# Benchmark Sample Design for the Validation of Residual Stress Measurements by Diffraction: Insights and Practicalities

Gowtham Nimmal Haribabu, David Canelo-Yubero, Emad Maawad, Guilherme Abreu Faria, Peter Staron, Norbert Schell, Ranggi Sahmura, Ramadhan, Sandra Cabeza, Arnold Paecklar, Thilo Pirling, et al.

<figure><img src="image_1.png" /></figure>

## To cite this version:

Gowtham Nimmal Haribabu, David Canelo-Yubero, Emad Maawad, Guilherme Abreu Faria, Peter Staron, et al.: Benchmark Sample Design for the Validation of Residual Stress Measurements by Diffraction: Insights and Practicalities. Integrating Materials and Manufacturing Innovation, 2024, 13 (4), pp.955–968. 10.1007/s40192-024-00385-z. hal-05330162

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# Benchmark Sample Design for the Validation of Residual Stress Measurements by Diffraction: Insights and Practicalities

Gowtham Nimmal Haribabu<fnref n="1" />,<fnref n="5" /> · David Canelo-Yubero<fnref n="2" /> · Emad Maawad<fnref n="2" /> · Guilherme Abreu Faria<fnref n="2" /> · Peter Staron<fnref n="2" /> · Norbert Schell<fnref n="2" /> · Ranggi Sahmura Ramadhan<fnref n="3" />,<fnref n="4" /> · Sandra Cabeza<fnref n="3" /> · Arnold Paecklar<fnref n="3" /> · Thilo Pirling<fnref n="3" /> · Philip J. Withers<fnref n="1" /> · Matthew J. Roy<fnref n="5" />

Received: 12 August 2024 / Accepted: 6 November 2024 / Published online: 22 November 2024
© The Author(s) 2024

## Abstract

Non-destructive determination of residual stresses can present significant challenges. Two benchmark samples having the same geometry have been developed which can be measured by diffraction techniques and adapted to other material systems to validate RS measurements. While having the same geometry, one was solely elastically loaded while the other had stresses generated by significant plasticity. Measurements by a neutron diffraction and two synchrotron X-ray diffraction instruments (white beam and monochromatic beam) have been made in the ferrite phase at identical locations on S355 ferrite/pearlite steel benchmark samples. Broadly consistent results (± 30–50 MPa) are obtained across the three diffractometers. These results were corroborated by finite element predictions, with standard deviation on differences between predictions and measurements being ± 22 MPa for benchmarks with elastic sources of stress and ± 77 MPa for the plastically deformed case. Differences in the stresses were found when making near-surface measurements but only when around 80% of the gauge volume was outside the sample. The larger discrepancies between the measurements and the FE model in the plastic case may be due to plastic load transfer from the softer ferrite to the harder cementite during plastic deformation.

**Keywords** Inter-laboratory measurements · Residual stresses · Neutron diffraction · Synchrotron X-ray diffraction · Uncertainties · Finite element modelling

## Introduction

Residual stresses (RS) are those that remain in materials and structures even when no external loads are present [1–3]. They can arise during manufacture, or in service, and can arise from misfit strain generated plastically or elastically. In the case of the VAMAS shrink-fit ring and plug neutron diffraction benchmark sample, they were solely elastic

(generated by thermal contraction) [4]. Residual stress measurement techniques can be broadly categorised into ‘non-destructive’ techniques, for example, those that measure the elastic strain by diffraction, and destructive methods that infer the stress from displacements associated with destructive mechanical relaxation achieved by cutting the sample in some way. Regarding the former, the determined strains are translated to stresses employing elastic material properties. What is often considered a ‘stress measurement’ is, in practice, the determination of strain from which stresses are calculated. This distinction is crucial for understanding the methodology and accuracy of RS measurement techniques. Depending on their sign, position and direction RS may be advantageous or detrimental to a variety of failure modes, predominantly fatigue and fracture resilience [5].

High-energy synchrotron X-ray beamlines and neutron diffraction instruments available at large research infrastructure (LRIs) have been developed for strain scanning over the last few decades. These have been deployed to map RS in relatively thick samples (~ 10–40 mm) and with good

<footnote>Matthew J. Roy
matthew.roy@manchester.ac.uk</footnote>

<footnote n="1">¹ Department of Materials, Henry Royce Institute, University of Manchester, Manchester, UK</footnote>

<footnote n="2">² Helmholtz-Zentrum Hereon, Geesthacht, Germany</footnote>

<footnote n="3">³ Institute Laue-Langevin (ILL), Grenoble, France</footnote>

<footnote n="4">⁴ ISIS Neutron and Muon Source, Chilton, UK</footnote>

<footnote n="5">⁵ Department of Mechanical, Civil and Aerospace Engineering, Henry Royce Institute, University of Manchester, Manchester, UK</footnote>
spatial resolution across a wide range of industrial components [6, 7]. This has facilitated the validation of computational models, verification of stress-relief processes, fitness for service ratifications, deeper failure analysis investigations, and, amongst other applications, more robust product development.

A number of round-robin exercises for the measurement of RS in specific components have been conducted employing a variety of techniques to ascertain the RS in crystal-line materials. Prominent exercises include the round-robin activities carried out under VAMAS TWA 20 [4, 7, 8] which focused specifically on neutron diffraction and proposed four different benchmark sample configurations for round-robin measurements of which an aluminium ring with a shrink-fit plug [9] to generate a residual stress that could be estimated with a simple analytical approach. The NeT programme [10] focused specifically on residual stress quantification in weldments using a variety of techniques, including computational simulation. These approaches have been applied in the study of dissimilar metal welding (DMW), improving X-ray stress analyses [12, 13], residual stress predictions in welds [11–14], and more recently for additive manufacturing [15].

In the context of measuring reliable RS by diffraction, it is important to be able to infer the macrostress; i.e. the type I residual stress, equilibrating over the component length scale. This is complicated by the fact that different crystallographic planes have different stiffnesses (elastic anisotropy) and respond differently to the effects of plastic strain (plastic anisotropy). Both of these anisotropies generate type II and type III microstresses in addition to the type I macrostress [16]. Furthermore, many materials are either textured, which can affect the representativeness and strength of certain hkl planes in certain directions, or multi-phase introducing mean phase type II stresses that can further complicate interpretation in terms of the overall macrostress [17]. While there is an ISO standard for neutron RS measurement [18], synchrotron X-ray diffraction (SXRD) RS measurement does not have an internationally recognised standard.

There is therefore a need to compare ND and SXRD diffraction-based techniques, helping to standardise the latter. In order to do so, two benchmark sample configurations have been devised, which have approximately the same geometry, but differ in sources of RS. One contains RS generated only by elastic misfits and the other contains RS generated by plastic and elastic misfits. The study explored using the common mixed-phase (ferrite-pearlite) structural steel (S355) to determine the RS in mixed-phase materials

for both techniques. Finite element (FE) modelling has been employed to develop a predictive framework for the stress states for each benchmark sample. By establishing a comparative framework, the present work seeks to further establish the reliability and consistency of residual stress determination by ND and SXRD techniques through their direct comparison.

## Methods

### Material and Microstructure

A single 25-mm-thick Grade S355J0Z35 + N hot-rolled structural steel plate obtained from the Posco Daewoo Corporation was used to manufacture the benchmark samples. It is a low carbon manganese steel, with a minimum yield strength of 355 MPa, a Charpy toughness of 27 J at 0 °C, with a minimum 35% reduction in area at fracture and received in normalised condition. The chemical composition according to the obtained mill certificate is summarised in Table 1. Steel is the most commonly used metal in the world, with this specific grade being used extensively in the built environment and across many other engineering applications.

The microstructure of the steel plate from which the samples were manufactured is depicted in Fig. 1 which demonstrates a strong rolling texture. The average grain size measured using the intercept method [19] is $15 \pm 3$ µm for the ND plane configuration. The steel comprises 61% ferrite (light) and 39% pearlite (dark), as determined by image analysis.

### Design of Benchmark Samples

Two sample types were designed to measure the elastically and plastically induced RS, namely U-flexures (F) and U-bend (B) samples, respectively. The former could be loaded in compression (FC), tension (FT), or left unloaded in a neutral state (FN). These samples were devised as part of the European Activity for Standardization of Industrial Residual Stress Characterization (EASI-STRESS) round-robin activity [20]. A FE model was constructed to i) establish the limits of elastic (recoverable) deformation that could be imposed, ii) ensure that the reference positions remained undeformed, and iii) permit a rapid estimation of the residual stress distribution for direct comparison with any measurements obtained.

<table><caption>**Table 1** Summary of chemical composition (certificate) for S355J0Z35 + N (Tr-Trace)</caption><thead><tr><th>C, %</th><th>Si, %</th><th>Mn, %</th><th>P, %</th><th>S, %</th><th>Cr, %</th><th>Mo, %</th><th>Ni, %</th><th>Al, %</th><th>Cu, %</th><th>Nb, %</th><th>Ti, %</th><th>N, %</th><th>V, %</th><th>Fe, %</th></tr></thead><tbody><tr><td>0.15</td><td>0.34</td><td>1.45</td><td>0.014</td><td>0.003</td><td>0.02</td><td>0.001</td><td>Tr</td><td>0.036</td><td>0.02</td><td>0.025</td><td>0.004</td><td>0.0052</td><td>0.005</td><td>Balance</td></tr></tbody></table>

<figure><img src="image_3.png" /></figure>
<figure><img src="image_4.png" /><figcaption>**Fig. 1** Microstructure: Optical micrographs of the S355J0Z35 + N hot-rolled plate in RD, TD, and ND planes showing the anisotropic and mixed-phase microstructure</figcaption></figure>

The FC, FT, and FN flexures were electrical discharge machined directly to the geometry shown in Fig. 2. The U-bend samples started as an electrical discharge machined 10×20×160 mm blank which was subsequently subjected to three-point bending according to ISO 5173 [21]. This standard specifies a testing procedure for certifying weld ductility by bending strips of a butt-welded plate through 180° to form a U-bend, which elastic springback leads to slightly angled legs. The ductility specification of the subject steel lead resulted in a uniform bend without any cracking.

For the stress to be accurately determined, a stress-free reference sample ($d_0$) must be employed [22]. The manufactured components are shown in Fig. 3. Two specimen kits were prepared as shown in Fig. 3a, including all the key samples. The unloaded sample (FN) served as reference for the U-flexures and a pin was used for the U-bend (Fig. 3f) for neutron diffraction.

<figure><img src="image_5.png" /><figcaption>**Fig. 2** Front view (left) and side view (right) of the U-flexure samples, showing the geometric dimensions and tolerances in millimeter</figcaption></figure>

## Experimental Setup for the U-Shape Benchmarks

The U-flexure and U-bend samples are depicted in Fig. 4a. For the U-flexures, the two legs of the sample can be brought closer together, or pushed further apart by means of a 316L stainless steel locking turnbuckle and their position inferred by a dial/digital test indicator (DTI) mounted to a fixture as shown in Fig. 4b. Coordinate measurement machine styli (Renishaw A-5000–3604) are used as the fiducial markers to ascertain sample positioning during diffraction. The relative displacement of fiducial features on the loaded U-flexures can also be employed to determine the specific loading as boundary conditions for FE analysis. The maximum displacement of the U-flexures was determined such that they remained solely elastically loaded. The target deflection is ±0.4 mm for FT and FC specimens at the measurement position indicated with
<figure><img src="image_6.png" /><figcaption>**Fig. 4** Sample configurations and coordinate system. a) Samples with fiducial reference points for FT (tension), FC (compression) U-flexure and B U-bend samples ('RD' indicates the rolling direction of hot-rolled plate)</figcaption></figure>

from which they were cut. b) dial-test indicator (DTI) fixture readings for the different FT (tension), FC (compression), and FN (neutral/unloaded) U-flexure samples

<figure><img src="image_7.png" /></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 5 FE model—showing the setup and boundary conditions for a U-flexure, b initial configuration and c final configuration of the U-bend</figcaption></figure>

the devised loading fixture. This was determined by FE analysis.

## Finite Element (FE) Analysis

In order to predict the stress profiles, two FE models incorporating elastic–plastic deformation were implemented using ABAQUS<fnref n="1" /> Standard: one corresponding to the flexures and one for the bent samples. The material properties employed were taken from Forni et al. [23]. For the U-flexure, the YZ half-plane at x=0 as shown in Fig. 5a is constrained in the x direction and the XZ half-plane is constrained in the y direction, the displacement at the point at which the force is applied to the legs is constrained in the z direction. The load was increased at the location of the turnbuckle until the displacement recorded by the DTI was achieved. In the case of the U-bends, the constraints are applied, as shown in Fig. 5b. A displacement of 100 mm was applied progressively to the upper roller in order to plastically deform the U-bend to shape. The three rollers which are used for the bending were considered rigid bodies. The contact between the roller and the blank was considered to be frictional, with a coefficient μ=0.1. A mesh sensitivity study was conducted, and a mesh size of 0.8 mm and 1 mm were chosen for the U-flexure and U-bend samples, respectively. For the U-flexures, a static stress analysis is used with a single linear perturbation step. For the U-bend, a static stress analysis is used with a maximum number of 1000 increments in a step, with stabilisation where the dissipated energy fraction is 0.0002, and the adaptive automatic damping algorithm was used.

## Diffraction Stress Measurement

For the round-robin study, a sample kit (Fig. 3) was provided to SALSA (Stress Analyser for Large-Scale engineering Applications) instrument at the Institute Laue-Langevin

(ILL), and another kit to P07 and P61A beamlines at the Deutsches Elektronen-Synchrotron (DESY). The normal elastic strains (ε<sub>xx</sub>, ε<sub>yy</sub>, ε<sub>zz</sub>) were determined for the FT, FC, and B samples at each facility along the mid-thickness line from the inner to outer curvature (Fig. 2). The interplanar distances, d<sup>hkl</sup>, were calculated using Bragg’s law from the beam wavelength and the diffraction angle.

RS were inferred from the change in lattice (d) spacing relative to the d<sub>0</sub> unstressed lattice spacing. In order to determine the change in lattice spacing (d<sup>hkl</sup> - d<sup>hkl</sup>) from Bragg’s equation, one can either use a monochromatic beam (wavelength, λ) and record the peak in terms of scattering angle (2θ) referred to as the angle dispersive technique, or hold θ constant and record the diffraction peak as a function of λ using the energy-dispersive method. In either case, the crystallographic strain is given by:

$$ \epsilon^{hkl} = \frac{d^{hkl} - d_0^{hkl}}{d_0^{hkl}} \qquad (2.2) $$

The strains were used along with the lattice plane specific diffraction elastic constants to find the stress for a particular direction by applying the generalised Hooke’s law:

$$ \sigma_{xx} = \frac{E^{hkl}}{(1 + v^{hkl})(1 - 2v^{hkl})} [(1 - v^{hkl})\epsilon_{xx} + v^{hkl}(\epsilon_{yy} + \epsilon_{zz})] \qquad (2.4) $$

$$ \sigma_{yy} = \frac{E^{hkl}}{(1 + v^{hkl})(1 - 2v^{hkl})} [(1 - v^{hkl})\epsilon_{yy} + v^{hkl}(\epsilon_{xx} + \epsilon_{zz})] \qquad (2.5) $$

$$ \sigma_{zz} = \frac{E^{hkl}}{(1 + v^{hkl})(1 - 2v^{hkl})} [(1 - v^{hkl})\epsilon_{zz} + v^{hkl}(\epsilon_{xx} + \epsilon_{yy})] \qquad (2.6) $$

where E<sup>hkl</sup> is the hkl-specific Young’s modulus, and v<sup>hkl</sup> is the hkl-specific Poisson’s ratio for an untextured polycrystal. In accordance with the VAMAS guidelines, the Fe-211 plane was used for the ferrite phase as it is less prone to elastic and

<footnote n="1">¹ Registered trademark of Dassault Systèmes.</footnote>
<figure><img src="image_9.png" /><figcaption>**Fig. 6** Gauge volume and stress determination configurations for a neutron diffraction at SALSA and synchrotron X-ray diffraction at b P07 and c P61A beamlines. IB: incoming beam, DB: diffracted beam, GV: gauge volume, LD: long diagonal</figcaption></figure>

plastic anisotropy than other planes and has a high multiplic-
ity [4, 18, 24–26] with E²¹¹ = 228 GPa and ν²¹¹ of 0.28.

**Neutron Diffraction (ND) Measurements**

All samples belonging to the first sample kit were meas-
ured on SALSA [27]. The instrument description and data
analysis workflow are briefly summarised below. A mono-
chromatic wavelength of ~0.159 nm was selected, giving a
2θ of 91° for the 211 peak. The sample positioning using
the hexapod table was accomplished with the aid of a cam-
era system. For the measurement of the radial component,
the U-shapes were placed with the long side parallel to the
hexapod table with a ω-rotation axis position of –44.5°
(Fig. 6a). The radial collimators were chosen to give a gauge
volume (GV) of 0.6×0.6×2 mm (for the x strain meas-
urement direction). For the normal (z) and transversal (y)
measurement directions, the orientation of the samples on
the hexapod table had to be changed, and the GV was also
changed to 2×2×0.6 mm.

The raw data, including metadata, were saved in the
NeXus format, and the raw data reduction (calibration, inte-
gration, peak fitting, etc.) was performed using an in-house
data analysis routine called LAMP (Large Array Manipu-
lation Program), providing the reduced output data in an

ASCII format. For each measurement point, the 2θ posi-
tion in the 211 diffraction peak was obtained. The residual
strain and stress values were then calculated using a Math-
cad<fnref n="2" /> script devised for SALSA. The data were corrected to
account for incomplete gauge filling near surface [28].

High-Energy Synchrotron X-ray Diffraction (SXRD)
Measurements

Samples from the second kit were characterised by both
monochromatic (narrow spectrum) and white beam (broad
spectrum) diffraction. A monochromatic X-ray beam of
wavelength 0.14235 nm was selected from an undulator
source on the Hereon P07 beamline at the PETRA III stor-
age ring at DESY (Deutsches Elektronen-Synchrotron) and
LaB₆ standard powders were used to calibrate the sample-
detector distance. The incident and diffracted beams were
defined by slits and a conical slit [29], respectively, giving
a 0.2×0.2 mm cross section, and is elongated (–2.0 mm)
along the direction of the beam. The sin<fnref n="2" />Ψ method was
applied to find the associated strain components for the
211 peak in x and z before the sample was rotated 90° and
–90°, applying double immersion in the beam direction to
eliminate pseudo-strain, such that the final y direction was
obtained. The 2D diffraction patterns obtained were inte-
grated using the software Fit2D [30, 31] to ascertain peak
shifts for strain components. These strain components were
then used to find the associated stress in the three coordinate
directions using Eqs. 2.4–2.6.

<figure><img src="image_10.png" /></figure>

<footnote n="2">² Mathcad is a trademark of Mathsoft PTC.</footnote>
The U-shape samples were also measured on the Hereon P61A White Beam Engineering Materials Science beamline at PETRA III (DESY-Hereon) in transmission. The beam dimension was shaped with the incident and diffracted slit apertures of 0.15×0.15 mm² using a diffraction angle, 2θ, of ~5.02°, providing a rhombohedral-shaped gauge volume with its long diagonal of ~3.4 mm and standard Si powders were used for the 2θ calibration. The diffraction profiles were collected using a high-purity Ge energy-dispersive detector. The samples were aligned via the beam-stop photodiode current. A normal pattern usually covers a range between 0 and 200 keV, including first- and second-order reflection peaks. All the scanned lines were measured with the sample positioned at different ψ angles (0°, 90°, 180°, and 270°), aiming at performing a double immersion of the sample for the pseudo-strain correction. A ψ scan between 0° and 90° (with steps of 4.5°) was also performed at each ψ angle to apply the sin²ψ method. In-house software, P61A:Viewer, was used for the data reduction and peak position determination and a pseudo-Voigt function was used for the peak fitting. The gauge volumes are shown in Fig. 6. As neither the FN sample nor the d₀ pin was deemed reliable for the SXRD, equilibrium conditions were used for d₀ calculations.

## Results and Discussion

The following sections provide a detailed analysis of the various aspects of stress measurement, modelling and comparison with experimental results in U-shaped samples.

### Error Estimation

Determining RS can be very challenging with many opportunities for errors to arise [16]. The most important sources of error associated with the techniques applied here are

briefly considered below. Sources of errors mainly originate from sample alignment and peak fitting and from the sin²ψ method. Errors in positioning are mainly affected by the accuracy of the sample alignment and the positional accuracy of the beam. This depends on the alignment and metrology system employed, as well as the type of sample stage used. It is required to have the reference point coincident with the instrument centre of the ω-rotation axis within an accuracy of 10% of the gauge volume width [32]. The peak fitting error, calculated by error propagation, determines the final stress error. This stems from fitting the diffraction peaks depending on the peak width and shape, as well as on the peak-to-background ratio. Depending on the material, the ability to scatter or absorb neutrons can vary. In the case of synchrotron, a white beam usually provides diffraction peaks with higher signal-to-background ratio when compared to a monochromatic beam coupled with a conical slit cell (CSC), i.e. errors are smaller for the former [29]. Additionally, the detected signal depends on the microstructure (e.g. large or non-uniform grain size). Finally, a general source of error relates to measuring the diffraction angle relative to a stress-free reference.

## Modelling Results

Contour maps of the σyy stress predicted by FEA for the three cases considered are shown in Fig. 7. The following equation is used below to quantify the disparity, e_pq, between the determined stress σ^m and the predicted (by FE) stress σ^p:

$$e_{pq} = \sigma^m - \sigma^p$$

A kernel density error (KDE) plot was used to visualise the difference between the model and experimental measurement. The average and standard deviation of differences was calculated and is tabulated below.

<figure><img src="image_11.png" /><figcaption>Fig. 7 Contour maps showing the stress distribution (σyy) of residual stress in the a U-flexure tension (FT), b U-flexure compression (FC), and c U-bend</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**Fig. 8** Comparison between model and diffraction results for the FC U-flexure-residual stress distribution along z direction from outer radius for the three directions **a** $\sigma_{xx}$, **b** $\sigma_{yy}$, **c** $\sigma_{zz}$, and **d** KDE plot showing distribution of deviation in determined stress values by diffraction and model</figcaption></figure>

<figure><img src="image_13.png" /></figure>
## Residual Stress in U-Flexures

The U-flexures were expected to be the simplest case, having only elastically generated stresses. The results for the ND, SXRD stress measurements and the FEA predictions for the corresponding locations are summarised in Figs. 8 and 9 for the FT and FC U-flexures, respectively. Here, the measured stresses were corrected for differences in loading (the stresses applied by the turnbuckles were slightly different for the 2 sample kits). As a result, there are two model predictions in the figures (one for the SXRD and one for the ND) such that the model reflects the measured conditions. The difference between the FEA model and the measurements is plotted in (d) in each case. On the whole, the agreement between the different diffractometers was excellent, as well as with the predictions of the FE model. An average difference of 15 ± 35 MPa can be seen between the SXRD and ND, considering all three directions along the measurement line. However, slight differences are observed at the extremities, namely very close to the external radius surface (z=0) and internal radius z ≈ 10. This could be because a) the electrical discharge recast layer affects the near-surface regions, b) this is where the samples were most heavily loaded (e.g. 85% of yield) and small-scale plasticity/type I yielding could have taken place, or c) because of errors that arise as when the gauge volume is partially filled [33]. Similar to the FC sample, the stress prediction in the FT sample at the extremes was very close to the residual stress measurements by SXRD and ND, as seen in Fig. 9.

As shown by the KDE plots in Figs. 8 and 9, the predictions from the model were generally in good agreement with the measurements, being in the range of -50 to +50 MPa. The disparities were greatest for the ND measurements close to the surface (within 500 µm) of the surface—given that the gauge volume was > 3 mm across its diagonal the gauge of only fractionally filled at this location, which could cause a pseudo-strain effect [28, 33–35]. Overall, the measurements were scattered due to measurement uncertainty (by < 10 MPa) about the model. For the case of U-flexure tension (FT), the measurements ($\sigma_{yy}$ & $\sigma_{zz}$) of SXRD were seen to be consistently higher than the model by an average value of 25–40 MPa. The fact that the stresses as determined from the ferrite phase match well to those determined by the FE model for the steel as a whole was probably because the ferrite and cementite (making up the ferrite and pearlite regions) have similar elastic properties [36–40].

## Residual Stress in U-Bend Samples

The predictions for the U-bend samples are shown in Fig. 10 alongside the measurements. Generally, there was a good agreement between both diffraction techniques and the model for all stress components. When compared with the U-flexures, a greater disparity was observed for the ND measurements near the surfaces. It is generally not recommended to make measurements by neutron diffraction in regions where the gauge volume protrudes by more than 50% out of the sample, and the near-surface measurements shown here extend well beyond that point [41]. The fact that this difference is not seen in the SXRD measurements would suggest that this deviation is not due to machining effects or due to plastic anisotropy between the ferrite and pearlite. The difference in the model predictions is slightly greater than for the all elastic U-flexure samples, as shown in the KDE plots as shown in Fig. 10d, being around ~100 MPa. However, the deviation was symmetric about zero. The difference could be in part due to poorer predictions when plasticity occurs, with the actual level of plastic deformation being different from that predicted. Another reason for the difference could be the strain partitioning effect in the multi-phase steel consisting of ferrite and pearlite [42]. In these steels, plastic deformation begins in the soft ferrite regions, creating a strain mismatch between the soft ferrite and the hard pearlite regions. Consequently, the ferrite grains would be expected to transfer load to the harder pearlite during loading leaving the ferrite less tensile or less compressive than expected at maximum load. Upon unloading the stresses in the plastically deformed regions might therefore be expected to reverse into greater compression or tension relative to the FE model which neglects the two-phase nature of the steel.

Comparison of the SXRD measurements for $\sigma_{xx}$ with those made by ND and FE modelling suggest that the positional alignment might be in error by around 0.7 mm. In addition, there is some evidence in the $\sigma_{yy}$ plot to suggest that the plastic strain in the two U-bend samples is slightly different with the sample measured by SXRD having a slightly greater plastic deformation. The average and standard deviation on differences between model and measurements are detailed in Table 2. Here, we have developed benchmark samples for RS on S355 steel. Such samples could be simply made to benchmark RS measurements by diffraction focusing on other metals and alloys.
<figure><img src="image_14.png" /><figcaption>**Fig. 9** Comparison between model and diffraction results for the FT U-flexure-residual stress distribution along *z* direction from outer radius for the U-flexure in tension, along three directions *a*, *σ*<sub>xx</sub>, *b*, *σ*<sub>yy</sub>, *c*, *σ*<sub>zz</sub>, and *d* KDE plot showing distribution of deviation in determined stress values by diffraction and model</figcaption></figure>

<figure><img src="image_15.png" /></figure>
<figure><img src="image_16.png" /><figcaption>**Fig. 10** Comparison between model and diffraction results for U-bends (B)-residual stress distribution along z direction from outer radius for the U-bend sample, along three directions **a** $\sigma_{xx}$, **b** $\sigma_{yy}$, **c** $\sigma_{zz}$, and **d** KDE plot showing distribution of deviation in determined stress values by diffraction and model. The measurements do not start</figcaption></figure>

at zero, as the starting point is with respect to the edge on face (YZ plane) of the legs which are protruded towards -z direction due to plastic deformation and the measurement point inside sample starts at an offset
<table><caption>**Table 2** Summary of differences between model and measurement for different samples and techniques</caption><thead><tr><th>Sample</th><th>Statistics</th><th>ND-FEA</th><th>SXRD P61A-FEA</th><th>SXRD P07-FEA</th></tr></thead><tbody><tr><td rowspan="2">FC</td><td>Average</td><td>4</td><td>8.4</td><td>1</td></tr><tr><td>Standard deviation</td><td>38</td><td>21</td><td>21</td></tr><tr><td rowspan="2">FT</td><td>Average</td><td>-8</td><td>20</td><td>-4</td></tr><tr><td>Standard deviation</td><td>24</td><td>21</td><td>26</td></tr><tr><td rowspan="2">B</td><td>Average</td><td>-21</td><td>4</td><td>9</td></tr><tr><td>Standard deviation</td><td>59</td><td>89</td><td>84</td></tr></tbody></table>

## Conclusions

This study establishes two benchmark sample designs which can easily be reproduced for validating neutron or synchrotron residual stress determination arrangements. The U-flexure and U-bend designs set residual stress measurements challenges that include elastic and plastic anisotropy effects, respectively. In our case, the choice of a ferrite/pearlite steel also raises the issue of stress partitioning between the phases.

Two synchrotron X-ray diffraction (SXRD) beamlines (one wavelength dispersive and one angle dispersive) and one neutron diffraction (ND) instrument have been used to make a series of measurements along the line of maximal residual stress for both samples. All the instruments show broadly the same stress measurement profiles for the ferrite phase through demonstrating a high tensile stress ($\sigma_{yy} \approx 200$ MPa) at the extrados U-flexure tension (FT) sample and a similarly high compressive stress ($\sigma_{yy} \approx -200$ MPa) for the U-flexure compression (FC) sample. Generally, all the measurements performed by SXRD and ND were in good agreement ($\pm 21-38$ MPa). FE modelling was able to predict the stress states with good agreement with the stress measurements which is in part due to the fact that the elastic properties of ferrite and cementite are similar.

The comparison between the diffraction measurements was also good in the case of the plastically deformed U-bends. A small difference between the diffraction and FE results may be due to plastic load transfer between the softer ferrite and harder cementite [42].

The experimentally determined stresses in this work were evaluated with the generalised Hooke's law (Eqs. 2.4–2.6) on the assumption that the illuminated gauge volumes behave isotropically and the 211 peak is representative of the sample as a whole. As the raw material exhibits a strong initial rolling texture, i.e. anisotropy, the use of different crystallographic elastic constants for each direction may have merit. In this respect, while the (211) peak is recommended by VAMAS TWA20 [18], if possible, it is advised

to assess multiple hkl peaks [43]. Full pattern refinements (e.g. Pawley) based on many peaks would be able to assess the degrees of elastic and plastic anisotropy for different hkl peaks and across multiple phases.

**Acknowledgements** The authors acknowledge the funding received from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 953219 (EASI-STRESS). GNH, MJR, and JW are grateful for the Henry Royce Institute for Advanced Materials, established through EPSRC grants EP/R00661X/1, EP/P025498/1, and EP/P025021/1. The authors also acknowledge DESY (Deutsches Elektronen-Synchrotron) for the beamtime provision at the P07 and P61A beamlines.

**Author Contributions** Gowtham Nimmal Haribabu¹ was involved in writing—original draft, visualization, formal analysis. David Canelo Yubero² helped in investigation, writing—review and editing, methodology, formal analysis. Emad Maawad² helped in investigation. Guilherme Abreu Faria² contributed to investigation. Peter Staron² took charge of investigation, writing—review and editing, methodology. Norbert Schell² contributed to investigation. Ranggi Sahmura Ramadhan³,⁴ took charge of investigation, formal analysis. Sandra Cabeza³ contributed to investigation. Arnold Paecklar³ helped in investigation, writing—review and editing, formal analysis. Thilo Pirling³ took charge of investigation. Philip J. Withers¹ was involved in writing—review and editing, conceptualization. Matthew Roy¹ contributed to visualisation, writing—review and editing, conceptualization.

## Declarations

**Conflict of 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.

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## References

1. Schajer GS, Ruud CO (2013) Overview of residual stresses and their measurement. In: Practical residual stress measurement methods. pp 1–27

2. Withers PJ, Bhadeshia HKDH (2001) Residual stress. Part 1 – measurement techniques. Mater Sci Technol 17:355–365. https://doi.org/10.1179/026708301101509980

3. Withers PJ, Bhadeshia HKDH (2001) Residual stress. Part 2 – nature and origins. Mater Sci Technol 17:366–375. https://doi.org/10.1179/026708301101510087

4. Daymond MR, Johnson MW, Sivia DS (2002) Analysis of neutron diffraction strain measurement data from a round robin sample. J Strain Anal Eng Des 37:73–86. https://doi.org/10.1243/0309324021514844
5. Withers PJ (2007) Residual stress and its role in failure. Rep Prog Phys 70:2211–2264. https://doi.org/10.1088/0034-4885/70/12/ R04

6. Petry W (2015) Neutrons for industry. EPJ Web Conf 104:01001. https://doi.org/10.1051/epjconf/201510401001

7. Suzuki M (2014) Recent trends in industrial applications. Synchrotron Radiat News 27:2. https://doi.org/10.1080/08940886. 2014.908696

8. Ohms C, Youtsos AG, Webster GA, Wimpory RW (2002) Poly-crystalline materials determination of residual stresses by neutron diffraction, TWA 20 ISO/TTA3 Euro Commission, ISSN 1018– 5593 20265

9. Webster GA, Youtsos AG, Ohms C, Wimpory RC (2002) Draft standard for the measurement of residual stresses by neutron diffraction. In: Gdoutos, EE (ed) Recent advances in experimental mechanics. Springer Netherlands, Dordrecht, pp 467–476

10. Smith MC, Smith AC (2018) Advances in weld residual stress prediction: a review of the NeT TG4 simulation round robins part 2, mechanical analyses. Int J Press Vessels Pip 164:130–165. https:// doi.org/10.1016/j.ijpvp.2018.03.007

11. Rathbun HJ, Fredette LF, Scott PM, et al (2011) NRC welding residual stress validation program international round robin program and findings. In: Volume 6: Materials and Fabrication, Parts A and B. ASMEDC, pp 1539–1545

12. Kerr M, Prime MB, Swenson H et al (2013) Residual stress characterization in a dissimilar metal weld nuclear reactor piping system mock up. J Press Vessel Technol. https://doi.org/10.1115/1. 4024446

13. Fredette LF, Broussard JE, Kerr M, Rathbun HJ (2011) NRC/EPRI welding residual stress validation program-Phase III details and findings. In: American Society of Mechanical Engineers, Pressure Vessels and Piping Division (Publication) PVP. ASMEDC, pp 1547–1555

14. Goldak JA, Tchernov S, Zhou J, Downey D (2012) A sensitivity analysis of NRC welding residual stress validation program international round Robin program. In: American Society of Mechanical Engineers, Pressure Vessels and Piping Division (Publication) PVP. pp 1245–1254

15. Phan TQ, Strantza M, Hill MR et al (2019) Elastic residual strain and stress measurements and corresponding part deflections of 3D additive manufacturing builds of IN625 AM-bench artifacts using neutron diffraction, synchrotron x-ray diffraction, and contour method. Integr Mater Manuf Innov 8:318–334. https://doi. org/10.1007/s40192-019-00149-0

16. Schajer GS, Prime MB, Withers PJ (2022) Why is it so challenging to measure residual stresses? Exp Mech 62:1521–1530. https://doi.org/10.1007/s11340-022-00879-x

17. Fitzpatrick ME, Hutchings MT, Withers PJ (1997) Separation of macroscopic, elastic mismatch and thermal expansion misfit stresses in metal matrix composite quenched plates from neutron diffraction measurements. Acta Mater 45:4867–4876. https://doi. org/10.1016/S1359-6454(97)00209-7

18. British Standards Institution (2020) BS EN ISO 21432:2020: Non-destructive testing. Standard test method for determining residual stresses by neutron diffraction

19. ASTM E112–10 (2010) Standard test methods for determining average grain size ASTM International. 13:1 27. https://doi.org/ 10.1520/E0112-13

20. Cui W, Sánchez Poncela M, Badyka R, et al (2022) EASI-STRESS standardization of industrial residual stress measurement: benchmark specimen design. ICRS 11-The 11th International Conference of Residual Stresses, Nancy, France

21. (2023) ISO 5173:2023 Destructive tests on welds in metallic materials — Bend tests British Standards Institute. 24

22. Withers PJ, Preuss M, Steuwer A, Pang JWL (2007) Methods for obtaining the strain-free lattice parameter when using diffraction

to determine residual stress. J Appl Crystallogr 40:891–904. https://doi.org/10.1107/S0021889807030269

23. Forni D, Chiaia B, Cadoni E (2016) Strain rate behaviour in tension of S355 steel: base for progressive collapse analysis. Eng Struct 119:164–173. https://doi.org/10.1016/j.engstruct.2016.04. 013

24. Youtsos AG, Ohms C (2002) European standardization activities on residual stress analysis by neutron diffraction. Appl Phys A: Mater Sci Process. https://doi.org/10.1007/s003390201841

25. Johnson MW, Daymond MR, Ohm C, Youstos AG (2000) Neutron diffraction measurements of residual stress in a shrink-fit ring and plug VAMAS Report no 38 (ISSN 1016–2186) Technical Working Area 20

26. Pang JWL, Holden TM, Mason TE (1998) The development of intergranular strains in a high-strength steel. J Strain Anal Eng Design 33:373–383. https://doi.org/10.1243/0309324981513075

27. Pirling T, Bruno G, Withers PJ (2006) SALSA—A new instrument for strain imaging in engineering materials and components. Mater Sci Eng, A 437:139–144. https://doi.org/10.1016/J.MSEA. 2006.04.083

28. Pirling T (2011) Precise analysis of near surface neutron strain imaging measurements. Procedia Eng 10:2147–2152. https://doi. org/10.1016/j.proeng.2011.04.355

29. Staron P, Fischer T, Eims EH et al (2014) Depth-resolved residual stress analysis with conical slits for high-energy x-rays. Mater Sci Forum 772:3–7. https://doi.org/10.4028/www.scientific.net/MSF. 772.3

30. Hammersley AP (2016) {vit FIT2D}: a multi-purpose data reduction, analysis and visualization program. J Appl Crystallogr 49:646–652. https://doi.org/10.1107/S1600576716000455

31. Hammersley AP, Svensson SO, Hanfland M et al (1996) Two-dimensional detector software: from real detector to idealised image or two-theta scan. High Press Res 14:235–248. https://doi. org/10.1080/08957959608201408

32. Ramadhan RS, Cabeza S, Pirling T et al (2021) Quantitative analysis and benchmarking of positional accuracies of neutron strain scanners. Nuclear Instr Methods Phys Res Sect A: Accel, Spectrom, Detect Assoc Equip 999:165230. https://doi.org/10. 1016/j.nima.2021.165230

33. Suzuki H, Harjo S, Abe J et al (2013) Effects of gauge volume on pseudo-strain induced in strain measurement using time-of-flight neutron diffraction. Nucl Instrum Methods Phys Res, Sect A 715:28–38. https://doi.org/10.1016/j.nima.2013.03.031

34. Creek SR, Santisteban JR, Edwards L (2005) Modelling pseudo-strain effects induced in strain measurement using time-of-flight neutron diffraction. Frg/2005/02/08 1–16

35. Rebelo Kornmeier J, Hofmann M, Luzin V et al (2019) Fast neutron surface strain scanning with high spatial resolution. Mater Charact 154:53–60. https://doi.org/10.1016/j.matchar.2019.05.031

36. Isavand S, Kardan-Halvaei M, Assempour A (2021) Crystal plasticity modeling and experimental characterization of strain localization and forming limits in ferrite-pearlite steels. Int J Solids Struct 233:111205. https://doi.org/10.1016/j.ijsolstr.2021.111205

37. Isavand S, Assempour A (2021) Strain localization and deformation behavior in ferrite-pearlite steel unraveled by high-resolution in-situ testing integrated with crystal plasticity simulations. Int J Mech Sci 200:106441. https://doi.org/10.1016/j.ijmecsci.2021. 106441

38. Isavand S, Assempour A (2021) Effects of microstructural morphology on formability, strain localization, and damage of ferrite-pearlite steels: experimental and micromechanical approaches. Metall Mater Trans A 52:711–725. https://doi.org/10.1007/ s11661-020-06115-2

39. Basantia SK, Bhattacharya A, Khutia N, Das D (2021) Plastic behavior of ferrite-pearlite, ferrite-bainite and ferrite–martensite
steels: experiments and micromechanical modelling. Met Mater Int 27:1025–1043. https://doi.org/10.1007/s12540-019-00519-5

40. Gadalińska E, Baczmański A, Braham C et al (2020) Stress localisation in lamellar cementite and ferrite during elastoplastic deformation of pearlitic steel studied using diffraction and modeling. Int J Plast 127:102651. https://doi.org/10.1016/j.ijplas.2019.102651

41. Taran YV, Balagurov AM (2012) Correction of a neutron diffraction peak shift due to a partial immersion of a gauge volume in an unstressed sample. Nucl Instrum Methods Phys Res, Sect A 679:19–24. https://doi.org/10.1016/j.nima.2012.03.009

42. Oliver EC, Daymond MR, Withers PJ (2004) Interphase and intergranular stress generation in carbon steels. Acta Mater 52:1937–1951. https://doi.org/10.1016/j.actamat.2003.12.035

43. Daymond MR, Bourke MAM, Von Dreele RB et al (1997) Use of Rietveld refinement for elastic macrostrain determination and for evaluation of plastic strain history from diffraction spectra. J Appl Phys 82:1554–1562. https://doi.org/10.1063/1.365956

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