ASTM E2860-20

Standard Test Method for Residual Stress Measurement by X-Ray Diffraction for Bearing Steels

SIGNIFICANCE AND USE
5.1 This test method covers a procedure for experimentally determining macroscopic residual stress tensor components of quasi-isotropic bearing steel materials by XRD. Here the stress components are represented by the tensor σij as shown in Eq 1 (1,5 p. 40). The stress strain relationship in any direction of a component is defined by Eq 2 with respect to the azimuth phi(φ) and polar angle psi(ψ) defined in Fig. 1 (1, p. 132).
5.1.1 Alternatively, Eq 2 may also be shown in the following arrangement (2, p. 126):
5.2 Using XRD and Bragg’s law, interplanar strain measurements are performed for multiple orientations. The orientations are selected based on a modified version of Eq 2, which is dictated by the mode used. Conflicting nomenclature may be found in literature with regard to mode names. For example, what may be referred to as a ψ (psi) diffractometer in Europe may be called a χ (chi) diffractometer in North America. The three modes considered here will be referred to as omega, chi, and modified-chi as described in 9.5.  
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 4. Post-measurement corrections may be applied to account for possible σ33 influences (12.12). Since the σij values will remain constant for a given azimuth, the s1{hkl} term is renamed C.
FIG. 2 Omega Mode Diagram for Measurement in σ11 Direction  
FIG. 3 Chi Mode Diagram for Measurement in σ11 Direction
Note 1: Stress matrix is rotated 90° about the surface normal compared to Fig. 2 and Fig. 14.  
5.3.1 The measured interplanar spacing values are converted to strain using Eq 24, Eq 25, or Eq 26. Eq 4 is used to fit the strain versus sin2ψ data yiel...
SCOPE
1.1 This test method covers a procedure for experimentally determining macroscopic residual stress tensor components of quasi-isotropic bearing steel materials by X-ray diffraction (XRD).  
1.2 This test method provides a guide for experimentally determining stress values, which play a significant role in bearing life.  
1.3 Examples of how tensor values are used are:  
1.3.1 Detection of grinding type and abusive grinding;  
1.3.2 Determination of tool wear in turning operations;  
1.3.3 Monitoring of carburizing and nitriding residual stress effects;  
1.3.4 Monitoring effects of surface treatments such as sand blasting, shot peening, and honing;  
1.3.5 Tracking of component life and rolling contact fatigue effects;  
1.3.6 Failure analysis;  
1.3.7 Relaxation of residual stress; and  
1.3.8 Other residual-stress-related issues that potentially affect bearings.  
1.4 Units—The values stated in SI units are to be regarded as standard. No other units of measurement are included in this standard.  
1.5 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of regulatory limitations prior to use.  
1.6 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.

General Information

• Status: Published
• Publication Date: 31-Oct-2020
• ICS: 77.040.20 - Non-destructive testing of metals
• Technical Committee: E28 - Mechanical Testing
• Drafting Committee: E28.13 - Residual Stress Measurement

Relations

• Refers: ASTM E1426-14(2019)e1 - Standard Test Method for Determining the X-Ray Elastic Constants for Use in the Measurement of Residual Stress Using X-Ray Diffraction Techniques
• Effective Date: 01-Oct-2019
• Refers: ASTM E915-16 - Standard Test Method for Verifying the Alignment of X-Ray Diffraction Instrumentation for Residual Stress Measurement
• Effective Date: 01-Aug-2016
• Refers: ASTM E7-15 - Standard Terminology Relating to Metallography
• Effective Date: 01-Jun-2015
• Refers: ASTM E1426-14 - Standard Test Method for Determining the X-Ray Elastic Constants for Use in the Measurement of Residual Stress Using X-Ray Diffraction Techniques
• Effective Date: 01-Dec-2014
• Refers: ASTM E7-14 - Standard Terminology Relating to Metallography
• Effective Date: 01-Nov-2014
• Refers: ASTM E177-14 - Standard Practice for Use of the Terms Precision and Bias in ASTM Test Methods
• Effective Date: 01-May-2014
• Refers: ASTM E691-13 - Standard Practice for Conducting an Interlaboratory Study to Determine the Precision of a Test Method
• Effective Date: 01-May-2013
• Refers: ASTM E177-13 - Standard Practice for Use of the Terms Precision and Bias in ASTM Test Methods
• Effective Date: 01-May-2013
• Refers: ASTM E691-11 - Standard Practice for Conducting an Interlaboratory Study to Determine the Precision of a Test Method
• Effective Date: 01-Nov-2011
• Refers: ASTM E177-10 - Standard Practice for Use of the Terms Precision and Bias in ASTM Test Methods
• Effective Date: 01-Oct-2010
• Refers: ASTM E915-10 - Standard Test Method for Verifying the Alignment of X-Ray Diffraction Instrumentation for Residual Stress Measurement
• Effective Date: 01-Jun-2010
• Refers: ASTM E7-03(2009) - Standard Terminology Relating to Metallography
• Effective Date: 01-Oct-2009
• Refers: ASTM E1426-98(2009)e1 - Standard Test Method for Determining the Effective Elastic Parameter for X-Ray Diffraction Measurements of Residual Stress
• Effective Date: 01-Jun-2009
• Refers: ASTM E6-09be1 - Standard Terminology Relating to Methods of Mechanical Testing
• Effective Date: 15-May-2009
• Refers: ASTM E6-09b - Standard Terminology Relating to Methods of Mechanical Testing
• Effective Date: 15-May-2009

Overview

ASTM E2860-20 is the established standard test method for the experimental determination of macroscopic residual stress tensor components in quasi-isotropic bearing steel materials using X-ray diffraction (XRD). Developed by ASTM International, this method enables the accurate assessment of residual stresses, which are critical factors influencing bearing life, component performance, and failure risks. X-ray diffraction is recognized as a reliable, non-destructive testing technique for residual stress measurement, ensuring precision and repeatability for a range of bearing applications.

Key Topics

• Residual Stress Tensor Measurement: This standard provides a procedural approach for quantifying the principal components of the residual stress tensor (σij) in bearing steels using XRD. The focus is on macrostress levels (average stresses over many grains or crystals).
• X-Ray Diffraction Techniques: The method leverages Bragg’s law to perform interplanar strain measurements for different sample orientations. The three principal XRD measurement modes covered are:
  • Omega (iso-inclination) Mode
  • Chi (side-inclination) Mode
  • Modified-Chi Mode
• Measurement Process: The procedure includes specimen placement in a goniometer, data acquisition across multiple angular positions, application of instrument-specific corrections (such as background, Lorentz-polarization, absorption), and calculation of stress from collected diffraction profiles.
• Surface Preparation: The standard details requirements for specimen preparation, emphasizing the need for clean, smooth, and oxide-free surfaces to achieve accurate results.

Applications

ASTM E2860-20 is of significant practical value in the bearing industry and related sectors where high-precision stress analysis in steels is essential. Its main applications include:

• Detection of Manufacturing Defects: Identify and analyze residual stresses arising from abusive grinding or improper machining processes.
• Process and Quality Monitoring: Evaluate the effects of heat treatments such as carburizing and nitriding, as well as surface enhancement techniques like shot peening and sand blasting.
• Component Health Assessment: Track changes in residual stress for life prediction, rolling contact fatigue analysis, and bearing reliability assessment over time.
• Failure Analysis: Investigate the residual stress states contributing to premature bearing failures.
• Tool Wear Monitoring: Determine tool wear effects in turning or machining operations.
• Relaxation Studies: Analyze how stress-relief processes (thermal or mechanical) impact residual stress relaxation.
• Research and Development: Support the development of new processes or materials by providing accurate stress measurements.

Related Standards

Users of ASTM E2860-20 may also benefit from familiarity with the following key related standards, which provide additional guidance on terminology, procedures, and complementary testing methods:

• ASTM E6: Terminology Relating to Methods of Mechanical Testing
• ASTM E7: Terminology Relating to Metallography
• ASTM E915: Test Method for Verifying the Alignment of X-Ray Diffraction Instrumentation for Residual Stress Measurement
• ASTM E1426: Test Method for Determining the X-Ray Elastic Constants for Use in the Measurement of Residual Stress Using X-Ray Diffraction Techniques
• SAE HS-784/2003: Residual Stress Measurement by X-Ray Diffraction
• ANSI N43.2/N43.3: Radiation Safety for X-ray Diffraction and Fluorescence Analysis Equipment
• DIN EN 15305: Non-destructive Testing-Test Method for Residual Stress Analysis by X-Ray Diffraction

Practical Value

Implementing the ASTM E2860-20 method allows for:

• Consistent quality assurance throughout the bearing manufacturing process
• Enhanced understanding of how various treatments affect material performance
• Informed decision-making during failure investigations and product development
• Alignment with international best practices and regulatory requirements for material testing

This standard is essential for engineers, laboratory specialists, and quality assurance professionals engaged in the analysis, manufacture, or research of bearing steels and related components. By following ASTM E2860-20, organizations can substantially improve reliability, extend service life, and reduce the risk of premature failures in rolling and sliding bearing applications.

Standards Content (Sample)

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 Steels
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.Anumber 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
popularityinthematerialstestingfieldoverthepasthalfcenturyandhasbecomeamandatorytestfor
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
inthisevolution;forexample,systemsthatstillusethesingle-anddouble-exposuretechniquesaswell
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 sin ψ 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
achievenotonlythemostreliablemeasurementresultsbutalsotoenableidentificationandevaluation
of potential measurement errors, thus paving the way for future developments.
1. Scope* 1.3.8 Other residual-stress-related issues that potentially
affect bearings.
1.1 This test method covers a procedure for experimentally
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 asstandard.Nootherunitsofmeasurementareincludedinthis
(XRD). standard.
1.5 This standard does not purport to address all of the
1.2 This test method provides a guide for experimentally
safety concerns, if any, associated with its use. It is the
determining stress values, which play a significant role in
responsibility of the user of this standard to establish appro-
bearing life.
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 Monitoringofcarburizingandnitridingresidualstress
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;
2.1 ASTM Standards:
1.3.7 Relaxation of residual stress; and
E6Terminology Relating to Methods of Mechanical Testing
E7Terminology Relating to Metallography
This test method is under the jurisdiction of ASTM Committee E28 on
Mechanical Testing and is the direct responsibility of Subcommittee E28.13 on
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 [e-mail]. 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
E2860 − 20
E177Practice for Use of the Terms Precision and Bias in α =Linear thermal expansion coefficient
L
ASTM Test Methods β=Angle between the incident beam and σ or surface
E691Practice for Conducting an Interlaboratory Study to normal on the σ σ plane
33 11
Determine the Precision of a Test Method χ=Anglebetweenthe σ directionandthenormaltothe
φ+90°
E915Test Method for Verifying the Alignment of X-Ray
diffracting plane
Diffraction Instrumentation for Residual Stress Measure- χ =Fixed χ offset used in modified-chi mode
m
ment
d=Interplanar spacing between crystallographic planes;
E1426Test Method for Determining the X-Ray Elastic also called d-spacing
Constants for Use in the Measurement of Residual Stress
d =Interplanar spacing for unstressed material
o
Using X-Ray Diffraction Techniques
d =Perpendicular spacing
'
2.2 ANSI Standards:
∆d=Change in interplanar spacing caused by stress
N43.2Radiation Safety for X-ray Diffraction and Fluores-
ε =Strain component i, j
ij
cence Analysis Equipment
E=Modulus of elasticity (Young’s modulus)
{hkl}
N43.3For General Radiation Safety—Installations Using
E =Effective elastic modulus for X-ray measurements
eff
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:
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 σ 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.
σ =Normal stress component i, j
ij
{hkl}
s =X-ray elastic constant of quasi-isotropic material
3.2 Definitions of Terms Specific to This Standard: 1
2ν
3.2.1 interplanar spacing, n—perpendicular distance be-
equal to
hkl
$ %
E
eff
tween adjacent parallel atomic planes.
τ =Shear stress component i, j
ij
3.2.2 macrostress, n—average stress acting over a region of
θ=Bragg angle
the test specimen containing many gains/crystals/coherent
ν=Poisson’s ratio
domains.
Mode
x =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
4. Summary of Test Method
3.3.4 MSDS—Material safety data sheet
4.1 Atest specimen is placed in a XRD goniometer aligned
3.3.5 XEC—X-ray elastic constant
as per Test Method E915.
3.3.6 XRD—X-ray diffraction
4.2 The diffraction profile is collected over three or more
{hkl}
3.4 Symbols: ⁄2 S =X-ray elastic constant of quasi-
angles within the required angular range for a given {hkl}
11ν
isotropic material equal to
plane, although at least seven or more are recommended.
$hkl%
E
eff
4.3 The XRD profile data are then corrected for LPA,
3 background, and instrument-specific corrections.
Available fromAmerican National Standards Institute (ANSI), 25 W. 43rd St.,
4th Floor, New York, NY 10036, http://www.ansi.org.
4.4 The peak position/Bragg angle is determined for each
AvailablefromSAEInternational(SAE),400CommonwealthDr.,Warrendale,
XRD peak profile.
PA 15096, http://www.sae.org.
FIG. 1 Stress Tensor Components
E2860 − 20
4.5 The d-spacings are calculated from the peak positions 1
$hkl% $hkl% 2 2 2 2 2
ε 5 s @σ cos φ sin ψ1σ sin φ sin ψ1σ cos ψ#
φψ 2 11 22 33
via Bragg’s law. 2
4.6 The d-spacing values are plotted versus their sin ψ or
$hkl% 2
1 s τ sin 2φ sin ψ1τ cosφsin 2ψ 1τ sinφsin 2ψ
@ ~ ! ~ ! ~ !#
2 12 13 23
sin β values, and the residual stress is calculated using Eq 4 or
$hkl%
Eq 8, respectively.
1s @σ 1σ 1σ # (2)
1 11 22 33
4.7 TheerrorinmeasurementisevaluatedasperSection14.
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
$hkl% $hkl% 2 2 2
ε 5 s @σ cos φ1τ sin~2φ!1σ sin φ 2 σ # sin ψ
reported results.
φψ 2 11 12 22 33
4.8.1 Depth of penetration correction (see 12.12) and
1 1
hkl hkl hkl
$ % $ % $ %
4.8.2 Relaxation as a result of material removal correction
1 s σ 2 s σ 1σ 1σ 1 s τ cosφ
@ # @
2 33 1 11 22 33 2 13
2 2
(see 12.14).
1τ sinφ#sin~2ψ!
5. Significance and Use
5.2 UsingXRDandBragg’slaw,interplanarstrainmeasure-
mentsareperformedformultipleorientations.Theorientations
5.1 This test method covers a procedure for experimentally
are selected based on a modified version of Eq 2, which is
determining macroscopic residual stress tensor components of
dictated by the mode used. Conflicting nomenclature may be
quasi-isotropicbearingsteelmaterialsbyXRD.Herethestress
found in literature with regard to mode names. For example,
components are represented by the tensor σ as shown in Eq 1
ij
what may be referred to as a ψ (psi) diffractometer in Europe
(1, p. 40). The stress strain relationship in any direction of a
may be called a χ (chi) diffractometer in North America. The
component is defined by Eq 2 with respect to the azimuth
three modes considered here will be referred to as omega, chi,
phi(φ) and polar angle psi(ψ) defined in Fig. 1 (1, p. 132).
and modified-chi as described in 9.5.
σ τ τ
11 12 13
τ σ τ 5.3 Omega Mode (Iso Inclination) and Chi Mode (Side
σ 5 where τ 5 τ (1)
21 22 23
ij ij ji
F G
Inclination)—Interplanarstrainmeasurementsareperformedat
τ τ σ
31 32 33
multiple ψanglesalongone φazimuth(let φ=0°)(Figs.2and
3), reducing Eq 2 to Eq 3. Stress normal to the surface (σ )is
assumed to be insignificant because of the shallow depth of
penetration of X-rays at the free surface, reducing Eq 3 to Eq
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 σ Direction
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 σ Direction
4.Post-measurementcorrectionsmaybeappliedtoaccountfor slopeofthedata,while τ isrelatedtothedirectionanddegree
possible σ influences(12.12).Sincethe σ valueswillremain of elliptical opening. Fig. 4 shows a simulated d versus sin ψ
33 ij
{hkl}
constant for a given azimuth, the s term is renamed C. profile for the tensor shown. Here the positive 20-MPa τ
1 13
stress results in an elliptical opening in which the positive psi
1 1
hkl hkl 2 2 hkl hkl
$ % $ % $ % $ %
ε 5 s σ sin ψ1σ cos ψ 1 s τ sin 2ψ 1s σ
@ # @ ~ !# @ range opens upward and the negative psi range opens down-
φψ 2 11 33 2 13 1 11
2 2
ward.Ahigher τ value will cause a larger elliptical opening.
1σ 1σ # (3)
22 33
Anegative20-MPa τ stresswouldresultinthesameelliptical
opening only the direction would be reversed with the positive
hkl hkl 2
$ % $ %
ε 5 s σ sin ψ1τ sin 2ψ 1C (4)
@ ~ !#
φψ 2 11 13
psi range opening downwards and the negative psi range
opening upwards as shown in Fig. 5.
5.3.1 The measured interplanar spacing values are con-
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 sin ψ data yielding the values σ , τ , and are performed at multiple β angles with a fixed χ offset,
11 13
C. The measurement can then be repeated for multiple phi χ (Fig. 6). Measurements at various β angles do not provide a
m
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, σ , will influence the overall in the same manner as for omega and chi mode.
FIG. 4 Sample d (2θ) Versus sin ψ Dataset with σ = -500 MPa and τ = +20 MPa
11 13
E2860 − 20
FIG. 5 Sample d (2θ) Versus sin ψ Dataset with σ = -500 MPa and τ = -20 MPa
11 13
FIG. 6 Modified Chi Mode Diagram for Measurement in σ Direction
5.4.1 Eq 2 shall be rewritten in terms of β and χ . Eq 5 and 5.4.2 Substituting φ and ψ in Eq 2 with Eq 5 and 6 (see
m
6 are obtained from the solution for a right-angled spherical
X1.1), we get:
triangle (3).
ψ 5 arccos cos βcosχ (5)
~ !
m
sin βcosχ
m
φ 5 arccos (6)
S D
sin ψ
E2860 − 20
FIG. 7 ψ and φ Angles Versus β Angle for Modified Chi Mode with χ = 12°
m
1 5.4.5 The τ influences on the d versus sin β plot are more
ij
hkl hkl 2 2 2 2 2
$ % $ %
ε 5 s @σ sin β cos χ 1σ sin χ 1σ cos β cos χ #
βχ 2 11 m 22 m 33 m
m
complex and are often assumed to be zero (3). However, this
may not be true and significant errors in the calculated stress
$hkl% 2
1 s @τ sinβsin~2χ !1τ sin~2β! cos χ 1τ cosβsin~2χ !# 2
2 12 m 13 m 23 m
may result. Figs. 9-13 show the d versus sin β influences of
$hkl% individual shear components for modified chi mode consider-
1s @σ 1σ 1σ # (7)
1 11 22 33
ing two detector positions (χ = +12° and χ = -12°). Compo-
m m
5.4.3 Stress normal to the surface (σ ) is assumed to be
nents τ and τ cause a symmetrical opening about the σ
12 13 11
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, σ 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 τ and
sible σ influences(see12.12).Sincethe σ valuesand χ will
33 ij m
τ termsmaybepossible,althoughdistinguishingbetweenthe
{hkl}
remain constant for a given azimuth, the s term is
two influences through regression is not normally possible.
renamed C, and the σ term is renamed D.
5.4.6 The τ value affects the d versus sin β slope in a
1 1
hkl hkl 2 2 hkl
$ % $ % $ %
similar fashion to σ for each detector position (Figs. 12 and
ε 5 s σ sin β cos χ 1D 1 s τ sinβsin 2χ
@ # @ ~ !
βχ 2 11 m 2 12 m
m
2 2
13). This is an unwanted effect since the σ and τ influence
11 23
1τ sin 2β cos χ 1τ cosβsin 2χ #1C (8)
~ ! ~ !
13 m 23 m
cannot be resolved for one χ position. In this instance, the τ
m 23
shear stress of -100 MPa results in a calculated σ value of
5.4.4 The σ 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 χ = +12° or -527.5 MPa for χ = -12°, while
m m
slopeshallbedividedbycos χ .Thisincreasestheeffective ⁄2 the actual value is -500 MPa. The value, σ can still be
m 11
{hkl} 2
s by a factor of 1/cos χ for σ .
determined by averaging the β data for both χ positions.
2 m 11 m
FIG. 8 Sample d (2θ) Versus sin β Dataset with σ = -500 MPa
E2860 − 20
FIG. 9 Sample d (2θ) versus sin β Dataset with χ = +12°, σ = -500 MPa, and τ = -100 MPa
m 11 12
FIG. 10 Sample d (2θ) Versus sin β Dataset with χ = -12°, σ = -500 MPa, and τ = -100 MPa
m 11 12
FIG. 11 Sample d (2θ) Versus sin β Dataset with χ = +12 or -12°, σ = -500 MPa, and τ = -100 MPa
m 11 13
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 σ but shall be approached with caution using one sources used for XRD.
χ positionbecauseofthepossiblepresenceofa τ stress.The
6.1.2.1 Conventional Sealed Tube—This is by far the most
m 23
combination of multiple shear stresses including τ 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.
6.1.2.3 Synchrotron—Particle accelerator that is capable of
following main components:
6.1.1 Goniometer—An angle-measuring device responsible producing a high-intensity X-ray beam.
forthepositioningofthesource,detectors,andsamplerelative 6.1.2.4 Sealed Radioactive Sources—Although not com-
to each other. monly used, they may be utilized.
E2860 − 20
FIG. 12 Sample d (2θ) Versus sin β Dataset with χ = +12°, σ = -500 MPa, τ = -100 MPa, and Measured σ = -472.5 MPa
m 11 23 11
FIG. 13 Sample d (2θ) Versus sin β Dataset with χ = -12°, σ = -500 MPa, τ = -100 MPa, and Measured σ = -527.5 MPa
m 11 23 11
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).
8.1 This guide is intended for materials with the following
6.1.4.3 Data processing—Responsible for all data fitting
characteristics:
and calculations.
8.1.1 Fine grain size and
6.1.4.4 Data management—Responsible for data file man-
8.1.2 Nearrandomcoherentdomainorientationdistribution.
agementaswellasoverallrecordkeeping.Individualmeasure-
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
andshouldbefreeofvisiblesignsofoxidation,materialdebris,
database of key measurement values and file names.
and coatings such as oil and paint.
8.3 Sample surfaces shall be free of any significant rough-
7. Hazards
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.
governmentregulationsorguidelinesshallalwaysbefollowed.
21).
Examples include ANSI N43.2-2001 and ANSI N43.3.
8.4 Samplesurfacesmaybepreparedusingelectropolishing
7.2 Theaslowasreasonablyachievable(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).
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 ⁄2 s
mm is recommended. and s may also be determined with Test Method E1426. The
depth of penetration (x) for omega and chi mode based on Eq
8.6 Sample curvatures should not exceed the acceptable
9 and 10 are included (DIN En 15305, p. 22), (1, p. 106). Note
limits for the goniometer setup used (see 9.1.2).
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 Measurementofthincoatingsmaynotberepresentative 2 2
1 sin θ 2 sin ψ
Ω
x 5 (9)
of the bulk material. Diffraction of the substrate may create ψθ
2µ sin θcosψ
interfering diffraction lines.
χ
x 5 sinθcosψ (10)
ψθ
2µ
9. Preparation of Apparatus
9.1 Primary Beam Size—Theprimarybeamsizeistypically
χ 2
m
x 5 cosβ~1 2 cot θ! (11)
βθ
2µ
adjustable using a primary beam aperture. To ensure the best
counting statistics, the largest beam size should be used that
9.3 Filters—Filters may be used to suppress K peak inter-
β
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 K value, which should lie between the target K and K
edge α β
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%and10%error,respectively,intheaxialdirection.Incases
9.5 Modes—Three modes are described in 9.5.1 – 9.5.3.
inwhichthebeamsizecannotbesufficientlysmall,corrections
Each has specific advantages and disadvantages. Some goni-
can be applied (5, p. 107), (6, pp. 327-336).
ometers offer multiple modes.
9.2 Target/Plane Combination—The characteristic wave-
9.5.1 Omega Mode—Alsoknownasiso-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 σ -σ plane. Multiple ψ angles are observed by
φ 33
wavelengths, and K filters are shown in Table 1.
β
rotation about the ω, Ω, θ,or σ axis while χ remains equal
φ+90°
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;
higherstrainresolutionthusimprovingmeasurementprecision.
(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.Shallowpenetrationreducesstressgradienteffects
recommended complete ψ range; and
but limits the number of coherent domains contributing to the
(4)Conducivetoslitopticsandimprovedparticlestatistics.
diffraction profile. When performing residual stress measure-
9.5.1.2 Disadvantages:
mentsinmartensiticbearingsteels,theCrKαtargetistypically
(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 σ axis to realize the full
recommended ψ range while avoiding low incidence angle
TABLE 1 Target Wavelengths and Appropriate K Suppression
β
errors; and
Filers
(3)Alignment issues may negate advantages of using two
K K K Filter K
Target α1 α2 β edge
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
observed by rotation about the χ or σ axis while χ remains
25 Mn 2.101 820 2.105 78 1.910 21 Cr 2.070 20 φ+90°
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
(1)Lorentz-polarization-absorption (LPA) is not affected
42 Mo 0.709 300 0.713 590 0.632 288 Nb 0.652 98
with varying ψ angle and
E2860 − 20
TABLE 2 Commonly Used Target, Plane, and 2θ Combinations
Ω
x
ψθ
Ω χ
and x x
{hkl} {hkl} ψθ ψθ
2θ ⁄2 s s
2 1 χ
x ψ = 45° ψ = 60°
Target {hkl} Alloy ψθ
ψ =0°
-6 -1
[degrees] [10 MPa ] [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
sample. It is recommended that round robin methodologies be used to
appreciably and
determine the residual stress values of such reference samples. Specifica-
(2)Usually requires spot focus and collimators that reduce
tionDINEN15305providesamethodologyforcreatingastress-reference
particle statistics.
specimen.
9.5.3 Modified Chi Mode—With modified chi mode, the
sourcepositioning,samplepositioning,andaxisofrotationare
11. Procedure
the same as omega mode.The detector positions, however, are
11.1 Positiontestspecimenformeasurementinthegoniom-
rotated 90° about the incident beam creating a fixed χ offset
eter. Ensure that specimen-positioning devices such as clamps
(χ ). Conflicting nomenclature may be found in literature with
m
do not create an applied load because the XRD method does
regard to axis names. For example, the χ and ω names may be
not differentiate between applied and residual stress but rather
reversed such that multiple angles are observed by rotation
measures the summation of the two.
about the χ axis. Since modified chi mode is typically used by
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—Capableofaccessingdeepgroovespar-
Iftherangeisfurtherlimitedbyspecimengeometry,thelargest
allel to axis of rotation.
possible range should be used where no shadowing effects
9.5.3.2 Disadvantages:
occur.
(1)Values τ and τ cannot be resolved (see 5.4) and
12 13 11.2.1 Iso Inclination—ψ max = 645° (sin ψ = 0.5). (9,p.
(2)Values τ and σ cannot be resolved (see 5.4).
23 11 121)
11.2.2 Side Inclination—ψ max = 677° (sin ψ = 0.95). (9,
10. Calibration and Standardization
p. 121)
10.1 InstrumentalignmentcanbeverifiedwithTestMethod 11.2.3 Modified Chi—β max = 678° (sin β = 0.96). (1,p.
E915 by the measurement of a stress-free powder. 179)
E2860 − 20
11.3 In the case of single detector configurations other than diffraction peaks at high 2θ angles. These effects can be
chimode,therangecanberestrictedtothepositiveornegative compensated for using the following equation (8, p. 131):
ψ range depending on the goniometer used. The sample can
I
actual
I 5 (16)
then be rotated 180° about the σ axis and remeasured to
corrected 2
11 cos ~2θ!
realize the full ψ range while avoiding low-incident angle S D
sin θcosθ
errors.
11.6.6 Modified Lorentz-Polarization Correction—For
11.4 At least three ψ or β angles are to be used, although
broaddiffractionpeaksathigh2θangles,themodifiedLorentz-
sevenormorearerecommended.Whenpossible, ψor βangles
polarization correction can result in a more symmetrical peak
should be chosen such that they are evenly distributed through
profile (8, p. 463):
2 2
thesin ψorsin βrangeused.Formodifiedchimode,identical
I
actual
positive and negative angles should be chosen to simplify
I 5 (17)
corrected
11 cos 2θ
~ !
averaging.
S D
sin θ
11.5 Collect each profile with sufficient exposure time to
ensure accurate intensity information is collected. Random
12. Calculation and Interpretation of Results
error as a result of counting statistics from insufficient collec-
12.1 ThepositionofthecorrectedXRDpeakprofilesshould
tion times may result in an inaccurate peak position determi-
be determined using an appropriate method. Historically,
nation.
popularpracticessuchasstrippingtheK peakandthenusing
α2
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:
beenshowntobesubjecttosignificanterrors(10,pp.103-111).
11.6.1 Gain Correction—Multichannel detectors may offer
Theparabolicfitiscapableofproducingsatisfactoryresultsfor
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
picbearingsteelscanbesubjecttoadditionalerrorsandshould
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-
onlywithcaution,asoversmoothingwillaffecttheaccuracyof
intensity format.
peak position determination. If smoothing is used, Fourier
12.3 To ensure accurate peak position determination, the
smoothing is recommended since the threshold between major
entire peak profile including background should be included.
peak contributions and smaller noise frequencies are much
Peak truncation caused by collection over an insufficient 2θ
more distinguishable in the frequency domain.
range has been shown to cause inaccurate peak position
11.6.3 Absorption Correction—The intensity of a diffracted
determination without the use of advanced numerical methods
beam may be subject to a θ-dependent absorption effect
(11,pp.524-525).Generally,thedetectorwidthshouldbethree
causing a distortion of the peak profile. This effect can be
times the FWHM value.
compensated for using the following equation (1, p. 90):
12.4 Selection of Peak Position Determination Method—
I
measured
I 5 (12)
corrected
Thereareanumberofmethodstodeterminethepositionofthe
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 position-sensitive detectors assuming the peak is approxi-
measured
I 5 (13)
corrected
mately centered in the detector window. Commonly used peak
~1 2 tan ψcotθ!
functions are listed in Table 3.
11.6.3.2 Chi mode absorption correction—η
...

This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
Designation: E2860 − 12 E2860 − 20
Standard Test Method for
Residual Stress Measurement by X-Ray Diffraction for
Bearing Steels
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 sin ψ 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 Scope*
1.1 This test method covers a procedure for experimentally determining macroscopic residual stress tensor components of
quasi-isotropic bearing steel materials by X-ray diffraction (XRD).
1.2 This test method provides a guide for experimentally determining stress values, which play a significant role in bearing life.
1.3 Examples of how tensor values are used are:
1.3.1 Detection of grinding type and abusive grinding;
1.3.2 Determination of tool wear in turning operations;
1.3.3 Monitoring of carburizing and nitriding residual stress effects;
1.3.4 Monitoring effects of surface treatments such as sand blasting, shot peening, and honing;
This test method is under the jurisdiction of ASTM Committee E28 on Mechanical Testing and is the direct responsibility of Subcommittee E28.13 on Residual Stress
Measurement.
Current edition approved April 1, 2012Nov. 1, 2020. Published May 2012February 2021. Originally approved in 2012. Last previous edition approved in 2012 as
E2860–12. DOI: 10.1520/E2860–12.10.1520/E2860–20.
*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
E2860 − 20
1.3.5 Tracking of component life and rolling contact fatigue effects;
1.3.6 Failure analysis;
1.3.7 Relaxation of residual stress; and
1.3.8 Other residual-stress-related issues that potentially affect bearings.
1.4 Units—The values stated in SI units are to be regarded as standard. No other units of measurement are included in this
standard.
1.5 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility
of the user of this standard to establish appropriate safety safety, health, and healthenvironmental practices and determine the
applicability of regulatory limitations prior to use.
1.6 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.
2. Referenced Documents
2.1 ASTM Standards:
E6 Terminology Relating to Methods of Mechanical Testing
E7 Terminology Relating to Metallography
E177 Practice for Use of the Terms Precision and Bias in ASTM Test Methods
E691 Practice for Conducting an Interlaboratory Study to Determine the Precision of a Test Method
E915 Test Method for Verifying the Alignment of X-Ray Diffraction Instrumentation for Residual Stress Measurement
E1426 Test Method for Determining the X-Ray Elastic Constants for Use in the Measurement of Residual Stress Using X-Ray
Diffraction Techniques
2.2 ANSI Standards:
N43.2 Radiation Safety for X-ray Diffraction and Fluorescence Analysis Equipment
N43.3 For General Radiation Safety—Installations Using Non-Medical X-Ray and Sealed Gamma-Ray Sources, Energies Up
to 10 MeV
2.3 SAE standard:
HS-784/2003 Residual Stress Measurement by X-Ray Diffraction, 2003 Edition
3. Terminology
3.1 Definitions—Many of the terms used in this test method are defined in Terminologies E6 and E7.
3.2 Definitions of Terms Specific to This Standard:
3.2.1 interplanar spacing, n—perpendicular distance between adjacent parallel atomic planes.
3.2.2 macrostress, n—average stress acting over a region of the test specimen containing many gains/crystals/coherent domains.
3.3 Abbreviations:
3.3.1 ALARA—As low as reasonably achievable
3.3.2 FWHM—Full width half maximum
3.3.3 LPA—Lorentz-polarization-absorption
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM Standards
volume information, refer to the standard’s Document Summary page on the ASTM website.
Available from American National Standards Institute (ANSI), 25 W. 43rd St., 4th Floor, New York, NY 10036, http://www.ansi.org.
Available from SAE International (SAE), 400 Commonwealth Dr., Warrendale, PA 15096, http://www.sae.org.
E2860 − 20
3.3.4 MSDS—Material safety data sheet
3.3.5 XEC—X-ray elastic constant
3.3.6 XRD—X-ray diffraction
11ν
{hkl}
3.4 Symbols: ⁄2S = X-ray elastic constant of quasi-isotropic material equal to
2 $hkl%
E
eff
α = Linear thermal expansion coefficient
L
β = Angle between the incident beam and σ or surface normal on the σ σ plane
33 33 11
χ = Angle between the σ direction and the normal to the diffracting plane
φ+90°
χ = Fixed χ offset used in modified-chi mode
m
d = Interplanar spacing between crystallographic planes; also called d-spacing
d = Interplanar spacing for unstressed material
o
d = Perpendicular spacing
'
Δd = Change in interplanar spacing caused by stress
ε = Strain component i, j
ij
E = Modulus of elasticity (Young’s modulus)
{hkl}
E = Effective elastic modulus for X-ray measurements
eff
μ = Attenuation coefficient
η = Rotation of the sample around the measuring direction given by φ and ψ or χ and β
ω or Ω = Angle between the specimen surface and incident beam when χ = 0°
φ = Angle between the σ direction and measurement direction azimuth, see Fig. 1
“hkl” = Miller indices
σ = Normal stress component i,j
ij
2ν
{hkl}
s = X-ray elastic constant of quasi-isotropic material equal to
1 $hkl%
E
eff
τ = Shear stress component i,j
ij
θ = Bragg angle
ν = Poisson’s ratio
Mode
x = Mode dependent depth of penetration
ψ = Angle between the specimen surface normal and the scattering vector, that is, normal to the diffracting plane, see Fig. 1
4. Summary of Test Method
4.1 A test specimen is placed in a XRD goniometer aligned as per Test Method E915.
4.2 The diffraction profile is collected over three or more angles within the required angular range for a given {hkl} plane,
although at least seven or more are recommended.
4.3 The XRD profile data are then corrected for LPA, background, and instrument-specific corrections.
4.4 The peak position/Bragg angle is determined for each XRD peak profile.
FIG. 1 Stress Tensor Components
E2860 − 20
4.5 The d-spacings are calculated from the peak positions via Bragg’s law.
2 2
4.6 The d-spacing values are plotted versus their sin ψ or sin β values, and the residual stress is calculated using Eq 4 or Eq 8,
respectively.
4.7 The error in measurement is evaluated as per Section 14.
4.8 The following additional corrections may be applied. The use of these corrections shall be clearly indicated with the reported
results.
4.8.1 Depth of penetration correction (see 12.12) and
4.8.2 Relaxation as a result of material removal correction (see 12.14).
5. Significance and Use
5.1 This test method covers a procedure for experimentally determining macroscopic residual stress tensor components of
quasi-isotropic bearing steel materials by XRD. Here the stress components are represented by the tensor σ as shown in Eq 1 (1,
ij
p. 40). The stress strain relationship in any direction of a component is defined by Eq 2 with respect to the azimuth phi(φ) and
polar angle psi(ψ) defined in Fig. 1 (1, p. 132).
σ τ τ
11 12 13
τ σ τ
σ 5 where τ 5 τ (1)
21 22 23
ij ij ji
F G
τ τ σ
31 32 33
1 1
hkl hkl 2 2 2 2 2 hkl 2 hkl
$ % $ % $ % $ %
ε 5 s σ cos φ sin ψ1σ sin φ sin ψ1σ cos ψ 1 s τ sin 2φ sin ψ1τ cosφsin 2ψ 1τ sinφsin 2ψ 1s σ 1σ 1σ (2)
@ # @ ~ ! ~ ! ~ !# @ #
φψ 2 11 22 33 2 12 13 23 1 11 22 33
2 2
5.1.1 Alternatively, Eq 2 may also be shown in the following arrangement (2, p. 126):
1 1 1
$hkl% $hkl% 2 2 2 $hkl% $hkl% $hkl%
ε 5 s @σ cos φ1τ sin~2φ!1σ sin φ2 σ # sin ψ1 s σ 2 s @σ 1σ 1σ #1 s @τ cosφ1τ sinφ#sin~2ψ!
φψ 2 11 12 22 33 2 33 1 11 22 33 2 13 23
2 2 2
5.2 Using XRD and Bragg’s law, interplanar strain measurements are performed for multiple orientations. The orientations are
selected based on a modified version of Eq 2, which is dictated by the mode used. Conflicting nomenclature may be found in
literature with regard to mode names. For example, what may be referred to as a ψ (psi) diffractometer in Europe may be called
a χ (chi) diffractometer in North America. The three modes considered here will be referred to as omega, chi, and modified-chi
as described in 9.5.
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 (σ ) is assumed
to be insignificant because of the shallow depth of penetration of X-rays at the free surface, reducing Eq 3 to Eq 4.
Post-measurement corrections may be applied to account for possible σ influences (12.12). Since the σ values will remain
33 ij
{hkl}
constant for a given azimuth, the s term is renamed C.
1 1
$hkl% $hkl% 2 2 $hkl% $hkl%
ε 5 s @σ sin ψ1σ cos ψ#1 s @τ sin~2ψ!#1s @σ 1σ 1σ # (3)
φψ 2 11 33 2 13 1 11 22 33
2 2
$hkl% $hkl% 2
ε 5 s @σ sin ψ1τ sin~2ψ!#1C (4)
φψ 2 11 13
5.3.1 The measured interplanar spacing values are converted to strain using Eq 24, Eq 25, or Eq 26. Eq 4 is used to fit the strain
versus sin ψ data yielding the values σ , τ , and C. The measurement can then be repeated for multiple phi angles (for example
11 13
0, 45, and 90°) to determine the full stress/strain tensor. The value, σ , will influence the overall slope of the data, while τ is
11 13
related to the direction and degree of elliptical opening. Fig. 4 shows a simulated d versus sin ψ profile for the tensor shown. Here
The boldface numbers in parentheses refer to the list of references at the end of this standard.
E2860 − 20
FIG. 2 Omega Mode Diagram for Measurement in σ Direction
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 σ Direction
the positive 20-MPa τ stress results in an elliptical opening in which the positive psi range opens upward and the negative psi
range opens downward. A higher τ value will cause a larger elliptical opening. A negative 20-MPa τ stress would result in the
13 13
same elliptical opening only the direction would be reversed with the positive psi range opening downwards and the negative psi
range opening upwards as shown in Fig. 5.
E2860 − 20
FIG. 4 Sample d (2θ) Versus sin ψ Dataset with σ = -500 MPa and τ = +20 MPa
11 13
FIG. 5 Sample d (2θ) Versus sin ψ Dataset with σ = -500 MPa and τ = -20 MPa
11 13
5.4 Modified Chi Mode—Interplanar strain measurements are performed at multiple β angles with a fixed χ offset, χ (Fig. 6).
m
Measurements at various β angles do not provide a constant φ angle (Fig. 7), therefore, Eq 2 cannot be simplified in the same
manner as for omega and chi mode.
5.4.1 Eq 2 shall be rewritten in terms of β and χ . Eq 5 and 6 are obtained from the solution for a right-angled spherical triangle
m
(3).
ψ5 arccos cos βcosχ (5)
~ !
m
sin βcosχ
m
φ5 arccos (6)
S D
sin ψ
5.4.2 Substituting φ and ψ in Eq 2 with Eq 5 and 6 (see X1.1), we get:
1 1
$hkl% $hkl% 2 2 2 2 2 $hkl% 2 $hkl%
ε 5 s @σ sin β cos χ 1σ sin χ 1σ cos β cos χ #1 s @τ sinβsin~2χ !1τ sin~2β! cos χ 1τ cosβsin~2χ !#1s @σ 1σ 1σ # (7)
βχ 2 11 m 22 m 33 m 2 12 m 13 m 23 m 1 11 22 33
m
2 2
5.4.3 Stress normal to the surface (σ ) is assumed to be insignificant because of the shallow depth of penetration of X-rays at the
free surface reducing Eq 7 to Eq 8. Post-measurement corrections may be applied to account for possible σ influences (see 12.12).
{hkl}
Since the σ values and χ will remain constant for a given azimuth, the s term is renamed C, and the σ term is renamed
ij m 1 22
D.
1 1
hkl hkl 2 2 hkl 2
$ % $ % $ %
ε 5 s σ sin β cos χ 1D 1 s τ sinβsin 2χ 1τ sin 2β cos χ 1τ cosβsin 2χ 1C (8)
@ # @ ~ ! ~ ! ~ !#
βχ 2 11 m 2 12 m 13 m 23 m
m
2 2
5.4.4 The σ influence on the d versus sin β plot is similar to omega and chi mode (Fig. 8) with the exception that the slope shall
2 {hkl} 2
be divided by cos χ . This increases the effective ⁄2s by a factor of 1/cos χ for σ .
m 2 m 11
E2860 − 20
FIG. 6 Modified Chi Mode Diagram for Measurement in σ Direction
FIG. 7 ψ and φ Angles Versus β Angle for Modified Chi Mode with χ = 12°
m
5.4.5 The τ influences on the d versus sin β plot are more complex and are often assumed to be zero (3). However, this may not
ij
be true and significant errors in the calculated stress may result. Figs. 9-13 show the d versus sin β influences of individual shear
components for modified chi mode considering two detector positions (χ = +12° and χ = -12°). Components τ and τ cause
m m 12 13
a symmetrical opening about the σ slope influence for either detector position (Figs. 9-11); therefore, σ can still be determined
11 11
by simply averaging the positive and negative β data. Fitting the opening to the τ and τ terms may be possible, although
12 13
distinguishing between the two influences through regression is not normally possible.
5.4.6 The τ value affects the d versus sin β slope in a similar fashion to σ for each detector position (Figs. 12 and 13). This
23 11
is an unwanted effect since the σ and τ influence cannot be resolved for one χ position. In this instance, the τ shear stress
11 23 m 23
E2860 − 20
FIG. 8 Sample d (2θ) Versus sin β Dataset with σ = -500 MPa
FIG. 9 Sample d (2θ) versus sin β Dataset with χ = +12°, σ = -500 MPa, and τ = -100 MPa
m 11 12
FIG. 10 Sample d (2θ) Versus sin β Dataset with χ = -12°, σ = -500 MPa, and τ = -100 MPa
m 11 12
of -100 MPa results in a calculated σ value of -472.5 MPa for χ = +12° or -527.5 MPa for χ = -12°, while the actual value
11 m m
is -500 MPa. The value, σ can still be determined by averaging the β data for both χ positions.
11 m
5.4.7 The use of the modified chi mode may be used to determine σ but shall be approached with caution using one χ position
11 m
because of the possible presence of a τ stress. The combination of multiple shear stresses including τ results in increasingly
23 23
complex shear influences. Chi and omega mode are preferred over modified chi for these reasons.
6. Apparatus
6.1 A typical X-ray diffractometer is composed of the following main components:
6.1.1 Goniometer—An angle-measuring device responsible for the positioning of the source, detectors, and sample relative to each
other.
E2860 − 20
FIG. 11 Sample d (2θ) Versus sin β Dataset with χ = +12 or -12°, σ = -500 MPa, and τ = -100 MPa
m 11 13
FIG. 12 Sample d (2θ) Versus sin β Dataset with χ = +12°, σ = -500 MPa, τ = -100 MPa, and Measured σ = -472.5 MPa
m 11 23 11
FIG. 13 Sample d (2θ) Versus sin β Dataset with χ = -12°, σ = -500 MPa, τ = -100 MPa, and Measured σ = -527.5 MPa
m 11 23 11
6.1.2 X-Ray Source—There are generally three X-ray sources used for XRD.
6.1.2.1 Conventional Sealed Tube—This is by far the most common found in XRD equipment. It is identified by its anode target
element such as chromium (Cr), manganese (Mn), or copper (Cu). The anode is bombarded by electrons to produce specific X-ray
wavelengths unique to the target element.
6.1.2.2 Rotating Anode Tube—This style of tube offers a higher intensity than a conventional sealed tube.
6.1.2.3 Synchrotron—Particle accelerator that is capable of producing a high-intensity X-ray beam.
6.1.2.4 Sealed Radioactive Sources—Although not commonly used, they may be utilized.
E2860 − 20
6.1.3 Detector—Detectors may be of single channel, multi-channel linear, or area design.
6.1.4 Software—Software is grouped into the following main categories:
6.1.4.1 Goniometer control—Responsible for positioning of the sample relative to the incident beam and detector(s) in automated
goniometers.
6.1.4.2 Data acquisition—Responsible for the collection of diffraction profile data from the detector(s).
6.1.4.3 Data processing—Responsible for all data fitting and calculations.
6.1.4.4 Data management—Responsible for data file management as well as overall record keeping. Individual measurement data
is typically stored in a file format that can later be reopened for reevaluation. It is often beneficial to keep a database of key
measurement values and file names.
7. Hazards
7.1 Regarding the use of analytical X-ray equipment, local government regulations or guidelines shall always be followed.
Examples include ANSI N43.2-2001 and ANSI N43.3.
7.2 The as low as reasonably achievable (ALARA) philosophy should always be used when dealing with radiation exposure.
7.3 Always follow the safety guidelines of the equipment manufacturer.
7.4 Refer to material safety data sheets (MSDS) sheets for handling of dangerous materials potentially found in XRD equipment
(that is, beryllium and lead).
7.5 The high voltage used to generate X-rays is very dangerous. Follow the manufacturer’s and local guidelines when dealing with
high-voltage equipment.
8. Test Specimens
8.1 This guide is intended for materials with the following characteristics:
8.1.1 Fine grain size and
8.1.2 Near random coherent domain orientation distribution.
8.2 Test specimens shall be clean at the measured location and should be free of visible signs of oxidation, material debris, and
coatings such as oil and paint.
8.3 Sample surfaces shall be free of any significant roughness. Grooves produced by machining perpendicular to the measurement
direction may affect measurement results (4, p. 21).
8.4 Sample surfaces may be prepared using electropolishing as this method does not impart stress within the sample; however,
removal of stressed layers may influence the subsurface residual stress state. Corrections are available to estimate the true stress
that existed when the specimen was intact (see 12.13).
8.5 If material removal methods other than electropolishing (that is, grinding or sanding) are necessary, subsequent electropol-
ishing is required to ensure the cold-worked region is removed. For light grinding or sanding, the removal of 0.25 mm is
recommended.
8.6 Sample curvatures should not exceed the acceptable limits for the goniometer setup used (see 9.1.2).
E2860 − 20
8.7 Measurement of a single-phase stress in multiphase materials may not be representative of the bulk material when significant
amount of additional phases are present.
8.8 Measurement of thin coatings may not be representative of the bulk material. Diffraction of the substrate may create interfering
diffraction lines.
9. Preparation of Apparatus
9.1 Primary Beam Size—The primary beam size is typically adjustable using a primary beam aperture. To ensure the best counting
statistics, the largest beam size should be used that does not exceed the following limitations:
9.1.1 Preferably beam divergence should not exceed 1° (2, p. 107). Divergence may be limited by devices such as Soller slits and
sample masking.
9.1.2 For cylindrical specimens of radius, R, the maximum incident X-ray spot size to use shall be R/6 for 5 % error and R/4 for
10 % error in the hoop direction and R/2.5 and R/2 for 5 % and 10 % error, respectively, in the axial direction. In cases in which
the beam size cannot be sufficiently small, corrections can be applied (5, p. 107), (6, pp. 327-336).
9.2 Target/Plane Combination—The characteristic wavelengths available for diffraction are determined by the target element. A
list of common target elements, their K line wavelengths, and K filters are shown in Table 1.
β
9.2.1 There are several possible target-plane combinations for a given bearing steel that will produce a diffraction peak. When
choosing a combination, there are many factors to take into account including the relative peak versus background intensity, mass
absorption coefficient, possible interfering peaks, and strain resolution. Higher 2θ values will have a higher strain resolution thus
improving measurement precision. A higher mass absorption coefficient reduces the depth of penetration. Shallow penetration
reduces stress gradient effects but limits the number of coherent domains contributing to the diffraction profile. When performing
residual stress measurements in martensitic bearing steels, the Cr Kα target is typically used with the {211} plane with a 2θ angle
of approximately 154 to 157º. When performing residual stress measurements in austenitic bearing steels, the Mn Kα target is
typically used with the {311} plane with a 2θ angle of approximately 152 to 155º. Table 2 shows a list of target-plane combinations
commonly shown in literature. X-Ray elastic constants ⁄2 s and s may also be determined with Test Method E1426. The depth
2 1
of penetration (x) for omega and chi mode based on Eq 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 mode. The ψ = 0 values are, therefore, listed in the same column. The depth
of penetration for modified chi mode is given by Eq 11.
2 2
1 sin θ2 sin ψ
Ω
x 5 (9)
ψθ
2μ sin θcosψ
χ
x 5 sinθcosψ (10)
ψθ
2μ
χ 2
m
x 5 cosβ~12 cot θ! (11)
βθ
2μ
9.3 Filters—Filters may be used to suppress K peak interference and fluorescence. A filter material is chosen based on the K
β edge
value, which should lie between the target K and K wavelength values. See Table 1.
α β
TABLE 1 Target Wavelengths and Appropriate K Suppression
β
Filers
K K K Filter K
Target α1 α2 β edge
Element
[Å] = [nm × 10]
22 Ti 2.748 51 2.752 16 2.513 91 {
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
26 Fe 1.936 042 1.939 980 1.756 61 Mn 1.896 43
27 Co 1.788 965 1.792 850 1.620 79 Fe 1.756 61
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
E2860 − 20
TABLE 2 Commonly Used Target, Plane, and 2θ Combinations
Ω
x
ψθ
Ω χ
and x x
{hkl} {hkl} ψθ ψθ
2θ ⁄2s s
2 1 χ
x ψ = 45° ψ = 60°
Target {hkl} Alloy ψθ
ψ = 0°
-6 -1
[degrees] [10 MPa ] [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)
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
9.4 Monochromators—Monochromators(s) may be used to eliminate spectral components including the Kβ and the Kα2 line,
although they will reduce the beam intensity and increase measurement time significantly.
9.5 Modes—Three modes are described in 9.5.1 – 9.5.3. Each has specific advantages and disadvantages. Some goniometers offer
multiple modes.
9.5.1 Omega Mode—Also known as iso-inclination, ω, or Ω method. With omega mode, the incident beam and ψ angle(s) remain
on the σ -σ plane. Multiple ψ angles are observed by rotation about the ω, Ω, θ, or σ axis while χ remains equal to zero.
φ 33 φ+90°
9.5.1.1 Advantages:
(1) Keeps experiment two dimensional (2D), which is useful for thin coatings, films, and layers;
(2) Capable of accessing deep grooves perpendicular to axis of rotation;
(3) Using two detectors (if available) simultaneous observation of both Debye ring locations is possible over the recommended
complete ψ range; and
(4) Conducive to slit optics and improved particle statistics.
9.5.1.2 Disadvantages:
(1) Absorption varies with ψ angle;
(2) The use of single detector systems may require 180° rotation of the sample about the σ axis to realize the full
recommended ψ range while avoiding low incidence angle errors; and
(3) Alignment issues may negate advantages of using two detectors.
9.5.2 Chi Mode—Also known as side-inclination or χ method. With ω, Ω, or θ equal to 2θ/2, multiple ψ angles are observed by
rotation about the χ or σ axis while χ remains equal to ψ.
φ+90°
E2860 − 20
9.5.2.1 Advantages:
(1) Lorentz-polarization-absorption (LPA) is not affected with varying ψ angle and
(2) Capable of accessing deep grooves parallel to axis of rotation.
9.5.2.2 Disadvantages:
(1) Beam spot on sample is pseudo elliptical and spreads appreciably and
(2) Usually requires spot focus and collimators that reduce particle statistics.
9.5.3 Modified Chi Mode—With modified chi mode, the source positioning, sample positioning, and axis of rotation are the same
as omega mode. The detector positions, however, are rotated 90° about the incident beam creating a fixed χ offset (χ ). Conflicting
m
nomenclature may be found in literature with regard to axis names. For example, the χ and ω names may be reversed such that
multiple angles are observed by rotation about the χ axis. Since modified chi mode is typically used by omega mode diffractometers
with detector positioning rotated 90° about the incident beam, omega axis labeling is used for consistency.
9.5.3.1 Advantage—Capable of accessing deep grooves parallel to axis of rotation.
9.5.3.2 Disadvantages:
(1) Values τ and τ cannot be resolved (see 5.4) and
12 13
(2) Values τ and σ cannot be resolved (see 5.4).
23 11
10. Calibration and Standardization
10.1 Instrument alignment can be verified with Test Method E915 by the measurement of a stress-free powder.
10.2 Additionally, a nonzero known residual stress proficiency reference sample should be measured to verify that hardware and
software are working correctly.
NOTE 1—No national reference sample exists other than a stress-free sample. It is recommended that round robin methodologies be used to determine
the residual stress values of such reference samples. Specification DIN EN 15305 provides a methodology for creating a stress-reference specimen.
11. Procedure
11.1 Position test specimen for measurement in the goniometer. Ensure that specimen-positioning devices such as clamps do not
create an applied load because the XRD method does not differentiate between applied and residual stress but rather measures the
summation of the two.
11.2 The angular range over which measurements are carried out is limited by the mode used. Measurements should always be
performed over the maximum permissible ψ range. If the range is further limited by specimen geometry, the largest possible range
should be used where no shadowing effects occur.
11.2.1 Iso Inclination—ψ max = 645° (sin ψ = 0.5). (9, p. 121)
11.2.2 Side Inclination—ψ max = 677° (sin ψ = 0.95). (9, p. 121)
11.2.3 Modified Chi—β max = 678° (sin β = 0.96). (1, p. 179)
11.3 In the case of single detector configurations other than chi mode, the range can be restricted to the positive or negative ψ range
depending on the goniometer used. The sample can then be rotated 180° about the σ axis and remeasured to realize the full ψ
range while avoiding low-incident angle errors.
11.4 At least three ψ or β angles are to be used, although seven or more are recommended. When possible, ψ or β angles should
2 2
be chosen such that they are evenly distributed through the sin ψ or sin β range used. For modified chi mode, identical positive
and negative angles should be chosen to simplify averaging.
11.5 Collect each profile with sufficient exposure time to ensure accurate intensity information is collected. Random error as a
result of counting statistics from insufficient collection times may result in an inaccurate peak position determination.
E2860 − 20
11.6 For each of the profiles collected, apply the following applicable corrections in the following order:
11.6.1 Gain Correction—Multichannel detectors may offer a gain correction intended to correct for intensity variations caused by
the detector itself. This is performed by collecting the profile of a sample that is nondiffracting in the observed 2θ region with a
similar background intensity.
11.6.2 Data Smoothing—Smoothing may be applied, but only with caution, as over smoothing will affect the accuracy of peak
position determination. If smoothing is used, Fourier smoothing is recommended since the threshold between major peak
contributions and smaller noise frequencies are much more distinguishable in the frequency domain.
11.6.3 Absorption Correction—The intensity of a diffracted beam may be subject to a θ-dependent absorption effect causing a
distortion of the peak profile. This effect can be compensated for using the following equation (1, p. 90):
I
measured
I 5 (12)
corrected
12 tan ψcotθcosη
11.6.3.1 Omega mode absorption correction—η = 0°.
I
measured
I 5 (13)
corrected
~12 tan ψcotθ!
11.6.3.2 Chi mode absorption correction—η = 90°.
I 5 I (14)
corrected measured
11.6.3.3 Modified chi mode absorption correction
I 5 I (15)
corrected measured
11.6.4 Background Correction—To account for sloping peak backgrounds, choose two reference points on either side of the
diffraction profile. This is typically achieved by fitting a selected range within background using a linear least squares regression.
The line intensity drawn between these two points is subtracted from the profile giving the peak a level background of zero.
11.6.5 Lorentz-Polarization Correction—The intensity of a diffracted beam is subject to additional θ dependent effects known as
Lorentzian and polarization effects. This causes a further distortion of the peak profile particularly for wide diffraction peaks at high
2θ angles. These effects can be compensated for using the following equation (8, p. 131):
I
actual
I 5 (16)
corrected 2
11 cos 2θ
~ !
S D
sin θcosθ
11.6.6 Modified Lorentz-Polarization Correction—For broad diffraction peaks at high 2θ angles, the modified Lorentz-polarization
correction can result in a more symmetrical peak profile (8, p. 463):
I
actual
I 5 (17)
corrected 2
11 cos ~2θ!
S 2 D
sin θ
12. Calculation and Interpretation of Results
12.1 The position of the corrected XRD peak profiles should be determined using an appropriate method. Historically, popular
practices such as stripping the K peak and then using a parabolic peak fit to the top 20 % of the peak profile have been shown
α2
to be subject to significant errors (10, pp. 103-111). The parabolic fit is capable of producing satisfactory results for standards such
as Test Method E915 for the measurement of fine-grained, isotropic materials; however, its use for anisotropic bearing steels can
be subject to additional errors and should be used with caution.
12.2 The profiles may either be in 2θ or channel-versus-intensity format.
E2860 − 20
12.3 To ensure accurate peak position determination, the entire peak profile including background should be included. Peak
truncation caused by collection over an insufficient 2θ range has been shown to cause inaccurate peak position determination
without the use of advanced numerical methods (11, pp. 524-525). Generally, the detector width should be three times the FWHM
value.
12.4 Selection of Peak Position Determination Method—There are a number of methods to determine the position of the XRD
peak. If a function is used, the function that best describes the corrected peak shape should be used when using position-sensitive
detectors assuming the peak is approximately centered in the detector window. Commonly used peak functions are listed in Table
3.
12.4.1 Many factors affect peak shape including material properties and goniometer configuration. Fig. 14 shows an example of
incident beam size effect on peak shape in which an increasing incident beam size results in a transition from a Pearson VII profile
with a distinguishable K K contribution to a well overlapped Gaussian profile.
α1 α2
12.5 Relative peak positions are usually determined using the absolute or cross-correlation method. If the detectors are not
calibrated for actual 2θ positioning, an assumed stress-free 2θ value may be used if periodically checked for accuracy. Note,
however, that stress-free 2θ values can significantly change with depth in case-hardened steels.
12.5.1 Cross-Correlation Method (DIN EN 15305)—Diffraction angles are determined relative to a chosen reference peak. For
instance, in omega or chi mode, the 2θ position is given by:
2θ 5 2θ 1δ (18)
ψ ref ψ
12.5.1.1 It is recommended to use the strongest peak in the series for the reference peak. The shift is calculated as the value for
which the cross section between the actual and reference profile becomes maximal.
F~δ !5 I ~2θ!I ~2θ2 δ !d~2θ!5 max (19)
*
ψ ref ψ ψ
12.5.1.2 If texture effects are present, the use of cross correlation may cause larger errors than other methods.
12.5.2 Absolute Method—Diffraction angles are determined relative to each detector.
12.6 The 2θ values are converted to d spacing using Bragg’s law.
$hkl% $hkl%
nλ 5 2d sinθ (20)
Kα1 φψ φψKα1
12.7 The measurement d value shall be determined for the calculation of strain.
o
TABLE 3 Peak Distribution Functions
Values
Name Equation a = Intensity
b = Center
Gaussian c = Peak width constant
s d
x 2 b 2
f x 5ae 2
s d
2c
Pearson VII c = Peak width constant
2 2m
x 2b
s d
m = Tail curvature constant
fsxd 5a 11
H 2 J
mc
Cauchy c = Peak width constant
x 2b
s d
fsxd 5a 11
H J
c
Generalized q,r = Left and right side peak shape constants
A
Fermi function (For symmetrical peak, q = r)
f x 5
s d
2qsx2bd rsx2bd
e 1e
A = Intensity × 2
Parabolic c = Peak width constant
fsxd 52csx 2bd 1a
E2860 − 20
FIG. 14 Example of Incident Beam Size Effect on Peak Shape
12.7.1 d Omega and Chi Mode—When using the plane stress model, there is a negligible error for the substitution of d with d ψ
o o '
= 0 (HS-784). When d versus sin ψ plots exhibit large oscillations or random deviations, an elliptical regression of the data using
Eq 21 may produce a more accurate estimate of d (HS-784).
'
hkl 2
$ %
d 5 A sin ψ1Bsin 2ψ 1d (21)
~ !
φψ '
where:
A and B = regression variables.
12.7.1.1 In the case of single-detector omega mode in which software does not support overlapping of separate positive and
negative ψ range collections, and such oscillations or deviations are present, a linear regression may be performed separately for
the positive and negative range to determine each d value, respectively.
'
$hkl% 2
d 5 A sin ψ1d (22)
φψ '
where:
A = regression variable.
12.7.2 d Modified Chi Mode—When using the plane stress model, there is a negligible error for the substitution of d with d β
o o '
= 0 (HS-784). When d versus sin β plots exhibit large oscillations or random deviations, an elliptical regression of the data may
produce a more accurate estimate of d .
'
$hkl% 2
d 5 A sin β1Bsinβ1Csin~2β!1Dcosβ1E (23)
βχ
m
where:
d 5 D1E see X1.2
~ !
'
A,B,C and D = regression values.
12.8 Strain Calculation—The strain value for each data point is determined using one of the following methods.
12.8.1 Linear Variation—Also known as Cauchy or engineering strain.
d 2 d Δd sin θ
φψ o o
$hkl%
ε 5 5 5 2 1 (24)
φψ
d d sin θ
o φψ
12.8.2 Differentiating Bragg’s Law:
cotθΔ2θ
hkl
$ %
ε 52 (25)
φψ
E2860 − 20
12.8.3 True Strain Definition:
d sin θ
φψ o
$hkl%
ε 5 ln 5 ln (26)
F G F G
φψ
d sin θ
o φψ
12.9 Stress Calculation (Omega and Chi Mode):
12.9.1 The ε versus sin ψ data is fit using Eq 4 and the values for σ , τ , and C may then be determined.
11 13
12.9.2 In the case of single-detector configurations other than chi mode in which software does not support overlapping of separate
positive and negative ψ range collections, a linear regression may be performed separately for both the positive and negative ranges
using Eq 27. The two resulting σ values may then be averaged. This method should be used with caution as it ignores the shear
stresses that may be present.
$hkl% $hkl% 2
ε 5 s @σ sin ψ#1C (27)
φψ 2 11
12.10 Stress Calculation (Modified Chi Mode):
12.10.1 The ε versus sin β data is fit using Eq 28 and the values for σ , C, and D may then be determined.
$hkl% $hkl% 2 2
ε 5 s @σ sin β cos χ 1D#1C (28)
βχ 2 11 m
m
where:
180°22θ
βχ
m
χ 5
m
12.10.2 In the case of single-detector configurations other than chi mode in which software does not support overlapping of
separate positive and negative β range collections, a linear regression may be performed separately for both the positive and
negative ranges. The two resulting σ values may then be averaged. This method should be used with caution as it ignores the
shear stresses that may be present.
12.11 Stress Error Calculation—Various sources may contribute to the error in stress measurement values and can be considered
statistical or systematic.
12.11.1 Statistical errors include detector-counting statistics and the repeatability of the peak position determination method.
Neglecting statistical error, a repetition of a measurement will always give the same result.
12.11.2 Systematic errors include goniometer alignment and the errors in parameters used for the measuring and evaluation
procedure. The systematic errors of a single measurement cannot be determined.
12.11.3 Ideally, the error of all sources should be considered and combined through error propagation, although this is not always
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possible or practical. The linear or elliptical regression errors of the d versus sin ψ or d versus sin β data provide only an indication
of measurement error. The error in peak position determination as well as X-ray elastic constant(s) may also be included through
propagation.
12.12 Gradient Correction—Also known as transparency correction. Differences in effective layer thickness with orientation and
target-plane combinations can affect the measured stress of samples when a stress gradient versus depth is present. The gradient
correction determines the true 2θ values for recalculation. (See HS-784, p. 75 for procedure.)
12.13 Material Removal Correction—Material removal via electropolishing does not impart any stress in the sample; however,
relaxation or redistribution of residual stresses in the component may occur if a stressed layer is removed. There are models
available for simple geometries such as a solid cylinder, hollow cylinder, and infinite flat plate for determining what the stresses
were before material removal (see HS-784, p. 76). A spherical model is also available based on the cylindrical model (12, p. 1372).
These models should be used with caution as they assume a material removal from entire surfaces, which is frequently not the case.
If finite element model solutions are available, these should be used for best accuracy.
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12.14 Relaxation as a Result of Sectioning—It is commonly necessary to section samples to gain access to the measurement
location thus potentially altering the stress state of the sample. It is advantageous to monitor the change in stress using strain
gauge(s) in the intended direction of measurement. Relaxation through a section may be estimated by placing a strain gauge on
either side of certain geometries. The relaxation profile between the two gauges may be considered linear or calculated via other
analytical or numerical methods, assuming the material properties remain consistent throughout the section and radial stress is
disregarded (see HS-784, p. 43). XRD measurements before and after sectioning are also an acceptable means for approximating
relaxation. If an accessible area is adjacent to the desired measurement area, then before and after XRD measurements should
determine if relaxation has occurred.
13. Report
13.1 Reported data may include the following main items to ensure that sufficient information is available for comparison of
results as well as record-keeping purposes.
13.1.1 General Information:
13.1.1.1 Specify the operator(s) who performed all aspects of the measurement;
13.1.1.2 Sample identification such as part number, lot number, and so forth;
13.1.1.3 Specifications used for sample preparation, measurement, and reference to result requirements; and
13.1.1.4 Measurement location and direction.
13.1.2 Results—The following values are placed in a table using international standardized units (MPa, mm) or Imperial units (ksi,
inches) or both. In the case of stress profiles, graphs should also be included displaying SI or U.S. customary units or both.
13.1.2.1 Normal and shear stress values (if available) including errors and direction of measurement relative to the sample
reference frame;
13.1.2.2 FWHM values (if required or available) including errors. In the case of two detector setups, the average values may also
be included if required; and
13.1.2.3 Integrated intensity ratio values if required or available (see 14.2.114.4.2.1).
13.1.3 Verification of Equipment Used:
13.1.3.1 Current Test Method E915 results including date;
13.1.3.2 Current measurement results of stress free standard other than 13.1.3.1 (that is, single daily measurement); and
13.1.3.3 Current measurement results of nonzero known residual stress proficiency reference sample if available.
13.1.4 Measurement Parameters:
13.1.4.1 Equipment used including manufacturer and model;
13.1.4.2 Goniometer mode;
13.1.4.3 Goniometer radius;
13.1.4.4 Software and version used for goniometer control, data acquisition, and data processing;
13.1.4.5 Target and wavelength used, for example, Cr Kα 2.289 70 [Angstroms];
13.1.4.6 Target power used, for example, 30.00 mA × 30.00 kV = 900 W = 69 % (percent of maximum power);
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13.1.4.7 Filters used and whether the filter is lo
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