French Round-robin Test of X-ray Stress Determination on a Shot-peened Steel by M. Fran;ois, F. Convert and S. Branchu ABSTRACT--A round-robin test of X-ray residual stress determination was performed by the Groupement Fran(;ais d'Analyse des Contraintes and involved 16 laboratories. The standard deviation of the measurements was about 36 MPa, but a posttreatment of the raw data with the same software reduced the dispersion to 19 MPa. Analysis of the uncertainty sources (goNometer alignment, operator, counting

statistics, nonlinearity of the material, etc.) revealed that the main sources come from the data treatment and the operator. KEY WORDS--Residual stresses, X-ray analysis, uncertainty analysis, interlaboratory measurements

Introduction In France, people from both industrial and university laboratories involved in stress determination by X-ray or neutron diffraction regularly meet in a workgroup called Groupement Fran~ais d'Analyse des Contraintes, a thematic committee of Socirt6 Frangaise de Mrtallurgie et Matrriaux and of GAMAC. Along with general meetings, there are several specialized subcommittees working on specific questions. One of them, led by the first two authors, was activated in March 1992 to work on experimental procedures. It comprises 27 people, of whom 15 meet regularly twice per year. The purpose of this paper is to present the results of a round-robin test organized within this group during three years from 1992 to 1995. This was not the first test of this kind performed in France. A test on nickel-based alloys was organized on the initiative of an important French company. Because the results of this first test were disappointing in terms of dispersion, it was decided to start a new one on a different basis. To reduce the dispersions between laboratories, two ideas were proposed: the first idea was to impose on each participating laboratory as many experimental conditions as possible, and the second idea was to keep all the experimental diffraction patterns and to treat them with the same software. It should then be possible to separate experimental dispersions from dispersions coming from treatment algorithms and parameters.

M. Francois is a Professor, LAMM/CRTZ, Bd de l'Universit~. 44602 SaintNazaire Cedex, France. F. Convert is an Engineer, CETIM, 60304 SeMis Cedex, France. S. Branchu is an Associate Professor, LAMM/CRI~, Bd de l'Universitd, 44602 Saint-Nazaire Cedex, France. Original manuscript submitted: July 29, 1999. Final manuscript received." July 13, 2000.

Round-robin tests are conducted in the United States and in Europe. In Refs. 1 and 2, the aim was to obtain bulk-zero stress samples. A preparatory work to organize round-robin tests on samples with a known applied stress is described in Ref. 3, but to our knowledge measurements have not been published. A round-robin test was performed on a steel bearing track with a high-level compressive residual stress. 4 The results obtained in these references will be discussed in the Dispersion Analysis section.

Brief Review of X-ray Stress Analysis The basis of residual stress analysis by X-ray diffraction is to measure the elastic strain of the crystallographic lattice Edp~ in several directions defined by the azimuth angle dpand the inclination (tilt) angle ~ (see Fig. 1). For a given value of the dOangle, the normal stress cr~ and the shear stress xqb can be obtained from the sin2~ relation: 1

033 ) sin 2 q~

= ~ S2{hkl} (Gdp -

1

+ ~Sa{hkl}xrb sin2~ + $1 {hkl}Tr (5) 1 -t- ~ S2{hkl}o33, where 89Sa{hkl} and Sl {hkl} are the X-ray elasticity constants of the material for the measured lattice plane family {hkl}. Tr (5) is the trace of the stress tensor and t:r33 the normal stress component in direction x3p. To obtain the complete stress tensor, at least six independent directions d ~ must be measured:

6 Ec~xlI = ~ eij0-ij, i,j=l

(1)

where the F coefficients are the generalized X-ray elasticity constants. In the case of an isotropic homogeneous material, they depend on the X-ray elasticity constants and the ~b~ direction. The d~ and ~ angles only define the measurement direction x~; however, the incident and diffracted beams could lie either in the (x L, x3L) plane or in the (xL, x L ) plane. The first case is called the f2 setting [Fig. 2(a)] because inclination is achieved by the [2 rotation of a two-circle goniometer. The second case is called the X setting [Fig. 2(b)] because ~ is obtained with the X rotation of a four-circle goniometer. It

Experimental Mechanics 9 361

Imposed Parameters

Fig. 1--Definition of reference systems and measurement direction

is sometimes called the ~ setting in the literature. Another setting called the E setting is sometimes used on portable goniometers. In this case, the rotation axis E lies in the (xL, x~') plane, making an angle of ~ - 0 with the x~-axis. This induces systematic errors that are partly compensated for by the use of two detectors. 5 Goniometers can be either laboratory equipment or a portable (in situ) apparatus. In the first case the sample is rotated; in the second case the X-ray tube and the counter are rotated together around the sample. The detectors can be classified according to their geometry or their physical principle. They can be position sensitive (i.e., they acquire the whole diffraction pattern at one time without movement) or punctual (i.e., a rotation 20 is needed to scan the diffraction range). Most of the time, position sensitive detectors are proportional counters, but few are composed of an array of CCDs. Punctual detectors can be scintillation counters or solid-state detectors. Further information can be found in the literature. 6'7

Material and Experimental Conditions Our goal was to analyze dispersions due to experimental conditions rather than investigate difficulties coming from the material itself. A sample for which the assumptions on which the sin2~ method is based 6'7 should hold: the material should be macroscopically isotropic, with negligible surface or depth gradients and a small size of coherently diffracting domains, and it should lead to diffraction peaks with a reasonable breadth so that there is almost no truncation and an easy background removal is possible. A plain, medium, carbon steel shot-peened with a fiat surface was chosen. Details of the chemical composition and the treatments of the sample are not available but are not relevant for the present study. The stress gradient in the depth of the sample was not considered a problem, since the radiation and the crystallographic plane was imposed. Stress relaxation was not anticipated, since the specimen had been used for several years by CETIM as an internal reference sample with a stable value of the stress. 362 9 VoL 40, No. 4, December2000

To avoid potential problems of spatial or orientational heterogeneity, the measurement area and the direction r = 0 were clearly marked on the sample. Measurements had to be performed with CrKa radiation on {211} plane family at a Bragg angle of 20 = 156.3 deg. A vanadium K[3 filter had to be used. Eleven values of inclination angle ~ should be taken uniformly over the range [ - 4 0 deg, +40 deg]. The exact values were not imposed because some software did not allow a free choice; however, operators had to choose them uniformly in terms of sin2qr if possible. The limit of 40 deg was chosen so that measurements could be achieved on all the equipment. To make stress tensor analysis, six values of azimuth angle r were imposed: 0 deg, 45 deg, 90 deg, 180 deg, 225 deg and 270 deg. These directions are not independent, since negative and positive values of ~ were taken but were chosen to detect potential missettings of the specimen surface normal at ~ = 0. To explore the same volume of material as much as possible, a spot size of about 4 mm 2 (at = 0) was required, but its shape could be chosen freely by the operator. To minimize the dispersion due to counting statistics, a minimum height above the background of the diffraction peak at ~ = 0 of 3000 counts was imposed; the same counting time could be kept for other ~ values. To ensure a correct alignment of the goniometer, a stressfree iron powder fixed to a glass plate with collodion was circulated along with the shot-peened specimen. Operators could use any adequate method to align their equipment but were required to make a measurement on the powder sample before the measurement on the steel specimen. Whether the collodion caused stresses in the powder was determined at Ecole Nationale Sup6rieure des Arts et Mrtiers on a Seifert PTS goniometer. This goniometer had been aligned independently with a stress-free sample made of an iron powder of another origin fixed to a glass plate with Vaseline. The goniometer was considered well aligned when the measurement was 0 + 1 MPa with an adequate number of ~ angles and counting time. A measurement on the collodion sample in the conditions of the round-robin test was performed and was 3 -4- 4 MPa. The collodion sample was thus considered stress free. Free Parameters

To enable as many laboratories as possible to join the round-robin test, some parameters could not be imposed: the goniometer setting (X, f2 or others), the goniometer radius (distance from sample to detector), the detector geometry and its physical principle, the optical setting (parallel or focusing) and the acquisition range and step in 20.

Participating Laboratories Sixteen industry and university laboratories participated in the round-robin and performed a total of 21 independent measurements. By independent, we mean that each measurement was performed on a different goniometer. The list of laboratories and their equipment is shown in Table 1. Some goniometers were in situ goniometers; others were two- or four-circle laboratory goniometers. All possible kinds of detectors were used, with a majority of position-sensitive proportional counters. Three goniometers were equipped with parallel beam optics, while the others were equipped with focusing optics.

(a)

(b)

Fig. 2--Laboratory goniometer: (a) ~ setting (f2 = 0 + ~), (b) ~. setting (~ = 0 and X = ~) TABLE lwLIST OF THE PARTICIPATING LABORATORIES AND THEIR EQUIPMENT Name of Laboratory and Detector Type Person(s) in Charge Goniometer Setting Elphyse Set-X X PSPC Aarospatiale Suresnes, C. Not Homemade in situ X PSPC Cetim Senlis, F. Convert Seifert MZ IV PTS X PSPC ENSAM, M. Francois, J. L. Lebrun Four-circle goniometer ~. PSPC Ec. des Mines de Nancy, M. Zandona CGR with Dosophatex X SC Ec. des Mines de Paris, J. P. Hanon Siemens D5000 with )(. SC Ec. des Mines de St-Etienne, R. Fillit Dosophatex Elphyse Set-X )~ PSPC IRSID (1), H. Michaud Siemens Dh000 with ~. SC IRSID (2), H. Michaud Dosophatex Rigaku Strainflex f2 SC IRSID (3), H. Michaud Siemens D500 ~2 SiLi SSD IUT de St-Nazaire (1), M. Francois Siemens DhO0 f2 SiLi SSD IUT de St-Nazaire (2), M. Frangois CGR with Dosophatex X PSPC Universit~ de Limoges, B. Dionnet Seifert MZ Vl ~ Ge SSD Renault S.A., J. Leflour Siemens Dh000 X PSPC SNECMA, M. AIIouard Siemens Dh000 X PSPC SNFA, H. Carrerot Stresstech AST 2002 ~, PS CCD SONATS, P. Jacob Elphyse Set-X X PSPC Sollac, M. Friedrich Elphyse Set-X X PSPC Unimetal, H. Michaud Homemade in situ ~2 PSPC Universita de Bourgogne, F. Bernard SC = scintillation counter, PS = position sensitive, PC = proportional counter, SSD = solid-state detector, CCD = chargecoupled device

Results

Raw Results Given by Each Laboratory Each laboratory was asked to give the stress obtained from the collected data with the software that it usually uses. The results are reported in Table 2. Twenty-one values of 1711 and 19 values of 1722 are available, which is more than in other sections of this paper because some data sets were not available on floppy disks. The values of 1711range from - 3 8 5 MPa to - 5 0 4 MPa; the values for 1722 range from - 3 6 3 to - 4 7 7 MPa. The total scatter is 119 MPa for 1711and 114 MPa for 1722. The average value of the stresses is - 4 6 9 MPa for 1711 and - 4 3 6 MPa for 1722 with a standard deviation of 32 MPa and 30 MPa, respectively. To compare with the following analysis, the average values and standard deviations have been computed for the 15 laboratories for which computer files are available.

Note that the error bars given by each laboratory cannot be directly compared with the standard deviation between the laboratories because their meaning may be very different from one laboratory to another: standard deviation or uncertainty at the 95 percent confidence level, counting statistics or least squares residue. Two laboratories gave values that are smaller (in magnitude) than the others. To determine whether these values should be discarded, a Dixon test was performed according to standard ISO 5725. 8 It was determined that the values should be kept. The magnitude of 1722 was lower than that of 1711for all laboratories. The difference A17 =1 1711 - 1 7 2 2 I can be computed, and its average value is 32 MPa with a standard deviation of 13 MPa. Note that the dispersion on A17 is lower than on 1711 and 1722, which means that the two components of the stress tensor are not independent measurements and that some systematic errors such as goniometer misalignment can be compensated for when estimating Am

ExperimentalMechanics 9 363

TABLE 2--RAW VALUES tMPa/GIVEN BY EACH LABORATORY cr(powder) "~(powder) Laboratory Cr]l and ACrtl er22 and Ao22 ICrl] - o221 1" -463 4- 21 -448 4- 21 15 3 4- 4 1+ I 2 -489 + 4 -438 4- 4 51 10 4- 4 - 4 4- 3 3* -458 4- 11 -432 4- 11 26 3 4- 4 1 4- 1 4 -482 4- 40 -459 4- 30 23 1 4- 3 1 4- 3 5* - 4 9 0 4- 10 -468 4- 10 22 4 4- 5 1 -I- 4 6* -494 4- 31 -442 4- 28 52 1 4- 5 - 2 4- 1 7 -494 4- 20 -464 4- 21 30 5 4- 1 1 4- 1 8* - 4 6 4 4- 17 -422 4- 17 42 2 4- t7 0 4- 2 9* -491 4- 12 -453 4- 12 38 3 4- 6 0 4- 1 10 -458 4- 32 . . . . 11" -488 4- 18 -477 4- 36 11 0 4- 18 - 2 8 4- 18 12" -493 4-44 -441 4- 45 52 - 5 4- 31 13* -497 4- 8 -454 4- 8 43 14 -481 4- 26 4 4- 31 - 7 4- 5 15" -385 4- 33 -366 4- 33 19 0 4- 5 - 3 4- I 16 -463 4- 11 -436 4- 13 27 17" - 4 6 6 -4- 12 -437 4- 12 29 0 4- 26 0 4- 5 18" -397 4- 5 -363 4- 20 34 - 1 4- 3 19" -431 4- 12 -4164- 13 15 94- 10 1 4- 1 20* -504 4- 16 -452 4- 15 52 10 4- 18 1 4- 2 21" -451 4- 30 -422 4- 13 29 - 7 4- 5 Average (all) -469 -436 32 2.3 3.4 SD (all) 32 30 13.5 4.5 7 Average (15 lab.) -464.8 -432.9 31.9 1.6 -2.5 SD (15 lab.) 36.3 32.4 14,1 4.6 8.5 An asterisk denotes the 15 laboratories for which computer files are available. The meaning of the value after the + sign varies from one laboratory to another. It can be based on counting statistics or least squares residual, and the confidence level is not specified. SD = standard deviation.

The values obtained for the powder sample are small, leading to an average value of 3.2 MPa with a standard deviation of 4.5 MPa for the apparent normal stress and of 3.5 MPa with a standard deviation of 7 MPa for the apparent shear stress. This means that goniometer misalignment errors are small.

Treatment of Experimental Data In the previous subsection, the data were treated in each laboratory with different software. To remove the influence of the various software, the data were treated homogeneously with Stress/AT software (version 2). The elasticity constants of the steel were E ---- 210,000 MPa and ~ = 0.29 with an X-ray anisotropy factor of Anx = 1.49, which leads to X-ray elasticity constants 89 = 5.76510 -6 MPa -1 and $1{211} = - 1 . 2 5 5 1 0 -6 MPa -1. Besides the three peak localization algorithms (parabola, middle of chord at 40 percent or 50 percent height, centered centroid9), the software offers many possibilities of treatment--Lorentzpolarization-absorption corrections (LPA), background removal, Ka2 stripping and weighted least squares regression analysis (so all the peak positions have the same variance). The data can be treated as a whole to obtain the stress tensor for each separate qb value. In this case, the graph of peak positions versus sin2~ is an elliptic curve, the slope of which is proportional to the normal stress value and the opening proportional to the shear stress value. The software gives a stress component with two uncertainty values. The first value represents an estimation of the repeatability of the measurement. It is estimated from some parameters describing the diffraction patterns such as the net height or the peak-to-background ratio.l~ 11 The second value is estimated from the residue of the least squares optimization and represents the scatter o f 364 9 VoL 40, No. 4, December2000

the experimental peak positions from the model assumed to calculate the stress. It The software gives uncertainties that are broadened to represent confidence intervals of 95 percent. However, to make comparisons easier, the standard uncertainty has been recalculated and reported in every table but Table 2.

Elliptical Regression Analysis The results for treatments with the centered centroid 9 are presented in Table 3 and Fig. 3. It can be seen that the stress values in the direction qb = 0 range from - 4 4 1 MPa to - 5 1 3 MPa with a standard deviation of 18.5 MPa. The average value is slightly higher than for the raw results, but the standard deviation is significantly lower. This means that the treatment software does have an influence on the dispersion of the results. It was shown by Noyan and Cohen 12 that a shift A ~ of inclination angle qr, due for instance to a wrong position of the sample, can lead to apparent stress components given, for the case in which the true shear stress is null, by (r~ parent = o~)ue cos 2 A ,

and

(2) 1;~)pparent -- !otrue sin 2 A ~ .

-2~

If A ~ is small (e.g., 1 deg), its effect on oqb is negligible while its effect on the shear stress can easily be measured. However, the apparent shear stress is not a real stress and, if the specimen is rotated 180 deg around its normal, its sign remains constant. It is therefore possible, from measurements with positive and negative values of ~ for a given qb value and for ~b-I-180 deg, to obtain the true stress components and

TABLE 3---RESULTS OF ELLIPTICAL REGRESSION ANALYSIS FOR ~b = 0 AND ~b= 90 DEG Laboratory Number crn (MPa) o22 (MPa) 1 -462 -428 2 -454 -351 3 -441 -413 4 -513 -438 5 -466 -409 6 -485 -433 7 -496 -445 8 -496 -449 9 -477 -446 10 -463 -433 11 -491 -436 12 -477 -425 13 -480 -458 14 -490 -387 15 -474 -422 Average without A ~ correction -477.7 -424.9 Average with A ~ correction -475.9 -423.3 SD without A ~ correction 18.56 27.07 SD with A ~ correction 18.54 26.74 SD from counting statistics 4.64 5.92 SD from least squares residue 8.41 9.33 SD = standard deviation.

[O11 -- 0221 34 103 28 75 57 52 51 47 31 30 55 52 22 103 52 52.8 52.6 24.65 24.66

-430 -440

03

-450

9

-460

9

G" lk -470 9

-

~13- 9

-480 -490 -500 -510

9

12

50t~

I'f5 . . . . . . . . .

06 9 it # 14

- ---- ---O'---O- . . . . . . . . 7 8 04

-520 Axerage ----A~erage • - - - - A,,erage+s.d. fromco~mtingstatisfies Fig. 3--Dispersion of the measurements of the 15 laboratories after treatment of the raw data by the same software

the value of AO. The largest apparent shear stress obtained is 46 MPa, which leads to 2x~ = 5.6 deg and a correction of the normal stress component of 9.4 MPa. On the global result, this A ~ correction decreases the average value of less than 2 MPa and the standard deviation of less than 1 MPa. The raw data obtained by each laboratory were treated in the same way with three localization methods: the centered centroid, the parabola and the middle of chord (with a chord taken at 40 percent of the net height of the peaks). Prior to the localization, an LPA correction and a background removal treatment were done, and no Kct2 stripping was performed. The results are shown in Table 4, It can be seen that the average stress values agree fairly well (within less than 13 MPa). The dispersion between the laboratories, however, is much larger for the parabola method than for the two other methods. The centered centroid exhibits the smallest value of standard deviation. In fact, this method presents an uncertainty due to counting statistics that is small enough to estimate the nonlinearities due to the material from the least squares regression analysis (see the Dispersion Analysis section). Because of

its lower dispersion, we chose in other parts of the paper to present only results obtained with the centered centroid. As can be seen in Table 5, the influence of background removal on the average stress and on the interlaboratory dispersion is low: without background removal, the average stress level is slightly higher than about 10 MPa and the standard deviation is slightly increased. This is consistent with the results given by Convert. 9 As can also be seen, the influence of two other treatment parameters (weighting of the least squares regression analysis and K~t2 stripping) is not significant. In X-ray stress analysis, the absolute position 200 of diffraction peaks appears three times: first for the LPA correction, second for the calculation of peak shifts with the gr angle and third in the tan(00) term to convert peak shifts into strains. However, it is not necessary to know the value of 200 accurately, and it can be shown that replacing 200 by the position 20• of the peak at ~ = 0 is sufficient for stress levels less than 1 GPa. When examining the data sent by the participating laboratories, however, it was found that the Experimental Mechanics 9 365

TABLE 4--INFLUENCE OF THE LOCALIZATION METHOD Centered Centroid iii

I I

Average

SD SD from counting statistics SD from least squares residue SD = standard deviation.

II

]

II II

0"11

0"22

-475.9 19.8 4.6 7.8

-427.7 25.5 5.0 9.3

II I !

Middle of Chord

011 -474.8 21.8 11.4

12.9

TABLE 5--INFLUENCE OF THE TREATMENT PARAMETERS i

I

I

Average SD SD from counting statistics SDfrom least squares residue SD = standard deviation.

Standard Treatment

0.11 (I22 -475.9 -427.7 19.8 25.5

0"11 (r22 -486.5 -437.1 24.2 36.0

5.0 9.3

dispersion in the 20• values was rather large, ranging from 155.036 to 157.007 deg 20 (at do = 0) with a standard deviation of 0.502 deg 20. This means that the operators do not pay strong attention to the adjustment of the 20 angle origin. We have estimated the influence of this dispersion on the LPA correction to about 1.7 MPa of standard deviation, which is negligible. However, its influence on the tan(00) term is not negligible and leads to a standard uncertainty by about 10.3 MPa. Tensorial Regression Analysis The 66 diffraction peaks obtained by each laboratory were treated for background removal and LPA corrections. Their position was estimated by the centered centroid method, and the stress tensor was calculated with the assumption that 033 is equal to zero. The results are reported in Table 6. Compared with elliptical regression analysis, it can be seen that the average value is lower in magnitude (about 31 MPa for 1711and 20 MPa for O22 ). The dispersion of the results is also significantly higher, with a standard deviation of about 40 MPa instead of 20 MPa in elliptical regression analysis. The standard deviation estimated from counting statistics is lower because the number of detected photons per stress component is higher. More interesting, the standard deviation estimated from the least squares residue is much higher, which means that the peak positions obtained from the different azimuth angles are not as consistent with each other as those obtained for different ~ values at one azimuth value. This can be done by examining the 20• values for each azimuth angle dO: it should always be the same, except for small fluctuations due to counting statistics. For each laboratory, the standard deviation on 20• was computed for the six dos. Its value ranged from 0.007 deg 20 to 0.06 deg 20 with an average of 0.027 deg. The largest dispersions are obtained with goniometers that are not really equipped for tensor analysis and for which the dorotation is performed with an extra rotating stage or by hand, while the smallest are obtained on actual four-circle goniometers. Therefore, a significant portion of the dispersion obtained on tensor components could be attributed to a misalignment of the d0-axis with the other axis of the goniometers. From the average uncertainty of 0.027 deg, the 366 9 Vol.40, No. 4, December2000

0.11 -463.7 45.9 27.3 17.4

0.22 -430.5 44.6 41.2 18.7

ii

Without Background Removal

4.6 7.8

Parabola

0.22 -426.0 32.8 12.2 17.9

5.2 11.6

5.6 10.0

Weighted Least Squares

0.11 -475.3 19.7 4.7 8.1

022 -427.0 25.4 5.1 9.6

With K(~2 Stripping

0.11 -477.4 22.4 5.1 8.7

~22 -429.0 25.6 5.5 9.3

standard uncertainty on the stress value can be evaluated to 16 MPa. However, this value is probably underestimated because it explains only part of the difference between the dispersion on tensorial regression analysis and the dispersion on elliptical regression analysis. Peak Breadth and Other Results Along with residual stress values, the average peak width is an important parameter in an analysis of the mechanical state of the material. It is composite information that reflects the size and the elastic distortion of the coherently diffracting domains, the second-order broadening and the instrumental broadening. The separation of these effects is beyond the scope of the present paper; however, it is commonly known that it leads to a broadening of diffraction peaks when the material is cold worked. Although the experimental conditions are very different (beam divergence, parallel or focusing geometry, spot size, receiving slits, etc.) from one laboratory to another, we analyzed the dispersion of peak width estimation. Two indicators were used: the full width at half maximum (FWHM) and the integral breadth B computed from peak intensity and height. The mean value for each laboratory is computed from the width of the 66 peaks. It can be seen that the intralaboratory dispersion of B and FWHM is about 1 percent of the average value, while the interlaboratory dispersion is much higher: 11 percent for FWHM and 20 percent for B. This means that the variations in experimental conditions lead to uncertainties that are too large to compare peak width from one laboratory to another if the instrumental broadening is not corrected. The relative stability of FWHM compared with B may be explained by the sensitivity of the latter to background removal. The peak-to-background ratio varies dramatically from 0.45 to 25.44. It strongly depends on the detection technique used and on the optical setup of the goniometer. The lowest value is obtained with a focusing geometry and a positionsensitive proportional counter (PSPC), and the highest value is obtained with a parallel geometry and a SiLi diode. Measurements made with an in situ goniometer with a PSPC resulted in a peak-to-background ratio around 9, and those made with a PSPC on a laboratory goniometer resulted in ratios around 5. This means that acquisition conditions other

TABLE 6~RESULTS OF TENSORIAL Laboratory 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Average SD SD SD

from counting statistics from least squares residue

SD

= standard deviation.

REGRESSION ANALYSIS IMPa/ cq I O22 o12 -452 -423 5 -433 -389 21 -444 -421 6 -485 -453 11 -462 -426 7 -481 -436 12 -475 -449 -8 -492 -452 23 -382 -349 -7 -475 -356 57 -390 -334 7 -375 -347 5 -475 -452 -17 -455 -401 -7 -396 -369 2 -444, 8 -403, 8 7, 8 40, 3 43, 2 17, 3 3.5 3.7 2.0 22.5 22.8 13.0

than counting time do have an influence on the statistical repeatability of measurements.

Dispersion Analysis All the results presented in the previous sections enable one to separate the various sources of uncertainty in the measurements. Consider the following model of the measurement. The global variance u~ of the interlaboratory measurements is the sum of six variances due to data treatment software u~ (including the value of X-ray elasticity constants used), goniometer alignment errors u~, counting statistics u 2, heterogeneity of the material u 2 , errors in the origin of 20 rotation u 2 and goniometer setup and operator influence u~, respectively:

u~ = u2 + u 2 + u~, + u~t + u S + u 2.

(3)

The variance due to counting statistics is estimated by the software with a method based on the work of Sprauel and colleagues: 10,11 u 2 = 4.642 = 21.5 MPa 2. The global variance is obtained directly from the raw results given by the laboratories: u 2 = 36.32 = 1317.7 MPa 2. The variance due to the software is obtained by comparing the same set of data treated by different software (raw results in Table 2) and treated by the same software (see Table 3): u 2 = 2 = 36.32 18.62 = 31.22 MPa 2. The variance U2 _ Uss due to goniometer alignment is estimated by considering two effects: 1. Shifts in the beam and sample position with respect to the goniometer center that are reflected on values obtained on the powder sample and give a standard uncertainty ul = 4.6 MPa (see Table 2). 2. Shifts in the origin of the ~ angle (the so-called missetting) that are estimated from the apparent shear stress on the shot-peened sample itself. The maximum uncertainty coming from this problem is u2 = I MPa (see Table 3).

(~13 -3 -10 -6 -2 -4 -4 4 -6 2 -20 5 -3 -23 -4 0 4, 9 7, 8 0.6 3.3

O23 1 -t -1 3 0 1 -1 -6 -1 -5 -3 -3 0 11 -1 -0, 4 3, 9 0.6 3.3

ICr11-- Cr221 29 44 23 32 36 45 26 40 33 119 56 28 23 54 27 41 24.0

From //1 and u2, we can compute u~ = ul2 + u 2 = 4.7 2 MPa 2. The uncertainty UM due to the material can be obtained from the uncertainty uR calculated with the residue of the least squares regression analysis, uR is estimated from the experimental variance of peak positions due to counting statistics and to nonlinearities of the material (due for instance to texture effects, stress gradients or other inhomogeneities in the diffracting volume). Note that goniometer misalignments can cause deviations from an elliptic curve and strictly speaking are included in the estimation of uR; however, the values obtained on the powder sample being small, we will neglect the influence of goniometer alignment on the least squares residue. The variance u~ is thus the sum of u 2 and u 2 . It is given in Table 3, and we can deduce u 2 = 8.412 -- 4.642 = 7.02 MPa 2. The uncertainty u o due to errors in 20o has been discussed--it is equal to 10.3 MPa. The last term u~, is not as well defined as the others. It is the difference between the observed global dispersion u~ and all the other terms described above. We obtain u 2 = 15.92 MPa 2. It covers several factors that are not easy to describe quantitatively: influence of the operator, beam divergence, goniometer setting ( ~ or X), shape of the irradiated surface, detector type and/or width of the receiving slits. The analysis can be repeated on the results for o22, and the results are given in Table 7, where it can be seen that the main sources of dispersion are the software and the operator. For tensor regression analysis, another source of uncertainty u~b = 162 MPa 2 should be added to the global uncertainty to account for variations of 20• with the ~ angle. The dispersion obtained in the present study can be compared with that obtained in other round-robin tests. In Ref. 1, eight samples of AISI 1018 steel were heat treated to bulkzero stress. Measurements were performed by eight laboratory goniometers and four pieces of portable equipment. Sixty-eight measurements made in the axial direction had an average value of - 7 MPa, and 23 measurements in the transverse direction had an average value of - 1 5 MPa. The Experimental Mechanics

9

367

TABLE 7--ANALYSIS OF THE ORIGINS OF THE OBSERVED GLOBAL DISPERSION ON ~]l AND cr22 I

Uncertainty (MPa) all ~22

Global 36.3 32.4

Software 31.2 17.8

Alignment 4.7 4.7

standard deviations were 50 MPa and 57 MPa, respectively. In Ref. 2, 13 X-ray measurements were performed on heattreated 304 stainless steel specimens. The values obtained in two directions were - 1 4 MPa and - 4 9 MPa with a standard deviation of 34 MPa and 55 MPa, respectively. The dispersions obtained in these two works were slightly higher compared with our study, which may be due to the greater variability in measurement conditions; in particular, the number of ~ inclinations used by the operators is very low (from one to six). It should be mentioned that some alignment errors have influence only when the stress level is high. In Ref. 4, measurements were performed in 10 laboratories on high-level compressive stress-bearing tracks. Only the X-ray wavelength was imposed; other conditions were left free. In the hoop direction, the values obtained on 10 measurements ranged from - 6 0 0 MPa to - 8 9 0 MPa with a standard deviation of 80 MPa. In the radial direction, 7 measurements ranged from - 4 4 0 MPa to - 8 9 0 MPa with a standard deviation of 130 MPa. These very high values of dispersion may again be due to the variability in measurement conditions but also to the geometry and the metallurgical state of the samples, which made the measurement difficult.

Conclusion This round-robin test performed among 16 French laboratories led to several interesting results because a significant amount of postprocessing was performed. Twenty-one measurements were performed on various equipment. The standard deviation for the obtained values is about 36 MPa, but with a homogeneous processing of the data it can be decreased to about 18 MPa. This means that a significant source of dispersion between laboratories comes from the software used to localize the peaks and calculate the stress. The difference between the X-ray elasticity constants used by each

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Material 7.0 7.2

Counting Statistics 4.6 5.9

Origin in 20o 10.3 10.3

Operator 12.1 22.8

laboratory can explain part of the dispersion but not all of it. Other sources of dispersion have been analyzed and estimated quantitatively, but some sources remain unidentified. One interesting result is that the origin of the 20 angle is often not adjusted accurately enough, which leads to significant systematic errors, The dispersion obtained with tensor analysis is larger than with elliptical analysis because some laboratories are not equipped with actual four-circle goniometers.

References 1. "Bulk-zero Stress Standard AISI 1018, Carbon Steel Specimens, Round-robin Phase 1," Exp. Tech., 38-41 (Apr. 1985). 2. Flaman, M.T. and Herring, J.A., "SEM/ASTM Round-robin Residual Stress Measurement Study--Phase 1," Exp. Tech., 23-25 (May 1986). 3. Yerman, J.A., Kroenke, W.C., and Long, W.H., "Accuracy Evaluation of Residual Stress Measurements," Proceedings of ECRS4, Cluny, France (June 1996). 4. Paty, G., "Brite Euram 2," Elabomm BE 5406, Second Year Progress Report No. ELAB A025 (1994). 5. Jacquot, T., "Caractgrisation de l'endommagement des aciers pour emballage par diffraction des rayons X," Doctoral thesis, Universitg de Nantes (1997). 6. Noyan, L C. and Cohen, J.B., Residual Stress, Springer-Verlag, New York (1987). 7. Lu, J. and James, M.R., eds,, Handbook on Techniques of Residual Stresses Measurement, SEM, Bethel, CT(1996). 8. NF ISO 5725-2, "Exactitude (justesse etfid~litd) des r~sultats et m~thodes de mesure" (Dec. 1994). 9. Convert, E, "Mesure des d~placements des pics de diffraction tr~s larges dans l'analyse des contraintes par diffractomdtrie X," Revue Fran9aise de Mdcanique, 81-92 (1991). 10. Sprauel, J.M. and Barrallier, L., "Utilisation d'outils statistiques pour l'~valuation des contraintes par diffractom~trie X," Colloque de Rayons X Siemens, 2, Paris (Apr. 1992). 11. Sprauel, ,I.M., "dtude par diffraction X des facteurs mdcaniques influenqant la corrosion sous contraintes d' aciers inoxydables," Thbse de doctorat es sciences, Universit# de Paris (1988). 12. Noyan, L C, and Cohen, J.B., "Determining Stresses in the Presence of Nonlineari~es in lnterplanar Spacing vs. sin2xlr,'' Adv. X-ray Anal., 27, 129-148 (1984).