>* = 3 tan2 θ/(2*bh*<sup>∗</sup> ) and *k* = 3π/2. Thereinto, *h*∗ is a characteristic length related to the bulk hardness [41]. It is indicated by Equation (4) that *v*∗ *<sup>S</sup>* is independent of *h* but characterizes the intrinsic creeping properties of materials without size effect.

When simultaneously addressing the contribution of GNDs and SSDs, one may consider the relation 1/*v*∗ = 1/*v*∗ *<sup>G</sup>* + 1/*v*<sup>∗</sup> *<sup>S</sup>* [27], and then the general expression of the activation volume yields

$$v^\* = \frac{3\pi b}{2} \frac{1}{\rho\_S + \rho\_G} \tag{5}$$

which can be reduced to Equation (3) when there exists an obvious indentation size effect (i.e., ρ*<sup>G</sup>* >> ρ*S*), or be degraded into Equation (4) when ρ*<sup>G</sup>* → 0 at deep indents. Given the expressions of ρ*<sup>G</sup>* and ρ*<sup>S</sup>* as mentioned above, the activation volume can be recast as

$$w^\* = \frac{\pi b^2 h^\* H\_0^2}{\tan^2 \theta} \frac{1}{H^2} \tag{6}$$

and

$$
\ln(\frac{\upsilon^\*}{b^3}) = \ln k\_2 - 2\ln H \tag{7}
$$

where *H* = *H*<sup>0</sup> √ 1 + *h*∗/*h* is the depth-dependent indentation hardness [41], and *k*<sup>2</sup> = π*h*<sup>∗</sup> *H*2 <sup>0</sup>/(*<sup>b</sup>* tan<sup>2</sup> <sup>θ</sup>) is a constant related to the bulk hardness *H*<sup>0</sup> and characteristic length *h*<sup>∗</sup> [44]. As indicated by Equation (7), ln(*v*∗ /*b*3) scales linearly with ln *H*, which is consistent with the experimental observations for most FCC materials, like aluminum, silver and nickel [40]. In addition, the decrease in *v*∗ with increasing *H* can be ascribed to the accumulation of dislocations at shallow indents that leads to a small activation area swept out by gliding dislocations during the thermal activation event [30].
