# Investigating grain-resolved evolution of lattice strains during plasticity and creep using 3DXRD and crystal plasticity modelling

Farhan Ashraf, Ranggi S. Ramadhan, Abdullah Al Mamun, James A.D. Ball, Eralp Demir, Thomas Connolley, David M. Collins, Mahmoud Mostafavi, David Knowles

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Farhan Ashraf, Ranggi S. Ramadhan, Abdullah Al Mamun, James A.D. Ball, Eralp Demir, et al. Investigating grain-resolved evolution of lattice strains during plasticity and creep using 3DXRD and crystal plasticity modelling. Acta Materialia, 2024, 278, pp.120250. 10.1016/j.actamat.2024.120250. hal-05095792

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# Investigating grain-resolved evolution of lattice strains during plasticity and creep using 3DXRD and crystal plasticity modelling

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Farhan Ashraf<sup>a, *</sup>, Ranggi S. Ramadhan<sup>a</sup>, Abdullah Al Mamun<sup>a, b</sup>, James A.D. Ball<sup>c</sup>, Eralp Demir<sup>e</sup>, Thomas Connolley<sup>f</sup>, David M. Collins<sup>d</sup>, Mahmoud Mostafavi<sup>a</sup>, David Knowles<sup>a, g</sup>

## ARTICLE INFO

**Keywords:**

3D-X-Ray Diffraction

Crystal Plasticity Modelling

Creep

Grain Families

Local Elastic Strain

## ABSTRACT

Microstructure-informed crystal plasticity finite element models have shown great promise in predicting plastic and creep deformation in polycrystalline materials. These models can provide substantial insight into the design, fabrication and lifetime assessment of critical metallic components during operation, for instance, in thermal power plants. However, to correctly incorporate damage prediction into models, the microstructural strain simulated at the grain level must be accurately predicted with suitable validation. For this reason, a 3D X-ray Diffraction (3DXRD) experiment was carried out on 316H stainless steel, a material commonly used in thermal power plants, to obtain the per-grain strain response during plastic and creep deformation at 550°C. Several hundred grains within a probed X-ray volume were tracked and measured whilst loading in-situ, obtaining per-grain centre-of-mass positions, crystallographic orientations, and average lattice strain over individual grains. These data were used to calibrate a crystal plasticity model to study the plastic and creep deformation using macroscopic stress-strain and stress-relaxation data. Subsequently, the model was used to predict the average elastic strain in different grains during the cyclic creep experiment, which was validated by 3DXRD datasets. The model results reveal that {100} or {311} grain families are strongly sensitive to microstructure, thereby a polycrystal model that describes specific orientation and neighbourhood characteristics is essential to predict the local response of these grain families. Whereas, self-consistent models are suitable for {110} and {111} grain families. This study shows that only with a suitable calibration of subsurface grain behaviour, crystal plasticity models reveal grain characteristic-dependent micromechanical behaviour.

## 1. Introduction

Metallic structural components used in nuclear power plants operate in challenging environments including prolonged periods at high temperatures in conjunction with complex thermo-mechanical cycling. These cyclic-creep loads cause damage at a microstructural level, which often determines the life-limiting factors when used in applications. This is pertinent for the UK's ageing advanced gas-cooled reactors (AGRs) and advanced modular reactors (AMRs) to be used in the future. Continuous efforts are being made to improve the damage prediction accuracy applied to alloys used in demanding nuclear power

applications by integrating advanced material models into the design, fabrication and lifetime assessment. This approach represents a step change over historical strategies that were heavily reliant on large-scale macro-mechanical experimental and continuum modelling [1].

To simulate the deformation behaviour in a polycrystalline material at a grain length scale, advanced micro-mechanical modelling approaches may be adopted. Crystal Plasticity Finite Element (CPFE) is attractive for this purpose, predicting the aggregate response of individual grains deforming, governed by constitutive descriptions of crystallographic slip [2,3]. To provide new grain-scale insight into deformation whilst validating CPFE-based micro-mechanical models,

<footnote><sup>a</sup> Department of Mechanical Engineering, University of Bristol, Bristol, BS8 1TR, UK</footnote>

<footnote><sup>b</sup> Nuclear Futures Institute, Bangor University, Gwynedd, LL57 2DG, UK</footnote>

<footnote><sup>c</sup> European Synchrotron Radiation Facility (ESRF), 71 Avenue des Martyrs, 38000 Grenoble, France</footnote>

<footnote><sup>d</sup> Department of Materials Science & Metallurgy, 27 Charles Babbage Road, Cambridge, CB3 0FS, UK</footnote>

<footnote><sup>e</sup> Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK</footnote>

<footnote><sup>f</sup> Diamond Light Source, Fermi Ave, Didcot, OX11 ODE, UK</footnote>

<footnote><sup>g</sup> Henry Royce Institute, Oxford Road, Manchester, M13 9PL, UK</footnote>

<footnote>* Corresponding author.
E-mail address: f.ashraf@bristol.ac.uk (F. Ashraf).</footnote>
<table><caption>**Table 1**

Chemical composition of the 316H stainless steel used in the experiment.</caption><thead><tr><th>Element</th><th>C</th><th>S</th><th>N</th><th>Co</th><th>Cr</th><th>Cu</th></tr></thead><tbody><tr><td>wt.%</td><td>0.068</td><td>0.006</td><td>0.049</td><td>0.22</td><td>17.4</td><td>0.41</td></tr><tr><td>Element</td><td>Fe</td><td>Mn</td><td>Mo</td><td>Ni</td><td>P</td><td>Si</td></tr><tr><td>wt.%</td><td>67.25</td><td>1.38</td><td>2.08</td><td>10.16</td><td>0.04</td><td>0.45</td></tr></tbody></table>

experimental approaches such as neutron diffraction or 2D Synchrotron X-ray diffraction have proved valuable [4–6]. Such experiments are, however, typically limited to describing the average response of grains with a common crystallographic orientation (grain family), whereas the deformation formulation employed in a CPFE model is calculated at the individual grain level. The elastic and plastic deformation of an individual grain is not only dependent on its own orientation but also on the orientation and morphology of its neighbouring grains. Grains having the same idealised crystallographic orientation but embedded in different surroundings will respond differently when stresses are applied [7] because the deformation behaviour of an individual grain must adapt to the state of its neighbours. Measures of the average grain family response cannot capture this grain-to-grain variation of deformation within a grain family; a per-grain understanding is crucial to improve current micromechanical models. A suitable strategy for model validation can only be obtained by utilising experimental deformation data that includes individual grain measurements within a polycrystal. Several efforts [8–10] have studied local deformation using surface-based imaging techniques such as high-resolution Digital Image Correlation HR-DIC and high-resolution Electron Back Scattered Diffraction HR-EBSD. However, such measurements may contain surface preparation-related artefacts, and observe features arising from a plane stress condition (near the free surface). This must differ subsurface, within the bulk of material.

Three-Dimensional X-ray Diffraction (3DXRD) is an expansion from a conventional Debye-Scherrer/ 2D X-ray diffraction. The experimental setup is similar to its 2D counterpart where, in transmission mode, a diffraction pattern is captured on a 2D detector. For 3DXRD, a limited number of grains (fewer than 1000) are illuminated, producing diffraction spots (instead of continuous rings) corresponding to each grain. The sample is then rotated about its vertical axis whilst acquiring diffraction images. The resultant data are indexed by associating diffraction spots together, then assigning them to individual grains [11]. Using reconstruction algorithms, information such as crystal orientation, position, volume, and elastic strain tensor can be calculated for each grain [12–16], making this method suitable for validating CPFE models. Several studies [7,17,18] have found a good agreement between CPFE calculations and 3DXRD measurements for average stresses within each grain, though creep stress relaxation at high temperatures has not been measured by this technique before. Using 3DXRD experiments on 316L austenitic stainless steel, Juul et al. [17] showed significant heterogeneity in the stress state of individual grains even for those with a similar orientation. Their CPFE simulations qualitatively reproduced the experimental stress states, however, an individual grain-level agreement could not be obtained [17].

Measuring the local deformation and predicting the accurate response using crystal plasticity models at a grain-scale is crucial and is still unresolved. In this study, an in-situ far-field 3DXRD [19] creep experiment was performed at an elevated temperature to validate a CPFE simulation developed for modelling grain-average strain. This was performed not only during time-independent deformation but also during stress relaxation at high temperatures. The 3DXRD measurements including grain orientation were used as an input to a single-crystal and a full-scale polycrystalline crystal plasticity model. The model was calibrated to predict the plastic and creep deformation using macroscopic stress-strain data and stress relaxation data for orientation-specific grain families. Subsequently, these parameters were used to predict the average elastic strain response from individual grains

from different grain families and were compared to measured values for simulation validation.

## 2. Experiments

The material investigated in this study was 316H austenitic stainless-steel processed via electric arc melting and vacuum degassing, followed by forging and solution annealing. Its chemical composition conformed to the ASTM A478/479M-17 [20] specification, given in Table 1. The steel matrix comprised an austenite FCC phase with a weak crystallographic texture. From Electron Back Scatter Diffraction (EBSD) measurements shown in Fig. 1, the material had an average grain size of 114 µm with evidence of annealing twins. The EBSD analysis was conducted on a mechanically polished sample using a field emission gun scanning electron microscope (FEG-SEM) equipped with a Symmetry-S detector. The data was collected at 20 kV using a step size of 1 µm. Matchstick specimens with gauge dimensions of 6.0 × 0.7 × 0.7 mm³ were extracted from the bar; the geometry is shown in Fig. 2(a). The full dimensions of the sample are shown in Fig. 2(a).

The far-field 3DXRD experiment was carried out at the I12: Joint Engineering, Environmental, and Processing (JEEP) beamline at the Diamond Light Source, UK [21]. An INSTRON 3 kN Electro-Thermo-Mechanical Tester (ETMT) was used to simultaneously heat and deform the specimen whilst diffraction data were collected. The specimen was mounted into the ETMT via a custom-built screw-grip set-up, as shown in Fig. 2(b). An R-type thermocouple was spot-welded to the centre of the specimen gauge to both control and monitor the specimen temperature. A Linear Variable Differential Transformer (LVDT) was used to measure the macroscopic strain of the specimen. The specimen was heated up via an electric-resistant heating system at a preload of 5 MPa to a target temperature of 550°C and then soaked for 30 minutes to achieve thermal equilibrium before acquiring the first 3DXRD scan.

The data collection strategy implemented in this experiment is described in [14], which utilises the experiment configuration schematically in Fig. 3(a). During the experiment, the ETMT was placed onto a rotational sample stage at the I12 beamline and was precisely aligned so that the specimen gauge centre was coaxial with the centre of the stage rotation axis (ω-rotation). An incident synchrotron X-ray beam with a cross-section of 0.5 × 1.1 mm² and a monochromatic energy of 81 keV, calibrated with a CeO₂ standard, illuminated the centre of the specimen gauge. The sample was placed ~1 m away from the Dectris Pilatus 2M CdTe 2D area detector, permitting 4 complete diffraction rings to be acquired. From the sample thickness of 0.7 mm, the measured volume was ~0.25 mm³ which illuminated approximately 500 grains. A single 3DXRD scan comprises a sample rotation about the vertical axis (ω-rotation) at increments of 0.1° while taking diffraction patterns at each step with an exposure time of 1 to 2 s. For this study case, the total maximum rotation angle was 46°, with the range determined by the orientation range through which the beam path was not obstructed by the components of the ETMT. The scanning strategy adopted here is hereon referred to as limited angle 3DXRD. The duration of each 3DXRD scan was approximately 15 minutes.

Before the first scan, the sample was pre-loaded to 5 MPa; measurements at this point were used to define the reference lattice parameter. The subsequent scans were then performed at different points in the loading cycle, which were interrupted under load control. Only the stress relaxation dwells were carried out under the strain-control by switching the mode at a dwell stress of around 160 and 205 MPa for ~30 minutes as indicated in Fig. 3(b). A total of 13 3DXRD scans were performed. Fig. 3(b) shows the loading cycle and the position where 3DXRD scans were performed. The amber circles here indicate scans in which grains were successfully reconstructed scans and the red squares (scan 8th-10th) were those where grain reconstruction was unsuccessful.

Data analysis carried out in this work was made using the software
<figure><img src="image_4.png" /><figcaption>Fig. 1. An inverse pole Fig. grain map, with respect to the z-direction, of a 316H specimen in the reference condition measured using Electron Backscattered Diffraction (EBSD).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2. (a) Geometry of the specimen; (b) Specimen shown in the in-situ testing configuration on the I12 beamline, here mounted on the ETMT machine.</figcaption></figure>

package ImageD11 [22]. Diffraction peaks were extracted from raw detector images that were then filtered and corrected for detector distortion as well as errors arising from module gaps. Using ImageD11, the X-ray scattering vectors were calculated and indexed, assigning grains, followed by refinement of their centre-of mass positions, orientations, and elastic Biot strain tensors. From the latter, the measured unit cell lengths were obtained. It is noted that the limited angle 3DXRD approach restricts the measurements to only a subset of grains from each sample location. Full details of the processing pipeline and the data analysis method are described in more detail elsewhere [23]. For data analysis, the strain tensor for each grain in the crystal reference frame was transformed into the orientation of the sample reference frame. The strain state of each grain at each load step was then normalized with respect to the reference lattice strain, i.e., at a pre-load of 5 MPa at 550°C, according to Eq. (1), where $\epsilon_i$ is the normalised lattice strain, $\epsilon$ is the lattice strain of the corresponding load step, and $\epsilon_0$ is the reference lattice strain.

$$ \epsilon_i = \frac{\epsilon - \epsilon_0}{\epsilon_0} \qquad (1) $$

To compare trends from the measured data to simulated predictions, the detected grains were grouped into grain families based on orientation. Here, each grain family comprise normal vectors of their {hkl} planes within a specified tolerance angle to the scattering vector (in this case the loading direction (z-axis)). For a sufficient number of grains within each grain family to calculate a representative measure of the average lattice strain, the specified tolerance was set to ±15° from vectors normal to the planes: {100}, {110} and {111}, and ±9° for {311}. Eq. (2) calculates the weighted average lattice strain for each

grain family $\epsilon_{hkl}$, where $\epsilon_i$ and $V_i$ are the lattice strain and volume of the grain (from a corresponding hkl grain family), and $N_{hkl}$ is the number of grains within the family.

$$ \epsilon_{hkl} = \frac{\sum_{i=1}^{N_{hkl}} \epsilon_i V_i}{\sum_{i=1}^{N_{hkl}} V_i} \qquad (2) $$

### 3. Crystal Plasticity Model

#### 3.1. Constitutive law

Full details of the crystal plasticity solver: BRISTOL employed in this work can be found elsewhere [5,24]. For the single-phase austenitic steel studied, slip systems for FCC materials were used in the simulation. The inelastic shearing rate of each slip system was computed using a power law-based crystal plasticity flow rule following the Asaro and Needleman work [25].

$$ \dot{\gamma}^{\alpha} = \left[ \dot{\gamma}_{0.1} \left( \frac{|\tau^{\alpha} - X^{\alpha}|}{\tau_{c}^{\alpha}} \right)^{1/m_{1}} \right] \operatorname{sgn}(\tau^{\alpha} - X^{\alpha}) \qquad (3) $$

Here, $\dot{\gamma}_{0.1}$ is the reference strain rate and $m_1$ is the strain rate sensitivity factor for the plasticity. The $\tau^\alpha$, $\tau_c^\alpha$ and $X^\alpha$ are resolved shear stress, critical resolved shear stress and backstress, respectively, on a particular slip system $\alpha$.

Creep caused by the diffusion-controlled dislocation climb is a two-stage process [26]. At higher temperatures, an immobile edge dislocation in a slip plane can overcome obstacle resistance in that plane by climbing. By doing so, a dislocation can now glide onto a new slip plane. However, the time required for a dislocation to glide between obstacles is significantly less than the time required to climb to a new plane. Hence, creep occurs at very low strain rates that differentiate it from the plastic deformation during monotonic or cyclic loading. To account for two different deformation processes i.e., plasticity and creep, a superposition of rate-insensitive (associated with dislocation glide and cross slip) and rate-sensitive (associated with glide and climb) power laws were used. The combination of two power laws representing the deformation in plasticity and creep regimes is a phenomenological approach postulated by several researchers [27–30].

$$ \dot{\gamma}^{\alpha} = \left[ \dot{\gamma}_{0.1} \left( \frac{|\tau^{\alpha} - X^{\alpha}|}{\tau_{c}^{\alpha}} \right)^{1/m_{1}} + \dot{\gamma}_{0.2} \left( \frac{|\tau^{\alpha} - X^{\alpha}|}{\tau_{c}^{\alpha}} \right)^{1/m_{2}} \right] \operatorname{sgn}(\tau^{\alpha} - X^{\alpha}) \qquad (4) $$

In the second term, $\dot{\gamma}_{0.2}$ and $m_2$ are the reference strain rate and strain rate sensitivity for the creep regime. The critical resolved shear stress evolves with hardening, according to the expression [31,32]:
<figure><img src="image_6.png" /><figcaption>Fig. 3. (a) Geometry of the 3DXRD experiment; (b) Loading cycle applied to the specimen during the experiment. Markers show where 3DXRD scans were performed: amber circles indicate successful grain reconstruction, and red boxes indicate otherwise.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 4. a) Full-scale model: RVE created from Dream3D with 148 grains. The IPF shows the orientation assigned to grains in the RVE. b) IPF plot with respect to loading direction, of grains calculated from 3DXRD. The inset IPF show all measured grains. Different colours indicate grain families: red, green, blue, and magenta for {100}, {110}, {111} and {311} families, respectively. The loading direction in IPF is normal to the plane.</figcaption></figure>

$$ \dot{\tau}_c^a = h_0 \sum \left( 1 + \frac{h_0 \Gamma}{n \tau_{c0}} \right)^{n-1} |\dot{\gamma}^a| - A (\tau_c^a)^d \exp\left(-\frac{Q}{R T}\right) \quad (5) $$

where $h_0$ is the initial hardening modulus, $\tau_{c0}$ is the initial slip hardening, $n$, $A$, and $d$ are fitting parameters, $Q$ is the creep activation energy,

$R$ is the universal gas constant, and $T$ is the temperature. $\Gamma$ is the cumulative slip on all slip systems following Eq. (6). The second term in Eq. (5) corresponds to the temperature-dependent recovery process during creep as suggested by Evans et al. [33].
<figure><img src="image_8.png" /><figcaption>Fig. 5. Comparison of model and experiments for a) uniaxial stress-strain response [48] and b) cyclic stress-strain response of 316H at 550 °C [49].</figcaption></figure>

$$ \Gamma = \sum_{a} \int_{0}^{t} \dot{\gamma} dt \qquad (6) $$

Several studies [34,35] showed that the back stress significantly contributes to the macroscopic stress response of stainless steel. Crystal plasticity finite element models have an intrinsic capability to generate and incorporate the intergranular stress induced by grain-to-grain interactions. However, intragranular back stress that originates from dislocation interactions within grains is required to be included in the constitutive model to accurately predict the macroscopic stress response of polycrystals [36]. In this work, the backstress ($\dot{X}^a$) evolves according to the Armstrong-Fredrick formulation [37] as,

$$ \dot{X}^a = h \dot{\gamma}^a - h_D X^a |\dot{\gamma}^a| \qquad (7) $$

where $h$ is the hardening coefficient and $h_D$ is related to the dynamic recovery. The ratio of $(h/h_D)$ controls the saturation level of backstress in a given $\alpha$ slip system [38]. The Armstrong-Fredrick formulation, as shown in Eq. (7), is a phenomenological approach proposed by several researchers [39,40]. It is considered to provide a relatively robust solution unlike other methods that are either computationally complex or require iterative loops [41–43].

<figure><img src="image_9.png" /><figcaption>Fig. 6. Comparison of the CPFE model and experiment for stress relaxation of 316H steel, for the {100} grain family at 550°C.</figcaption></figure>

<table><caption>Table 2
Calibrated model parameters used for CPFE simulations.</caption><thead><tr><th>Constant</th><th>Description</th><th>Value</th></tr></thead><tbody><tr><td>c<sub>11</sub>, c<sub>12</sub>, c<sub>44</sub></td><td>Elastic modulus</td><td>183.9 GPa,<br>123.4 GPa,<br>91.5 GPa</td></tr><tr><td>h<sub>0</sub></td><td>Initial hardening modulus</td><td>200 MPa</td></tr><tr><td>τ<sub>c0</sub></td><td>Initial slip hardening</td><td>47 MPa</td></tr><tr><td>n</td><td>Rate of saturation parameter</td><td>0.90</td></tr><tr><td>γ̇<sub>0.1</sub>, γ̇<sub>0.2</sub></td><td>Slip rate evolution for plasticity & creep</td><td>0.1 s<sup>-1</sup>, 3 × 10<sup>-6</sup> s<sup>-1</sup></td></tr><tr><td>m<sub>1</sub>, m<sub>2</sub></td><td>Strain rate sensitivity factor</td><td>0.02, 0.2</td></tr><tr><td>Q</td><td>Creep activation energy</td><td>418 kJ mol<sup>-1</sup></td></tr><tr><td>R</td><td>Universal gas constant</td><td>8.31 J mol<sup>-1</sup> K<sup>-1</sup></td></tr><tr><td>T</td><td>Temperature</td><td>823 K</td></tr><tr><td>A, d</td><td>Fitting parameter</td><td>3 × 10<sup>16</sup>, 3.0</td></tr><tr><td>h, h<sub>D</sub></td><td>Hardening coefficients</td><td>3555 MPa, 245</td></tr></tbody></table>

## 3.2. Parameter Calibration

The calibrated modelling parameters used in a previous study (e.g., [5]) were related to 316H steel, which had thousands of hours of aging in service, hence are invalid for the present heat-treated material. Therefore, the model was recalibrated for the as-received condition of 316H steel to study the plastic and creep deformation at 550°C using macroscopic stress-strain data and creep relaxation data respectively. For this purpose, a crystal plasticity framework [5] was implemented along with a representative volume element (RVE) in Abaqus v.2018 [44]. Further details of the finite element implementation can be found in previous work [5]. The RVE was created using Dream3d [45] with a size of 1.2 × 0.9 × 0.6 mm³, 192000 C3D8 elements, and 148 grains as shown in Fig. 4. The grains in the RVE were assigned the same crystallographic orientations that were measured from the 3DXRD experiment. Fig. 4(b) shows the inverse pole Fig. (IPF) with respect to the loading direction of a selection of grains indexed from the 3DXRD analysis. The four different colours represent grain families, where grains in the approximate {100}, {110}, {111}, and {311} families are labelled with
<figure><img src="image_10.png" /><figcaption>Fig. 7. a) Single crystal-J2 model contains a single grain (green) surrounded by an isotropic medium (grey) and, b) an IPF showing selected grains from the extremes of the {311} grain family.</figcaption></figure>

red, green, blue, and magenta data points, respectively. The inset of Fig. 4(b) shows the IPF containing all grains detected in all scans, indicating random grain orientations from the measured subset.

Displacement-control loading was applied on the front face along the z-axis and the opposite face, along the z-axis, was constrained as a boundary condition to avoid a rigid body motion. The constitutive law (Eq.s 3, 4, and 6) contains 11 parameters, which can be further divided into three categories, (i) slip rate parameters ($\dot{\gamma}_{0,1}$, $m_1$), (ii) strain and kinematic hardening parameters ($h_0$, $\tau_{c0}$, $n$, $h$, $h_D$), (iii) parameters related to creep deformation ($\dot{\gamma}_{0,2}$, $m_2$, A, d). The slip rate parameter $m_1$ is sensitive to higher strain rates [46], therefore, it does not affect the stress-strain evolution in the present quasi-static analysis. Hence, the parameter $m_1$ was increased from 0.002 to 0.02 to speed up the simulation.

The parameters that required calibration to predict the macroscopic quasi-static stress-strain response were $h_0$, $\tau_{c0}$, $n$, and $h$. These strain and kinematic hardening parameters were calibrated by fitting the model with uniaxial cyclic stress-strain experimental data. The values of $\tau_{c0}$ and $h_0$ were modified to fit the macroscopic yield stress and stress-strain evolution beyond the yield point, respectively. The back stress hardening coefficient ($h$) was optimised to accurately predict the kinematic hardening evident in the Bauschinger factor evolution. Finally, $n$ was modified to fit the stress response at larger strains. It is noted that these parameters were optimised using bounds determined from previous studies [5,6,47]. Fig. 5(a) shows that the optimised parameters provide a good fit to the experimentally determined uniaxial stress-strain response [48]. The same calibration was employed to predict the cyclic stress-strain response of 316H at 550°C for the first cycle. Fig. 5(b) shows that the simulated macroscopic cyclic response agrees well with experiments, however, cyclic loading was not tested here [49].

For stress relaxation analysis, the parameters: $\dot{\gamma}_{0,2}$ and $m_2$ were calibrated using the creep data related to the {100} grain family measured from the 3DXRD experiment. Strain-controlled simulations were run using the same RVE as shown in Fig. 4. A displacement corresponding to 4% total strain (~205 MPa experimental dwell stress) was applied along the z-axis. The x-y plane was constrained to avoid rigid body motion. The values for $\dot{\gamma}_{0,2}$ and $m_2$ were fitted to predict the stress relaxation from the {100} grain family by assigning values within the bounds obtained from prior studies [50,51]. Fig. 6 shows the simulation results after the optimisation of $\dot{\gamma}_{0,2}$ and $m_2$; good agreement to experimental values is shown. The full list of calibrated parameters is given in Table 2; these parameters are used for the remaining analysis.

### 3.3. Finite Element Models

Following the model calibration, the crystal plasticity framework was employed to simulate the grain behaviour in a polycrystal. Two modelling approaches were investigated: i) a Single crystal/J2 model and ii) a Full-scale model.

#### 3.3.1. Single Crystal/ J2 Model

The single crystal/J2 model consisted of a grain surrounded by an isotropic media to simulate the average elastic lattice response of a grain. The isotropic media was used to represent the equilibrium and compatibility of a grain in a high-resolution polycrystal aggregate. This way, modelling a large and computationally expensive polycrystal representative volume element (RVE) was avoided.

A truncated octahedron of the grain geometry was created in Abaqus v2018 [44] as shown in Fig. 7. The coordinate system used in the model matched the data analysis, where the applied load was along the vertical z-axis. Stress-controlled simulations were run with 220 MPa stress applied without any interruptions or stress relaxation dwells along the z-axis. To avoid rigid body motion, the opposite face along the z-axis was constrained as a boundary condition. To evaluate the variation in elastic strains as a function of crystal orientation, the orientations of the selected grains from the extremes of each grain family, as measured by 3DXRD, were inserted into the grain model as a user material. For example, Fig. 7(b) shows the selected grains from the extremes of the {311} grain family. The grain model deformed according to the abovementioned constitutive law, while the isotropic J2 model uses the same constitutive laws homogenized with the Taylor factor, $M$, where the total strain rate and stresses are computed using the following Eq.

$$ \dot{\epsilon} = \frac{1}{M} \dot{\gamma} \qquad (8) $$

$$ \sigma = M\tau \qquad (9) $$

The Taylor factor of 3.1 was used for assuming an isotropic equiaxed polycrystal and the J2 solution was performed using the radial return method as described in detail in [2].

#### 3.3.2. Full-Scale Model

For simulating the average elastic lattice response of each grain in full-scale polycrystal, stress-controlled simulations were run with the full-scale model using the same RVE as shown in Fig. 4. A load of 220 MPa was applied without any interruptions or stress relaxation dwells on the front x-y plane. To avoid rigid body motion, the opposite x-y plane was constrained as a boundary condition as shown in Fig. 4.
<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Elastic lattice strain response for selected grains from each grain family during tensile loading up to 220 MPa. Open markers show the individual grain response, while filled markers show the average behaviour of each grain family. Dashed lines show the linear response. The colour in each plot corresponds to the orientation plotted on the IPF.</figcaption></figure>

The deformation of each grain was calculated using the crystal plasticity constitutive model described above. The elastic lattice strain averaged on each grain in both models was extracted using the elastic deformation component from the total deformation gradient [5]. Average values for each grain family were also calculated. For the analysis of stress-relaxation, only the full-scale model was used in which displacement-controlled simulations were run. A 30 min dwell period was applied following 4% total strain on the front x-y plane.

# 4. Results

## 4.1. 3DXRD Measurement of Average Elastic Lattice Strain of Grains

Using the current data analysis pipeline, a total of 366 grains were measured. This represents a subset of the total number of grains in the probed volume; those shown were those that were reliably indexed within the constraints of the 3DXRD scanning strategy. There are, however, sufficient grains in each scan to permit the per-grain lattice strain response and their variation to be observed as the load was incrementally increased.
<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Elastic lattice strain response from individual grains predicted by the single crystal/ J2 model. The inset shows an IPF plot with respect to the loading direction, showing the direction of the grains of interest.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Fig. 10.** Elastic lattice strain versus stress response of individual grains predicted by the full-scale model. The colour in each plot corresponds to the orientation plotted on the IPF.</figcaption></figure>

**Fig. 8** shows the elastic lattice strain response of selected grains from different grain families (open symbol) during loading to an equivalent of ~ 220 MPa, excluding the unloading and dwell data, i.e., the 7th and 12th scan. The selected grains are those that were successfully indexed on the first scan at 5 MPa. The individual grain responses have a 10-15% uncertainty of around ± 0.025% as indicated in **Fig. 8**. The elastic deformation of the selected grains from the {100} grain family for an applied stress of 220 MPa was measured to be around 0.15-0.2%, which is noticeably higher compared to grains from the {110}, {111}, and {311} families which were measured mostly below 0.1%. There is a noticeable inhomogeneity of the elastic lattice strain response between grains within the same grain family. This spread is likely due to a combination of factors such as variation in grain orientation, grain size, experimental uncertainty, and grain neighbourhood interactions.

The average strain responses from the grain families were calculated

using Eq. (2) and are plotted as the filled markers in **Fig. 8**. The average responses are compared against the linear response of each grain family (dashed lines in **Fig. 8**), representing the behaviour if all the grains were to transition from elastic to plastic deformation under the same applied stress. These were obtained by fitting the averaged experimental data up to the yield point, which is around 120 MPa at 550°C. Despite the limited number of data points, the deviation from the linear response beyond the yield point can be observed for the different grain families. Beyond yielding, the {100} grain family accumulated a much higher elastic strain compared to the linear response. This indicates the grains sustained greater elastic deformation (plastically stronger). In contrast, the {110} grain family evidently stopped accumulating elastic strain early during loading, indicating the grains yielded before those in other grain families (plastically weaker). Meanwhile, the {111} and {311} grain families did not show significant deviations from the linear responses. Similar findings were reported in other studies [4,6], where the average grain family response was obtained using 2D Synchrotron X-ray or neutron diffraction.

## 4.2. Simulated average elastic lattice strain of grains

The elastic lattice strain response of grains simulated by the single crystal model is shown in **Fig. 9**. A clear trend can be observed where grains in the {100} family elastically deformed to above 0.15% and the {311} grains deformed to 0.11-0.13%, while the deformation of {110} and {111} grains were below 0.1%. The single crystal simulations predict the grain-average elastic strain variation within a grain family, however, not to the extent measured in experiments. Since all single crystal models have an identical grain environment and grain volume, the simulations only consider the effect of grain orientation with the variation in grain response.

**Fig. 10** shows the simulated elastic lattice strain averaged over individual grains from the full-scale model. The grains along the {100} and {311} families elastically deform to 0.2%, while the deformation of {110} and {111} grains are above 0.1%. The inhomogeneity of grain-average elastic strain within a grain family is found to be significantly larger in the full-scale model as compared to the single crystal model; a variation in lattice strain was observed to increase as the macroscopic deformation builds for all grain families.

**Fig. 11** compares all simulation results with the 3DXRD measurements for the elastic lattice strain averaged over individual grains. The full-scale model results are presented using the upper and lower bounds of results mentioned in **Fig. 10** to avoid the congestion. Deviation of average elastic strain from an elastic linear behaviour for each grain family can be observed as well, either a positive (mainly {100} family) or a negative ({110} and {111} family) deviation. The results show that a full-scale model has a large variation of grain-average elastic strain within each grain family. Considering the experimental uncertainty, the full-scale model agrees better with 3DXRD experiments than the single-crystal model.

## 4.3. Average grain response to stress relaxation

**Fig. 12** compares the 3DXRD and simulated results for the stress relaxation dwell of different grain families. The 3DXRD measurements for the evolving elastic lattice strain are taken from selected grains at the beginning and end of the dwell as shown with the markers in **Fig. 12**. The simulated results are estimated for each grain in IPF and shown with continuous lines. In the 3DXRD experiments, all grains, except one outlier in the {110} grain family, show a decreased elastic lattice strain at the end of the dwell i.e., around 0.02% and 0.1%. A similar trend for the decrease in elastic lattice strain is found in simulations except for the one outlier in the {110} grain family. However, there is no clear trend between the decrease of the elastic strain of the different grain families, with the average value measure to be approximately 0.05% for {110} and {111} grain families, and 0.07% for the {311} grain family. A
<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Comparison of individual grain responses during plastic loading, between measured values from 3DXRD and simulated values using the J2, and full-scale model.</figcaption></figure>

comparison of the simulated results to the measured values shows good agreement within experimental uncertainty of 0.05% for all orientations.

## 5. Discussion

A 3DXRD measurement has the capability to resolve the deformation localisation at the grain scale during elastic and plastic deformation [52]. The measured grain behaviour should then reflect the combined effect of its crystal orientation and the orientation and deformation state of the neighbouring grains. The current 3DXRD experiment successfully resolved many individual grains providing the grain-average elastic strain with an error bar. Interestingly, the 3DXRD measurements show a large variation in the elastic lattice strain within a grain family. A similar trend has been observed for resolved shear stresses measured with 3DXRD [17], providing an independent validation of the measurements given here.

In terms of comparison between the 3DXRD experiment and finite element models, the J2 model shows substantially less inhomogeneity compared the full-scale model and measured responses. As previously mentioned, the inhomogeneity within the grain family observed in the J2 model only relates to the variation of grain orientations. This indicates the effect of grain orientation is not the dominant factor influencing the grain response, and the effect of the grain environment is critical to accurately predict the strain response of a grain embedded in a real polycrystal. However, the single crystal/J2 model allows efficient computation of the overall response of the grain in actual loading, creep

simulations with long time intervals with load-unload cycles in between. It is an effort related to mapping a few individual grains that can also be experimentally traced to investigate in-grain response with high resolution that cannot be achieved efficiently with the full-scale polycrystal models.

The full-scale model shows higher inhomogeneity compared to the J2 model due to a larger set of grains and grain environment. Thus, considering microstructural variability is expected to improve the prediction of local deformation and hence local damage processes. However, an exact grain-to-grain comparison between 3DXRD measured values and simulations is still challenging.

Rather than describe all grains including their neighbourhood, this study instead provides insight from a random subset of subsurface grains. We evaluated the effect of the environment on the elastic response of grains independently by running eight simulations with a different RVE (0.8 × 0.8 × 0.8 mm³) having ~150 grains and a different random environment. Fig. 13 shows the effect of the environment on the maximum elastic strain as a function of grain orientations. All simulations present a consistent distribution of grain-average elastic strain regardless of the environment i.e. {001} grain family deforms the most, followed by the {311}, {110}, and {111}. A similar distribution of elastic strain as a function of grain orientations in stainless steel has been found in different 3DXRD experiments [17], which validates our simulation data independently. Instead of describing an entirely 3D space-filled microstructure, it is argued that a statistically significant number of grains and a random environment within an RVE is sufficient to represent the polycrystal response for 316H steel, as shown in Fig. 13.
<figure><img src="image_15.png" /><figcaption>**Fig. 12.** The drop of elastic lattice strain in individual grains at the start and the end of stress relaxation dwell, measured by 3DXRD and predicted by the full-scale model. The colour for the simulated data in each subplot corresponds to the orientation shown in the IPF.</figcaption></figure>

One of the important contributions of this study was to quantify the effect of the grain environment on the local elastic strain and the underlying uncertainty in accurately predicting it using the crystal plasticity model. Table 3 compares the variability of the local grain-averaged elastic strain estimated with 3DXRD experiments and simulations. The mean value of the local elastic strain was calculated by averaging the elastic strain of all grains, and the uncertainty was estimated using the upper and lower bounds of the elastic strains considering all grains. The quantitative analysis from Table 3 shows that the {100} and {311} grain families are sensitive to the environment. Therefore, a full-scale model is essential for predicting the local response of these grains. Whereas {110} and {111} grain families are insensitive to the environment, thereby a J2 or self-consistent model is sufficient to predict the local response of these grains with 10-20% uncertainty.

Further to our findings in Table 3, the explicit effect of the environment is quantified for the {100} grain-averaged elastic strain. For this purpose, an RVE was employed as shown in Fig. 13, with the centre of the grain fixed and oriented along a (100) directions aligned close to the tensile axis. In total, 20 simulations were tested with different grain environments.

Fig. 14(a) shows the schematic of models and grain-average elastic strain for the {100} grain family calculated from these simulations. Fig. 14(b) shows the explicit contribution of orientation (calculated from the J2 model) and environment (full-scale model with fixed-centred (100) grain). The results show that in addition to the effect of crystal orientation on the elastic strain deformation of a grain from the {100} family i.e., 1.7 × 10⁻³, different environments can increase it up to 7.5 × 10⁻⁴. Fig. 14(b) also guides the choice of the model based on the accuracy for suitable prediction. If one is flexible in estimating the local response with 20-30% error, the J2 model can provide a computational-friendly solution. Otherwise, a full-scale model provides an accurate prediction of the local deformation.

Since the experiments completed in this work considered cyclic creep loading sequences, estimates of the effect of type-II residual stress were considered for grain-averaged elastic strain. The residual stresses measured from the 3DXRD experiment were incorporated as an initial stress state in ABAQUS and the average elastic response of the grain was simulated using the J2 model. The full details of the analysis are presented in Appendix D. The results show that type-II residual stress does not affect grain-average elastic strain. Previous work also confirmed that
<figure><img src="image_16.png" /><figcaption>**Fig. 13.** Effect of the grain environment on the average elastic lattice strain. Each subplot shows an RVE (0.8 × 0.8 × 0.8 mm³), with ~150 grains and a random environment. The maximum elastic lattice strain on the loading face corresponding to 220 MPa applied stress is plotted as a function of the crystal orientation as shown with the colourbar inside the IPF, next to each respective RVE.</figcaption></figure>

<table><caption>**Table 3**
Uncertainty quantification of average elastic lattice strain in experiments and simulations</caption><thead><tr><th rowspan="2">Grain families</th><th colspan="3">Effect of environment on average elastic lattice strain</th></tr><tr><th>3DXRD Experiments</th><th>Simulation (Full-Scale Model)</th><th>Simulation (J2 Model)</th></tr></thead><tbody><tr><td>{100}</td><td>1.98 × 10<sup>-3</sup> ± 5 × 10<sup>-4</sup></td><td>1.90 × 10<sup>-3</sup> ± 5.5 × 10<sup>-4</sup></td><td>1.50 × 10<sup>-3</sup> ± 2 × 10<sup>-4</sup></td></tr><tr><td>{110}</td><td>0.75 × 10<sup>-3</sup> ± 5 × 10<sup>-4</sup></td><td>1.20 × 10<sup>-3</sup> ± 1.7 × 10<sup>-4</sup></td><td>0.95 × 10<sup>-3</sup> ± 2 × 10<sup>-4</sup></td></tr><tr><td>{111}</td><td>0.62 × 10<sup>-3</sup> ± 5 × 10<sup>-4</sup></td><td>1.10 × 10<sup>-3</sup> ± 2.0 × 10<sup>-4</sup></td><td>0.82 × 10<sup>-3</sup> ± 2 × 10<sup>-4</sup></td></tr><tr><td>{311}</td><td>0.90 × 10<sup>-3</sup> ± 5 × 10<sup>-4</sup></td><td>1.50 × 10<sup>-3</sup> ± 5.5 × 10<sup>-4</sup></td><td>1.10 × 10<sup>-3</sup> ± 2 × 10<sup>-4</sup></td></tr></tbody></table>
<figure><img src="image_17.png" /><figcaption>**Fig. 14.** Effect of grain environment on the elastic lattice strain. The first subplot shows an RVE (0.8 × 0.8 × 0.8 mm³), with a fixed central grain for orientations common to the {100} grain family, tested at 20 random environments.</figcaption></figure>

the inclusion of type-III residual stress has minimal effect after loading [53] for 316H steel.

In addition to the full-scale model created by Dream3D, a background study was performed considering a different full-scale model: Voronoi tessellation, in which the RVE microstructure was created using the coordinates of detected grains (as seed points) and grain orientations obtained from 3DXRD. It is noted that this model RVE is an approximated microstructure as only a subset of grains from a real sample was measured. Full details of the Voronoi tessellation model and results are presented in Appendix A. Regarding the average elastic strain response of grains with and without stress relaxation dwell (Fig. 16 and Fig. 17 in Appendix A), the Voronoi tessellation model shows similar results as predicted by the full-scale model. The comparison of the Voronoi and full-scale model can be used to reveal the combined effect of the element type (quadratic tetrahedral - C3D10 vs. hexahedral - C3D8), grain boundaries as straight lines or as a staircase representation, as well as the differences in the geometry of grains (e.g. triple junction in Voronoi). The relative difference in the grain-averaged strain between the two

models for grains-A, grain-B, and grain-C is 9.1% (maximum). Therefore, the uncertainty related to the misrepresentation of triple and multiple junctions for the Voronoi model is believed to have a lower effect on the grain-averaged strain. Moreover, the Voronoi tessellation model takes longer (15 hours) to complete due to the larger mesh size as compared to the full-scale model, which completes in only 8 hours. Therefore, a full-scale model presents a more computationally viable solution over a Voronoi tessellation-derived model. Regarding the creep deformation, the current heat-treated material shows a drop in elastic strain of around 0.1%. The stress relaxation data related to each grain family measured from current 3DXRD experiments is limited and conveys higher uncertainties.

## 6. Conclusion

This study demonstrated the combined use of a 3DXRD experiment and CPFE modelling to predict plastic and creep deformation of 316H-type stainless steel at 550°C. The following conclusions can be drawn from this study:

1. A limited angle 3DXRD setup was employed to measure the grain-averaged strain during plastic and creep deformation at elevated temperatures. The measured elastic strains exhibit large variations within grains of common crystallographic orientation.

2. A single crystal/J2 model presented the effect of crystal orientation on the grain-average strain response. Since grains with vectors normal to the {110} and {111} planes that are aligned close to the tensile axis were not significantly affected by the grain environment, the J2 model provides a reasonable prediction of the elastic response of these grains.

3. Within the measurement uncertainty, the full-scale model effectively captures the inhomogeneity of the average grain response during plastic and creep deformation. The {100} and {311} grain families were found to be sensitive to the environment, indicating a full-scale model that includes the effect of orientation and grain environment is essential to accurately predict the local elastic strain of these grains.

4. Instead of describing an entirely 3D space-filled microstructure, a statistically significant number of measured subsurface grains is sufficient to draw comparisons to elevated temperature crystal plasticity and creep simulations.

## CRediT authorship contribution statement

**Farhan Ashraf:** Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Validation, Writing – original draft, Writing – review & editing. **Ranggi S. Ramadhan:** Data curation, Formal analysis, Investigation, Methodology, Writing – original draft. **Abdullah Al Mamun:** Data curation, Formal analysis, Investigation, Methodology. **James A.D. Ball:** Data curation, Formal analysis, Investigation. **Eralp Demir:** Data curation, Formal analysis, Investigation, Methodology, Software, Writing – original draft, Writing – review & editing. **Thomas Connolley:** Data curation, Formal analysis, Investigation, Methodology. **David M. Collins:** Conceptualization, Writing – review & editing, Formal analysis, Methodology. **Mahmoud Mostafavi:** Conceptualization, Funding acquisition, Resources, Supervision, Writing – review & editing, Formal analysis, Methodology, Project administration. **David Knowles:** Conceptualization, Formal analysis, Funding acquisition, Methodology, Project administration, Supervision, Writing – review & editing.
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.

Acknowledgment

The authors are grateful to the Engineering and Physical Sciences Research Council (EPSRC) for funding the project through contract no.

EP/V038079/1 and EP/R020108/1. The Authors thank the support from Royce Institute through contract no. EP/X527257/1 and Dr. Julio Spadotto for performing EBSD. The University of Manchester at Harwell is acknowledged for accessing the ETMT, to Diamond Light Source for awarding beamtime (MG24084-1), and to I12 staff, in particular Dr Stefan Michalik, who are thanked for their support in conducting the 3DXRD experiments. MM acknowledges the support of the Royal Academy of Engineering through a Research Chair. EDF is thanked for providing part funding for the work and Dr Marc Chevalier and Dr Nicolo Grilli are gratefully thanked for valuable discussions.

Appendix A

Full-Scale Voronoi Tessellation Model

Since the Voronoi tessellation model provides an idealized representation of grain boundaries [54], it has been proposed to improve the use of weighted Voronoi tessellation given by finite size or radius using Laguerre tessellation [55]. A modified Voronoi tessellation method was developed to produce grains while satisfying statistical dimensions and interface properties [56]. Similarly, the surface tension and grain boundary formation energy determine the angle of triple connections, so more realistic grain shapes were generated using the inverse Monte Carlo method [57]. Crystal orientation or texture significantly influences macro scale response and grain average stress [58]. The grain boundary geometry is important for failure modelling, therefore it is considered in the cohesive modelling of grain boundaries that considers the triple and multiple junctions of a polycrystalline [59,60].

Here, the Voronoi tessellation model consists of an RVE with a size of 1.2 × 0.9 × 0.6 mm³, which is comparable with the illuminated gauge volume of the specimen in the 3DXRD experiment, approximately 1.1 × 0.7 × 0.5 mm³. 148 grains were reconstructed from the first 3DXRD scan (5 MPa at 550°C), and the grain information was used to build the RVE. The coordinates of the detected grains were used as seed points and became the centre of gravity of the simulated grains in the RVE. 3D Voronoi tessellation was then applied to fill the grain volume [61] as shown in Fig. 15. The measured grain orientations were then registered to the corresponding grains.

Please note that the grain volume in the simulation is the product of Voronoi tessellation and not the accurate representation of the actual grain volume. Also, there are grains that were not successfully reconstructed by 3DXRD, hence the grain environment for the simulated grains was different compared to the actual ones. The accuracy of the full-scale Voronoi tessellation model in replicating the grain environment is however limited by the capability of the 3DXRD scans and analysis to reconstruct the grains in the gauge volume.

For simulating the average elastic lattice response of each grain, stress-controlled simulations were run with 220 MPa applied along the z-axis, and the opposite face along the z-axis was constrained to avoid rigid body motion as shown in Fig. 15. The deformation was calculated using the crystal plasticity constitutive model. The elastic lattice strain averaged on each grain was extracted using the elastic deformation component from the total deformation gradient [5]. The grains in the model were indexed the same way as the measured ones, therefore direct comparison can be made for each grain ID.

Regarding the stress-relaxation analysis, displacement-controlled simulations were run. A 30 min dwell period was applied following 4% total strain along the z-axis and the opposite face along the z-axis was constrained to avoid rigid body motion.

<figure><img src="image_18.png" /><figcaption>Fig. 15. Full-scale Voronoi Tessellation model with 359040 elements (C3D10) constructed using grain position and orientation from the 3DXRD measurement.</figcaption></figure>

Fig. 16 shows the simulated elastic lattice strain response of selected grains from the Voronoi tessellation model. The grains have the same ID as the ones highlighted in the 3DXRD analysis, as previously shown in Fig. 8. Qualitatively, the average response of each grain family extracted from the simulation agrees well with that of the 3DXRD measurement. Inhomogeneity of grain response within the grain family is observed, and the variation in lattice strain increased with increasing macroscopic deformation for all grain families, which are also observations made in [6].
<figure><img src="image_19.png" /><figcaption>**Fig. 16.** Elastic lattice strain response from individual grains predicted by the full-scale Voronoi tessellation model. Each line shows an average elastic lattice strain in a grain family and each subplot shows a different grain family.</figcaption></figure>

**Fig. 17** shows the evolution of elastic lattice strain on selected grains during the stress relaxation dwell simulated by the full-scale Voronoi tessellation model. As observed in the measured data, the simulated behaviour of all grains also showed a decrease in elastic lattice strain. Overall, the simulated results agree well with measured values within experimental uncertainty. The average decrease in elastic strains is around 0.085% for {100} family, around 0.06% for {311} family, and around 0.05% for both {110} and {111} family.

<figure><img src="image_20.png" /><figcaption>**Fig. 17.** The drop of elastic lattice strain at the start and the end of stress relaxation dwell predicted using the full-scale Voronoi tessellation model. Each line shows an average response of individual grains from different grain families.</figcaption></figure>
# Appendix B

## Full-Scale Single Crystal Model

The full-scale single-crystal RVE was created using Dream3d [45] with a size of 0.1 × 0.1 × 0.1 mm³ (comparable to average grain size in experiments), and 15625 C3D8 elements as shown in Fig. 18. Each element of RVE is assigned the same crystal orientations and is oriented such that the loading direction lies along the crystal directions corresponding to IPF as shown in Fig. 18.

A stress-controlled loading was applied on the front face along the z-axis and the opposite face along the z-axis was constrained as a boundary condition to avoid a rigid body motion.

<figure><img src="image_21.png" /><figcaption>**Fig. 18.** Full-scale single crystal RVE (100 × 100 × 100 µm³). Inset shows the IPF plot with respect to the loading direction, showing the direction of the grain of interest.</figcaption></figure>

Fig. 19 compares the elastic strain predicted by the single crystal surrounded by a homogenised medium and full-scale single crystal. The full-scale single-crystal model shows large deformation across all orientations compared to the single-crystal model constrained by the homogenised medium. This might be due to a lack of constraint in the full-scale single crystal. However, both models show the same trends across all orientations within the grain families.

<figure><img src="image_22.png" /><figcaption>**Fig. 19.** Comparison between single crystal models: marker shows results from a single crystal surrounded by a homogenized media, and dashed lines show results from a full-scale single crystal model where each element is assigned the same crystal orientation. The label shows the index of the grain from each grain family for comparing models.</figcaption></figure>

Fig. 20 shows the inhomogeneity of elastic strain within the {001} grain family predicted by the full-scale single crystal model. Regardless of their different deformation response, both models do not predict inhomogeneity.
<figure><img src="image_23.png" /><figcaption>Fig. 20. Spread/inhomogeneity of elastic strain within the {001} grain family.</figcaption></figure>

# Appendix C

## Effect of the number of grains on the elastic lattice strain

The RVE was created using Dream3d [45] with a size of 1.2 × 1.2 × 1.2 mm³, 216000 C3D8 elements, and 271 grains as shown in Fig. 21. A stress-controlled loading was applied on the front face along the z-axis and the opposite face along the z-axis was constrained as a boundary condition to avoid a rigid body motion.

<figure><img src="image_24.png" /><figcaption>Fig. 21. Full-scale model: RVE created from Dream3D with 271 grains.</figcaption></figure>

Fig. 22 shows the simulated elastic strain averaged over individual grains from the full-scale model with 271 grains. The grains along the {100} and {311} families elastically deform to 0.23% and 0.2% respectively, while the deformation of {110} and {111} grains are above 0.1%. These results are in good agreement with the full-scale model results in Fig. 10, justifying the sufficiency of 148 grains in the full-scale model.
<figure><img src="image_25.png" /><figcaption>**Fig. 22.** Elastic lattice strain versus stress response of individual grains predicted by full-scale model with 271 grains. The colour in each plot corresponds to the orientation plotted on the IPF.</figcaption></figure>

# Appendix D

## Effect of type II residual stress on the elastic lattice strain

This analysis was performed using the J2 model as explained in section 3.3.1. The residual stress measured from the 3DXRD experiment in tensor form was applied to the grain as an initial stress state in ABAQUS. Fig. 23 shows the effect of type II residual stress on the average elastic lattice strains. The results show that regardless of crystal orientations, type II residual stress does not significantly affect the elastic strains.
<figure><img src="image_26.png" /><figcaption>Fig. 23. Effect of type-II residual stress on average elastic strains. The colour in each plot corresponds to the orientation plotted on the IPF.</figcaption></figure>
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