Designation: E2860 − 12Standard Test Method forResidual Stress Measurement by X-Ray Diffraction forBearing Steels1This standard is issued under the fixed designation E2860; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (´) indicates an editorial change since the last revision or reapproval.INTRODUCTIONThe measurement of residual stress using X-ray diffraction (XRD) techniques has gained muchpopularity in the materials testing field over the past half century and has become a mandatory test formany production and prototype bearing components. However, measurement practices have evolvedover this time period. With each evolutionary step, it was discovered that previous assumptions weresometimes erroneous, and as such, results obtained were less reliable than those obtained usingstate-of-the-art XRD techniques. Equipment and procedures used today often reflect different periodsin this evolution; for example, systems that still use the single- and double-exposure techniques as wellas others that use more advanced multiple exposure techniques can all currently be found inwidespread use. Moreover, many assumptions made, such as negligible shear components andnon-oscillatory sin2ψ distributions, cannot safely be made for bearing materials in which the demandfor measurement accuracy is high. The use of the most current techniques is, therefore, mandatory toachieve not only the most reliable measurement results but also to enable identification and evaluationof potential measurement errors, thus paving the way for future developments.1. Scope1.1 This test method covers a procedure for experimentallydetermining macroscopic residual stress tensor components ofquasi-isotropic bearing steel materials by X-ray diffraction(XRD).1.2 This test method provides a guide for experimentallydetermining stress values, which play a significant role inbearing 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 stresseffects;1.3.4 Monitoring effects of surface treatments such as sandblasting, shot peening, and honing;1.3.5 Tracking of component life and rolling contact fatigueeffects;1.3.6 Failure analysis;1.3.7 Relaxation of residual stress; and1.3.8 Other residual-stress-related issues that potentiallyaffect bearings.1.4 Units—The values stated in SI units are to be regardedas standard. No other units of measurement are included in thisstandard.1.5 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:2E6 Terminology Relating to Methods of Mechanical TestingE7 Terminology Relating to MetallographyE915 Test Method for Verifying the Alignment of X-RayDiffraction Instrumentation for Residual Stress Measure-mentE1426 Test Method for Determining the Effective ElasticParameter for X-Ray Diffraction Measurements of Re-sidual Stress1This test method is under the jurisdiction of ASTM Committee E28 onMechanical Testing and is the direct responsibility of Subcommittee E28.13 onResidual Stress Measurement.Current edition approved April 1, 2012. Published May 2012. DOI: 10.1520/E2860–12.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at

[email protected] For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States12.2 ANSI Standards:3N43.2 Radiation Safety for X-ray Diffraction and Fluores-cence Analysis EquipmentN43.3 For General Radiation Safety—Installations UsingNon-Medical X-Ray and Sealed Gamma-Ray Sources,Energies Up to 10 MeV3. Terminology3.1 Definitions—Many of the terms used in this test methodare defined in Terminologies E6 and E7.3.2 Definitions of Terms Specific to This Standard:3.2.1 interplanar spacing, n—perpendicular distance be-tween adjacent parallel atomic planes.3.2.2 macrostress, n—average stress acting over a region ofthe test specimen containing many gains/crystals/coherentdomains.3.3 Abbreviations:3.3.1 ALARA—As low as reasonably achievable3.3.2 FWHM—Full width half maximum3.3.3 LPA—Lorentz-polarization-absorption3.3.4 MSDS—Material safety data sheet3.3.5 XEC—X-ray elastic constant3.3.6 XRD—X-ray diffraction3.4 Symbols:1⁄2 S2{hkl}= X-ray elastic constant of quasi-isotropic materialequal to11νEeff$hkl%αL= Linear thermal expansion coefficientβ = Angle between the incident beam and σ33or surfacenormal on the σ33σ11planeχ = Angle between the σφ+90°direction and the normal to thediffracting planeχm= Fixed χ offset used in modified-chi moded = Interplanar spacing between crystallographic planes;also called d-spacingdo= Interplanar spacing for unstressed materiald = Perpendicular spacing∆d = Change in interplanar spacing caused by stressεij= Strain component i, jE = Modulus of elasticity (Young’s modulus)Eeff{hkl}= Effective elastic modulus for X-ray measurementsµ = Attenuation coefficientη = Rotation of the sample around the measuring directiongiven by φ and ψ or χ and βω or Ω = Angle between the specimen surface and incidentbeam when χ =0°φ = Angle between the σ11direction and measurement di-rection azimuth, see Fig. 1“hkl” = Miller indicesσij= Normal stress component i, js1{hkl}= X-ray elastic constant of quasi-isotropic materialequal to2νEeff$hkl%τij= Shear stress component i, jθ = Bragg angleν = Poisson’s ratioxMode= Mode dependent depth of penetrationψ = Angle between the specimen surface normal and thescattering vector, that is, normal to the diffracting plane, seeFig. 14. Summary of Test Method4.1 A test specimen is placed in a XRD goniometer alignedas per Test Method E915.4.2 The diffraction profile is collected over three or moreangles 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 eachXRD peak profile.4.5 The d-spacings are calculated from the peak positionsvia Bragg’s law.4.6 The d-spacing values are plotted versus their sin2ψ orsin2β values, and the residual stress is calculated using Eq 4 orEq 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 thereported results.4.8.1 Depth of penetration correction (see 12.12) and4.8.2 Relaxation as a result of material removal correction(see 12.14).3Available from American National Standards Institute (ANSI), 25 W. 43rd St.,4th Floor, New York, NY 10036, http://www.ansi.org.FIG. 1 Stress Tensor ComponentsE2860 − 1225. Significance and Use5.1 This test method covers a procedure for experimentallydetermining macroscopic residual stress tensor components ofquasi-isotropic bearing steel materials by XRD. Here the stresscomponents are represented by the tensor σijas shown in Eq 1(1,4p. 40). The stress strain relationship in any direction of acomponent is defined by Eq 2 with respect to the azimuthphi(φ) and polar angle psi(ψ) defined in Fig. 1 (1, p. 132).σij5Fσ11τ21τ31τ12σ22τ32τ13τ23σ33Gwhere τij5 τji(1)εφψ$hkl%512s2$hkl%@σ11cos2φ sin2ψ1σ22sin2φ sin2ψ1σ33cos2ψ#112s2$hkl%@τ12sin~2φ! sin2ψ1τ13cosφsin~2ψ!1τ23sinφsin~2ψ!#1s1$hkl%@σ111σ221σ33# (2)5.1.1 Alternatively, Eq 2 may also be shown in the follow-ing arrangement (2, p. 126):εφψ$hkl%512s2$hkl%@σ11cos2φ1τ12sin~2φ!1σ22sin2φ 2 σ33# sin2ψ112s2$hkl%σ332 s1$hkl%@σ111σ221σ33#112s2$hkl%@τ13cosφ1τ23sinφ#sin~2ψ!5.2 Using XRD and Bragg’s law, interplanar strain measure-ments are performed for multiple orientations. The orientationsare selected based on a modified version of Eq 2, which isdictated by the mode used. Conflicting nomenclature may befound in literature with regard to mode names. For example,what may be referred to as a ψ (psi) diffractometer in Europemay be called a χ (chi) diffractometer in North America. Thethree modes considered here will be referred to as omega, chi,and modified-chi as described in 9.5.5.3 Omega Mode (Iso Inclination) and Chi Mode (SideInclination)—Interplanar strain measurements are performed atmultiple ψ angles along one φ azimuth (let φ = 0°) (Figs. 2 and3), reducing Eq 2 to Eq 3. Stress normal to the surface (σ33)isassumed to be insignificant because of the shallow depth ofpenetration of X-rays at the free surface, reducing Eq 3 to Eq4. Post-measurement corrections may be applied to account forpossible σ33influences (12.12). Since the σijvalues will remainconstant for a given azimuth, the s1{hkl}term is renamed C.εφψ$hkl%512s2$hkl%@σ11sin2ψ1σ33cos2ψ#112s2$hkl%@τ13sin~2ψ!#1s1$hkl%@σ111σ221σ33# (3)εφψ$hkl%512s2$hkl%@σ11sin2ψ1τ13sin~2ψ!#1C (4)5.3.1 The measured interplanar spacing values are con-verted to strain using Eq 24, Eq 25,orEq 26. Eq 4 is used tofit the strain versus sin2ψ data yielding the values σ11, τ13, andC. The measurement can then be repeated for multiple phiangles (for example 0, 45, and 90°) to determine the full4The boldface numbers in parentheses refer to the list of references at the end ofthis standard.FIG. 2 Omega Mode Diagram for Measurement in σ11DirectionE2860 − 123stress/strain tensor. The value, σ11, will influence the overallslope of the data, while τ13is related to the direction and degreeof elliptical opening. Fig. 4 shows a simulated d versus sin2ψprofile for the tensor shown. Here the positive 20-MPa τ13stress results in an elliptical opening in which the positive psirange opens upward and the negative psi range opens down-ward. A higher τ13value will cause a larger elliptical opening.Anegative 20-MPa τ13stress would result in the same ellipticalopening only the direction would be reversed with the positivepsi range opening downwards and the negative psi rangeopening upwards as shown in Fig. 5.5.4 Modified Chi Mode—Interplanar strain measurementsare performed at multiple β angles with a fixed χ offset,χm(Fig. 6). Measurements at various β angles do not provide aconstant φ angle (Fig. 7), therefore, Eq 2 cannot be simplifiedin the same manner as for omega and chi mode.5.4.1 Eq 2 shall be rewritten in terms of β and χm. Eq 5 and6 are obtained from the solution for a right-angled sphericaltriangle (3).ψ 5 arccos~ cos βcosχm! (5)NOTE 1—Stress matrix is rotated 90° about the surface normal compared to Fig. 2 and Fig. 14.FIG. 3 Chi Mode Diagram for Measurement in σ11DirectionFIG. 4 Sample d (2θ) Versus sin2ψ Dataset with σ11= -500 MPa and τ13= +20 MPaE2860 − 124φ 5 arccosSsin βcosχmsin ψD(6)5.4.2 Substituting φ and ψ in Eq 2 with Eq 5 and 6 (seeX1.1), we get:εβχm$hkl%512s2$hkl%@σ11sin2β cos2χm1σ22sin2χm1σ33cos2β cos2χm#112s2$hkl%@τ12sinβsin~2χm!1τ13sin~2β! cos2χm1τ23cosβsin~2χm!#1s1$hkl%@σ111σ221σ33# (7)5.4.3 Stress normal to the surface (σ33) is assumed to beinsignificant because of the shallow depth of penetration ofX-rays at the free surface reducing Eq 7 to Eq 8. Post-measurement corrections may be applied to account for pos-sible σ33influences (see 12.12). Since the σijvalues and χmwillremain constant for a given azimuth, the s1{hkl}term isrenamed C, and the σ22term is renamed D.εβχm$hkl%512s2$hkl%@σ11sin2β cos2χm1D#112s2$hkl%@τ12sinβsin~2χm!1τ13sin~2β! cos2χm1τ23cosβsin~2χm!#1C (8)FIG. 5 Sample d (2θ) Versus sin2ψ Dataset with σ11= -500 MPa and τ13= -20 MPaFIG. 6 Modified Chi Mode Diagram for Measurement in σ11DirectionE2860 − 1255.4.4 The σ11influence on the d versus sin2β plot is similarto omega and chi mode (Fig. 8) with the exception that theslope shall be divided by cos2χm. This increases the effective1⁄2s2{hkl}by a factor of 1/cos2χmfor σ11.5.4.5 The τijinfluences on the d versus sin2β plot are morecomplex and are often assumed to be zero (3). However, thismay not be true and significant errors in the calculated stressmay result. Figs. 9-13 show the d versus sin2β influences ofindividual shear components for modified chi mode consider-ing two detector positions (χm= +12° and χm= -12°). Compo-nents τ12and τ13cause a symmetrical opening about the σ11slope influence for either detector position (Figs. 9-11);therefore, σ11can still be determined by simply averaging thepositive and negative β data. Fitting the opening to the τ12andτ13terms may be possible, although distinguishing between thetwo influences through regression is not normally possible.5.4.6 The τ23value affects the d versus sin2β slope in asimilar fashion to σ11for each detector position (Figs. 12 and13). This is an unwanted effect since the σ11and τ23influencecannot be resolved for one χmposition. In this instance, the τ23shear stress of -100 MPa results in a calculated σ11value of-472.5 MPa for χm= +12° or -527.5 MPa for χm= -12°, whilethe actual value is -500 MPa. The value, σ11can still bedetermined by averaging the β data for both χmpositions.5.4.7 The use of the modified chi mode may be used todetermine σ11but shall be approached with caution using oneχmposition because of the possible presence of a τ23stress. Thecombination of multiple shear stresses including τ23results inincreasingly complex shear influences. Chi and omega modeare preferred over modified chi for these reasons.6. Apparatus6.1 A typical X-ray diffractometer is composed of thefollowing main components:6.1.1 Goniometer—An angle-measuring device responsiblefor the positioning of the source, detectors, and sample relativeto each other.6.1.2 X-Ray Source—There are generally three X-raysources used for XRD.6.1.2.1 Conventional Sealed Tube—This is by far the mostcommon found in XRD equipment. It is identified by its anodetarget element such as chromium (Cr), manganese (Mn), orcopper (Cu). The anode is bombarded by electrons to producespecific X-ray wavelengths unique to the target element.FIG. 7 ψ and φ Angles Versus β Angle for Modified Chi Mode with χm= 12°FIG. 8 Sample d (2θ) Versus sin2β Dataset with σ11= -500 MPaE2860 − 1266.1.2.2 Rotating Anode Tube—This style of tube offers ahigher intensity than a conventional sealed tube.6.1.2.3 Synchrotron—Particle accelerator that is capable ofproducing a high-intensity X-ray beam.6.1.2.4 Sealed Radioactive Sources—Although not com-monly used, they may be utilized.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 followingmain categories:6.1.4.1 Goniometer control—Responsible for positioning ofthe sample relative to the incident beam and detector(s) in