## 3 Results and Discussion
### Change in Average Height of Grains under Elastic and Plastic Conditions  
The change in the average height of each grain is shown in Figure 5. The initial average height was varied, because of the surface treatment before the tensile test. Small increases and decreases occurred until the fourth elastic load step, but \(h_{\mathrm{m}}\) values changed substantially at, and after, the fifth load step, which corresponded to the macroscopic plastic conditions. With respect to the change in height between loading and unloading, all grains showed opposite movements (rising and then sinking or vice versa) during loading and unloading, respectively, and this continued under the macroscopic plastic loading conditions. The extent of these movements increased substantially at the beginning of the plastic deformation stage. These random height-change behaviors of the grains were caused by microscopic inhomogeneous elastic and plastic deformations. The microscopic inhomogeneity was attributed to the crystal orientation and geometry of both surface and subsurface grains.  
Figure 4: Height distribution at four load steps (1 pixel = 4.443 \(\upmu\)m) [24].  
Figure 5: _Cont_.  
The change in the average height per unit load, \(\Delta h_{\text{m}}/\Delta P\), is shown in Figure 6 for different combinations of load steps. Figure 6a shows the relationship between the first (\(P=317\) N) and third (\(P=423\) N) load steps, and Figure 6b shows the relationship between the third and seventh (\(P=741\) N) load steps. The term \(\Delta h_{\text{m}}\) is the difference between the height at the tension-hold and during the unloaded conditions immediately prior to that. Although there was a certain amount of scatter in the plots, there was a positive correlation in \(\Delta h_{\text{m}}/\Delta P\) between the two load steps represented in Figure 6a. The relatively large scatter was due to measurement error and inhomogeneous deformation within the grains. The following outcomes arose from this result: (a) Grains on the surface did not deform uniformly, even under elastic conditions, which was caused by anisotropic elasticity based on crystal orientation and grain geometry; and (b) each grain deformed linearly under elastic conditions, which was easily predicted by linear elastic theory. Figure 6b shows there was a similar strong correlation between \(\Delta h_{\text{m}}/\Delta P\) at the third and seventh load steps. Considering that the abscissa and the ordinate represent the data under elastic and plastic conditions, respectively, it can be concluded that microscopic inhomogeneity under plastic conditions was related to that under elastic conditions. Similar results were obtained for monotonic tensile tests of pure titanium [19] and pure copper [20] specimens. The average height change per unit tensile load is thought to reflect all the microstructural factors influencing microscopic deformation, such as the geometries and crystal orientation of surface and subsurface grains, and grain boundary properties. In addition, considering that localization of plastic deformation and fracture first occur in a stress-strain concentrated area, the strong relationship between the microscopic height change under elastic and plastic conditions seems to be reasonable. This positive relationship between elastic and plastic height change will be useful for predicting plastic behavior or fatigue crack initiation from elastic height change [25].  
Figure 5: Change in height distribution with load \(P\) at each load step. (**a**) Grains 1 to 12; (**b**) grains 13 to 23; (**c**) grains 25 to 41; (**d**) grains 42 to 55; and (**e**) grains 56 to 81.