Quantifying nanoscale 3D stresses in zirconium and hydrides

Abstract

Zirconium alloys are widely used to manufacture the core components of nuclear reactors. Despite having good mechanical properties, they are susceptible to hydrogen embrittlement and hydride formation. In this study, scanning three-dimensional synchrotron X-ray diffraction is employed to determine, for the first time, the 3D internal stresses within hydrides and the surrounding zirconium matrix at 200 nm spatial resolution. The three-dimensional measured grain morphologies are imported into a crystal plasticity finite element model to simulate the evolution of localized stresses induced by hydride precipitation. Both experimental and numerical results consistently reveal significant compressive hydrostatic stresses within hydrides, with magnitudes exceeding 1 GPa, accompanied by pronounced variations in the individual stress components. Stresses and dislocation densities are localized at the hydride-zirconium boundaries. A force equilibrium analysis demonstrates that the measured stresses in the surrounding zirconium matrix are sufficient to recover internal stresses within hydrides, further validating the measured values.

Combining high resolution 3D X-ray diffraction with crystal plasticity modeling, this manuscript highlights the micromechanics of transformation-induced stress/strain in hydride-zirconium boundaries.

Introduction

Due to their low thermal neutron absorption cross section, zirconium (Zr) alloys have been used to manufacture structural components in water-cooled nuclear reactor cores. During service, hydrogen generated by the Zr-water reaction diffuses into the zirconium matrix and precipitates as zirconium hydrides. It is widely recognized that these hydrides reduce the fracture resistance of the alloys and may lead to a phenomenon known as delayed hydride cracking¹. Fracture behavior has been shown to depend on hydride orientation and the interaction between hydrides and local dislocation structures^2–4. While hydrogen embrittlement is commonly attributed to the intrinsic brittleness of the hydride phase⁵, recent studies have highlighted the role of internal stresses associated with hydride precipitation⁶. These stresses arise due to the ~17% volumetric mismatch between δ-hydrides, with face-centered cubic (FCC) crystals, and the α-Zr matrix, with hexagonal close-packed (HCP) crystals, which produces localized plastic deformation and stress concentrations within both the hydrides and the surrounding α-grains.

The measurement of internal stresses within hydrides is extremely challenging, yet scientifically crucial for validating numerical models that predict crack initiation and assess the integrity of zirconium components in nuclear reactors. Significant effort has been made to quantify the average elastic strains and stresses associated with the formation of δ-hydrides^7–10. This is of particular importance as the shear energy density within hydrides has recently been identified as a driving force for crack initiation^3,6, and this energy is related to the state of deformation caused by the hydride mismatch strain. Despite the critical role of these internal stresses, a direct experimental measurement resolving microscale hydrides and the nanoscale stresses within has remained elusive. The small size of hydrides and the resolution limitations of conventional measurement techniques, including diffraction-based methods, have prevented the full-field quantification of these nanoscale stresses. In this work, we present the first experimental measurement of three-dimensional stress fields within δ-hydrides and the surrounding α-Zr matrix using high-energy scanning three-dimensional X-ray diffraction (S3DXRD). These measurements offer new insights into the true mechanical state of hydrides during formation and interaction with neighboring grains, with direct implications for their fracture response in service.

Hydride transformation (mismatch) strain induces significant lattice rotation, misorientation gradients, and localized geometrically necessary dislocation (GND) density in both the hydride and surrounding α-Zr. While these effects have been quantified using electron backscatter diffraction (EBSD) and transmission electron microscopy (TEM), these analyses have been limited to surface observations. Neither the three-dimensional distribution of GND density nor the heterogeneous distribution of the stress field in the vicinity of hydrides has been reported previously. Hydride formation can generate highly localized deformation zones¹¹. Several studies have shown that GND density within δ-hydrides can exceed that of the parent α-Zr during the phase transformation, contributing to high local stress accumulation^11–15. Previous numerical simulations based on crystal plasticity finite element (CPFE)¹¹ or dislocation dynamics¹⁶ have suggested that the hydrostatic pressure in hydrides may reach 1 GPa, however, no direct experimental validation has been reported to date. This raises an important question about the reliability of the simulated values, especially for hydrides as small as 1 μm, given their three-dimensional morphology, a question that has never been addressed.

Three-dimensional synchrotron X-ray diffraction (3DXRD) enables the measurement of internal stress and orientation fields in polycrystalline materials^17–19. Three-dimensional observations of strain and microstructure have become possible in recent years using various synchrotron-based X-ray diffraction and imaging techniques^20–26. In particular, 3DXRD is used to quantify grain shapes and average grain orientations to study deformation and cracking in polycrystals^22,27–32. While initially developed for measuring grain-average stress and orientation^33–35, recent advancements in 3DXRD have improved the spatial resolution to sub-micron levels. These capabilities have enabled three-dimensional mapping of intragranular stresses³⁶, strain evolution in deformed materials³⁷, lattice rotations and intragranular orientation spreads^38,39, and grain growth during annealing⁴⁰.

CPFE is a class of materials constitutive models that has been used for numerical modeling of polycrystalline materials⁴¹. The state of stress and plastic strain at the sub-grain scale can be investigated through CPFE modeling, allowing the analysis of critical phenomena such as stress concentrations near grain boundaries, phase interfaces, and secondary phases^42–44. Importantly, several studies have provided direct comparisons between CPFE simulations and 3DXRD measurements, showing good agreement in trends for lattice strain, rotation fields, and stress evolution^29,45–48.

In this study, we provide the first direct experimental measurement of 3D internal stresses within δ-hydrides and adjacent α-Zr grains using S3DXRD. The measured microstructure, including hydride morphology and grain orientations, is mapped into a CPFE model to study the effects of hydride-induced transformation strain. The experimental and numerical results are compared, and a force-equilibrium analysis is performed to validate the physical consistency of the measured data. This work provides the first three-dimensional nanoscale quantification of phase transformation–induced stress variations within an individual zirconium hydride and establishes a robust framework for quantifying internal stress in small, experimentally inaccessible phases using three-dimensional diffraction data. The developed methodology is generic and can be applied to other phenomena involving phase transformation.

Results

Details of sample preparation, experiments, and numerical modeling are provided in the Methods section. A summary is provided here. A Zircaloy-2 sample was first heat-treated to recrystallize the grains, hydrided cathodically, and homogenized to produce δ-hydrides. As shown in the Methods section, these hydrides mainly precipitate at grain boundaries, a phase transformation process which is accompanied by a ~ 17% volumetric expansion. No external mechanical load was applied to the sample so that the sole effects of hydride-induced strain could be studied. After measuring grain orientations using EBSD, a triple point hydride was selected for further analysis. A pillar of X = 31 µm $\times$ Y = 35 µm $\times$ Z = 77 µm was milled out from the same region. The coordinate system used in this paper is shown in Fig. 1a. The orientation of grains and the state of stress and strain within this pillar were measured using S3DXRD at the ID-11 beamline of the European Synchrotron Radiation Facility. A 200 nm point-focus X-ray beam was used to collect diffraction patterns while scanning transverse to the beam direction and rotating the sample 360° around the pillar’s axis. The sample was then translated vertically between horizontal scan layers, enabling 3D measurements. As explained in the Methods section, the collected diffraction patterns were used to produce sinograms, from which grain shapes were determined. Stress, strain, orientation, and their variations were subsequently quantified using a suite of programs developed in ImageD11^49,50.

Fig. 1. Geometrically necessary dislocation (GND) densities.

3D GND maps calculated from a Scanning three-dimensional X-ray diffraction (S3DXRD) and b Crystal Plasticity Finite Element (CPFE) modeling. c GND map on the YZ plane measured by EBSD, pre-fabrication of the pillar. d S3DXRD-phase map on the YZ plane after pillar fabrication, with corresponding GND densities from e S3DXRD and f CPFE. g Examples of some slices in the XY plane from S3DXRD and CPFE. The first two rows of (g) show the S3DXRD results, while the last row shows the CPFE results. The differences between EBSD and S3DXRD results for the location of grain boundaries in (c) and (d) are due to surface removal during sample preparation. In phase maps, Zr matrix is in yellow and hydrides are shown in turquoise. Source data are provided as a Source Data file.

Following the methodology described in the Methods section, the measured grain shapes and average grain orientations were used to reconstruct a 3D microstructure for CPFE modeling, enabling the application of a biased mesh towards hydrides and grain boundaries. Two different strategies were used for the numerical modeling. The first strategy is used in the Results section, in which the hydride misfit strain was applied to the hydride domain so that the CPFE-calculated fields, e.g., GND or stress fields, can be compared with the measured ones. Results for the second strategy are provided in the Discussion section, where the stresses measured by S3DXRD are mapped onto the α-Zr grains in the model, and force equilibrium is enforced to determine the stress state within the hydrides. In this second strategy, the hydrides are modeled as a homogeneous stiff elastic phase with zero initial stress, while the measured stresses are prescribed only in the α-Zr grains. The stress distribution inside the hydrides is then obtained through the equilibrium solution, such that traction continuity at the hydride–matrix interfaces and global mechanical equilibrium are satisfied. These two modeling strategies provide two independent checkpoints for examining the quality of measured stresses within hydrides.

GND density

The orientations of the grains on two faces of the pillar were measured by EBSD, and the results are compared in the Supplementary Fig. 1. After accounting for rigid body rotations, a good agreement was achieved between measured orientations by S3DXRD and those from the two EBSD maps. These orientations, along with the calculated ones from CPFE after applying hydride-induced transformation strain, were used for producing the GND maps shown in Fig. 1. Two different methods were used for determining GND densities. First, GND densities were estimated from EBSD maps using the method proposed by Pantleon⁵¹, which accounts for the incomplete curvature tensor in general two-dimensional EBSD data. As described in the Methods section, in the second method, GND densities were determined from S3DXRD-measured elastic strain and orientation fields or CPFE-calculated ones using the full Nye tensor; i.e., no assumption was made for the out-of-plane components of the Nye tensor due to the nature of 3D data sets.

Figure 1a, b show the calculated GND density within the α-Zr matrix in 3D. For direct comparison between EBSD, S3DXRD, and CPFE, a layer close to the side surface in the YZ plane was selected from the S3DXRD dataset. The GND density presented in Fig. 1c is from the EBSD map measured prior to the fabrication of the micropillar; therefore, the grain boundaries differ from those in the S3DXRD and CPFE results shown in Fig. 1d–f. In addition, some features, such as the presence of small intragranular hydrides, might affect the GND map of Fig. 1c. Such small features were removed during pillar fabrication and were not imported into the CPFE model. Despite these differences, all three approaches consistently show higher GND densities at the hydride-Zr interfaces, showing the effects of the deformation field induced by the hydride transformation strain. Although the magnitude of the GND densities differ, with the highest in CPFE and the lowest in S3DXRD, the overall trends are consistent. In the CPFE results, GND densities exceed 10¹⁴m⁻² near hydride interfaces, while the same regions in S3DXRD results show lower values. This difference arises from the lower lattice misorientations measured by S3DXRD compared to EBSD and CPFE. It has been reported previously that, in comparison to EBSD, lattice curvatures are underestimated in measurements based on S3DXRD^52,53. This is mainly due to the spatial averaging in S3DXRD and the way diffraction peaks are merged during postprocessing. When diffraction patterns about the pillar axis (ω-frames) are combined to improve strain accuracy and reduce noise, the uncertainty in the peak position decreases, and small variations in peak position are smoothed. This improves the stability of the strain fit but reduces the measured orientation spread. As a result, the measured lattice curvature is systematically lower, resulting in smaller GND values in S3DXRD than in EBSD and CPFE. The spatial averaging effect is numerically examined and further discussed in the Discussion section.

Figure 1g presents examples of internal slices in the XY plane, at Z = 31.5μm and Z = 40.5 μm, along with the corresponding phase maps. Both simulation and S3DXRD results show that GND density is localized near hydride interfaces- some of those regions are highlighted by black ovals in Fig. 1. The presence of localized GND distribution across methods reinforces the role of hydride misfit strain in generating local GND density.

Stresses within α-Zr grains

Following the orientation and strain refinement step, the components of the stress tensor in the α-Zr matrix were determined from the S3DXRD data and compared with those from CPFE simulations. Figure 2a shows the 3D distribution of σ_xx from S3DXRD, while some other stress components are provided in the Supplementary Fig. 2. Figure 2b, c show stress distribution and phase maps of representative layers in the XZ and YZ planes, where localized stresses are observed at the boundaries containing hydrides. These boundaries are specified by the black ovals, and hydrides are shown in turquoise. Figure 2d, e show the distributions of σ_xx at four selected depths of Z = 13.5, 25.5, 58.5, and 67.5 μm. The corresponding phase maps are shown in Fig. 2f. The stresses from S3DXRD are shown in Fig. 2d, while those calculated by CPFE for the same location are provided in Fig. 2e. At Z = 13.5 μm, both S3DXRD and CPFE results show a highly localized stress field near the hydride. At Z = 25.5 μm, the tensile region becomes more localized at the tip of the hydride. At Z = 58.5 μm, tensile stress can be seen at the hydride tips (regions A and B), showing good agreement between the simulation and S3DXRD results. However, some differences are evident, particularly at a very thin boundary near the hydride, indicated by a dotted red oval, which will be discussed in the Discussion section. At Z = 67.5 μm, although no hydride is directly present, localized stress concentrations are still observed, which are attributed to the interaction of hydrides located in adjacent layers.

Fig. 2. Measured and calculated σ_xx stress fields in α-Zr grains.

a 3D σₓₓ field from S3DXRD. b Measured σₓₓ in the XZ plane at Y = 2.0 μm and Y = 25.0 μm. c Measured σₓₓ cross-sections in the YZ plane at X = 1.2 μm and X = 12.0 μm. d Measured σₓₓ maps from S3DXRD at four selected depths of Z = 13.5, 25.5, 58.5, and 67.5 μm. e The corresponding CPFE results for σₓₓ, calculated by applying the hydride transformation strain. f The corresponding measured phase maps. The Zr matrix is in yellow, and the hydrides are shown in turquoise. Source data are provided as a Source Data file.

Higher stress magnitudes are calculated by CPFE in some cases, especially where hydride interactions are present, such as in Fig. 2e at Z = 25.5 µm and Z = 58.5 µm, with values reaching up to approximately 400 MPa. For the same locations, S3DXRD results show lower stresses, around 250 MPa. This difference is mainly due to spatial averaging in an S3DXRD experiment and to the effect of the X-ray diffracting volume element (see the Discussion section). Despite the difference in magnitude, the trends in the distribution of stress are in good agreement, indicating that the hydride transformation strain governs the local stress state in both cases. These stress variations confirm that the transformation strain associated with hydride formation induces stress fields extending through multiple cross-sectional planes.

Stress distribution within δ-hydrides

Although the hydrides are thin compared with the surrounding matrix, with measured thicknesses below ~2.5 µm, it was possible to index their orientation and extract their internal stresses using S3DXRD. The interaction between precipitated hydrides and the surrounding α-Zr grains can lead to the development of internal stresses within hydrides, even in the absence of any external mechanical loads. The subsequent effects of internal stresses are reflected in the variation of measured lattice parameters, as shown in Supplementary Fig. 3. The lattice parameter fluctuations indicate non-uniform internal strains resulting from the mechanical constraint imposed by the surrounding α-Zr grains. The measured and calculated components of the elastic strain tensor for the layer at z = 28.5 μm are shown in Fig. 3, where the strain heterogeneity is clearly evident in both S3DXRD and CPFE results.

Fig. 3. Variation of elastic strain components within δ-hydrides measured by S3DXRD and calculated using CPFE.

The contours show the elastic strain tensor components for the layer at Z = 28.5 μm. S3DXRD measurements in the left panel are compared with the corresponding CPFE calculations in the right panel for each strain component. The color scale represents the magnitude of elastic strain for each component, revealing significant strain heterogeneity within the hydrides. Source data are provided as a Source Data file.

Figure 4a shows the 3D configuration of δ-hydrides, color-coded with respect to IPF-Z, revealing that the hydrides possess a range of crystallographic orientations. After conducting hydride-parent orientation analysis, the transformation strain was applied to the hydride domain in the CPFE model following the methodology proposed by Taherijam et al. ¹¹. Figure 4b presents a 3D schematic of hydrostatic stress (σ_H) within hydrides, calculated by CPFE. Figure 4c compares the hydrostatic stress distributions obtained from CPFE with those measured by S3DXRD. In both cases, the hydrides are noticeably under significant compressive hydrostatic stresses, with magnitudes exceeding 1 GPa. Such tensile or compressive hydrostatic stresses have direct effects on further hydrogen diffusion within the α-Zr matrix^11,54, which affects further hydride formation in service conditions.

Fig. 4. Stress fields within hydrides.

a Measured hydride orientations in 3D. b CPFE results for hydrostatic stresses (σ_H). c A comparison between S3DXRD-measured and CPFE-calculated hydrostatic stresses within all hydrides. Measured and calculated stress components for two different layers, d,e Z = 28.5 μm, and f,g Z = 40.5 μm. First to fourth columns respectively show σₓₓ, σ_yy, σ_zz, and ${\mathrm{\sigma }}_{\mathrm{H}}$. d,f show S3DXRD results, while e,g show CPFE results after applying hydride transformation strain. The first columns of d,f show the measured Inverse Pole Figure map along Z (IPF-Z) for the two layers. Source data are provided as a Source Data file.

Figure 4d–g present σ_xx, σ_yy, σ_zz$,$ and hydrostatic stress components within hydrides at two depths, Z = 28.5 μm and Z = 40.5 μm. The distributions of σₓ_y and σ_yz stress components are shown in Supplementary Fig. 4. As shown in their corresponding IPF–Z maps of the first column, different hydride orientations were measured. Figure 4d, f show the S3DXRD measurements, while Fig. 4e, g show the corresponding CPFE calculations. In both layers, all normal stress components within the hydrides are predominantly compressive, and hydrostatic stress exceeds 1 GPa in both experimental and numerical results- regardless of the hydride orientation. This is consistent with the previously published numerical results^11,16, and represents the first direct measurement of three-dimensional stress fields within δ-hydrides. While numerical and experimental results show the same trends, the magnitudes of stresses calculated by CPFE are generally higher, which may be attributed to spatial averaging effects in the measurement or model assumptions (see Discussion). Despite the differences, the agreement in both magnitude and compressive stress state confirms the dominant effects of hydride transformation strain.

While the presented results are obtained from a limited number of grains in the pillar, the subsequent effect of hydride transformation strain is highly local and decays within a few micrometers from the hydrides¹¹. Therefore, the measured and calculated stress fields remain representative of the local micromechanical behavior expected in the bulk material.

Discussion

Analyzing S3DXRD measured stress fields using a force equilibrium approach

To further assess the quality of the S3DXRD-measured stress fields for hydrides, a finite element method was employed that enforces mechanical force equilibrium (∇·σ = 0) to determine the state of stress within hydrides. The S3DXRD stresses were interpolated from voxel centers to the FE integration points and applied as initial stresses via the Abaqus subroutine SIGINI. A static analysis with linear elasticity, no external loads, and only constraints to remove rigid-body motion was then performed, allowing the stresses to adjust through the standard FE equilibrium iterations. Full details of the mapping and equilibrium-enforcement procedure are given in the supplementary file, in the Supplementary Discussion section and Supplementary Fig. 5. To evaluate the quality of the measured stress fields, the developed model was first used to examine whether the S3DXRD-measured stresses in a hydride-free layer satisfy force equilibrium. Supplementary Fig. 6 shows the S3DXRD stress contours at Z = 7.5 µm, and the corresponding FE result after enforcing equilibrium. It is shown that after enforcing equilibrium, 90% of voxels show stress changes of less than 20, 30, and 40 MPa for the stress components of ${\mathrm{\sigma }}_{\mathrm{xx}}$, ${\mathrm{\sigma }}_{\mathrm{yy}}$, and ${\mathrm{\sigma }}_{\mathrm{zz}}$, respectively. More importantly, the localized high-stress regions at grain boundaries are preserved. It should be noted that the deviations observed from a “perfect” match are due to the lower spatial resolution used in the Z-direction during the experiment.

Generally, if the state of stress in the surrounding medium is known, the same procedure can be applied to precipitates or unindexed regions without requiring prior knowledge of their lattice parameters or material properties. Here, the developed force-equilibrium method was subsequently used independently to determine internal stresses in hydrides, using the values measured in the surrounding α-Zr matrix. That is, the S3DXRD stresses in the α-Zr matrix were mapped to the corresponding FE mesh and applied as initial stresses, while no initial (or zero) stresses were assigned to hydride elements, which are considered as stiff elastic inclusions. During the static step, the stress field adjusts to satisfy traction force continuity at the hydride-matrix interfaces while maintaining global equilibrium, which generates the internal hydride stresses shown in Fig. 5a–c. For comparison, Fig. 5d–f show the as-measured S3DXRD hydrostatic stresses, and Fig. 5g–i present the CPFE calculation obtained by applying hydride transformation strain. All three independent approaches reveal significant compressive stresses within hydrides, exceeding 1 GPa. Local tensile regions near some interfaces (red ovals) arise from stress assignment across sharp phase boundaries and the finite voxel and element sizes. Other stress components for Z = 13.5 µm are also provided in the Supplementary Fig. 7. Overall, these comparisons indicate that the measured and calculated stress fields are consistent with each other and with the presence of substantial compressive stresses within hydrides.

Fig. 5. Assessing as-measured S3DXRD stress fields by enforcing force-equilibrium conditions.

a–c Hydrostatic stresses within hydrides determined from surrounding α-Zr matrix stresses by enforcing equilibrium, d–f corresponding as-measured S3DXRD results, and g–i CPFE results for incorporating the effects of hydride transformation strain. All three independent approaches reveal significant compressive stress within hydrides. Local tensile regions near some interfaces are indicated by red ovals and arise from stress assignment across sharp phase boundaries and from finite voxel and element sizes. Source data are provided as a Source Data file.

The effects of representative volume element

Although S3DXRD measures the spatial variation of stress fields, the measured stress magnitude was lower than that calculated by CPFE. While the numerical calculations from CPFE may serve as an upper bound limit for the calculated fields, EBSD results for GND densities yet provided another checkpoint that further confirmed the lack of sharper peaks in S3DXRD measurements. Such differences may arise due to the size of the diffracting volume, i.e., representative volume element (RVE), which affects the magnitude of measured values by averaging sharp gradients. To quantify these averaging effects, a parametric study was performed using a CPFE model with a refined mesh, which increases the sampling density and enables assessment of the impact of spatial smoothing. Figure 6a shows the CPFE-calculated stresses, computed based on hydride transformation strain, which were used as the reference solution and  artificially smoothed using the following Gaussian filter of:

$${\mathit{\sigma }}^0\left(\mathbf{x}\right)=\frac{{\sum }_i{\mathit{\sigma }}^{\mathrm{tr}}\left({\mathbf{x}}_i\right).{\omega }_i}{{\sum }_i{\omega }_i}\mathrm{where}{\omega }_i=\exp \left(-\frac{{\mid \mid \mathbf{x}-{\mathbf{x}}_i\mid \mid }^2}{r_0^2}\right)$$ 1

where ${\mathit{\sigma }}^0\left(\mathbf{x}\right)$ is the mapped stress tensor at location x, ${\mathit{\sigma }}^{\mathrm{tr}}\left({\mathbf{x}}_{\mathrm{i}}\right)$ is the calculated stress tensor at neighboring point ${\mathbf{x}}_i$ induced by hydride transformation strain, and r₀ is the smoothing radius. To understand the role of RVE, smoothing was done using radii (r₀) of 100, 50, and 30 nm, to simulate varying degrees of spatial averaging. Since an X-ray beam and step size of 200 nm were used in the experiment, numerical results for r₀ = 100 nm resemble the measurements. While the results shown in Fig. 6a have already satisfied force-equilibrium, the smoothed results may no longer be in a force-equilibrium condition. To further understand the subsequent effects of force equilibrium enforcement on the smoothed data, such as those measured in an S3DXRD experiment, force equilibrium is re-enforced on the smoothed data and results are shown in Fig. 6. This is to critically analyze the approach taken in the previous section for re-determination of stress fields.

Fig. 6. The effect of representative volume element (RVE) size and spatial averaging on stress magnitudes.

a The “exact” CPFE solution after applying hydride transformation strain (${\mathrm{\sigma }}_{\mathrm{xx}}^{\mathrm{tr}}$). b–d Smoothed stress fields (${\mathrm{\sigma }}_{\mathrm{xx}}^0$), obtained by Gaussian averaging of the CPFE results using averaging radii of r₀ = 100 nm, 50 nm, and 30 nm, respectively. Increasing the averaging radius reduces peak stress magnitudes and weakens local stress gradients, especially near hydride-matrix interfaces and hydride-hydride interaction zones, as highlighted by the black ovals in (b) and (c). e–g Corresponding stress fields after static relaxation (${\mathrm{\sigma }}_{\mathrm{xx}}^{\mathrm{eq}}$), used to enforce force equilibrium on the smoothed data. The smallest averaging radius, r₀ = 30 nm, best preserves the localized stress features observed in the exact CPFE solution while maintaining mechanical equilibrium. The hydrides are shown in gray. Source data are provided as a Source Data file.

Figure 6b–d illustrate the ${\mathrm{\sigma }}_{\mathrm{xx}}^0$ after smoothing for the different averaging radii. At the largest radius of 100 nm (Fig. 6b), stress gradients and local peaks are significantly weaker. Peak stresses drop below ~300 MPa, while critical features such as stress concentrations at hydride–matrix interfaces or at hydride–hydride interaction zones are less pronounced- see the highlighted areas in the black ovals of Fig. 6b, c. As the size of RVE decreases, the smoothed stress fields progressively approach the “exact” CPFE solution. At the smallest radius of 30 nm (Fig. 6d), localized stress features near hydrides are the same as the exact CPFE solution. Figure 6e–g display the corresponding stress fields after static relaxation (${\mathrm{\sigma }}_{\mathrm{xx}}^{\mathrm{eq}}$) for enforcing force-equilibrium. Among the three cases considered, only the smallest averaging radius preserves both mechanical equilibrium and key localized stress features. This analysis highlights the critical impact of spatial averaging on stress magnitudes and local gradients in S3DXRD measurements, even when a fine spatial resolution of 200 nm is used. Despite employing the highest achievable resolution, further advancements in S3DXRD spatial resolution and data processing are being pursued.

Overall, these results confirm that while spatial averaging reduces the magnitude of peak stresses, the measured stress fields in the zirconium matrix remain sufficiently accurate. They provide reliable boundary conditions to numerically determine internal stresses in regions with less confidence or poor diffraction data. In addition, the CPFE results of this section further confirm that the size of RVE and spatial averaging are the primary reasons for the observed underestimation of stress magnitudes in the S3DXRD results. Additional CPFE results showing the effects of averaged volume size on GND densities are provided in the Supplementary Fig. 8. It is shown that increasing the size of the averaged volume leads to a reduction in peak GND density and a smoothing of local variations.

The role of thermal residual stresses

When cooling the sample from the recrystallization temperature, thermal residual stresses will develop in zirconium grains. The grain-average stresses resulting from this process can be higher than 150 MPa⁴⁵. These stresses develop as the coefficients of thermal expansion and the elastic tensor of HCP α-Zr crystals are highly anisotropic. Following the methods previously suggested for modeling such effects in anisotropic polycrystals^55–57, these thermal residual stresses are introduced by simulating the cooling process. To isolate the effect of hydride transformation strain from thermal residual stresses, the thermal step was removed so that only the effect of the hydride transformation strain could be observed.

Figure 7 compares the stress distributions between layers with and without hydrides. In the layers at Z = 6.0 and 70.5 μm, where no hydrides are present in either the current or adjacent layers, both the experimental and numerical results for stress fields are of the same order. These experimental and numerical results are respectively shown in Fig. 7a, b, e, f. CPFE results presented in Fig. 7i, j reveal that when thermal residuals are excluded from the simulations, the stresses in these two hydride-free layers drop to nearly zero, confirming that the observed stresses originate from thermal effects. In contrast, in layers containing hydrides, e.g., Z = 25.5 and 58.5 μm, the hydride-induced transformation strain is the dominant factor determining the magnitude and sign of local stress fields. For example, a comparison of the results shown in the last two columns, i.e., Fig. 7g, h vs. Fig. 7k, l, shows that the stresses induced by hydride formation are significantly higher than those caused by thermal strains. Although considering thermal residual stresses may provide a better match with the measurements, as shown in the regions highlighted by the black ovals of Fig. 7c, g, k, the transformation-induced stresses are on the order of several hundred MPa and therefore dominate the overall stress state.

Fig. 7. The effects of thermal residual strains on the distribution of σ_xx.

The maps compare hydride-free layers far from hydrides and layers containing hydrides at different depths. The first row shows the S3DXRD-measured stress fields, the second row shows the CPFE results including both hydride-induced transformation strain and thermal residual strains, and the third row shows the CPFE results without thermal residual strains. a,b,e,f,i,j correspond to hydride-free layers at Z = 6.0 and 70.5 μm. In these layers, the stresses predicted by CPFE decrease to nearly zero when thermal residual strains are excluded, indicating that the measured stresses mainly arise from thermal effects. c,d,g,h,k,l correspond to layers containing hydrides at Z = 25.5 and 58.5 μm. In these layers, the transformation-induced stresses are on the order of several hundred MPa and dominate the overall stress state. The black ovals highlight regions where including thermal residual strains improves the agreement between the S3DXRD measurements and CPFE predictions. Source data are provided as a Source Data file.

It is noteworthy that, in the Results section, $\mathrm{\Delta }{\mathit{\epsilon }}^{\mathbf{th}}$ is first applied to mimic the quasi-static cooling to room temperature, followed by the hydride transformation strain applied at the same temperature. The modeling can be performed using other sequences, too. For example, the hydride transformation strain can be applied at a high temperature, followed by cooling to room temperature. The resulting stress components for a representative layer (Z = 28.5 μm) are shown in the Supplementary Fig. 9. The results indicate that although changing the transformation path introduces minor changes in the local stresses, the overall trends remain the same.

The shear components of hydride transformation strain

The transformation from α-Zr (HCP) to δ-hydride (FCC) requires a change in the close-packed stacking from ABAB to either ABCABC or the reversed sequence CBACBA. This change occurs through shear associated with basal partial dislocations with Burgers vectors of the type $\frac{1}{3}⟨10\overline{1}0⟩$^58–60. Six such directions exist in the basal plane, forming three pairs separated by 120°. The transformation proceeds through shear on every second basal plane. As illustrated in Supplementary Fig. 10, shear along three of these directions produces the ABCABC sequence, whereas shear along the opposite directions produces CBACBA. These two stacking sequences correspond to two FCC orientations that can originate from a single HCP parent grain, following the orientation relationship ${\left\{0002\right\}}_{\text{HCP}}\parallel {\left\{111\right\}}_{\text{FCC}}$ and $⟨11\overline{2}0{⟩}_{\text{HCP}}\parallel ⟨1\overline{1}0{⟩}_{\text{FCC}}$.

Previous studies report that intragranular hydrides form on a habit plane slightly inclined to the basal plane, where the stress field during hydride formation may favor one of the three partial-dislocation directions and promote autocatalytic growth^60–64. However, some simulations suggest that shear may occur along all three directions, resulting in little or no net shear at the hydride length scale^65–67. In the present study, the hydrides precipitate along grain boundaries, where the boundary plane controls the hydride morphology rather than a crystallographic habit plane. Under these conditions, the transformation may involve a mixture of the three possible shear directions within a hydride, whose relative contributions cannot be resolved from the current experimental measurements. To assess the influence of this effect on the predicted stress fields, additional CPFE simulations were performed, including a shear component in the transformation strain. Figure 8 compares the S3DXRD-measured strain components with CPFE calculations obtained with and without the shear term for a representative hydride. For simulations that include shear, the three possible shear directions associated with the observed orientation relationship are considered. The corresponding shear components of the transformation strain are taken from previous works^64,67 and applied to the hydride domain. It is evident that including the shear term affects the distribution of elastic strain within the hydrides but does not improve the agreement with the S3DXRD-measured strain components. In fact, simulations without the shear component show better agreement with the experimental ones. Another example is shown in Supplementary Fig. 11, where a better agreement is achieved when only the normal components of the misfit strain are considered. These results indicate that, for hydrides of the size examined here and located at grain boundaries, the transformation strain can be effectively represented by the normal components of the misfit strain. The good agreement obtained without the shear term suggests that contributions from different shear directions at the atomistic scale are averaged out at the length scale considered in the present study, resulting in an effective shear close to zero.

Fig. 8. Effect of shear component of the hydride transformation strain.

a The IPF-X map of hydride H2 for the layer at Z = 28.5 μm, and the corresponding crystallographic orientation relationship (COR) between the hydride and its parent grain (G3). Strain tensor components measured by S3DXRD are shown in (b) and compared with CPFE results obtained by applying the hydride transformation strain considering c only the normal components and d both normal and shear components in the three possible directions ($\mathrm{b}=\frac{1}{3}⟨10\overline{1}0⟩$). Check marks and crosses indicate whether the predicted elastic strain distribution is in agreement with the S3DXRD measurements. Source data are provided as a Source Data file.

In this study, S3DXRD was used to investigate stress development in a specimen containing α-Zr grains and δ-hydride precipitates. The measurements were done at 200 nm spatial resolution to resolve small hydride grains. The measured orientations and morphologies of α-grains and hydrides were used to construct a crystal plasticity finite element model to simulate the stresses induced by hydride precipitation. Both numerical and experimental measurements confirm the presence of geometrically necessary dislocation densities induced in the surrounding α-Zr grains by the precipitation of hydrides. Further, upon precipitation and in the absence of external mechanical loads, hydrides are shown to be subject to significant compressive internal stresses. The magnitude of normal or hydrostatic stress within hydrides exceeds 1 GPa. The high stresses within hydrides are counterbalanced by highly localized stresses that develop within the surrounding α-Zr grains, in the vicinity of hydrides.

Two independent finite element methods were used. In the first method, CPFE modeling showed that localized stresses develop due to the transformation strain accompanied by hydride precipitation. In the second method, measured stresses from α-Zr grains were used to enforce force equilibrium and determine the state of stress within hydrides. The second method confirmed the magnitude and sign of measured stresses, further reinforcing that hydrides are stressed upon precipitation. It was further shown that while thermal residual stresses are non-negligible, they are locally overcome by the high stresses induced by hydride precipitation.

Comparisons between S3DXRD measurements and CPFE simulations showed that, for the grain-boundary hydrides examined in this study, the net macroscopic shear component of the transformation strain is negligible at the length scale used. Accordingly, the transformation strain can be adequately represented by the normal components alone.

Methods

Experimental setup

A Zircaloy-2 plate with chemical composition comprising 1.43–1.45 wt.% Sn, 0.13–0.14 wt.% Fe, 0.10 wt.% Cr, 0.05 wt.% Ni, 1400 ppm O, and 150–160 wt. ppm C was heat treated at 837 K and subsequently air-cooled to produce an average grain size of ~50 μm. Hydrogen charging was conducted cathodically in a 0.2 mol solution of H₂SO₄ at 353 ± 5 K for 48 h. The specimen was then homogenized at 607 K for 40 h to achieve a nominal hydrogen concentration of approximately 100 wt. ppm, followed by cooling at a rate of 1 K/min⁵⁴. Post-charging, the specimen was mechanically polished down to 1200 grit, followed by polishing with 1 μm Alumina suspension. Final surface preparation was performed via electro-polishing in a − 30 °C solution of 10% perchloric acid in methanol at 25 V for 60 s. Using an Oxford instrument, EBSD mapping with a step size of 1 μm was conducted to identify regions of interest for micropillar fabrication. The EBSD data were post-processed using MTEX to analyze the crystallographic orientations and Euler angles⁶⁸. Figure 9a presents the inverse pole figure map along the Z’-direction (IPF-Z’) in the EBSD specimen coordinate system, where a δ-hydride located at a triple point was selected for further analysis. A micropillar with dimensions of 35 × 31 × 77 μm was fabricated at this location using a Helios G4 plasma focused ion beam (PFIB), as shown in Fig. 9a.

Fig. 9. The experiment and model set up.

a IPFZ’ of the Zircaloy-2 specimen and SEM image of the micropillar, b A schematic of the S3DXRD measurement setup, c reconstructed 3D grain shapes. d EBSD measured IPF-X maps (in global coordinate system) for surfaces S1 and S2. e CPFE input model and boundary conditions. f IPF-X for an example layer measured by S3DXRD showing both α-Zr and δ-hydride grains and g the corresponding CPFE model and mesh. X, Y, and Z are  the global coordinate system. Source data are provided as a Source Data file.

Scanning three-dimensional X-ray diffraction measurements were carried out at beamline ID11 of the European Synchrotron Radiation Facility (ESRF). A monochromatic X-ray beam with an energy of 65.35 keV and a spot size of 200 nm was focused onto the micropillar. An Eiger2 detector with pixel size of 75 µm was used, which was positioned at a fixed distance of 138.4 mm from the sample. For each of the 48 layers spaced by 1.5 μm along the Z-axis, the specimen was rotated 360° in 0.05° increments, yielding 7200 diffraction frames per layer with 2 ms exposure time per frame. Between each full rotation, the stage was translated by 200 nm in the Y-direction to collect diffraction patterns for each layer. A schematic of the setup is provided in Fig. 9b. The collected diffraction data for each layer were used to determine grain parameters and stress fields for each layer, as described below. These layers were stitched to make the 3D grain map in Fig. 9c. The ImageD11 program⁵⁰ was used to extract the elastic strain fields from the collected diffraction data. After peak segmentation, mask reconstruction, and grain indexing, orientation and strain refinement were performed⁶⁹. From these results, grain morphologies, local misorientation maps, and strain distributions were extracted in the global coordinate system³⁷. The global stress tensor was calculated using ${\mathit{\sigma }}^{\mathbf{global}}=\tilde{\mathbf{C}}{\mathit{ϵ}}^{\mathbf{global}}$ where $\tilde{\mathbf{C}}$ is the fourth-rank stiffness tensor in the global coordinate system and is obtained from the transformation ${\tilde{C}}_{rsmn}=R_{rp}{R}_{sq}{R}_{mi}{R}_{nj}{C}_{pqij}$. Here, $\mathbf{R}$ represents the rotation matrix derived from the crystal orientation, and $\mathbf{C}$ is the stiffness tensor in crystal coordinates. A detailed description of the S3DXRD analysis is available elsewhere³⁷ and in the Supplementary Methods section. While all stresses in this work are reported in the global coordinate system, they can also be expressed in other reference frames, such as the parent grain coordinate system of each hydride. As an example, the stress field was evaluated in the parent coordinate systems for a representative layer at Z = 28.5 μm, and the corresponding results are shown in Supplementary Fig. 12.

Grain indexing

Details of grain indexing procedures for α-Zr and δ-hydrides are provided in the supplementary file, in the Supplementary Methods section and Supplementary Figs. 13, 14. The measured diffraction peaks were used to determine lattice parameters of a = 3.2327 Å, c = 5.1471 Å for HCP α-Zr grains and a₀ = 4.779 Å for FCC δ-hydrides. In a few regions, secondary intermetallic phases such as FeNiZr₄ and ZrCrFe were also detected, primarily at grain boundaries. These phases were submicron in size and present in only a small number of layers, with most layers containing only α-Zr and δ-hydride. The identified phases are consistent with known second-phase particles in Zircaloy-2^70,71. Supplementary Fig. 15a shows the IPF-X map at Z = 12 μm, where the concentrations of other phases were highest. The corresponding phase map is provided in Supplementary Fig. 15b. To further confirm the presence of these phases, wavelength-dispersive spectroscopy (WDS) analysis was performed in the SEM on the same bulk in regions adjacent to the micropillar. WDS results from neighboring regions (Supplementary Fig. 15c, d) confirm the compositions of the precipitates. In Supplementary Fig. 15c, the WDS-detected Cr, Fe, and Ni elements are consistent with the presence of Zr-Fe-Ni particles. Supplementary Fig. 15d confirms Zr-Cr-Fe particles through the co-localization of Cr and Fe. These particles, however, are not included in the CPFE model because their size is very small compared to that of the hydrides.

Crystal plasticity modeling

Following grain indexing of α-Zr and δ-hydride phases, the 3D measured microstructure was imported into a crystal plasticity finite element model. Any area with fewer than two measurement points was removed and merged with the neighboring α-Zr or δ-hydride. Points that were not indexed, for example at some grain boundaries, were also assigned to the closest indexed grain. The model comprises 48 layers, each 1.5 μm thick, connected through tie constraints along the Z-direction. Grain boundaries within the X-Y plane were imported, and a refined mesh was applied. Each layer contains two elements through the thickness, with a biased mesh used towards grain boundaries and hydrides, resulting in a total of over 600,000 C3D6 elements. Figure 9d shows the IPF-X maps obtained from EBSD conducted on two orthogonal surfaces of the pillar, labeled S1 and S2. These surfaces were used to approximate the 3D morphology of the pillar and to compare EBSD-measured orientations with those from S3DXRD. Figure 9e shows the full CPFE model built from the S3DXRD data with assigned grain orientations for α-Zr and δ-hydride. The top and side surfaces were traction-free. The bottom was fixed in the Z direction, with edges normal to X fixed in the X-direction and edges normal to Y fixed in the Y-direction to prevent rigid body motions. An example layer from Z = 40.5 μm is shown in Fig. 9f, which has both α-Zr and hydride phases. The corresponding CPFE model and mesh for this layer are provided in Fig. 9g. The crystallographic relation between each hydride and its surrounding α-Zr grains was examined, and the parent grain for each hydride was identified accordingly. An example is shown in Fig. 9f where one of the hydrides is highlighted with a red oval and its α-Zr parent denoted as P. This relationship was used to apply the appropriate transformation strain tensor to the hydride domain.

A User MATerial (UMAT) subroutine⁷² was used to calculate stress increments and update the solution-dependent state variables based on the inputs provided by the FE solver, i.e., strain, rotation, and time increments. The total strain increment ($\mathit{△}\mathit{\epsilon }$) can be written as:

$$\mathrm{\Delta }\mathit{\epsilon }=\mathrm{\Delta }{\mathit{\epsilon }}^{\mathbf{el}}+\mathrm{\Delta }{\mathit{\epsilon }}^{\mathbf{pl}}+\mathrm{\Delta }{\mathit{\epsilon }}^{\mathbf{trH}}+\mathrm{\Delta }{\mathit{\epsilon }}^{\mathbf{th}}$$ 2

where $\mathit{\Delta }{\mathit{\epsilon }}^{\mathbf{el}}$, $\mathit{\Delta }{\mathit{\epsilon }}^{\mathbf{pl}}$, $\mathit{\Delta }{\mathit{\epsilon }}^{\mathbf{trH}}$, and $\mathit{\Delta }{\mathit{\epsilon }}^{\mathbf{th}}$ denote the elastic, plastic (due to crystallographic slip), transformation, and thermal strain increments, respectively. Previous studies have shown that the transformation from α-Zr (HCP) to δ-hydride (FCC) involves a change in the close-packed stacking sequence. This transformation can be described by a shear on the basal plane together with normal strain associated with the change in interplanar spacing ^60–64. Accordingly, the transformation strain tensor may include both shear and normal components. At the atomic scale, the stacking change can be accommodated through the glide of basal Shockley partial dislocations along three symmetry-related directions in the basal plane. In a hydride composed of many atomic layers, a combination of these shear directions may occur, which can reduce the net macroscopic shear at the experimental length scale. In the present work, formulations with and without shear components were evaluated. Comparison with the measured elastic strain within the hydrides showed that the formulation including only the normal misfit components provides better agreement with the S3DXRD measurements for the hydrides studied here, as discussed in the Discussion section. Therefore, the CPFE results presented in this paper are based on a transformation strain tensor containing only the normal misfit components, consistent with the original work by Carpenter⁷³. The components of this unconstrained misfit strain associated with the formation of δ-hydrides (${\mathit{\epsilon }}^{\mathbf{trH}}$) were calculated to be 4.58% and 7.2% along the HCP crystal a-axis and c-axis, respectively⁷³.

Following the methodology explained in Taherijam et al.¹¹, this transformation strain was applied incrementally and slowly to mimic the quasi-static condition of the domain that undergoes phase transformation¹¹. Under these conditions, and for evaluating the final GND distribution, residual stress state, and stress localization trends, the implementation path of the transformation strain does not influence the conclusions of this study. This is demonstrated in Supplementary Fig. 16, where the hydride transformation strain was applied progressively from the parent grain along two different paths, resulting in no observable differences in the GND distribution. The increment of thermal residual strain is defined as $\mathit{\Delta }{\mathit{\epsilon }}^{\mathbf{th}}=\mathit{\alpha }\mathrm{\Delta }\mathrm{T}$, where $\mathit{\alpha }$ is the thermal expansion tensor and $\mathrm{\Delta }\mathrm{T}$ is the temperature increment. For HCP zirconium, $\mathit{\alpha }$ is 5.7 × 10⁻⁶ °C⁻¹ along the a-axis and 10.3 × 10⁻⁶ °C⁻¹ along the c-axis⁷⁴. The shear components of hydride transformation strain were only included to produce the results presented in Fig. 8 and Supplementary Fig. 11.

The crystal plasticity equations used for calculating plastic strain increments are provided in the supplementary file. The material properties and CPFE parameters for both Zircaloy-2 and hydrides are provided in the Supplementary Table 1^75,76. Measured grain average orientations were assigned to each grain of the CPFE model, but these orientations were updated during the steps when thermal strain or transformation strain was applied. The updated orientations were used to calculate GND densities.

GND densities

Geometrically necessary dislocation densities were calculated using lattice curvatures derived from the Nye tensor⁷⁷. The tensor ${\alpha }_{ij}$ is defined as: ${\alpha }_{ij}={\kappa }_{ji}-{\delta }_{ji}{\kappa }_{mm}-{\epsilon }_{jlk}\frac{\partial {ϵ}_{ik}^{el}}{\partial x_l}$ and can also be written as:

$${\alpha }_{ij}={\sum }_{t=1}^N{b}_i^t{l}_j^t{\rho }^t\mathrm{where}i,j=1,2,3$$ 3

Here, ${\kappa }_{ji}=\frac{\partial {\theta }_j}{\partial x_i}$ represents the lattice curvature, ${\theta }_j$ is the lattice rotation vector, ${\delta }_{ji}$ is the Kronecker delta, ${\epsilon }_{jlk}$ is the permutation tensor, and ${ϵ}_{ik}^{el}$ is the elastic strain tensor component, which is either measured from S3DXRD or calculated from CPFE simulations. For GND densities obtained from EBSD, the elastic strain gradient term is not available and is therefore not included. In this work, only the total GND density is reported, i.e. ${\rho }_{GND}={\sum }_{t=1}^N{\rho }^t$. The same calculation procedure was applied to both CPFE and S3DXRD datasets to ensure consistency in GND density evaluation. The 3D data sets were used to fully resolve the Nye tensor, following the methodology explained in Witzen et al.⁷⁷. Due to using a biased mesh density, a binning method was used in CPFE results so that spatial step sizes can be consistent with the ones used in the S3DXRD experiment.

Author contributions

H.A. and J.P.W. designed and planned the experiment. J.P.W., H.A. and A.T. conducted the experiment and collected the diffraction data. S.M., J.P.W., A.H., J.B., S.H., A.T. and B.K. contributed to the experimental data processing. M.T., S.M. and H.A. contributed to numerical modeling. H.A. and A.T. measured EBSD maps. S.M. and M.T. prepared the first draft while all authors contributed to reviewing the draft. H.A. supervised, acquired funding, and led the research.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.

Funding

This work was supported by a Discovery Grant (RGPIN- 2022-02955) as well as Alliance Grants (ALLRP 588372-23, and ALLRP 601891-24) from the Natural Sciences and Engineering Research Council of Canada (NSERC). In addition, this research was undertaken, in part, thanks to funding from the Canada Research Chairs Program.

Data availability

The raw S3DXRD data generated in this study have been deposited in the ESRF data repository at https://doi.org/10.15151/ESRF-DC-2476816001. The processed S3DXRD data and numerical results generated in this study have been deposited in Zenodo under accession code https://doi.org/10.5281/zenodo.21365418⁷⁸. The data underlying the main and Supplementary Figs. are provided in the Source Data file. Source data are provided with this paper.

Code availability

The computer codes used for processing the diffraction data are available on GitHub: https://github.com/FABLE-3DXRD/ImageD11 and have been archived on Zenodo under accession code https://doi.org/10.5281/zenodo.21398853. The finite element subroutines used for enforcing force equilibrium have been deposited in Zenodo under accession code: https://doi.org/10.5281/zenodo.21365418.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41467-026-77243-3.