BSSM WorkshopPART IIThe sin2 Method Using Laboratory X-RaysJudith ShackletonSchool of Materials, University of ManchesterThe sin2 Method What are We Measuring? We measure the ELASTIC Strain. We measure the ELASTIC Strain. We can determineWe can determine Magnitude of the stress,Magnitude of the stress, Its directionIts direction Its natureIts nature Compressive or tensileCompressive or tensile We use the planes of the crystal lattice as an atomic We use the planes of the crystal lattice as an atomic scale “strain gauge”scale “strain gauge”The sin2 Method How Does it Work?We measure We measure STRAINSTRAIN ( ( ) ) not not STRESS STRESS ( ( ) ) We We CALCULTE STRESSCALCULTE STRESS from the from the STRAINSTRAIN & the & the ELASTIC CONSTANTSELASTIC CONSTANTS We use the planes We use the planes d dhklhkl , of the crystal lattice as a , of the crystal lattice as a strain gaugestrain gauge We can measure the change in We can measure the change in d d-spacing, -spacing, d dStrain = Strain = = = d/dd/dChanges in d-spacing with StressConsider a bar which is in tensionConsider a bar which is in tension The The d d-spacings of the planes normal to the applied stress increase, as -spacings of the planes normal to the applied stress increase, as the stress is tensilethe stress is tensile The The d d-spacings of the planes parallel to the applied stress decrease, due -spacings of the planes parallel to the applied stress decrease, due to Poisson strainto Poisson strainMeasuring Elastic & InelasticStrainPrimarily we are measuring macro stressesPrimarily we are measuring macro stresses This is a uniform displacement of the lattice planesThis is a uniform displacement of the lattice planes These cause a VERY SMALL shift in the position, the These cause a VERY SMALL shift in the position, the Bragg angle 2Bragg angle 2, , of the reflection & we can measure of the reflection & we can measure this (Only Just!)this (Only Just!)Inelastic stresses cause peak broadening, which Inelastic stresses cause peak broadening, which can be measured. This is an extensive subject, not can be measured. This is an extensive subject, not covered here.covered here.Which Materials Can We Measure?Works on any poly-crystalline solid which Works on any poly-crystalline solid which gives a high angle Bragg reflectiongives a high angle Bragg reflection MetalsMetals Ceramics (not easy!)Ceramics (not easy!) Multi-phase materials Multi-phase materials Not usually applied to polymers, as no suitable Not usually applied to polymers, as no suitable reflections, can add a metallic powder, reported reflections, can add a metallic powder, reported in the literaturein the literatureWhy use the sin2 MethodThe AdvantagesMost ImportantA stress free A stress free d d-spacing is NOT required for the -spacing is NOT required for the bi-axial case which is almost always usedbi-axial case which is almost always usedOther advantagesOther advantages Low cost (compared with neutrons & synchrotrons, Low cost (compared with neutrons & synchrotrons, but not hole drilling)but not hole drilling) Non-destructive, unlike hole drillingNon-destructive, unlike hole drilling Easy to do & fairly fool proof (if you are careful!)Easy to do & fairly fool proof (if you are careful!)DisadvantagesMost ImportantSurface method only, X-ray beam penetration Surface method only, X-ray beam penetration depth 10 to 20 microns, at bestdepth 10 to 20 microns, at bestFor depth profiling must electro-polish, gives For depth profiling must electro-polish, gives 1-1.5mm1-1.5mmOther DisadvantagesOther Disadvantages Affected by grain size, texture (preferred Affected by grain size, texture (preferred orientation) & surface roughnessorientation) & surface roughness Doesnt work on amorphous materials (obviously!)Doesnt work on amorphous materials (obviously!)Basic TheoryConsider a unit cube (quite a big one!) embedded in a componentNotation, (ij) the stress component acting on face i in direction (parallel to axis) jBasic Theory The normal stresses act normal to the cube faces & the two The normal stresses act normal to the cube faces & the two subscripts are the same subscripts are the same e.g. e.g. (22)(22) The shear stresses (twisting forces) act parallel to the cube The shear stresses (twisting forces) act parallel to the cube faces & the two subscripts are differentfaces & the two subscripts are different e.g. e.g. (31) (31) or in the general case or in the general case (ij)(ij) We measure normal stresses & shear stresses, but thats not We measure normal stresses & shear stresses, but thats not what we want, we dont get all of the information! Why?what we want, we dont get all of the information! Why?Basic TheoryNormal StressesFrom elastic theory of isotropic materials, the 3 normal strains are given by, 1111 = = 1 1 1111 - - ( ( 2222 + + 3333) ) E E 2222 = = 1 1 2222 - - ( ( 3333 + + 1111) ) E E 3333 = = 1 1 3333 - - ( ( 1111 + + 2222) ) E E The strain in The strain in anyany direction is a function of the direction is a function of the stressstress in the in the others!. Ideally, we should measure more than one others!. Ideally, we should measure more than one directiondirectionPrincipal StressesWe should measure We should measure more than one directionmore than one direction to get to get a complete picture of the stress in the component a complete picture of the stress in the component If we measure 3 directions or more we can If we measure 3 directions or more we can calculate the PRINCIPAL STRESSESS, these are calculate the PRINCIPAL STRESSESS, these are the directions on which no shear stress actsthe directions on which no shear stress actsWe do this by rotating the sample through an angle We do this by rotating the sample through an angle , in its own plane, exact details & diagrams later, in its own plane, exact details & diagrams laterHow the Sin2 Method WorksSample in “Bragg Condition”Diffraction Diffraction vector, normal to vector, normal to sample surfacesample surfacednWe measure the We measure the d d- -spacing with the spacing with the angle of incidence angle of incidence ( ( ) ) & the angle of & the angle of reflection of the X-reflection of the X-ray beam (with ray beam (with respect to the respect to the sample surface) sample surface) equal. These planes equal. These planes are parallel to the are parallel to the free surface & free surface & unstressed, but not unstressed, but not unstrainedunstrained Also called focussed Also called focussed geometrygeometryHow the Sin2 Method WorksDiffraction vector, titled with respect to sample surfaceTilt the sample through an angle and measure the d-spacing again. These planes are not parallel to the free surface. Their d-spacing is changed by the stress in the sample.dDefocused geometryHow the Sin2 Method WorksWe tilt the sample through an angle psi,We tilt the sample through an angle psi, to to measure magnitude the normal & shear stressesmeasure magnitude the normal & shear stresses We use a range of values of We use a range of values of (called offsets) for (called offsets) for example, from 0 to 45example, from 0 to 45 in steps of 5 in steps of 5 NEVERNEVER use the “Double Exposure Method” which use the “Double Exposure Method” which uses just one uses just one offsetsoffsets. Not enough data points!. Not enough data points!We rotate the the sample through an angle, We rotate the the sample through an angle, to to determine the directions of the principle stressesdetermine the directions of the principle stressesNo Stress Free d-Spacing Needed The Approximation The depth of penetration of the X-ray beam in the sample The depth of penetration of the X-ray beam in the sample is small, typically 20is small, typically 20 We can say that there is no stress component We can say that there is no stress component perpendicular to the sample surface, that is perpendicular to the sample surface, that is 3333 = 0 = 0 We can use the We can use the d d-spacing measured at -spacing measured at = 0 as the stress = 0 as the stress free free d d-spacing-spacing This is the d-spacing of the planes parallel to the sample surfaceThis is the d-spacing of the planes parallel to the sample surface A reasonable approximation! The error is 2%, certainly A reasonable approximation! The error is 140 2 The change in The change in d d-spacing, due to strain, is very small, -spacing, due to strain, is very small, typically in the third decimal placetypically in the third decimal place The dispersion of the diffraction pattern is much greater The dispersion of the diffraction pattern is much greater at high 2at high 2 angles. angles. The small changes in d-spacing The small changes in d-spacing can only be detected at angles can only be detected at angles 125 2 125 2 Choice of X-Ray Tube(1) DispersionAn ExampleIf we have a reflection from ferrite 211 at 156 2. Using radiation from a chromium anode X-ray tube of wavelength 2.2897If we introduce a stress of 200 MPa, given Youngs modulus of 220 GPa, what is the change in the 2 angle? Answer, the new 2Answer, the new 2 angleis155.51 angleis155.51 The difference is 0.48The difference is 0.48 NOT MUCH! NOT MUCH!Choice of X-Ray Tube(2) Sample FluorescenceIf the K-1 component of the incident X-ray beam causes the sample emit its own fluorescent X-rays, DO NOT USE ITX-ray penetration depth will be very small 5 X-ray penetration depth will be very small 5 microns & microns & NOT representative of the bulk NOT representative of the bulkPeak to background ratio will be terriblePeak to background ratio will be terribleMay damage sensitive detectorsMay damage sensitive detectorsChoice of X-Ray Tube(3) Choice of Crystallographic PlaneFor accurate comparison with other peoples For accurate comparison with other peoples data CHECK which planes have been used data CHECK which planes have been used historically!historically! Measurements made on planes with different Miller Measurements made on planes with different Miller hkl indices are not usually comparable.hkl indices are not usually comparable.If the sample is textured (preferred orientation) If the sample is textured (preferred orientation) select a set of planes with a high multiplicityselect a set of planes with a high multiplicityChoice of Measurement Conditions: SummaryAsk someone who has experience with that particular materialDont re-invent the wheelChoose radiation type carefullyAvoid X-ray tubes which cause K -1 fluorescenceLots of “tricks of the trade” see the NPL Good Practice Guide for Residual Stress Measurements using the sin2 MethodData CollectionPositioning the SampleSample must be centre of rotation of the Sample must be centre of rotation of the goniometer, most instruments have depth gauge goniometer, most instruments have depth gauge or a pointeror a pointerBe careful that the sample is as flat as possible, Be careful that the sample is as flat as possible, bent samples will give artificial shear stressesbent samples will give artificial shear stressesFor curved and uneven samples restrict the For curved and uneven samples restrict the irradiated areairradiated area Hoop direction, Spot size R/4, where R= radius of Hoop direction, Spot size R/4, where R= radius of curvaturecurvature Axial direction, Spot size R/2Axial direction, Spot size 1000 count at the top of the statistics, need 1000 count at the top of the peak if possiblepeak if possibleData Processing ALWAYS CHECK THIS STAGEALWAYS CHECK THIS STAGE Need a program with good graphicsNeed a program with good graphics Stages in the data processingStages in the data processing Background strippingBackground stripping K-K- 2 stripping (only if K2 stripping (only if K- - 2 peak is visible)2 peak is visible) Lorentz Polarisation CorrectionLorentz Polarisation Correction Peak fitting to locate maximumPeak fitting to locate maximum Critical Stage, check the results on the screen.Critical Stage, check the results on the screen. A variety of peak models are available most of which will work. A variety of peak models are available most of which will work. Usually use Gaussian, dont use parabolaUsually use Gaussian, dont use parabola Good quality data can be fitted with most models, this is a Good quality data can be fitted with most models, this is a good test!good test!How Precise are the Results Generally theres a lot of scatter on sinGenerally theres a lot of scatter on sin2 2 plots! plots! The error bars printed out by most PCs are just the standard deviation The error bars printed out by most PCs are just the standard deviation of the points from the fitted line and tend under estimate the errorsof the points from the fitted line and tend under estimate the errors Large error bars are not necessarily unacceptable and Large error bars are not necessarily unacceptable and are due to,are due to, Texture, large grain size, poor peak fitting etcTexture, large grain size, poor peak fitting etc For example, 200 For example, 200 50MPa is quite normal 50MPa is quite normal Check the peaks on the screen!Check the peaks on the screen! Values of less than Values of less than 50MPa, can usually be thought of 50MPa, can usually be thought of as zero, this depends on the instrumentas zero, this depends on the instrument To confirm such low readings make several measurements & see if To confirm such low readings make several measurements & see if they all come out with the same sign (i.e. all compressive)they all come out with the same sign (i.e. all compressive)Instrument Misalignment- Omega-2 misalignments-Omega- misalignments (side inclination method)Instrument misalignment causes, Shifts in the positions of the reflections and incorrect stress valuesThe positive and negative measurements give different peak positions, this is called splittingWe must measure at least two standards to verify that the machine is working correctlyInstrument Misalignment- Recommended StandardsA stress free powderNot an easy thing to makeBeware stresses due to filing and oxidationCan be combined with resin for ease of useA stressed standardBe careful, always measure in the same directionShot peened samples are goodUsually verified by Round-Robin tests No certified standards (?)One set for each tube anodeA typical Example of a Stress Profile in a Shot Peened Sample A shot peened surface, depth profiled by Electro-A shot peened surface, depth profiled by Electro-polishingpolishing Problems! Good one, Good one, the material the material has a small has a small grain size grain size ( ( 100 100 ) ) is isotropic, is isotropic, rather than rather than textured & textured & theres no theres no shear stress. shear stress. Ideal!Ideal!Texture, the “wiggle” Texture, the “wiggle” Our sample is not Our sample is not isotropicisotropicShear Stress, the positive & negative plots splitShear Stressessin2 Splitting Positive and negative Positive and negative give different results when a give different results when a shear stress is present (or sample is not correctly shear stress is present (or sample is not correctly positioned, always check!)positioned, always check!) Function of the direction of the measurementFunction of the direction of the measurementConclusionsThe sinThe sin2 2 method works well if you are carefulmethod works well if you are careful Check to see whats been done by othersCheck to see whats been done by others Dont reinvent the wheelDont reinvent the wheel Choose you X-ray tube with careChoose you X-ray tube with care Position the sample carefullyPosition the sample carefully Think about the directions you wish to measureThink about the directions you wish to measure Measure a sufficient range of 2Measure a sufficient range of 2 & count for a & count for a sufficient timesufficient time Check you peak fittingCheck you peak fitting Do the results make sense?Do the results make sense?Thank You & Happy Landings!