## 3 Results and Discussion
### Hardness  
Hardness measurements were performed at different heights in the samples cross-section to evaluate whether the thermal history and consequent variations in grain width along Z impact on hardness is relevant or not. The resulting average hardness values for all the conditions considered were 398 \(\pm\) 20 HV for ArS samples, 311 \(\pm\) 27 HV and 325 \(\pm\) 11 HV for SE-CB and SE-HB samples respectively. The markedly higher hardness for the first sample type is probably the result of the finer and different microstructure (as shown in Figures 2 and 7), deriving from the increase in interstitial elements content. Microstructure refinement can also explain the slightly higher hardness value for SE-HB samples (finer martensite) with respect to SE-CB ones [61]. Figure 7: Martensite width relative and cumulative frequency distribution for all the deposition conditions considered. The average results for different Z values are plotted in Figure 8a, in the same way used previously for grain width, in Section 3.3. Important variations in hardness along Z are explained by confirming the initial hypothesis that the sample height has a direct impact on grain width. As previously stated, while ArS sample hardness values showed an upward-sloping variation as Z increases, SE-CB and SE-HB samples were characterized by an opposite trend, displaying a decreasing HV trend as the distance from the platform increases. This markedly different behaviour might be caused by the different microstructures involved. Considering the correlation between baseplate distance and grain width/hardness in the specimens built in a sealed environment, it can be derived that the intrinsic variation in cooling rate typical of this process and related to the baseplate temperature has a direct impact on the microstructure of the material. It then leads to the formation of anisotropic samples. On the other hand, this sensitive change was detected in small samples and in an area relatively close to the baseplate. This phenomenon might be negligible for greater values of Z (bigger samples). In fact, significant grain width/hardness variations in bigger samples seem unlikely. Moreover, the first millimetres from the surface are usually removed anyway due to machining. Considering that, this variation might be negligible for large samples. These phenomena might be critical for repairing applications, which usually involve a small area. To assess all these scenarios, further studies must be carried out on bigger samples to evaluate when and if the growing trend eventually reaches a plateau or at least a negligible variation. To further investigate the above-mentioned behaviour, being able to distinguish between the influence of the microstructure itself and the grain size effect (grain boundary strengthening) is important. Both phenomena in fact play an important role in determining the mechanical properties of the alloy, including hardness. The Hall-Petch equation [62; 63] is a well-known correlation between grain size and yield strength, that can be correlated to hardness [64; 65], for polycrystalline metallic materials:  
\[\sigma_{y}=\sigma_{0}+\frac{k}{\sqrt{d}} \tag{5}\]  
where \(\sigma_{y}\) is the material yield strength, \(\sigma_{0}\) and k are material constants and d is the grain diameter. The evaluation of this last parameter is particularly complex in AM materials, since the grains are usually strongly anisotropic. Furthermore, the correlations between \(d\) and \(\sigma_{y}\) (or HV) are acceptable for comparison only if the same type of microstructure is considered. In fact, different samples of the same material with similar grain size, but completely different microstructures (e.g., due to different thermal history) might show very different Hall-Petch curves. Figure 8b provides the results obtained evaluating the Hall-Petch curves for the samples in all the building conditions.