*2.2. Size E*ff*ect of the Strain Rate Sensitivity*

Next, following the Taylor relation, the shear strength of FCC materials is determined by the dislocation density, i.e.,

$$
\pi = \mu b \alpha \sqrt{\rho\_S + \rho\_G} \tag{8}
$$

where μ is the shear modulus and α is the dislocation strength coefficient. The lattice friction τ<sup>0</sup> is ignored in the expression of τ as it is usually very small for FCC materials [45]. By further considering the von Mises flow rule [46] and Tabor's factor [47], the hardness can be given as

$$H = 3\sqrt{3}\tau = 3\sqrt{3}\mu b\alpha \sqrt{\rho\_S + \rho\_G} \tag{9}$$

Submitting Equations (5) and (9) into the definition of SRS, it yields

$$m = \frac{3\sqrt{3}k\_B T}{v^\* H} = \frac{2k\_B T}{3\pi\mu\alpha b^2} \sqrt{\rho\_S + \rho\_G} \tag{10}$$

where *kB* and *T* are the Boltzmann constant and testing temperature, respectively. Recalling the expressions of ρ*<sup>S</sup>* and ρ*<sup>G</sup>* as mentioned above, one can further have

$$m = \frac{2kgT}{3\pi\mu a b^2} \sqrt{\frac{3\tan^2\theta}{2b\hbar^\*} + \frac{3\tan^2\theta}{2b\hbar}} = m\_0 \sqrt{1 + \frac{h^\*}{h}}\tag{11}$$

where *m*<sup>0</sup> = *kBT* tan θ/(πμα*b*2) ) 2/(3*bh*∗) is the SRS without size effect for bulk materials that depends on the density of SSDs through *h*∗ . Equations (10) and (11) indicate that both GNDs and SSDs contribute to the SRS measured by indentation creep tests. However, the variation of SRS with respect to the indentation depth is determined by the contribution of GNDs.

When one further takes the relation between the loading force *P* and *h*<sup>2</sup> in a proportional form [48], i.e., *P* = *Kh*<sup>2</sup> and *P*<sup>∗</sup> = *K*(*h*<sup>∗</sup> ) 2 , where *K* is a proportionality factor and *P*∗ is the characteristic loading force corresponding to *h*∗ , then the expression of Equation (11) can be recast as

$$m = m\_0 \sqrt{1 + \sqrt{\frac{P^\*}{P}}} \tag{12}$$

It is interesting to note that Equations (11) and (12) offer a characteristic form for the depth dependence or loading force dependence of the SRS so that the square of the SRS scales linearly with the reciprocal of the indentation depth or of the square root of the loading force. When the raw experimental data of polycrystals are drawn in this way, a straight line is anticipated so that the intercept informs the value of *m*<sup>0</sup> and the slope yields *h*<sup>∗</sup> or *P*<sup>∗</sup> . In order to verify this proposed scaling law, four different sets of experimental data are considered in the following, including annealed and work-hardened alpha brass [8,49], annealed aluminum [8] under CLH tests, austenitic steel [38] under SRJ tests and annealed alpha brass [35] under CSR tests.
