## 3 Results and Discussion
### Elastic and Plastic Deformation and Height Changes of Grains  
Figure 10a schematically illustrates the microscopic deformation of surface grains. Under small stress-strain conditions, all grains (including both surface and subsurface grains) deformed elastically. The deformation of surface grains depended on their own geometries and crystal orientation. It is also easily understood that the geometries and crystal orientation of the subsurface and interior grains exerted a large influence on the deformation of the surface grains. Under elastic conditions, grain orientation and anisotropic elastic properties, which are usually provided by the elastic constants (\(c_{11}\), \(c_{33}\), \(c_{44}\), \(c_{12}\), and \(c_{13}\)), were major causes of elastic inhomogeneity. The result was that inhomogeneous sinks and rises in the surface grains appeared. The linear correlation of the increase in average height per unit load \(\Delta h_{\rm m}/\Delta P\) between two small load steps, shown in Figure 6a, suggested that linear elasticity held in each crystal grain of polycrystalline pure titanium. The location of each  
Figure 9: Digital holographic-microscope intensity image of grains 1, 3, 5, 6, 7, 8, 18, and 19, at \(P\) = 635 N.  
grain on the plot indicated its own elastic properties, considering all factors such as the geometries and crystal orientation of neighboring, subsurface, and interior grains. However, localized plastic deformation may take place below the macroscopic yield stress, where the stress and strain are highly concentrated. It should be noted that the macroscopic yield stress was different from the microscopic one, as suggested by previous researches [4; 14], and the localized plastic region gradually extended with increasing external load. Figure 10b schematically illustrates the elastic and plastic deformation of a single-crystal grain, which was initially shown by [PERSON] et al. [19]. Under a small stress-strain, each grain in the polycrystalline titanium deformed according to the anisotropic elastic properties of a single crystal, while the surface generally sank, because of the Poisson effect. By increasing the stress-strain, plastic deformation occurred in a direction similar to the elastic deformation, and enhanced its elastic properties. It was therefore reasonable to obtain a strong correlation between the increase in average height per unit load \(\Delta h_{\text{m}}/\Delta P\) occurring in both small elastic and large plastic load steps (Figure 6b). The large values of \(\Delta h_{\text{m}}/\Delta P\) under plastic deformation compared with those under elastic deformation (i.e., the large scale of the ordinate axis in Figure 6b) corresponded to large plastic strain in the single-crystal grains. For reference, the change in average height per unit load, \(\Delta h_{\text{m}}/\Delta P\), that was obtained for pure titanium under monotonic tension, is shown in Figure 11[19]. Figure 11a is the correlation between two small load ranges and Figure 11b is that between small and large load ranges. Figures 6a and 11a show similar positive correlations, but the scatter was larger for cyclic than for monotonic tension. One possible reason for this is that the change in average height under cyclic tension was evaluated between a certain tension-hold load and a zero load immediately prior to that. When the load was released, the position of the specimen may have changed slightly. In contrast, the change in average  
Figure 10: Height change of surface grains as a result of elastic and plastic deformation. (**a**) Rise and sink of surface grains, and deformation of subsurface grains; and (**b**) elastic and plastic deformation of each grain [19]. height under monotonic tension was evaluated without unloading, and the position of the specimen was extremely stable during the test. Concerning the correlation between small and large load ranges, we obtained similar results under monotonic and cyclic tension, as shown in Figures 6b and 11b.