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Mechanical and Materials Engineering Faculty Publications Mechanical and Materials Engineering

1982

Measurement of Neck Development in Tensile Testing Using Projection Moire

Raghavan Srinivasan Wright State University - Main Campus, [email protected]

C. S. Hartley

C. Raju

J. Clave

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Repository Citation Srinivasan, R., Hartley, C. S., Raju, C., & Clave, J. (1982). Measurement of Neck Development in Tensile Testing Using Projection Moire. Optical Engineering, 21 (4), 655-662. https://corescholar.libraries.wright.edu/mme/123

This Article is brought to you for free and open access by the Mechanical and Materials Engineering at CORE Scholar. It has been accepted for inclusion in Mechanical and Materials Engineering Faculty Publications by an authorized administrator of CORE Scholar. For more information, please contact [email protected]. Measurement ofof neck development in tensile testing using projection moire

R. Srinivasan Abstract. TheThe projectionprojection moiremoire techniquetechnique affordsaffords anan accurateaccurate methodmethod of State University of New YorkYork measuring the geometry of the neckednecked region inin aa tensiletensile specimen.specimen. Correc-Correc­ DeptDept. ofof MaterialsMaterials Science and Engineering tions toto the measured stress, necessary toto accountaccount forfor thethe triaxialitytriaxiality introducedintroduced Stony Brook, New York 11794 by thethe neck, requirerequire valuesvalues forfor thethe radius of curvature of the specimen at the minimum section. The spacing of fringes resultingresulting fromfrom thethe interferenceinterference C. S. Hartley between a grating projected on the specimen surface andand aa master grating notnot Louisiana StateState UniversityUniversity only provide information which permitspermits calculationcalculation ofof thethe radiusradius butbut alsoalso Dept. ofof MechanicalMechanical Engineering demonstrates any changes in symmetry resulting fromfrom plasticplastic deformation.deformation. Baton Rouge, LouisianaLouisiana 7080370803 Results obtained by testingtesting specimens of copper and mild steel having initially straight andand initially curved curved profilesprofiles indicateindicate that that Bridgman's Bridgman's approach approach providesprovides B. B. Raju a self-consistentself- consistent correctioncorrection forfor triaxiality.triaxiality. However, comparison of incremental J. ClaveClave tests with continuouscontinuous teststests indicatesindicates thatthat thethe developmentdevelopment ofof neckneck geometry may be sensitivesensitive to the strain rate.rate. University of Florida Dept. ofof MaterialsMaterials Science and Engineering Keywords: incoherent optical techniques in experimentalexperimental mechanics; mechanics; projection moire; Gainesville, Florida 32611 plasticity; necking.necking. Optical Engineering 2121(4), (4), 655-662655 -662 (July/August(July /August 1982).

CONTENTS the neck; 2) thethe stress distribution distribution across thethe minimum crosscross section 1. Introduction changes from from aa uniform,uniform, uniaxial statestate of stressstress toto a nonuniform, 2. PrinciplesPrinciples ofof measurement measurement multiaxial statestate of stress.stress. A correction for thethe crosscross sectionsection of the 2.1. Projection moire method deforming regionregion requiresrequires onlyonly an an instantaneousinstantaneous measurement,measurement, but 2.2. TheoryTheory of thethe methodmethod correction forfor triaxiality ofof the the stressstress distribution introducedintroduced by the 2.3. MeasurementMeasurement of profile radiusradius neck demandsdemands more more carefulcareful analysis.analysis. Additional Additional complications maymay 3. ExperimentalExperimental procedureprocedure arise if plastic anisotropy causes causes the specimen shape toto deviatedeviate from from its 3.1. Apparatus initialinitial axisymmetry. 3.2. Calibration of thethe instrument Two approachesapproaches havehave been used to correct stressstress-strain -strain datadata forfor 3.3. ProcedureProcedure necking. Direct methods rely on elimination ofof the the neckneck by machining 4. ResultsResults or cold workingworking specimensspecimens atat various stages of of thethe testtest afterafter flowflow 5. DiscussionDiscussion localizationlocalization hashas occurred.1occurred. 1 "-33 IndirectIndirect measurements utilizeutilize analytical 5.1. Calibration expressions relatedrelated toto the shape ofof thethe specimenspecimen to to correctcorrect thethe appar­appar- 5.2. MeasurementsMeasurements onon specimensspecimens ent stress for for triaxiality.triaxiality.44 "-77 AllAll indirect methodsmethods requirerequire simultaneous 6. ConclusionsConclusions measurements ofof thethe dimensions dimensions ofof thethe minimumminimum cross section normalnormal 7. AcknowledgmentsAcknowledgments to thethe loadingloading axisaxis andand R,R, thethe radiusradius ofof curvaturecurvature ofof the the specimenspecimen 8. AppendixAppendix A:A: generalgeneral equations of the moire fringes profile measured atat thethe minimumminimum section. The most widely used method,method, 9. ReferencesReferences suggested by by Bridgman,Bridgman,66 statesstates that that forfor an axisymmetric specimenspecimen

1. INTRODUCTIONINTRODUCTION aatt = aa { {[1[1 ++ (R/ (R/2a)] 2a)] Znfa [1[l ++ (a(a/2R)]}~ /2R)]}-1, (1) Uniaxial tensile tests of of ductile ductile materialsmaterials oftenoften exhibitexhibit a periodperiod of flow of flow where aatt is thethe correctedcorrected uniform true true axial axial stress stress and and a a is is thethe measured localization, called "necking, "prior to fracture of the specimen. Because called "necking," prior to fracture of the specimen. Because true axial stressstress on the minimum cross cross section,section, respectively;respectively; aa repre-repre­ of this phenomenon, of this phenomenon, aa specimen having having an an initiallyinitially uniformuniform gaugegauge sents thethe radius radius ofof thethe minimumminimum crosscross-sectional -sectional dimension. dimension. AnAn exten­exten- section, suchsuch as as a a right right circularcircular cylinder, developsdevelops a locally reducedreduced sion to specialspecial casescases of nonaxisymmetrynonaxisymmetry hashas beenbeen proposedproposed byby cross section,section, whichwhich decreases in in size size until until failure. failure. Necking introduces introduces Eisenberg.?Eisenberg.7 two important considerationsconsiderations intointo thethe analysisanalysis of tensile deformation: Obtaining accurate experimental values of R presentspresents aa difficultdifficult 1) thethe volumevolume ofof thethe deformingdeforming region decreases from from thatthat of the entireentire problem. MostMost techniquestechniques relyrely onon incrementalincremental tests, tests, betweenbetween whichwhich gauge sectionsection to to a a smaller, smaller, generally generally unknown unknown value value in in the the vicinity vicinity ofof measurements areare made by optical8optical8 oror mechanical mechanical methods.2.3,6,9 methods.2' 3'6'9 Continuous measurements requirerequire calibrationcalibration with anan incrementalincremental Invited Paper 10IO-109 -109 receivedreceived Mar.Mar. 31,1982;31,1982; revisedrevised manuscriptmanuscript received Apr. 14,1982;14,1982; accepted for publication Apr. 14,1982;14, 1982; receivedreceived by Managing Editor Apr.Apr. 26,26, 1982.1982. technique toto ensure accuracy.accuracy.'°10 Finally, choicechoice of anan appropriateappropriate © 1982 SocietySociety of of Photo-Optical Photo -Optical Instrumentation Instrumentation Engineers. analytical correction requires detaileddetailed knowledgeknowledge of of thethe evolution evolution of of

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specimenSpecimen

Projected lide Fig. 2. ImageImage ofof aa specimen:specimen: a) without referencereference grid;grid; b)b) with reference reference grid Projector Carera grid.

Reference grid

Projection axi s Viewing axis

Fig. 1. Schematic diagramdiagram of the projectionprojection moiremoire apparatus.apparatus.

the specimen shape during the test.test. The present work describes anan application of the projection moire technique to determine R and toto verifyverify axialaxial symmetry.symmetry. ThisThis proce-proce­ dure offers aa powerfulpowerful methodmethod ofof obtainingobtaining measurementsmeasurements required for analytical corrections and for quantitative measurementsmeasurements ofof thethe spec-spec­ imen shape. 2.2, PRINCIPLESPRINCIPLES OFOF MEASUREMENTMEASUREMENT 2.1. ProjectionProjection moiremoire methodmethod The projection moire method isis a modification ofof the shadowshadow moiremoire method. Shadow moire requires the grating toto bebe at leastleast as large as the surface being studied, making it inconvenient for largelarge struc-struc­ tures. Projection moiremoire overcomesovercomes this limitationlimitation byby projectingprojecting anan image of aa grid on thethe surface being studied, using a slide projector. This method is, therefore, capable of handling both largelarge andand smallsmall structures. The projection moire method discussed here is aa singlesingle projector, projector, single exposureexposure method. AA slide projector projectsprojects anan imageimage ofof aa gridgrid on the specimenspecimen surface. A camera then forms an imageimage ofof thethe specimen onon aa second grid,grid, the reference grid.grid. Photographic filmfilm Fig. 3.3. Relation between neck profile radiusradius R, spacingspacing of isostathmicisostathmic mounted against the referencereference grid is used to record the final image. planes 6<5 and the distances h1h-, and rfeh2 of of thethe first first twotwo moirernoire fringes from thethe Figure 11 shows thethe schematicschematic drawingdrawing ofof thethe projectionprojection moire setup. minimum crosscross section.section. Moire fringes are seen on the final image of the specimen, as shown in Fig. 2. and thethe specimenspecimen surface.surface. Projection moire has been extensively investigated by others.1 -16 investigated by others. 1 ' ~ 16 ifIf thethe pitches of thethe grids areare smallsmall comparedcompared to the distances of The approach to the derivations in this paper is substantially differ- paper is substantially differ­ thethe camera and projectorprojector fromfrom thethe specimen,specimen, thesethese surfacessurfaces cancan bebe ent from that in other analyses. However,However, thethe resultsresults "are consistantconsistant approximated asas aa setset ofof equallyequally spacedspaced parallelparallel planes,planes, thethe spacingspacing with previous investigations. between which is given by 2.2. TheoryTheory of the method PiPlP P22 d = ClaveClave"17 showed that thethe moiremoire fringesfringes are thethe intersectionintersection lineslines 6 = 1/2 (3)(3) between a set of surfaces given by an equation ofof thethe form*form* p2 +P2+ p2 - 2pi p2cosal2

Ax2+By2+Cxy+Dx+Ey+FAx2 + By2 + Cxy + Dx + Ey + F == 00 (2) where 86 equals the spacing between planes, p,pi andand p2p2 representrepresent pitches of the projectedprojected andand referencereference gratingsgratings atat thethe specimenspecimen surface, and a isis the angle between projection and viewingviewing axes. The *Derivation of equations used in this section comprises Appendix A. effective pitches P]pl and p2p2 dependdepend on thethe pitchespitches of thethe projectedprojected andand

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reference grids, thethe focalfocal lengthslengths of projector and camera lenses,lenses, and the distances of the lenseslenses from thethe specimen.specimen. The above equations assume the grid lineslines to bebe parallel.parallel. AnyAny rotation ofof one grid with respectrespect toto thethe otherother changeschanges thethe valuevalue ofof S,<5, the sensitivitysensitivity of the instrument.instrument. TheThe sensitivitysensitivity ofof thethe measure-measure­ ments, therefore, depends on the actual physical dimensions of the optical setup, specifically the pitches of the grids, the angle between them, and thethe angle between the projection andand viewingviewing axes.

2.3. Measurement of profile radius To measure the profile radius of curvature, thethe specimenspecimen forms thethe target of the projection moire apparatus. TheThe curvedcurved surfacesurface of the a specimen intersectsintersects severalseveral ofof thethe parallelparallel planes resulting in moire fringes asas shown inin Fig. 2(b). Approximating thethe profile of the specimen atat the minimum cross section by an arc ofof aa circle,circle, thethe geometry illustrated inin Fig.Fig. 33 givesgives

hZhv, 2~ - hZhu i 2 1 _ (4) 26

where hjhl and h2h2 are the distances of the first two moire fringesfringes from the minimum cross section.section. 3. EXPERIMENTALEXPERIMENTAL PROCEDURE 3.1. Apparatus The moire optical apparatus consisted of two RonchiRonchi rulings,rulings, a Beseler SlideSlide KingKing 31/431/4 in. X4X4 in.in. lantern slide projector, and aa bellows typetype AreaArca SwissSwiss camera camera mounted mounted onon aa rackrack andand pinion so both lens and ground glassglass screenscreen couldcould bebe movedmoved independently.independently. Pictures were takentaken on Polaroid PN 5555 film using a PolaroidPolaroid 545545 Land film holder mounted betweenbetween thethe groundground glassglass screenscreen ofof thethe camera and the reference grid. The projected grid had 200 lines perper inch. The referencereference gridgrid couldcould bebe changedchanged dependingdepending on the sensitiv­sensitiv- ity required.

3.2. CalibrationCalibration of the instrument The sensitivity ofof thethe instrumentinstrument can be determined usingusing Eq.Eq. (3) if pj,p , Fig. 4. CalibrationCalibration specimen.specimen. Ppz,2, andand a are known. An experimental calibrationcalibration isis alsoalso possiblepossible by observing thethe moiremoire fringefringe patternpattern on specimens of knownknown shape. The latter procedure latter procedure was adopted. TABLE I. Specimens Tested The moire fringe pattern on a specimenspecimen shaped as shownshown inin Fig.Fig. 4 was usedused toto calibratecalibrate thethe instrument.instrument. IfIf thethe diametersdiameters aa andand b of the Initial crosscross-section -section Initial profileprofile specimen at at thethe twotwo endsends ofof the tapered zone are knownknown and thethe Material Specimen no.no. diameter (cm) radius (cm)(cm) number of fringes N inin thisthis regionregion areare counted, the sensitivitysensitivity is simply Copper A 1.27 00 B 1.27 11.15 b-ab -a C 1.25 5.84 6 = (5) 2N Mild steel 1 0.980 oo 2 0.925 7.14 For the example illustratedillustrated inin Fig. 4, the diameters a and b were 3 0.932 5.99 0.599 and 1.1911.191 cmcm respectively, andand 17 fringes were counted on thethe specimen. This gavegave anan experimentalexperimental value of S8 = 0.01741 cm, which the projection moire apparatus toto measuremeasure thethe filefile radius.radius. FiguresFigures 6 was about 10%10% higher than the sensitivity calculated from Eq. (3). the and 7 show the growth of the neck in the two types of specimensspecimens thatthat reasons for this are discusseddiscussed inin Sec.Sec. 5.1.5.1. were tested. Continuous tests werewere also conductedconducted onon samplessamples ofof thethe twotwo 3.3. Procedure materials for comparison withwith thethe interruptedinterrupted tests. tests. Specimens Specimens werewere Incremental tensiontension tests were performed on twotwo materials,materials, corn-com­ tested at a constant truetrue plastic plastic strain strain rate rate of of 0.05 0.05/ / secsec on thethe system mercially pure copper and mildmild steelsteel (0.05%(0.05% C) on a 10,00010,000 Iblb Instron described by Hartley andand Jenkins.10Jenkins. 10 testing machine. Figure 5 showsshows thethe twotwo typestypes ofof specimens specimens used:used: straight specimensspecimens with a uniform initialinitial crosscross sectionsection andand specimensspecimens 4. RESULTSRESULTS with an initial curved profile machined in them. Table II listslists thethe Figure 8 shows the uncorrected flowflow curvescurves ofof twotwo coppercopper speci-speci­ specimens tested. mens, A and B. Specimen B, which had an initialinitial profileprofile radius of The specimens werewere pulledpulled atat aa constant crosshead speedspeed ofof 0.050.05 11.15 cm,cm, supportedsupported a larger stressstress thanthan specimenspecimen A,A, atat allall strains.strains. in.in./min. /min. Load was recordedrecorded on a stripstrip chartchart recorder.recorder. TestsTests werewere Figure 9 shows the corrected flow curves of the threethree coppercopper speci-speci­ interrupted at different stages to measure the diameter at thethe min-min­ mens, and Fig. 1010 shows the corresponding curvescurves forfor thethe mildmild steelsteel imum cross section using vernier calipers. In thethe casecase ofof neckednecked specimens. InIn each case the corrected flowflow curves coincide. specimens, i.e.,i.e., specimensspecimens inin whichwhich thethe cross section was not uni-uni­ Figures 11 andand 12 show the results of the continuous tests. A few form, they were removedremoved fromfrom thethe testingtesting machinemachine andand mounted on points from the interrupted tests tests have have beenbeen markedmarked forfor comparison.comparison.

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Fig. 7.7. GrowthGrowth ofof the the neck neck inin aa profiledprofiled specimenspecimen (specimen(specimen C). Pictures were taken at strainsstrains ofof a)a) 0.00; 0.00; b)b) 0.12; 0.12; c) 0.16;0.16; d) 0.32;0.32; e)e] 0.98;0.98; f) 1.47. Fig. 5.5. Shapes of the twotwo typestypes ofof specimens specimens tested:tested: a)a) Straight Straight-sided -sided (uniformuniform crosscross section)section) specimen; b) Profiled (nonuniform(nonuniform crosscross section)section) specimen.

O Specimen A Specimen B

I I I I I I I I I I I

True Strain

(e)

Fig. 6.6. Growth ofof thethe neckneck in in a astraight straight-sided -sided specimenspecimen (specimen(specimen 1).1). Pictures werewere taken at strainsstrains ofof a)a) 0.45; 0.45; b) 0.57;0.57; c) 0.71; d)d) 11.00; .00; e) 1.23;1.23; Fig. 8.8. Comparison of thethe uncorrecteduncorrected flowflow curvescurves ofof a astraight straight-sided -sided f) 1.46 (at(at fracture).fracture). specimen A and a profiled specimen B. Flow curvecurve of BB isis higher.higher.

658 / /OPTICAL OPTICAL ENGINEERING ENGINEERING / / July/AugustJuly /August 1982 / Vol.Vol. 2121 No.No. 4

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E 02 6.00

5.00 -

.- D D 4.00 ,»""' Continuous test 4.00 ,..cf

/' * 3.00 Q'

4->8 C/3 A 3 £ _ x" Z.002.00 A /

1.00 j?

00.00 . 00 1 1 1 1 1 I ! 0.0O . 0 00.20 . 20 0.400 . 40 0.6 0.800 . 80 1.001 . 00 1.201 . 20 1.401 . 40 E 00 True Strain

Fig. 11. Comparison between the flow curvescurves obtainedobtained by a continuous testtest and interruptedinterrupted tests for copper.

Fig. 9. Corrected flow curvescurves for coppercopper specimensspecimens A, B,B, andand C. E 02 8.8.0000

900 7.7.0000 A

800 0 0O A 6.6.0000 700 _..-'-"' Continuous test « D -••""""' o ,.-•• 600 1? 5.5.0000 ~~ R*-'"r* ex ^"' s00 - /* 8 4.004. 00 400 •M

3.3.0000 / 30 1

200 D Specimen 11 / £a Specimen 2 — O Specimen 33 2.2.0000 00 - 1.001. 00 0.0 0.2 0.4 0.60.0 0.!8 0 0.2 1

True Strain 0.0.0000 ———————————————————————————————i I I i I i i 0.0 0.20 0.40 0.6 0.80 1.001.00 1.201.20 1.401.40 Fig. 10. CorrectedCorrected flow curvescurves forfor mildmild steel steel specimensspecimens 1,1, 2,2. andand 3.3. E 00 True Strain

The results Fig. 12.12. Comparison between the flow curvescurves obtained by a continuous The results of the two tests coincide for copper, while for mild steel test and interrupted teststests forfor mildmild steel.steel. the flow curve of thethe continuous testtest lieslies below thethe curvecurve forfor thethe interrupted tests.tests. Figure 13 shows thethe variation of the neck profile radius withwith 5. DISCUSSIONDISCUSSION strain for specimen B, aa behavior typical of all specimensspecimens with an 5.1. CalibrationCalibration initial nonuniform cross section. The profile radius increasedincreased and The experimental value of the sensitivity 35 was found to be higher decreased a few timestimes inin thethe initialinitial stagesstages of thethe test.test. Later, when flow than thethe value calculated usingusing Eq.Eq. (3).(3). OneOne ofof thethe reasonsreasons forfor thisthis is became localized, thethe radiusradius decreased continuously. the approximations usedused toto simplify simplify thethe analysis.analysis. TheThe moremore generalgeneral All copper specimens failedfailed atat anan apparentapparent strain ee =— 22 in£n(a (ao/0/a af)f) equation (Eq. (2) and Eq.Eq. (A.6))(A.6)) would yieldyield better valuesvalues of 6.8. = 2.4,2.4, withwith aoa0 and ofaf representingrepresenting the diametersdiameters ofof thethe minimumminimum cross section initially (o)(o) and atat fracturefracture (f),(0, respectively.respectively. TheyThey 5.2. Measurements on specimens exhibited a deep double-cupdouble -cup fracture.fracture. The steel specimensspecimens failedfailed withwith Application of the BridgmanBridgman correctioncorrection toto thethe measuredmeasured stress-stress - a cup and cone fracturefracture at a strain of about 1.45.1.45. strain curvescurves of incrementallyincrementally tested copper andand steelsteel resultedresulted inin

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Fig. 13.13. Variation of the neck profile radius withwith strain for specimen B.

consistent results for eacheach materialmaterial regardlessregardless ofof thethe originaloriginal speci-speci­ men profile. ThisThis resultresult doesdoes notnot supportsupport thethe assumptionassumption underlying the correctioncorrection butbut merelymerely indicatesindicates thatthat it ityields yields self- self-consistent consistent results. InitialInitial yieldingyielding inin specimensspecimens having curved profilesprofiles ledled toto aa period of uniform deformationdeformation inin whichwhich thethe effectiveeffective value of RR increased above that machinedmachined intointo thethe originaloriginal specimen.specimen. WhenWhen Fig. A.I.A.1. OpticalOptical modelmodel toto thethe projectionprojection moiremoire method.method. flow localization began,began, thethe value of R decreased with strain in approximately the samesame manner asas forfor initiallyinitially straightstraight specimens.specimens. Comparison of incremental tests with continuous teststests revealedrevealed that thethe variation of R with strain apparently depends on the strain rate. The more raterate-sensitive -sensitive steelsteel specimensspecimens did did notnot exhibitexhibit thethe same correction for incrementally loaded and continuouslycontinuously testedtested material, whilewhile the less raterate-sensitive -sensitive material,material, copper,copper, showed sub-sub­ stantially the samesame results for both typestypes ofof tests.tests. ThisThis suggestssuggests that the correction term mustmust bebe modifiedmodified toto accountaccount forfor strainstrain-rate -rate sensitivity of the material. Analysis of thethe fringefringe patternpattern provides a simpler determinationdetermination ofof R than measurements mademade directlydirectly on a silhouettesilhouette ofof thethe specimenspecimen profile. This condition arises because of the difficulty in illuminatingilluminating the specimen uniformlyuniformly over thethe regionregion ofof interestinterest toto provideprovide sharpsharp edges of thethe silhouettesilhouette image.image. InIn addition, addition, thethe fringefringe patternspatterns provide a more vivid display of the symmetrysymmetry of the specimenspecimen shape.shape. 6. CONCLUSIONSCONCLUSIONS An accurate method for the determination of profile radii of curva­curva- tureture ofof necked tensile specimens has been described. The projection moire method is a powerful technique for studying thethe geometrygeometry ofof the neck.neck. The method would be very useful to studystudy anisotropicanisotropic materials, like Zircaloy, which neck nonsymmetrically. It should be noted that thethe analysisanalysis of projection moire as pre­pre- sented in this paper can only be applied to thethe determinationdetermination of necking radii of tensile test specimens. ItIt cannot be directly applied to the determination ofof general surface topography,topography, forfor which other analyses'analyses11 -16" 16 are more appropriate. Fig. A.2.A.2. IsostathmicIsostathmic planes.planes. 7. ACKNOWLEDGMENTSACKNOWLEDGMENTS The work presented in this paper is from the thesestheses submittedsubmitted byby 8. APPENDIX A:A: GENERALGENERAL EQUATIONS OFOF THETHE JérômeJerome Clavel7Clave17 andand Raghavan Srinivasan18Srinivasan18 inin partialpartial fulfillmentfulfillment of MOIRE FRINGES the requirements for the degree of Master of Engineering inin Mate­Mate- Figure A.1A.I shows a schematicschematic drawing of the projectionprojection moiremoire rials Science andand Engineering,Engineering, University ofof Florida.Florida. Support forfor thethe apparatus. O isis thethe originorigin ofof aa CartesianCartesian coordinatecoordinate frameframe XOY.XOY. work by the U. S. Nuclear Regulatory Commission, Contract No. Light and dark raysrays whichwhich makemake upup thethe imagesimages ofof thethe referencereference and NRCNRC-04-77-039, -04- 77 -039, is is gratefully gratefully acknowledged.acknowledged. projected grids radiate from points CC and R,R, respectively, and inter-

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sect inin aa regionregion aroundaround the origin O. Planes 7r_1,n_{ , TT7r0,O, and and n 7r1{ are are planes planes corresponding corresponding to to k k= = — -1, 1,0, 0, andand M (x,(x ,y), y), a apoint point on on the the surface surface being being studied, studied, isis the the intersection intersection of 1, respectively.respectively. 10(0,0,0)10(0,0,0) and the nn-th -th lineline ofof thethe projected grid image with thethe mm-th -th line of thethe reference gridgrid image.image. TheThe pointpoint M(x,M(x,y) y) subtends subtends angles angles On n andand Om i//m -P1P2-PiP2 at the optical nodal points R and C:C: JI(0> 001 pl - pecosap2 cosa //I np1 tan = (A.I)(A.1) are pointspoints onon ir0TTO andand 7r1.TTJ. NN intersectsintersects 70TTO andand 7r1TTJ at IDI0 and II,Ij, OR respectively. We have and lol,IDI1 = = S=8 = 10J1 VjCos/3 COS ß, , mpgmp2 ~oc~ (A.2) tan Om =OC where N-7Nj where p,p1 and and pp22 are are the the pitchespitches ofof thethe imagesimages ofof thethe projectedprojected andand cosßcos/3 =- reference gridsgrids inin thethe vicinityvicinity ofof thethe origin O. Also, PIPlP2 P2 — -xsina~xsina + ycosaycosa IDJ1V, = —— tan ,;b= (A.3) Pip1 —- pecosap2 cosa n OR —-xcosa -f+ ysina

and pi+p2p2 -2p- 2plP2cosa)1/21 p2 cosa)1 / 2 . ININ =_ p1 - p2cosap2 cosa Y tan Om =OCOC-x -x (A.4) PiP2PlP2 By thethe equationequation for moire fringes, point MM wouldwould lielie onon thethe kk-th -th 8 = (A.9) fringe of aa subtraction moire fringe pattern, wherewhere 6=(P¡(p2 ++ PZP2 --2 2P1P2cos01/2Pl p2 cosa)1 / 2 k = m —-n n ,, k,m,n, == 0,0, ±1, ±2,....t2,.... (A.5) where S8 is the spacing between the planes.planes. If the grid lineslines areare notnot parallel,parallel, the equations are toto be modified. Substituting the values of m and n from Eqs. (A.1)(A.I) to (A.4)(A.4) in Eq.Eq. If the referencereference gridgrid isis rotatedrotated by an angle y withwith respectrespect to thethe (A.5), we get projected grid about the xx axis,axis, thenthen thethe equationsequations describingdescribing thethe intersecting rays change from AxeAx2 + +By2 By2 + +CxyCxy + +DxDx -f +Ey Ey + +F F = 00 ,, (A.6)

where mp1 y = -x—xtana tan a + A == p2(ORsin«p2 (ORsina -— kp1 kj cos a) , Cosacos a B = OCp1 sin , and C == (ORp2(ORp2 -- OCp1)cosa OCp,)cosa ++ kplp2 sin a , D = p2[kp1(ORp2 [kp,(OR -f + OCcosa) OCcosa) - -OR OR - OCsina] , y =— np2nP2 E == OC[OR(p1OC[OR(pj -— p2cosa) p2 cosa) -— kplp2 kp,p2 sinsina] a] ,, and to F == -kpip2OCXOR-kp,p2 OCXOR . Each moire fringe is the intersectionintersection line betweenbetween the surfacesurface mPi y = -xtana—xtana ++ being studied andand a cylindricalcylindrical surface, the axis of which is parallelparallel toto cos a the two verticalvertical gratings.gratings. CertainCertain approximationsapproximations can be made that make Eq. (A.6) easier to analyze. SinceSince OROR andand OC OC (---10 (~10 cm) are and much larger than p1PJ andand pp22 (~10~ (--10 -22 cm), cm), higher higher order order terms terms dropdrop out of Eq. (A.6), which cancan then be rewritten as np2np2 y=ztany+y = ztany + coscosy y y = ——————pp22 smasin a x - kp1p2 Y x pjp1 —- p2cosap2 cosa p1 —- p2cosap2 cosa The resulting isostathmic planes are defined by

k = 0,0, ±1,±2,....±1, f2,.... (A.7) p2p2 sinasin a pi sin y y = x + z This is equivalentequivalent toto assumingassuming thatthat the two beams that intersect at pl cos y -- p2cosap2 cosa pi cos -— p2cosa p2 cosa O are made up ofof parallelparallel rays.rays. TheThe moiremoire fringesfringes can nownow bebe considered as the intersection lines between a set of parallel equally kpkp1p2lP2 + (A.10)(A. 10) spaced planes, thethe isostathmicisostathmic planes, and the surface being studied. pecosasa —- p2cosap2 cosa The spacing S<5 betweenbetween thethe planes is the sensitivity of the projection moire method. The spacing between thesethese planesplanes cancan be shown to be In Fig. A.2, thethe normal vector to the isostathmic planes is PlP2P1P2 8 = pp22 sinsin aa __* (5= (A.11)(A.ll) N = i j (A.8) (p + p2P2 - -2p1p2cosacosy) 2pjp2 cosacosy) 1/21/2 P1- p2cosap2 cosa

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The sensitivity ofof the system, therefore, is a functionfunction ofof thethe NUREG/CR-0235,NUREG /CR -0235, U.S.U.S. NuclearNuclear RegulatoryRegulatory Commission. pitches, the angle y between the grid lines, 8. Trozera,Trozera, T.T. A., A., Trans. Trans. ASM ASM 56, 56, 780(1963). 780(1963). projected viewing gridgrid pitches, the angle y between the grid lines, C, Trans. ASM 44, 705(1952). and viewing axes. The 9. Marshall,Marshall, E.E. R.R. and and Shaw, Shaw, M. M. C., Trans. ASM 44, 705(1952). and the angle a betweenbetween thethe projectionprojection and viewing axes. The 10. Hartley, C. S.S. and Jenkins,Jenkins, D.D. A., A., J. J. Metals Metals 32(7), 32(7), 23(1980). 23(1980). sensitivity increasesincreases (6 (6 decreases) decreases) with with increasing increasing a a andand y. 11. Khetan,Khetan, R. P., "Theory and Applications of Projection Moire Methods," Ph.D. Thesis,Thesis, Dept.Dept. ofof Mechanics,Mechanics, StateState UniversityUniversity ofof NewNew York at Stony Brook (May 1975).1975). 12. Wasowski,Wasowski, J.,J., Opt.Opt. Commun.Commun. 2(7), 2(7), 3213(1970). 3213(1970). 9. REFERENCES 13. Hovanesian,Hovanesian, J.J. Der,Der, and and Hung, Hung, Y. Y. Y., Y., Appl. Appl. Opt. Opt. 10(12), 10(12), 273(1971). 273(1971). R. E., and Powell, R. L., Proc. Eng. Applica­ 1. Kuntze, W., Mitteilung aus dem MaterialprüfungsamtMaterialpriifungsamt undund demdem KaiserKaiser- - 14. Hovanesian,Hovanesian, J.J. Der,Der, Haskell,Haskell, R. E., and Powell, L., Proc. Eng. Applica- Wilhelm Institut furfür Metallforshung, 42,42, 31(1924). tions of Holography,Holography, LosLos Angeles Angeles 377(Feb.377(Feb. 1972).1972). 2. Thomsen,Thomsen, E. G., Shabaik, A. H., and Sohrabpour, S.,S., Trans. ASME, J.J. 15. Khetan, R. P.P. andand Chiang,Chiang, F. P.,P., Developments in Mechanics, Vol. 8, Engineering Materials and Technology,Technology, 252252 (July(July 1977). 1977). Proc. 15th15th MidwesternMidwestern MechanicsMechanics Conf.Conf. UniversityUniversity ofof Illinois, Illinois, ChicagoChicago 3. Thomsen,Thomsen, E.E. G., Shabaik, A. H., andand Sohrabpour, S., Fifth North Ameri-Ameri­ Circle, 1616 (March 1977).1977). can Metalworking Research Conf. Proc., 139139 (May 1977).1977). 16. Idesawa,Idesawa, M.,M., Yatagai,Yatagai, T.,T., and and Soma, Soma, T., T., Appl. Appl. Opt. Opt. 16(8), 16(8), 2152(1977). 2152(1977). by a Moire 4. Siebel, E., Verein deutcher Eisenhüttenleute,Eisenhuttenleute, BerlinBerlin 75,75, (1925).(1925). 17. Clave, J., "Fullfield OutOut-of-Plane -of -Plane DisplacementDisplacement Measurement by a Moire 5. Davidenkov,Davidenkov, N.N. N.N. andand Spiridonova,Spiridonova, N.N. I.,I., Proc.Proc. ASTM ASTM 32, 32, 553(1944). 553(1944). Method," M.S. Thesis, Dept. of Materials ScienceScience and Engineering,Engineering, Uni-Uni­ 6. Bridgman,Bridgman, P.P. W.,W., Rev.Rev. Mod.Mod. Phys. Phys. 17, 17, 3(1945). 3(1945). versity of Florida (Dec.(Dec. 1978).1978). 7. Eisenberg,Eisenberg, M.M. A.,A., "Stress"Stress DistributionsDistributions in in the the Necks Necks ofof Tension SpecimensSpecimens 18. Srinivasan, R.,R., "Correction"Correction of of the the TensileTensile FlowFlow CurveCurve forfor TriaxialTriaxial StressesStresses of Anisotropic andand Isotropic Materials," PropertiesProperties of Reactor Materials at thethe NeckNeck by thethe ProjectionProjection MoireMoire Method,"Method," M.M. E.E. Thesis,Thesis, Dept.Dept. ofof at ConstantConstant TrueTrue StrainStrain Rates,Rates, FinalFinal ReportReport July July 1978 1978-June -June 1980,1980, Materials Science andand Engineering, University of Florida (June 1980).1980). 3

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