Precision and accuracy of stress measurement with a portable X-ray machine using an area detector

research papers J. Appl. Cryst. (2017). 50, 131–144 https://doi.org/10.1107/S1600576716018914 131 Received 27 July 2016 Accepted 26 November 2016 Edited by K. Chapman, Argonne National Laboratory, USA Keywords: X-ray diffraction; stress measurement; area detectors; portable X-ray devices. Precision and accuracy of stress measurement with a portable X-ray machine using an area detector Seung-Yub Lee, a Jinjing Ling, a Shenghe Wang a and Joaquin Ramirez-Rico b * a Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA, and b Fisica de la Materia Condensada – ICMS, Universidad de Sevilla – CSIC, 41012 Sevilla, Spain. *Correspondence e-mail: [email protected] The use of portable X-ray stress analyzers, which utilize an area detector along with the newly adopted ‘cos’ or full-ring fitting method, has recently attracted increasing interest. In laboratory conditions, these measurements are fast, convenient and precise because they employ a single-exposure technique that does not require sample rotation. In addition, the effects of grain size and orientation can be evaluated from the Debye ring recorded on the area detector prior to data analysis. The accuracy of the measured stress, however, has been questioned because in most cases just a single reflection is analyzed and the sample-to-detector distances are relatively short. This article presents a comprehensive analysis of the uncertainty associated with a state-of-the-art commercial portable X-ray device. Annealed ferrite reference powders were used to quantify the instrument precision, and the accuracy of the stress measurement was tested by in situ tensile loading on 1018 carbon steel and 6061 aluminium alloy bar samples. The results show that the precision and accuracy are sensitive to the instrument (or sample) tilt angle ( 0 )aswellastothe selected hkl reflection of the sample. The instrument, sample and data analysis methods all affect the overall uncertainty, and each contribution is described for this specific portable X-ray system. Finally, on the basis of the conclusions reached, desirable measurement/analysis protocols for accurate stress assessments are also presented. 1. Introduction X-ray diffraction techniques have been used to determine the near-surface stress state of crystalline materials for nearly a century, and significant advances have been made in both hardware technologies and analysis methods (Noyan & Cohen, 1987). The former includes high-resolution detectors, modern optical devices and associated electronics. The latter corresponds to the advanced grain interaction models (Kroner, 1958; Hashin & Shtrikman, 1962; Do ¨lle, 1979), as well as to the integration of finite element simulations (Chen & Kovacevic, 2003; Clausen, Lee et al., 2003). High precision and accuracy of the measured stress must be balanced with other important criteria, such as measurement speed and portability for industrial applications. Formerly, small devices with portable films provided less reliable results compared to a laboratory diffractometer, but advances in the reliability of portable equipment have opened the door to in situ and in-line measurements, which are of great industrial interest. The idea of miniaturization of the X-ray system itself is not new at all. Early reports of portable devices from the 1960s described residual stress measurement in aircraft parts using back-reflection photographic film techniques (Bolstad, 1967; Homicz, 1967). Later generations of portable devices (James & Cohen, 1978a; Ruud et al., 1984; Araki, 1989; Brauss et al., 1996; Monin et al., 2003; Farrell, 2010) utilized position ISSN 1600-5767 #2017 International Union of Crystallography sensitive detectors (PSDs) to minimize or completely remove detector rotation because non-digitized film reading is laborious (Monin et al., 2000) and any motion of sample or detector can introduce significant errors due to the short sample-to- detector distance. For example, the double exposure technique is ten times more sensitive to the displacement error than a single exposure (Ruud et al., 1984). Two-dimensional image plates, also known as storage phosphors, were developed in the 1980s, first for diagnostic radiography (Sonoda et al., 1983) and later for X-ray diffraction (Miyahara et al., 1986; Amemiya & Miyahara, 1988; Yoshioka & Ohya, 1992). This laser-stimulated luminescence technique replaced the old films and even one-dimensional PSDs in portable X-ray systems. Other types of area detectors, such as two-dimensional PSDs (multi-wire PSDs) and charge coupled devices (CCDs), also competed with the image plate. Detailed descriptions of the advantages and disadvantages of each detector (Eathough et al., 1999) and a performance comparison between image plates and CCDs (Kiss et al., 2002) can be found elsewhere. Another critical component in the success of new portable devices utilizing area detectors is the development of stress determination methods that take full advantage of what has been termed ‘two-dimensional’ diffraction: cos(Taira et al., 1978; Taira & Tanaka, 1979), XRD 2 (He, 2011) and full-ring fitting methods (Kampfe et al., 2000), 1 all of which make use of the contour of the Debye ring. Regardless of detector type, the sin 2 method was ubiquitous until the 1980s; since the 1990s, two-dimensional methods have become more available along with high-resolution area detectors. Currently, there are many competing efforts developing high-resolution portable X-ray devices 2 for industrial applications. Initially these were targeted for the steel industry (Ganesh et al., 2013; Farrell, 2010), but applications are expanding alongside the shrinking scale of functional materials. Despite the obvious benefits of a portable device, there have been no objective evaluations of the uncertainty budget of advanced, commercial, portable X-ray units from third parties without conflict of interest. Furthermore, there has been no agreement among experts about the most efficient, precise and accurate stress determination methods. For these reasons, we chose one of the state-of-the-art portable X-ray devices equipped with an image plate and carried out a series of systematic tests on both precision and accuracy of the stress measurement. A schematic of the portable device used in this study is displayed in Fig. 1(a), and a typical Fe 211 powder pattern is shown in Fig. 1(b). We previously reported that the strain precision of this portable machine is about 9 microstrain (m") with ferrite reference powders (Ling & Lee, 2015), and that in essence the sin 2 and cosmethods are theoretically identical (Ramirez-Rico et al., 2016) and yield statistically indistinguishable experimental results (Ramirez-Rico et al., 2016; Ling & Lee, 2015). The mathematical equivalency between sin 2 and coswas also shown very recently (Miyazaki & Sasaki, 2016). Stress measurement with an image plate and the cosmethod is now known to be fast (90 s for steel), convenient (no center of rotation alignment) and precise (2 MPa repeatability in ferritic steel), in a well controlled environment. However, measured stress values often differ from the known applied ones, and only a narrow range of measurement conditions have been tested. Thus it is necessary to investigate measurement accuracy in more relaxed environments that resemble field or production line conditions. In this article, we report comprehensive uncertainty analyses of a portable X-ray device, including the measurement parameter dependency of both stress precision (x4.1) and accuracy (x4.2), as well as the effect of the choice of a particular reflection (x4.3). Uncertainties from each error source are discussed in x5. 2. Stress determination methods Diffraction-based stress determination is an indirect conversion process from a strain, ", to a stress, , tensor via a constitutive law such as ij ¼Cijkl"kl;ð1Þ where C ijkl are the components of the elastic stiffness tensor. If a material is elastically isotropic, equation (1) can be simplified to research papers 132 Seung-Yub Lee et al. Precision and accuracy of stress measurement J. Appl. Cryst. (2017). 50, 131–144 Figure 1 (a) A schematic diagram of the portable X-ray device used in this study. The detector (shaded in blue) has a diameter of 60 mm with a 6 mm hole at the center for beam penetration. (b) A typical diffraction pattern of Fe 211 viewed from the top. Data used for the sin 2 and cosmethods are marked with hollow circles and dashed lines, respectively. 1 XRD 2 and the full-ring fitting method are basically the same, but only part of the ring is used in the XRD 2 system. 2 m-X360 (Pulstec), SmartSite RS (Rigaku), mXRD (Proto), Xstress 3000 (Stressteck), MAX (TEC), Xsolo (Inel) etc. are currently available portable X- ray devices for stress measurements. "ij ¼1þ Eij ij  Ekk;ð2Þ with two independent parameters, Young’s modulus Eand Poisson’s ratio . Here ij ¼0ifi6¼ jor ij ¼1ifi¼j.The interplanar spacing d hkl is used as a built-in strain gauge and calculated from the diffraction peak position in the laboratory coordinate system ( 0 ,’ 0 ); corresponding strains are transformed into the sample coordinates ( ,’) before stress conversion. Starting from the basic formula of equation (2), the next two sub-sections briefly describe the equations used in the sin 2 and cosmethods. 2.1. The sin 2 wmethod The strain measured along the scattering vector at angles ( ,’) from the zand xaxes can be written in terms of the strain component "ij in the sample coordinate system as " ’ ¼d ’ d0 d0 ¼"11 cos2’þ"12 sin 2’þ"22 sin2’"33  sin2 þ"33 ð3Þ by assuming a bi-axial plane stress condition without steep stress gradients in the near-surface region. Here d ’ is the interplanar spacing at and ’,andd 0 is the unstressed interplanar spacing. If the material’s elastic behavior is isotropic and homogeneous, equation (3) can be expressed in terms of stress via equation (2), as follows: d ’ d0 d0 ¼1þ E’sin2  E11 þ22 ðÞ;ð4Þ where ’¼11 cos2’þ12 sin 2’þ22 sin2’. If the in-plane stress is not dependent on angle ’, the equation can be further simplified as 11 ’¼0ðÞ¼ E 1þ @" @sin2  ;ð5Þ which is the well known sin 2 equation. The bi-axial in-plane stress can then be determined from the slope of a d versus sin 2 plot. If the d versus sin 2 plot is linear, only two measurements at = 0 and , the so-called ‘two-tilt’ method, are required, as expressed in equation (6) with ’=0: 11 ’¼0ðÞ¼ E 1þ 1 sin2 cot  2202  ¼K2; ð6Þ where the stress constant K¼E 1þ 1 sin2 cot  2:ð6aÞ is the Bragg angle, which is about ( 0 + )/2, where  0 and  are the diffracting angles at =0and , respectively. This two-tilt sin 2 method using an area detector is very useful because the displacement error due to sample rotation is eliminated. 2.2. The cosamethod The two-dimensional cosmethod was first proposed in the late 1970s (Taira et al., 1978; Taira & Tanaka, 1979) using photographic film. It was applied to the image plate in the early 1990s (Yoshioka & Ohya, 1992), and has been further developed for advanced use since the late 1990s (Sasaki et al., 1997; Sasaki & Kobayashi, 2009; Sasaki et al., 2014). Suppose that a diffraction ring forms on the image plate through which the incident beam passes with an angle of 0 and ’ 0 from the z and xaxes, as shown in Fig. 1. The strain "projected along a direction with angle can be expressed as the following, under the same assumptions as in the sin 2 method: "¼11 1 E n2 1n2 2þn2 3  þ22 1 E n2 2n2 1þn2 3  þ12 21þðÞ E  n1n2 ðÞ;ð7Þ where n1,n2,n3are the directional cosines of the normal of the diffracting plane with respect to the axes in the sample coordinates and depend on ,, 0and ’0. Substituting ,þ and for in equation (7), we can then obtain the following relations: a1’0 ðÞ¼ 1 2ð""þÞþð""Þ  ;ð8Þ a2’0 ðÞ ¼1 2ð""þÞð""Þ  :ð9Þ Expressing equations (8) and (9) in terms of stress after reexpressing the directional cosines leads to the final relationships for this method: 11 ’0¼0ðÞ¼ E ð1þÞ 1 sin 2 1 sin 2 0 @a1 @cos   ;ð10Þ 12 ’0¼0ðÞ¼ E 2ð1þÞ 1 sin 2 1 sin 0 @a2 @sin   ;ð11Þ where is defined in Fig. 1. For a given 0 and , we can get the in-plane normal stress from the slope of the a1versus cos  plot when ’0¼0 from equation (10). There are other two-dimensional methods such as XRD 2 or full-ring fitting. However, only cosis chosen in this study as a representative of two-dimensional methods because the fundamental equation of the XRD 2 method is basically identical to equation (7), and it measures only part of the Debye ring (Miyazaki & Sasaki, 2016), while the cosand full-ring methods use all of the information. We have also shown experimentally that, when using the same dataset, there was no difference between the cosand full-ring fitting methods (Ramirez-Rico et al., 2016). In summary, equations (5) and (10) are the basic equations for the sin 2 and cosanalysis in isotropic, homogeneous and bi-axial stress conditions. For anisotropic, heterogeneous or multi-axial stress cases, these equations have to be modified as reported previously by Noyan & Cohen (1987) and Do ¨lle (1979) for the sin 2 method and by Sasaki and co-workers (Sasaki et al., 2014; Sasaki & Kobayashi, 2009) for the cos method. research papers J. Appl. Cryst. (2017). 50, 131–144 Seung-Yub Lee et al. Precision and accuracy of stress measurement 133 3. Experiment and data analysis 3.1. Samples Three samples were tested: (1) Fe reference powders (E: 224 GPa, : 0.28) provided by the manufacturer of the portable X-ray device, (2) solid bars of 1018 mild carbon steel (E: 203 GPa, : 0.28) and (3) solid bars of 6061 aluminium alloy (E: 69 GPa, : 0.33) purchased from the McMaster–Carr supply company. The Fe reference powders were annealed to remove any pre-existing defects and residual stress. With average grain size > 1 mm, size or strain broadening effects were not expected in the diffraction patterns of these powders. Measurements of the reference powders were therefore used to characterize the intrinsic instrument precision of the portable X-ray device. The solid bar samples of Fe and Al, both with dimensions 0.25 0.25 2400 (100 = 25.4 mm), were loaded in tension for accuracy testing. Since the Fe is the main testing sample, we measured the microstructure of the carbon steel bar by optical microscopy (ZEISS Axio Scope) and electron backscatter diffraction (EBSD) (JEOL 5600 equipped with HKL Nordlys and Channel 5 software; Oxford Instruments, Abingdon, Oxfordshire, UK), as shown in Fig. 2. These measurements clearly display elongated grains along the vertical (Z) direction. X-ray diffraction from 20 to 110in 2with a Cu source (D500 diffractometer) probing both the top cross section (XY plane) and the side wall (XZ plane) revealed that some (110) texture was developed in the Z direction, similar to wire extrusion, while grains parallel to the side wall (XZ plane) are distributed quite isotropically. The 110 peak intensity ratio [I 110 /(I 110 +I 200 +I 211 +I 220 )] is 0.74, 0.66 and 0.63 for the top cross section, side wall and JCPDS reference, respectively. The EBSD analysis from the top cross section also shows a larger population of (110) grains than seen in the side walls. 3.2. In situ diffraction measurements in tensile loading The 1018 mild carbon steel and 6061 Al alloy bar samples were placed in an Instron 5984 Universal Testing Machine for tensile loading up to 300 and 200 MPa, respectively. Experiments were carried out under load control to keep the elastic stress constant during diffraction measurements, and samples were held by pneumatic grips as shown in Fig. 3. A miniature portable X-ray apparatus was mounted next to the sample (sample-to-detector distance 30–60 mm) with various tilt angles ( 0 in the range 5–48). The load was increased in 25 or 50 MPa steps. At each load we performed diffraction measurements for about 90 s for Fe and 120 s for Al. The specifications of the portable X-ray unit are the following: tube voltage (30 kV), current (1 mA), source (Cr Kwith  filter), X-ray energy (=2.29A ˚,E= 5.4 keV), collimator (1 mm diameter), beam size (2 mm diameter), sensor unit weight (5 kg), power supply weight (6 kg), energy consumption (80 W in operation, 30 W in standby mode). The specific model and company information about this portable device is not disclosed here for objective evaluation. 3.3. Measurement parameters Instrumental parameters which influence the precision and accuracy of the stress measurement include beam size and divergence, use of a Kfilter, sample-to-detector distance, choice of tilt angle 0 , detector resolution, peak fitting research papers 134 Seung-Yub Lee et al. Precision and accuracy of stress measurement J. Appl. Cryst. (2017). 50, 131–144 Figure 2 Optical (a), (b) and EBSD (c), (d) images for the 1018 carbon steel sample from the top cross section (a), (c) and side surface (b), (d), which show elongated grains in the vertical direction. The measured average grain size is about 15–20 mm. Figure 3 (a) Experimental setup for the in situ tensile loading test with a 2400 long rectangular bar. (b) Schematic illustration with diffraction geometry. The portable device was tilted to probe various diffraction vectors. Note that 2=22, = 0 . function, scanning time etc. Among these parameters, the most important are sample tilt angle ( 0 ) and sample-to-detector distance (abbreviated as ‘SD’ hereafter). Fig. 4 shows the one-dimensional diffraction profiles integrated from the Debye rings of Fe reference powders. If the displacement error is autocorrected by SD adjustment so that the peak centers are shifted to the position expected from Bragg’s law, there is no noticeable difference due to sample tilt angle ( 0 ), as shown in Fig. 4(a). However, the 2range and intensity are directly influenced by the SD. The use of large SDs results in a narrower 2range and thus in higher resolution, but at the cost of sacrificing intensity. Meanwhile, as the SD becomes smaller, the peak intensity increases with increasing 2coverage. For example, for a given detector configuration (29.7 mm outer radius with 3 mm inner radius, 50 mm pitch, 534 radial pixels), a 60 mm SD covers 23in 2 resulting in 0.04resolution, whereas coverage doubles when the SD is reduced to 20 mm at the cost of increasing the resolution to 0.08. In general, there is a trade-off between intensity and resolution in the chosen sample-to-detector distance. This effect is somewhat mitigated because, as seen in Fig. 4(b), the peak-to-background ratio remains about the same. Thus it is better to keep the SD large as long as the reflection peak of interest falls within the range covered by the image plate. 3.4. Instrument resolution and beam center calibration Before addressing questions of precision and accuracy under field measurement conditions, the intrinsic instrument research papers J. Appl. Cryst. (2017). 50, 131–144 Seung-Yub Lee et al. Precision and accuracy of stress measurement 135 Figure 4 Fe 211 powder diffraction patterns obtained at = 0, with various (a) 0 at SD = 35 mm and (b)SDat 0 =35 . The intensity and 2range are strongly dependent on SD. Figure 5 (a) Peak center position versus angle (blue solid line) and calibration curve (red dashed line) resulting from the 25 mm beam center offset. (b) Calibrated peak center plot obtained by adding the two curves from (a). This corrected profile is used for the stress calculations expressed in equation (10). resolution, i.e. the best attainable precision in laboratory conditions, and raw data calibration procedure must be established. The spatial resolution of each detector pixel is determined by the pitch of the laser scan, which can be adjusted from 20 to 100 mm; a default value of 50 mm was used throughout all the measurements in this study. For a given detector size (59.4 mm diameter with 6 mm diameter hole in the center for beam penetration and collimator installation), 50 mm radial pitch and 0.72angular step in a spiral reading generates 534 data points in 2and 500 data values at each 2in terms of the  angle. We performed preliminary tests with annealed Fe reference powders at one of the recommended settings: 35 0 ,38mm SD. A total of 80 measurements were repeated over the course of 48 h with multiple cycles, turning the apparatus on/off, but without moving any part of the sample or machine for the best true instrument precision. In order to quantify every source of uncertainty, all data analyses were done fully manually in the following steps: (1) Raw intensity versus 2data were exported for 500 angles. (2) Peak center positions in 2were found via pseudo-Voigt peak fitting for 40 000 datasets (80 scans 500 angles). (3) Residual stresses were calculated via the sin 2 and cosmethods for five selected datasets, both with and without beam center correction using equations (5) and (10), and then compared with cosstress values obtained from the instrument’s own analysis software. Fig. 5(a) shows the averaged peak center from 80 measurements versus azimuthal angle, , as a solid blue line. Error bars (one standard deviation, about 0.005) are displayed only in a few places, but they were fairly consistent for all angles. The blue curve with 0.025amplitude in 2 indicates that the beam center is offset by 25 mm from the exact center point, which is half of the pixel resolution. Owing to the difficulty of the X-ray beam alignment and the offset being less than the pitch resolution, this device has the internal calibration curve shown as a red dashed line in Fig. 5(a). Adding those two curves gives the corrected final peak positions shown in Fig. 5(b). Since these calibration curves change according to the SD, the measurement error in sample height (SD) introduces error in the stress measurement. The typical way to calculate the SD is to spread a mixture of reference Fe powders in vacuum grease on the sample and let the machine calculate the SD from the Fe 211 peak. Afterwards, the grease is wiped off, exposing the sample surface for measurement. Since the wavelength and Fe 211 Bragg angle are known, the accurate SD can be calculated and users can employ this value for their actual samples. The precision in SD measurement is less than 10 mm, so the stress error associated with SD uncertainty is negligible. During the 50 s period of data reading and analysis after the 40 s beam exposure, the portable machine goes through fitting and calibration processes, and then provides an in-plane residual stress of 0 2 MPa, which is identical to that provided by our manual cosanalysis. However, if the calibration step is missed, i.e. the blue line in Fig. 5(a) is used, the stress from the cosmethod is 35 2 MPa. In the sin 2 method, 3 the measured stresses are 22 MPa for noncalibrated data (Fig. 5a, blue) and 0 2 MPa for the calibrated data (Fig. 5b), which proves that the sin 2 method is not sensitive to the calibration process because it probes -dependent strain at a fixed , while the cosmethod uses -dependent strain at a fixed (Table 1). However, the sin 2 method requires samples to be at the exact center of rotation for an accurate measurement, which is challenging and time consuming for such a small portable X-ray device. One can use the ‘two-tilt’ sin 2 method to avoid the sample rotation issue by using only two points at = 0 and 180from the Debye ring, but its accuracy is also sensitive to the calibration process, just as in the cosmethod, when non-calibrated data are used. Uncertainty arises at each analysis step and possible error sources are summarized in Table 2. The overall uncertainty is a combination of the contributions related to (1) diffraction peak fitting error (Type 1), (2) the repeatability of the peak center calculation (Type 2), and (3) linear fitting of equation (10) or (5) for stress determination in the cos(Type 3) or sin 2 (Type 4) methods, respectively. As shown in Table 1, without sample heterogeneity or displacement errors, the overall stress error can be as good as 2 MPa for ferritic steel. Types 5–7 in Table 2 are listed as bounds of maximum errors. Note that these uncertainties are measured at a fixed 0 and SD, and the precision was obtained at the best static laboratory conditions, but those parameters are not often achievable in field measurements. Therefore, the effects of 0 and SD on the precision and accuracy need to be measured for practical applications. research papers 136 Seung-Yub Lee et al. Precision and accuracy of stress measurement J. Appl. Cryst. (2017). 50, 131–144 Table 1 Measured stress from the sin 2 or cosmethod from different data types. The error (2 MPa) is the standard deviation of the five measurements. Scans were done at 35 0 and 38 mm SD. Data type sin 2 cos Non-calibrated data (Fig. 5a, blue solid line) 22 MPa 35 2MPa Calibrated data (Fig. 5b, black solid line) 0 2MPa 02MPa Table 2 List of uncertainty types and values. Measurement precision can be represented by the Type 3 error, which is about 2 MPa in the case of Fe. All stress values calculated in x3.4 are based on the reference powder modulus, E= 224 GPa. Type Uncertainty 2 () d/d (m")  (MPa) 1 Diffraction peak fitting error (2, Fig. 4) 0.003 5 1 2 Average standard deviation from all angles (Fig. 5b) 0.005 9 2 3 cosfitting error [equation (10)] NA NA 2 4 sin 2 fitting error [equation (5)] NA NA 4 5 Maximum deviation among angles (Fig. 5b) 0.015 28 6 6 Detector spatial resolution (50 mm) 0.05 90 20 7 FWHM of the peak (Fig. 4) 2.3 4000 NA 3 For the sin 2 method, six 0 counterclockwise rotations (0, 5, 10, 15, 20, 25) were made in laboratory coordinates, which corresponds to the six angles (11.8, 16.8, 21.8, 26.8, 31.8, 36.8) in sample coordinates at = 180. 4. Results 4.1. Precision test with ferrite reference powders The dependence of the precision on the choice of measurement parameters ( 0 and SD) was investigated using annealed ferrite reference powders. The sample tilt ( 0 ) angle varied from 5 to 60for each sample-to-detector distance (SD), which itself ranged from 20 to 60 mm. For the statistical analysis, this whole process was repeated five times (for a total of 540 measurements) at different sample positions. Since equivalency between cosand sin 2 was shown in x3.4 and by other authors (Ling & Lee, 2015; Ramirez-Rico et al., 2016; Miyazaki & Sasaki, 2016) under a homogeneous bi-axial stress state, we used stress values calculated with the instrument software by the cosmethod. Reference powders were expected to exhibit no stress irrespective of the chosen value for the aforementioned parameters. The averaged stress values versus tilt angles ( 0 ) are plotted in Fig. 6, showing that there is a proper range of sample tilt angle for precise and accurate measurements: 20  0 50. Outside that range, measurements become imprecise and inaccurate. At lower tilt angles, 0 15, the measured stresses strongly depend on SD, while monotonic decreases are observed at 0 55. This is consistent with our previous simulation results (Ramirez-Rico et al., 2016) in that orientation error,  0 , is more significant at lower 0 , and that beam defocusing becomes important at higher 0 . The inset of Fig. 6 shows that for this particular apparatus an SD of 35 mm results in good accuracy regardless of tilt angle. The same data as in Fig. 6 are plotted in Fig. 7, as a function of SD for given 0 angles. The calculated stress tends to increase with increasing SD, and its effect becomes large at lower tilt angles. Fig. 8 displays one standard deviation from five measurements at each measurement condition ( 0 and SD) as a representative parameter quantifying precision. At intermediate ranges (20  0 50,20SD 60 mm), the mean stress is 3 MPa; the precision is about 4 MPa for five different spots and 3 MPa for five continuous scans at one spot. Fig. 8 also illustrates that the sample-to-detector distance is not a critical parameter unless 0 is too low or high. More errors at very low tilt angles are inevitable because below 0 = 11.8 4 we lose the independent information volume that the Debye ring captures for cosstress calculation. The wide beam spread at high 0 angle also causes errors in both precision and accuracy. In general, these undesirable measurement conditions should be avoided, although in field conditions this might not be possible. For such cases, it is necessary to know the uncertainty budgets caused by measurement parameters as shown in Figs. 6, 7 and 8. 4.2. Accuracy test with solid samples under in situ tensile loading Obtaining the true stress state is a challenging task for both diffraction-based techniques and destructive mechanical research papers J. Appl. Cryst. (2017). 50, 131–144 Seung-Yub Lee et al. Precision and accuracy of stress measurement 137 Figure 7 Measured stress values versus SD. The SD effect is minor around intermediate tilt angles (20  0 50), but becomes significant at lower 0 . Figure 6 Measured stress values versus 0 angles. Measured stresses deviate from zero considerably at 0 15and 0 55. Figure 8 Contour plot of standard deviation (precision) out of five measurements at each 0 and SD. The precision error is less sensitive to SD around intermediate tilt angles (20  0 50), while 0 shows a wider variation in all SD ranges. 4 = 0 ,= 11.8for Fe 211, so the information becomes redundant at 0 11.8. methods; thus they are often used together as complementary tools. The best way to evaluate measurement accuracy is to compare measurements with known applied stresses. Two rectangular solid bars (1018 steel and 6061 aluminium) were loaded under tensile stress using the apparatus and methodology described in xx3.1 and 3.2; a tensile stress was applied at several tilt angles ( 0 = 12, 25, 35 and 45)in50or25MPa steps. Fig. 9 shows raw two-dimensional diffraction profiles and the Debye ring distortion from each sample at various 0 . The carbon steel shows no preferred orientation, at least in the 211 reflections within the diffracting volume, and the peak center is very stable throughout all angles. However, the research papers 138 Seung-Yub Lee et al. Precision and accuracy of stress measurement J. Appl. Cryst. (2017). 50, 131–144 Figure 9 Normalized two-dimensional diffraction patterns of (a) mild carbon steel 1018 and (b) aluminium 6061 alloy at zero applied stress. The Debye ring moves outside as 0 increases owing to the corresponding SD increase. The carbon steel shows a homogeneous grain distribution, while the aluminium alloy is textured. The left column displays how much the Debye ring is distorted by the peak center positions (polar scale from 2000 to 2000 m"). The sample- to-detector distances are 34, 34, 50 mm for steel and 41, 41, 52 mm for aluminium. Figure 10 Measured versus applied stress from (a) the 1018 carbon steel 211 peak and (b) the 6061 aluminium alloy 222 peak. For steel, both sin 2 and cosshow identical results for a given 0 , while aluminium does not show reliable results. Initial residual stresses were offset to zero. The steel data at 0 =12 are plotted instead of at 0 =25 , because the data at 25are nearly the same as those at 35. aluminium 222 intensity is weak and inhomogeneous. The Debye ring is also distorted as peak centers fluctuate around the ring. Therefore, steel samples are expected to show more reliable results than aluminium. Also note that the SD was rather large (about 50 mm) for 0 =45 , because the minimum sample-to-detector distance increases with increasing tilt angle owing to the fact that the sample is very long and interferes with the instrument at high tilt angle (see Fig. 3). Measured stresses from both the cosand the two-tilt sin 2 methods are plotted against applied stress in Fig. 10. As a measure of accuracy, the slope (ratio of measured over applied stress) was calculated for each set of tilt angle conditions and noted in Fig. 10. The steel sample shows that both analysis methods yield similar results and there is a clear 0 dependency in terms of accuracy. For example, measurement was most accurate and precise at 0 =45 . As for the aluminium sample, 0 dependency can be inferred, but cosand sin 2 generate quite different results, neither of which is accurate or precise for the tested angles. Knowing that the accuracy depends on the tilt angle and that the steel sample is much more reliable than the aluminium sample, a series of new in situ tensile loading experiments (0, 100, 200, 300 MPa) were performed with a 2400 long steel bar sample for a 0 -dependent accuracy test. A total of ten 0 angles were tested from 5 to 48 0 with 5intervals as 0 =0 does not give a cosstress and 48is the maximum tilt angle to capture the 211 peak for a given sample geometry. This set of measurements was repeated five times (4 steps  10 0 5 = 200 measurements). Since the two-tilt sin 2 results shown in Fig. 10(a) are very similar to the cosresults as expected, cosvalues reported by the apparatus software were used throughout. The ratios of measured over applied stress are plotted in Fig. 11 for all five measurements and the average stresses are listed in Table 3 along with standard deviations. As in the previous precision analysis, the measurements are quite repeatable at 20  0 48, but the accuracy changes from 70 to 100% of the actual stress values. In other words, the conditions for the best precision ( 0 =35 , with both Fe powders and the solid bar) and accuracy ( 0 =19or45 ) are different. This result does not mean that the machine calibration is perfect at 45or offset by 30% at 35because there may be other factors affecting the accuracy, which are discussed in the next section. The measured/applied stress ratios at each value of 0 are summarized in Table 4. research papers J. Appl. Cryst. (2017). 50, 131–144 Seung-Yub Lee et al. Precision and accuracy of stress measurement 139 Table 3 Applied and measured cosstresses (MPa) with carbon steel at each tilt angle ( 0 ). Average values and error bars were obtained from five repeated measurements. Note that average values (accuracy) can be different if the internal calibration profile changes, while error bars (precision) are more sensitive to measurement conditions. For the accuracy evaluation, the measured/applied stress ratio, as shown in Table 4, is the better way because the true residual stress state at zero load is unknown. The best precision is achieved at 0 =35 . Measured stress (MPa) at each tilt angle ( 0 ) Applied stress (MPa) 5101520253035404548 0 67 (9) 30 (11) 27 (12) 25 (9) 20 (10) 1 (13) 19 (8) 50 (15) 98 (26) 138 (22) 100 236 (23) 173 (14) 143 (8) 123 (12) 98 (8) 70 (13) 54 (7) 39 (15) 5 (21) 38 (23) 200 403 (37) 323 (16) 258 (6) 218 (12) 170 (10) 138 (13) 127 (7) 125 (15) 102 (21) 65 (23) 300 558 (68) 463 (28) 372 (7) 311 (9) 254 (30) 205 (9) 198 (6) 208 (17) 198 (22) 166 (23) Figure 11 0 angle effect on the ratio of measured stress (cos) over applied stress for the 1018 carbon steel sample. A ratio of 1 indicates accurate stress measurement. Each ratio (slope) was determined from the four tensile loading steps (0, 100, 200, 300 MPa) and is summarized in Table 4. Figure 12 SD effect on the ratio of measured stress (cos)versus applied stress for the 1018 carbon steel sample. A ratio of 1 indicates accurate stress measurement. The accuracy of stress measurements is not sensitive to the SD except for the low 0 angle regions where the measured stress increases with increasing SD.