Validating texture and lattice strain evolution models via in-situ neutron diffraction and shear tests

Validating micromechanical models under different loadings is challenging due to limited availability of experimental setups that can probe complex stress states. In this work, we perform in situ neutron diffraction experiments during shear loading to study the evolution of crystallographic texture and lattice strains that develop in different grain families. The experiments are conducted on 316 stainless steel samples, having a flat shear sample geometry suitable for testing in uniaxial rigs. The complex stress state in the gauge volume is estimated using an elasto-plastic finite element (FE) simulation, which is validated via experimentally measured gauge strains. The FE simulation reveals the generation of non-negligible in-plane normal stress components together with the in-plane shear stress component. The predicted macroscopic stresses are used as boundary conditions to drive an elasto-viscoplastic fast Fourier transform (EVP-FFT) based crystal plasticity model, which predicts lattice strain evolution. In addition, the evolution of the diffraction intensity is simulated using the Taylor model. The predictions of both models match very well with the experimental results. The experiments reveal that the {220} and {311} grain families have higher lattice strains than the {200} family. This surprising result is explained with the help of simulations.


Introduction
Developing and validating accurate models and constitutive equations that are able to predict the macroscopic and microscopic mechanical behavior of metals and alloys under complex loading conditions requires a synergistic combination of multiscale material modeling and experimental characterization. One such synergetic combination can be between multiscale material models to thoroughly understand the micromechanical response and either microscopy techniques to follow slip activity and strain localization [1][2][3][4], or neutron and/or synchrotron X-ray diffraction techniques to follow the collective response of families of crystallographic planes [5][6][7][8][9][10][11]. Between microscopy and diffraction studies, the latter have the advantage of providing spatially-averaged information over relatively large volumes that are representative of the studied microstructure. However, spatially resolved information within the averaging volume such as intergranular interactions, strain gradients or strain concentrations is lost. Multiscale materials models that complement such experiments entail capturing the complex stress states arising from the geometry of the sample using finite element (FE) simulations and using the predicted stress components as macroscopic boundary conditions to drive mesoscale crystal plasticity models [8,9,12,13] that predict the micromechanical behavior. For texture prediction [9,[14][15][16], crystal plasticity models can be used, however, simpler alternatives, such as the Taylor model of polycrystal plasticity [17], can be similarly effective [16].
In situ (neutron or synchrotron X-ray) diffraction methods are wellestablished techniques to study internal stress, load sharing in multiphase materials, mechanical anisotropy and the microstructure evolution during the course of a deformation test in engineering materials. Compared to X-ray diffraction, the high penetration depth of neutrons allows probing the lattice structure within the bulk of many crystalline aggregates and averaging a statistically relevant number of grains when the grain size of the material is considered small with respect to typical interaction volumes of X-ray diffraction investigations [18]. Furthermore, when a time-of-flight neutron instrument is used, it allows simultaneous in situ access to several different {hkl} grain families having one of their {hkl} lattice planes perpendicular to the scattering vector, q → [19].
Deformation under shear and shear strength determination are very relevant to service conditions of engineering parts and manufacturing processes. Although torsion tests are well-established methods for studying the shear response of materials, they are not ideal for diffraction studies due to the strong variation in shear strain along the radial direction [20][21][22][23]. Alternatively, flat shear sample geometries have been developed to obtain uniform shear within the gauge volume. The main driving factor for their development was the need to find simple geometries that are compatible with the more commonly used uniaxial tensile test rigs [24][25][26][27]. As shown in this work, these geometries are also ideal for investigating samples with relatively large grain sizes using diffraction methods, where a large volume of the material needs to be probed, in order to achieve satisfactory grain diffraction statistics. In the present work, a flat shear sample is mechanically tested in situ during neutron diffraction. Lattice strain and texture evolution are obtained along two orthogonal directions and they are used to validate two different models. For simulating the evolution of the crystallographic texture, the Taylor model implemented in the MTEX tool box [28] is used. For simulating the lattice strain evolution, a multiscale modeling approach is used. This approach involves first simulating the complex macroscopic stress state in the sample using the FE method. Next, these macroscopic stresses as a function of time are used to drive an elasto-viscoplastic fast Fourier transform (EVP-FFT) crystal plasticity model [29,30], which is used to simulate the lattice strain evolution.

Materials and methods
Shear probes based on the modified ASTM geometry proposed in [24] were machined from an AISI316 cold-rolled and annealed sheet; the geometry is shown in Fig. 1a. The biaxial machines developed at the Paul Scherrer Institute have been extensively used by the authors and provided useful experimental data for validating and at times further developing existing multiscale models [8,9,12,13] and for obtaining new insights on micromechanisms such as martensitic phase transformations and twinning [31][32][33][34][35] under in-plane biaxial loadings and strain path changes. For the in situ uniaxial deformation and neutron diffraction experiments, the shear probes were deformed with 3 μm/s displacement rate using the biaxial rig of POLDI, SINQ, PSI [36][37][38]. The neutron diffraction measurements were performed by stopping and holding the displacement at pre-defined force values followed by a waiting time of 180 s to allow stress stabilization, which results in stress relaxation. The in situ neutron diffraction measurements were performed in the central region of the sample using a beam size of 6 × 3.1 mm 2 (width × height) for the horizontal setup (see Fig. 1b) and of 3.6 × 8 mm 2 (width × height) for the vertical setup (see Fig. 1c). Two tests with different sample orientations were conducted. During one test, a sample was placed horizontally in the test rig such that q → was parallel to the displacement direction x (hereafter denoted as longitudinal direction -LD) as shown in Fig. 1b. During the other test, the sample was placed vertically in the test rig such that q → was perpendicular to the displacement direction (hereafter denoted as transverse direction -TD) as shown in Fig. 1c. The neutron data were analyzed and fitted using the Mantid software [39] to obtain the peak position and integrated intensity. The average position of each hkl peak determines the average interplanar spacing of the lattice planes of an {hkl} grain family, whose normal is parallel to q → . Then, the average lattice strain, ε hkl , is calculated as: where d 0 hkl and d hkl is the d-spacing of the {hkl} grain family before and during deformation, respectively.
The applied macroscopic strain was measured by a 3-dimensional (3D) digital image correlation (DIC) system (GOM Aramis®), which is a full-field optical method that provides high-resolution 3D displacements on a 20 × 20 mm 2 surface of a sample. The DIC setup consists of two 5 MP CMOS cameras mounted at an angle of 15 • to the surface normal. A speckle pattern is created on the sample surface by spraying fine droplets of white and black paint; the method and an example speckle pattern are presented in our previous work [40]. The Aramis analysis software correlates each pair of images with an initial set of images and generates 3D displacement maps. The macroscopic strain tensor is then evaluated and the calculated in-plane normal and shear strain components are averaged over the rectangle of the beam footprint on the sample at the center of the shear probe in both measurement setups shown in Fig. 1b and Fig. 1c.
EBSD investigations were undertaken on the as-received sheet and after the deformation test of the LD sample. Electrical discharge This region was used to determine the average strains and stresses for both experiments and simulations.
E. Polatidis et al. machining was used to extract a microscopy sample from the deformed sample at the location where neutron diffraction was performed. Stress relaxation due to cutting, if any, does not alter the texture. This sample was ground with 1200 grit SiC paper and then electropolished for 10 s with a 16:3:1 (by volume) ethanol, glycerol and perchloric acid solution at 48 V. A field emission gun scanning electron microscope (FEG SEM) Zeiss ULTRA 55 equipped with EDAX Hikari Camera operated at 20 kV in high current mode with 120 µm aperture was used. The pole figures were produced from two 0.5 × 0.4 mm 2 EBSD maps that were stitched together to improve grain statistics and reliability of texture analysis. The EBSD raw data was post-processed using either the EDAX OIM Analysis 8.1 or the open-source Matlab toolbox MTEX [28]. To evaluate the crystallographic texture of the as-received sheet, 7 EBSD maps covering an area of 3.5 × 0.4 mm 2 were stitched together to improve the grain statistics.

Simulations
The initial crystallographic texture of the material is nearly random, as shown by the EBSD map and the inverse pole figure (IPF) maps in Fig. 2; which is coherent with our previous investigations on the same material [8,40]. The evolution of the deformation texture was modelled using the Taylor model implementation in MTEX [28], which is known to well capture the evolution of deformation texture under uniaxial tension [41,42]. The Taylor model [17,43] assumes that under an applied loading (strain or stress), all grains experience the same plastic strain. Furthermore, within a grain, a combination of five independent slip systems, having minimum accumulated slip, is required to accommodate the five independent strain components of plastic deformation. The MTEX code calculates the orientation-dependent Taylor factor, the slip rate tensor and the anti-symmetric part of the plastic deformation tensor, which is used to update the crystallographic orientation of a grain.
The random texture of the sample was simulated using 100000 random orientations representing 100000 grains. While the MTEX code allows simulating EBSD datasets, we have not adopted this approach for the following reason. The grain size is ~24 μm, which entails that an EBSD map of an area of 3.5 × 0.4 mm 2 only includes a few thousand grains. On the other hand, the neutron gauge volume includes a few million grains. Therefore, considering 100000 randomly oriented grains gives a good approximation of the texture shown in Fig. 2. It also provides sufficient grain statistics to compare the results with the neutron measurements in a computationally efficient manner.
We imposed a pure shear loading with shear strain increments of 0.003 in the following manner:   However, it was observed that when pure shear is applied, the final simulated pole figures have a rigid body rotation of 0.3 rad with respect to the experimentally measured ones. This difference arises from the fact that during the experiment, the gauge region is subjected to simple shear. In order to account for the rigid body rotation under simple shear [44] in the simulation, after each pure shear loading step, a rigid body rotation equal to 0.003 rad was imposed to compensate for the aforementioned difference. The orientation of each grain is updated at the end of each iteration; thus, the orientation matrix of the aggregate is determined as a function of each deformation step.
For modeling the lattice strain evolution, we follow a multi-scale experiment-modeling synergistic procedure used in some of our previous works [8,12]. This procedure first involves estimating the macroscopic stress state in the gauge region of the sample, which is done with the aid of FE simulations. To that end, the ABAQUS/Standard (time implicit) software is used. The FE simulations are performed on 1/2 of the sample by applying appropriate symmetric boundary conditions. A structured hexahedron mesh is employed with linear 8-node mesh elements (C3D8 in ABAQUS). The FE simulations are performed under force control, and following [8,12], the simulations are validated by comparing the predicted Von Mises strains with those obtained from DIC. Following this, the macroscopic gauge stresses are used as boundary conditions to drive a full-field EVP-FFT model for the prediction of micromechanical fields in polycrystals; more details on this model can be found in [8,29]. Since the material is the same one that was used in [8], the material and other model parameters used in the present study are the same as those in [8]. The evolution of the average shear stress within the gauge volume as a function of the von Mises strain (calculated from the nominal strain) is shown in Fig. 3a. The FE model predicts  Fig. 5. Evolution of integrated diffraction intensity for various {hkl} families (data points) and simulated ratio of an orientation that is close to an orientation <hkl>, by a misorientation tolerance of 7.5 degrees to the volume of the entire ODF (dashed lines), when measuring in two setups as shown in Fig. 1: (a) along x (i.e., displacement parallel to q → ) and (b) along y (i.e., displacement perpendicular to q → ).
non-negligible normal stress components in the in-plane directions, which are a consequence of the sample geometry (see Fig. 3b); all the out-of-plane stress components are zero. Therefore, the stress state in the gauge volume is not a simple shear state, but rather a complex in-plane stress state.

Results and discussion
The Taylor model-predicted texture evolution yields an excellent agreement with the measured texture from EBSD as shown in the pole figures in Fig. 4. To further test the predictive capability of this model, we compared the evolution of the normalized neutron diffraction intensity to the one predicted from the simulations. From the experiments, the normalized intensity is calculated as I hkl /I 0,hkl , where I 0,hkl and I hkl denote the integrated intensity of the hkl peak before and during deformation, respectively. From the simulation, I hkl /I 0,hkl is calculated from the evolving orientation distribution function (ODF) at each simulated strain step. Here I 0,hkl (before deformation) and I hkl (during deformation) is the ODF intensity that is close to an orientation (center), hkl, by a tolerance (7.5  the displacement direction is perpendicular to q → i.e., along y direction, as shown schematically in Fig. 1b and Fig. 1c, respectively. It is seen that the simulation and experiment are in excellent agreement with each other; the minor differences are due to the fact that the simulation mimics simple shear but the experiment has a slightly more complex stress state. Therefore, one can conclude that a simple shear simulation using the Taylor model can very well model the evolving crystallographic texture during this test. The FE simulations reveal that the sample geometry results in nonnegligible in-plane normal stress components [25] in comparison to the in-plane shear stress. The normal components do not affect the plastic deformation of the material significantly, and hence the deformation texture evolution (as evidenced in Figs. 4 and 5), but they affect the evolution of lattice strain. Fig. 6 shows the experimental and EVP-FFT simulated lattice strain evolution of four {hkl} families of the fcc crystal structure in the direction parallel to the applied force, x, as a function of von Mises stress.
In general, the lattice strain evolution trends of all grain families are well captured by the simulations. The simulation captures particularly well the behavior of the {220} and {311} families. However, the stiffness of the {111} family and the compliance of the {200} family are underestimated at von Mises stress higher than 300 MPa. Interestingly, the {220} and {311} grains exhibit high lattice strain build up. The grains which contribute to a given {hkl} reflection have one of their <hkl> directions parallel to the scattering vector (i.e., along the x axis), but they have a variety of orientations with respect to the second axis, y. As such, the grains that have a specific {hkl} plane perpendicular to the scattering vector can be divided into grain subfamilies according to their orientation in the transverse direction (i.e., along the y axis). The lattice strain evolution of the stiffest and most compliant of these subfamilies is depicted by the shaded area which surrounds the average lattice strain of each grain family in Fig. 6. Amongst all the grain families, the {111} grains have the least spread between their subfamilies, while the highest spread is observed amongst the {311} grain subfamilies. This prediction can be better understood by studying the normalized resolved shear stress histograms (Fig. 7) of the four slip systems carrying the highest resolved shear stress per grain family; the normalization was performed by dividing the resolved shear stress with the von Mises stress and the calculations were performed in MTEX assuming 10000 grains. It is observed that the mean value of the resolved shear stress amongst the {111} grain subfamilies is the highest and the spread is relatively low. More specifically, the calculation suggests that the {111} grain family deforms plastically earlier than the other grain families, which is made apparent by the calculated highest normalized resolved shear stress, and sheds its load due to the activation of slip systems with the highest Schmid factor earlier than the other grain families. Consequently, the spread in the simulated lattice strain is relatively small (Fig. 7-a). In contrast, the {200} grain subfamilies exhibit relatively low normalized resolved shear stresses with a narrower spread than those in the {111} grain subfamilies. This result implies that once the critical resolved shear stress is reached, more than 1 and up to 4 slip systems are activated in the {200} grain subfamilies. Finally, the {220} and {311} grains have one slip system with relatively high normalized resolved shear stress, but still with a relatively wide spread amongst the grain subfamilies. The secondary, tertiary and quaternary slip systems exhibit significantly low mean value and a wide spread in the normalized resolved shear stress. Thus, plasticity is delayed in these grains and they take up the load shed by the {111} grain family and later on by the {200} grain family.
In order to compare the lattice strain evolution obtained from the  complex stress state shown in Fig. 3b and a pure shear stress state, another EVP-FFT simulation was performed by applying a pure shear stress state in such way that the macroscopic von Mises stress was the same for both simulations. Fig. 8

Conclusions
A new experimental approach extending the capabilities of in situ shear tests with diffraction measurements is presented that capitalizes on the usage of simple uniaxial tensile test rigs. A flat shear sample geometry was deformed in situ during neutron diffraction and the evolution of the integrated intensity and lattice strain from different hkl families was followed. Two types of simulations were undertaken to compare with the experimental results: (i) the Taylor model implemented in MTEX was used to simulate texture (neutron diffraction intensity) evolution and EBSD at the end of test, and (ii) an FE-FFT multiscale elasto-plastic modeling approach was used to predict the lattice strain evolution. Macroscale FE simulations suggest that the normal stress components in the in-plane directions are non-negligible in comparison to the in-plane shear stress. While they affect the lattice strain evolution, they do not have any appreciable effect on the plastic deformation and texture evolution. Assuming a simple shear stress state i.e., neglecting the normal stress components, the Taylor model is able to capture the texture evolution very well. The combined FE-FFT simulations capture well the lattice strain evolution of different hkl grain families, which are seen to be strongly affected by the presence of the inplane normal stress components. The load sharing and the spread of lattice strain between grain families is explained by the differences in the normalized resolved shear stress of the different grain families. The results show that grains in the {111} and {200} families plastically deform earlier than the grains of the {220} and {311} families. Consequently, the {111} and {200} families shed their load to the {220} and {311} families, which exhibit higher lattice strains in the macroscopic plastic regime.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.