**3. Experimental Verifications and Results**

In this section, four sets of experimental data for single crystal Cu [**?** ], Al [**?** ], CaF2 [**?** ] and W [**?** ] are applied to verify the rationality and accuracy of the developed model. Before the calibration of model parameters, it should be noted that *H*<sup>f</sup> for Cu and Al is so small that is generally not considered [**? ?** ]. For CaF2, *H*<sup>f</sup> is informed to be 1.2 GPa at room temperature [**?** ], and vanishes to be zero around 473 K [**?** ]. As to W, *H*<sup>f</sup> can be theoretically calculated by referring to Equation (**??**) with *τ*p0 = 1038 MPa, *τ*f0 = 2035 MPa, *<sup>T</sup>*<sup>0</sup> = 580 K, *<sup>k</sup>*<sup>B</sup> = 1.38 × <sup>10</sup>−<sup>23</sup> J/K, *<sup>H</sup>*<sup>k</sup> = 1.65 × <sup>10</sup>−<sup>19</sup> J, *<sup>γ</sup>*˙ p0 = 3.71 × 1010 <sup>s</sup>−<sup>1</sup> and *<sup>ε</sup>*˙ = 0.02 s−<sup>1</sup> [**? ?** ]. Then, the experimental data can be plotted in the form as *<sup>H</sup>*˜ 1/*<sup>m</sup>* <sup>n</sup> <sup>−</sup> 1/*<sup>h</sup>* for the four materials, as illustrated in Figure **??**. Following the calibration procedure as mentioned in Section **??**, the model parameters are obtained, as listed in Table **??**.

**Figure 2.** (Color online) Comparison of the *<sup>H</sup>*1/*<sup>m</sup>* <sup>n</sup> <sup>−</sup> 1/*<sup>h</sup>* relationships at various temperatures between theoretical results (lines) and experimental data (dots) for (**a**) Cu [**?** ], (**b**) Al [**?** ], (**c**) CaF2 [**?** ] and (**d**)W[**?** ].


**Table 1.** Parameter calibration of the proposed theoretical model for single crystal Cu [**?** ], Al [**?** ], CaF2 [**?** ] and W [**?** ].

With these calibrated parameters, the *<sup>H</sup>*1/*<sup>m</sup>* <sup>n</sup> − 1/*<sup>h</sup>* relationships are compared between the theoretical results (lines) and experimental data (dots), and a reasonable agreement is achieved, as presented in Figure **??**. It shows that both the intercept and slope of the fitting line decrease with increasing temperature, which mutually contribute to the decrease of *H*<sup>0</sup> and ¯ *h*∗ at elevated temperatures. The former is mainly ascribed to the weakened elastic constants and dislocation strength coefficient with the increase of temperature, and the latter originates from the stimulated expansion of the plasticity affected region at high temperatures that results in the decrease of the intrinsical length scale and limited indentation size effect. Moreover, the hardening coefficient *m* of Cu and CaF2 is noticed to decrease with increasing temperature when compared with that of Al and W. Similar experimental data has also been observed for OFHC copper that *m* decreases from 0.48 to 0.41 when *T* increases from 293 K to 698 K [**?** ], which indicates the weakened work hardening behavior at elevated temperatures that resembles an elastic-ideally plastic material [**?** ]. In order to further characterize the thermally activated deformation mechanisms resulting in the decrease of *m* with *T*, it could be addressed by the nano-indentation strain jump tests [**?** ] or long term creep tests [**?** ], especially in terms of the strain-rate sensitivity.

In Figure **??**, we present the *H* − *h* relationships obtained from the calibrated theoretical model and experimental data of single crystal Cu [**?** ], Al [**?** ], CaF2 [**?** ] and W [**?** ]. As one can see the results match reasonably well for the four materials, and an obvious indentation size effect is informed at different temperatures. However, the increasing rate of *H* with the decrease of *h* tends to decrease with the increase of temperature, which is determined by the decrease of ¯ *h*∗ at elevated temperatures, as illustrated in Figure **??**. In addition, the bulk hardness at the deep indentation depth also decreases with increasing temperature as both the hardness components *H*f(*T*) and *H*0(*T*) get weakened at high temperatures.

**Figure 3.** (Color online) Comparison of the *H* − *h* relationships at different temperatures between theoretical results (lines) and experimental data (dots) for (**a**) Cu [**?** ], (**b**) Al [**?** ], (**c**) CaF2 [**?** ] and (**d**)W[**?** ].

Based on the developed model, the effect of temperature on the evolution of different microstructures can be further analyzed, for example, the expansion of the plasticity affected region and evolution of dislocation density. Take W for an example, the temperature dependent shear modulus μ(*T*) and α(*T*) can be informed in previous works [**? ?** ], and it is known that *b* = 0.274 nm and tan θ = 0.358 for the Berkovich nano-indentation of W [**? ?** ], as summarized in Table **??**. Therefore, according to Equation (**??**), one can calculate *h*∗(*T*) and *M*(*T*) = <sup>3</sup> *h*∗(*T*)/¯ *h*∗(*T*) at 160 K, 230 K and 300 K, respectively. Figure **??** illustrates the evolution of *h*∗(*T*) and *M*(*T*) as a function of *T* for single crystal W, which indicates that *h*∗(*T*) decreases while *M*(*T*) increases with *T*. The former is rational as *ρ*S(*T*), determined by *h*∗(*T*) as expressed in Equation (**??**), is considered to increase due to the high internal strain stored in the materials under high temperatures [**?** ]. As a comparison, the latter is ascribed to the weakened impediment of slipping dislocations that the expansion of the plasticity affected region becomes comparatively easy at high temperatures [**?** ].

**Table 2.** Material properties for single crystal W with temperature effect.


*b*: the magnitude of Burgers vector; μ: shear modulus; α: dislocation strength coefficient; *C*11, *C*<sup>12</sup> and *C*44: elastic constants; α0: dislocation strength coefficient when the temperature equals zero; *k*0: proportional coefficient; *T*: temperature.

**Figure 4.** (Color online) Evolution of *M* and *h*∗ as a function of *T* for single crystal W.

Last but not the least, the evolution of *ρ*G(*T*) and *ρ*S(*T*) as a function of *h* at different temperatures is compared for single crystal W, as illustrated in Figure **??**. According to Equation (**??**), *ρ*G(*T*) is determined by both *M*(*T*) and *h*. Thereinto, the inverse scaling law between *ρ*G(*T*) and *h* indicates the fundamental mechanism for the indentation size effect. Whereas, this scaling law tends to get weakened at high temperatures due to the increase of *M*(*T*) with *T*. As a comparison, *ρ*S(*T*) is independent with *h* but only increases with *T*. Moreover, one should note that *ρ*G(*T*) is generally more than one order of magnitude higher than *ρ*s(*T*) at the shallow indentation region, indicating the dominant dislocation hardening mechanism originates from the contribution of GNDs.

**Figure 5.** (Color online) Comparison of the *ρ*<sup>G</sup> − *h* and *ρ*<sup>S</sup> − *h* relationships at various temperatures for single crystal W.

## **4. Conclusions**

In this work, a mechanistic model is proposed for the hardness-depth relationships of single crystals with temperature effect. Fundamental hardening mechanisms, including the lattice friction and network dislocation interaction, are considered in the hardness model. Four sets of experimental data are applied to verify the rationality and accuracy of the proposed model, and a reasonable agreement is achieved. Moreover, it is realized that the moderated indentation size effect at elevated temperatures is ascribed to the accelerated expansion of the plasticity affected region, which results in the decrease of the density of GNDs.

**Author Contributions:** Conceptualization, H.L.; Writing—Original draft preparation, H.L.; Writing—Review and editing, L.Y. and X.X.; Project administration, X.X.; Funding acquisition, H.L. and X.X. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work is supported by the National Nature Science foundation of China (NSFC) under Contract No. 11802344, 11872379 and 11805061, and Natural Science Foundation of Hunan Province, China (Grant No. 2019JJ50809 and 2019JJ50072). H.L. thanks the Fundamental Research Funds for the Central Universities.

**Acknowledgments:** The author acknowledges Terentyev for the useful discussion.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
