**1. Introduction**

Determining microstructure-property relationships is an essential engineering problem and is directly linked to our ability to observe both the microstructure and the deformation/failure mechanisms concurrently. Electron back scattered diffraction (EBSD), which provides crystal orientation maps with sub-micrometer spatial resolution, remains a key tool, but is limited to the specimen surface [1]. To measure and interpret the strain field produced within individual grains, digital image correlation can be used provided a small-enough speckle can be produced at the specimen surface [2,3]. In this regard, subsequent analysis using numerical methods such as finite elements has also proved to be a powerful tool to interpret experimental results [4–6], but ultimately remains limited if the underlying material volume is not known [7]. In this paper, a new method combining in situ mechanical testing, three-dimensional (3D) bulk X-ray inspections and the crystal plasticity finite elements method (CPFEM) is used to study how plasticity proceeds in individual grains of a polycrystalline sample.

In the last 10 years, one particular focus of the 3D imaging community has been on obtaining reliable three-dimensional grain maps. Since most structural materials are polycrystalline and the mechanical properties are determined by their internal microstructure, this is a critical issue. For instance, considering slip transmission in crystal plasticity problems or small tortuous cracks evolving in a 3D grain network, it is a recurring challenge to assess the bulk mechanisms from surface observations only. Knowing how the different grains of the microstructure are arranged below the

surface is therefore essential. Therefore, there has been considerable effort to develop characterization techniques at the meso-scale, which can image typically 1 mm<sup>3</sup> of material with a spatial resolution in the order of the micrometer.

Among 3D characterization, an important distinction exists between destructive and non-destructive techniques. Serial sectioning relies on repeated 2D imaging (which may include several modalities) of individual slices where a thin layer of material is removed between each observation [8]. Considerable progress on that side has been made in the last decade, bringing high quality measurements in 3D of grain sizes and orientations, but also detailed grain shapes and grain boundary characteristics. The price to pay with serial sectioning remains, however, the destruction of the sample. In parallel, accessing crystallographic information in the bulk of polycrystalline specimens was subsequently achieved by using the high penetration power of hard X-rays and leveraging diffraction contrast. This led to the development of a variety of 3D X-ray diffraction techniques (3DXRD; see [9,10]) enabling the characterization of millimeter-sized specimens by tracking the diffraction of each individual crystal within the material volume while rotating the specimen. Among them, the near field variant called X-ray diffraction contrast tomography (DCT) uses an extended box beam to illuminate the specimen and allows for simultaneous reconstruction of both the sample microstructure visible in X-ray absorption contrast and the crystallographic grain microstructure as determined from the diffraction signals with a single tomographic scan provided the grains have a limited orientation spread [11–13]. The typical acquisition time is one hour on a high brilliance beam line such as ID11at ESRF, which makes it the fastest non-destructive grain mapping technique.

Non-destructive imaging allows one to observe both the microstructure and the deformation/failure mechanisms in situ (4D studies). However, resolving 3D grain shapes by near-field diffraction imaging requires reducing the sample to detector distance to a few millimeters, which has for a long time drastically limited mechanical 4D studies due to the space constraints. Recent progresses with mechanical stress rigs solved this issue and opened new perspectives to study the deformation and fracture of polycrystalline materials [14,15].

Another key challenge is to link 3D microstructure characterization tools with computational models in order to predict engineering mechanical properties. This can be done using synthetically-generated images, but it requires using sophisticated models to ensure that the microstructure is representative, in particular of the tail distributions (critical when looking at fracture processes) [16]. Another approach is to use as-measured 3D microstructures [17]. The major advantage of this route is directly comparing experiment and mechanical simulation at the grain scale. For mechanical problems, continuum crystal plasticity (either using the finite elements or the spectral method to solve the equilibrium) has proved to be a powerful tool to interpret experimental results obtained in the deformation of metallic polycrystals [4,18–20]. Large-scale 3D polycrystalline simulations can be performed with sufficient local discretization to predict the transgranular plastic strain fields. One of the issues of this type of models is the material parameters' identification. This is mostly due to the fact that identification is classically done by minimizing a cost function versus macroscopic tests' response, which makes the problem ill posed. Some recent attempts directly used local strain measurements in the identification dataset in order to identify material model parameters [21], but have remained limited to surface studies so far.

The mechanical behavior of polycrystalline alloys is especially important for structural materials. A few key examples are Al alloys used for transports in general, Ti alloys used in aerospace where specific strength and high performance is particularly needed or Ni base alloys when looking for very high temperature performances. In this regard, advancing our understanding on the deformation of such materials will lead to lighter, stronger and more reliable parts.

In this paper, we present a new 4D study, using a combination of X-ray diffraction contrast tomography, topotomography and phase contrast tomography to study polycrystal plasticity in an aluminum-lithium alloy. Experimental and simulation methods are detailed in Section 2. The main experimental results are then presented in Section 3.1 for the observed slip system activation and

Section 3.2 for the measured crystal lattice orientation. Simulation results and comparison with the experiments are described in Section 3.3. In the Conclusion Section, the main results are summarized, and some future directions of research are suggested.

#### **2. Materials and Methods**

#### *2.1. Experimental Setup for Topotomography*

A plate made of Al-Li 2.5% wt purchased from MaTeck was rolled to 45% reduction and recrystallized 20 min at 530 ◦C to tune the grain size at 100 μm. This step ensured the number of grains in the illuminated section was well below the DCT limit (typically a few thousands) and that the initial state was as defect free as possible to study the onset of plasticity. The material was then aged 4 h at 100 ◦C to form very small Al3Li precipitates with an expected size of 8 nm [22]. At this size, the precipitates remain shearable by the dislocations, which will promote a planar and localized slip as the main deformation mechanism. Small dog-bone tomographic tension specimens (0.5 mm × 0.5 mm minimal cross-section) were cut using EDM in the middle of the plate for the experiment. A 8.5-mm radius in the gauge length produces a small stress concentration and ensures that the first plastic events will occur in the observed region.

The experiment was carried out at the material science beam line (ID11) at the ESRF. The X-ray beam was produced by an in-vacuum undulator and collimated to a size of about 0.7 × 0.7 mm by means of an X-ray transfocator [23]. The beam energy was set to 40 keV with a relative bandwidth of 3 × 10−3. The diffractometer installed on ID11 has been designed for this particular variant of diffraction imaging experiments, which requires the alignment of a scattering vector parallel to the tomographic rotation axis [24,25]. The scattering vector *G* is defined as *G* = *K* − *X* with *X* the incident wave vector and *K* the diffracted wave vector, both with a norm of 1/*λ*. Let us call (*p*, *q*,*<sup>r</sup>*) the components of the scattering vector *G s* expressed in the sample coordinate system for a set of (*hkl*) planes: 

$$\mathbf{G}\_s = \mathbf{g}^{-1} \mathbf{B} \begin{pmatrix} h \\ k \\ l \end{pmatrix} = \begin{pmatrix} p \\ q \\ r \end{pmatrix} \tag{1}$$

where *g* is the orientation matrix of the crystal and *B* accounts for the lattice geometry (for cubic structures like aluminum, it reduces to the lattice parameter times the identity).

Following [26,27] and accounting for the four diffractometer rotation angles ( Θ, *ω*, *φ*, *χ*), via the four associated rotation matrices ( *T*, **Ω**, **Φ**, *X*), the Bragg diffraction condition in 3D can be written as:

$$\frac{2\sin^2(\theta\_{Rrag\,\%})}{\lambda} = -\left[T\Omega\Phi X \begin{pmatrix} p \\ q \\ r \end{pmatrix}\right]\_1\tag{2}$$

where the rotation matrix *X* associated with the diffractometer angle *χ* should not be confused with the incident wave vector *X* .

In a topotomographic experiment, the scattering vector is aligned with the rotation axis of the tomographic rotation stage *ω* (see Figure 1a). Circles *χ* and *φ* are used to set *G* parallel to the rotation axis *ω* (matrices *X* and **Φ**, respectively), while Θ is used to tilt the whole setup (including the tomographic rotation axis) around *Y* by the Bragg angle *<sup>θ</sup>Bragg* (matrix *T*). For a known crystal structure and orientation (i.e., *B* and *g* are known), it is straightforward to rework Equation (2) to derive the two tilt rotation values (*<sup>χ</sup>*, *φ*):

$$\chi = \arctan\left(-\frac{p}{r}\right) \tag{3}$$

$$\phi = \arctan\left(\frac{q}{-p\sin(\chi) + r\cos(\chi)}\right) \tag{4}$$

Equation (2) is then fulfilled for all possible values of *ω*. Note that in the DCT case, *T*, **Φ** and *X* vanish, and the equation is only verified for 2 particular values, solutions of a quadratic equation in *ω* known as the rotating crystal problem [26].

**Figure 1.** Schematics of the topotomographic alignment: (**a**) a scattering vector is put on the rotation axis (*ω*), and the whole setup is tilted by the nominal value Θ = *<sup>θ</sup>Bragg*; (**b**) example of integrated detector intensity to build the rocking curve.

It is important to understand clearly the difference between Θ and *<sup>θ</sup>Bragg*, as they differ in nature. Θ is the value of the tilt applied by the base tilt motor, whereas *<sup>θ</sup>Bragg* is a material parameter associated with a given *dhkl*spacing and wavelength *λ*: *<sup>θ</sup>Bragg* = arcsin *λ*/2*d*.

Now, in the ideal case of a perfect crystal and quasi-monochromatic plane wave illumination, the entire grain would diffract for the position of the base tilt Θ = *<sup>θ</sup>Bragg*, and only a simple rotation around *ω* would be needed to collect the topotomographic dataset (see Figure 1a). This is in practice never the case, as the inner mosaicity of the grain and dispersion effects require rocking the base tilt (this is again a rotating crystal problem, which is covered in more detail in Section 2.5). A topotomographic (TT) dataset can therefore be seen as a collection of rocking curves and associated stacks of 3D projection data and can be used in two different ways. First, integrated projection topographs corresponding to projections of the entire crystal volume are obtained by summing the intensity over the base tilt Θ. Inspection of these topographs (see Figure 1b) allows for direct identification of the presence of crystalline defects such as slip bands. Second, the width of the rocking curve *I* = *f*(Θ) for a given *ω* position is a measure of the lattice rotation around the base tilt axis (*Y* here; see Figure 1c).

#### *2.2. Details of the In Situ Experiment*

A small tomographic tension specimen was mounted in the Nanox stress rig, specifically designed to be compatible with both DCT and topotomography acquisition geometries [15]; see Figure 2, left. The machine has a very limited size and weight and, thanks to the load bearing quartz tube, allows 360◦ visibility in the DCT configuration with the detector as close as 3 mm to the rotation axis. Full visibility is also achieved in the TT configuration with *<sup>θ</sup>Bragg* ≤ 10◦, with the detector as close as 10 mm from the rotation axis, for the complete range of motion of the two inner diffractometer circles (±20◦ and ±15◦ for Φ and *χ*, respectively). For given values of *<sup>θ</sup>Bragg*, Φ and *χ*, it is possible to move the TT detector even closer, but this has to be checked manually.

With the specimen inside the stress rig, it is possible to analyze the initial undeformed bulk microstructure (positions, shapes and grain orientations) by a DCT scan. The DCT data are then processed to extract all grain orientations and positions within the gauge length. Later on, the DCT reconstruction is also used as input to perform 3D CPFEM calculations (see Section 2.3). For now, these data are used to select a series of grains for further analysis using topotomography imaging during mechanical loading. Here, the selection was based on the following criteria: (i) a low order reflection must be accessible (note that the two circles Φ, *χ* used for the topotomographic alignment only have a

limited range of motion); (ii) the grain must be located in the bulk of the specimen; (iii) all the selected grains must form a small neighborhood. Grain Numbers 4, 10 and 18 (see Figure 3), located in the central region of the specimen, fit these constraints and were selected for the present study to carry out a series of topotomographic scans during a (interrupted) tensile test; see Figure 2, right.

**Figure 2.** Sequence of the in situ topotomography experiment: the initial microstructure is characterized by DCT and analyzed after mounting the specimen into the Nanox device [15], then 23 complete sequences, each comprising three topotomography and one phase contrast tomography scan, have been recorded at increasing levels of load (the color code used in the tension curve is used later on to plot results at a given load level).

At each of the 23 load steps, a phase contrast tomography of the gauge length (1.5 mm in height) is also recorded and used to extract macroscopic strain information. The essential information (grain orientation, aligned *hkl* reflection, Bragg angle, diffractometer angles) for each grain is reported in Table 1. Grains 10 and 18 have a similar orientation, as seen in Figure 3c).

**Figure 3.** Details on the 3-grain cluster (**a**) 3D visualization of the grains (**b**); XZ slice through the 3 grains; (**c**) inverse pole figure of the gauge length with the 3 grain orientations of interest highlighted; (**d**) Θ integration range automatically determined at each loading step.

**Table 1.** Details of the 3 grains selected for TT imaging; the orientation convention is consistent with [28]; and angle values are given in ◦.


DCT scans were composed of 3600 equally-spaced projections over 360◦, recorded on a high-resolution detector with a transparent luminescent screen optically coupled by a 10× microscope objective to a 2048 × 2048 pixel ESRF Frelon camera, giving an effective pixel size of 1.4 μm. A 0.3-s exposure time has been used, resulting in an acquisition time of about 40 min. The same camera was used to record the PCT scans; the camera traveled back and forth along the *X*-axis from the DCT position (about 5 mm behind the specimen) to the PCT position, about 105 mm downstream. A second detector system with a 20× objective and an effective pixel size of 0.7 μm was used for topotomographic scan acquisition. The angular range of the rocking curve scans was automatically adjusted after each load increment in order to cover the entire width of the crystal reflection curve for any *ω* rotation position of the sample. Moreover, the X-ray flux density was further increased by focusing the beam on the area covered by the 3-grain cluster. A continuous motion acquisition procedure with a fixed integration range of 0.1◦ and 0.5-s exposure time per image was used. Integration gaps caused by the readout time of the CCD detector could be eliminated by operating the system in frame transfer mode. In this mode, only half of the active area is available for image acquisition, whilst the other half is used for temporary storage and readout of the previous frame. This procedure was repeated every 4◦, and a complete topotomographic acquisition comprising 90 such rocking scans per grain typically lasted from 10 min up to an hour as the Θ integration range increased during loading. In this experiment, the integration range determination after each load step has been automated by acquiring a coarse TT scan at two values of *ω* separated by 90◦, post-processing the intensity in the image and taking the largest bounds of the two rocking curves, increased by a small amount not to miss any intensity. From the beginning to the end of the experiment, the integration range has been multiplied by a factor of 10 (from 0.05◦ to 0.5◦; see Figure 3d).

#### *2.3. Crystal Plasticity Finite Element Simulations*

A finite strain crystal plasticity model, fully described in [29], is used here to compute the mechanical response of the polycrystalline sample under tension. It is based on the multiplicative decomposition **F**∼ =**E**∼**P**∼ of the deformation gradient, **F**∼, into an elastic part, **E**∼, and a plastic part, **P**∼. The multiplicative decomposition is associated with the definition of an intermediate configuration for which the elastic part of the deformation gradient is removed. The intermediate released configuration is uniquely determined up to a rigid body rotation, which is chosen such that the lattice orientation in the intermediate configuration is the same as the initial one. Mandel called it the isoclinic intermediate configuration [30]. As a result, lattice rotation and distortion during elastoplastic deformation are contained in the elastic deformation part **E** ∼ . The transformation **E** ∼ has a pure rotation part **R** ∼ *e* and a pure distortion part **U**∼ *e*, which can be obtained by the polar decomposition:

$$
\underline{\mathbf{E}} = \underline{\mathbf{R}}^{\varepsilon} \underline{\mathbf{U}}^{\varepsilon} \tag{5}
$$

Plastic deformation is the result of slip processes according to a collection of *N* slip systems, each one characterized by the slip direction *m s* and the normal to the slip plane *n s*. Note that here, plastic slip is the only deformation mechanism considered. This has been double checked up to 10% strain by performing an in situ tensile test in a scanning electron microscope (not reported here for brevity). In other cases, mechanical twinning or grain boundary sliding may need to be considered. In the intermediate configuration, **P** ∼ verifies:

$$\mathbf{P}\mathbf{P}^{-1} = \sum\_{s=1}^{N} \dot{\gamma}^{s} \underline{\mathbf{m}}^{s} \otimes \underline{\mathbf{u}}^{s} \tag{6}$$

In order to analyze the microplastic behavior of the studied AlLi polycrystal, an elasto-visco-plastic crystal plasticity model was selected. Numerical computations were performed using the Z-set software (http://www.zset-software.com) (see [31]). The slip rate on a given slip system *s* depends, via a phenomenological power law with two parameters *K* and *n*, on how much the resolved shear stress *τ<sup>s</sup>* exceeds the threshold *τ*0 + *rs* :

$$\gamma^s = \left\langle \frac{|\mathbf{r}^s| - \mathbf{r}\_0 - r^s}{K} \right\rangle^n \text{sign}(\mathbf{r}^s) \tag{7}$$

Here, *τ*0 is the critical resolved shear stress, and *<sup>r</sup>s*, initially zero, increases with increasing plastic strain and hardens the system *s* through a non-linear isotropic Voce hardening rule, as developed in [32]:

$$r^s = Q \sum\_{r=1}^{N} h^{sr} (1 - \exp(-bv^s)) \tag{8}$$

*v<sup>s</sup>* is the cumulative slip and *hsr* denotes the interaction matrix taking into account the relative influence of slip systems on each other. It includes the self and latent hardening, and only indirect and estimated quantitative information is available about the components of this matrix (see for instance [33,34]). *Q* and *b* are 2 material parameters to be determined.

Monotonic tensile tests were performed on five different macroscopic samples, at three strain rates, and a numerical optimization using Z-set implementation was performed to find a suitable parameter set (*<sup>τ</sup>*0, *Q*, *b*), as presented in Table 2. The yield stress of the material exhibits an inverse strain rate sensitivity, which prevented identifying the viscosity parameters *K* (not to be confused with the norm of the diffraction vector *K* ) and *n*; instead, sensible values for aluminum alloys have been used. Modeling this effect requires a more complex model, including dynamic strain aging as in [35]. For the interactions between dislocations, coefficients *hrs* from [36] have been used.

**Table 2.** Material parameters identified from the macroscopic tensile tests.


The experimental grain map is used as input to produce a high fidelity digital clone of the specimen. The entire *L* = 1.57 mm zone, where 3 DCT scans were merged, was used to ensure the boundary condition application is far enough from the grains of interest to avoid any boundary layer effect [37]. Details on how the mesh was produced can be found in [17]. The initial grain boundary surface generated contains a very large number of triangles, and an iterative decimation approach using an edge collapsing algorithm is applied. The surface mesh is filled with tetrahedra controlling the mesh density as a function of the euclidean distance *d* from the three-grain cluster. This allowed minimizing the computational cost while preserving a rich description of the mechanical fields in the region of interest. The final mesh is composed of 341,687 linear tetrahedra with a gradient in element size (the ratio between the maximum and minimum tetrahedron size is about 4000) visible in Figure 4.

**Figure 4.** Comparison between the DCT data (**a**) and the mesh generated (**b**); the colors denote the grain numbers, which are consistent from the experiment to the simulation. Note the specimen shape with a radius in the gauge length as mentioned in Section 2.

Dirichlet boundary conditions (*uz* = 0 on the lower face and *uz* = 15.7 microns on the upper face for the final deformation step) were imposed to deform the specimen in tension up to 1% total strain in 100 steps. Suitable boundary conditions have been set on the lower surface of the sample to prevent any rigid body motion, and lateral surfaces were free of stress. The steps corresponding to the experimentally-measured strain (for instance 0.32% total strain) can be used for comparison.

#### *2.4. Lattice Rotations*

The continuum mechanical approach used here makes it possible to distinguish between the transformation of material and lattice directions. Material lines are made of material points that are subjected to the motion field *u*. In contrast, lattice directions are not material insofar as they are not necessarily made of the same material points (atoms) in the initial and current configurations due to the passing of dislocations, but keep the same crystallographic meaning. According to the concept of isoclinic configuration, lattice directions are unchanged from the initial to the intermediate configuration. Dislocations passing through a material volume element do not distort nor rotate the lattice, although material lines are sheared. According to the continuum theory of dislocations, statistically-stored dislocations accumulating in the material volume element affect material hardening, but do not change the element shape. Accordingly, an initial lattice direction *d* - is transformed into *d* by means of the elastic deformation:

$$
\underline{\underline{d}} = \underline{\underline{\underline{E}}} \,\, \underline{\underline{d}}\,\, \,\tag{9}
$$

This important distinction allows one to precisely compute both the local rotation and distortion of the crystal lattice, which will be further used to derive the 3D rocking curve of a grain from its deformed state in the simulation (see Section 2.5).

#### *2.5. Rocking Curves Simulations from CPFEM Data*

As explained in Section 2.1, the rocking curve represents the intensity diffracted by the illuminated grain at a given Θ angle. As soon as the crystal deforms, the exact Bragg condition is violated, and the *I*(Θ) curve will widen. In crystal plasticity, geometrically necessary dislocations (GND) give rise to gradients of crystal orientation, leading to local modification of the Bragg condition. The problem is therefore to solve the 3D diffraction condition stated in Equation (2) for the angle Θ, with a locally deformed crystal lattice. In Equation (2), the values of (*p*, *q*,*<sup>r</sup>*), as well as *<sup>θ</sup>Bragg* need to be updated for the new lattice geometry. It is therefore possible to use the mechanical fields computed in each element (namely **E**∼ and **R**∼ *e*) to evaluate locally the Bragg condition (for a given *ω* value) and to find the corresponding Θ. Building the volume weighted histogram for all elements within the grain will produce a simulated rocking curve (for this value of *ω*).
