**3. Results**

#### *3.1. Topography Results*

X-ray topographs are 2D oblique projections of a crystal. At the onset of plasticity, slip system activity may modify the local Bragg condition within the grain and produce orientation contrasts on the detector. In this section, topographs over a full *ω* turn for three neighboring grains as a function of the load are analyzed.

As we shall see, the perturbations of the crystal lattice are localized within the slip plane and may only be visible at certain *ω* angles, called edge-on configuration, when the diffraction direction is contained in the plane (i.e., the tilted slip plane normal *n t* is perpendicular to *K* ). Knowing the grain orientation and the tilt geometry, it is straightforward to obtain the two edge-on *ω* angles for a given slip plane by solving (here, we do not account for the base tilt, as both the left and right side of the equation would be equally affected):

$$(\Omega.\underline{\mathfrak{u}}\_t).\underline{\mathbb{K}} = 0 \quad \text{with} \quad \underline{\mathbb{K}} = \frac{1}{\lambda} \begin{pmatrix} \cos(\theta\_{Brag}) \\ 0 \\ \sin(\theta\_{Brag}) \end{pmatrix} \tag{10}$$

This means solving [*nt*[0] cos(*ω*) − *nt*[1] sin(*ω*)] cos(*θ*) + *nt*[2] sin(*θ*) = 0. The two *ω* values, for a given slip plane, are separated by close to 180◦, a value depending on *<sup>θ</sup>Bragg*.

Table 3 gathers the *ω* values for the two slip systems with the highest Schmid factor in the 3three grains calculated with Equation (10). These values will be used to show the topographs in edge-on configuration where the contrast is expected to be maximal for a given slip plane. Note that the values for the two observed slip planes (not to be confused with the two values (*<sup>ω</sup>*1, *ω*2) for a given slip plane) are exactly separated by 180 degrees. For Grains 4 and 18, the aligned reflection is (202). Rotating around this axis, the edge-on configurations for the slip planes with the two highest Schmid factors (11¯1) and (111) are 180◦ apart (they share the [1¯01] zone axis, which is perpendicular to the scattering vector). For Grain 10, the aligned reflection is (002); rotating around this axis, the edge-on configurations for the (111) and (111¯) planes are also 180◦ apart (they share the [1¯10] zone axis, which is perpendicular to the scattering vector).

The Schmid factor allows estimating the resolved shear stress and is here calculated in the single crystal approximation. It should be noted that it cannot rigorously be applied to the polycrystal case (this value neglects the effect of neighboring grains and any other heterogeneity), although it is very often used to predict slip system activation. It is thus interesting to see in this case how well this

indicator performs. In contrast to the estimation using the Schmid factor, the full field finite element simulations performed in this work (see Section 3.3) take into account the interaction of individual grains with their neighbors. This results in strongly non-homogeneous fields of plastic strain and lattice rotation inside the grains. The advantage of the Schmid factor is clearly that it depends only on the crystal orientation (and is thus easily computed on the fly during this type of diffraction experiment). However, the material scientist doing an experiment may rely on a more detailed stress estimation, either using a supplementary far field detector and track the motion of diffraction spots [19] or inferring the stress tensor value from mean field or full field computations using the actual 3D microstructure, as done in the present work.


**Table 3.** Edge-on *ω* values (in ◦) for the two slip systems with the highest Schmid factor in the 3 grains; *ω* values in bold are used to show the topographs in edge-on configuration in Figure 5.

Integrated topographs at the recorded *ω* closest to the edge-on values and for selected load levels are shown in Figure 5. Contrast forming bands within the grains are clearly visible and appear first for Grain 10, then Grain 18 and finally for Grain 4.

**Figure 5.** Topographs, integrated over Θ, in edge-on configuration for each grain of the cluster and different load levels; videos of the complete *ω* set are available as Supplementary Material for Grain 4 in the initial and deformed states.

These sets of bands were parallel; most of the time, they extended through the whole grain, and their number increased as the deformation increased. Going through the whole projection set for each grain shows that two sets of bands were visible at different *ω* values. For Grain 10, the crystallographic configuration was such that both sets were visible at around 40◦. This is, to the best of the author knowledge, the first in situ observation of bulk plasticity in a millimeter sized polycrystalline specimen. From there, the orientation of the bands, their number and location within the grain can be further studied.

Using the initial grain orientation and the tilt geometries for each aligned reflection, it is possible to correlate the angle of those bands to specific crystallographic planes (see Figure 6). For this, a 3D geometrical representation of the grain was built using the DCT reconstruction, and the relevant slip planes had been added inside according to the measured grain orientation (the open source library pymicro [38] was used to this end). The grain can be tilted in the topotomographic condition and rotated to the given edge-on omega angle. Using a parallel projection mode and setting the view in the diffracted beam direction *K* = (1, 0, tan(2*θBragg*)) produced the same conditions as when collecting the topographs. All the observed bands had been identified without any ambiguity as projections of the {111} planes; see Figure 6. In each case, the exact slip plane could be identified and further related to the Schmid factor. This demonstrates that the onset of plasticity, from the first slip band to the more advanced state where several slip systems are active, was indeed captured in situ during this experiment.

**Figure 6.** Identification of the bands as slip plane traces visible in the edge-on geometrical configuration in the topographs; here, the two observed active slip planes are shown for one of the two *ω* angles; the active slip plane locations within the grains have been measured manually and displayed in 3D.

For Grains 4 and 18, the two observed slip planes correspond to the two highest Schmid factors (see Table 3), whereas for Grain 10, they correspond to the first and third highest Schmid factors. The slip system corresponding to the second highest Schmid was not observed to be active in this grain.

Rocking curves are presented in Figure 7, for each grain at the strain levels and for the same *ω* values as in Figure 5. All three grains exhibited a consistent behavior, with a narrow curve at the beginning, which first shifted to the lower |*<sup>θ</sup>Bragg*| values due to elastic loading (increase of the *dhkl* interplanar spacing) and then widened considerably when plasticity took place.

**Figure 7.** Evolution of the rocking curves for each grain of the cluster during in situ loading; the curve colors (green, red, black, yellow) refer to the load levels in Figure 5, respectively (undeformed, 0.09%, 0.16% and 0.32%).

#### *3.2. 3D Rocking Curves' Results*

As grains are aligned in a topotomographic sense, it is possible to measure rocking curves at every *ω* position. This measure is therefore sensitive not only to the amount of curvature of a crystal, but also to the orientation of this curvature in real space. To quantify the intragranular orientation spread revealed by a rocking curve, we introduced the width of the rocking curve at 10% of the peak of the normalized intensity, denoted as full width of the effective misorientation (FWEM). Although the effective misorientation describes the change in Bragg condition due to both orientation and lattice spacing variations, in practice, the orientation effect is largely predominant. Therefore, this value is a direct (qualitative) measure of the orientation spread, around the axis defined by the base tilt.

The FWEM was measured every 4◦ (for each *ω* position) and is plotted at all different load levels in Figure 8 to observe its evolution with increasing plasticity. An interesting dumbbell-shaped curve was consistently obtained for the three grains. The curve widened in a preferential direction linked to the active slip systems within the grain. One can observe that the FWEM was similar for Grains 4 and 18, which have the same combination of active slip systems. The orientation of the curves for Grain 10 was different and exhibited a clear reorientation of the preferential direction towards the end of the loading sequence. This may be linked to changes in the relative activity of the dominant slip system(s) during deformation, but would require further analysis.

The shape of the curve can be understood more clearly considering the idealized case of a strain-free crystal bent by an amount ΔΘ by geometrically necessary dislocations. In the kinetic approximation, this configuration would produce a figure made of two tangent circles giving a very limited FWEM in the bending axis direction (almost no lattice rotation) and of exactly ΔΘ perpendicular to it. The FWEM can be seen as the limit of the Bragg condition when rocking the base tilt Θ and can be computed using the rotating crystal equation solving for Θ for a known *ω* (see Figure 8, bottom right).

**Figure 8.** In situ FWEM as a function of *ω*, for each grain of the cluster (the color code matches the one used for the tensile curve in Figure 2) and example of the FWEM in the case of a crystal ideally bent around a single axis by 0.2◦ (bottom right).

#### *3.3. Comparison with CPFEM Simulations*

The first comparison is the slip activity computed with the numerical model (seen Section 2.3) for each grain with the activity visible on the topographs for a particular *ω* position (see Section 3.1). The CPFEM simulations give access to the active slip systems as opposed to the slip planes only, which were identified experimentally on the topograph. The slip activity was captured by averaging the amount of slip for a given slip system within the grain and can be compared to the qualitative information obtained on the topographs (intensity of the contrast and number of bands).

Accumulated plasticity (all slip system contributions) within the bulk is shown in Figure 9a, and the slip activity for each of the three grains is detailed in Figure 9b. For Grains 4 and 18, which have close orientations and behave similarly, the model predicted double slip with systems (111)[0-11] and (1-11)[011], in agreemen<sup>t</sup> with the experimentally observed slip planes, and no other slip activity. The situation is different in Grain 10, which also showed double slip experimentally, but with planes (111) and (11-1), whereas the model predicts plastic activity on three planes (111), (1-11) and (11-1). It is interesting to see that despite the general agreement, the details of the slip system activation are far from perfect. The discrepancy may be attributed to the parameters of the constitutive laws, especially regarding the interaction matrix coefficients *hrs*. These parameters were determined from macroscopic tensile curves and from the literature. A detailed parametric study is necessary to analyze the impact of these parameters values on the activation of slip systems in that grain. This can be seen as an opportunity to use such experimental data collected at the length scale of the grains and in the bulk to enrich identification datasets and solve the long-standing issue of crystal plasticity material parameter identification [39].

**Figure 9.** Predicted plastic activity: (**a**) view of the interior of the specimen showing the accumulated plasticity at the load level corresponding to the end of the experiment; (**b**) average plastic activity in the three grains of the cluster; the most active slip system is represented with a solid line and the second most active with a dashed line, while the grain color code is used.

Using the simulated mechanical fields within the grains, rocking curves can be simulated as described in Section 2.5. The generated rocking curves for each grain of the cluster are plotted in Figure 10, at *ω* = 165◦. The obtained behavior was consistent with the experimental observations. The rocking curve first shifted to lower Θ values due to the increase of the interplanar distance during loading. A very small amount of broadening due to the heterogeneous elastic strain field within the grain was also observed. When plasticity sets in and slip systems start to be active, the curve widening was more pronounced, and the shape changed, which can be related to the tendency to form subgrain-like regions. This effect had been studied both theoretically and experimentally, for instance by [40–42].

**Figure 10.** (**a**) Simulated rocking curves for Grain 4 at *ω* = 165◦ at four different strain levels; (**b**) comparison between experimental and simulated rocking curves at *ε* = 0.34 × 10−2.

Finally, a quantitative comparison between experimental FWEM and simulated FWEM is presented in Figure 11. Simulated curves are plotted for an applied strain of *ε*33 = 0.0034, and experimental surfaces are plotted for the last step of the topotomography experiment. Both curves were in very good agreemen<sup>t</sup> for Grain 4 and Grain 18, for both shape and orientation. For Grain 10, the surface had the right amplitude, but not the right direction. As explained previously, the direction of the dumbbell-shaped curve was linked to the precise combination of active slip systems within the grain, and the discrepancy between the predicted and observed slip systems was presumably the cause of the mismatch in orientation for this case.

**Figure 11.** Comparison between experimentally simulated full width of the effective misorientation (FWEM) at *ε* = 0.34 × 10−<sup>2</sup> for the three investigated grains.
