^{1}

^{*}

^{1}

In order to optimize the conditions for residual stress measurement using a two-dimensional X-ray diffraction (2D-XRD) in terms of both efficiency and accuracy. The measurements have been conducted on three stainless steel specimens in this study. The three specimens were processed by annealing, a cavitating jet in air and a disc grinder, with each method introducing different residual stresses at the surface. The specimens were oscillated in the ω-direction, representing a right-hand rotation of the specimen about the incident X-ray beam. The range of the oscillation, Δω, was varied and optimum Δω was determined. Moreover, combinations of the tilt angle between the specimen surface normal and the diffraction vector, ψ, with the rotation angle about its surface normal, f, have been studied with a view to find the most optimum condition. The results show that the use of ω oscillations is an effective method for improving analysis accuracy, especially for large grain metals. The standard error rapidly decreased with increasing range of the ω oscillation, especially for the annealed specimen which generated strong diffraction spots due to its large grain size.

X-ray diffraction (XRD) is an effective method for the measurement of residual stress in polycrystalline metals. Such measurements are important since residual stress affects the mechanical property of metallic materials, e.g., resistance to fatigue strength [1-3], stress corrosion cracking [

The sin^{2}ψ method is one of the most simple and common XRD methods for residual stress measurements and standardized approaches for this method have already been suggested [8,9]. On problem for the sin^{2}ψ method, however, is that it is not easy to conduct accurate residual stress measurements for large grain metals when employing only a one-dimensional position sensitive proportional counter (PSPC) or scintillation counter. With such a detector only a few grains are irradiated resulting in a sparse X-ray diffraction pattern and shortage of ψ angular information. In addition, the conventional sin^{2}ψ method can measure residual stresses only in one direction per measurement and is therefore only really suitable for metals which do not have a complex stress field, i.e., not including shear stresses. In order to solve this problem, several studies have been conducted on the conventional sin^{2}ψ method and based on these a useful additional methodology has been proposed for evaluating the biaxial stress including shear stress [10,11].

Recently, an XRD method for residual stress measurements using a two-dimensional PSPC (2D-PSPC) has been developed. The method focuses on the direct relationship between the stress tensor and diffraction conic section distortion [^{2}). The 2DXRD method can measure six components of the stress tensor, i.e., biaxial stress, including shear stress, from the direct relationship between the stress tensor and the distortion of the diffraction conic section obtained directly by the 2D-PSPC. Moreover, this method can also measure the local residual stress in textured material, weakly diffracting material and in regions of small area down to mm^{2}, since the diffracted X-ray are efficiently detected by the 2D-PSPC.

The fundamentals and details of 2D-XRD have been described well in the literature [13,14] and in a book [

The diffracted X-ray from large grain materials, such as annealed specimens, produces strongly scattered spots in the X-ray diffraction profile and, consequently, a lack of accuracy in the residual stress measurement. This occurs even if the measurement time is prolonged. This can be a problem for the measurements using 2D-XRD. In order to solve this problem, an oscillatory method should be effective and indeed in the conventional sin^{2}ψ method, an oscillation of the ψ angle is generally used. However when the ψ angle is oscillated, the range of this oscillation needs to be as small as possible because the incident angle about the lattice plane directly changes during the ψ oscillation. This affects the relationship between the diffraction angle and the ψ angle, i.e., the 2θ-sin^{2}ψ relationship. For this reason an oscillation which keeps the ψ angle constant is more preferable.

In this paper residual stress measurements using the 2D-XRD method have been carried out on three specimens made of stainless steel in order to optimize several of the conditions for residual stress measurements, e.g., sample rotation angles and exposure time, with respect to both measurement efficiency and accuracy. The specimens had three types of residual stress as follows: a low stress value with a large grain (by Annealing), an equibiaxial compressive residual stress (by Cavitating jet in air) and an anisotropic tensile residual stress (by Disc grinder). The conditions for the stress measurements were optimized, in terms of the detection time and the number of diffraction ring measurements taken from various angles of the specimen. The optimization process took into account the achievement of both high accuracy and good measurement efficiency.

The material under test was made of JIS SUS316L austenitic stainless steel. The geometry and dimensions of the specimen were 35 mm square and 3 mm thick. Three types of treatment were chosen which introduce different residual stresses into the specimens. The specimens were prepared by annealing, the use of a cavitating jet in air and by using a disc grinder. The annealing treatment releases residual stresses introduced by shape forming and increases the grain size. The use of a cavitating jet in air introduces high equibiaxial compressive residual stresses into the surface layer by impacts due to cavitation bubble collapse. The method is highly effective and has now been applied for practical usage. The disc grinder is one of a number of surface finishing methods that are frequently applied to metals. Its characteristics are that it introduces high anisotropic tensile residual stress at the near surface of the specimen. For the annealed specimen, the sample was heated at 1000 degree Celsius for 1 h, and then furnace cooled. For the specimen prepared using a cavitating jet in air, the conditions for the treatment were same as previously reported [16,17] and for this treatment the processing time per unit length was 1 s/mm. For the sample prepared using the disc grinder, the rotation speed was set to 11,000 rpm and the disc, manufactured by NIPPON RESIBON COOPRATION (A/W36P), had a diameter of 100 mm. The direction of residual stress caused by the disc grinder was defined by the rotating and the scanning direction of the disc as x and y, respectively. After the preparation of the specimens, residual stress measurements were carried out in accordance with following conditions.

The X-ray diffraction measurements were carried out using Cr Ka X-rays from a tube operated at 35 kV and 40 mA through a 0.5 mm diameter total reflection collimetor and with an incident monochromator (D8 DISCOVER, Bruker AXS Inc.). The lattice plane, (h k l), used was the γ-Fe (2 2 0) plane and the diffraction angle without strain was 128 degrees. The specimen was placed in a diffractometer, the geometry of which is shown in

An initial ω angle of 110 degrees was chosen and the specimen was oscillated over a range of Δω during the measurement. The parameter Δω was varied between 0 (without oscillation) and 2, 4, 6, 8 and 10 degrees in order to verify the effect of the ω oscillation on the diffraction ring and ultimately on the stress calculation. In addition, the exposure time per frame for the detection of the diffraction ring at a single position (f, ψ), t_{e}, was also varied as t_{e} = 60, 90, 120, 150 and 180 sec for each value of

(a) Diffractometer

(b) Axes of the specimen

(c) Coordinates of the diffraction ring

Δω and for each of the three specimens. The f and ψ angles were chosen as shown in

were obtained from 33 frames of this measurement. The biaxial stress was calculated from these data points. From these measurements, the optimum range of ω oscillation, Δω_{opt}, and exposure time, t_{e}_{ opt}, were determined.

For the biaxial residual stress measurement at least six different diffraction rings are needed, taken in appropriate directions [_{opt} and t_{e} = t_{e}_{ opt}. The black points in

Table. 2. Conditions of the specimen rotation angles.

by the conditions shown in

Figures 4-6 plot the biaxial residual stress, σ_{Rx}, σ_{Ry}, and the standard errors, Δσ_{Rx}, Δσ_{Ry}, varying with the exposure time, t_{e}, as a function of the range of the ω oscillation, Δω, for the annealed specimen (AN), the specimen treated by means of a cavitating jet in air (CJA) and the sample processed using a disc grinder (DG), respectively. The negative value on the plots represents a compressive stress. The standard error values Δσ_{Rx} and Δσ_{Ry} are almost the same for all the specimens, since the condition for these measurements can obtain the diffraction ring distortion from multi-symmetrical directions. In _{Rx} and σ_{Ry} are quite variable, for instance, σ_{Rx} varies from −150 to 30 MPa without oscillation (Δω = 0 degree) and the standard error is over 200 MPa. This error does not decrease in spite of an increase in the exposure time, e.g., in the case of t_{e} = 180 s. In contrast, the values of σ_{Rx} and σ_{Ry} gradually converge with σ_{Rx} = −20 MPa and σ_{Ry} = −25 MPa with an increase in the range of the oscillation, Δω. Moreover, the standard error, Δσ_{Rx}, and, Δσ_{Ry}, rapidly decreases along with increasing Δω and then saturates at Δω = 8 degrees for each t_{e}. These results will be discussed later.

In Figures 5 and 6 it can be seen that high equibiaxial compressive residual stresses, e.g., σ_{Rx} = −381 and σ_{Ry} = −354 MPa at Δω = 8 degree for t_{e} = 120 s, were introduced by CJA processing and high anisotropic tensile residual stresses, e.g., σ_{Rx} = 657 and σ_{Ry} = 196 MPa at Δω = 8 degree for t_{e} = 120 s, were introduced by DG processing into each specimen. The values of σ_{Rx} and σ_{Ry} for both the CJA and DG specimens does not fluctuate regardless of the values of Δω and t_{e}, unlike those of the AN specimen in _{Rx} and Δσ_{Ry} slightly decrease with increasing Δω for each t_{e}. The standard error exhibits a small value, e.g., Δσ_{Rx} = 6 MPa at Δω = 8 degree for t_{e} = 120 s without oscillation, except for the case of Δσ_{Rx} = 28 MPa at t_{e} = 60 s. For the DG specimen as shown in _{Rx} and Δσ_{Ry} values stay constant regardless of an increase in Δω for each t_{e}. The standard error also exhibits a low value, e.g., Δσ_{Rx} = 6 MPa at Δω = 8 degree for t_{e} = 120 s in the case without oscillation. The ω oscillation has a favorable effect on the residual stress measurements, considerably improving those for the AN specimen and then showing a decreasing improvement for the CJA and DG specimens, respectively. From these results, the optimum range of the ω oscillation, Δω_{opt}, and of the exposure time, t_{e}_{ opt}, can be determined as Δω_{opt} = 8 degree and t_{e}_{ opt} = 120 s, respectively in this

(a) f angle for baseline and Condition 5 (b) f angle for Condition 1 and 6 (c) f angle for Condition 2 and 7

(d) f angle for Condition 3 and 8 (e) f angle for Condition 4 and 9

(f) ψ angle for baseline and Condition 1, 2, 3 and 4 (g) ψ angle for Condition 5, 6, 7, 8 and 9

(a) Residual stress in the x direction (b) Residual stress in the y direction

(c) Standard error in the x direction (d) Standard error in the y direction

(a) Residual stress in the x direction (b) Residual stress in the y direction

(c) Standard error in the x direction (d) Standard error in the y direction

(a) Residual stress in the x direction (b) Residual stress in the y direction

(c) Standard error in the x direction (d) Standard error in the y direction

study.

The optimum exposure time needs to be generalized in order to apply it to other metals. The standard error, Δσ_{Rx}, is plotted as a function of the number of X-ray counts, N, for the AN specimen in _{Rx} decreases with an increasing number of counts, but then saturates at around N = 200,000 for at t_{e} = 120 s. For these measurements it can be concluded that an exposure time which realizes N ≥ 200,000 is enough to calculate accurately the distortion of the diffraction ring due to

residual stress. This rule can be applied just as much to other metals as it can be to SUS316L.

_{e} = 60 and 180 s. For the AN specimen shown in _{e} = 60 and 180 s cases. This makes the distortion of diffraction ring due to residual stresses much easier to detect, leading to an improved accuracy for the residual stress measurements as shown

(a) Annealed specimen (AN)

(b) Specimen treated using the cavitating jet in air (CJA)

(c) Specimen treated using the disc grinder (CJA)

the surface and has a tendency to minimize the grain size, which can be instantly visible through the ease of formation of the diffraction ring.

Summarizing the points above, the use of ω oscillation is quite an effective way of suppressing the large grain size effect, which occurs from annealed metals. The use of ω oscillation does not require an additional time and helps to detect X-rays diffracted from other grains, which have same crystal orientation. In general terms the use of ω oscillation is effective in residual stress measurements to improve the accuracy.

It is important to verify the effect of choosing different angles for specimen rotation, i.e., for the f and ψ angles, on the residual stress value and on the standard error value. Figures 9-11 plot the ratio of the residual stress value and the standard error to the baseline value, σ_{R}/σ_{R}_{ B}, and, Δσ_{R}/Δσ_{R}_{ B}, for conditions 1 to 9 for the AN, CJA and DG specimens, respectively. The baseline value was calculated from the 33 frames shown in _{Rx} and σ_{Ry} values, respectively. The baseline values are −38 ± 9 MPa in σ_{Rx} and −34 ± 9 MPa in σ_{Ry} for the AN specimen, −381 ± 6 MPa in σ_{Rx} and −354 ± 6 MPa in σ_{Ry} for the CJA specimen and 657 ± 6 MPa in σ_{Rx} and 196 ± 7 MPa in σ_{Ry} for the DG specimen, respectively. The value which nearly equals to 1 represents that the stress calculation has been precisely done as well as the baseline data.

The σ_{R}/σ_{R}_{ B} and Δσ_{R}/Δσ_{R}_{ B} values for each specimen are highly dependent upon the combination of f and ψ angles chosen. Using conditions 2, 4, 7 and 9 shows inaccurate results for σ_{R}/σ_{R}_{ B}, especially for the AN and the DG specimens. For instance, the σ_{R}/σ_{R}_{ B} and Δσ_{R}/Δσ_{R}_{ B} values calculated using condition 7 are 3.4 and 5.8, respectively with regard to σ_{Rx} for the AN specimen. These values are quite large. The reason for this is that these

(a) Biaxial residual stress

(b) Standard error

(a) Biaxial residual stress

(b) Standard error

(a) Biaxial residual stress

(b) Standard error

conditions can obtain a diffraction ring from only one side, e.g., for f = 0, 45, 90, 135 and 180 degrees in conditions 2 and 7. In contrast, accurate results for σ_{R}/σ_{R}_{ B} for each specimen are obtained using conditions 1, 3, 5, 6 and 8. This occurs since several angles of f, which are not one-sided, were employed in these conditions, e.g., f = 0, 90, 180, and 270 degrees for conditions 3 and 9. The general rule is that a lack of diffraction information from a opposite side causes variability in the determination of the stress vector. This effect appears in any stress direction and depends precise combination of ω, f and ψ angles used [

Comparing conditions 3 and 5, both having same number of frames of 17, the Δσ_{R}/Δσ_{R}_{ B} for σ_{Rx} values obtained were 2.0 and 1.6 for the AN specimen, 2.7 and 1.3 for the CJA specimen and 2.4 and 1.3 for the DG specimen, respectively. Overall therefore, the standard error obtained using condition 5 is smaller than that using condition 3. Focusing on the combination of f and ψ angles, the intervals between these, Δf and Δψ, are 45 and 30 degrees, respectively, in condition 5 and are 90 and 15 degrees, respectively, in condition 3. A Δψ value of 15 degrees is more than enough, since a large area can be covered with regard to the ψ direction using the 2D-PSPC. The area in the ψ direction covered by the 2D-PSPC in the case of both Δψ = 15 and Δψ = 30 degrees is plotted in

In order to determine the optimum conditions, _{R}/Δσ_{R}_{ B}, as a function of total measurement time, t_{T}, for the AN specimen. In _{R}/Δσ_{R}_{ B} value of close to 1 represents a measurement result that shows the same high accuracy as the baseline. The lower t_{T} time indicates that the meas-

(a) Δψ = 15 degree

(b) Δψ = 30 degree

(a) Ratio of standard error to the baseline data in the x direction

(b) Ratio of standard error to the baseline data in the y direction

urements become quicker. Therefore, the condition realized by the lower left point in

In order to optimize the conditions for residual stress measurement using 2D-X-ray diffraction (2D-XRD) in terms of both efficiency and accuracy, residual stress measurements were conducted on three specimens made of austenitic stainless steel. The specimens were processed by annealing, a cavitating jet in air and a disc grinder introducing different residual stresses at the surface. The specimens were oscillated in the ω direction, representing a right-hand rotation of the specimen about incident X-ray with a varying range of the oscillation, Δω and taken for several exposure times, te. Moreover, the combination of the tilt angle between the specimen surface normal and the diffraction vector, ψ, with the rotation angle about its surface normal, f, has been optimized in terms of the measurement efficiency and accuracy. The conclusions obtained in the present study are summarized as follows.

1) The standard error rapidly decreases with an increase in the range of the ω oscillation, Δω. This occurs especially for the annealed specimen, which creates strongly diffracted spots due to the presence of large grain sizes within the sample. The use of ω oscillations is quite an effective way of suppressing the problems posed by large grain metals. For the specimens processed by means of the cavitating jet in air and the disc grinder, the effectiveness of the ω oscillation method is less than that for the annealed specimen due to the minimization of grain size caused by these processes. However the oscillation still helps by detecting additional X-rays diffracted from other grains with the same crystal orientation. Therefore, the use of ω oscillations is generally effective for improving the accuracy of residual stress measurements. The optimum range of the ω oscillation, Δω, is around 8 degrees.

2) The greater the number of diffraction rings, the greater the accuracy of the result. When the number of diffraction rings is same, the detection of the diffraction ring at several f angles makes the measurements more accurate than detecting at several ψ angles. The combination of using ψ and f values of f = 0 degree and ψ = 0 degree with f = 0, 45, 90, 135, 180 and 270 degrees and ψ = 30 and 60 degrees can lead to an optimized quick and accurate result.

This work was partly supported by the Japan Society for the Promotion of Science under the Grant-in-Aid for Scientific Research (B) 24360040, the Young Scientists (Startup) 24860004.