This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles

for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.

Designation: E2860 − 20

Standard Test Method for

Residual Stress Measurement by X-Ray Diffraction for
Bearing Steels1
This standard is issued under the fixed designation E2860; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.

INTRODUCTION

The measurement of residual stress using X-ray diffraction (XRD) techniques has gained much
popularity in the materials testing field over the past half century and has become a mandatory test for
many production and prototype bearing components. However, measurement practices have evolved
over this time period. With each evolutionary step, it was discovered that previous assumptions were
sometimes erroneous, and as such, results obtained were less reliable than those obtained using
state-of-the-art XRD techniques. Equipment and procedures used today often reflect different periods
in this evolution; for example, systems that still use the single- and double-exposure techniques as well
as others that use more advanced multiple exposure techniques can all currently be found in
widespread use. Moreover, many assumptions made, such as negligible shear components and
non-oscillatory sin2ψ distributions, cannot safely be made for bearing materials in which the demand
for measurement accuracy is high. The use of the most current techniques is, therefore, mandatory to
achieve not only the most reliable measurement results but also to enable identification and evaluation
of potential measurement errors, thus paving the way for future developments.

1. Scope* 1.3.8 Other residual-stress-related issues that potentially

1.1 This test method covers a procedure for experimentally affect bearings.
determining macroscopic residual stress tensor components of 1.4 Units—The values stated in SI units are to be regarded
quasi-isotropic bearing steel materials by X-ray diffraction as standard. No other units of measurement are included in this
(XRD). standard.
1.2 This test method provides a guide for experimentally 1.5 This standard does not purport to address all of the
determining stress values, which play a significant role in safety concerns, if any, associated with its use. It is the
bearing life. responsibility of the user of this standard to establish appro-
priate safety, health, and environmental practices and deter-
1.3 Examples of how tensor values are used are:
mine the applicability of regulatory limitations prior to use.
1.3.1 Detection of grinding type and abusive grinding;
1.6 This international standard was developed in accor-
1.3.2 Determination of tool wear in turning operations;
dance with internationally recognized principles on standard-
1.3.3 Monitoring of carburizing and nitriding residual stress
ization established in the Decision on Principles for the
effects;
Development of International Standards, Guides and Recom-
1.3.4 Monitoring effects of surface treatments such as sand
mendations issued by the World Trade Organization Technical
blasting, shot peening, and honing;
Barriers to Trade (TBT) Committee.
1.3.5 Tracking of component life and rolling contact fatigue
effects; 2. Referenced Documents
1.3.6 Failure analysis;
1.3.7 Relaxation of residual stress; and 2.1 ASTM Standards:2
E6 Terminology Relating to Methods of Mechanical Testing
E7 Terminology Relating to Metallography
1
This test method is under the jurisdiction of ASTM Committee E28 on
Mechanical Testing and is the direct responsibility of Subcommittee E28.13 on
2
Residual Stress Measurement. For referenced ASTM standards, visit the ASTM website, www.astm.org, or
Current edition approved Nov. 1, 2020. Published February 2021. Originally contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
approved in 2012. Last previous edition approved in 2012 as E2860–12. DOI: Standards volume information, refer to the standard’s Document Summary page on
10.1520/E2860–20. the ASTM website.

*A Summary of Changes section appears at the end of this standard

Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States

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 E2860 − 20
E177 Practice for Use of the Terms Precision and Bias in αL = Linear thermal expansion coefficient
ASTM Test Methods β = Angle between the incident beam and σ33 or surface
E691 Practice for Conducting an Interlaboratory Study to normal on the σ33 σ11 plane
Determine the Precision of a Test Method χ = Angle between the σφ+90° direction and the normal to the
E915 Test Method for Verifying the Alignment of X-Ray diffracting plane
Diffraction Instrumentation for Residual Stress Measure- χm = Fixed χ offset used in modified-chi mode
ment d = Interplanar spacing between crystallographic planes;
E1426 Test Method for Determining the X-Ray Elastic also called d-spacing
Constants for Use in the Measurement of Residual Stress do = Interplanar spacing for unstressed material
Using X-Ray Diffraction Techniques d' = Perpendicular spacing
2.2 ANSI Standards:3 ∆d = Change in interplanar spacing caused by stress
N43.2 Radiation Safety for X-ray Diffraction and Fluores- εij = Strain component i, j
cence Analysis Equipment E = Modulus of elasticity (Young’s modulus)
N43.3 For General Radiation Safety—Installations Using Eeff{hkl} = Effective elastic modulus for X-ray measurements
Non-Medical X-Ray and Sealed Gamma-Ray Sources, µ = Attenuation coefficient
Energies Up to 10 MeV η = Rotation of the sample around the measuring direction
2.3 SAE standard:4 given by φ and ψ or χ and β
HS-784/2003 Residual Stress Measurement by X-Ray ω or Ω = Angle between the specimen surface and incident
Diffraction, 2003 Edition beam when χ = 0°
φ = Angle between the σ11 direction and measurement di-
3. Terminology
rection azimuth, see Fig. 1
3.1 Definitions—Many of the terms used in this test method “hkl” = Miller indices
are defined in Terminologies E6 and E7. σij = Normal stress component i, j
3.2 Definitions of Terms Specific to This Standard: s1{hkl} = X-ray elastic constant of quasi-isotropic material
3.2.1 interplanar spacing, n—perpendicular distance be- 2ν
equal to E $ hkl%
tween adjacent parallel atomic planes. eff

3.2.2 macrostress, n—average stress acting over a region of τij = Shear stress component i, j
the test specimen containing many gains/crystals/coherent θ = Bragg angle
domains. ν = Poisson’s ratio
xMode = Mode dependent depth of penetration
3.3 Abbreviations: ψ = Angle between the specimen surface normal and the
3.3.1 ALARA—As low as reasonably achievable scattering vector, that is, normal to the diffracting plane, see
3.3.2 FWHM—Full width half maximum Fig. 1
3.3.3 LPA—Lorentz-polarization-absorption
3.3.4 MSDS—Material safety data sheet 4. Summary of Test Method
3.3.5 XEC—X-ray elastic constant 4.1 A test specimen is placed in a XRD goniometer aligned
as per Test Method E915.
3.3.6 XRD—X-ray diffraction
{hkl} 4.2 The diffraction profile is collected over three or more
3.4 Symbols: 1⁄2 S2 = X-ray elastic constant of quasi-
11ν
angles within the required angular range for a given {hkl}
isotropic material equal to E $ hkl% plane, although at least seven or more are recommended.
eff
4.3 The XRD profile data are then corrected for LPA,
3
Available from American National Standards Institute (ANSI), 25 W. 43rd St., background, and instrument-specific corrections.
4th Floor, New York, NY 10036, http://www.ansi.org.
4
Available from SAE International (SAE), 400 Commonwealth Dr., Warrendale,
4.4 The peak position/Bragg angle is determined for each
PA 15096, http://www.sae.org. XRD peak profile.

FIG. 1 Stress Tensor Components

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4.5 The d-spacings are calculated from the peak positions $ hkl% 5
1 $ hkl%
via Bragg’s law.
ε φψ s
2 2
@ σ 11 cos2 φ sin2 ψ1σ 22 sin2 φ sin2 ψ1σ 33 cos2 ψ #

4.6 The d-spacing values are plotted versus their sin2ψ or 1

1 s $2hkl% @ τ 12sin~ 2φ ! sin2 ψ1τ 13cosφsin~ 2ψ ! 1τ 23sinφsin~ 2ψ ! #
sin2β values, and the residual stress is calculated using Eq 4 or 2
Eq 8, respectively. 1s $1hkl% @ σ 111σ 221σ 33# (2)
4.7 The error in measurement is evaluated as per Section 14. 5.1.1 Alternatively, Eq 2 may also be shown in the follow-
4.8 The following additional corrections may be applied. ing arrangement (2, p. 126):
The use of these corrections shall be clearly indicated with the 1 $ hkl%
$ hkl% 5
reported results. ε φψ s
2 2
@ σ 11 cos2 φ1τ 12sin~ 2φ ! 1σ 22 sin2 φ 2 σ 33# sin2 ψ
4.8.1 Depth of penetration correction (see 12.12) and
1 1
4.8.2 Relaxation as a result of material removal correction 1 s $2hkl% σ 33 2 s $1hkl% @ σ 111σ 221σ 33# 1 s $2hkl% @ τ 13cosφ
2 2
(see 12.14).
1τ 23sinφ # sin~ 2ψ !
5. Significance and Use 5.2 Using XRD and Bragg’s law, interplanar strain measure-
5.1 This test method covers a procedure for experimentally ments are performed for multiple orientations. The orientations
determining macroscopic residual stress tensor components of are selected based on a modified version of Eq 2, which is
quasi-isotropic bearing steel materials by XRD. Here the stress dictated by the mode used. Conflicting nomenclature may be
components are represented by the tensor σij as shown in Eq 1 found in literature with regard to mode names. For example,
(1,5 p. 40). The stress strain relationship in any direction of a what may be referred to as a ψ (psi) diffractometer in Europe
component is defined by Eq 2 with respect to the azimuth may be called a χ (chi) diffractometer in North America. The
phi(φ) and polar angle psi(ψ) defined in Fig. 1 (1, p. 132). three modes considered here will be referred to as omega, chi,
and modified-chi as described in 9.5.
σ ij 5 F σ 11 τ 12 τ 13
τ 21 σ 22 τ 23
τ 31 τ 32 σ 33
G where τ ij 5 τ ji (1) 5.3 Omega Mode (Iso Inclination) and Chi Mode (Side
Inclination)—Interplanar strain measurements are performed at
multiple ψ angles along one φ azimuth (let φ = 0°) (Figs. 2 and
3), reducing Eq 2 to Eq 3. Stress normal to the surface (σ33) is
assumed to be insignificant because of the shallow depth of
penetration of X-rays at the free surface, reducing Eq 3 to Eq
5
The boldface numbers in parentheses refer to the list of references at the end of
this standard.

FIG. 2 Omega Mode Diagram for Measurement in σ11 Direction

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 E2860 − 20

NOTE 1—Stress matrix is rotated 90° about the surface normal compared to Fig. 2 and Fig. 14.
FIG. 3 Chi Mode Diagram for Measurement in σ11 Direction

4. Post-measurement corrections may be applied to account for slope of the data, while τ13 is related to the direction and degree
possible σ33 influences (12.12). Since the σij values will remain of elliptical opening. Fig. 4 shows a simulated d versus sin2ψ
constant for a given azimuth, the s1{hkl} term is renamed C. profile for the tensor shown. Here the positive 20-MPa τ13
1 $ hkl% 1 stress results in an elliptical opening in which the positive psi
$ hkl% 5
ε φψ s
2 2
@ σ 11 sin2 ψ1σ 33 cos2 ψ # 1 2 s $2hkl% @ τ 13sin~ 2ψ ! # 1s $1hkl% @ σ 11 range opens upward and the negative psi range opens down-
ward. A higher τ13 value will cause a larger elliptical opening.
1σ 221σ 33# (3)
A negative 20-MPa τ13 stress would result in the same elliptical
$ hkl% 5
1 $ hkl% opening only the direction would be reversed with the positive
ε φψ s
2 2
@ σ 11 sin2 ψ1τ 13sin~ 2ψ ! # 1C (4)
psi range opening downwards and the negative psi range
5.3.1 The measured interplanar spacing values are con- opening upwards as shown in Fig. 5.
verted to strain using Eq 24, Eq 25, or Eq 26. Eq 4 is used to 5.4 Modified Chi Mode—Interplanar strain measurements
fit the strain versus sin2ψ data yielding the values σ11, τ13, and are performed at multiple β angles with a fixed χ offset,
C. The measurement can then be repeated for multiple phi χm (Fig. 6). Measurements at various β angles do not provide a
angles (for example 0, 45, and 90°) to determine the full constant φ angle (Fig. 7), therefore, Eq 2 cannot be simplified
stress/strain tensor. The value, σ11, will influence the overall in the same manner as for omega and chi mode.

FIG. 4 Sample d (2θ) Versus sin2ψ Dataset with σ11 = -500 MPa and τ13 = +20 MPa

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 E2860 − 20

FIG. 5 Sample d (2θ) Versus sin2ψ Dataset with σ11 = -500 MPa and τ13 = -20 MPa

FIG. 6 Modified Chi Mode Diagram for Measurement in σ11 Direction

5.4.1 Eq 2 shall be rewritten in terms of β and χm. Eq 5 and 5.4.2 Substituting φ and ψ in Eq 2 with Eq 5 and 6 (see
6 are obtained from the solution for a right-angled spherical X1.1), we get:
triangle (3).
ψ 5 arccos~ cos βcosχ m ! (5)

φ 5 arccos S sin βcosχ m

sin ψ D (6)

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FIG. 7 ψ and φ Angles Versus β Angle for Modified Chi Mode with χm = 12°

$ hkl% 5
1 $ hkl% 5.4.5 The τij influences on the d versus sin2β plot are more
ε βχ s @ σ 11 sin2 β cos2 χ m 1σ 22 sin2 χ m 1σ 33 cos2 β cos2 χ m #
m 2 2 complex and are often assumed to be zero (3). However, this
1 may not be true and significant errors in the calculated stress
1 s $2hkl% @ τ 12sinβsin~ 2χ m ! 1τ 13sin~ 2β ! cos2 χ m 1τ 23cosβsin~ 2χ m ! # may result. Figs. 9-13 show the d versus sin2β influences of
2
1s $1hkl% @ σ 111σ 221σ 33# (7) individual shear components for modified chi mode consider-
ing two detector positions (χm = +12° and χm = -12°). Compo-
5.4.3 Stress normal to the surface (σ33) is assumed to be nents τ12 and τ13 cause a symmetrical opening about the σ11
insignificant because of the shallow depth of penetration of slope influence for either detector position (Figs. 9-11);
X-rays at the free surface reducing Eq 7 to Eq 8. Post-
therefore, σ11 can still be determined by simply averaging the
measurement corrections may be applied to account for pos-
positive and negative β data. Fitting the opening to the τ12 and
sible σ33 influences (see 12.12). Since the σij values and χm will
τ13 terms may be possible, although distinguishing between the
remain constant for a given azimuth, the s1{hkl} term is
renamed C, and the σ22 term is renamed D. two influences through regression is not normally possible.
5.4.6 The τ23 value affects the d versus sin2β slope in a
1 $ hkl% 1
$ hkl% 5
ε βχ s @ σ 11 sin2 β cos2 χ m 1D # 1 2 s $2hkl% @ τ 12sinβsin~ 2χ m ! similar fashion to σ11 for each detector position (Figs. 12 and
m 2 2
13). This is an unwanted effect since the σ11 and τ23 influence
1τ 13sin~ 2β ! cos2 χ m 1τ 23cosβsin~ 2χ m ! # 1C (8) cannot be resolved for one χm position. In this instance, the τ23
2 shear stress of -100 MPa results in a calculated σ11 value of
5.4.4 The σ11 influence on the d versus sin β plot is similar
to omega and chi mode (Fig. 8) with the exception that the -472.5 MPa for χm = +12° or -527.5 MPa for χm = -12°, while
slope shall be divided by cos2χm. This increases the effective 1⁄2 the actual value is -500 MPa. The value, σ11 can still be
s2{hkl} by a factor of 1/cos2χm for σ11. determined by averaging the β data for both χm positions.

FIG. 8 Sample d (2θ) Versus sin2β Dataset with σ11 = -500 MPa

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 E2860 − 20

FIG. 9 Sample d (2θ) versus sin2β Dataset with χm = +12°, σ11 = -500 MPa, and τ12 = -100 MPa

FIG. 10 Sample d (2θ) Versus sin2β Dataset with χm = -12°, σ11 = -500 MPa, and τ12 = -100 MPa

FIG. 11 Sample d (2θ) Versus sin2β Dataset with χm = +12 or -12°, σ11 = -500 MPa, and τ13 = -100 MPa

5.4.7 The use of the modified chi mode may be used to 6.1.2 X-Ray Source—There are generally three X-ray
determine σ11 but shall be approached with caution using one sources used for XRD.
χm position because of the possible presence of a τ23 stress. The 6.1.2.1 Conventional Sealed Tube—This is by far the most
combination of multiple shear stresses including τ23 results in common found in XRD equipment. It is identified by its anode
increasingly complex shear influences. Chi and omega mode target element such as chromium (Cr), manganese (Mn), or
are preferred over modified chi for these reasons. copper (Cu). The anode is bombarded by electrons to produce
specific X-ray wavelengths unique to the target element.
6. Apparatus 6.1.2.2 Rotating Anode Tube—This style of tube offers a
6.1 A typical X-ray diffractometer is composed of the higher intensity than a conventional sealed tube.
following main components: 6.1.2.3 Synchrotron—Particle accelerator that is capable of
6.1.1 Goniometer—An angle-measuring device responsible producing a high-intensity X-ray beam.
for the positioning of the source, detectors, and sample relative 6.1.2.4 Sealed Radioactive Sources—Although not com-
to each other. monly used, they may be utilized.

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 E2860 − 20

FIG. 12 Sample d (2θ) Versus sin2β Dataset with χm = +12°, σ11 = -500 MPa, τ23 = -100 MPa, and Measured σ11 = -472.5 MPa

FIG. 13 Sample d (2θ) Versus sin2β Dataset with χm = -12°, σ11 = -500 MPa, τ23 = -100 MPa, and Measured σ11 = -527.5 MPa

6.1.3 Detector—Detectors may be of single channel, multi- 7.4 Refer to material safety data sheets (MSDS) sheets for
channel linear, or area design. handling of dangerous materials potentially found in XRD
6.1.4 Software—Software is grouped into the following equipment (that is, beryllium and lead).
main categories: 7.5 The high voltage used to generate X-rays is very
6.1.4.1 Goniometer control—Responsible for positioning of dangerous. Follow the manufacturer’s and local guidelines
the sample relative to the incident beam and detector(s) in when dealing with high-voltage equipment.
automated goniometers.
6.1.4.2 Data acquisition—Responsible for the collection of 8. Test Specimens
diffraction profile data from the detector(s).
6.1.4.3 Data processing—Responsible for all data fitting 8.1 This guide is intended for materials with the following
and calculations. characteristics:
6.1.4.4 Data management—Responsible for data file man- 8.1.1 Fine grain size and
agement as well as overall record keeping. Individual measure- 8.1.2 Near random coherent domain orientation distribution.
ment data is typically stored in a file format that can later be 8.2 Test specimens shall be clean at the measured location
reopened for reevaluation. It is often beneficial to keep a and should be free of visible signs of oxidation, material debris,
database of key measurement values and file names. and coatings such as oil and paint.
7. Hazards 8.3 Sample surfaces shall be free of any significant rough-
ness. Grooves produced by machining perpendicular to the
7.1 Regarding the use of analytical X-ray equipment, local measurement direction may affect measurement results (4, p.
government regulations or guidelines shall always be followed. 21).
Examples include ANSI N43.2-2001 and ANSI N43.3.
8.4 Sample surfaces may be prepared using electropolishing
7.2 The as low as reasonably achievable (ALARA) philoso- as this method does not impart stress within the sample;
phy should always be used when dealing with radiation however, removal of stressed layers may influence the subsur-
exposure. face residual stress state. Corrections are available to estimate
7.3 Always follow the safety guidelines of the equipment the true stress that existed when the specimen was intact (see
manufacturer. 12.13).

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 E2860 − 20
8.5 If material removal methods other than electropolishing austenitic bearing steels, the Mn Kα target is typically used
(that is, grinding or sanding) are necessary, subsequent elec- with the {311} plane with a 2θ angle of approximately 152 to
tropolishing is required to ensure the cold-worked region is 155º. Table 2 shows a list of target-plane combinations
removed. For light grinding or sanding, the removal of 0.25 commonly shown in literature. X-Ray elastic constants 1⁄2 s2
mm is recommended. and s1 may also be determined with Test Method E1426. The
8.6 Sample curvatures should not exceed the acceptable depth of penetration (x) for omega and chi mode based on Eq
limits for the goniometer setup used (see 9.1.2). 9 and 10 are included (DIN En 15305, p. 22), (1, p. 106). Note
that when ψ = 0, the depth of penetration is the same for either
8.7 Measurement of a single-phase stress in multiphase mode. The ψ = 0 values are, therefore, listed in the same
materials may not be representative of the bulk material when column. The depth of penetration for modified chi mode is
significant amount of additional phases are present. given by Eq 11.
8.8 Measurement of thin coatings may not be representative 1 sin2 θ 2 sin2 ψ
Ω
of the bulk material. Diffraction of the substrate may create x ψθ 5 (9)
2µ sin θcosψ
interfering diffraction lines.
χ
1
x ψθ 5 sinθcosψ (10)
9. Preparation of Apparatus 2µ

9.1 Primary Beam Size—The primary beam size is typically 1

χm
x βθ 5 cosβ ~ 1 2 cot2 θ ! (11)
adjustable using a primary beam aperture. To ensure the best 2µ
counting statistics, the largest beam size should be used that 9.3 Filters—Filters may be used to suppress Kβ peak inter-
does not exceed the following limitations: ference and fluorescence. A filter material is chosen based on
9.1.1 Preferably beam divergence should not exceed 1° (2, the Kedge value, which should lie between the target Kα and Kβ
p. 107). Divergence may be limited by devices such as Soller wavelength values. See Table 1.
slits and sample masking. 9.4 Monochromators—Monochromators(s) may be used to
9.1.2 For cylindrical specimens of radius, R, the maximum eliminate spectral components including the Kβ and the Kα2
incident X-ray spot size to use shall be R/6 for 5 % error and line, although they will reduce the beam intensity and increase
R/4 for 10 % error in the hoop direction and R/2.5 and R/2 for measurement time significantly.
5 % and 10 % error, respectively, in the axial direction. In cases
in which the beam size cannot be sufficiently small, corrections 9.5 Modes—Three modes are described in 9.5.1 – 9.5.3.
can be applied (5, p. 107), (6, pp. 327-336). Each has specific advantages and disadvantages. Some goni-
ometers offer multiple modes.
9.2 Target/Plane Combination—The characteristic wave- 9.5.1 Omega Mode—Also known as iso-inclination, ω, or Ω
lengths available for diffraction are determined by the target method. With omega mode, the incident beam and ψ angle(s)
element. A list of common target elements, their K line remain on the σφ-σ33 plane. Multiple ψ angles are observed by
wavelengths, and Kβ filters are shown in Table 1. rotation about the ω, Ω, θ, or σφ+90° axis while χ remains equal
9.2.1 There are several possible target-plane combinations to zero.
for a given bearing steel that will produce a diffraction peak. 9.5.1.1 Advantages:
When choosing a combination, there are many factors to take (1) Keeps experiment two dimensional (2D), which is
into account including the relative peak versus background useful for thin coatings, films, and layers;
intensity, mass absorption coefficient, possible interfering (2) Capable of accessing deep grooves perpendicular to
peaks, and strain resolution. Higher 2θ values will have a axis of rotation;
higher strain resolution thus improving measurement precision. (3) Using two detectors (if available) simultaneous obser-
A higher mass absorption coefficient reduces the depth of vation of both Debye ring locations is possible over the
penetration. Shallow penetration reduces stress gradient effects recommended complete ψ range; and
but limits the number of coherent domains contributing to the (4) Conducive to slit optics and improved particle statistics.
diffraction profile. When performing residual stress measure- 9.5.1.2 Disadvantages:
ments in martensitic bearing steels, the Cr Kα target is typically (1) Absorption varies with ψ angle;
used with the {211} plane with a 2θ angle of approximately (2) The use of single detector systems may require 180°
154 to 157º. When performing residual stress measurements in rotation of the sample about the σ33 axis to realize the full
recommended ψ range while avoiding low incidence angle
TABLE 1 Target Wavelengths and Appropriate Kβ Suppression errors; and
Filers
(3) Alignment issues may negate advantages of using two
Target Kα1 Kα2 Kβ Filter Kedge detectors.
Element [Å] = [nm × 10] 9.5.2 Chi Mode—Also known as side-inclination or χ
22 Ti 2.748 51 2.752 16 2.513 91 { method. With ω, Ω, or θ equal to 2θ/2, multiple ψ angles are
24 Cr 2.289 70 2.293 606 2.084 87 V 2.269 1
25 Mn 2.101 820 2.105 78 1.910 21 Cr 2.070 20 observed by rotation about the χ or σφ+90° axis while χ remains
26 Fe 1.936 042 1.939 980 1.756 61 Mn 1.896 43 equal to ψ.
27 Co 1.788 965 1.792 850 1.620 79 Fe 1.756 61 9.5.2.1 Advantages:
29 Cu 1.540 562 1.544 390 1.392 218 Ni 1.488 07
42 Mo 0.709 300 0.713 590 0.632 288 Nb 0.652 98 (1) Lorentz-polarization-absorption (LPA) is not affected
with varying ψ angle and

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 E2860 − 20
TABLE 2 Commonly Used Target, Plane, and 2θ Combinations
xψθΩ
{hkl} {hkl} and xψθΩ xψθχ
2θ ⁄ s2
12 s1
Target {hkl} Alloy xψθχ ψ = 45° ψ = 60°
ψ = 0°
[degrees] [10-6 MPa-1] [µm]
Ferritic and Martensitic Steels—BCC
Cr Kα {211} – 156.07 (1) 5.76 (7) -1.25 (7) 5.60 3.78 2.80
6.35 (HS-784) -1.48 (HS-784)
(0.73 %C) (0.73 %C)
4340 156.0 (8) 5.92 (8) – 5.46 3.69 2.73
(50 Rc)
SAE 52100 ~156 5.7504 (7) -1.327 (7) 5.58 3.77 2.79
100Cr6
M50 ~156 5.577 (7) -1.287 (7) 4.98 3.36 2.49
M50-Nil 5.4645 (7) -1.261 (7) 4.37 2.95 2.18
Fe Kα {220} – 145.54 (1) 5.63 (HS-784) -1.32 (HS-784) 8.65 5.55 4.33
(0.39 %C) (0.39 %C)
Co Kα {310} – 161.32 (1) 6.98 (7) -1.66 (7) 11.14 7.67 5.57
7.48 (HS-784) -1.84 HS-784)
(0.73 %C) (0.73 %C)
{220} – 123.9 (7) 5.76 (7) -1.25 (7) 9.97 5.05 4.98
{211} – 99.7 (7) 5.76 (7) -1.25 (7) 8.63 1.76 4.32
Mo Kα {732+651} (1) – 153.88 (7) 6.05 (7) -1.34 (7) 16.88 11.30 8.44
Ti Kα {200} – 146.99 (1) – – 16.40 10.97 8.20
Austenitic Steels—FCC
Mn Kα {311} – 152.26 (1) 6.98 (7) -1.87 (7) 7.02 4.66 3.51
Cr Kβ {311} – 148.74 (1) 6.98 (7) -1.87 (7) 5.50 3.57 2.75
Cr Kα {220} – 128.84 (1) 6.05 (7) -1.56 (7) 5.16 2.81 2.58
128 ± 1 (HS-784)
304 SS 129.0 (8) 7.18 (8) – 5.06 2.76 2.53
Cu Kα {420} – 147.28 (1) 1.94 1.27 0.97
150 ± 3 (HS-784)
Incoloy 800 147.0 6.75 (8) 2.90 1.87 1.45
Cu Kα {331} – 138.53 (1) 1.92 1.23 0.96
146 (HS-784)
Mo Kα {884} – 150.87 (1) 16.29 10.74 8.15

(2) Capable of accessing deep grooves parallel to axis of 10.2 Additionally, a nonzero known residual stress profi-
rotation. ciency reference sample should be measured to verify that
9.5.2.2 Disadvantages: hardware and software are working correctly.
(1) Beam spot on sample is pseudo elliptical and spreads NOTE 1—No national reference sample exists other than a stress-free
appreciably and sample. It is recommended that round robin methodologies be used to
(2) Usually requires spot focus and collimators that reduce determine the residual stress values of such reference samples. Specifica-
tion DIN EN 15305 provides a methodology for creating a stress-reference
particle statistics. specimen.
9.5.3 Modified Chi Mode—With modified chi mode, the
source positioning, sample positioning, and axis of rotation are 11. Procedure
the same as omega mode. The detector positions, however, are
rotated 90° about the incident beam creating a fixed χ offset 11.1 Position test specimen for measurement in the goniom-
(χm). Conflicting nomenclature may be found in literature with eter. Ensure that specimen-positioning devices such as clamps
regard to axis names. For example, the χ and ω names may be do not create an applied load because the XRD method does
reversed such that multiple angles are observed by rotation not differentiate between applied and residual stress but rather
about the χ axis. Since modified chi mode is typically used by measures the summation of the two.
omega mode diffractometers with detector positioning rotated 11.2 The angular range over which measurements are car-
90° about the incident beam, omega axis labeling is used for ried out is limited by the mode used. Measurements should
consistency. always be performed over the maximum permissible ψ range.
9.5.3.1 Advantage—Capable of accessing deep grooves par- If the range is further limited by specimen geometry, the largest
allel to axis of rotation. possible range should be used where no shadowing effects
9.5.3.2 Disadvantages: occur.
(1) Values τ12 and τ13 cannot be resolved (see 5.4) and 11.2.1 Iso Inclination—ψ max = 645° (sin2ψ = 0.5). (9, p.
(2) Values τ23 and σ11 cannot be resolved (see 5.4). 121)
11.2.2 Side Inclination—ψ max = 677° (sin2ψ = 0.95). (9,
10. Calibration and Standardization p. 121)
10.1 Instrument alignment can be verified with Test Method 11.2.3 Modified Chi—β max = 678° (sin2β = 0.96). (1, p.
E915 by the measurement of a stress-free powder. 179)

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11.3 In the case of single detector configurations other than diffraction peaks at high 2θ angles. These effects can be
chi mode, the range can be restricted to the positive or negative compensated for using the following equation (8, p. 131):
ψ range depending on the goniometer used. The sample can I actual
then be rotated 180° about the σ33 axis and remeasured to I corrected 5 (16)
realize the full ψ range while avoiding low-incident angle
errors.
S 11 cos2 ~ 2θ !
sin2 θcosθ D
11.6.6 Modified Lorentz-Polarization Correction—For
11.4 At least three ψ or β angles are to be used, although
broad diffraction peaks at high 2θ angles, the modified Lorentz-
seven or more are recommended. When possible, ψ or β angles
polarization correction can result in a more symmetrical peak
should be chosen such that they are evenly distributed through
profile (8, p. 463):
the sin2ψ or sin2β range used. For modified chi mode, identical
positive and negative angles should be chosen to simplify I actual
I corrected 5 (17)
averaging.
11.5 Collect each profile with sufficient exposure time to
S 11 cos2 ~ 2θ !
sin2 θ D
ensure accurate intensity information is collected. Random 12. Calculation and Interpretation of Results
error as a result of counting statistics from insufficient collec-
tion times may result in an inaccurate peak position determi- 12.1 The position of the corrected XRD peak profiles should
nation. be determined using an appropriate method. Historically,
popular practices such as stripping the Kα2 peak and then using
11.6 For each of the profiles collected, apply the following a parabolic peak fit to the top 20 % of the peak profile have
applicable corrections in the following order: been shown to be subject to significant errors (10, pp. 103-111).
11.6.1 Gain Correction—Multichannel detectors may offer The parabolic fit is capable of producing satisfactory results for
a gain correction intended to correct for intensity variations standards such as Test Method E915 for the measurement of
caused by the detector itself. This is performed by collecting fine-grained, isotropic materials; however, its use for anisotro-
the profile of a sample that is nondiffracting in the observed 2θ pic bearing steels can be subject to additional errors and should
region with a similar background intensity. be used with caution.
11.6.2 Data Smoothing—Smoothing may be applied, but
12.2 The profiles may either be in 2θ or channel-versus-
only with caution, as over smoothing will affect the accuracy of
intensity format.
peak position determination. If smoothing is used, Fourier
smoothing is recommended since the threshold between major 12.3 To ensure accurate peak position determination, the
peak contributions and smaller noise frequencies are much entire peak profile including background should be included.
more distinguishable in the frequency domain. Peak truncation caused by collection over an insufficient 2θ
11.6.3 Absorption Correction—The intensity of a diffracted range has been shown to cause inaccurate peak position
beam may be subject to a θ-dependent absorption effect determination without the use of advanced numerical methods
causing a distortion of the peak profile. This effect can be (11, pp. 524-525). Generally, the detector width should be three
compensated for using the following equation (1, p. 90): times the FWHM value.
I measured 12.4 Selection of Peak Position Determination Method—
I corrected 5 (12) There are a number of methods to determine the position of the
1 2 tan ψcotθcosη
XRD peak. If a function is used, the function that best
11.6.3.1 Omega mode absorption correction—η = 0°. describes the corrected peak shape should be used when using
I measured position-sensitive detectors assuming the peak is approxi-
I corrected 5 (13)
~ 1 2 tan ψcotθ ! mately centered in the detector window. Commonly used peak
functions are listed in Table 3.
11.6.3.2 Chi mode absorption correction—η = 90°. 12.4.1 Many factors affect peak shape including material
I corrected 5 I measured (14) properties and goniometer configuration. Fig. 14 shows an
example of incident beam size effect on peak shape in which an
11.6.3.3 Modified chi mode absorption correction
increasing incident beam size results in a transition from a
I corrected 5 I measured (15) Pearson VII profile with a distinguishable Kα1 Kα2 contribution
11.6.4 Background Correction—To account for sloping to a well overlapped Gaussian profile.
peak backgrounds, choose two reference points on either side 12.5 Relative peak positions are usually determined using
of the diffraction profile. This is typically achieved by fitting a the absolute or cross-correlation method. If the detectors are
selected range within background using a linear least squares not calibrated for actual 2θ positioning, an assumed stress-free
regression. The line intensity drawn between these two points 2θ value may be used if periodically checked for accuracy.
is subtracted from the profile giving the peak a level back- Note, however, that stress-free 2θ values can significantly
ground of zero. change with depth in case-hardened steels.
11.6.5 Lorentz-Polarization Correction—The intensity of a 12.5.1 Cross-Correlation Method (DIN EN 15305)—
diffracted beam is subject to additional θ dependent effects Diffraction angles are determined relative to a chosen reference
known as Lorentzian and polarization effects. This causes a peak. For instance, in omega or chi mode, the 2θ position is
further distortion of the peak profile particularly for wide given by:

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TABLE 3 Peak Distribution Functions
Values
Name Equation a = Intensity
b = Center
Gaussian sx 2 b d2 c = Peak width constant
f s x d 5 ae2 2c
2

Pearson VII c = Peak width constant

H
f s x d 5 a 11
sx 2 bd2
mc2 J 2m
m = Tail curvature constant

Cauchy c = Peak width constant

H
f s x d 5 a 11
sx 2 bd2
c 2 J
Generalized q,r = Left and right side peak shape constants
A
Fermi function fsxd 5 2q s x2b d
(For symmetrical peak, q = r)
e 1e r s x2b d A = Intensity × 2

Parabolic c = Peak width constant

f s x d 5 2c s x 2 b d 2 1a

FIG. 14 Example of Incident Beam Size Effect on Peak Shape

2θ ψ 5 2θ ref1δ ψ (18) large oscillations or random deviations, an elliptical regression

of the data using Eq 21 may produce a more accurate estimate
12.5.1.1 It is recommended to use the strongest peak in the
of d' (HS-784).
series for the reference peak. The shift is calculated as the value
$ hkl% 5 A sin2 ψ1Bsin 2ψ 1d
for which the cross section between the actual and reference d φψ ~ ! ' (21)
profile becomes maximal. where:
F ~ δ ψ! 5 *I ref ~ 2θ ! I ψ ~ 2θ 2 δ ψ ! d ~ 2θ ! 5 max (19) A and B = regression variables.

12.5.1.2 If texture effects are present, the use of cross 12.7.1.1 In the case of single-detector omega mode in which
correlation may cause larger errors than other methods. software does not support overlapping of separate positive and
12.5.2 Absolute Method—Diffraction angles are determined negative ψ range collections, and such oscillations or devia-
relative to each detector. tions are present, a linear regression may be performed
separately for the positive and negative range to determine
12.6 The 2θ values are converted to d spacing using Bragg’s each d' value, respectively.
law. $ hkl% 5 A sin2 ψ1d
d φψ ' (22)
$ hkl% sinθ $ hkl%
nλ Kα1 5 2d φψ (20)
φψKα1
where:
12.7 The measurement do value shall be determined for the
calculation of strain. A = regression variable.
12.7.1 do Omega and Chi Mode—When using the plane 12.7.2 do Modified Chi Mode—When using the plane stress
stress model, there is a negligible error for the substitution of model, there is a negligible error for the substitution of do with
do with d'ψ = 0 (HS-784). When d versus sin2ψ plots exhibit d'β = 0 (HS-784). When d versus sin2β plots exhibit large

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oscillations or random deviations, an elliptical regression of the method. Neglecting statistical error, a repetition of a measure-
data may produce a more accurate estimate of d' . ment will always give the same result.
$ hkl% 5 A sin2 β1Bsinβ1Csin 2β 1Dcosβ1E
d βχ ~ ! (23) 12.11.2 Systematic errors include goniometer alignment and
m
the errors in parameters used for the measuring and evaluation
where: procedure. The systematic errors of a single measurement
d ' 5 D1E ~ see X1.2! cannot be determined.
12.11.3 Ideally, the error of all sources should be considered
A, B, C and D = regression values. and combined through error propagation, although this is not
12.8 Strain Calculation—The strain value for each data always possible or practical. The linear or elliptical regression
point is determined using one of the following methods. errors of the d versus sin2ψ or d versus sin2β data provide only
12.8.1 Linear Variation—Also known as Cauchy or engi- an indication of measurement error. The error in peak position
neering strain. determination as well as X-ray elastic constant(s) may also be
included through propagation.
$ hkl% 5
d φψ 2 d o ∆d sin θ o
ε φψ
do
5
d
5
sin θ φψ
21 (24) 12.12 Gradient Correction—Also known as transparency
correction. Differences in effective layer thickness with orien-
12.8.2 Differentiating Bragg’s Law: tation and target-plane combinations can affect the measured
cotθ∆2θ stress of samples when a stress gradient versus depth is present.
$ hkl% 5 2
ε φψ (25)
2 The gradient correction determines the true 2θ values for
recalculation. (See HS-784, p. 75 for procedure.)
12.8.3 True Strain Definition:
12.13 Material Removal Correction—Material removal via
ε $ hkl%
φψ 5 lnF G F
d φψ
do
5 ln
sin θ o
sin θ φψ G (26)
electropolishing does not impart any stress in the sample;
however, relaxation or redistribution of residual stresses in the
12.9 Stress Calculation (Omega and Chi Mode): component may occur if a stressed layer is removed. There are
12.9.1 The ε versus sin2ψ data is fit using Eq 4 and the models available for simple geometries such as a solid
values for σ11, τ13, and C may then be determined. cylinder, hollow cylinder, and infinite flat plate for determining
what the stresses were before material removal (see HS-784, p.
12.9.2 In the case of single-detector configurations other
76). A spherical model is also available based on the cylindrical
than chi mode in which software does not support overlapping
model (12, p. 1372). These models should be used with caution
of separate positive and negative ψ range collections, a linear
as they assume a material removal from entire surfaces, which
regression may be performed separately for both the positive
is frequently not the case. If finite element model solutions are
and negative ranges using Eq 27. The two resulting σ11 values
available, these should be used for best accuracy.
may then be averaged. This method should be used with
caution as it ignores the shear stresses that may be present. 12.14 Relaxation as a Result of Sectioning—It is commonly
necessary to section samples to gain access to the measurement
1 $ hkl%
$ hkl% 5
ε φψ s @ σ 11 sin2 ψ # 1C (27) location thus potentially altering the stress state of the sample.
2 2
It is advantageous to monitor the change in stress using strain
12.10 Stress Calculation (Modified Chi Mode): gauge(s) in the intended direction of measurement. Relaxation
12.10.1 The ε versus sin2β data is fit using Eq 28 and the through a section may be estimated by placing a strain gauge
values for σ11, C, and D may then be determined. on either side of certain geometries. The relaxation profile
1 $ hkl% between the two gauges may be considered linear or calculated
$ hkl% 5
ε βχ m
s
2 2
@ σ 11 sin2 β cos2 χ m 1D # 1C (28) via other analytical or numerical methods, assuming the
material properties remain consistent throughout the section
where: and radial stress is disregarded (see HS-784, p. 43). XRD
180°22θ βχ m measurements before and after sectioning are also an accept-
χm 5
2
able means for approximating relaxation. If an accessible area
is adjacent to the desired measurement area, then before and
12.10.2 In the case of single-detector configurations other after XRD measurements should determine if relaxation has
than chi mode in which software does not support overlapping occurred.
of separate positive and negative β range collections, a linear
regression may be performed separately for both the positive 13. Report
and negative ranges. The two resulting σ11 values may then be
13.1 Reported data may include the following main items to
averaged. This method should be used with caution as it
ensure that sufficient information is available for comparison of
ignores the shear stresses that may be present.
results as well as record-keeping purposes.
12.11 Stress Error Calculation—Various sources may con- 13.1.1 General Information:
tribute to the error in stress measurement values and can be 13.1.1.1 Specify the operator(s) who performed all aspects
considered statistical or systematic. of the measurement;
12.11.1 Statistical errors include detector-counting statistics 13.1.1.2 Sample identification such as part number, lot
and the repeatability of the peak position determination number, and so forth;

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13.1.1.3 Specifications used for sample preparation, 14. Precision and Bias
measurement, and reference to result requirements; and 14.1 The precision of this test method is based on an
13.1.1.4 Measurement location and direction. interlaboratory study of ASTM E2860, ILS for Test Method for
13.1.2 Results—The following values are placed in a table Residual Stress Measurement by X-Ray Diffraction for Bear-
using international standardized units (MPa, mm) or Imperial ing Steels, conducted from 2016 to 2020. Each of nine
units (ksi, inches) or both. In the case of stress profiles, graphs volunteer laboratories were asked to test eight samples con-
should also be included displaying SI or U.S. customary units sisting of two different bearing steels. Every “test result”
or both. represents an individual determination, and all participants
13.1.2.1 Normal and shear stress values (if available) in- were instructed to report five replicate test results for each
cluding errors and direction of measurement relative to the sample. Practice E691 was followed for the design and analysis
sample reference frame; of the data; the details are given in ASTM Research Report No.
13.1.2.2 FWHM values (if required or available) including RR:E28-2001.6 The results provided are based on “as mea-
errors. In the case of two detector setups, the average values sured” residual stress not corrected for gradient (12.12),
may also be included if required; and material removal (12.13) or relaxation as a result of sectioning
13.1.2.3 Integrated intensity ratio values if required or (12.14) as this would have been a destructive testing process.
available (see 14.4.2.1). While “as measured” residual stress values may be inaccurate
13.1.3 Verification of Equipment Used: (14.5) they should be consistent provided the same target and
13.1.3.1 Current Test Method E915 results including date; similar configuration are used. A mock gradient and material
13.1.3.2 Current measurement results of stress free standard removal study is pending in which different samples mimic the
other than 13.1.3.1 (that is, single daily measurement); and depths of a single sample.
13.1.3.3 Current measurement results of nonzero known 14.1.1 Repeatability limit (r)—The difference between re-
residual stress proficiency reference sample if available. petitive results obtained by the same operator in a given
laboratory applying the same test method with the same
13.1.4 Measurement Parameters:
apparatus under constant operating conditions on identical test
13.1.4.1 Equipment used including manufacturer and material within short intervals of time would in the long run, in
model; the normal and correct operation of the test method, exceed the
13.1.4.2 Goniometer mode; following values only in one case in 20.
13.1.4.3 Goniometer radius; 14.1.1.1 Repeatability can be interpreted as maximum
13.1.4.4 Software and version used for goniometer control, difference between two results, obtained under repeatability
data acquisition, and data processing; conditions, that is accepted as plausible due to random causes
13.1.4.5 Target and wavelength used, for example, Cr Kα1 under normal and correct operation of the test method.
2.289 70 [Angstroms]; 14.1.1.2 Repeatability limits are listed in Table 4.
13.1.4.6 Target power used, for example, 30.00 mA × 30.00 14.1.2 Reproducibility limit (R)—The difference between
kV = 900 W = 69 % (percent of maximum power); two single and independent results obtained by different
13.1.4.7 Filters used and whether the filter is located be- operators applying the same test method in different laborato-
tween the source and sample or sample and detector; ries using different apparatus on identical test material would,
13.1.4.8 Sample material, for example, M50; in the long run, in the normal and correct operation of the test
13.1.4.9 Miller indices of crystallographic plane used, for method, exceed the following values only in one case in 20.
example, {211}; 14.1.2.1 Reproducibility can be interpreted as maximum
difference between two results, obtained under reproducibility
13.1.4.10 The do Bragg angle in degrees;
conditions, that is accepted as plausible due to random causes
13.1.4.11 Detector type used;
under normal and correct operation of the test method.
13.1.4.12 The 2θ region scanned; 14.1.2.2 Reproducibility limits are listed in Table 4.
13.1.4.13 Step size or channel size in degrees 2θ; 14.1.3 The above terms (repeatability limit and reproduc-
13.1.4.14 Counting time per step for single-channel detec- ibility limit) are used as specified in Practice E177.
tors or exposure time × number of exposures for linear array
14.2 Bias—At the time of the study, there was no accepted
detector;
reference material suitable for determining the bias for this test
13.1.4.15 The β angles used in degrees; method, therefore no statement on bias is being made.
13.1.4.16 The ψ angles used in degrees;
13.1.4.17 The φ angles used in degrees for triaxial measure- 14.3 The precision statement was determined through sta-
ment; tistical examination of 250 results, from 8 of the 9 laboratories,
on 7 samples.
13.1.4.18 Primary and secondary aperture size;
13.1.4.19 X-ray elastic constant 1⁄2 S2—Include S1 for tri- 14.4 The precision of this method will be dependent upon
axial measurement; the sources of error described in this section.
13.1.4.20 Data smoothing—If used specify method/formula
used; and 6
Supporting data have been filed at ASTM International Headquarters and may
13.1.4.21 Estimated depth of penetration at ψ = 0° and I = be obtained by requesting Research Report RR:E28-2001. Contact ASTM Customer
0.63 Io. Service at service@astm.org.

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TABLE 4 Normal “As Measured” Stress σ11 (MPa)
Material Number AverageA Repeatability Reproducibility Repeatability Reproducibility
of Standard Standard Limit Limit
Laboratories Deviation Deviation
n x̄ sr sR r R
R2 M50 Roller 0.003 mm 6 -328.24 11.73 22.31 32.84 62.46
R3 M50 Roller 0.006 mm 6 -129.59 10.43 19.61 29.21 54.90
R4 M50 Roller 0.012 mm 6 -65.08 9.25 19.42 25.89 54.38
P8 52100 8 -1102.59 8.35 27.70 23.37 77.55
P10 52100 8 -19.61 9.99 23.44 27.97 65.63
P11 52100 8 -359.11 6.05 35.12 16.94 98.34
P12 52100 8 -978.86 6.51 30.33 18.23 84.92
A
The average of the laboratories’ calculated averages

14.4.1 Uncertainty in Peak Fitting—The collected peak (b) Maximize ψ tilt—It has been suggested that lineariza-
profile is a summation of multiple overlapping peaks. tion of the d versus sin2ψ distribution may be achieved by
Therefore, not properly accounting for all contributions will maximizing the ψ tilt range (1, p. 189).
affect the accuracy of deconvoluting individual contributions 14.4.2.2 Large Grain Size—For a given homogeneous
using profile regression. Possible deviations include the fol- volume, a larger grain size will reduce the number of grains
lowing: available for diffraction at a given orientation. This is often
14.4.1.1 Peak Asymmetry—This is commonly observed in associated with random deviations in the sin2ψ or sin2β plot.
bearing steel materials. Possible sources of asymmetry include Intensity ratios above 1.6 may require corrective actions that
dislocation effects, separation of peaks as a result of slight include the following:
tetragonalities in cubic materials, and incomplete heat treat- (1) Different hkl plane—It has been shown that some hkl
ment causing a mixture of ferrite and martensite. planes are more susceptible to nonelliptical sin2ψ distributions
14.4.1.2 Interfering Peak(s)—Diffraction peaks free of in- than others for cubic materials (13, pp. 899-906). The selection
terfering peaks from the measured phase are typically used. of a high-multiplicity hkl plane may also reduce the effects of
The presence of other phases such as carbides may create large grain size.
interfering peaks. (2) Oscillation(s)—The φ and β oscillations as well as
14.4.1.3 Counting Statistics—Random error as a result of sample translation may be introduced to minimize the effects of
counting statistics may result from failure to take sufficient large grains (14).
time during the measurement to obtain accurate intensity (3) Increase aperture size—Where possible, the use of a
information and, thus, to determine accurately the diffraction larger aperture will increase the number of grains sampled and
peak positions. Methods are available (HS-784) for estimating minimize the effects of large grain size.
the standard deviation of the measured stress as a result of the (4) Increase number of tilts and range—Increasing the
errors involved in counting and curve fitting to determine peak number of ψ tilts and the ψ tilt range can also reduce errors
positions. caused by large grain size.
14.4.2 Uncertainty in d Versus sin2ψ or sin2β Fitting—There (5) Deeper penetrating radiation—A deeper penetration
are many possible sources of error in the d versus sin2ψ or may increase the number of grains sampled and reduce the
sin2β plot. Some of these errors may create an elliptical offset effects of large grain size.
causing an apparent shear influence, while others create oscil-
latory or seemingly random patterns. Deviations often become 14.4.2.3 Temperature Gradient—Changes in sample tem-
more apparent at lower σ11 values because of the lower perature will cause a change in strain according to the material
d-spacing axis scaling. Distinguishing between sources of error coefficient of thermal expansion. If the sample material is
can be difficult and multiple influences may overlap. Some gradually heated or cooled during measurement, a change in
possible sources are listed in the following. strain will occur. For example, the air in a goniometer
14.4.2.1 Texture—A nonrandom coherent domain orienta- enclosure may be warmer than the outside ambient air causing
tion distribution is found in all bearing materials in varying the sample to warm up gradually when placed in the enclosure
degrees. Texture may be manifested by a significant change in for measurement. Fig. 15 shows d-spacing deviations caused
peak intensity versus orientation as well as oscillatory sin2ψ by a 5 °C increase in temperature during measurement. The
distributions. The degree of texture observed in a measurement rate of sample temperature change will depend on its thermal
is indicated by the peak integrated intensity ratio (Eq 30). conductivity.
14.4.2.4 Stability of Residual Stress/Strain—Residual
I max
I.R. $ hkl% 5 (29) stresses may cause the strain to change over time through creep
I min
mechanisms. The strain rate of change is largest in the primary
(1) Intensity ratios above 1.6 may require corrective ac-
stage after the stresses are created or altered and will slow over
tions that include the following:
time with increasing strain. Larger strain rates may cause a
(a) Different hkl plane—It has been shown that some hkl significant change in strain over the course of the measure-
planes are more susceptible to nonelliptical sin2ψ distributions ment. The effects of aging may also contribute to the stability
than others for cubic materials (12, pp. 899-906) and of residual stress over time.

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FIG. 15 Simulated Two Detector Omega Mode sin2ψ Offset for 5°C Increase during Measurement where σ11 = -100 MPa, τ13 = +20 MPa,
and αL = 11 × 10-6 mm/mm/°C (αL Applied to do{hkl} Value)

14.4.2.5 Plastic Deformation—Oscillatory sin2ψ or sin2β 14.4.4.2 Sample Alignment—The sample measurement lo-
plots may occur as a result of plastic deformation (1, p. 400). cation shall remain aligned with the goniometer center of
14.4.3 Maximum Acceptable Errors: rotation. This is especially true for curved surfaces where
14.4.3.1 Intensity ratio shall not exceed 3.0. Corrective measurement of misaligned samples can result in d-spacing
actions may be required if the intensity ratio exceeds 1.6. offsets that appear to be stress component influences.
14.4.3.2 Stress Values—Alternatively, strain values may be 14.4.4.3 Sample Curvature—Sample curvature should not
used. This avoids error as a result of the selection of inappro- exceed limits specified in 9.1.2.
priate elastic constants. 14.4.4.4 Surface Condition—Sample surface should be pre-
(1) Normal stress error not to exceed 10 % of the normal pared in accordance with 8.2.
stress value or 35 MPa, whichever is larger. Corrective actions 14.4.4.5 X-Ray Optics—X-ray optics should be used accord-
may be required if error exceeds 20 MPa. ing to the manufacturer’s specifications.
(2) Normal strain error not to exceed 10 % of the normal 14.4.4.6 Errors in Peak Fitting—See 14.4.1.
strain value or 250 ppm, whichever is larger. Corrective actions 14.4.4.7 Errors in sin2ψ or sin2 β Fitting—See 14.4.2.
may be required if the error exceeds 150 ppm.
(3) Shear stress errors not to exceed 10 % of the shear 14.5 “As measured” residual stress values not corrected for
stress value or 35 MPa, whichever is larger. Corrective actions gradient (12.12), material removal (12.13) or relaxation as a
may be required if the error exceeds 20 MPa. result of sectioning (12.14) are potentially subject to high
(4) Shear strain errors not to exceed 10 % of the shear inaccuracies.
strain value or 125 ppm, whichever is larger. Corrective actions 14.5.1 Gradient Effect—A significant stress gradient within
may be required if the error exceeds 75 ppm. the diffracted volume will affect the dependence of measured
14.4.4 Uncertainty in Stress Values—Sources of experimen- strain with incident beam orientation because of a difference in
tal error in the residual stress measurement result may include volume penetration. In chi and omega mode, large gradients
the following: may be indicated by a curved d versus sin2ψ distribution.
14.4.4.1 X-Ray Elastic Constant(s)—Factors affecting Unlike curvatures caused by shear stress, the curvatures caused
Young’s modulus will also affect the X-ray elastic constants. by a stress gradient bend in the same direction for the positive
X-ray elastic constants may be determined with Test Method and negative ψ range (see Fig. 16) (1, p. 181). Corrections are
E1426. Some examples of X-ray elastic constant influences available for the stress gradient effect (HS-784, p. 75)
include the following: Surface and near surface residual stress values for ground
(1) Stability of residual stress/strain—Material instabilities surfaces may be over 200 % greater in magnitude than “as
such as creep and aging may influence the stress-strain measured” values due to a residual stress gradient versus depth.
relationship over time. Factors that effect the depth of penetration such as diffracto-
(2) Chemistry—Material chemistry can have a large influ- meter mode and target radiation may influence the “as mea-
ence on X-ray elastic constants. For example, in case-hardened sured” value. If initial measurement depths of a ground surface
steels, a carbon gradient versus depth can cause a significant are significantly larger than the depth of penetration the near
material properties gradient with depth. surface residual stress gradient versus depth may be
(3) Grain structure—In martensitic steels, two structures underestimated, causing the gradient correction (12.12) to
are commonly observed: “lath” and “plate” martensite (15). under-correct. Depths should be chosen that capture turning
The percentage of each structure is largely dependent on the points and inflections of the residual stress versus depth curve.
carbon content and where the carbon is concentrated, that is, in 14.5.2 Material Removal—The redistribution of residual
solution/in carbides. stress’ as a result of material removal may cause the freshly

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 E2860 − 20

FIG. 16 Sample d Versus sin2ψ Dataset with Large Stress Gradient

exposed surface to have a different residual stress than existed 15. Keywords
before electropolishing. See 12.13 for correction. 15.1 bearing; residual stress; X-ray diffraction; XRD
14.5.3 Relaxation as a Result of Sectioning—Sample sec-
tioning may partially release and redistribute the residual stress
in a sample. See 12.14 for correction methods.

APPENDIX

(Nonmandatory Information)

X1. MODIFIED χ CALCULATION

X1.1 The values, ψ and φ, are converted to β and χ reference { sin2 s arccoss x dd 512x 2
using Eq 5 and 6.
5σ 22s 12cos2 βcosχ m 2sin2 βcos2 χ m d
ψ 5 arccos~ cos βcosχ m ! ~5!
5σ 22s 12cos2 x s cos2 β1sin2 β dd
φ 5 arccos S sin βcosχ m
sin ψ D ~6!
{ cos2 x1sin2 x51

X1.2 Eq 5 and 6 are applied to each term in Eq 2. 5σ 22s 12cos2 χ m d

$ hkl% 5
1 $ hkl% { 12cos2 x5sin2 x
ε φψ s
2 2
@ σ 11 cos2 φ sin2 ψ1σ 22 sin2 φ sin2 ψ1σ 33 cos2 ψ #
5σ 22sin2 χ m
1
1 s $2hkl% @ τ 12sin~ 2φ ! sin2 ψ1τ 13cosφsin~ 2ψ ! 1τ 23sinφsin~ 2ψ ! # σ 33cos2 ψ
2
1s $1hkl% @ σ 111σ 221σ 33# ~2! 5σ 33cos2 s arccoss cosβcosχ m dd
2 2
σ 11cos φsin ψ
{ cos2 s arccosx d 5x 2
2 2
sin βcos χ m
5σ 11 sin2 ψ 5σ 33cos2 βcos2 χ m
sin2 ψ
τ 12sins 2φ d sin2 ψ
5σ 11sin2 βcos2 χ m
σ 22sin2 φsin2 ψ
S
5τ 12sin 2arccos S sinβcosχ m
sinψ DD sin2 ψ

5σ 22sin2 F arccos S
sinβcosχ m
sin2 ψ DG sin2 ψ {sins 2arccosx d 52x œ12x 2

{ sin2 s arccoss x dd 512x 2

5τ 12
2sinβcosχ m
sinψ Œ 12
sin2 βcos2 χ m
sin2 ψ
sin2 ψ

S
5σ 22 12
sin2 βcos2 χ m
sin2 ψ D sin2 ψ
5τ 122sinβcosχ m sinψ Œ 12
sin2 βcos2 χ m
sin2 ψ
5σ 22s sin2 ψ2sin2 βcos2 χ m d

5σ 22s sin f arccoss cosβcosχ m d g 2sin βcos χ m d

2 2 2
5τ 122sinβcosχ m sins arccoss cosβcosχ m dd Œ 12
sin2 βcos2 χ m
sin2 s arccoss cosβcosχ m dd

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 E2860 − 20

{sins arccosx d 5 œ12x 2 and sin2 s arccosx d 512x 2

5τ 23Œ 12
sin2 βcos2 χ m
sin2 ψ
sins 2ψ d

5τ 122sinβcosχ m œ12cos
2
βcos2 χ m Œ 12
sin2 βcos2 χ m
12cos2 βcos2 χ m 5τ 23Œ 12
sin2 βcos2 χ m
sin2 s arccoss cosβcosχ m dd
sins 2arccoss cosβcosχ m dd

5τ 122sinβcosχ m Œ 12cos2 βcos2 χ m 2

s 12cos2 βcos2 χ m d s sin2 βcos2 χ m d
12cos2 βcos2 χ m
{ sin2 s arccosx d 512x 2 andsins 2arccosx d 52x œ12x 2

5τ 122sinβcosχ m œ12cos
2
βcos2 χ m 2sin2 βcos2 χ m
5τ 23Œ 12
sin2 βcos2 χ m
12cos2 βcos2 χ m
2cosβcosχ m œ12cos
2
βcos2 χ m
5τ 122sinβcosχ m œ12cos
2
χ m s cos β1sin β d
2 2

{ cos2 x1sin2 x51

5τ 232cosβcosχ m Œ 12cos2 βcos2 χ m 2
s 12cos2 βcos2 χ m d s sin2 βcos2 χ m d
12cos2 βcos2 χ m
2
5τ 232cosβcosχ m œ12cos βcos2 χ m 2sin2 βcos2 χ m
2
5τ 122sinβcosχ m œ12cos χm
5τ 232cosβcosχ m œ12cos
2
χ m s cos2 β1sin2 β d
{ 12cos2 x5sin2 x
{ cos2 x1sin2 x51
5τ 122sinβcosχ m sinχ m
2
5τ 232cosβcosχ m œ12cos χm
1
{cosxsinx5 sins 2x d
2 { 12cos2 x5sin2 x
5τ 122sinβsins 2χ m d
5τ 232cosβcosχ m sinχ m
τ 13cosφsins 2ψ d
1
{cosxsinx5 sins 2x d
S
5τ 13cos arccos S
sinβcosχ m
sinψ DD sins 2ψ d
2

5τ 232cosβsins 2χ m d
sinβcosχ m
5τ 13 sins 2ψ d
sinψ

{
sins 2x d
52cosx
X1.3 Bringing the terms together we obtain:
sinx
1 $ hkl%
5τ 132sinβcosχ m cosψ $ hkl% 5
ε βχ s @ σ 11 sin2 β cos2 χ m 1σ 22 sin2 χ m 1σ 33 cos2 β cos2 χ m #
m 2 2
5τ 132sinβcosχ m coss arccoss cosβcosχ m dd 1
1 s $2hkl% @ τ 12sinβsin~ 2χ m ! 1τ 13sin~ 2β ! cos2 χ m 1τ 23cosβsin~ 2χ m ! #
5τ 132sinβcosβcos χ m 2 2
1s $1hkl% @ σ 111σ 221σ 33# ~7!
1
{cosxsinx5 sins 2x d
2
X1.4 Modified χ d' Calculation:
5τ 13sins 2β d cos2 χ m
τ 23sinφsins 2ψ d X1.4.1 For β = 0, Eq 23 becomes:
$ hkl%
d β50χ 5 A sin2 01Bsin01Csin~ 2·0 ! 1Dcos01E 5 D1E
S
5τ 23sin arccos S sinβcosχ m
sinψ DD sins 2ψ d
m

Substituting d β50χ
$ hkl% with d':
m
{sins arccosx d 5 œ12x 2
d ' 5 D1E

REFERENCES

(1) Hauk, V., Structural and Residual Stress Analysis by Nondestructive (6) Dionnet, B., Francois, M., Lebrun, J. L., and Nardou, F., “Influence of
Methods, Elsevier, Amsterdam, The Netherlands, 1997. Tore Geometry on X-Ray Stress Analysis,” in Proceedings of the
(2) Noyan, I. C. and Cohen, J. B., Residual Stress Measurement by Fourth European Conference on Residual Stresses, Vol 1, France,
Diffraction and Interpretation, Springer-Verlag, 1987. 1996.
(3) Londsdale, D. and Doig, P., “The Development of a Transportable (7) Residual Stress Measurement by X-Ray Diffraction, Schaeffler Group
X-Ray Diffractometer for Measurement of Stress,” in International Standard, 2008.
Conference on Residual Stresses ICRS2, Elsevier Applied Science, (8) Cullity, B. D., Elements of X-Ray Diffraction–Second Edition,
London and New York, 1989. Addison-Wesley Publishing Company Inc., Reading, MA, 1978.
(4) Taira, S. and Arima, J., “X-Ray Investigation of Stress Measurement (9) Sue, Albert J. and Schajer, Gary S., Handbook of Residual Stress and
(on the Effect of Roughness of Specimen Surface),” in Proceedings of Deformation of Steel – Stress Deformation in Coatings, ASM
the Seventh Japan Congress on Testing Materials, The Society of International, 2002.
Materials Science, 1964. (10) Prevey, P., “The Use of Pearson VII Distribution Functions in X-Ray
(5) SEM, Handbook of Measurement of Residual Stress, Fairmont Press, Diffraction Residual Stress Measurement,” Advances in X-Ray
1996. Analysis, Vol 29, 1986.

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 E2860 − 20
(11) Belassel, M., Bocher, E., and Pineault, J., “Effect of detector width textured materials by X- and neutron-rays,” in Residual Stresses-III,
and peak location technique on residual stress determination in case Science and Technology, ICRS3, Vol 2, H. Fujiwara, T. Abe, and K.
of work-hardened materials,” Materials Science Forum, 2006. Tanaka, Eds., Applied Science, London and New York, 1992.
(12) Kikuo, M., Nakashima, H., and Tsushima, N., “The Influence of (14) Pineault, J. A. and Brauss, M. E., “Measuring Residual and Applied
Residual Stress in Radial Direction upon Rolling Contact Fatigue Stress Using X-ray Diffraction on Materials With Preferred Orien-
Life,” in Residual Stresses-III, Science and Technology, ICRS3, H. tation and Large Grain Size,” Advances in X-Ray Analysis, Vol 36,
Fujiwara, T. Abe, and K. Tanaka, Eds., Applied Science, London and 1993.
New York, 1992. (15) Parrish, G., Carburizing: Microstructures and Properties, ASM
(13) Behnken, H. and Hauk, V., “The evaluation of residual stresses in International, 1999, p. 108.

SUMMARY OF CHANGES

Committee E28 has identified the location of selected changes to this standard since the last issue (E2860–12)
that may impact the use of this standard.

(1) Section 14 Precision and Bias, was revised.

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