Designation: E1426 − 14Standard Test Method forDetermining the X-Ray Elastic Constants for Use in theMeasurement of Residual Stress Using X-Ray DiffractionTechniques1This standard is issued under the fixed designation E1426; 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.INTRODUCTIONWhen a crystalline material is strained, the spacing between parallel planes of atoms, ions, ormolecules in the lattice changes. X-Ray diffraction techniques can measure these changes and,therefore, they constitute a powerful means for studying the residual stress state in a body. Thecalculation of macroscopic stresses using lattice strains requires the use of x-ray elastic constants(XEC) which must be empirically determined by x-ray diffraction techniques as described in this testmethod.1. Scope1.1 This test method covers a procedure for experimentallydetermining the x-ray elastic constants (XEC) for the evalua-tion of residual and applied stresses by x-ray diffractiontechniques. The XEC relate macroscopic stress to the strainmeasured in a particular crystallographic direction in polycrys-talline samples. The XEC are a function of the elastic modulus,Poisson’s ratio of the material and the hkl plane selected for themeasurement. There are two XEC that are referred to as1⁄2 S2hkland S1hkl.1.2 This test method is applicable to all x-ray diffractioninstruments intended for measurements of macroscopic re-sidual stress that use measurements of the positions of thediffraction peaks in the high back-reflection region to deter-mine changes in lattice spacing.1.3 This test method is applicable to all x-ray diffractiontechniques for residual stress measurement, including single,double, and multiple exposure techniques.1.4 The values stated in SI units are to be regarded asstandard. The values given in parentheses are mathematicalconversions to inch-pound units that are provided for informa-tion only and are not considered standard.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:2E4 Practices for Force Verification of Testing MachinesE6 Terminology Relating to Methods of Mechanical TestingE7 Terminology Relating to MetallographyE1237 Guide for Installing Bonded Resistance Strain Gages3. Terminology3.1 Definitions:3.1.1 Many of the terms used in this test method are definedin Terminology E6 and Terminology E7.3.2 Definitions of Terms Specific to This Standard:3.2.1 interplanar spacing—the perpendicular distance be-tween adjacent parallel lattice planes.3.2.2 macrostress—an average stress acting over a region ofthe test specimen containing many crystals.3.3 Symbols:3.3.1 x = dummy parameter for Sum(x) and SD(x).3.3.2 c = ordinate intercept of a graph of ∆d versus stress.3.3.3 d = interplanar spacing between crystallographicplanes; also called d-spacing.3.3.4 d0= interplanar spacing for unstressed material.3.3.5 ∆d = change in interplanar spacing caused by stress.1This 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 Dec. 1, 2014. Published March 2015. Originallyapproved in 1991. Last previous edition approved in 2009 as E1426 – 98(2009)ε1.DOI: 10.1520/E1426-14.2For 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 States13.3.6 E = modulus of elasticity.3.3.7 ν = Poisson’s ratio.3.3.8 XEC = x-ray elastic constants for residual stressmeasurements using x-ray diffraction.3.3.9 hkl = Miller indices.3.3.101⁄2 S2hkl= (1+v)/E for an elastically isotropic body.3.3.11 S1hkl=–v/E for an elastically isotropic body.3.3.12 i = measurement index, 1 ≤ i ≤ n.3.3.13 m = slope of a plot of ∆d versus stress.3.3.14 n = number of measurements used to determine slopem.3.3.15 SD(x) = standard deviation of a set of quantities “x”.3.3.16 Sum(x) = sum of a set of quantities “x”.3.3.17 Ti= Ximinus mean of all Xivalues.3.3.18 Xi= i-th value of applied stress.3.3.19 Yi= measurement of ∆d corresponding to Xi.3.3.20 ψ = angle between the specimen surface normal andthe normal to the diffracting crystallographic planes.3.3.21 ϕ = the in-plane direction of stress measurement.3.3.22 ij = in-plane directions of the sample referenceframe.3.3.23 σij= calculated stress tensor terms.3.3.24 εϕψhkl= measured lattice strain tensor terms at agiven ϕψ tilt angle.3.3.25 σA= applied stress.3.3.26 εmax= maximum strain.3.3.27 δmax= maximum deflection.3.3.28 h = specimen thickness.3.3.29 b = width of specimen.3.3.30 AX= cross sectional area of specimen.3.3.31 L = distance between outer rollers on four-point bendfixture.3.3.32 a = distance between inner and outer rollers onfour-point bend fixture.3.3.33 F = known force applied to specimen.3.3.34 εϕψ0= the intercept value for each applied forcenecessary for S1calculation.4. Theory4.1 The sin2ψ method is widely used to measure stresses inmaterials using x-ray diffraction techniques. The governingequation can be written as follows:3,4εϕψhkl512S2hkl~σ11cos2ϕ 1 σ12sin 2 ϕ 1 σ22sin2ϕ 2 σ33!sin2ψ1(1)12S2hklσ331S1hkl~σ111σ221 σ33!112S2hkl~σ13cos ϕ 1 σ23sin ϕ !sin2ψwhere:1⁄2 S2hkland S1hkl= are the XEC.For a body that is elastically isotropic on the microscopicscale,1⁄2 S2hkl= (1+ v)/E and S1hkl=–(v/E) where E and v arethe modulus of elasticity and Poisson’s ratio respectively forthe material for all hkl.4.2 When a uniaxial force is applied along e.g. ϕ =0,Eq 1becomes:εϕψhkl512S2hklσAsin2ψ1S1hklσA(2)where:σA= the applied stress due to the uniaxial force.Therefore:12S2hkl5]2εϕψhkl] ~sin2ψ!·]σA(3)S1hklis embedded in the intersection term for each appliedforce increment and is necessary when performing triaxialmeasurements.5. Summary of Test Method5.1 A test specimen is prepared from a material that isrepresentative of the object in which residual stress measure-ments are to be performed.NOTE 1—If a sample of the same material is available it should be used.5.2 The test specimen is instrumented with an electricalresistance strain gauge, mounted in a location that experiencesthe same stress as the region that will be subsequentlyirradiated with x-rays.5.3 The test specimen is calibrated by loading it in such amanner that the stress, where the strain gauge is mounted, isdirectly calculable, and a calibration curve relating the straingauge reading to the applied stress is developed.5.4 The test specimen is mounted in a loading fixture in anx-ray diffraction instrument and sequentially loaded to severalforce levels.5.4.1 The change in interplanar spacing is measured foreach force level and related to the corresponding stress that isdetermined from the strain gauge reading and the calibrationcurve.5.5 The XEC and its standard deviations are calculated fromthe test results.6. Significance and Use6.1 This test method provides standard procedures forexperimentally determining the XEC for use in the measure-ment of residual and applied stresses using x-ray diffractiontechniques. It also provides a standard means of reporting theprecision of the XEC.6.2 This test method is applicable to any crystalline materialthat exhibits a linear relationship between stress and strain inthe elastic range, that is, only applicable to elastic loading.6.3 This test method should be used whenever residualstresses are to be evaluated by x-ray diffraction techniques andthe XEC of the material are unknown.7. Apparatus7.1 Any x-ray diffraction instrument intended for the mea-surement of residual macrostress that employs measurementsof the diffraction peaks that are, ideally and for best accuracy,in the high back-reflection region may be used, including filmcamera types, diffractometers, and portable systems.3Evenschor P.D., Hauk V. Z., Metallkunde, 1975, 66 pp. 167–168.4Dölle H., J. Appl. Cryst, 1979, 12, pp. 489-501.E1426 − 1427.2 A loading fixture is required to apply a force to the testspecimen while it is being irradiated in the x-ray diffractioninstrument.7.2.1 The fixture shall be designed such that the surfacestress applied by the fixture shall be uniform over the irradiatedarea of the specimen.7.2.2 The fixture shall maintain the irradiated surface of thespecimen at the exact center of rotation of the x-ray diffractioninstrument throughout the test with sufficient precision toprovide the desired levels of precision and bias in the mea-surements to be performed.7.2.3 The fixture may be designed to apply tensile orbending forces. A four-point bending technique such as thatdescribed by Prevey5is most commonly used.7.3 Electrical resistance strain gauges are mounted upon thetest specimen to enable it to be accurately stressed to knownlevels.8. Test Specimens8.1 Test specimens should be fabricated from material withmicrostructure as nearly the same as possible as that in thematerial in which residual stresses are to be evaluated. It ispreferred for superior results to use the same material with afine grain structure and minimum cold work on the surface tominimize measurement errors.8.2 For use in tensile or four-point bending fixtures, speci-mens should be rectangular in shape.8.2.1 The length of tensile specimens, between grips, shallbe not less than four times the width, and the width-to-thickness ratio shall not exceed eight.8.2.2 For use in four-point bending fixtures, specimensshould have a length-to-width ratio of at least four. Thespecimen width should be sufficient to accommodate straingauges (see 8.5) and the width-to-thickness ratio should begreater than one and consistent with the method used tocalculate the applied stresses in 9.1.NOTE 2—Nominal dimensions often used for specimens for four-pointbending fixtures are 10.2 × 1.9 × 0.15 cm (4.0 × 0.75 × 0.06 in.).8.3 Tapered specimens for use in cantilever bendingfixtures, and split-ring samples, are also acceptable.8.4 Specimen surfaces may be electropolished or as-rolledsheet or plate.8.5 One or more electrical resistance strain gauges areaffixed to the test specimen in accordance with Guide E1237.The gauge(s) shall be aligned parallel to the longitudinal axisof the specimen, and should be mounted on a region of thespecimen that experiences the same strain as the region that isto be irradiated. The gauge(s) should be applied to theirradiated surface of the beam either adjacent to, or on eitherside of, the irradiated area in order to minimize errors due tothe absence of a pure tensile or bending force.NOTE 3—In the case of four-point bending fixtures the gauge(s) shouldbe placed well inside the inner span of the specimen in order to minimizethe stress concentration effects associated with the inner knife edges of thefixture.9. Calibration9.1 Calibrate the instrumented specimen using forces ap-plied by dead weights or by a testing machine that has beenverified according to Practices E4. Use a loading configurationthat produces statistically determinate applied stresses in theregion where the strain gauges are mounted and where x-raydiffraction measurements will be performed (that is, such thatstresses may be calculated from the applied forces and thedimensions of the specimen and the fixture). In the case of purebending using a four-point bending apparatus, the strain gaugemay be calibrated by measurement of applied strains viadeflection of the specimen and calculated using the followingequation:εmax5δmax12h3L22 4a2(4)where:εmax= maximum applied strain to the strain gaugeδmax= maximum applied deflectionh = specimen thicknessL = distance between outer rollers on four-point bendfixturea = distance between inner and outer rollers on each sideof the four-point bend fixtureIf the modulus of elasticity E for the material being tested isknown, the applied stress on the specimen may then becalculated using Hooke’s law.σA5 Eεmax(5)If the modulus of elasticity E for the material being tested isnot known, the applied stress on the specimen may becalculated using known applied forces in the case wherebending is being used:σA53Fabh2(6)where:b = the width of the specimen, andF = known total force applied by the rollers to the specimenFor uniaxial loading, if the modulus of elasticity E for thematerial being tested is not known, the applied stress on thespecimen may be calculated using known applied forces:σA5 F⁄Ax(7)where:Ax= cross sectional area of specimen9.2 Pre-stress the specimen by loading to a level of approxi-mately 75 % of the force that is calculated to produce amaximum applied stress equal to the nominal yield strength ofthe material, then unload. This will minimize drift in thegauges and creep in the strain gauge adhesive during thesubsequent testing procedure.9.3 Apply forces in increasing sequence at increments ofapproximately 10, 20, 30, 40, 50, 60, and 70 % of the force thatwould produce a maximum applied stress equal to the nominal5Prevey, P. S., “A Method of Determining the Elastic Properties of Alloys inSelected Crystallographic Directions for X-Ray Diffraction Residual StressMeasurement,” Advances in X-Ray Analysis 20, 1977, pp. 345–354.E1426 − 143yield strength of the material (a minimum of five (5) equallyspaced points is recommended).9.4 At each force level record strain gauge readings andcalculate the applied stress.9.5 Repeat measurement at 70, 60, 50, 40, 30, 20, and 10%of the yield strength, record strain gauge readings and calculatethe applied stress at each level (a minimum of five (5) equallyspaced points is recommended).9.6 Repeat 9.3 – 9.5 at least once to ensure reproducibilityof the measurement data.9.7 Examine the data for repeatability and linearity. Devia-tions from either may indicate the failure of a strain gaugebond, stressing beyond the proportional limit of the material, oran imperfect loading configuration. If the deviations exceed theacceptable degree of uncertainty in the subsequent measure-ments of residual stress, the source of the deviations should belocated and corrected before proceeding further.9.8 If the repeatability and the linearity of the data areacceptable, plot a graph of strain gauge reading versus appliedstress, draw a straight line through the data, and extrapolate itdown to zero applied stress and up to 75 % of the nominal yieldstrength of the material. This is the calibration curve. (See Fig.1.)NOTE 4—For bending specimens, the