1 Introduction
Braggedge neutron transmission techniques provide a means for obtaining lower dimensional (one or twodimensional) tomographic strain images from higher dimensional (two or threedimensional) strain fields within polycrystalline materials. The success of these techniques, and the development of associated instruments and detectors has prompted research into the tomographic reconstruction of strain — i.e. strain tomography. This research aims to provide methods analogous to conventional Computed Tomography (CT) whereby the complete triaxial strain distribution within a sample could be reconstructed from a sufficient set of strain images. As the strain field is a tensor, this is a more complex task than conventional scalar CT.
Tomographic reconstruction of strain fields from these images can be used to study the residual elastic strain (and hence stress) within engineering components. Residual stresses are those which remain after applied loads are removed (e.g. due to heat treatment, or plastic deformation), and may have a significant and unintended impact on a component’s effective strength and service life (in particular fatigue life). A full field analysis of these stress and strains could have a significant impact on several areas of experimental mechanics. In particular, it could be used to study the residual stress within additively manufactured, laserclad, preened, welded, cast, forged and/or otherwise processed components. This full field analysis will have significant advantages over current destructive and semidestructive techniques (Prime, 2001; ASTM Standard, ), and pointwise xray and neutron diffraction methods (Hauk, 1997; Noyan and Cohen, 1987; Fitzpatrick and Lodini, 2003).
Strain tomography falls into the class of ‘rich’ tomography problems where the tomographic strain image is related to an unknown tensor field. The acquisition and analysis of these strain images is described in detail elsewhere (Santisteban et al., 2002a, 2001, b). Briefly summarising the process: the relative transmission of neutron pulses with known wavelengthintensity spectra through the sample is measured at a pulsed neutron source (e.g. the Japan Proton Accelerator Research Complex (JPARC) in Japan, ISIS in the United Kingdom, or the Spallation Neutron Source in the USA). For example, current stateoftheart MicroChannel Plate detectors (Tremsin et al., 2011) are capable of measuring the transmitted spectra over an array of pixels with a resolution of . From this data, the position of a given Braggedge (a sudden increase in transmitted intensity as a function of wavelength) is observed. The relative position of a Braggedge provides a measure of strain at each pixel of the form
(1) 
where is the wavelength at which the Braggedge occurs, is the corresponding Braggedge wavelength in a stressfree sample, and the following applies:

As with all diffraction measurements, only the elastic component of strain is measured.

The measured strain is the normal component in the direction of the transmitted neutron beam.

The measurement corresponds to a throughthickness average along the path of the corresponding ray.
The strain measured at each pixel can be related to the strain field using the Longitudinal Ray Transform (LRT) (Abbey et al., 2009; Lionheart and Withers, 2015). With the inclusion of measurement error this gives a measurement model as
(2) 
where the LRT geometry is defined in Figure 1 and
is the measurement error term, which is assumed to be zeromean Gaussian with standard deviation
. Estimating the strain field given a set of LRT measurements is made more complex as the LRT mapping is noninjective
(Lionheart and Withers, 2015; Sharafutdinov, 1994). This means that if the strain field components are considered independent, then infinitely many fields could produce the same set of measurements.This problem has been overcome in a number of ways, allowing the successful reconstruction of twodimensional strain fields. Several special cases have been considered including axisymmetric systems (Abbey et al., 2009, 2012; Kirkwood et al., 2015; Gregg et al., 2017) and granular systems (Wensrich et al., 2016a). More arbitrary strain fields caused by insitu loadings have been successfully reconstructed by assuming compatibility (Wensrich et al., 2016b; Hendriks et al., 2017). Methods for reconstructing a broader class of strain fields (i.e. residual strains) have been developed by ensuring the strain field satisfies equilibrium (Gregg et al., 2018; Jidling et al., 2018; Hendriks et al., in press).
Recently, an approach suitable for modelling and estimating threedimensional strain fields has been presented for simulated data (Hendriks et al., 2019). While this method deals with the reconstruction of strain from highenergy xray diffraction measurements, it requires only limited modification to be applied to the problem presented here.
1.1 Contribution
This paper details the first experimental demonstration of triaxial strain field reconstruction using Braggedge neutron transmission strain imaging. These strain images were acquired during an experiment at JPARC (January 2019) of a hollow EN26 steel cube sample loaded by an interference fit with a titanium plug. The reconstruction approach follows the method presented in Hendriks et al. (2019) which was previously demonstrated on simulated highenergy xray strain measurements. This method is adapted for the LRT measurement model and to include boundary conditions. The resulting reconstruction is compared with conventional constantwavelength strain measurements performed with the KOWARI engineering diffractometer at the Australian Nuclear Science and Technology Organisation (ANSTO) as well as a Finite Element Analysis (FEA) model.
2 Reconstruction Approach
In this paper, we present the first practical demonstration of a method capable of reconstructing the full triaxial strain field from Braggedge neutron transmission measurements. The reconstruction approach is modified from the method presented by Hendriks et al. (2019); which was used to reconstruct 3D strain field from simulated highenergy xray measurements. This approach models the strain field by a Gaussian process (Rasmussen and Williams (2006) provides a good introduction), and ensures that the reconstructed strain field satisfies the physical properties of equilibrium; assuming the sample to be linearly elastic and isotropic (i.e. without texture). Ensuring the strain field satisfies equilibrium is critical as the LRT mapping (2) has a nontrivial null space (Lionheart and Withers, 2015) (i.e. without these constraints a unique solution to the inverse problem does not exist). By enforcing equilibrium the null space is reduced to contain only the trivial field, allowing a unique solution to be (Hendriks et al., in press).
Gaussian processes are suitable for the modelling and estimation of spatially correlated phenomena. The use of Gaussian processes to model and estimate strain fields was pioneered in Jidling et al. (2018). By modelling the Airy stress function (a scalar potential field) by a Gaussian process a solution to the twodimensional stress (and hence strain) could be given that satisfies equilibrium in the absence of body forces. This method can be extended to three dimensions for which the Beltrami stress functions are used instead of the Airy stress function. The Beltrami stress functions consist of six unique potential fields from which a complete solution to the equilibrium equations in threedimensions can be given (Sadd, 2009). These potential fields are each modelled by a Gaussian process allowing a triaxial strain field satisfying equilibrium to be reconstructed.
In addition to equilibrium, boundary conditions can also be included following the work by Hendriks et al. (in press). Knowledge of unloaded surfaces for which the distribution of applied forces (known as tractions) is known to be zero can be incorporated. This can be done by including artificial measurements of zero traction at points on the surface not subject to an applied load.
A detailed description of the method and its implementation is given by Hendriks et al. (2019). This requires only minor modification to the measurement model, and the inclusion of traction measurements. The measurement model requires a slight modification due to the difference between highenergy xray and neutron transmission strain measurements. For highenergy xray strain measurements, the measured strain direction, denoted , is almost perpendicular to the ray direction . Whereas, for neutron transmission strain measurements the direction of measured strain is aligned with the ray, and so . Additionally, a large variation in strain measurement uncertainty is observed and therefore the method is modified so that each measurement can be assigned its own standard deviation. In essence this weights the importance of each measurement according to it’s confidence given by the strain imaging process.
In this paper, this approach is used to reconstruct the strain field within an EN26 steel sample from a set of strain images measured at JPARC. The sample, load case, and strain image acquisition are described in Section 3.1. The resulting strain field is validated by comparison with conventional diffraction strain scans and FEA results, which are described in Section 3.2. The reconstructed strain field and a comparison with the validation data is given in Section 3.3; potential sources of error are also discussed in this section.
3 Demonstration
3.1 Sample Design and Strain Imaging
The method is applied to a real world example using a data set collected on the RADEN energyresolvedneutronimaging instrument at JPARC (Shinohara and Kai, 2015; Shinohara et al., 2016). The experiment tomographically imaged the strain within an EN26 steel (medium carbon, low alloy) sample. The sample consisted of a steel cube with a precision ground hole of diameter along the diagonal. A load was applied by a interference fit (i.e. shrink fit) with a titanium plug. The sample and plug are shown in Figure 2.
Prior to assembling, the sample was heat treated to relieve stress and provide a uniform temperedmartensite (i.e. ferritic) structure with minimal texture, and with a final hardness of . The sample was assembled by first inserting the plug into a cylinder with an interference fit of , after which a cube with sides was milled from the cylinder and plug.
This sample and loading setup was designed to provide a smooth threedimensional strain field suitable for the first demonstration of threedimensional strain tomography. This has additional advantages for validation as FEA of strain resulting from interference fits is a more straight forward process than FEA of strain fields resulting from plastic deformation.
To this end, a titanium plug was chosen as strain within the plug was ‘invisible’ to the strain imaging process. This is because titanium does not have a Braggedge near to the steel Braggedge that was chosen for analysis. As a result, the titanium plug can be ignored during the reconstruction process and considered as imparting an insitu load on the interior face of the hollow cube.
Strain images were measured on the RADEN instrument using a microchannel plate detector ( pixels, per pixel)(Tremsin et al., 2011) at a distance of from the source. At the time of the experiment (January 2019) the source power was . Pixels were grouped together into macro pixels of giving sufficient neutron counts to provide a reasonable edge fit; giving a final strain image of macro pixels, each with a resolution of . It is important to note that this does not directly correspond to the resolution of the final reconstruction. The Gaussian process method used reconstructs a continuous (smooth) strain field where the maximum rate of change is automatically adapted to suit the available data.
Each macro pixel provides a strain measurement of the form (1) where the Braggedge position was found following the procedure given by Santisteban et al. (2001) applied to the (110) Braggedge. The undeformed location was determined from a stressfree sample. The resulting strain measurements had, on average, an uncertainty of standard deviation . This uncertainty is higher than previously achieved in twodimensional strain tomography experiments (Hendriks et al., 2017; Gregg et al., 2018), however further increasing the macro pixel size did not provide a sufficient decrease in uncertainty to warrant the loss in strain image resolution. A systematic bias in the edge fit as a function of the measurement path length was observed. This effect was previously described in Vogel (2000) and Gregg et al. (2018); although the exact mechanism is yet to be established. Following Gregg et al. (2018), a linear correction was applied to .
In total 70 tomographic strain images were collected. For these images the sample was positioned using a two axis goniometer, which allowed rotation in azimuth and elevation. Limitations of the experimental setup restricted the achievable elevation angles to the range from to . Therefore, in order to cover the full range of measurement directions, the sample was repositioned by rotating about the axis for the final 11 strain images.
The exact geometry of each neutron ray passing through the sample before reaching the detector is required to model the measurements by the LRT (2). In addition to designing a sample holder to carefully position the sample, an optimisation routine was run to determine the remaining orientation offsets and the offsets between the centre of rotations. The optimisation maximised the sum of Braggedge heights associated with rays that would pass through the sample for a given choice of offsets.Gregg et al. (2018)
Strain fields were reconstructed from this set of strain images using the Gaussian process method described earlier with the inclusion of 400 measurements of zero traction evenly distributed on each of the exterior faces.
3.2 Validation Data
Validation relies on comparison with conventional strain scans (Kisi and Howard, 2012; Fitzpatrick and Lodini, 2003; Noyan and Cohen, 1987) from the KOWARI constantwavelength strain diffractometer at the Australian Nuclear Science and Technology Organisation (Kirstein et al., 2009; Brule and Kirstein, 2006)
and Finite Element Analysis (FEA). The strain scans provide measurements of the 6 components of strain on two section planes (33 points on plane 1 and 45 points on plane 2). As with all diffraction methods these measurements correspond to the average strain inside gauge volumes. These gauge volume locations were chosen on two section planes that were expected to exhibit strong skew symmetry and therefore help to validate a larger region of the reconstruction. The section planes and gauge volume locations are shown in Figure
2.These measurements were based on the relative shift of the (211) diffraction peak measured with neutrons of wavelength ( geometry) and a gauge volume. The and lattice planes have effectively the same diffraction elastic constants (Daymond and Priesmeyer, 2002), therefore the results from the transmission and diffraction experiments can be directly compared without rescaling or recalculation to stress. Sampling times with the KOWARI diffractometer were based on providing uncertainty in strain around : which required 60 hours. Together with sample setup and alignment, a total of 4 days of beam time was required. The long sampling times required for the comparatively small gauge volumes meant that only a portion of each section plane could be measured.
While only a small amount of the strain field can be verified using the strain scanning measurements, the full reconstructed strain field can be compared to FEA results to provide further validation. Comparison of the reconstruction to the FEA results is made for the three section planes shown in Figure 2.
3.3 Results and Discussion
The reconstructed strain field is shown together with strain scans made on the KOWARI diffractometer and FEA results in Figures 4 and 5. As the KOWARI strain scans cover only a region of section plane 1 and 2, comparison between all three sets is made in these regions only. Example Matlab code to perform this reconstruction and produce the strain plots shown in this paper is available on Github (Hendriks, 2019).
Comparison of the KOWARI strain scans, the RADEN reconstruction and the FEA results shows good agreement in general. However, some specific differences can be noted:

In the component of plane 1, the reconstruction shows a region of tension in the bottom right that is also present in the strain scans but is not seen in the FEA.

In the and components of plane 1, the reconstructed strain has the same shape but a greater magnitude than the FEA strain and it is not clear whether this is supported in the strain scans.

In the component of plane 2, the reconstructed strain field has a region of compression also present in the FEA but not seen in the strain scans.
When making these comparison it is important to remember that the KOWARI strain scans are themselves measurements which are relatively noisy ( standard deviation) and are averages over the gauge volumes; this in some cases makes comparison difficult. In particular, the and components of plane 1, and the and components of plane 2 appear particularly noisy.
The FEA strain fields and the reconstructed strain fields are available for the entire sample allowing comparison over the entire section planes 1, 2, and 3; these are shown in Figure 6, Figure 7 and Figure 8 respectively. A selection of strain components is shown for each section plane so that all components are shown at least once. This comparison shows that the reconstruction shows close agreement with the FEA strain fields on plane 2 and plane 3. The shape of the reconstructed strain field is very similar with the main difference being noted in slightly reduced magnitudes and the peak strains are less concentrated. There is a slightly greater difference between the reconstruction and FEA results observable in plane 1; particularly in the and components. However, at least for the component some of this difference is supported by the KOWARI strain scans which also showed the region of tension present in the bottom right. Additionally, as the strain fields are skew symmetric this would lend some support to the region of tension present in the top left of this component.
From these results a quantitative assessment of the discrepancies between the the KOWARI strain scan measurements and the reconstruction as well as between the reconstruction and the FEA results was carried out. In both cases the differences are mean zero and Gaussian, implying that there is no systematic error or bias resulting from the reconstruction technique. The differences between the KOWARI strain scans and the reconstruction are calculated by taking gauge volume style averages of the reconstructed strain field and comparing these to the KOWARI measurements. The resulting differences have an average magnitude of . The discrepancies between the FEA and reconstructed strain fields have an average magnitude of .
Although the reconstruction shows, in general, good agreement with the FEA results and KOWARI strain scans some differences have been noted and these differences may be attributed to several sources. Firstly, the number and quality of strain measurements acquired is less than has been previously achieved in twodimensional strain tomography experiments. In total, 14000 strain measurements were acquired with an average standard deviation of compared to approximately with a standard deviation of in previous twodimensional experiments (Hendriks et al., 2017; Gregg et al., 2018).
This is in part due to the trade off between macro pixel size and uncertainty in the Braggedge fits. While smaller macro pixels ( pixels) would give better resolution in the strain image, the measurement standard deviations would be increased to around . Conversely, larger macro pixels () could be used to decrease the measurement standard deviation to around , however the resolution in the strain image would be made worse. This is in contrast to twodimensional geometry where the assumption of no out of plane strain variation meant the pixels could be binned into columns without affecting the resolution of the resulting onedimensional strain image.
The effect of this is particularly significant in regions near the sample boundary – i.e. in the corners where the plug and sample intersect. These regions contain a small amount of material and so the averaging effect of the LRT means that the strains are poorly sampled. Additionally, the smaller amount of material means that the Braggedge height of any strain measurements passing predominately through these regions is reduced, resulting in poorer measurement confidence. When combined with the macro pixel averaging, this results in the peak strains in these regions being somewhat obscured. This issue may be somewhat alleviated in future experiments at JPARC with an expected source power increase to over the next few years.
Differences between reality and the FEA model may also account for some of the observed discrepancies. The sample was milled into a cube from a typical ring and plug. During this process it was not possible to ensure the plug was perfectly on the diagonal of the cube. This would account for the measured strain fields (KOWARI and RADEN) not being entirely skew symmetric and for some of the difference between these fields and the FEA strain field. Additionally, the milling process itself may have introduced residual stresses not accounted for in the FEA model. Finally, the peak stresses are around the yield strength of the material which could result in effects such as hardening and account for differences between the measured strain fields and the FEA strain field.
4 Conclusion and Future Work
This paper has presented the first threedimensional fullfield strain reconstruction from neutron transmission strain images. Strain images were collected using the RADEN energyresolvedneutronimaging instrument at JPARC. The reconstructed strain field was validated by comparison with conventional strain scans from the KOWARI diffractometer and FEA results and shows very promising results.
The reconstruction was performed using a Gaussian process based method that ensures the resulting strain field satisfies equilibrium. This is achieved by using the Beltrami stress functions to provide a complete solution to the stress (and strain) fields in threedimensions. Such an approach provides a continuous strain field throughout the entire sample, whereas conventional strain scanning methods may struggle to cover a reasonable portion of a sample considering the implication of small gauge volumes required for threedimensional geometry.
The reconstructed strain field was developed within a hollowed EN26 cube by an ‘insitu’ loading created by interference fitting a titanium plug. Although this strain field is compatible, the method is applicable to a broader class of problems (i.e. residual strains) as it makes no assumption of compatibility. To this end, future work involves the planning of a threedimensional residual strain experiment. This could also involve adapting the Gaussian process model to be more suitable for strain fields exhibiting rapid changes or discontinuities.
Additionally, it was noted that the strain measurements were fewer and of poorer quality than what has been previously achieved and that this may have affected the accuracy of the reconstruction. Although this issue may be somewhat alleviated by an increase in source power, future work will also investigate full pattern fitting methods (Luzin et al., 2011; Sato et al., 2013, 2017) which could provide better measurement statistics by analysing multiple Braggedges. Full pattern fitting may also provide a path to extending the method to samples that contain significant texture.
Finally, future work should investigate methods for validating the results when other data sets are not available; such as cross validation (Devijver and Kittler, 1982).
Acknowledgements
This work was supported by the Australian Research Council through a Discovery Project Grant No. DP170102324. Access to the RADEN and KOWARI instruments was made possible through the respective useraccess programs of JPARC and ANSTO (JPARC Long Term Proposal No. 2017L0101 and ANSTO Program Proposal No. PP6050). The authors would also like to thank AINSE Limited for providing financial assistance (PGRA) and support to enable work on this project.
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