# Determination of Residual Stress in Additively Manufactured 316L Stainless Steel Benchmark Parts Through Synchrotron X-Ray Diffraction and Neutron Diffraction

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

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Robin C. Laurence, David Canelo-Yubero, Emad Maawad, Guilherme Abreu Faria, Peter Staron, et al. Determination of Residual Stress in Additively Manufactured 316L Stainless Steel Benchmark Parts Through Synchrotron X-Ray Diffraction and Neutron Diffraction. Strain, 2025, 61 (2), pp.e70005. 10.1111/str.70005. hal-05099700

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# Determination of Residual Stress in Additively Manufactured 316L Stainless Steel Benchmark Parts Through Synchrotron X-Ray Diffraction and Neutron Diffraction

Robin C. Laurence<fnref n="1" /> | David Canelo-Yubero² | Emad Maawad² | Guilherme Abreu Faria² | Peter Staron² | Norbert Schell² | Ranggi Sahmura Ramadhan³,⁴ | Sandra Cabeza³ | Arnold Paecklar³ | Thilo Pirling³ | Manuel Sánchez Poncela⁵ | Juan Manuel Martinez⁵ | M. F. Slim⁶ | Thomas Buslaps⁷ | Philip J. Withers<fnref n="1" /> | Matthew Roy<fnref n="1" />

**Correspondence:** Robin C. Laurence (robin.laurence@manchester.ac.uk)

**Received:** 16 September 2024 | **Revised:** 7 February 2025 | **Accepted:** 1 March 2025

**Funding:** This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 953219. RCL, MR and PJW are grateful to the Henry Royce Institute established through EPSRC Grants EP/R00661X/1, EP/P025498/1 and EP/P025021/1.

**Keywords:** additive manufacturing | neutron diffraction | strain mapping | residual stress | synchrotron X-ray diffraction | strain mapping

## ABSTRACT

Additive manufacturing (AM) of parts is typically associated with the generation of high residual stresses because of repeated exposure to high thermal gradients. Non-destructive means of mapping the residual stresses are required for optimising these processes and/or mitigating such stresses by thermal treatments. However, the reliable determination of residual stress in AM parts remains challenging. Here, neutron diffraction, as well as energy- and angle-dispersive synchrotron X-ray diffraction have been used to map the residual stress within a laser powder bed fusion (LPBF) additively manufactured stainless steel 316L arch. The arch was designed by the EASI-STRESS project as a residual stress benchmarking exercise in order to compare different residual stress analysis techniques. Residual stresses were determined along two scan lines deep within the bulk of the component. The results for the different neutron and synchrotron instruments are found to broadly agree with a standard deviation of 50 MPa or better, as well as with those predicted by an inherent strain finite element model. The results show near-yield level residual stresses and suggest that both synchrotron X-ray diffraction and neutron diffraction can be used to reliably determine the residual stress in LBPF parts.

## 1 | Introduction

Residual stress is that stress which remains in the material in the absence of any external load. The aim of this work is to improve the reliability of residual stress measurements within the bulk of additively manufactured components by diffraction. In this respect, neutron diffraction (ND) [1, 2] and high-energy synchrotron X-ray diffraction (SXRD) [3] are both diffraction-based

residual stress analysis techniques capable of probing the internal residual stresses within a polycrystalline material. Diffraction methods for determining residual stress rely on comparing the distance between the atomic planes in the part with a stress-free plane spacing. This comparison is done within a specific defined subvolume of the part, referred to as a gauge volume. This gives a measure of the elastic strain that can be combined with knowledge about other strain or stress components to calculate

<footnote n="1">¹Henry Royce Institute, University of Manchester, Manchester, UK | ²Helmholtz-Zentrum Hereon, Geesthacht, Germany | ³Institut Laue-Langevin (ILL), Grenoble, France | ⁴ISIS Neutron and Muon Source, Didcot, UK | ⁵ArcelorMittal Global R&D, Avilés, Spain | ⁶Mechanics, Surfaces & Materials, Arts et Métiers Institute of Technology, Paris, France | ⁷European Synchrotron Radiation Facility (ESRF), Grenoble, France</footnote>

<footnote>This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.</footnote>
the stress through the generalised form of Hooke's law. Although the measurement of the part is non-destructive, the techniques require a stress-free reference material if no assumptions can be made about the underlying residual stress distribution, for example, plane stress or strain. The reference material should have the same microstructure and Type II (intergranular) and Type III (atomistic) stresses as that at the location of interest but be free from any Type I macro stress. In order to achieve this, the stress-free reference often takes the form of a small section cut out of the part to mechanically relieve the Type I residual stresses [4]. In those cases where cutting is required [5], the techniques cannot be fully described as non-destructive.

Additive manufacturing (AM) is a layer-by-layer manufacturing technique that allows parts to be made in their final shape and is often used to produce parts with complex geometries, which can be otherwise difficult to manufacture. Low volume, but highly customised parts can also be made this way without dies or moulds [6]. The laser powder bed fusion (LPBF) manufacturing technique scans a focused laser to selectively melt a thin layer of metal powder, which then solidifies into a single layer. Over many layers, this produces a full 3D component, near net shape, minimising finishing steps [7, 8]. LPBF is the most common metal AM technique used within the aerospace industry [9].

This manufacturing technique typically results in large residual stresses within the finished part as it undergoes many localised high-gradient thermal cycles [10] as the laser scans over the part. The stress field induced will vary in a complex manner according to how the 3D shape was achieved, but in general, the part will be in net tension overall relative to the substrate, and the stresses will be most tensile in the last regions to be consolidated [11, 12].

Both ND [13] and SXRD [14] have been adopted as a way of determining the residual stress of additively manufactured parts. The uptake in ND and SXRD residual stress analysis for LPBF components has been directed at:

* evaluating how the final residual stress field varies as a function of changes in the manufacturing or processing parameters, such as a change in the scanning strategy [15–17], input energy density [18] or the effect of a post-production heat treatment [19, 20],

* confirming the outputs of lab-based residual stress measurements, such as through the contour method [13, 20, 21] by comparison of the stress at given locations',

* validating predictive computational models of the LPBF manufacture process [14, 22–24] by comparing model outputs with the stresses recorded at key locations over specific gauge volumes.

Although both ND and SXRD have been used to compare the strain in an LPBF component [21], there is a paucity of studies to date that examine the stress in the same LPBF part by two different diffraction techniques. Increasing industry uptake in diffraction-based residual stress characterisation of LPBF parts requires there to be confidence in the techniques. Although the principles underlying the ND and SXRD techniques are the same, there are differences in the specific practice such as the size and shape of gauge volume and the penetration depth of radiation.

The present work serves to address this by providing residual stress as determined in an LPBF part from four different SXRD and ND strain scanners. By conducting a series of measurements using different instruments, the reliability and repeatability of diffraction techniques for the determination of residual stress in additively manufactured parts can be validated confirming them as appropriate for use in determining residual stress in industrially relevant components. A similar round-robin style approach has been employed for ND through the VAMAS project [25] and weldments for NeT research activities [26].

Although there is a guiding ISO standard for determining residual stress by ND [27], there is no equivalent available for SXRD. This paper examines the issues associated with determining residual stresses in an exemplar AM component made by LPBF by ND and SXRD. This required a sample that is appropriate for both ND and SXRD techniques. The EASI-STRESS AM benchmark specimens were designed for round-robin testing with this proposed purpose in mind [28]. They are geometrically more complicated (realistic) than studied previously [13] and contain features close to the resolution limit of the LPBF process used to create them. They are appropriate for this comparative study as they are large enough to accommodate ND gauge volumes, but small enough for SXRD to penetrate the thickness of the sample. This has enabled both ND and SXRD to be fully compared for AM parts and key issues affecting their accuracy and suitability to be explored, which in turn can inform future studies.

## 2 | Experimental Methods

LPBF was used to produce the EASI-STRESS benchmark additively manufactured arches from 316L stainless steel powder using a MetalFAB1 LPBF machine. The arches consist of a 20-mm square top and are 10 mm in height, and the arch is central with its apex 2 mm below the top surface, formed by a 4-mm radius overhang. They were made using a 67° rotating scanning strategy, to reduce the texture of the resulting microstructure [17]. A schematic of the arch is shown in Figure 1 with the processing parameters used in Table 1. After manufacture, the arches did not undergo any further heat treatment and were measured in the as-built condition, still attached to a section of the build plate. The stress in the arches was investigated by diffraction using four distinct beamline strain scanners, utilising ND (SALSA at ILL [29]) and SXRD (P07, P61A at DESY, operated by Hereon [30], and ID15A at ESRF [31]) with each institution measuring a physically different but nominally identical arch. Dimensional variation between each arch was found to be below 0.1 mm for all features.

### 2.1 | Determination of Stress by Diffraction Techniques

The fundamental approach used to calculate strain by diffraction techniques exploits Bragg's law:

$$n\lambda = 2d^{hkl}\sin\theta \quad (1)$$

Here, the distance between lattice planes in a particular crystallographic direction, $d^{hkl}$, is found using a radiation source of wavelength $\lambda$ and by recording the diffracted radiation at an
<figure><img src="image_3.png" /><figcaption>**FIGURE 1** | AM Arch dimensions and coordinate system. The origin of the coordinates system used is in the centre of the top surface. The inset depicts the two stress measurement scan lines: a vertical Line 1 (orange) from the top surface to the apex of the arch starting at the coordinate origin at the centre of the arch and a horizontal Line 2 (blue) that bisects the previous line at a depth of 1 mm running along the x-axis. A photograph of an example arch is shown inset.</figcaption></figure>

<table><caption>**TABLE 1** | Processing parameters for the arches.</caption><thead><tr><th>Parameter</th><th></th></tr></thead><tbody><tr><td>Machine</td><td>MetalFAB1</td></tr><tr><td>Input volumetric energy density</td><td>50J/mm<sup>3</sup></td></tr><tr><td>No. of lasers</td><td>4</td></tr><tr><td>Layer thickness</td><td>50µm</td></tr><tr><td>Environment</td><td>Argon (Max 0.05% O<sub>2</sub>)</td></tr><tr><td>Substrate material and thickness</td><td>8-mm 304 stainless steel</td></tr><tr><td>Recoating time</td><td>12 s per layer</td></tr><tr><td>316L powder size</td><td>15–45 µm</td></tr></tbody></table>

angle $\theta$. To determine $d^{hkl}$, either a known wavelength is used, and the angle changed (angle-dispersive techniques) such that the diffraction peak corresponding to the crystallographic direction of interest can be observed, or a fixed angle is used with a beam containing many wavelengths (energy-dispersive techniques). The diffraction peak typically has a shape that can be represented by a Gaussian curve or a Voigt or pseudo-Voigt shape, depending on the instrument and the microstructure of the sample. The peak is fitted using a least squares fitting algorithm to find the location of the centre of the peak, plotted against wavelength or energy (energy-dispersive techniques) or diffraction angle (angle-dispersive techniques), which experimentally determines $2\theta$, and Equation (1) is used to find the corresponding lattice spacing, $d^{hkl}$. From a practical viewpoint, there must be sufficient lattice planes oriented to fulfil the diffraction condition within the gauge volume probed by the radiation such that a high peak-to-background ratio is collected for the analysed crystallographic plane. In practice, for crystalline materials, this means a large number of randomly orientated grains. That is, materials should have a low level of texture and

a small grain size (compared with the gauge volume size). This is not always the case for AM structures, which can often exhibit elongated grains in the build direction.

The stress-free lattice spacing for a given crystallographic direction is referred to as $d_0^{hkl}$. This can vary significantly with position within additively manufactured parts, much in the same way it can vary in conventionally manufactured parts. The difference between the measured lattice spacing and this stress-free reference spacing can then be expressed as a residual elastic strain, $\epsilon^{hkl}$, by the equation:

$$ \epsilon^{hkl} = \frac{d^{hkl} - d_0^{hkl}}{d_0^{hkl}} \quad (2) $$

Equation (2) along with the hkl-specific elastic constants permit the determination of the stress in a particular direction through the application of generalised Hooke's law:

$$ \sigma_{xx} = \frac{E^{hkl}}{(1 + v^{hkl})(1 - 2v^{hkl})} \left[ (1 - v^{hkl})\epsilon_{xx} + v^{hkl}(\epsilon_{yy} + \epsilon_{zz}) \right] \quad (3) $$

$$ \sigma_{yy} = \frac{E^{hkl}}{(1 + v^{hkl})(1 - 2v^{hkl})} \left[ (1 - v^{hkl})\epsilon_{yy} + v^{hkl}(\epsilon_{xx} + \epsilon_{zz}) \right] \quad (4) $$

$$ \sigma_{zz} = \frac{E^{hkl}}{(1 + v^{hkl})(1 - 2v^{hkl})} \left[ (1 - v^{hkl})\epsilon_{zz} + v^{hkl}(\epsilon_{xx} + \epsilon_{yy}) \right] \quad (5) $$

where $E^{hkl}$ is the plane-specific Young's modulus and $v^{hkl}$ is the plane-specific Poisson's ratio. For all measurements described, the FCC {311} plane for austenitic stainless steel was used with values for $E^{hkl}$ and $v^{hkl}$ of 183.97 GPa and 0.294, respectively [32]. This lattice plane is recommended for use in FCC materials [27] as it is weakly affected by intergranular strains, and therefore,
the determined stresses can be compared with macro stresses determined by other methods.

In principle, the above procedure can be employed for all diffraction measurements. However, for laboratory X-ray diffraction, the in-plane stresses are typically derived using the sin²Ψ method. This is also the case for the SXRD data obtained from the P07 and P61A beamlines. The sin²Ψ method relies on changing the angle between the scattering vector and the sample by some angle Ψ within the diffraction plane resulting in a shift in the measured lattice strain for a given direction governed by the equation [33]:

$$ \frac{d_{\psi^+} + d_{\psi^-}}{2} \propto \sin^2\Psi \quad (6) $$

where $d_{\psi^+}$ and $d_{\psi^-}$ represent the $d^{hkl}$ measured at the angle of positive Ψ and angle of negative Ψ respectively, which is applied to the plane orthogonal to the incident beam.

This change in Ψ can be achieved through the rotation of the sample or through the use of an area detector [14], recording at different Ψ angles without rotating the sample. In the angle-dispersive case, an area detector was used in combination with a conical slit cell [34], defining a three-dimensional gauge volume. In the energy-dispersive case, a solid-state semiconducting point detector was used, and the sample rotated. By plotting the interplanar distance versus sin²Ψ, the lattice spacing corresponding to the orthogonal directions can be derived. This can improve the quality of the result as the stress is calculated from a fit to a set of points as opposed to a single data point. Moreover, it significantly improves the grain statistics as not only x, y and z directions are sampled but also directions in between.

The shear stresses can also be determined through the sin²Ψ/ sin 2Ψ method [33], and this analysis was applied to the SXRD data here from P07 and P61A. The governing expression for d-spacing on the plane orthogonal to the beam can be expressed as

$$ d_{\psi^+} - d_{\psi^-} \propto \sin(2\Psi) \quad (7) $$

The generalised Hooke's law was then used to calculate the shear stresses as follows:

$$ \sigma_{xz} = G^{hkl} \varepsilon_{xz} \quad (8) $$

$$ \sigma_{yz} = G^{hkl} \varepsilon_{yz} \quad (9) $$

where $G^{hkl}$ is the plane-specific shear modulus. The G value for the FCC {311} austenite plane is here taken to be 71.09 GPa [32].

## 2.2 | AM Microstructure and Texture

To ascertain the potential degree of texture in the obtained components and determine the possible suitability for conventional diffraction-based residual stress analysis for all instruments, a component was subjected to electron backscatter diffraction (EBSD). An arch was cut along its centre at y=0, providing an

$x-z$ cross-section for mapping. Figure 2 shows pole figure plots indicating a generally weak texture in the part, especially for the {311} lattice planes, which were used in the diffraction strain calculation by all instruments. This is in part a consequence of the 67° rotation between successive laser depositions. This weak texture means that the diffraction gauge volume will probe grains that are almost randomly oriented, preventing a bias towards strain in any particular direction. However, this only remains true if the mean grain size is sufficiently small when compared with the gauge volume. The somewhat higher texture recorded for the {100} suggests that the lower multiplicity {100} plane might prove more problematic if it were used to evaluate the residual stress. Figure 3 shows the grain structure from which it is evident that the grains are elongated along the build direction extending over a distance much greater than the layer thickness (50 µm). Such elongation of grains in the build direction beyond the layer height is common for 316L LPBF [35, 36]. The average equivalent grain diameter is estimated to be ~14 µm, significantly smaller than the gauge volume sizes used in the diffraction measurements as will be presented in the next section.

## 2.3 | Experimental Set-ups

Four strain scanners were used to determine the residual stress in the arches by SXRD (P07, P61A and ID15A) and ND (SALSA). The experimental set-ups for each are summarised in Table 2. As each strain scanner uses different beam optics, the definition of a representative gauge volume varies. When slits are used to define the gauge volume, the maximum cross-section of the gauge volume is assumed to be the width of the slits and that all the signal is coming from between the slits, as

<figure><img src="image_4.png" /><figcaption>FIGURE 2 | Pole figure plots (using a xRandom colourbar) indicate a low level of texture in the arch for three lattice planes, as determined by EBSD over an area containing 625 grains where z corresponds to the build direction.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**FIGURE 3** | Grain structure as determined by EBSD. The grains are elongated in the build direction.</figcaption></figure>

<table><caption>**TABLE 2** | Summary of the four beamlines used as strain scanners and their set-ups used to determine the residual stress in the arch.</caption><thead><tr><th>Strain scanner</th><th>Technique</th><th>Wavelength</th><th>2θ</th><th>Gauge dimensions</th><th>Gauge method</th><th>Scan lines</th><th>Stress calculation method</th></tr></thead><tbody><tr><td>P07 [37]</td><td>SXRD</td><td>0.011 nm</td><td>&lt; 20° (variable)</td><td>200×200 μm², ~1.9 mm</td><td>Conical slit cell</td><td>1, 2</td><td>Sin²Ψ</td></tr><tr><td>P61A</td><td>SXRD</td><td>White beam (many wavelengths)</td><td>5.972°</td><td>150×150 μm², ~2.9 mm</td><td>Double slit collimators</td><td>1, 2</td><td>Sin²Ψ</td></tr><tr><td>ID15A [38]</td><td>SXRD</td><td>White beam (many wavelengths)</td><td>5°</td><td>Horizontal: 200×50 μm², ~3.4 mm<br>Vertical: 200×50 μm², ~1.7 mm</td><td>Beam and detectors slits</td><td>1</td><td>Hooke's law</td></tr><tr><td>SALSA [39]</td><td>ND</td><td>~0.159 nm</td><td>98.5°</td><td>0.6×0.6×2 mm³</td><td>Radial focusing collimators</td><td>1, 2</td><td>Hooke's law</td></tr></tbody></table>

*Note:* Where possible, a reference describing how the respective strain scanner has been recently applied for residual stress quantification has been included. The gauge dimensions refer to the maximal cross-section followed by length in the beam direction where the majority of the signal is detected from by each strain scanner and the gauge method is how that volume is defined.

there is negligible beam divergence. When the gauge volume is defined by radial focusing collimators, the quoted gauge cross section instead refers to the full width at half maximum of the beam at the focal point of the detector's focus. This means that the majority of the signal will come from within this focal region, though some will originate outside of this. The diffraction angles used also vary with strain scanner; therefore, the shape of the gauge volume also changes. A sketch of the general shape of the gauge volumes for comparison and associated beam paths are given in Figure 4. When data are taken along the scanned lines, the locations indicated are the centre of the gauge volume positions, but it is noted that the extent of the region within the part that will contribute to the signal will be different for each strain scanner. The lines scanned are represented in Figure 1.

### 2.3.1 | SXRD at Hereon P07 Beamline

A monochromatic X-ray beam of wavelength 0.011004 nm was selected from an undulator source on the PETRA III storage ring at DESY (Deutsches Elektronen-Synchrotron). The gauge volume had an elongated shape with a cross-section of 200×200 µm² as defined by conical slits [34] and the elongated length in the direction of the beam was ~1.9 mm, creating a similar shape to the sketch in Figure 4b. Data were acquired along both Line 1 and Line 2 at 0.1- and 0.5-mm intervals, respectively. The 2D diffraction images were integrated with the software Fit2D [40] using a full 360° azimuthal integration with 20° intervals. A Voigt function with a linear background was used for peak fitting for the selected crystallographic plane. The sin²Ψ/sin²Ψ method was applied
<figure><img src="image_6.png" /></figure>

**FIGURE 4** | Gauge volumes (yellow) as defined by the incident beam (I) and the portion of the diffracted beam (D) which is detected (a) representative of ND with $2\theta \approx 90^\circ$ and (b) representative of SXRD with $2\theta \ll 90^\circ$. The sample orientations needed to determine the strain in the *z* (top) and *y* (bottom) directions are shown in each case. This is a generalised sketch for comparison between SXRD and ND, and the true gauge volume will vary with the experimental set-up. The longest gauge direction was always set to be within the *x*–*y* plane for it to be fully enclosed within the thin ligament of the arch.

to find the associated strain components in *x*, *z* and *xz* before the sample was rotated $90^\circ$ so that the final *y* and *yz* strain components could be determined. These strain components were then used to find the associated stress in the three normal directions and the shear stress.

### 2.3.2 | SXRD at Hereon P61A Beamline

A white X-ray beam was used from a wiggler source at the PETRA III storage ring in combination with an energy-dispersive point detector. The gauge volume had an elongated diamond shape with a cross-section of $150 \times 150 \mu m^2$ and length of ~2.9 mm elongated in the direction of the beam due to the diffraction angle of $2\theta = 5.97^\circ$, resulting in a similar shape to the sketch in Figure 4b. Data were taken along both Line 1 and Line 2 at 0.2- and 0.5-mm intervals, respectively. In this case, a pseudo-Voigt function was used for peak fitting and a $10^\circ$ Chebyshev polynomial was used for the background. The $sin^2\Psi/sin2 \Psi$ method was applied to find the associated normal strain and stress components. The shear stresses were also calculated. Peak fitting was carried out with the open-source P61A Toolkit software [41].

### 2.3.3 | SXRD at ESRF ID15A Beamline

The high-energy beamline ID15A at ESRF in Grenoble, France, utilises a wiggler photon source, to produce a high-flux beam with a broad spectrum of energy up to several hundred keV. Two solid-state semiconducting germanium single-crystal detectors cooled with liquid nitrogen were used to record simultaneously the strain in two orthogonal directions. The two detectors were fixed approximately at a diffraction angle of $2\theta = 5^\circ$, and the multichannel analyser (MCA) was set to 4096 channels. The conversion calibration from channels to energy was carried out by fitting the fluorescence spectrum of a well-known barium radioactive source using a second-order polynomial [42, 43]. The exact diffraction angles of the two detectors were determined using a cerium dioxide powder diffraction standard from NIST (CeO$_2$–SRM 674b). An incident beam with dimensions of $200 \mu m \times 50 \mu m$ was defined using a pair of vertical and horizontal slits. The diffracted beams recorded by the detectors were selected using two sets of slits with an aperture of $100 \mu m$ positioned after the sample and before the detectors. The dimensions of the gauge volume are given in Table 2, with the resulting shape being similar to that in Figure 4b. The peak fitting routine was facilitated by the EASI-Strain package [44], and the peak
fitting was done using a pseudo-Voigt function with a linear background function. The three principal strain components were measured along Line 1 with a step size of 100 µm. Then, the stress was calculated for three orthogonal directions using Equations (3–5).

### 2.3.4 | ND at ILL SALSA Beamline

A monochromatic neutron beam with a wavelength ~0.159 nm was selected such that the diffraction peak was at 98.5°, giving a cuboidal gauge volume similar to that in Figure 4a of 0.6×2×0.6 mm³ as defined by radial collimators. Data were taken along Line 1 and Line 2 at non-uniform intervals. The ILL in-house software LAMP [45] was used to do the peak fitting, using a Voigt function and a linear background, in order to find the precise angle of the diffraction peak, which was then used to calculate the strains. The three orthogonal stresses were then calculated with Equations (3–5).

## 2.4 | Determination of Stress-Free Lattice Spacing

Determining what constitutes an appropriate stress-free reference sample for an additively manufactured part is not a simple matter. Although the feedstock powder is naturally free of Type 1 macro stresses, employing it as the stress-free reference is not appropriate given the potential for loss of alloying elements during AM and the severe non-equilibrium thermal excursions experienced by the part during manufacture. These can fundamentally change the elemental composition and microstructure of the alloy when compared with the raw feedstock powder [46]. Additionally, the alloy composition may change because of dilution from the substrate [47]. In this respect, even small variations in the elemental composition of the alloy can have large effects on the lattice spacing [48]. As a result, the repeated thermal excursions from subsequent depositions can affect the solute concentrations in many alloys (especially heat-treatable ones) and thereby the stress-free lattice parameter [49]. As a result, different locations in the part can potentially have very different lattice spacings.

Though there are a number of ways to obtain a good measure of the stress-free lattice parameter, mechanical relief of Type 1 residual stresses through cutting an identical component (with the same processing history) into small stress-free cuboids is the preferred option [4]. In principle, this should be done in such a way that no further stresses are introduced. Although it is often impossible to completely relieve all stress components acting in all orientations with this approach, it significantly mitigates the influence of variations in texture and composition as it is functionally the same part with the same processing history. For ease of handling, a 'comb' consisting of a series of cuboids that are still connected by a small bridge was produced by electrical discharge machining (EDM), as shown in Figure 5a, where the internal solid lines indicate a cut into the cross-section but not completely through it. This comb allows the stresses to be largely relieved while preserving the relative location and orientation of the cubes in case the stress-free lattice spacing varies within the sample. The comb consisted of cubes with a depth of 3 mm, width of 2.5 mm and height of

<figure><img src="image_7.png" /><figcaption>**FIGURE 5** | (a) Sketch of comb used as stress-free reference cut from an identically made arch by EDM. With indication of the teeth used for Line 1 (orange) and Line 2 (blue) measurements. (b) Photograph of the actual EDM cut comb. The internal lines in 5a correspond to the cuts to create the teeth seen in 5b.</figcaption></figure>

2 mm. These dimensions ensure that the gauge volumes can be completely enclosed, preventing shifts in diffraction peaks caused by partially filled gauge volumes, referred to as pseudo strains [50]. Measurements along the scan Lines 1 and 2 were taken within each tooth to provide a $d_0$ value for that measurement location.

## 2.5 | Finite Element Model

The normal stresses ($\sigma_{xx}$, $\sigma_{yy}$ and $\sigma_{zz}$) within the arch have also been predicted using the inherent strain method (IHS) using the Amphyon [51] finite element package. The model geometry that was used was the same as the input for the LPBF print file. This was modelled as an individual part laid down on the base plate, whereas experimentally many parts were manufactured on the same base plate. The material properties for 316L are manufacturing dependant and calibrated for the printing parameters through several cantilever specimens. The modeling approach, materials parameters used and a description of the inherent strain method have previously been described by Sanchez-Poncela et al. [39].

# 3 | Uncertainties When Determining Residual Stress by Diffraction

The uncertainties in diffraction experiments can be numerous. Small shifts in peak position can have a large effect on the perceived strain, so it is important that the fitting is accurate and consistent. This includes corrections for any background effects and accounting for possible asymmetric peaks caused by peak overlap of two phases or attenuation broadening. The fitting of the data from the different strain scanners was completed by following each institution's standard approach within their recommended software. This ensured consistency within any given campaign, limiting the effect of
systematic offsets in the stress calculation. Although the general steps were the same, this means that the implementation was not completely identical between campaigns. Regardless, calculations carried out are such that any user of these institutions would be encouraged to carry out their analysis and therefore comparing the reproducibility of the results underpins the aim of the present work.

Given that high-energy diffraction techniques can penetrate into the bulk to determine the residual stress, it is very important to know exactly where in the bulk the stress is being probed, especially in areas of high-stress gradients. Slightly different methods for locating the part within the beam can be used, but the most common is the use of an 'entry scan'. The uncertainty as to the exact position of the sample is ~10% of the size of the gauge volume width used for the entry scan [52]. A photodiode was used to determine the sample position in a plane orthogonal to the beam at the SXRD beamlines. The centre of the gauge volume in the beam direction was set mechanically to a known focal distance at P07 (a 1-mm-thick piece of the same material was previously scanned to determine the focal distance). A similar approach relies on absorption, placing a detector on the through path of the beam and moving the part in front of it; as the part enters the beam path, the measured intensity will drop, providing location information in a similar manner. This approach was used at P61A and at ID15A to determine the centre of the gauge volume. Optical alignment can also be employed. At SALSA, this approach was used and has been shown in the past for the present gauge volume employed to have an uncertainty < 100 µm [52].

Although the centroid positions of the gauge volumes representative of the various instrument set-ups are along the same linear path, because of the differing size and shape of the gauge volumes, the stress calculated might differ in areas of the component where there are stress gradients. The gauge volumes used for SXRD are much smaller than that for ND. This may result in different average levels of residual stress being determined for the same nominal location across the different instruments.

The choice of the diffraction elastic constants used directly impacts the stress calculated but will not affect the trends in the residual stress when compared with other areas in the part. As such having some knowledge of the possible uncertainty caused by the choice of elastic constants is useful, as it, along with the other uncertainties, provides a bound on the residual stress in the part. Here, $E^{311}=183.97$ GPa and $\nu^{311}=0.294$ were assumed, which were taken from an X-ray elastic constant software package ISODEC [32]. For comparison, when measured experimentally at an X-ray synchrotron source, Chen found $E^{311}=179.6$ GPa [53] for an AM 316L stainless steel. For wrought 316L, Clausen obtained $E^{311}=183.2$ GPa [54] and Zhang $E^{311}=183.7$ GPa [55] through theoretical calculations. The actual diffraction elastic constant may also be affected by any texture. The variation in these values is all in the region of a few percent which is considerably less than the other uncertainties in the measurement procedure. A similar level of uncertainty is also expected for $G^{311}$.

A measure of the uniformity of the build, and therefore the validity of assuming these singular elasticity constants, can be

inferred from the $d_0$ measured across the comb at a constant height in the build. For the comb teeth measured along Line 2 at P61A indicated in Figure 5, this variation in $d_0$ is $1.2 \times 10^{-4}$ Å. This is a relatively small variation, which would correspond to a maximum stress variation of around 20 MPa.

Given the consistent nature of the diffraction elastic constants and the positional uncertainties across all instruments being similar, the error bars that appear in the reported stress figures presented in the results consist only of uncertainties propagated from the fitting of the diffraction peaks.

## 4 | Results

As shown in Figure 6a,b,c along scan Line 1 (see Figure 1), the in-plane residual stress ($\sigma_{xx}$) in the as-built condition is typified by high tensile stresses at the top surface of the arch, balanced by compressive stresses deeper into the arch [21, 56]. Indeed, the highest residual stress (near the top surface) is essentially representative of the stress in the last layers to be deposited and approaching the yield stress. The FE model also predicts that the $\sigma_{xx}$ stress should transition from tension on the top surface to compression on the underside of the arch (the first layers laid down in this region), although it predicts much higher compressive stresses there. This is also consistent with other experimental investigations of residual stress in AM parts [11]. The residual stress in the y direction, $\sigma_{yy}$, is also tensile at the top surface but reduces to a low tensile or near zero stress towards the apex of the arch. The in-plane stresses on the top surface $\sigma_{xx}$ (380 ± 30 MPa) and $\sigma_{yy}$ (360 ± 40 MPa) suggest an equi-biaxial stress state, which, given the rotation of the laser tracks, is not surprising. Towards the underside of the arch, the in-plane stresses begin to diverge significantly, which has been attributed to the geometry and is also captured by the FE mode, albeit the compressive stresses are overpredicted. In this respect, it is worth noting that in LPBF, the overhanging arch is built upon supporting metal powder, whereas the model assumes the arch contains free space. In this respect, it is worth noting that in LPBF, the overhanging arch is built upon supporting metal powder, whereas the model assumes the arch contains free space. Finally, the stress in the build direction, $\sigma_{zz}$, is near-zero and fairly constant along this path indicating an approximately plane stress condition, in agreement with the FEM.

The residual stress components along scan Line 2 are shown in Figure 7. The in-plane stress is essentially constant in the middle region of the part equating to around $230 \pm 40$ MPa and $270 \pm 30$ MPa in the x and y directions, respectively. Within ~4 mm from the ends, $\sigma_{xx}$ falls to zero as it must from boundary conditions. Within around 1 mm of the ends, $\sigma_{yy}$ is predicted by FEM to rise; however, this was not observed because the scans extend only to within 1–3 mm from the ends. In practice, a small step size would be needed to observe the sharp stress profile in this region. As was observed for the central ligament, the stress in the direction of the build, $\sigma_{zz}$, is almost zero ($16 \pm 10$ MPa) across the whole line for SXRD. The fact that the stress measurements vary within $\pm 50$ MPa of the predictions gives confidence that this represents the measurement uncertainty of the stress measurements. The FEM model mainly overpredicts $\sigma_{zz}$, when close to the external surface.
<figure><img src="image_8.png" /><figcaption>**FIGURE 6** | Residual stress in (a) x, (b) y and (c) z directions along Line 1 determined by the four strain scanners. ND from SALSA (green), SXRD from P07 (black), P61A (red) and ID15A (blue) and as predicted by the inherent strain finite element prediction (orange). Error bars represent uncertainties propagated from the peak fitting only.</figcaption></figure>

As was the case for Line 1, this effect can be attributed to the fact that the simple FE model does not accurately account for heat transfer.

The $\sigma_{xz}$ and $\sigma_{yz}$ shear stresses, as measured by P07 and P61A, are shown in Figure 8 for Lines 1 (a) and 2 (b). Along Line 1, the shear stresses are low (close to or below 40 MPa) in all measured

<figure><img src="image_9.png" /><figcaption>**FIGURE 7** | Residual stresses in the (a) x, (b) y and (c) z directions along Line 2 as measured by ND at SALSA (green) and SXRD at P07 (black) and P61A (red) with the inherent strain finite element prediction (orange). Error bars represent uncertainties propagated from the peak fitting only.</figcaption></figure>

locations. This indicates that at the centre of the arch, the measured orthogonal directions are close to the principal directions of the arch, as one would expect. Along Line 2, for the majority of the path, the shear stresses are also low (~ 40 MPa or less). Within 4 mm of the ends, the shear stresses ($\sigma_{xz}$) begin to rise, reaching values of around 80 MPa close to the edges, being positive at one side and negative at the other, reflecting the shedding of the in-plane $\sigma_{xx}$ stress as it falls to zero at the ends.
<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>**FIGURE 8** | Shear stress in the arch along Line 1 (a) and Line 2 (b) determined by P07 (black) and P61A (red).</figcaption></figure>

**5 | Discussion**

It is reassuring that although one X-ray strain scanner is energy dispersive and the other is angle dispersive, the results for P07 and P61A, which employ similar gauge volumes, give very similar estimates of the stress with the differences between them being on the order of 20–30 MPa. For Line 1, the in-plane SXRD stress data from the ID15A follow the same trends as the other SXRD experiments and lies only marginally above or below the P07 and P61 curves. Except for the $\sigma_{xx}$ stresses along Line 2, which are around 70 MPa more compressive than for the other measurements, the ND results show good agreement with the SXRD in the bulk of the AM sample, with similar uncertainties. The data range for the ND presented is smaller, to ensure that the larger gauge volume is completely enclosed, preventing the influence of partially filled gauge volume-induced pseudo strains [50] from being reported as a true stress. This would appear as an S curve function convoluted within the data set, artificially reducing the reported tensile stress close to the top surface and also reducing the reported compressive stress at the apex of the arch.

The standard deviation is ~50 MPa for $\sigma_{xx}$ when averaged along Line 1 and lower for all other lines and directions. This variation is low when considering the uncertainties when performing residual stress analysis, especially in this case where each facility has measured a physically separate sample with different gauge volume shapes and sizes. The extent to which the gauge volume size is likely to affect the reported stress can be inferred from the FE prediction. When effective gauge volumes of 0.6 and 0.2 mm are created by applying a bounding box encompassing the integration points of these sites and then comparing the stress at 0.1-mm intervals along Line 1, the mean variation in predicted stress is $4 \pm 4$ MPa, with a maximum of 17 MPa. This is low when compared with the other factors and indicates that the choice of gauge volume in these parts is not a defining factor in the differences in stress reported at the centre of the gauge volume. The shear stresses have also been shown to be comparable when measured at different SXRD beamlines. As a result, both ND and SXRD have been found to provide excellent results for this benchmark component.

It is worth noting that the SXRD method has the potential to provide a better spatial resolution in two directions, with less susceptibility to near-surface effects. However, strain measurements on coarse-grained samples remain very challenging. Typically, an SXRD strain scanner presents a highly parallel incident beam. When coupled with the smaller gauge volumes typical of SXRD versus ND means that in many cases the number of grains diffracting within the gauge volume can be too small for powder diffraction analysis. This can be evident in the data as big variations in peak intensity as the sample is traversed through the gauge volume and can give point-to-point variations in the recorded strain, which are much greater than those expected in terms of the diffraction peak fit. Although the likelihood of this issue occurring is reduced by the application of the $\sin^2\Psi$ method, this can still be a significant detractor for AM parts because of the potential for strong texture and elongated grains. This remains a concern for both energy-dispersive and angle-dispersive techniques. Further, synchrotron beams may not have the penetration needed for longer path lengths in larger, thicker parts. ND can therefore give potentially more reliable results in cases where the application of SXRD is challenging by utilising a larger gauge volume. As a result, the most appropriate method will depend on the AM part to be measured.

Although this paper has focused on single peak fitting, it is possible to conduct full diffraction profile fitting of multiple diffraction peaks [57], which can also improve grain sampling statistics reducing texture effects, and as many diffraction planes are being considered, it is less susceptible to elastic and plastic anisotropy. Full diffraction profile acquisition and fitting can be undertaken by energy-dispersive X-ray, or time of flight neutron, diffraction. Given the wide range of diffraction peaks observable, these approaches can also be more beneficial when investigating multiphase materials.

Given that AM is particularly advantageous for the building of complex 3D shapes, it is not always easy to determine the principal directions a priori. In this case, the relatively simple geometry and the locations at which the line scans have been
undertaken mean that the three perpendicular stresses measured here are in most locations aligned with the principal stresses. Consequently, this leads to low residual shear stress magnitudes as confirmed by P07 and P61A measurements for the majority of scan Lines 1 and 2. Only near the ends of the samples where the in-plane stress $\sigma_{xx}$ must fall to zero, are there significant shear stresses as the principal axis rotates away from the x-direction as would be expected. It should also be borne in mind that the AM method can introduce considerable crystallographic texture that can further complicate the measurement of stress and the collection of *hkl* diffraction peaks in the required directions. The AM processing strategy used in the present work has proven effective in minimising texture.

## 6 | Summary

Given the propensity for large thermally driven residual stresses in additively manufactured parts, it is important to be able to measure them and the efficacy of any mitigating thermal treatments non-destructively. Although diffraction methods have potential, making accurate measurements is no easy task. There are a number of considerations that must be taken into account if reliable measurements are to be made. This paper has sought to identify some of these and to evaluate their importance for an exemplar additively manufactured stainless steel arch. These include grain size and texture, experimental challenges associated with the accurate location of the gauge volume, choice of appropriate *hkl*-specific elastic constants and the determination of the location-specific stress-free lattice spacing.

Three synchrotron X-ray and one neutron diffraction beamline have been used to determine the residual stress along two profiles within an LPBF additively manufactured 316 stainless steel arch. Despite the measurements being made on different benchmark samples, good agreement is found between the four strain scanners revealing a high tensile (380 ± 50 MPa in both x and y directions) in-plane stress at the top surface and a more compressive stress (–130 ± 30 MPa and 60 ± 40 MPa in the x and y directions, respectively) on the underside of the arch. Despite the presence of the arch, the tensile stresses along the top of the arch were fairly constant all the way across with $\sigma_{yy}$ (270 ± 30 MPa) being only slightly more tensile than $\sigma_{xx}$ (230 ± 30 MPa). As it must, $\sigma_{xx}$ fell to zero at each end with the fall occurring within 4mm of the ends. A simple FE model was in broad agreement with the residual stress distributions albeit overestimating the compressive stresses on the underside of the arch.

This comparative study demonstrates that under favourable conditions, neutron and synchrotron measurements are equally suitable for measuring stresses to within ±50 MPa. As a result, high-energy synchrotron diffraction holds an advantage for fine-grained materials when the path lengths are not prohibitive because of the higher flux and smaller gauge cross sections (lateral spatial resolutions). However, ND with millimetric spatial resolution is more suited to large AM parts and cases whether the grain size is somewhat larger. Full diffraction profile fitting of peaks can be advantageous over single peak fitting both in terms of improved grain sampling statistics and less sensitivity texture as well as to elastic and plastic anisotropy.

## Acknowledgements

This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 953219. The Deutsches Elektronen-Synchrotron DESY (Hamburg, Germany) is acknowledged for the provision of beam time at the Hereon P07 High Energy Materials Science Beamline and at the Hereon P61A White Beam Engineering Materials Science beamline. The authors gratefully acknowledge the European Synchrotron Radiation Facility for beamtime allocation at ID15A beamline and the Institut Laue-Langevin for beamtime allocation at SALSA. RCL, MR and PJW are grateful to the Henry Royce Institute established through EPSRC Grants EP/R00661X/1, EP/P025498/1 and EP/P025021/1.

## Conflicts of Interest

The authors declare no conflicts of interest.

## Data Availability Statement

The data that support the findings of this study are openly available in zenodo at https://zenodo.org/records/13890709.

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