Lattice curvatures and dislocation density tensor components

Figure 3.5 shows the evolution of the lattice curvature components upon cycling in mrad/μm. The same scale bar (|8 mrad/μm|) has been given to all the components for better comparison. Most of the lattice curvature components mark the perimeter of the abovementioned misoriented regions (see Figure 3.3-a) and exhibit changes during fatigue.

- The maps of Ɉ_௫௫ and Ɉ_௫௬ components are given in the first two rows. Before cycling (step 0) they already mark the perimeter of regions C, D’ and E. Upon cycling, these traces of Ɉ௫௫ and Ɉ௫௬still mark the boundaries of the three misoriented regions and none of the curvature components vanish (e.g. one can follow the expansion of regions B and E).

There is another region, named H, which is not distinguishable in the misorientation map (Figure 3.3-a) and ߱_௫,߱_௬ and߱_௭ maps (Figure 3.4-a-c) but shows Ɉ_௫௫ and Ɉ_௫௬ curvatures. This heterogeneous region is relatively invariant during fatigue as its physical location and features do not exhibit major changes.

At map 80 few boundaries are marked with arrows in the Ɉ_௫௫ and Ɉ_௫௬ maps. These boundaries are initially discontinuous, whereas after 80 cycles and until the last cycle they become connected.

The onset of region F is visible at map 100 and is clearly defined at step 120. There is another region (G) with identical features below it. Nevertheless, this region is not created during fatigue as its traces already exist from the beginning (G’).

- The Ɉ_௬௫and Ɉ_௬௬maps are shown in the third and fourth rows. Initially there are areas without Ɉ_௬௫and Ɉ_௬௬. Mainly the perimeter of regions B, G’ and E are recognizable before cycling.

At the beginning there is a faint mark of region A that vanishes upon cycling. Region B, on the other hand, is always present and the fragmentation observed in the misorientation map (Figure 3.3-a) can be tracked.

Regions G’ and E follow the same evolution as described above. Both maps are relatively clean of traces during the homogenization process, until step 80, after which region D’ is noticeable.

In the last maps the formation of region F and expansion of D’ can be followed. Region G is recognizable at step 120 too.

- The footprints of Ɉ௭௫and Ɉ௭௬are the less pronounced among all the lattice curvature components and are shown in the last two rows. In the initial maps mainly region B and C can be detected. There are also a few lines marked with arrows that are present during the 120 cycles.

After 100 cycles, the onset of region F is visible and after 120 cycles, the boundaries of the region are clearly marked.

The lower parts of these maps do not show any strong Ɉ_௭௫and Ɉ_௭௬curvature changes.

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Figure 3.5 Evolution of lattice curvatures and dislocation density components of sample S1-L

The dislocations corresponding to the primary slip system for single slip orientation have their Burgers vector parallel to Y=[ͳത01] and line direction in the Z-axis. These straight dislocations cause Ɉ_௭௫and Ɉ_௭௬lattice curvatures. As already mentioned mainly regions B, C and F show Ɉ௭௫and Ɉ௭௬footprints. The former two have these traces from the beginning; the latter has evolved from the low misoriented background. This suggests that region F is result of fatigue deformation.

All the lattice curvatures that define the evolved region F observed in Figure 3.5 can be interpreted following Figure 2.27. Their geometrical meaning is given in Figure 3.6. On the left wall, region F has negative torsion around X-axis, and the plane perpendicular to X-axis has positive flexion around the Y-axis and negative around Z-axis. Reverse signs are found in the right wall. On the upper side, the region has positive torsion around Y-axis and the plane perpendicular to the Y-axis has negative flexion around both X-axis and Z-axis. The lower part has opposite values.

Figure 3.6 Representation of the observed lattice curvatures around region F

The Ƚ௭௭ and Ƚ௫௫െ Ƚ௬௬ maps are plotted in Figure 3.7 where most of the regions already mentioned are observable. Regions B, D’ and F are the most noticeable, but E and G can be easily distinguished too. Regions C is less sharply defined and region A is mainly invisible. In the Ƚ_௭௭ map the boundaries of some regions have alternating signs particularly on the curved areas (e.g. E and F at step 120). In both maps the traces are rather strong and evident during the 120 cycles. As in the curvature maps the density of the traces is higher in the upper part of the maps than in the lower part.

Figure 3.7 Evolution of Ƚ௭௭ and Ƚ௫௫െ Ƚ௬௬components of sample S1-L

The statistical distribution of each lattice curvature and dislocation density tensor components at cycle 0 and cycle 120 are graphically depicted as boxplots in Figure 3.8. Briefly, each box represents the middle fifty (interquartile = IQR) of the population and the whiskers and arms extend till ȁͳǤͷ ൈ ȁ. The points plotted as '+' symbol are outliers that are values outside theȁͳǤͷ ൈ ȁ range. The median of all distributions, the central mark of the box that is almost invisible, is ~0 mrad/μm and it does not change after 120 cycles. Visual description of boxplots is given in Appendix D.

Before deformation, all the lattice curvature components are roughly below |10 mrad/μm|.Ɉ_௬௫, Ƚ_௭௭ and Ƚ_௫௫െ Ƚ_௬௬have some outliers higher than |10 mrad/μm|. In the later some values are as high as |15 mrad/μm|. At cycle 120 the boxplots show that outliers of Ɉ_௫௫, Ɉ_௫௬, Ɉ_௬௫, Ɉ_௬௬ and Ƚ_௭௭exceed |10 mrad/μm|, whereas the limits of Ɉ_௭௫ and Ɉ_௭௬ stay below |10 mrad/μm|. The distribution of Ƚ_௫െ Ƚ_௬௬ broads considerably and values almost as high as |30 mrad/μm| are reached.

Figure 3.8 Statistical distribution of the lattice curvature and Nye tensor components of sample S1-L