COMPARISONOFCPFEMANDSPECTRALSOLUTIONMETHODSINPREDICTIONOFSTRAINSNEARGRAINBOUNDARIESINAUNIAXIALLYLOADEDOLIGOCRYSTALLINETENSILESPECIMENBySiddharthSankarRathATHESISSubmittedtoMichiganStateUniversityinpartialoftherequirementsforthedegreeofMaterialsScienceandEngineeringŒMasterofScience2016ABSTRACTCOMPARISONOFCPFEMANDSPECTRALSOLUTIONMETHODSINPREDICTIONOFSTRAINSNEARGRAINBOUNDARIESINAUNIAXIALLYLOADEDOLIGOCRYSTALLINETENSILESPECIMENBySiddharthSankarRathInthiswork,wehavebuiltupontheworkdonebyLimetal[1]todeterminewhetherthediscrep-ancyinexperimentallyobservedgrainboundarystrainsandthosecalculatedbyCrystalPlasticityFiniteElementMethod(CPFEM)isaresultofinherentshortcomingsofF.E.M.Wealsodeter-mineifaspectralsolutionmethodwouldprovehelpfulandrelativelymoreaccurateatpredictingstrainsnearthesegrainboundaries.IntheworkdonebyLimetal,ahighlyresolvedCPFEMsimulationofanoligocrystallineuniaxiallyloadedtantalumtensionsamplecapturedmostoftheexperimentallyobservedfeaturesbutmarkedlyunderestimatedthegrainboundarystrainconcen-trationswhentheresultsofthesimulationwerecomparedwithexperimentalresults.SimulationswereperformedforthesameoligocrystallineuniaxiallyloadedtensionsampleusingDüsseldorfAdvancedMaterialsSimulationKit(DAMASK)andusedthebuiltinelementandspectralsolverstocalculatestrainsandgrainrotationsnearthegrainboundaries.Varyingresolutionscom-parabletothoseusedbyLimetal,wereused.InvestigationhasbeendoneonwhethertheinabilityofusingresolutionswithFEMandaninherentstiffnessintheFEsolutionmethod,wasthereasonorthediscrepanciesobservedascomparedtoexperimentaldata.Thespectralsolversuseapolarisationsolutionmethodwhichconvergesrapidlyandallowsustouseevenresolutionswithoutbeingtoocomputationallyexpensive.Italsohasmuchlowerstiffnessasregardstheso-lutionstrategy.Thedistributionofequivalentstrainasafunctionofgrainboundarydistancewasplotted.Comparisonsandequivalencieswiththeresultsfromelementsolutionsweredoneandestablishedtoascertainwhetherornottheobserveddiscrepanciesinthecaseofelementsimulationwereinherentinthemethoditself.ACKNOWLEDGMENTSIexpressmygratitudetomyadviserDr.PhilipEisenlohrwhoguided,helped,andinstructedmethroughoutmyresearchjourney.IwouldliketousethisopportunitytothankDr.CarlJBoehlertandDr.ThomasRBielerfortheirinsight,help,corrections,andinstructions.IwouldalsoliketoextendmythankstomyfellowresearchersandcolleaguesDavidSmiadak,TiasMaiti,AritraChakraborty,ZhuowenZhao,andChenZhangfortheirassistance.AheartfeltthanksalsogoestomyfriendsinsideandoutsidethedepartmentwhokeptmemotivatedandhelpedmeupwhenIwasdown.Finally,IwouldliketothankmyparentsforbelievinginmeandsupportingmeaswellasemotionallythroughoutmytimeatMichiganStateUniversity.iiiPREFACEThechaptersinthisthesisaresodesignedthatthereadercangetvaluablebackgroundinformationbeforegettingtothepointofanalyzingtheresults.Itbeginswithalistofwhichishelpfulforeasymaneuveringofthethesis.Thestchapterisanintroductionabouttheworkthatwehavedone,themotivationforthesameandtouchesuponsomeassumptions.Thesecondchapterdealswithpolycrystals,crystalstructure,grains,grainboundaries,orien-tationsandwaystodepictthem,andgrainmisorientations.Thethirdchapterisallaboutdislocationsandrelationshipsbetweendislocationconcentrationsandstraininuniaxiallyloadedtensionsamples.Grainboundariesandgrainboundary-dislocationinteractionsarealsodiscussed.Stressesandstrainsinsingleandpolycrystalshavebeenexplainedinthefourthchap-ter.Resolvedshearstress,andcriticalresolvedshearstressinsingleandpolycrystalsarealsodiscussed.Thechapterisabriefdescriptionoftheconstitutivemodelforcrystalplasticitythathasbeenusedforthesimulations.Thesearebasedoncontinuummechanicalmodelsofcrystalplas-ticityThefocusisonthePhenomenologicalpowerlawdescription.ThesixthchapterdiscussesthepolarisationSpectralSolutionmethod.Afterthebackgroundinformation,theseventhchapterdealswiththepresentresearchquestionofinterestandhowapreviousworkbyLimetal[1]atSandiaNationalLaboratories,wasinstru-mentalinraisingthisquestion,whatistheobjectiveoftheresearchworkandtheplanofactionaswellasprocedurethatwasfollowedtoanswertheresearchquestionofinterest.Theeighthandchapterdealswiththeresultsthatwereobtainedduringtheresearch,andtheconclusionsthatcanbesafelydrawnfromthem.Thethesisconcludesbylistingtheresearchpapersthatweconsultedandusedinourresearch.ivv  TABLE OF CONTENTS  LIST OF FIGURES .................................................................................................................. vii CHAPTER 1 INTRODUCTION ............................................................................................. 1 CHAPTER 2 CRYSTALS ................................................................................................................. 3 2.1 Perfect Single Crystal ................................................................................................... 3 2.1.1 Lattices and unit cells ....................................................................................... 4 2.1.2 Crystallographic directions and Planes ............................................................. 5 2.1.3 Crystal Orientation ........................................................................................... 8 2.1.3.1 Quaternions ...................................................................................... 9 2.1.3.2 Euler Angles ..................................................................................... 9 2.1.3.3 Rotation Matrices ............................................................................ 10 2.2 Polycrystals ................................................................................................................ 11 2.2.1 Misorientations and grain boundaries ............................................................. 11 2.2.2 Texture and its representation ......................................................................... 14 2.2.2.1 Pole-Figures.................................................................................... 15 2.2.2.2 Inverse Pole-Figures ....................................................................... 17 2.2.2.3 X-Ray Diffraction and Electron Back Scatter Diffraction ................. 19 CHAPTER 3 DISLOCATIONS ...................................................................................................... 22 3.1 Edge and Screw Dislocations ..................................................................................... 22 3.1.1 Edge Dislocations........................................................................................... 23 3.1.2 Screw Dislocations ......................................................................................... 24 3.2 Interaction of dislocations ........................................................................................... 26 3.2.1 Forces associated with dislocations ................................................................. 26 3.2.2 Some sources of Dislocations ......................................................................... 30 3.2.2.1 Frank Read Source .......................................................................... 31 3.2.2.2 Multiple Cross slip ................................................................................. 32 3.2.3 Dislocations near Grain boundaries ................................................................ 33 CHAPTER 4 STRESSES AND STRAINS IN CRYSTALS ......................................................... 36 4.1 Engineering and Instantaneous stress and strain ......................................................... 36 4.2 Flow curve and Yielding criteria ................................................................................ 38 4.2.1 The Flow Curve ............................................................................................. 38 4.2.2 Yielding criteria ............................................................................................. 39 4.3 stresses and strains in single-crystals and polycrystals ................................................ 41 4.3.1 Plastic Deformation in Single Crystals ............................................................ 42 4.3.2 Plastic Deformation in Polycrystals ................................................................ 44 CHAPTER 5 CONSTITUTIVE LAWS ................................................................................................ 46 5.1 Phenomenological Power Law for FCC crystals ......................................................... 46 vi  CHAPTER 6 THE SPECTRAL METHOD: POLARISATION49 6.1 The kinematics ........................................................................................................... 49 6.2 Static Equilibrium ...................................................................................................... 50 6.3 Numerical Implementation ......................................................................................... 52 6.3.1 polarisation scheme of discretization .............................................................. 52 6.3.2 Numerical solution strategy ............................................................................ 53 CHAPTER 7 PROCEDURE .................................................................................................. 55 7.1 Setting up the Geometry files in DAMASK environment ............................................ 56 7.2 Setting up the Material configuration .......................................................................... 60 7.3 Setting up the Numeric Configuration ........................................................................ 61 7.4 Simulation .................................................................................................................. 61 7.4.1 Simulation by the Spectral Method.................................................................. 61 7.4.2 Simulation by Finite element method ............................................................. 62 7.5 Analysis tools used to obtain and compare results....................................................... 62 CHAPTER 8 RESULTS AND CONCLUSIONS .................................................................... 64 APPENDICES .......................................................................................................................... 84 APPENDIX A AUTOMATOR CODE ................................................................................ 85 APPENDIX B MATERIAL CONFIGURATION FILE ........................................................ 91 APPENDIX C NUMERICS CONFIGURATION FILE...................................................... 100 APPENDIX D ALTERATIONS TO NUMERICS CONNFIGURATION FILE ................ 104 APPENDIX E LOADCASE FILE .................................................................................. 105 APPENDIX F QUEUE SUBMISSION FILE.................................................................... 106 APPENDIX G POST PROCESSING ............................................................................... 108 APPENDIX H EULER ANGLES FROM GEOMETRY FILE HEADER ......................... 110 BIBLIOGRAPHY .................................................................................................................. 112 vii  LIST OF FIGURES  Figure 2.1 Silar SC-9M silicon carbide whiskers. Each of these whiskers is a single crystal. Source: acm-usa.com/silar-sc-.4 Figure 2.2 The fourteen Bravais lattices in three dimensions. Source: D.V.Anghel, Bra- vais lattice table, 2003...6 Figure 2.3 Example planes and directions showing miller indices for each in cubic unit cells. Source: http://pms.iitk.ernet.in/wiki/index.php/..............................................8 Figure 2.4 The shape of the fundamental zone in Rodrigues space.  Source:  Drawn by the author using Microsoft Paint ..13 Figure 2.5 Grain boundaries showing mismatch between grains. Source: www.engineeringarchives.com/............................................................14 Figure 2.6    (100) pole figures for sheet material, illustrating (left) random orientation and                           (right) preferred orientation. RD (rolling direction) and TD (transverse direction)                             are reference directions in the plane of the sheet. Source: Elements of X-Ray   Diffraction, B.D.Cullity, 1956..16    Figure 2.7 Stereographic projection sphere with the plane on the north pole. It also                         shows the projection of the south pole onto this plane. Source: Mark Howison                         at English Wikipedia; https://en.wikipedia.org/wiki/Stereographic_projection16  Figure 2.8 Wulff net or stereonet. It is used for plotting stereographic projection of poles and/or points .17 Figure 2.9    Standard Inverse pole figure for a cubic crystal. The [100],[010] and                     [001] are parallel to the i,j, and k respectively. Source: Applied Mechanics                               of Solids by A. F. Bower, Chapter 3.18 Figure 2.10  EBSD micrographs of hot-rolled and annealed titanium samples,                     along with the IPF color coded triangle for hexagonal close packed metals.                      Scale bar= 200 µm. This is for representation purposes only; for details                     about the labels in the micrographs, please be referred to the source:                     "The importance of crystallographic texture in the use of Titanium as an                               orthopedic biomaterial", by Bahl et al, J. RSC Advances, August 2014.                     the purpose of this figure is to demonstrate how an EBSD texture map                     may look like..20 Figure 3.1   Visualization of an edge dislocation as an extra half plane of atoms                    in the lattice. The burgers vector(shown in black color) is perpendicular toviii                      the dislocation line(shown in ..23 Figure 3.2   Visualization of a screw .24 Figure 3.3  Visualization of a screw dislocation in a plane. The red atoms are already displaced, i.e, the dislocation has passed through; the blue ones are yet to  be displaced while the green ones are in the process of moving and can be together called as the screw dislocation. Source: https://www.nde-ed.org/EducationResources/...........................................25 Figure 3.4   Visualization of stress fields around an edge dislocation. The dislocation                     line itself is perpendicular to the plane of the paper. zz which is just the sum                     of xx and yy  has not been shown. -                     represents compression.Source: G.E.Dieter, Mechanical Metallurgy,                                3rd Edition................................................................................................................28 Figure 3.5   Visualization of stress contour (xz) around a screw dislocation. the dislocation                     line is perpendicular to the plane of paper, x to the right and y to the top in the                     plane of the paper. the stress is always ...29 Figure 3.6   Visualization of stress contour (yz) around a screw dislocation. the dislocation                      line is perpendicular to the plane of paper, x to the right and y to the top in the                      plane of the paper. the stress is always ...29 Figure 3.7 Visualization of stages of Dislocation multiplication by the Frank-Read mechanism under shear on a slip plane.   A and B are the pining points,  and         is the applied shear stress.   Source:  Jntf - Own work,  CC    BY-SA  3.0, https://commons.wikimedia.org/w/index.php?curid=7501825................................32 Figure 3.8   Visualization of stages of multiple cross slip of the screw component of a mixed dislocation. The planes (111) are the family of slip ...33 Figure 4.1   True stress vs true strain for a ductile solid. 0 is the yield stress, 1 2                  is the recoverable elastic strain, and 2 3 is the anelastic behavior                       which is mostly neglected in mathematical theories of plasticity.                     Source: Mechanical Metallurgy by George.E.Dieter, Chapter-3,                            Section 3-238 Figure 4.2    Resolved shear stress in a single crystal. Source: Mechanical                       Metallurgy by George.E.Dieter, Chapter-4, Section 4-7; E.Schimd,                             Z. Elektrochem., vol. 37,p. 447, 1931...42 Figure 7.1  Comparison of their IPF color maps, with those produced by our software.                     The figures on the left are those produced at SNL, experimentally in the                 EBSD. The ones at the right are produced by the imageDataRGB commandix                      .56 Figure 7.2 The 2 dimensional geometry file (thickness in z direction = 1) which will                      be extruded. The x axis is towards the right and the y axis is .57 Figure 7.3 The Oligocrystal, made by extruding the 2D structure in the z direction. The                      aspect ratios have been maintained.  Resolution:  1084x294x168.                        Voxel aspect ratio x:y:z ::1:..57 Figure 7.4 The Oligocrystal with the isotropic airlayer around ..58 Figure 7.5 The Oligocrystal with the rigid elastic baseplate added at both ends. This                      makes the baseplate continuous because of the periodic structure and                renders the air layer dis..59 Figure 7.6 The rigid baseplate is added first in this new assignment of boundary                         .59 Figure 7.7 The isotropic plastic air layer is added around the baseplate and the                               Oligocrystal. This makes the baseplate periodically discontinuous,                     while the airlayer becomes continuous.60 Figure 8.1    The HRDIC versus CPFEM strain maps.The column of images on the left                       shows the strain map of the oligocrystal obtained by high resolution digital                      image correlation, while those on the left are obtained by CPFEM. In our work                      we have focused only on the strain in the x direction,  i.e.,  xx. The                       positive x direction is towards the right, and positive y is .64 Figure 8.2 The strain map generated in three dimensions by DAMASK.The color scale                      has been modified so that anyting abbove 0.3 is the same shade of red as                         0.3, and anything below 0 is the same shade of blue as 65 Figure 8.3   The comparison of strain maps upon change of boundary conditions. Left side                     images show the results that had the older boundary conditions, while those                     on the right have the newer boundary .......66 Figure 8.4   The comparison of strain maps upon change of boundary conditions, shown                      as the cumulative probability of ratios of strains at each material point                        calculated with the two different periodic boundary 67 Figure 8.5   The comparison of strain maps upon increasing resolution in the z-                     direction. Left side images show the results that had been obtained with                     a lower z resolution, while the right shows the results obtained after                      increasing the z resolution by a factor of 2. As is evident just by visual                      inspection, the change in increasing the z resolution does not effect any                     major change, but the gradients become .68x  Figure 8.6   The quantitative comparison of strain maps upon change of z resolution.                      cumulative probability of the ratio of strains calculated at each material                        point for resolution 271x74x21 to those at 271x74x10 has been shown69 Figure 8.7   The comparison of grain rotations upon increasing resolution in the z direction.                     Left side images show the results that had been obtained with a higher z                     resolution. As is evident just by visual inspection, the change in increasing                     the z resolution does not effect any major 70 Figure 8.8   The quantitative comparison of grain rotation maps upon change of z                           resolution. Cumulative probability of the ratios of grain rotations calculated                          at each material point for the resolution 271x74x21 to those calculated                     at 271x74x10.71 Figure 8.9   The grain rotation spatial map after simulating a rotated geometry.                     Mesh resolution 271x10x74, voxel aspect ratio 1:4:1. The top face and front                     face are interchanged from previous simulations because y and z directions are                     now interchanged due to the rotation. The band-like structure still exists                     in the thickness direction.  The thickness direction was z in the previous ones,                     but y in the current .72  Figure 8.10 The grain rotation spatial map after simulating an unrotated geometry with                      equiaxed voxels. Mesh resolution 271x74x42, voxel aspect ratio 1:1:1.                      Little to no banding can be seen in the thickness direction. Thus, an equiaxed                            voxel is 72 Figure 8.11 The grain rotation spatial maps comparing the CPFEM and spectral                      solution estimations of grain rotations. Mesh resolution 271x74x10,                            voxel aspect ratio 1:1:4. Little to no banding can be seen in the CPFEM                     solution. Spectral solution estimates higher grain rotation than FEM near the                       grain boundaries, but most of the other places, they estimate similar values of                     grain rotation as seen in 73 Figure 8.12 The cumulative probability of ratios of grain rotations calculated by FEM to those calculated by spectral 74 Figure 8.13 The grain rotation spatial maps for different resolutions.  the calculations                       are done by spectral method.  There seems to be no effective change             except smoothing upon increasing 75 Figure 8.14 Quantitative comparisons of grain rotations at different resolutions. The range                     of the ratios is from 0.1 to 5.0 which indicates similarity between the spatial                     maps showing grain 76 Figure 8.15  Spatial maps for strains calculated by CPFEM and spectral solution.They look similar except for the regions near the grain boundaries77xi  Figure 8.16 Quantitative comparisons of strains calculated by FEM and spectral method.                     The range of the ratios of strains calculated by FEM to those by spectral                     method is from 0.1 to 5.0 which indicates similarity between the spatial                     maps showing the strains.77 Figure 8.17 Distance from closest grain boundary versus ratio of spectral solution method strains to fem 78 Figure 8.18  Spatial maps of spectrally calculated strains at various 78 Figure 8.19  Quantitative comparison of strains calculated at various resolutions by    the spectral 79 Figure 8.20  Distance from closest grain boundary versus normalized strains calculated                      spectrally for 271x74x10 re80 Figure 8.21  Distance from closest grain boundary versus normalized strains calculated                      spectrally for 542x147x21 81 Figure 8.22  Distance from closest grain boundary versus normalized strains calculated                     spectrally for 1084x294x21 82 Figure 8.23  Distance from closest grain boundary versus normalized strains   calculated by FEM at 271x74x10 83 CHAPTER1INTRODUCTIONExperimentalmethodsofstudyingdeformationbehaviorofpolycrystalshasalwaysbeenchalleng-ingandmoreoftenthannotdestructive.Heterogeneousgrainmorphology,variationinthecrystalorientationofeachgrainandmanytimeswithinagrain,discontinuitiesatthegrainboundaries,andneighboringeffectsduetopresenceofothergrainsonlyaddstothechallengeofunderstand-ingwhathappenswhenpolycrystalsdeform.Acomputationalmethodologyhasgrownpopularamongscientistsandresearchersbecauseitisnondestructive,andCPFEMsimulationshavebe-comemainstreaminpolycrystaldeformationresearch.Itthusbecomesimperativetoscrutinizethistechniqueandtodiscoveritslimitationsinordertoovercomethem,ortogoforothercom-putationalstrategiessuchastheSpectralSolutionmethod.Oneofthemainconsiderationswhilejudgingwhichcomputationalmethodtoemploy,iscomputationalcostwhichdirectlydependsonthespeedwithwhichthealgorithmprocessesdataandcomputestheresults,whichinturnisadirectresultofthegridsizeused:thethegrid,thehighertheresolution,thehigherthecom-putationalcost.Adesirabletraitinacomputationalmethodologywouldbeanoptimumbetweenfastcalculationsandhigherresolutions.Ideally,onewouldwantfasterandfastercomputationathigherandhigherresolutions.TheCPFEMsuffersfromashortcomingthatisinherentinthealgorithm:thecomputationalcostisprohibitivelyhighatratherhighresolutions.SpectralSolu-tionmethodthatworksbyemployingfastFouriertransformscanhandlehighresolutionswithoutaprohibitivelyhighcomputationalcost,andisalsomuchlessstifferthantheFEM.Thishasbeenex-ploredlaterwhenthespectralsolutionmethodhasbeendealtwith,ingreaterdetail.InvestigationhasbeendoneonwhetherthisshortcomingoftheFEMaffectsthecalculationofGrainBoundarystrainsinourOligocrystallinesample,andwhethertheunderestimationofthesestrainscanbeattributedtothisshortcoming.Tothisend,SpectralSolutionmethodincludedintheDAMASK(DüsseldorfAdvancedMaterialsSimulationKit)tosolveforthestrainsintheuniaxiallyloaded1Oligocrystallinemetallicsamplewasemployed.Anoligocrystallinesampleisamulti-crystalsheetmetalsampleinwhichthegrainmorphologyonboththetopandbottomfacesarealmostidentical,andthegrainsthemselvesarerelativelycoarseandcolumnar.Theyextendthroughoutthecrosssectionofthesample.Suchasystemisrelativelyeasytosimulateandanalyzeascomparedtoanevenmoregenericpolycrystalastheydoawaywithsubsurfacemicrostructures.Inthiscase,afacecenteredcubiccrystalsystemhasbeenassignedtothemetal.Wehaveassumedthatthereisnochangesinorientationinsideeachgrain,andwehaveuseda5degreestoleranceangle;i.e,onlyorientationsdifferingmorethan5degreesshouldbeconsideredasadifferentgrain.Thiswasdonetoreducecomplicationsinthecalculations,andtoremoveredundancies.Also,thenumberofgrainscomputedcomesdownto19or20(dependingonmeshresolutions),whichisthesameasthatcalculatedexperimentally.Ifatoleranceanglelessthanthatisused,thenthereisariskofincreasingthenumberofgrainsandgrainboundaries.SincetheearlierpaperbyLimetal[1]usesuniaxialtensioninthexdirection,thisworkhasusedthesameaswell.Thiswasanobviouschoicebecauseanuniaxialtensilestressanditseffectshavebeenmostwidelystudied,documented,andanalyzed.Bothinsinglecrystalsaswellaspolycrstals,deformationhappensunderuniaxialloadafteracriticalstresshasbeenexceeded.Thestress-straincurvesforoligocrystallinematerialsfollowtheusualshapeforallpolycrystals,i.e,linear,reversibleelasticdeformationatandthenirreversiblenonlinearplasticdeformation.Thesamplewasallowedtodeformonlyintheyandzdirectionsasitispulledinthexdirection.TheYZfacesonbothendsareattachedtoarigidstructurewithveryhighelasticmodulusandnoplasticity,whileanisotropicairlayer,whichallowsdilatationalchanges,surroundstheXYandXZfacesofthesample.Thisprovidesuswitharathersimple,andsomewhatrealisticsetofboundaryconditionstosetupourmodelanddothesimulations.Inthefollowingchapters,wedealwithmorebackgroundinformationoncrystals;grains,grainorientationsandmisorientations;stresses,strainsandtheireffects;computationalstrategies;andourexperimentalprocedures,resultsandconclusions,ingreaterdepthanddetail.2CHAPTER2CRYSTALSMostofthematerialsthatwedealwithregularlyinourdailylifearecrystallineinnature.Alongrangeorderinthespatialarrangementofatomsinamaterialistherootcauseofcrystallinityinthatmaterial.Amorphousmaterialshaveshortrangeorder.However,manycrystallinematerialsinnature,existasanaggregateofsmallercrystals.suchcrystalsaretermed`polycrystals'.Theindividualsinglecrystalsthatmakeuptheaggregatearecalled`grains'andtheboundarybetweentwosuchthree-dimensionalgrainsisatwodimensionalsurfacetermedas`grainboundaries'.Thegrainshavepointdefectslikevacanciesandonedimensionaldefectslikedislocations,andthesereacttoappliedstress.Theinteractionofthesedefectsamongthemselvesandwiththegrainboundaries,plustheeffectofthechangesinonegrainonanother,determinethestressresponseandstrainsinthepolycrystal.Wediscussthesedefectsandtheirinteractionsinotherchapters.Inthefollowingsectionsofthischapter,weconsidersinglecrystalswithoutanydefects,whichformthebulkoftheindividualgrains;unit-cells,latticesandcrystallographicdirectionsandplanes;ori-entations,misorientationsandtexturesinapolycrystal;waysofrepresentingtheseinpolycrystals,i.e,eulerangles,quaternions,rodriguesvectors,etc;andsomeexperimentaltechniquessuchasEBSDthatcanprovidetexturemapsfromphysicalsamples.2.1PerfectSingleCrystalAperfectsinglecrystalisacrystallinematerialwithlongrangespatialarrangementofatomsormoleculeswithoutanydefectsandgrainboundaries.Itisexactlyonecrystalwithanunbrokenlattice,exceptforthesurfacewhereitisexposedtothesurroundingmaterial(generallytheat-mosphere).Thesecrystalsarerarelyfoundinnature,becausepresenceofdefectsimpliesmoreentropy,andminimizationofoverallfreeenergyandhencedefectformationisfavoredinnature.3However,inlaboratories,underspeciallyenforcedconditions,singlecrystalscanbegrown.Someexamplesarenanorodsandwhiskers(showninFigure2.1).Figure2.1:SilarSC-9Msiliconcarbidewhiskers.Eachofthesewhiskersisasinglecrystal.Source:acm-usa.com/silar-sc-9mThesearegrowneitherasreinforcementsinspecialcompositematerialsorassamplestoouttheidealmaterialproperties.theapplicationsaremostlyresearchbased,however,andthetestsdoneonsinglecrystalsprovideuswiththestandardizedconstantssuchasshearstrength,elasticandshearmodulus,etc.2.1.1LatticesandunitcellsInmathematics,particularlyingrouptheory,alatticeinann-dimensionalrealvector-spaceRnisasubgroupofRnwhichisisomorphictoZn,andwhichspansRnfully.Simplyput,foranybasisofRn,thesubsetofalllinearcombinationswithintegralcoefformsalatticeinndimen-sions.Incrystallography,aBravaislattice,orsimplyalatticeisanarrayofdiscretepointsrepeatinginthreedimensionswhichisgeneratedbydiscretetranslationoperationsdescribedbyR=n1a1+n2a2+n3a3(2.1)whereniareintegersandaiaretheprimitivevectorswhichlieinarbitrarydirections(butlinearlyindependent)andspantheentirelattice.Oneconditionisthatthesediscretesetofvectorsmust4formaclosedtrianglewhenvectoradditionorsubtractionisoperatedonthem.Also,foranychoiceofthepositionvectorRintherealvectorspace,thelatticelooksidentical.ThisformaldiscussionaboutLatticesandBravaislatticesisimportantbecausethisconceptisusedtoacrystallinearrangementanditslimitations.oneormoreatomsofthecrystallinesubstanceformsthebasisandtheperiodicrepetitionofthisbasisateachlatticepointmakesupthecrystalstructure.Asaresult,thecrystalwilllookthesamewhenviewedfromanysimilar,orequivalentlatticepoint.Equivalentlatticepointsarethoseseparatedbyonetranslationoftheunitcell.Aunitcellisthesimplestarrangementofoneormoreatomsthatwhenreproducedperiodically,cangeneratetheentirecrystal;i.e,theunitcellsstackedinthreedimensions,willformthebulkarrangementoftheatomsinthecrystal.Aunitcellcanbecompletelydescribedbythelengthofitsedgesa,b,andcandtheanglesbetweenthem.Theanglebetweenaandbisg,thatbetweenbandcisaandbetweenaandcisb.Together,theseedgesandanglesaretermedaslatticeparameters.Inthreedimensionalspace,thereare14kindsofBravaislattices,andthesearegroupedintosevenlatticesystemsorcrystalsystems,namelyTriclinic,Monoclinic,Orthorhombic,Tetragonal,Rhombohedral(trigonal),Hexagonal,andCubic,asshowninFig.2.2Therearefourtypesofunitcells,namelyPrimitive(orsimple),volumeorBodycentered,Facecentered,andBaseorEndcentered.sometimes,thesimpleprimitiverhombohedral(trigonal)unticellissaidtobeaspecialkindofunitcellintisownright.Notallcombinationsoflatticesystemsandunitcelltypesareneededtodescribethepossiblecrystalsystemsinnaturebecausemostoftheseareequivalenttoeachotherandonlyoneofthoseisenough.Thatiswhy,eventhoughintheoryweshouldhave42possibleBravaislattices,wehavejust14.2.1.2CrystallographicdirectionsandPlanesIna3-dimensionalcrystallattice,crystallographicplanesanddirectionsareimportanttounder-standhowsymmetry,orthelackthereofplaysanimportantroleinthepropertiesofacrystallinesolid.Crystallinematerialsaregenerallynonisotropicsolidsandparticularlatticeplanesand5Figure2.2:ThefourteenBravaislatticesinthreedimensions.Source:D.V.Anghel,Bravaislatticetable,2003.directionsshowenhancedpropertiesthanothers.Crystallographicdirectionsarestraightlineslinkingatompositions,ornodes,whilecrystallographicplanesaretwodimensionalplaneslinkingthenodes.Certainplanesanddirectionshavehigherdensityandbetterpackingofatoms,andthushaveahigherdensityofnodes.Opticalpropertiessuchasrefractiveindexaredirectlyrelatedto6periodicindensity;surfaceadsorptionandreactivityaredependentonthedensityofnodes;higherdensityofnodescorrespondtolowersurfacetension;microstructuraldefectssuchasporesusuallyhavestraightboundariesimmediatelyafterhighdensityplanes;mechanicalbehaviorsuchasfracture,cleavage,slip,etcoccurpreferentiallyalonghighdensityplanesinacrystal,andsodoesdislocationglideandmovementbecausetheburgersvectorisalwaysalongaclosepackeddirectionandplane.TheplanesanddirectionsaregenerallyrepresentedintermsofMillerindices,orMiller-Bravaisindices,andthereexistfamiliesofequivalentplanesandfamiliesofequivalentdirectionsasweshalldiscussbelow.Afamilyofplanesordirectionsinalatticeorinanunitcellisdeterminedbythreeintegersh,k,andlforplanesoru,vandwfordirections.TheseareknownastheMiller-indicesforPlanesandDirectionsrespectively.Forplanes,theyarewrittenas(hkl)anddenotethefamilyofplanesthatareorthogonaltohb1+kb2+lb3,wherebiformsthebasisforvectorsinthereciprocallattice.Thereciprocallatticevectorsneednotbemutuallyorthogonalandhencetheplaneisnotnecessarilyorthogonaltoalinearcombinationofdirectlatticevectors.Conventionally,thenegativeintegersinasetofmillerindicesforeitherplanesordirections,isdenotedwithabarovertheinteger,andallintegersarewritteninlowestterms.Duetolatticesymmetry,allplanesequivalentto(hkl)aregroupedtogetherasafamilyofplanes:fhklg.Alternatively,theMillerindicesareproportionaltotheinverseoftheinterceptsoftheplane,inthebasisofthevectorsinthelattice.Thus,ifoneoftheindiceshasavalueof0,thentheplanedoesnotintersectthataxisastheintercept,ortheinverseof0,is.Forexample,iftheplaneinterceptstheLatticevectorsa,bandcat1;0:5and0:5,thentheMillerindicesare(211).Usually,incubiccrystalsystems,manyfamiliesofplanesexist,forexamplef100g,f110g,f111g,etc.Fordirections,theindicesareasmagnitudeoftheinterceptsofthedirectionwiththeorthogonalvectorsa,bandc,intheirlowestwholenumberforms.Forexample,ifacrystaldirectionhasintercepts0.5,-1and-0.5onthea,bandcaxesrespectively,thenthedirectioncanbedenotedas[121].Familiesofdirectionsaredenotedashuvwi.Infcclattice,thef111gfamilyofplanesaretheslipplanes.Alongwiththeh110ifamilyofdirections,wehave12slip-7systems.Millerindicesforthedirectionsandplanesarethusextremelyusefulincrystallographyforstudyingtheplanes,orientations,directionsofslip,etc.Figure2.3showsthe[010]directionwith(001)plane,andthe[110]directionwiththe(110)plane.Figure2.3:Exampleplanesanddirectionsshowingmillerindicesforeachincubicunitcells.Source:http://pms.iitk.ernet.in/wiki/index.php/2.1.3CrystalOrientationCrystalorientationreferstothemannerinwhichthecrystalisorientedinthreedimensionalspace.ItisdenotedgenerallyintermsofQuaternions,EulerAngles,orasRotationMatrices.Theseareangle-axisrepresentationsofcrystalOrientations.Alloftheseorientationsarerelativetoacoordinatesystemoutsideofthecrystal,knownasthe"LaboratoryCoordinates".Thereisacoordinatesysteminsidethecrystal,knownasCrystalCoordinates.82.1.3.1QuaternionsAccordingtoEuler'sRotationTheorem,anysinglerotation,orasequenceofrotationsofarigidbodyoracoordinatesystemaroundapointedinspace,isequivalenttoexactlyonerotationbyagivenangleqaboutanaxisedinspacethatrunsthroughthesaidedpoint.Thusanyrotationinspacecanbedenotedbyavector~uandascalarq.Quaternionsprovideaneasywayofrepresentingthisangleandaxisinformationintermsoffournumbers.Thesesetoffournumberscanbeusedtoapplyanyparticularorientationtoapositionvectorthatrepresentsapointrelativetoanorigininarealthreedimensionalspace.Inotherwords,arotationthroughanangleqaroundanaxisbyaunitvector~u=(ux;uy;uz)=ux‹i+uy‹j+uz‹kcanberepresentedbyaQuaternion:q=eq2(ux‹i+uy‹j+uz‹k)=cosq2+i(ux‹i+uy‹j+uz‹k)sinq2(2.2)Therealpartofthisquaternionisdenotedasw.Thisrotationcanbeassignedtoanordinaryposi-tionvectorp=px‹i+py‹j+pz‹ktogettheorientationp0=q:p:q1.Inacomputerprogram,thisisachievedbyconstructingaquaternionwhosevectorpartispandrealpart(w)equalszeroandthenperformingthequaternionmultiplication.Thevectorpartoftheresultingquaternionisthedesiredvectorp0.2.1.3.2EulerAnglesIncrystallography,theBungeEuleranglesareextensivelyused,andthesecanbeasasequenceofrotationsofthecrystallographicaxeswithreferencetothespatialaxes.TherotationisaroundtheZaxisbyananglef1,thesecondaroundthenewXaxisbyananglef,andthethirdisaroundthenewZaxisbyanangleF2.f1,f,andf2aretheBungeEuleranglesthatareusedtorepresentthecrystalorientations,aseachgraininapolycrystalwillhaveauniquesetoftheseangles.Computationally,whilemodeling,iftwopointshavethesamesetofangleswithinafewdegreestolerance(0:5°to3°usually)thentheyaresaidtobeinthesamegrain.Inasinglecrystal,theentirecrystalhasthesameorientation(withinallowedtolerance),despitepresenceofdislocations.92.1.3.3RotationMatricesRotationmatricesarethematrixversionofAngle-AxisrepresentationssuchasEuleranglesorquaternions,andcanbeusedtodenoteorientationsequallyskilfully.AlthoughEulerAnglesandquaternionsaremorewidespread,conciseandefRotationmatricesareusuallyobtainedasrawdata(asinourcase)andthenconvertedintoEulerAnglesorquaternionsasoperationsonthosearecomputationallyefandcostslessasregardsthetimetakenforcomputation.Therotationmatrixfromquaternionscanbewrittenas:Rq=26666412q2j2q2k2(qiqjqkqr)2(qiqk+qjqr)2(qiqj+qkqr)12q2i2q2k2(qjqkqiqr)2(qiqkqjqr)2(qjqk+qiqr)12q2i2q2j377775(2.3)whereqr=wandqijkaretheimaginarycomponentsofthequaternionrepresentation.Similarly,arotationmatrixfromanAngle-Axisrepresentation,whereinrotationhappensbyanangleqaroundanaxisalongthedirection‹uwouldlooklike:Raa=266664cosq+u2x(1cosq)uxuy(1cosq)uzsinquxuz(1cosq)+uysinquxuy(1cosq)+uzsinqcosq+u2y(1cosq)uzuy(1cosq)uxsinquxuz(1cosq)uysinquzuy(1cosq)+uxsinqcosq+u2z(1cosq)377775(2.4)Inasimplerform,R=cosq:I+sinq[u]+(1cosq)uuwhere[u]isthecrossproductmatrixofu,isthetensorproductandIistheidentitymatrix.ThisisthematrixformoftheRodrigues'rotationformula.AccordingtotheRighthandrule,rotationswillbeanticlockwisewhenupointstowardstheobserver,andshallbeconsideredaspositiverotations.Ifupointsawayfromtheobserverthentherotationwillbeclockwiseandconsiderednegative.BungeEulerAnglescanalsoberepresentedasarotationmatrixasfollows:Re=266664c1c3c2s1s3c1s3c2c3s1s1s2c3s1+c1c2s3c1c2c3s1s3c1s2s2s3c3s2c2377775(2.5)wherec1=cosf1;c2=cosf;c3=cosf2ands1=sinf1;s2=sinf;s3=sinf2,whereinf1;fandf2aretheBunge-Eulerangles.102.2PolycrystalsCrystallinematerials,donotexistasperfectsinglecrystalsinnature.Moreoftenthannot,dislo-cationsandotherdefectsexistincrystals.Also,duringmanynucleigrowsimulta-neously.Notallofthemhavethesamecrystalorientation,andupontwocrystalswithdifferentorientations(usuallybeyondacertaintolerancelimitcalledmisorientationangle)aretermedastwograinsinthesameoverallcrystallinesolid.Suchasolidwithmanysuchgrains,eachgraingenerallyhavingadifferentcrystalorientationthanitsneighbors,iscalledasapoly-crystals.Reasonsforabundanceofpolycrystalsisthatdislocations,preferentialgrowthofcertainnuclei,andinherentrandomnessincreasetheentropyofthesystemasawholeanddecreasetheoverallfreeenergy.Themaincharacteristicsofpolycrystalsthatsetthemapartfromsinglecrystalsarethepresenceofgrainboundaries,andthemisorientationsamongthegrainsthatmanytimesactasdislocationsourcesandmakepolycrystalseasilydeformablethansinglecrystals,atleastinitially.2.2.1MisorientationsandgrainboundariesManyunitcellsconstituteacrystal.Thereisgenerallyaslightgradientoforientationsoftheunitcellsinthelattice.Thismeans,thatorientationsofunitcells,orbetweenclustersofunitcells,arenotalwaysexactlythesame.Withinagrain,thesevariationsareusuallywithinafractionofadegreetoacoupledegrees.ThismismatchiscalledasMisorientation.Theboundariesbetweenthesecrystalsthatarewithinafractionofadegreetoacoupledegreesofmisorientation,arecalledlowangleboundaries.Whenthemisorientationexceedsthelimit,thenitcansafelybeassumedthatweareinanothergrain.TheboundariesbetweensuchcrystalsarecalledHighanglegrainboundaries.Inmodeling,wecanassignthemaximumallowedmisorientationwithinagrain,andifthatlimitisexceeded,thenweassignanothergrainidentitytothatorientation.Thisgivesuscontroloverourmodelandwecanmanipulatethenumberofgrainsinourvirtualpolycrystal,byusingthisparameter.11Thereisanotherwayofdescribingorientationsinspace,knownastheRodrigues'rotationformula.Itstatesthatvrot=vcosq+(‹kv)sinq+‹k(‹kv)(1cosq)(2.6)wherevisavectorinR3(realspace)and‹kisaunitvectorwhichrepresentstheaxisofrotationaboutwhichvrotatesthroughanangleqincompliancewiththerighthandrule.WhenwerepresentorientationsintheRodrigues'space,werepresentthemaspoints.Eachofthosepointshasapositionvectorvrot.Thisisusefultoshowmisorientation.Misorientationwithinagrainisoftheorderofafractionofanangletoacoupleangles,andallsuchorientationsclustertogetherbecausetheyareveryclosetoeachother,inRodriguesspace.Whenweexceedthosevalues,wemoveintoanothergrain,whichinRodriguesspace,isanotherclusterofpoints,whichmayormaynotbeclosetothecluster.Thus,ifweplotalltheorientationsofapolycrystalinRodriguesspace,thenumberofpointclustersgivesusthenumberofgrains.Whilemodelingapolycrystal,wecanmanipulatethemisorientationtolerance,andthuswehavecontroloverthenumberofgrainsthatwewantinourpolycrystal,andalsotosimplifyourmodelandtherebyreducingcomputationalcost.Inourwork,weusedthisconcepttomanipulatethetolerancewhilemodeling,inordertomatchthenumberofgrainsinourmodelwiththatintheactualOligocrystalthatLimetal[1]usedintheirexperiment.However,duetoinherentperiodicity,andaneedtorepresentorientationscompletelyrandomly,generallyafundamentalzoneinRodriguesspaceisconceptualized.Allthecrystalorientationsarevisualizedwithinthisfundamentalzone.Thiszonelookslikeacubewithallitscornerscutoff,asshowninFig.2.4Pointsthatlieoutsidethebordersofthiszone,appearontheothersideofthiszoneowingtoperiodicity.Itisthusimperativetobeabletoclearlydistinguishsuchmirrorimagesfromactualorientations,orwemightendupconcludingthatwehavemorenumberofgrainsthanwehadinitiallyintended.Grainboundaries,orrather,HighAngleGrainBoundariesaretwodimensionalentities,or12Figure2.4:TheshapeofthefundamentalzoneinRodriguesspace.Source:DrawnbytheauthorusingMicrosoftPaintprogramsurfaces,thatmarkthetransitionfromonegraintoanother,i.e.,whenthemisorientationexceedstheallowedtolerancewithinagrain.Physically,agrainboundary(showninFig.2.5)isanarrayofdislocations,thattracestheboundarybetweentwograins.Whenthemisorientationisgreaterthan15°thenthedislocationsbecomesoclosethattheircoresoverlapandindividualdislocationscannotbedistinguishedanymore.Theenergyofthisboundarybecomesindependentofthemis-orientationangle,withtheexceptionofafewspecialorientations.ThePresenceofalargenumber13ofdislocations,makethegrainboundariesundergomorestrain,andactasbarrierstoslip.Weshalldelvedeeperintothisintheforthcomingchapters.Figure2.5:Grainboundariesshowingmismatchbetweengrains.Source:www.engineeringarchives.com/2.2.2TextureanditsrepresentationAnobviousquestionthatcomestomindafterthediscussionaboutorientations,ishowdowerep-resentthemwithoutmakingittoomathematicallyinvolved.Apictorialrepresentationoftexturedataisveryimportantasitcansuccessfullyandefshowthetextureinformationwithoutbeingtooerudite.Tothisend,wehavenumeroustechniqueslikecolormaps,StereographicPro-jections,inversepoleElectronBackScatterDiffraction(EBSD)maps,etc,thathelpusinrepresentingthetextureofapolycrystalinasingleimageThecolormapshavelegendsassociatedwiththemthatcontaintheorientationdataofthegrains.Inmanycases,alongwithjustthecolorsrepresentingtheorientation,arotatedunitcellisalsoshowninsideeachgrainthat14correspondstothatparticularorientation.Themostpopular,andyet,conventionalmethodsofrep-resentingtexturedata,arehowever,poleandinversepoleEBSDmapscolorcodethedatafromthediffractionofbackscatteredelectronsfromanSEM,andcanbeusedeftorepresenttexturedata.2.2.2.1Pole-FiguresCrystallographictexturecanbedisplayedbymeansofapolee,whichisusuallyastere-ographicprojectionofthecrystaldirectionsofeachgraininthepolycrystal.thecrystalplanenormalsareknownaspolesandareplottedontheprojection.An(hkl)poledisplaysthedistributionofthefhklgpolesinthepolycrystallinesample.Individualorientationmeasurements,incaseofElectronBack-ScatterDiffraction,arerepresentedaspointsinthepoleThesein-dividualorientations(points),plottedtogetheronthepolegiveaglimpseintothetextureofthepolycrystal.Ifthereareclustersofpointsinthepolethenitmeansthatorientationsareclosertothosetexturecomponentsthatarerepresentedbythoseclusters.Thedifferencebe-tweenpoleofrandomlyorientedpolycrystals,andpolycrystalswithpreferredorientationisshowninFigure2.6forthe(100)poles.Ingeneral,polycrystallinematerialsseldomhaverandomlyorientedgrains.Moreoftenthannot,apreferredorientationexiststhataffectsmanypropertiesofpolycrystals,andintroducesananisotropy.Ifweconsideraplaneinacrystal,thenitsorientationcanbebyitsnormaldirection.Ifwedrawasphere,withitscenterontheplane,thentheintersectionofthenormaldirectionandthespherewillbeapoint.Thispointisknownasthepole.However,ifwerotatetheplanearounditsnormaldirection,thenitsnormaldoesn'tchangeandhencethepoledoesn'tchange.Therefore,twonon-parallelplanesarerequiredtoplottwopolesonthesphere,andtheniftheobjectrotates,andoneoftheplanesrotateswithit,thechangeinrelativepositionsofthepoleswilldescribetheneworientationcompletely.Astereographicprojection,isatechniqueinwhichweprojecttheupperhalfofthissphereuntoaplanethatgenerallyrestsonthenorthpole,asshowninFig.2.7:15Figure2.6:(100)poleforsheetmaterial,illustrating(left)randomorientationand(right)preferredorientation.RD(rollingdirection)andTD(transversedirection)arereferencedirectionsintheplaneofthesheet.Source:ElementsofX-RayDiffraction,B.D.Cullity,1956Figure2.7:Stereographicprojectionspherewiththeplaneonthenorthpole.Italsoshowstheprojectionofthesouthpoleontothisplane.Source:Mark.HowisonatEnglishWikipedia;https://en.wikipedia.org/wiki/Stereographic_projectionThepoles,orpointsonthisspherealsogetprojecteduntothisplane,anditresultsinacirclethatrepresentsthepoles,andtheplanetraces.Planetracesaretheintersectionsoftheplanesandthesphere.Theyappearasarcs.AWulffnet(Figure2.8)isacirclewitharcscorrespondingtoplaneswhichshareacommonaxisinthe(x,y)plane.ThisplaneisparalleltotheequatorialplaneofthesphereinFigure2.7.Wulffnetsarevitalinunderstandingandreadingstereographicprojections,eventhoughnowa-16Figure2.8:Wulffnetorstereonet.Itisusedforplottingstereographicprojectionofpolesand/orpointsmanuallydayswehavecomputersandspecializedsoftwaresthatcandoitforus.WerotatetheWulffnetsothattheplanetracewewanttostudycorrespondstoanarconthenet.Thepoleorthepointissituatedonanotherarc,andtheangulardistancebetweenthesetwoarcsisthentakentobe90°.Todrawapolewechooseaparticulardirection(usuallythenormaltooneofthef100gplanes)andthenplotthatpoleforallthecrystalsinapolycrystal,relativetoasetoforthogonaldirections,sometimesknownasrollingdirection,transversedirectionandrollingplanenormal(referFigure2.6).Ifwearetoplottheseforalargenumberofcrystals,thencontourpotsarepreferredasopposedtoindividualpoints.Tofullydeterminethetextureinapolycrystal,theplotoftwopoleeachcorrespondingtoaplaneoutofapairofnonparallelplaneswithdifferentinterplanarspacing.2.2.2.2InversePole-FiguresTexturecanalsoberepresentedbyusinganinversepoleInthiscase,theaxesofthespherearealignedwithanythreecrystaldirections(generally[100],[010]and[001])asopposedtothe17globaldirections:normal,rollingandtransverse,intheformercaseofThepointsplottedinaninversepolearecrystaldirectionsthatareparalleltoeitherthenormal,orthetransverseortherollingdirectioninthesample.Thus,onlycertaintypesoftexturescanbevisualizedbyaninverseDuetosymmetryconsiderations,however,aparticulardirectioncanbeassignedanyofthecrystallographicallysymmetricalequivalentdirections.ThisleadstotheconceptualizationofafundamentalorstandardtriangleasshownintheFig.2.9Figure2.9:StandardInversepoleforacubiccrystal.The[100],[010]and[001]areparalleltothei,j,andkrespectively.Source:AppliedMechanicsofSolidsbyA.F.Bower,Chapter3thethreevertexessymbolizetheh001i,h110iandtheh111idirections.ThetranslationofthistrianglealongthecrystalcoordinateswillgiverisetotheentireinversepoleThus,wecanmapallpointsintothisfundamentaltriangle,andaccountforperiodicity.Anytexturecomponentcanthusberepresentedinsidethistriangle.Foruniaxialtensiontestsofcubiccrystals,testsshouldthusberunonlyforthosecaseswheretheloadingdirectionisinsidethestandardtriangle.Inthecaseofapolycrystalwithmanygrains,thedistributionofpointswillgiveinformationabouttheorientation.Ifthedistributionisuniformthroughout,thenwecaninferthatthecrystalshavearandomuniformdistributionoforientation.Ifclustersform,thenthepositionoftheclusters,especiallytheirproximitytoeithervertexes,willrelayinformationontherelativedistributionoforientations,andalso,thepreferredorientationofthegrains.Acolorcodedfundamentaltriangle18canbeusedasalegendtoaccompanyacolormapofapolycrystal,andthusthetexturecanbeeffectivelydetermined.2.2.2.3X-RayDiffractionandElectronBackScatterDiffractionX-RayDiffraction(XRD)isthepreferredwayofmeasuringorientationsandtexturesinpolycrys-talsonamacroscopicscale,whileElectronBackScatterDiffraction(EBSD)isthepreferredwayofobtainingthedataatmicroscopicscales.XRDhaslargepenetrationdepthsofaround5mmandaspatialresolutionfromaminimumof25µmto1mm.Tensofthousandsofgrainsinarelativelylargerareaofabout10mm2canbescannedbyanXRDinonego,whichmakesitpreferableformacroscopicsamples.However,thespatialresolutionisverypooranditistherebyunsuitableformicroscopicsamples.Also,CalculationofOrientationDistributionFunctionfrompolecanbereallycomplexincaseofXRDbecauseofthelargeamountofgrainsscanned(whichleadsustoapolewithcontoursratherthanpointsandclusters),andthiscangiverisetoerroneousinformation.Plus,texturemeasurementbyXRDneedsabout30to60minutesperpoleandabout3to4poleareneededforacubicsymmetry(5to6forlowersymmetrycrystals)ifadistributionfunctionistobeobtained.Ontheotherhand,EBSDhasarathersmallpenetrationdepth(penetrationdepthislessbe-causeabeamofelectronscanpenetratemuchlessintoamaterialthanabeamofX-Rayphotons)ofaround20nmwhichmakesitsurfacesensitive.Moreover,ifaFieldEmissionScanningElec-tronMicroscopeisusedtoobtainEBSDdata,thenthespatialresolutionisoftheorderoftensofnanometerswhichmakesitsuperiortoXRDfortexturemeasurementofmicroscopicareas.TexturecanbedirectlyrelatedtomicrostructureandmeasurementofgrainsizescanbedonetooincaseofanEBSD(anexampleofanEBSDmapshowingmicrostructureofatitaniumsample,sourcedfromtheworkbyBahletal[2]isshowninFig.2.10)becauseallofthesecapabilitiesarebuiltintoanSEManditsaccompanyingsoftware.SEMalsohasdepthresolutionfeatureswhichareextremelyusefulwhileanalyzingsurfaces.ButEBSDmaynotworkonheavilydeformedmaterialsorpolymericmaterialsandneedsgreater19Figure2.10:EBSDmicrographsofhot-rolledandannealedtitaniumsamples,alongwiththeIPFcolorcodedtriangleforhexagonalclosepackedmetals.Scalebar=200mm.Thisisforrepresentationpurposesonly;fordetailsaboutthelabelsinthemicrographs,pleasebereferredtothesource:"TheimportanceofcrystallographictextureintheuseofTitaniumasanorthopedicbiomaterial",byBahletal,J.RSCAdvances,August2014.thepurposeofthisistodemonstratehowanEBSDtexturemapmaylooklike.samplepreparationtimes.Inheavilydeformedgrainswhereacertainslipsystemisoperating,anerroneouscalculationmightbemadewhichmaymanifestasunder-representationofaparticulartextureorplainlyinaccuratecalculations.However,acquisitiontimeofdataisprettylowbecause20crystalorientationscanbemeasuredatbetween50and100orientationspersecond.Thisgivesusaround30minutestocalculatedatacomparabletowhatanXRDmeasurementprovidesin3or4timesthattime.ThedimensionsoftheoligocrystallinesamplethatwasexperimenteduponatSandialaboratories,asdescribedinthepaperbyLimetalissuitableforanEBSDmeasurementandthustheyobtainedEBSDdatafortextureforthatsampleInthiswork,theirEBSDdatahasbeenusedtovirtuallymodelandreproducetheactualoligocrystalinthreedimensionsbyusingcertainfunctionalitiesprovidedbyDAMASKsuchasconversionoftabulateddata(EulerAnglesorGmatrix)fromanASCIIformattoageometrywhichcanthenbemanipulatedasregardsmeshsizesandresolutionstomakeitcomparabletotheeelementsimulationsdoneatSandiaNationalLaboratories,Albuquerque,NewMexico.21CHAPTER3DISLOCATIONSCrystalshavelongrangeorder.Anyirregularityinthatorderisknownasadefect.Dislocationsareonedimensionaldefectsthatarenaturallypresentinacrystalastheirformationisenergeticallyfavorablebecausetheylowerthefreeenergybyincreasingtheentropyofthesystem.Disloca-tionscanariseduringcrystallizationfromtheliquid(orgaseous)state,whensomeatomsdonottaketheirdesignatedplaceinthelatticeduetoheterogeneousnucleationatpreferredsitesanddirectionalion,asopposedtoahomogeneousnucleationandplanarDis-locationscanariseifmaterialsarestrainedorifastressisapplied,andthereareinclusions,orpointdefects,orgrainboundariespresent.Manytimesgrainboundariesthemselvesbehaveasthesourcesofdislocations.Wewilldiscussmoreaboutotherdislocationsourcesinsection3.2.2,andaboutwhathappenstothedislocationsnearthegrainboundariesinsection3.2.3.3.1EdgeandScrewDislocationsBroadly,alldislocationsoronedimensionalirregularitiesincrystalscanbecategorizedashavingeitheredgecharacter,screwcharacter,oramixtureofboth.Themixedtypeofdislocationsformthemajorityofdislocationsfoundinrealcrystals.tocharacterizeadislocation,avectorbisused,whichisknownastheBurgersVectoranditrepresentsthemagnitudeanddirectionofthedistortioninalatticeresultingfromthepresenceofadislocationinthelattice.Thedirectionofthevectordependsonthedislocation,andthecrystallographicplaneonwhichthedislocationexists.Generally,thoseplaneshavebeenobservedtobetheclosest-packedplanes.AlthoughaBurgersLoopisgenerallydrawntodeterminethedirectionandmagnitudeoftheburgersvector,mathematically,inacubiccrystallattice,themagnitudecanbegivenby:jjbjj=a2ph2+k2+l2(3.1)22whereaisthelatticeparameter,||b||isthemagnitudeoftheburgersvectorbandh,k,larethecomponentsofb3.1.1EdgeDislocationsThesedislocationscanbevisualizedasaonedimensionaldefectwherethereisanextrahalf-planeofatomsinthecrystal,whichdistortsthenearbyplanesofatoms,asshownintheFigure3.1SuchFigure3.1:Visualizationofanedgedislocationasanextrahalfplaneofatomsinthelattice.Theburgersvector(showninblackcolor)isperpendiculartothedislocationline(showninblue)adislocationhasalinedirectionthatisshowninblueandtheburgersvectorwhichisperpendiculartothelinedirectionthatisshowninblackinFigure3.1.Theburgersvectorbofanedgedislocationisalwaysperpendiculartotheedgedislocationline.Edgedislocations,uponapplicationofexternalstress,canmove,causingslip.Theplanesofslipareusuallyclosepackedplanes,andthedirectionsaretheclosepackeddirectionstoo(paralleltob).Theycannotchangeslipplanesandtheymovebycontinuousbreakingandformingofbonds.Theywillcontinuemovingifnoobstructionsexistuntiltheyemergeoutonthesurface,causingasliplinetobeformedonthesurface.Thesliplineheightwillbeexactlyequaltothemagnitudeoftheburgersvector.Ifhowever,thereexistsanotheredgedislocation(s)ofanoppositesign(imagineFig.3.1upsidedown),thentheycanceleachotherout.Withoutthepresenceofoppositedislocations,edgedislocationsdonotdisappearfromcrystals.Inthisbodyofwork,wehavenotexpoundeduponmanymoresuchpropertiesof23edgedislocations,rather,weshalljumpintothestressesexperiencedbydislocations,andstressassociatedwithdislocationsafterabriefdescriptionofanothertypeofdislocations,calledscrewdislocations.3.1.2ScrewDislocationsAscrewdislocationissomewhathardertoimaginethananedgedislocation.Ifwecutacrystalalongaplane,butonlyhalfwaythrough,andthenapplyashearstresssothatthetwohalvesmoveinoppositedirection,thentheatomsarrangethemselvesinahelicalfashionaroundthedislocationline,andwegetascrewdislocation,asshowninFig.3.2.Figure3.2:VisualizationofascrewdislocationAnothersimplerwayofvisualizingthescrewdislocationhasbeenshowninFigure3.3.Atomsaroundascrewdislocationcantraceahelicalpatharoundit,thusitisnamedassuch.ThepurescrewdislocationlineisalwaysparalleltotheBurgersvector,whilethedirectionofdislocationmotionduringslipisperpendiculartothelinedirection.asusual,sliphappensonclosepackedplanes(usuallythedensestplanes)alongtheclosestpackeddirectionsbecauseloweratomicspac-ingmakesiteasierforthedislocationtomoveatrelativelylowerstresses.screwdislocationscanhoweverchangeslipplanes,fromoneclosepackedplanetoanother(notnecessarilyparallel)byaphenomenoncalledcross-slip,owingtothefactthattheburgersvectorisparalleltothedislocation24Figure3.3:Visualizationofascrewdislocationinaplane.Theredatomsarealreadydisplaced,i.e,thedislocationhaspassedthrough;theblueonesareyettobedisplacedwhilethegreenonesareintheprocessofmovingandcanbetogethercalledasthescrewdislocation.Source:https://www.nde-ed.org/EducationResources/line.Thishasaneffectontheinteractionsbetweendislocationsofdifferentcharacters,asweshallseeinthenextsection.Atthispoint,itneedstobementionedthatmanytimes,dislocationssplitintopartialdisloca-tionstocircumventobstacles,givingrisetostackingfaults,andevenmorecomplicatedinteractionswithotherdislocations.Anin-depthdescriptionofpartialdislocationinteractionsandpartialdis-locationslipisoutofthescopeofthisbodyofwork.Thesepartialdislocationsmayhaveeitheredgeorscrewcharacter,ormixed,dependingontheirburgersvectors.Generally,whenadisloca-tionwithaburgersvectorbsplitsintotwopartials(namedasShockleypartialsiftheyaremobile25andcausestackingfaults)withburgersvectorsb1andb2,thesplitisfavoredenergetically.ifjbj2>jb1j2+jb2j2(3.2)Also,thecomponentsoftheShockleypartials,mustadduptotheoriginalvectorthatisbeingdecomposed,i.e,b!b1+b2(3.3)Ifthepartialdislocationsaresessile,theyaretermedasFrankpartials.ShockleyandFrankpar-tials,uponinteractioncanformastackingfaulttetrahedroncalledastheThompsonTetrahedron.Anothertypeofpartialdislocationinteraction,whentwoShockleypartialscombinetoformadis-locationoutsideoftheslipplane,isknownastheLomer-Cottrellbarrier.Asthenamesuggests,itissessileandactsasabarriertodislocationmotion.Barrierstodislocationmotionareoneofthemaincausesforstrainaccumulationingrainsandneargrainboundariesandleadtostrainhardening,asweshallseelater.3.2InteractionofdislocationsSinceatanygiventimeincrystals,numerousdislocationsarepresent,theyareboundtointeract.Theseinteractions,underappliedexternalstresses,leadtointerestingconsequences,suchasdefor-mation,strainhardening,formationofjogs,kings,dislocationsubstructures,etc.Lowanglegrainboundariesarealsoaconsequenceofdislocationinteractions,leadingtolocalorientationchanges.3.2.1ForcesassociatedwithdislocationsTheseaforementionedinterestingconsequencesofdislocationinteractionsoccurbecausealldis-locationsareassociatedwithcertainstressaroundthembecausethelatticeinthevicinityofdislocationsisstrained.Thesestressinteractwithappliedexternalstresses,andthiscauseseitherdislocationmotionorshapechangesdependingonthemagnitude,type,anddirectionoftheappliedforces.Thesestressalsointeractwiththestressesassociatedwithotherdislocations26inthevicinityandeitherrepelorattracttheotherdislocations.Thestresstensorassociatedwithanedgedislocationortheedgecomponentofamixeddislocationinisotropicmaterialswiththedislocationlinealongthezdirection,hasthexx,yy,zz,xyandyxcomponents,wherexx,yy,andzzcomponentsarenormalstresseswhereasxyandyxaretheshearstresses.thenormalstressesaregivenby:sxx=Gby(3x2+y2)2p(1n)(x2+y2)2(3.4)syy=Gby(x2y2)2p(1n)(x2+y2)2(3.5)szz=n(sxx+syy)=Gbnyp(1n)(x2+y2)2(3.6)txy=tyx=Gbx(x2y2)2p(1n)(x2+y2)2(3.7)whereGistheshearmodulus,bisthemagnitudeoftheBurger'svector,andnisthepoisson'sratio.Similarly,theelasticstraintensorforsuchanedgedislocationlookssomethinglike:e=266664e11e120e21e22000e33377775(3.8)whereexx;eyyandezzarenormalstrainswhileeyxandexyareshearstrains.Theseresultintensionwherethehalfplaneismissing,andcompressionofthelatticewherethereisanextrahalfplane.Whendepictedpictorially,thestresscontourslooksimilartothoseshowninFigure3.4.Itcanthusbesaidthatfortwoedgedislocationsthatiftheyareofoppositesigns,theywillattracteachothertominimizeoverallstrainandenergyofthelattice,whileiftheyareofthelikesign,thentheywillrepeleachother.Similarly,ifweconsiderascrewdislocationwiththedislocationlinealongthezdirection,inanisotropicmaterial,thenbytheinherentsymmetryofthescrewdislocation,thestresseswillbeonlyshearandexistonlyifthereisazcomponent.Thusthestressequationincaseofascrewdislocationare:sxz=szx=Gby2p(x2+y2)2(3.9)27Figure3.4:Visualizationofstressaroundanedgedislocation.Thedislocationlineitselfisperpendiculartotheplaneofthepaper.szzwhichisjustthesumofsxxandsyyhasnotbeenshown.'+'representstension,and'-'representscompression.Source:G.E.Dieter,MechanicalMetallurgy,3rdEditionsyz=szy=Gbx2p(x2+y2)2(3.10)Thustherearenonormalstresses,andshearexistsonlyinthezxandzyplanes.Thestraintensorthuslookslike:e=26666400e1300e23e31e320377775(3.11)Figure3.5and3.6depictthesestressespictorially.MATLABwasusedtogeneratethecontours.forcalculatingforcesbetweentwodislocations,'dislocation1'and'dislocation2',weusethePeach-KoehlerEquation:!F(2)=!x(2)x!b(2)sij(1)(3.12)where!x(2)isthesensevector(ordirection)ofdislocation2ataparticularpointofinterestonthedislocationline,!b(2)istheburger'svectorofdislocation2,ands(1)ijisthestressofdisloca-28Figure3.5:Visualizationofstresscontour(sxz)aroundascrewdislocation.thedislocationlineisperpendiculartotheplaneofpaper,xtotherightandytothetopintheplaneofthepaper.thestressisalwaysshear.Figure3.6:Visualizationofstresscontour(syz)aroundascrewdislocation.thedislocationlineisperpendiculartotheplaneofpaper,xtotherightandytothetopintheplaneofthepaper.thestressisalwaysshear.tion1atthatpoint.aPeach-Koehleranalysisofdislocationinteractionarrivesatconclusionsthatiftwoparalleledgedislocationsareofoppositesignsandonthesameplane,theycanceleachotherouttoformaperfectlattice;iftheyareofsamesigns,theyrepeleachotherandthestabledistancebetweenthemisdictatedbytheequilibriumbetweenrepulsionandappliedstress.Iftwooppositeedgesarenotinthesameplane,thentheattractiveforceswouldcausethemtocometogether,butinsteadoftotalannihilationofdislocations,vacanciesmightbeformed.Annihilationcanoccurat29highertemperatureswhendislocationclimbprocessesaremoreprevalent.Iftheyareoflikesignintwodifferentplanes,thentheywillrepeleachother,andmostlikelywillremainat45°toeachotherandatanequilibriumdistance.Similarly,whentwoscrewdislocations,parallel,andonthesameplaneinteract,thenlikescrews(Righthandedorlefthanded)willrepeleachotherandunlikescrewswillattract.However,ifthescrewsareorthogonal,thenlikescrewswillattracteachother.Whenedgedislocationsontwoorthogonalslipplanesinteractwitheachotherandpassthrougheachother,theneachacquiresajogequaltothemagnitudeanddirectionoftheburger'svectoroftheother.Thusbothjogsareofedgecharacter,andglissile.Ifjogsareformedonthesameslipplaneastheinitialdislocations,thenthosejogsarecalledkinks.However,iftheyareextendeddislocations,i.e,theyarecomposedoftwotrailingpartials,thenaLomer-Cottrellbarrierisformed,anditissessile.Whenanedgeandascrewdislocationonorthogonalslipplanespassthrougheachother,edgejogsarecreatedonbothdislocations.Thesejogsslowdownthemotionofdislocations,thuscaus-ingstrainbuildupandhardening.Thejogontheedgedislocationisinthedirectionofthescrewdislocation,whilethejogonthescrewdislocationisinthedirectionoftheburgersvectoroftheedge.Whentwoscrewdislocationsonorthogonalplanespassthrougheachother,theycreatejogswithedgecharacteronboththescrews,andtheseslowdownthemotionofthedislocations.Thesejogscanmoveonlybyclimborgeneratearowofvacanciesorinterstitials(pointdefects).Climbisnonconservativeandcostsenergy,andthusthesejogsseverelyimpedethescrewdislocationmovement.3.2.2SomesourcesofDislocationsSuchdislocationinteractions,keepongettingmoreandmorecomplicatedwithincreaseindis-locationdensity.Withmoreandmorenonparalleldislocationsonmanydifferentslipsystemsundergoingmovement,manylocks,barriersandjogs,alongwithvacanciesandinterstitialsareformedwhichactaspinningpointsorbarrierstodislocationmotion,whichcanleadtobuildupof30strains.However,toaccountforthesehugebuildupofplasticstrainsincrystals,therehastobeamechanismbywhichdislocationsregenerateandmultiplyandthusleadtoincreaseindislocationdensity.TwooftheimportantmechanismsareFrank-Readsourceandmultiplecrossslip.3.2.2.1FrankReadSourceIfweconsideradislocationlineonaslipplanepinnedatitsendsAandBbysomekindofbarrierorjog(maybecreatedasaresultofdislocationinteractions),asshowninFigure3.7.ifashearstresstisappliedonthisslipplane,thenaforcebeginstoactonthedislocationlineasaresultofthatstress.Thisforceisgivenby:F=tbx(3.13)ThisforceisperpendiculartothelineAB(showninredinthe3.7)andtriestobendthislineintoacurve,simultaneouslyincreasingthelinelength.thisisopposedbythedislocationlinetensionwhichactsalongthedislocationlinebutawayfromAandBwithamagnitudeofGb2ateachend.whenthedislocationbends,theverticalcomponentofthislinetensionactsoppositetotheappliedstress,andthus,F=tbx=2Gb2=)t=2Gbx(3.14)Equation3.14givestheamountofstressthatisatleastrequiredtogenerateadislocationfromaFrank-ReadsourcesuchasthatshowninFigure3.7.Ifthestressincreasesfurther,thenthedislocationlinewillcontinuetobendmoreandmoreuntilitreachesanequilibriumposition(theindigocoloredsemicircleinthewhereallofthelinetensionisnowutilizedagainsttheappliedstress.Iftheappliedstressstillincreases,thenthedislocationspontaneouslyexpandsandspirals(asshownbytheblueandthenthegreencurve)aroundthepinningpointsAandB.ItreachesapointwherethetwosegmentsspiralingaboutAandBcollidewitheachother.Sincetheyhaveoppositelinesense,theycanceleachotherout,andadislocationloop9showninyellow)isformedwhichsteadilymovesawayfromthecenter.Initswake,itleavesanewdislocationwhichfollowsthesamestepsasbefore,undercontinuedstress,andthemultiplicationcontinues.31Figure3.7:VisualizationofstagesofDislocationmultiplicationbytheFrank-Readmechanismundershearonaslipplane.AandBarethepiningpoints,andtistheappliedshearstress.Source:Jntf-Ownwork,CCBY-SA3.0,https://commons.wikimedia.org/w/index.php?curid=75018253.2.2.2MultipleCrossslipFigure3.8showsaschematicofthemultiplecrossslipprocesswhichleadstotwoedgedislocationsACandBDontheslipplane(111).Onlyscrewdislocationsorscrewcomponentsofmixeddislocationscancrossslip,fortoundergocrossslip,theburgersvectorneedstobeparalleltothedislocationline.Appliedstresscanbeeitherparallelorperpendiculartob.Thescrewcomponentofthemixeddislocationlinecrossslipsfromthe(111)slipplaneontothe(111)slipplane,andthenonceagaincrossslipsbackontothe(111)plane.ThetwosegmentsAcandBDwithedgecharacternowarepinnedattheirends.ThesethenstartbehavinglikeFrank-Readsourcesandmultiplyunderappliedshearstressontheslipplane(111).32Figure3.8:Visualizationofstagesofmultiplecrossslipofthescrewcomponentofamixeddislocation.Theplanes(111)arethefamilyofslipplanes3.2.3DislocationsnearGrainboundariesGrainBoundariesaretwodimensionalentitiesthatcanbevisualizedasanarrayofdislocationswhichmarkthetransitionofonegrainfromanother.WegenerallyrefertoHigh-AngleGrainBoundarieshere,i.e,GrainBoundariesbetweentwograinswithmorethan5-10°misorientation.TheGrainboundariesactassourcesofdislocationsbecausetheyhaveahigherenergythanthebulkofthegrain,havemorerandomnessthanorder.Thiscausesthemtobepreferredsitesforvacancynucleation,segregationofphases,inclusions,etcwhichcanleadtodislocationmultipli-cationand/orgeneration.Alsotheyalreadyhaveahighmismatchwhichcanbeginextendingintothegrain'sbulk,formingdislocations.However,themostimportantthingaboutgrainboundaries33isperhapsthewaydislocationsbehavenearthem,causinggrainboundarystrengthening.ThismeansthatGrainboundarieshavemuchhigherstrainconcentrationsthanthebulkofthegrain.Inthisbodyofwork,ourfocushasbeentodetermineifspectralsolutionstrategiesofcomputationgivesusgrainboundarystrainconcentrationsclosertowhatwefoundexperimentally,wheretheCPFEMmodelcouldnot.Inthissectionwediscusswhatthecauseofhigherstrainconcentrationsatgrainboundariesis.GrainBoundariesactasbarrierstodislocationmotionowingtorandomness,mismatch,andadisruptioninthelongrangeorder.Thelatticestructureofthegrainontheothersideisdifferentfromthecurrentgrainthedislocationisinanditcostsenergytochangedirectionsandslipplanes.Sincedislocationmotion,orslipisoneoftheprimarycausesofplasticdeformation,impedingthismotionwillcauseanincreaseintheyieldstrength,i.e,causedelayinplasticdeformation.Onebyone,thedislocationsstartcomingtograinboundariesfromthebulkofthegrain,andbeginpilingup.Sincedislocationsofoppositesignscanceleachotherout,thepileupswillbecausedbydislocationsoflikesigns.Thiscreatesrepulsionbetweenthedislocationsnearthegrainboundaries,whichisbythecloseproximityofthedislocationstoeachothernearthegrainboundaries,andthuscauseshigherstrainconcentrationsthaninthebulk.Eventually,thistotalrepulsiveforcebecomesenoughfordislocationstodiffusethroughtotheadjacentgrain,andcontinuedeformation.Thus,ifwehavesmallergrainsizes,thentheamountofpileupisnotenoughtoovercomethethresholdandstartdislocationdiffusion.Inotherwords,smallergrainsizesimplyhighernumberofgrainboundariesandhencegreaterobstructiontodislocationmotion,andthushigheryieldstrength.TherelationshipofyieldstressandgrainsizeisgivenbytheHall-PetchRelationship:sy=s0+Kypd(3.15)wheresyistheyieldstress,s0denotestheresistancetoslip,Kyisamaterialconstantknownasthestrengtheningcoefanddisthediameterofthegrain(averagegraindiameter).The34strengtheningcoefisgivenby:Ky=s8Gbscp(1n)(3.16)wherescisthecriticalstress.Theaboveequationsarehowever,derivedforcaseswheretheappliedstressistensileanduniaxial,thesametypeashasbeenconsideredinthisbodyofwork.Criticalstressesareaddressedinthenextchapter.35CHAPTER4STRESSESANDSTRAINSINCRYSTALSNodiscussionaboutpolycrystallineaggregatesundertensionwouldbecompleteifwedonotdiscussaboutthedifferentstressesandstrainsthataremeasuredtoquantitativelyandqualitativelycharacterizethedeformationbehavior.Itisknownfromexperiencethatifsufexternalloadisapplied,allmaterialscanbedeformed.However,upuntilacertainamountofloadisapplied,solidsgenerallyrecovertheiroriginaldimensions.Thisiscalledelasticbehavior.Beyondthiscertainamountoflimitingload,knownastheelasticlimitofthesubstance,thesolidexperiencesapermanentdeformation,slipbeingoneofwhosemoststudiedmechanisms.Thisbehavioristermedasplasticbehavior.Weshalldelvedeeperintothis,aswellasthestressversusstraingraphformaterialsintheupcomingsections.4.1EngineeringandInstantaneousstressandstrainEngineeringstressoraveragestressisastheForceappliedtoabody,perunitareaofcrosssection.Thisisjustanaveragevalueofthestressovertheentirebody,andassumesuniformdistributionofstressesinthecrosssection.Mathematically,ifweconsideracylinderofcrosssectionalareaA,underuniaxialtension,withtheforceappliedbeingF,thentheaveragestresssisexpressedas:s=FA(4.1)wherewehaveconsideredauniformdistributionofstressalongtheentirecross-section.Ifwedonotconsiderthisuniformity,andtakeintoaccountstresslocalizationatcertainareasinthebody,andusetheinstantaneousareaofcrosssection,thenthevaluewegetiscalledinstantaneousortruestress(s),anditisexpressedas:s=FRdA(4.2)36whereAistheinstantaneouscrosssectionalarea.Theunitsofstress(bothengineeringandinstan-taneous)areusuallypascalsormega-orgiga-pascals.Similarly,Engineeringstrainisasthechangeindimensionoveroriginaldimensionofthesolid.Inouronedimensionalcaseofuniaxialtension,averageorengineeringstrain,e,isgivenby:e=DLL=LL0L0(4.3)whereDListheChangeinlength,calculatedasLL0,LbeingthenewlengthafterdeformationandL0beingtheinitialdimension.Instantaneousstrain,e,however,isasthechangeinthedimensiondividedbytheinstantaneousvalueofthedimension.Mathematically,e=ZLfL0dLL=lnLfL0=ln(e+1)(4.4)whereLfisthelength,L0istheinitiallength,andListheinstantaneouslength;andeistheengineeringstrain.Thisisalsoknownasthetruestrain.Underthetrueelasticlimit,bothstrainsgivesimilarvalues(owingtotheverysmallstressesinvolved),howeverthetruestrainismoreimportantinplasticdeformation,especiallyinmetals.Strainisadimensionlessquantity.Upuntilnow,wehaveconsideredonlynormalstressesandstrains.However,shearstressesandstrainscanalsobeinsimilarways.Engineeringshearstressisgivenby:t=FAsinq(4.5)whereqistheanglebetweentheappliedforceFandthenormaltothecross-section.Instantaneousshearstressisgivenby:t=FRdAsinq(4.6)Similarly,ashearstraincanbeexpressedas:g=tanq(4.7)whereqistheangulardisplacementoftheoriginaldimension.374.2FlowcurveandYieldingcriteria4.2.1TheFlowCurveAwcurvedepictsplasticwwhenasolidisdeformedandisgenerallyastress-straincurveobtainedfromasimpletensiontestwhereasampleisuniaxiallyloaded.Thetruestresssandthetruestraineareplotted.AtypicalwcurveforaductilesolidisshowninFigure4.1.Figure4.1:Truestressvstruestrainforaductilesolid.s0istheyieldstress,'e1e2'istherecoverableelasticstrain,and'e2e3'istheanelasticbehaviorwhichismostlyneglectedinmathematicaltheoriesofplasticity.Source:MechanicalMetallurgybyGeorge.E.Dieter,Chapter-3,Section3-2Atthebeginning,thematerialdeformselasticallywithincreasingstress,andtherelationshipbetweenstressandstrainislinear.Thus,intheelasticregion,stressisdirectlyproportionaltostrain.ThisiscalledastheHooke'sLaw.Theratioofsandeintheelasticregionthusgivesus38thevalueoftheModulusofElasticity,E.sµe)se=E(4.8)Ifinsteadofnormalstresses,onewouldapplyshear,thenthemodulusiscalledasshearmodulusandisdenotedbyG.Thus,tg=G(4.9)wheretistheshearstress,andgistheshearstrain.Afters0,theyieldstress,hasbeenexceeded,Hooke'slawisnolongerfollowedandthemate-rialdeformsplastically.AsisevidentfromFigure4.1,stresseshigherthanyieldstressisrequiredtocausesimilarincreasesinstrainascomparedtotheelasticregion.Thisisaconsequenceofstrainhardening.AtpointA,whentheloadissuddenlyreleased,thestraindropsalmostimmedi-atelyfrome1toe2,byanamountequaltos=E.Thischangeinstrainiscalledrecoverableelasticstrain.Theremainingstrainisplastic.Butnotallplasticstrainispermanent.Somerelaxationofplasticstrainfrome2toe3occurs,dependingonthetemperature,orthematerialinvolved,andisknownasanelasticity.Ifhowever,nounloadingisdoneatA,andstressiscontinuouslyincreased,thensoon,neckingwilloccurandultimatelyfracture.Apowerequationoftheforms=Ken(4.10)canbeusedtodescribetheplasticregionuntilneckingbegins.Kisthestressate=1andnisthestrainhardeningcoefwhichisbasicallyfoundfromtheslopeofalog-logplotofequation4.10.4.2.2YieldingcriteriaForplasticdeformationtohappen,thematerialhastoyield.Thus,thestressandstrainmustgobeyondtheyieldpoint.Yieldingcriteriahelpusindevisingwaystopredictwhenamaterialwillyield,andthus,whenplasticdeformationwillbegin.Inthecaseofuniaxialtension(or39compression),plasticdeformationorplasticwbeginswhentheappliedstressreachesavalueofs0ortheyieldstress.Forthegeneralcase,yieldingmustbethereforerelatedtotheprincipalstresses,orsomecombinationofthem.Whenthebasisofastresstensorischangedinsuchawaythattheshearcomponentsbecomezero,theleadingdiagonalistheonlyonewithnonzerovalues.Thosethreestressesaretheprincipalstressess1;s2ands3.Mathematically,s1=J13+23qJ213J2cosjs2=J13+23qJ213J2cosj2p3s3=J13+23qJ213J2cosj4p3(4.11)wherej=13cos1 2J319J1J2+27J32(J213J2)3=2!J1=s11+s22+s33J2=s11s22+s22s33+s33s11s212s223s231J3=s11s22s33s11s223s22s231s33s212+2s12s23s31(4.12)whereJ1;J2andJ3areknownasstressinvariantsbecausetheirvaluesareindependentofthecoordinatesysteminwhichstressisexpressed,andsijarethecomponentsoftheappliedstresstensor.Becauseofthismathematicalcomplexity,mostyieldingcriteriaequationsareessentiallyempiricallyderived.TwoofthemostwidelyusedyieldcriteriaaretheTrescaandtheVon-Misesyieldcriteria.TheTrescacriterionisalsoknownastheMaximumShearStresstheory,anditpostulatesthatyieldingwilloccuronlywhenthemaximumshearstressexperiencedbyasolidwillreachacriticalvaluewhichisequaltotheshearyieldstressinanygivenuniaxialtensiontest.Thus,themaximumshearstressisgivenbytmax=s1s32(4.13)Forthecaseofuniaxialtension,s1=s0,andboths2ands3arezero(uniaxialmeansthereisstressinonlyonedirection).Thereby,tmax=s02(4.14)40wheres0istheyieldstressofthematerialundersimpleuniaxialtension.Thus,fromequations4.13and4.14,s1s3=s0(4.15)Sincehydrostaticcomponentsdonotnecessarilycausedeformationinin-compressiblematerials,weconsideronlythedeviatorcomponentsofappliedstress.Thus,equation4.15becomess01s03=2k(4.16)wheres01ands03arethedeviatorcomponentsoftheprincipalstresses,andkisthestressatwhich,undertorsion,yieldingoccurs,soitistheshearyieldstress.However,asdiscussedbyPragerHodgeetal[3],suchsimplemathematicalrelationscannotbesatisfactorilyappliedtomanyplas-ticityproblems,andthegeneralformoftheTrescacriterionisthenused:4J3227J2336k2J22+96k4J264k6=0(4.17)Thecomplexityofthisgeneralequationisovercomebyanotheryieldcriterion,knownastheVonMisesyieldcriterion.ThisisalsoknownastheDistortionenergycriterion.Accordingtothiscriterion,s0=1p2h(s1s2)2+(s2s3)2+(s3s1)2i12(4.18)Thus,yieldingoccurswhentheR.H.Sofequation4.18exceedss0whichistheyieldstressunderuniaxialtension.4.3stressesandstrainsinsingle-crystalsandpolycrystalsItisevidentthattheeffectsofstressesandstrainsandhowthematerialsdeform,aredifferentforsingleandpolycrystalsastheformerdonothavegrainboundaries.Thissectiondescribesthesedifferences,andthedifferenceinstrainhardeningaswell,insingleandpoly-crystals.414.3.1PlasticDeformationinSingleCrystalsSlipbymovementofdislocationsisthepredominantmechanismofdeformation.Howmuchdoesasinglecrystalslip,dependsontheamountofshearproducedbytheexternallyappliedload,thecrystalorientation(becauseslipoccursonpreferredplanesanddirections)withrespecttotheloads,andthegeometryofthecrystal.Slip,ordeformation,wouldbeginwhentheshearstressonthepreferredplaneinthepreferreddirection(Itisgenerallytheclosestpackedplaneandclosestpackeddirectiononthatplane,becausethelesstheatomshavetomovetomovethedislocations,theeasieritisforsliptohappen),justexceedsacertainthresholdvalue,knownasthecriticalresolvedshearstress.Ifwewouldplotanordinarystress-straincurve,thentheyieldstresswouldbeequaltothecriticalresolvedshearstress.Figure4.2:Resolvedshearstressinasinglecrystal.Source:MechanicalMetallurgybyGeorge.E.Dieter,Chapter-4,Section4-7;E.Schimd,Z.Elektrochem.,vol.37,p.447,1931LetusconsiderFigure4.2.Thecylindershownisasinglecrystal,withcross-sectionalareaA.42fistheanglebetweentheslipplanenormal,andthetensileaxis,whereaslistheanglebetweentheslipdirectionandthetensileaxis.Asisevident,thecomponentofstressactingalongtheslipdirectionisPcoslwherePistheapplieduniaxialtensileload.Similarly,theareaoftheslipplaneshownisgivenbyA=cosf.Therefore,theresolvedshearstressisgivenas:tR=PcosfA=(cosl)=PAcosfcosl:(4.19)wherecosfcoslisknownastheSchmidfactor.Ascanbeseen,thiswillreachamaximumvalueformostcaseswhentheschmidfactorhasavalueof0.5.Theresolvedshearstressisthusdependentontheorientationofthecrystal,itsslipplanesandslipdirections.Itisalsodependent,obviously,onthedislocationdensityinthesinglecrystalandotherdefectspresentthatinteractwiththedislocations.However,thecriticalresolvedshearstressistheminimumshearrequiredtostartorcauseslipinasinglecrystal.Aftersliphasbegun,thestressrequiredtocontinuethedeformationkeepsincreasingwithincreaseinstrain.Thisphenomenon,whereinmorestressisrequiredtocauseslipbecauseofpriorplasticdeformation,isknownasstrainhardening.Oneofthemostwidelyacceptedreasonsofwhythishappensisthatwithincreaseinplasticdeformation,moreandmoredislocationsareproduced(frompreviouslyexistingonesbymanymechanisms,oneofthembeingtheFrank-Readmechanism).Thiscausesthedislocationstointeractwitheachotherandformsessilejogsthatobstructdislocationmotion,orinteractwithvacancies,orinclusionspresentthatalsoimpedetheirmovement.Manytimes,Lomer-Cottrelllocksareproducedwhichalsocontributetoareducedmobilityofthedislocations.Manytimes,thedislocationspileupnearobstaclesandsincethesepileupsmostlyconsistofsimilardislocations,theyrepeleachotherwhichgivesrisetoabackstress.Thisbackstressopposestheappliedstress.Thus,whenonereversesthestressdirection,thebackstressalreadydevelopednowactsinthesamedirectionastheappliedloadandcausestheyieldstresstodecrease.SuchaphenomenonistermedasBauschingereffect.Apartfromthebackstress,sometimes,atlargedeformations,a[forest]ofdislocationsmovingononeslipplane,mightintersectwithanotherforestofdislocationonanotheractiveslipplane!43However,suchinteractionsoccuroverdistancesoflessthan510awhereaistheinteratomicdistance.Theyarethustemperaturedependent;becausethermalattemperaturescanovercomethehardeningeffectduetodislocationforestinteractions.Butdislocationpile-upinducedbackstressandthehardeningcausedduetothatisnottemperaturedependentandisneitherstrainratedependentbecausetheyoccuroverlongdistances!InworkdonebyBasinskietal[4],temperatureandstrainratedependencecanbeusedtodeterminewhichmechanism(thebackstressorthedislocationinteraction)ismoreprevalent,oristhemaincontributortostrainhardeninginsinglecrystals.inmanysinglecrystals,especiallywithfcclatticestructure,andifthematerialhashighstacking-faultenergy,thenratesofstrainhardeningareexpectedtobelowerascrossslipbecomesprevalent.4.3.2PlasticDeformationinPolycrystalsDeformationinpolycrystallineaggregatesisdifferentandmorecomplexthanthatinsinglecrystalsbecauseofthepresenceofgrainboundaries,sub-grainboundaries,preferredorientations,prefer-entialdispersionofsecondphases,etc.Grainboundariesarearguablythemostimportantofthese.Ordinaryhigh-anglegrainboundarieshavearatherhighsurfaceenergy,andthusactaspre-ferredsitesfordiffusion,phasetransformation(nucleationofotherphases)orsegregation,andprecipitateaggregation.However,dependingonthetemperaturerangeinwhichtheplasticde-formationistakingplace,thepurityofthemetal,andthestrainrate,grainboundariescaneitherstrengthenthesolidorweakenit.Atroomtemperatures,anduptohalfthemeltingpointoftheductilematerialwitharelativelyhighstrainrate,grainboundariesincreasethestrainhardeningbyactingasbarrierstodislocationmotion.However,undercreepconditions,i.e,highertemperaturesandslowstrainrates,mostofthedeformationhappensmostlyatgrainboundaries.Thislocalizationcauseseffectssuchasgrainboundaryslidingandgrainboundarymigrationwhichleadstofracturewhichalsotakesplaceatthegrainboundaries.Thetemperatureatwhichthishappens,i.e,grainboundariesbecomeweakerthanthebulkofthegrain,iscalledequicohesivetemperature.Butbelowthis44temperature,Grainboundarieswillalwaysbestrongerthanthegrainandwillalwayshaveahigherstrainthanthebulkbecausegrainboundariesarebarrierstoslip,andintroducediscontinuityinthelatticethatthedislocationscannotovercome.Thisalsocayusesmultipleslipsystemstoactivatewhichcausesevenfurthercomplexitiesinslipandleadstoformationofmorebarrierstoslip.Dislocationpileupsoccurinpolycrystallinematerialsatthegrainboundary,andnotjustatobstacles(asisthecasewithsinglecrystals).However,duetopresenceofmanyslipplanesinfccorbccmetals,dislocationpileupsareonlyatlesserstrains,buttheyaremoreeffectiveinthecaseofhcpmetalsbecausehcplatticehasonlyoneeasyslipplanef0001g.However,justdislocationpileupsatgrainboundaries,misorientation,andsliponmultipleplanesestimatestrainhardeningofpolycrystalswellbelowtheexperimentalvalues.Togetabetterestimate,grainsizeeffectshavetobetakenintoconsiderationbecausetheyieldstressdependsonthegrainsizesasdescribedbytheHallPetchequation(equation3.15).Apartfromthat,presenceoflowanglegrainboundaries,orsub-grainboundariesinsideindividualgrainsresultsinaveryincreaseinstrengthwithnottoomuchreductioninductility.Also,thereareprocessesofsolidsolutionhardening,strengtheningduetothepresenceofsecondphaseparticlesandhardeningduetointeractionofdislocationsandpointdefectswhichresultinoveralldeformationhardeningofpolycrystallineaggregatesandaffectplasticity.However,inourworkwehaveconsideredonlyapure,singlephasedfccpolycrystal.Eachofthegrainshavetheirownvaluesofcriticalresolvedshearstresses,andthusdeformatdifferentstresses.Butafterdeformationhasbegun,andextensivecoldworkhasbeendone,thepolycrystalsdevelopapreferredtexture.Grainsrotateunderstress,tomaketheirslipplanesparallel,andthusgrainrotationisoneoftheeffectsofpolycrsytallineplasticdeformation!45CHAPTER5CONSTITUTIVELAWSConstitutivelawsaretheestablishedlawsforelasticandplasticdeformations,representedasasetoffundamentalequationsthatdescribe,andparameterizethemicrostructure,dynamicsofelasticandplasticdeformation,andestablishtheconstitutionofthemathematicalmodelthatissolvedtoobtaintheresultsofthesimulations.5.1PhenomenologicalPowerLawforFCCcrystalsInthiscase,aphenomenologicalpowerlawconstitutivemodelforanfccstructurehasbeencon-sidered,thatintroducestheplasticwdynamicsintothecomputation.Thismakesthealgorithmcalculatetheplasticshearstrainrategaofeachofthetwelveslipsystemsa=f111gh110iforafacecenteredcubiclattice,asafunctionofthesecondPiolaŒKirchhoffstressS.ThegoverningequationsinthisPhenomenologicalpowerlawmodel(innoparticularorder)foranfccstructureareoutlinedbelow:ThedeformationgradientF,whendeformationoccursfromareferencecontoaresultingone,canbeseparatedintoelasticandplasticcomponents:F=FeFp)Fe=FFp1(5.1)whereFeistheelasticandFpistheplasticcomponent.Also,sinceweareinterestedintheplasticpart,wecanexpresstheelasticcomponentintermsoftheplasticcomponentasshownabove.thedeformationgradientsaretensorialquantities.TheelasticGreenstraintensorEeisexpressedas:Ee=12(FeTFeI)(5.2)whereFeTisthetransposeoftheelasticdeformationgradient,andIistheidentitymatrix.46TheSecondPiolaŒKirchhoffstresstensor[S]isgivenby:S=[C]Ee(5.3)where[C]istheelasticitytensor,ananisotropicelasticstiffnesstensorthatrelatestheelasticdeformationgradienttothesecondPiolaŒKirchhoffstress.Therateofchangeoftheplasticpartofthedeformationgradientisgivenby:Fp=LpFp(5.4)whereLpistheplasticwvelocitygradientgivenby:Lp=Nåa=1gaPa(5.5)whereaistheslipsystem,Nisthenumberofslipsystems,gaistheshearrateontheslipsystema,andPaisgivenby:Pa=mana(5.6)wheremaistheslipdirectionandnaistheslipplanenormalforthatslip-system.Theshearrateisgivenbytheequation:ga=g0taga1=nsgn(ta)(5.7)whereg0isaconstitutiveconstant,theinitialshearrate;taistheresolvedshearstressforthatslipsystema,gaistheresistanceagainstshearandnisthestrainratesensitivity.Sgnisthesignumfunctionthatgivesthesignofarealnumber.Theresolvedshearstressisgivenby:ta=S:Pa(5.8)orta=S(mana)(5.9)47thegradientofshearresistancegaisgivenas:ga=Nåb=1habgb(5.10)wherebisalsotheslipsystemindex,andNisthenumberofslipsystems.habisthehardeningmatrixthatrepresentstheinteractionbetweendifferentslipsystemsaandbandisgivenbyhab=qabh0"sgn 1gbgb¥!1gbgb¥a#(5.11)whereqabisthelatenthardeningmatrix,h0andaareconstitutiveconstants,gbistheresistancetoshearonthebslipsystem,andgb¥isthesaturatedshearresistanceontheslipsystemb.theshearresistancesasymptoticallyevolvetowardsthesaturatedvalue.Thus,thisphenomenologicalpowerlawconstitutivemodelhastenparametersforcomputation.Inthenextchapter,wedescribethemethodologyofcomputationbythePolarisationschemeofthespectralsolutionmethod.48CHAPTER6THESPECTRALMETHOD:POLARISATIONComputationalmethodshavebecomemorepopularowingtotheexpensesandtimeusageassoci-atedwithexperimentalprocedurestocharacterizestructure-propertyrelationshipsofmicrostruc-turedmaterials.Lebensohnetal[5]enumeratesthemanypowerfulnumericalsolutionmethodsandtoolsthathavebeendevelopedtopredictresponsesofsuchmaterialsundergoingplasticdeforma-tion.Finiteelementmethodshavealsobeenusedincrystalplasticityproblemsextensivelytopredictmoreaccuratebehaviorandtoobtainlocallyresolvedmicromechanicalstress/strain(MikaandDawson[6],1999,Cailletaudetal[7]andClaytonandMcDowell[8]).Howeverthesizeoftheelementorthevolumeelementthatrepresentsthemicrostructureataparticularmaterialpointthatonecanspecifyinaelementanalysis,islimitedbythetimetakentonumericallysolvepartialdifferentialequations.Thecomputationalcostassociatedisforbidding.TheSpectralmethodisanalternativetoit,andinthisbodyofwork,comparisonsandcontrastshavebeendrawnbetweentheresultsobtainedbyFEsimulationsandpolarisation.Eventhoughthespectralmethodisfairlynewandwasoriginallyintroducedin1994byMoulinecandsouquet[9],ithasbeenusedsuccessfullyusedwithpolycrystallinematerials(Gren-neratetal[10]andLebensohnetal[5]).It'ssuperiorityoveraelementapproachhasbeendemonstratedbyPrakashandLebensohn[11]andLiuetal[12].Inthischapter,wetalkaboutthepolarisationschemeofspectralsolution.6.1ThekinematicsLetusconsiderinarealspaceR3,ahexahedralvolumeelementormicrostructuraldomainB0inwhichweareinterestedandonwhichanexternalmacroscopicdeformationgradient¯Fhasbeen49imposed.ThisdomainwillchangeandresultinadeformedB.Abyc(x):x2B0!y2BmapspointsxinthereferenceB0topointsyinthedeformedB.ThismapofdeformationcanbesplitintothesumoftheappliedmacroscopicdisplacementandalocallydisplacementŸw:c(x)=¯Fx+Ÿw(x)(6.1)onsurfaces¶Band¶B+,aperiodicityofŸw=Ÿw+isimposed.Similarly,thetotaldeformationgradientFisthegradientofthedeformationmappingc(x),thatisF=¶c¶x=!Ñc=gradc.Itcanbedecomposedintothemacroscopicimposedgradient¯FandalocallygradientŸF:F=¯F+ŸF(6.2)whereŸF=ŸwÑ=!ÑŸw=gradŸw(6.3)6.2StaticEquilibriumTheresponseofthematerialisgovernedbythephenomenologicalconstitutivelaw(chapter6).ThisrelatestheFirstPiolaKirchoffstressP(orthesecondpiolaKirchoffstressbecauseS=F1P),ateverypointinthematerialintheinitialreferenceymeansofadensityfunctionalW,suchthatP(x)=dWdF(x)=fx;F;F;x(6.4)wherexiasthesetofcontinuouslyevolvingvariablesandparametersfromtheconstitutivedescrip-tion.Pmightalsobeafunctionofitsneighborhood,butthatisnotconsideredinourmodeling.ThestaticequilibriumcanbeformulatedbyeithertheDirectVariationalortheMixedVariationalformulation.Thepolarisationschemeusesthemixedvariationalformulationasdescribedbelow:ToobtaintheequilibriumdeformationweminimizeWoveralldeformationgradientsthatsatisfytheprescribedmacroscopicdeformationandthelocalperiodicityasmentionedin50equations6.1to6.3.OneoftheotherconstraintsimposedisthecompatibilityofthedeformationgradientminF[W(F)]!F=!Ñc(6.5)IfweintroduceaLagrangemultiplierL(x)andapenaltyterm,whereastiffnesstensorAisusedastheparameterforpenalty,thentheresultingequationbecomes:L[F;c;L]=W+ZB0F(x)L(x)[!Ñc(x)F(x)]dx+12ZB0[!Ñc(x)F(x)]A[!Ñc(x)F(x)]dx(6.6)AccordingtoFortinandGlowinski[13],thesaddlepointofthisequation,ortheminimum,isthestaticequilibriumcondition.Thisresultsinthefollowingthreestationaryconditions:dLdF(x)=P(x)F(x)L(x)+AF(x)!Ñc(x)=0(6.7)dLdc(x)=ÑhF(x)L(x)AF(x)!Ñc(x)=0(6.8)anddLdL(x)=!Ñc(x)F(x)=0(6.9)WhenaFouriertransformisappliedtotheseperiodicfunctions,wegettheequivalentequa-tions:P(k)LR(k)+A[F(k)c(k)ik]=0(6.10)c(k)=A(k)1(AF(k)LR(k))ik(6.11)andc(k)ikF(k)=0(6.12)whereLR(x)=F(x)L(x),andA(k)isanacoustictensor,asA(k)a(k)=A[a(k)ik]ik(6.13)51foravectora(k).Usingequation6.11in6.10and6.12toeliminatec(k)yieldstheofthedeformationgradientatequilibriumasthesolutionto:Rmixed[F(k);L(k)]:=8><>:P(k)LR(k)+AfF(k)G(k)fAF(k)LR(k)ggF(k)G(k)fAF(k)LR(k)g9>=>;=0:(6.14)wheretheG(k)operatorissuchthatG(k)T(k)=hA(k)1T(k)ikiikforatensorT(k).Aftertheformulationoftheconstitutivelaws,kinematicsandstaticequilibriumconditions,wenowtakealookatthenumericalimplementation.6.3NumericalImplementationTheframeworkforthenumericalimplementationisbasedonthemethodologyofdevelopedbyEisenlohretal[14],wherethedomainB0isdiscretizedintoaregulargridofNpoints(N=Nx+Ny+Nz)inthreedimensions.Thesolutionisapproximatedinadiscretizedfourierspaceassociatedwiththegrid.ThestiffnessAisobtainedastheaverageofthemaximumandminimumvalueofdPdF(x)F.Therearethreeschemesfordiscretization:Basicscheme,Lagrangemultiplierbasedscheme,andthePolarisationscheme.Inourbodyofwork,wehaveextensivelyusedthepo-larisationscheme.6.3.1polarisationschemeofdiscretizationInthisscheme,Rmixedfromequation6.14isexpressedintermsofthedeformationgradientF(x)andarescaledpolarisationFt(x)suchthatFt(x)=Fl(x)+F(x).Thus,usingacolloca-tionbasedapproachatthegridpointstodiscretizethemixedvariationalformulationofthestaticequilibrium,andtransformingtheequation6.14asmentioned,wearriveatawellscaledsystemof52equations:Rt[F(x);Ft(x)]:=A1fP(x)LR(x)g+bF(x)F1[G(k)fbAF(k)aLR(k)gifk6=0](6.15)Rt[F(x);Ft(x)]:=A1fP(x)LR(x)g+bF(x)F1[fbFBCgifk=0](6.16)andRt[F(x);Ft(x)]:=bF(x)F1[G(k)fbAF(k)aLR(k)gifk6=0](6.17)Rt[F(x);Ft(x)]:=bF(x)F1[fbFBCgifk=0](6.18)whereLR(x)=F(x)A[Ft(x)F(x)].ThecoefaandbaretherelativeweightsfortheequilibriumandcompatibilityconditionsrespectivelyandtheyhaveaneffectontheconvergencerateofthesolutionasexpoundeduponbyMoulinecandSilva[15].FBCisthedeformationgradientattheboundaryconditions.Theconstitutiveevaluationisperformedonthecompatibledeforma-tiongradientFc,whichisgivenby:Fc(x)=b1G(x)[bAF(x)aF(x)AfFt(x)F(x)g](6.19)Sincetheconstitutiveevaluationisperformedonthiscompatibledeformationtheconver-gencespeedincreases.6.3.2NumericalsolutionstrategyBalayetal[16]2013,describesaPETSclibraryofnon-linearsolutionmethodswhichareem-ployedtosolvethesystemofdiscretisedequations.Theresidualataninterationisobtainedbyevaluatingequation6.15,to6.18atthegridpoints.TheconvolutionoperationtoobtaintheGfunctional,usesfastfouriertransforms.InDAMASK,outofallthePETSclibrarysolutionmeth-ods,thenon-lineargeneralisedminimumresidualmethodisthedefault.Inthismethod,anup-datedsolutionfgn+1isfoundbylinearlycombiningmprevioussolutionsinaKrylovsubspaceKn=spanffgn;:::::;fgnm+1gforwhichthenormoftheresidualisminimum:minfgn+12Kn¶R¶fgn[fgn+1fgn]+Rfgn2(6.20)53ThisresidualiscalculatedwithoutJacobians(OosterleeandWashio[17]).Theupdateddeforma-tiongradientisaLinearcombinationofcompatibleDAMASKalsosupportstwoothernumericalsolutionstrategies,namelyNon-LinearRichardsonandInexactNewtongener-alisedminimumresidual,describedinShantharaj.Petal[18].InthenextchaptertheProcedurethatwasfollowedtoanswertheresearchquestionofinterest"whetherornottheelementmethodunderestimatesthegrainboundarystrainsascomparedtostrainsatthebulkbecauseofitsinherentstiffnessandinabilitytoincreasegridresolution,andifthespectralsolverdoesabetterjobatit"hasbeendescribed.54CHAPTER7PROCEDURETheDüsseldorfadvancedmaterialssimulationkit(F.Rotersetal[19])isthesoftwarethatwasusedtosimulatetheoligocrystalunderuniaxialtension.Itwasbuilttosimulatecrystalplasticitywithinastraincontinuummechanicalframework(Eisenlohretal[14]).Ithasbothelementcapability(MSCMarcMentat)andinbuiltspectralsolvers.Moredocumentationcanbefoundathttp://www.damask.mpie.deThemodelingoftheOligocrystalbeginswiththecreationofageometrythatcontains,asthenamesuggests,thegeometrydescriptionofthepolycrystallineaggregates,viz,thetexturedata,thegrainIDsallocatedtoeachtexture,thecoordinateswhereeachgrainexists,andthespatialarrangementoftheaggregates.Anyperipheralairlayersorrigidplatesarealsodescribedinhere.Whatthegeomcommandsdoisassignthestipulatedtexturestoeachmaterialpoint(pixel),searchtheneighborhood(globally)forsimilartextureswitha5°misorientationtolerance,andgroupsthemasonegrain;rather,assignsthesamegrainIDtothem.Thenextstepisthecreationofamaterialthatcontainsinitselftheconstitu-tivelawsanditsdescription,tobeappliedtotheairlayers,rigidbaseplates,theactualpolycrys-tallineaggregate,thetexturedata,thehomogenizationschemeifany,theoutputneededfromthesimulation,andeachofthosearethenallocatedtothegrains.Thethirdstepisthecreationofanumericalwhichthetolerances,theerrordivergences,numericalsolvingstrategies,discretizationschemesofthefourierspaceandalltheparametersdescribedinthepreviouschapterthatareneededtosolvetheequationsofthespectraldescriptionnumerically.Aloadcaseneedstobewhichcontainsinitselfthedeformationgradient,therateofstrainthenumberofincrements,andthenumberofincrementsafterwhichtheoutputiswrittentotheoutput55Inthenextsections,wedescribeinsomewhatdetail,thecreationoftheseforoursimula-tion.7.1SettinguptheGeometryinDAMASKenvironmentTheEBSDdatacontainingthexandycoordinates,theOrientationmatrixforthematerialpointsandthephasedatawasobtainedfromSandiaNationalLaboratories(SNL),Albuquerque,NewMexico,USA.ThestepwastocreateanIPFcolorimageseenalongthe(100),(010)andthe(001)directions,tocomparethemwiththeactualEBSDimagesobtainedfromSNL.Figure7.1showsthecomparison.Itwasobservedthattheimagesmatchforallthreedirectionalviews,andthusthedataobtainediscorrectandcomparable.Figure7.1:ComparisonoftheirIPFcolormaps,withthoseproducedbyoursoftware.TheontheleftarethoseproducedatSNL,experimentallyintheEBSD.TheonesattherightareproducedbytheimageDataRGBcommandinDAMASK56Thenextstep,wastocreateatwo-dimensionalgeometry(becausetheEBSDdatasetcon-tainsonlytwocoordinates)byusingthe'geom_fromTable'commandinDAMASK(Figure7.2),andextrudeitinthez-directiontoobtainthe3-dimensionaloligocrystal.Figure7.2:The2dimensionalgeometry(thicknessinzdirection=1)whichwillbeextruded.Thexaxisistowardstherightandtheyaxisisupwards.Theaspectratiosforsizeandgridresolution(1084x294x168)waskeptsimilartothatintheworkdonebyLimetal[1].ItisshowninFigure7.3.Figure7.3:TheOligocrystal,madebyextrudingthe2Dstructureinthezdirection.Theaspectratioshavebeenmaintained.Resolution:1084x294x168.Voxelaspectratiox:y:z::1:1:157Fromthis,threemoregeometrydescriptionswithdecreasingresolutionswerecreated,viz.,viz1084x294x21(voxelaspectratio:1:1:8),542x147x21(Voxelaspectratio:1:1:4),and271x74x10(voxelaspectratio:1:1:4).Tothisend,thecommandusedwasgeom_rescale.Thenextstepwastoattacharigidbaseplate,10%ofthelengthoftheoligocrystalinthicknesswithmicrostructureID1tothey-zplaneofthe3Doligocrystal.Thisactsaaperiodicboundaryconditionthatnodeformationhappensinthex-directionwhichisalsothedirectionoftheuni-axialtension.Aperiodicisotropicairlayerwasattachedlaterally(xzandxyplanes)aroundtheoligocrystal,10%ofthethicknessoftheoligocrystalinthickness,anditwasassignedagrainIDvalueof2.Initially,theairlayerwasattachedandthentherigidplate,andthisresultedintheairlayertobecomeperiodicallydiscontinuousandtherigidbase-platetobecomeperiodicallycon-tinuous.Thiswasachievedbyusingthecommandsofgeom_translateandgeom_canvas(Figure7.4and7.5)Figure7.4:TheOligocrystalwiththeisotropicairlayeraroundit.Thiswaslaterchangedinthesensethatweaddedthebaseplateandthentheairlayer,thusmakingtheairlayercontinuous,whilethebaseplatebecamediscontinuous.ThisnewgeometrywhenrenderedinParaview,lookslikethoseshowninFigure7.6and7.758Figure7.5:TheOligocrystalwiththerigidelasticbaseplateaddedatbothends.ThismakesthebaseplatecontinuousbecauseoftheperiodicstructureandrenderstheairlayerdiscontinuousFigure7.6:Therigidbaseplateisaddedinthisnewassignmentofboundaryconditions59Figure7.7:TheisotropicplasticairlayerisaddedaroundthebaseplateandtheOligocrystal.Thismakesthebaseplateperiodicallydiscontinuous,whiletheairlayerbecomescontinuous.AllthecolorassignmentsweredoneusingtheaddIPFcolorandimageDataRGBcommandsinbuiltinDAMASK.Paraviewwasusedforvisualization.Thecommandgeomcheckproducesawith.vtrextension,thatcanberenderedinparaview.Tothiswecanattachthecolordata,andbasicallyanyscalarvaluesbyusingthecommandvtkaddRectilinearGridDatathatweneedtovisualizeinParaview.ThedetailedstepbystepusageofthevariouscommandsisshowninAppendixA.7.2SettinguptheMaterialTheassupportedinDAMASKhasmanysections.Theoneishomoge-nization,towhichavalueofNONEwasassigned.ThesecondisCrystallite,wheretheoutputsweneed(deformationgradientf,elasticdeformationgradienttensorfe,piolakirchoffstressp,grainrotation,textureandorientation)wereandthateachcrystalliteisasinglecrystalby60itselfbuttheentiresampleisapolycrystal.TheninthephasesectionthreephaseswerePhenomenologicalpowerlawfccaluminumfortheoligocrystal,rigidelasticsolidforthebase-plateandanisotropicairlayer.Nextsectionshavethemicrostructuresandtextureswhichassigntheorientationstoeachgrain,andalsospecifywhichgrainhaswhatphase(s)andhowmuchofeachphaseispresent,whatcrystalliteandwhatgrainID.ThematerialhasbeenshowninAppendixB.7.3SettinguptheNumericThebuiltinlookslikethatshownintheAppendixC.Theerrortolerancedi-vergencewaschanged(itwasdecreasedapproximately10times)andtheSpectralsolutionmethodwasalsochanged(fromabasicmethodtoapolarisationmethod)asshowninAppendixD.7.4Simulation7.4.1SimulationbytheSpectralMethodTheloadcasewascreatedandisshowninAppendixE.Itthatthesimulationhastorununtil7%strainhasbeenachievedin70increments.Thefrequencywithwhichoutputwouldbewrittenoutintotheoutputwasalsoaswellasarestartcapabilitywasaddedsothatthesimulationcanberesumedfromthepreviousincrementifitstoppedinbetweenduetounforeseencircumstances.TheDAMASKspectralsolverwascalledbythecommand"DAMASK_spectral",insidea.qsubwhichsubmittedourjobtotheclusteratthehighperformancecomputingcenteratMichiganStateUniversity.The.qsubthewalltimeneededfortheprogramtorun,thenumberofnodesandnumberofcorespernodeneededforparallelcomputing,thememoryre-quiredforthejob,theenvironmentvariables,andtheactualcommandforsimulation.AnexemplarqsubisshowninAppendixF617.4.2SimulationbyFiniteelementmethodMSCMarcMentatwasinvokedtosetuptheelementmodel.Acommand'mentat_spectralBox'wasusedtotranslatethegeometryalreadymadeintoMentat.Otherparameterswerethensetupusingthefunctionalitiessuchasboundaryconditions,domaindecompositionetcprovidedinMentat.8domainswereusedtoreducecomputationaltime.Tomaketheoutputgivethede-siredcrystalliteresults,marc_addUserOutputcommandwasusedbeforethefemsimulationwasallowedtorunitscourse.7.5AnalysistoolsusedtoobtainandcompareresultsThepostResultscommandwasusedtoextractdatafromtheoutput(theoutputhasanextension.spectralOutincaseofspectralsolutionoutput,and.t16incaseofFEMoutput),andtheCauchystressandaleftlogarithmicstrainwasappendedtoitbyusingthecrystalliteoutputsfortheverylastincrement.Misesstressandstrainwasthencalculatedandappendedtothetable,andsowerethegrainrotations.A.vtrwasmadeforeachgeometrydescription,andtheMisesandgrain-rotationscalarsweremappedontoitandrenderedinparaviewtodrawconclusions.comparisonsweredonebetweenspatialmapsbyplottingcumulativefrequencyplotsfortheratioofvaluesateachspatialpoint.Toobtainthestrainsatthegrainboundariesandbulkofthegrain,inordertocomparethedifferencewiththosefoundbyFEMatSandia,theEuclideandistanceofeachmaterialpoint(distanceinnumberofvoxelsfromclosestgrainboundary)fromthenearestgrainboundarywasaddedtotheoutputtable,andthenaseparatetablewheretheaveragestressandstrainpergrainwerelistedwascreated.Thestrainateachmaterialpointwasthennormalizedbydividingthemwiththeaveragestrainpergrain.Thesenormalizedstrainswerethenplottedalongthey-axiswiththeeuclideandistancealongthex-axisforeachsuchgrain,andforeachgridresolutiontodrawconclusions.ThebinXYcommandwasusedtocreateaprobabilitydensitymapforthesetwoquantities,andthenitwasplottedinGri.TheactualcommandsusedforprocessingtheresultsareappendedinAppendixG.62Resultsobtainedaredescribedinthenextchapter,alongwiththecomparisonsmadeandcon-clusionsdrawnfromthestudy.Formoredetailedexplanationsofthecommands,thesoftwareused,andthevaluesthatareusedinthissimulation,pleasebereferredtotheofwebsiteofDAMASK:http://damask.mpie.de63CHAPTER8RESULTSANDCONCLUSIONSInthisbodyofworkonlythestrainsinsidethegrains,andintheoveralloligocrystal,andthegrainrotationsthathaveoccurredasaresultofthedeformationhavebeenanalyzed.Figure8.1showstheresultsthatSandiaNationalLaboratoriesobtainedfromtheCPFEMso-lution,andhowitcomparedwiththeexperimentalresults.Ascanbeclearlyseenuponcomparingtheimagesontheleftwiththoseontheright,theCPFEMsolutionunderestimatedthegrainbound-arystrainswhencomapredwiththeHighResolutionDigitalImageCorrelation(HRDIC)methodofobtainingstrainmapsexperimentally.ThismethodhasbeenelaboratedintheworkbyLimetal[1]2014.ThestrainsnearthegrainboundariesareofahighermagnitudefortheHRDICcaseascomparedtothoseintheCPFEMcase.Figure8.1:TheHRDICversusCPFEMstrainmaps.Thecolumnofimagesontheleftshowsthestrainmapoftheoligocrystalobtainedbyhighresolutiondigitalimagecorrelation,whilethoseontheleftareobtainedbyCPFEM.Inourwork,wehavefocusedonlyonthestraininthexdirection,i.e.,exx.thepositivexdirectionistowardstheright,andpositiveyisupwards.Inourwork,wefocusedonthestrainsinthexdirectiononly(exx),andhavetriedtoseeifjust64bychangingthesimulationsolutionmethodfromFEMtospectralsolution,ifthisproblemofgrainboundarystrainunderestimationcanbeovercome.ThiswouldalsorelatetothefactthatFEMisastiffersolutionmethodthanspectralsolution,andthatinherentproblemmightbetheplausiblecauseforthediscrepanciesobserved.First,astrainmapwasgeneratedbyspectralsolutionforthemeshwiththelowestresolution,andithasbeenshowninFigure8.2.Thiswasrenderedinparaview.Theboundaryconditionsforthisareshownin7.5.Figure8.2:ThestrainmapgeneratedinthreedimensionsbyDAMASK.Thecolorscalehasbeensothatanytingabbove0.3isthesameshadeofredas0.3,andanythingbelow0isthesameshadeofblueas0.WetriedtoseeifbychangingtheboundaryconditionsfromthatshowninFigure7.5,tothatinFigure7.7(i.e.,bycreatingaperiodicallycontinuousisotropicairlayerandaperiodicallydiscontinuouselasticbaseplate)whatwouldbetheeffectonthestrainsobserved.TheresultsareshowninFigure8.3ThischangeinboundaryconditionsdidnotnecessarilyhaveanyeffectontheresultsascanbegatheredfromFigure8.4.Next,meshresolutionwasincreasedjustinthezdirection,i.e,increasethezresolution2times65Figure8.3:Thecomparisonofstrainmapsuponchangeofboundaryconditions.Leftsideimagesshowtheresultsthathadtheolderboundaryconditions,whilethoseontherighthavethenewerboundaryconditions.more(271x74x21meshresolutionwithVoxelaspectratio:1:1:2)inanattempttoseeifmoreequiaxedvoxelswillhaveanyeffectonthestrainscalculated.TheresultsareshowninFigure8.5ThesimilaritybetweenthetwospatialmapsisquantitativelyshowninFigure8.6.Ifweconsiderthegrainrotations,asobtainedfromthesimulationofthelowestresolutionmesh,andtheonewithatwicelargerresolutioninthezdirection,wethatthegrainrotationsaregreaternearthegrainboundarieswhichalsogivesimpetustothefactthatstrainaccumulationsaregreatestnearthegrainboundaries.TheresultshavebeenshowninFigure8.7.Theinitialinputorientations(asQuaternions)andoutputorientations(asQuaternions)wereusedtoobtainthegrainrotationvalues.TheconjugateoftheinitialorientationwasmultipliedwiththeorientationandlastlyconvertedintoangleaxisrepresentationusingbuiltinDAMASKfunctionsandformulae,inordertoobtaintherotationdata.ThesimilaritybetweenbothgrainrotationresultshavebeenquantitativelyrepresentedinFig-ure8.8Eventhoughthegrainrotationsonthexyplaneareprettyencouragingtoourcauseandareconsistent,thoseinthexzplaneshowoscillationandabandlikestructure.Arotatedgeometry66Figure8.4:Thecomparisonofstrainmapsuponchangeofboundaryconditions,shownasthecumulativeprobabilityofratiosofstrainsateachmaterialpointcalculatedwiththetwodifferentperiodicboundaryconditions.withtheyandzdirectionsinterchanged,yieldedthefollowingresult(Figure8.9)Sincerotatingthegeometrydidnotprovideanyappreciablereductioninthebanding,andsincethenon-thicknessdirectionplanesdonotshowbanding,thentheoscillatingnatureofspectralsolutionasapossiblecandidateforcausingthebandingcanberuledout.Ahypothesiswasmadethatmaybethisishappeningbecausethevoxelsaretetragonalandnotcubic,withthezdirectionofeachvoxel2to8timesthedimensionsinthexandydirection(forex-ample,thevoxelaspectratiois1:1:4for271x74x10and1:1:2for271x74x21meshresolution).Toaddressthis,anunrotatedgeometrywithmeshresolution271x74x42wascreated,sothatthevoxelswereequiaxed(aspectratio1:1:1).Simulationswiththespectralmethodweredoneon67Figure8.5:Thecomparisonofstrainmapsuponincreasingresolutioninthezdirection.Leftsideimagesshowtheresultsthathadbeenobtainedwithalowerzresolution,whiletherightshowstheresultsobtainedafterincreasingthezresolutionbyafactorof2.Asisevidentjustbyvisualinspection,thechangeinincreasingthezresolutiondoesnoteffectanymajorchange,butthegradientsbecomesmootherthisnewgeometryandanappreciabledecreaseinbandingwasobserved(Figure8.10).Thus,forthespectralsolutionmethodtoprovideproperresultswithoutanysurprises,itisdesirabletohavethevoxelsasclosetoequiaxedaspossible.Owingtotheincreaseincomputationalcostifthegeomterywereallcreatedwithequiaxedvoxels,thegeomterydescriptionsinthisbodyofworkdonothaveappreciableequiaxedness.Grainrotationsofonlythexyplaneshavethusbeenconsidered.Afterthemysteryofthebandingwassolved,acomparisonwasmadebetweenthegrainrota-tionvaluescalculatedbyCPFEMwiththosecalculatedbyspectralsolutionforaconstantmeshresolutionof271x74x10.thevoxelaspectratioforbothcaseswere1:1:4.nobandingcanbeseenintheCPFEMsolution,evenwithnon-equiaxedvoxels,whereasthespectralsolutionshowsthebanding.Apartfromthat,mostofthegrainrotationsinthebulkofthegrainsappeartobesimilar.However,nearthegrainboundaries,thespectralsolutionseemstoestimatehigheramountofgrainrotationthanCPFEM,whichshowsthattheSpectralsolutionmethodestimateshigherstrainsneargrainboundariesascomparedtoCPFEMbecausehigherstrainconcentrationsimplyhighergrain68Figure8.6:Thequantitativecomparisonofstrainmapsuponchangeofzresolution.cumulativeprobabilityoftheratioofstrainscalculatedateachmaterialpointforresolution271x74x21tothoseat271x74x10hasbeenshown.rotations.TheresultsareshownintheFigure8.11.Aquantitativecomparisonbycalculatingthecumulativeprobabilityoftheratioofgrainrota-tionscalculatedbyFEMandthatcalculatedbyspectralsolutionmethod,isshownintheFigure8.12Thegrainrotationswerecalculatedforthedifferentresolutionsusingspectralsolution,toseeif69Figure8.7:Thecomparisonofgrainrotationsuponincreasingresolutioninthezdirection.Leftsideimagesshowtheresultsthathadbeenobtainedwithahigherzresolution.Asisevidentjustbyvisualinspection,thechangeinincreasingthezresolutiondoesnoteffectanymajorchange.achangeinresolutioneffectssubstantialchangeonthevaluescalculated.Whenthegrainrotationsfordifferentmeshresolutionsof271x74x10,542x147x21and1084x294x21areshownasspatialmaps,theydonotshowmuchvariationexceptforasmootherview(Figure8.13).Thequantitativecomparisonisdonebycalculatingtheratiosofgrainrotationsateachma-terialpointcalculatedat542x147x21and1084x294x21meshresolutiontothosecalculatedat271x74x10resolution,andthenplottingthecumulativeprobabilityonthesameplot(Figure8.14).Thisshowsthatthegrainrotationscalcuatedbythespectralmethodhavehighervaluesnearthegrainboundaries,andthatisnotjusttheeffectofresolutions,becauseachangeinresolutiondoesnoteffectmuchofachangeintheactualvaluescalculated.Movingonfromgrainrotationtostrains,theCPFEMandspectralsolutionspatialmapsforstrainsareshowninFigure8.15.70Figure8.8:Thequantitativecomparisonofgrainrotationmapsuponchangeofzresolution.Cumulativeprobabilityoftheratiosofgrainrotationscalculatedateachmaterialpointfortheresolution271x74x21tothosecalculatedat271x74x10TheFigure8.16showsthequantitativecomparisonbetweenthesespatialmaps.ThisshowsthateventhoughthestrainscalculatedarequiteclosetoeachotherinboththeCPFEMandspectralsolution,thereisaredistributionofstrain.whiletheCPFEMhasslightlyhigherstrainsinthebulkandlowerstrainsnearthegrainboundaries,thespectralsolutionhasslightlylowerstrainsinthebulkandhigherstrainsnearthegrainboundaries.Thisbecomeseven71Figure8.9:Thegrainrotationspatialmapaftersimulatingarotatedgeometry.Meshresolution271x10x74,voxelaspectratio1:4:1.Thetopfaceandfrontfaceareinterchangedfromprevioussimulationsbecauseyandzdirectionsarenowinterchangedduetotherotation.Theband-likestructurestillexistsinthethicknessdirection.Thethicknessdirectionwaszinthepreviousones,butyinthecurrentone.Figure8.10:Thegrainrotationspatialmapaftersimulatinganunrotatedgeometrywithequiaxedvoxels.Meshresolution271x74x42,voxelaspectratio1:1:1.Littletonobandingcanbeseeninthethicknessdirection.Thus,anequiaxedvoxelisdesirable.72Figure8.11:ThegrainrotationspatialmapscomparingtheCPFEMandspectralsolutionestimationsofgrainrotations.Meshresolution271x74x10,voxelaspectratio1:1:4.LittletonobandingcanbeseenintheCPFEMsolution.SpectralsolutionestimateshighergrainrotationthanFEMnearthegrainboundaries,butmostoftheotherplaces,theyestimatesimilarvaluesofgrainrotationasseeninFigure8.12moreclearwhenweobservetheGRImapwhichplotstheeuclideangrainboundarydistancewiththeratioofspectralsolutionstrainstoCPFEMstrains(Figure8.17).Thisisaprobabilitydensityplot,wherewhiteis0probabilityandblackis1probability.Thus,forablackdisc,forthateuclideandistance,thestrainratioshaveaprobabilityof1,i.e,suretooccur,and0ifitisawhitedisc.Thecolumnsofthismaphavebeennormalized,sothatthedarkestgrey(orblack)inthecolumnshowswherethestrainsaremostlylocatedatthatdistancefromthegrainboundary.Ananalysisofthisplotclearlyshowsthatatthecenterofthegrainthessmestimatesstrainslessthanfem,andthusthedarkercolorappearsbelow1,whileclosertograinboundaries,thedarkercolorsareabove1.Thus,thespectralsolutionindeedestimateshigherstrainsneargrainboundariesthananFEM.Uponconsideringspatialmapsforstrainscalculatedatvariousresolutionsbythespectralmethod,wenosubstantialchangeinthecalculatedstrainvalues(Figure8.18)withchangeinresolution.ThisfactiscorroboratedbythequantitativecomparisoninFigure8.19.differentGRImapsforeuclideandistancefromclosestgrainboundariesversusnormalizedstrains(strainateachmaterialpoint,dividedbytheaveragestrainofthegrainthhatpointbelongs73Figure8.12:ThecumulativeprobabilityofratiosofgrainrotationscalculatedbyFEMtothosecalculatedbyspectralsolutionto)intheoligocrystalsatvariousresolutionsshowthatstrainsclosertograinboundariesarehigherthantheaverageandaswegoclosertothebulkofthegrains,itapproachestheaveragestrainvalue.Figure8.20,Figure8.21andFigure8.22showtheseGRImaps.Sincetheeuclideandistanceisintermsofnumberofvoxels,itincreasesintegrallywithintegralincreaseinresolutionofthespectralmesh.However,inFigure8.23whichshowsthesamenormalizedstrainsintheoligocrystalversustheeuclideangrainboundarydistanceforthecaseofFEMcalculations,thehighestprobabilitynearthegrainboundaryisforanormalizedstrainbelow1.Thus,thespectralsolutionmethoddoesindeedestimatehigherstrainsneargrainboundariesthantheCPFEM.However,itisstillunclearwhetherornotitisclosertoexperimentaldataduetoalackofexperimentalanalysis.Thiscanbedoneinfurtherresearch,alongwithassigningabccphasetotheoligocrystalinplaceof74Figure8.13:Thegrainrotationspatialmapsfordifferentresolutions.thecalculationsaredonebyspectralmethod.Thereseemstobenoeffectivechangeexceptsmoothinguponincreasingresolution.fcc,todrawdirectcomparisonswithworkdonebyLimetal[1]atSNL..Toconclude,althoughitisevidentthattheSpectralsolutionestimateshigherstrainsnearthegrainboundariesascomparedtotheCPFEM,itisnotevidentwhichoftheseisclosertotheexper-imentaldata.Furtherresearchpertainingtodirectcomparisonstoexperimentaldataiswarranted.Also,equiaxedvoxelsaredesirableincaseofspectralsolutionwhereastheFEMdoesnothaveanissuewiththeaspectratioofthevoxels.Itisalsoclearthateventhoughtheoverallstrainisconstant,themaindifferencebetweenspectralmethodsandCPFEMliesinthedistributionofthestrains.IncaseofCPFEM,lowerstrainsatgrainboundariesandhigherstrainsatthebulkisobservedwhereasforaspectralsolution,thereisahigherstrainestimationneargrainboundaries,butlowerstrainsarecalculatedinthebulkofthegrain.75Figure8.14:Quantitativecomparisonsofgrainrotationsatdifferentresolutions.Therangeoftheratiosisfrom0.1to5.0whichindicatessimilaritybetweenthespatialmapsshowinggrainrotations.76Figure8.15:SpatialmapsforstrainscalculatedbyCPFEMandspectralsolution.TheylooksimilarexceptfortheregionsnearthegrainboundariesFigure8.16:QuantitativecomparisonsofstrainscalculatedbyFEMandspectralmethod.TherangeoftheratiosofstrainscalculatedbyFEMtothosebyspectralmethodisfrom0.1to5.0whichindicatessimilaritybetweenthespatialmapsshowingthestrains.77Figure8.17:Distancefromclosestgrainboundaryversusratioofspectralsolutionmethodstrainstofemstrains.Figure8.18:Spatialmapsofspectrallycalculatedstrainsatvariousresolutions.78Figure8.19:Quantitativecomparisonofstrainscalculatedatvariousresolutionsbythespectralmethod.79Figure8.20:Distancefromclosestgrainboundaryversusnormalizedstrainscalculatedspectrallyfor271x74x10resolution.80Figure8.21:Distancefromclosestgrainboundaryversusnormalizedstrainscalculatedspectrallyfor542x147x21resolution.81Figure8.22:Distancefromclosestgrainboundaryversusnormalizedstrainscalculatedspectrallyfor1084x294x21resolution.82Figure8.23:DistancefromclosestgrainboundaryversusnormalizedstrainscalculatedbyFEMat271x74x10resolution.8384                          APPENDICES 85  APPENDIX A AUTOMATOR CODE    # ! / b i n / bash   SOURCE=" ${BASH_SOURCE[ 0 ] } " w h i l e [  h   "$SOURCE" ] ; do DIR=" $ (   cd  P   " $ ( dirname  "$SOURCE" ) "   &&   pwd ) " SOURCE=" $ ( r e a d l i n k  "$SOURCE " ) " [ [  $SOURCE  != / * ] ]  &&  SOURCE="$DIR / $SOURCE" done DIR=" $ (   cd  P   " $ ( dirname  "$SOURCE" ) "   &&   pwd ) " echo  $DIR t h e F i l e =$DIR / . . / d a t a / Oligo 1 _ EBSD_ Initial_ xy G . csv o r i g i n a l D a t a =${ t h e F i l e %.*}. t x t t r  ,   \ t  <   $ t h e F i l e   >   $ o r i g i n a l D a t a r e L a b e l  l  X, Y, G11 , G12 , G13 , G21 , G22 , G23 , G31 , G32 , G33   \ s  1 _pos , 2 _pos , 1 _a , 2 _a , 3 _a , 1 _b , 2 _b , 3 _b , 1 _c , 2 _c , 3 _c \ < $ o r i g i n a l D a t a  >  w o r k i n g _ o r i g i n a l D a t a . t x t new File =$DIR / . . / d a t a / 5 _ d e g r e e s / w o r k i n g _ o r i g i n a l D a t a    > geom_ from Table  c o o r d i n a t e s pos t  5  d e g r e e s  phase \ > Phase  a   a  b   b  c   c   ${ new File } . t x t 86  Nmicro =  gr e p  m i c r o s t r u c t u r e s   \ ${ new File } . geom  |   awk p r i n t $2   show Table  n o l a b e l s   i n f o  ${ new File } . geom  | t a i l \ n  $ ( ( 2 + 6 * $Nmicro ) )   >   m a t e r i a l . c o n f i g  $DIR / . . / code / eulers_ f rom Geomheader . sh \ ${ new File } . geom g r a i n O r i e n t a t i o n s . t x t mv  ${ new File } . geom  1084 x294 . geom  g e o m _ r e s c a l e  g   0 . 5 x   0 . 5 x   1 . 0 x   <1084 x294 . geom  >542 x147 . geom g e o m _ r e s c a l e  g   0 . 5 x   0 . 5 x   1 . 0 x   <542 x147 . geom  >  271 x74 . geom g e o m _ r e s c a l e  g   1084  294  168 s 1 . 0 x   1 . 0 x  168 x \ <1084 x294 . geom  >1084 x294x168 . geom g e o m _ r e s c a l e  g   1 . 0 x   1 . 0 x   0 . 1 2 5 x  s 1 . 0 x   1 . 0 x   1 . 0 x \ <1084 x294x168 . geom  >1084 x294x21 . geom g e o m _ r e s c a l e  g   542  147  21  s 1 . 0 x   1 . 0 x  168 x \ <542 x147 . geom  >  542 x147x21 . geom g e o m _ r e s c a l e  g   271  74  10  s 1 . 0 x   1 . 0 x  168 x \ <271 x74 . geom  >271 x74x10 . geom    geom_ to Table p pos <1084 x294x168 . geom >1084 x294x168 . t x t geom_ to Table p pos <1084 x294x21 . geom >1084 x294x21 . t x t geom_ to Table p pos <542 x147x21 . geom >542 x147x21 . t x t geom_ to Table p pos <271 x74x10 . geom >271 x74x10 . t x t   a d d I P F c o l o r   e u l e r s e u l e r   d e g r e e s \ 87   p o l e 1 . 0 0 . 0 0 . 0 g r a i n O r i e n t a t i o n s . t x t a d d I P F c o l o r   e u l e r s e u l e r   d e g r e e s \  p o l e 0 . 0 1 . 0 0 . 0 g r a i n O r i e n t a t i o n s . t x t a d d I P F c o l o r   e u l e r s e u l e r   d e g r e e s \  p o l e 0 . 0 0 . 0 1 . 0 g r a i n O r i e n t a t i o n s . t x t   addMapped  map  m i c r o s t r u c t u r e   l a b e l \ IPF_ 100 _ cubic  a s c i i t a b l e g r a i n O r i e n t a t i o n s . t x t \ 271 x74x10 . t x t 542 x147x21 . t x t 1084 x294x21 . t x t 1084 x294x168 . t x t addMapped  map  m i c r o s t r u c t u r e   l a b e l \ IPF_ 010 _ cubic  a s c i i t a b l e g r a i n O r i e n t a t i o n s . t x t \ 271 x74x10 . t x t 542 x147x21 . t x t 1084 x294x21 . t x t 1084 x294x168 . t x t addMapped  map  m i c r o s t r u c t u r e   l a b e l \ IPF_ 001 _ cubic  a s c i i t a b l e g r a i n O r i e n t a t i o n s . t x t \ 271 x74x10 . t x t 542 x147x21 . t x t 1084 x294x21 . t x t 1084 x294x168 . t x t   # a d d I P F c o l o r   e u l e r s e u l e r   d e g r e e s   p o l e 1 . 0 0 . 0 0 . 0 \ 271 x74x10 . t x t 542 x147x21 . t x t 1084 x294x21 . t x t 1084 x294x168 . t x t # a d d I P F c o l o r   e u l e r s e u l e r   d e g r e e s   p o l e 0 . 0 1 . 0 0 . 0 \ 271 x74x10 . t x t 542 x147x21 . t x t 1084 x294x21 . t x t 1084 x294x168 . t x t # a d d I P F c o l o r   e u l e r s e u l e r   d e g r e e s   p o l e 0 . 0 0 . 0 1 . 0 \ 271 x74x10 . t x t 542 x147x21 . t x t 1084 x294x21 . t x t 1084 x294x168 . t x t   v t k _ r e c t i l i n e a r G r i d  m   p o i n t  p   pos \ <1084 x294x168 . t x t > mesh_ 1084x294x168 . v t k v t k _ r e c t i l i n e a r G r i d  m   p o i n t  p   pos \ 88  <1084 x294x21 . t x t > mesh_ 1084x294x21 . v t k v t k _ r e c t i l i n e a r G r i d  m   p o i n t  p   pos \ <542 x147x21 . t x t  >  mesh_ 542x147x21 . v t k v t k _ r e c t i l i n e a r G r i d  m   p o i n t  p   pos \ <271 x74x10 . t x t >  mesh_ 271x74x10 . v t k   v t k _ a d d R e c t i l i n e a r g r i d D a t a  m   p o i n t   v t k  mesh_ 1084x294x168 . v t k \ c  IPF_ 100 _ cubic <1084 x294x168 . t x t > mesh_ 1084 x 294 x 168 _ added 100 . v t r v t k _ a d d R e c t i l i n e a r g r i d D a t a  m   p o i n t   v t k  mesh_ 1084x294x21 . v t k \ c  IPF_ 100 _ cubic <1084 x294x21 . t x t > mesh_ 1084 x 294 x 21 _ added 100 . v t r v t k _ a d d R e c t i l i n e a r g r i d D a t a  m   p o i n t   v t k  mesh_ 542x147x21 . v t k \ c IPF_ 100 _ cubic <542 x147x21 . t x t >  mesh_ 542 x 147 x 21 _ added 100 . v t r v t k _ a d d R e c t i l i n e a r g r i d D a t a  m   p o i n t   v t k  mesh_ 271x74x10 . v t k \ c   IPF_ 100 _ cubic  <271 x74x10 . t x t > mesh_ 271 x 74 x 10 _ added 100 . v t r   v t k _ a d d R e c t i l i n e a r g r i d D a t a  m   p o i n t   v t k  mesh_ 1084x294x168 . v t k \ c  IPF_ 010 _ cubic <1084 x294x168 . t x t > mesh_ 1084 x 294 x 168 _ added 010 . v t r v t k _ a d d R e c t i l i n e a r g r i d D a t a  m   p o i n t   v t k  mesh_ 1084x294x21 . v t k \ c  IPF_ 010 _ cubic <1084 x294x21 . t x t > mesh_ 1084 x 294 x 21 _ added 010 . v t r v t k _ a d d R e c t i l i n e a r g r i d D a t a  m   p o i n t   v t k  mesh_ 542x147x21 . v t k \ c  IPF_ 010 _ cubic <542 x147x21 . t x t > mesh_ 542 x 147 x 21 _ added 010 . v t r v t k _ a d d R e c t i l i n e a r g r i d D a t a  m   p o i n t   v t k  mesh_ 271x74x10 . v t k \ c   IPF_ 010 _ cubic  <271 x74x10 . t x t > mesh_ 271 x 74 x 10 _ added 010 . v t r   v t k _ a d d R e c t i l i n e a r g r i d D a t a  m   p o i n t   v t k  mesh_ 1084x294x168 . v t k \ c   IPF_ 001 _ cubic <1084 x294x168 . t x t > mesh_ 1084 x 294 x 168 _ added 001 . v t r 89  v t k _ a d d R e c t i l i n e a r g r i d D a t a  m   p o i n t   v t k  mesh_ 1084x294x21 . v t k \ c  IPF_ 001 _ cubic <1084 x294x21 . t x t > mesh_ 1084 x 294 x 21 _ added 001 . v t r v t k _ a d d R e c t i l i n e a r g r i d D a t a  m   p o i n t   v t k  mesh_ 542x147x21 . v t k \ c  IPF_ 001 _ cubic <542 x147x21 . t x t > mesh_ 542 x 147 x 21 _ added 001 . v t r v t k _ a d d R e c t i l i n e a r g r i d D a t a  m   p o i n t   v t k  mesh_ 271x74x10 . v t k \ c   IPF_ 001 _ cubic  <271 x74x10 . t x t > mesh_ 271 x 74 x 10 _ added 001 . v t r   g e o m _ t r a n s l a t e  m   2   \ <1084 x294x168 . geom >1084 x 2 9 4 x 1 6 8 _ t r a n s l a t e d . geom g e o m _ t r a n s l a t e  m   2   \ <1084 x294x21 . geom >1084 x 2 9 4 x 2 1 _ t r a n s l a t e d . geom g e o m _ t r a n s l a t e  m   2   \ <542 x147x21 . geom >542 x 1 4 7 x 2 1 _ t r a n s l a t e d . geom g e o m _ t r a n s l a t e  m   2   \ <271 x74x10 . geom  >271 x 7 4 x 1 0 _ t r a n s l a t e d . geom   geom_ canvas  g   1084  324  186  o  0  15  9  f i l l 2  \ <  1084 x 2 9 4 x 1 6 8 _ t r a n s l a t e d . geom  >  1084 x 2 9 4 x 1 6 8 _ a i r . geom geom_ canvas g   1194  324  186  o  55  0  0  f i  l l 1  \ <1084 x 2 9 4 x 1 6 8 _ a i r . geom  >  1084 x 2 9 4 x 1 8 6 _ a i r _ s u p p o r t e d . geom   geom_ canvas  g   1084  324  23  o  0  15  1  f i l l 2  \ <  1084 x 2 9 4 x 2 1 _ t r a n s l a t e d . geom  >  1084 x 2 9 4 x 2 1 _ a i r . geom geom_ canvas g   1194  324  23  o  55  0  0  f i  l l 1  \ <1084 x 2 9 4 x 2 1 _ a i r . geom  >  1084 x 2 9 4 x 2 1 _ a i r _ s u p p o r t e d . geom 90  geom_ canvas  g   542  163  23  o  0  8  1  f i l l 2  \ <542 x 1 4 7 x 2 1 _ t r a n s l a t e d . geom  >  542 x 1 4 7 x 2 1 _ a i r . geom geom_ canvas g   596  163  23  o  27  0  0  f i  l l 1  \ <542 x 1 4 7 x 2 1 _ a i r . geom  >  542 x 1 4 7 x 2 1 _ a i r _ s u p p o r t e d . geom   geom_ canvas  g   271  82  12  o  0  4  1  f i l l 2  \ <271 x 7 4 x 1 0 _ t r a n s l a t e d . geom  >  271 x 7 4 x 1 0 _ a i r . geom geom_ canvas g   299  82  12  o  14  0  0  f i  l l 1  \ <271 x 7 4 x 1 0 _ a i r . geom  >  271 x 7 4 x 1 0 _ a i r _ s u p p o r t e d . geom   geom_ check  1084 x 2 9 4 x 1 6 8 _ a i r . geom  1084 x 2 9 4 x 1 6 8 _ a i r _ s u p p o r t e d . geom geom_ check  1084 x 2 9 4 x 2 1 _ a i r . geom  1084 x 2 9 4 x 2 1 _ a i r _ s u p p o r t e d . geom geom_ check 542 x 1 4 7 x 2 1 _ a i r . geom 542 x 1 4 7 x 2 1 _ a i r _ s u p p o r t e d . geom geom_ check  271 x 7 4 x 1 0 _ a i r . geom  271 x 7 4 x 1 0 _ a i r _ s u p p o r t e d . geom 91  APPENDIX B MATERIAL CONFIGURATION FILE    < homogenization >   [ SX]   type none     < c r y s t a l l i t e >   [ e s s e n t i a l ]   ( o u t p u t ) o r i e n t a t i o n ( o u t p u t ) g r a i n r o t a t i o n ( o u t p u t ) f ( o u t p u t ) f e ( o u t p u t ) p    < phase >   ###  $ Id :   Phase_ Phenopowerlaw_ Aluminum . c o n f i g \ 3063 20140402  1 0 : 5 9 : 1 4 Z  MPIE \m. d i e h l  $  ### [ Aluminum ] 92  e l a s t i c i t y hooke p l a s t i c i t y phenopowerlaw   ( o u t p u t ) r e s i s t a n c e _ s l i p ( o u t p u t ) s h e a r r a t e _ s l i p ( o u t p u t ) r e s o l v e d s t r e s s _ s l i p ( o u t p u t ) a c c u m u l a t e d s h e a r _ s l i p ( o u t p u t ) t o t a l s h e a r    l a t t i c e _ s t r u c t u r e N s l i p f c c 12   #   per   f a m i   l y Ntwin 0 # per f a m i l y  c11 c12 c44  106 . 75 e9 6 0 . 4 1 e9 2 8 . 3 4 e9      g d o t 0 _ s l i p  0 . 0 0 1     n _ s l i p t a u 0 _ s l i p 20 31 e6    #   per   f a m i l y t a u s a t _ s l i p a _ s l i p 63 e6 2 . 2 5  # per f a m i l y g d o t 0 _ t w i n 0 . 0 0 1     n_ twin t a u 0 _ t w i n s _ p r 20 31 e6 0    #   per   f a m i l y 93  twin_ b 0 t w i n _ c 0 twin_ d 0 t w i n _ e 0 h 0 _ s l i p s l i p 75 e6 h 0 _ s l i p t w i n 0 h 0 _ t w i n s l i p 0 h 0 _ t w i n t w i n 0 i n t e r a c t i o n _ s l i p s l i p 1   1   1 . 4 1 . 4 1 . 4 1 . 4  i n t e r a c t i o n _ s l i p t w i n 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  i n t e r a c t i o n _ t w i n s l i p 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i n t e r a c t i o n _ t w i n t w i n 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 a t o l _ r e s i s t a n c e 1                        [ r i g i d ] e l a s t i c i t y hooke p l a s t i c i t y none l a t t i c e _ s t r u c t u r e i s o t r o p i c c11 1 1 0 . 9 e10 c12 5 8 . 3 4 e10   [ a i r ] e l a s t i c i t y hooke p l a s t i c i t y i s o t r o p i c / d i l a t a t i o n / 94  l a t t i c e _ s t r u c t u r e i s o t r o p i c c11 1 1 0 . 9 e6 c12 0 . 0 t a y l o r f a c t o r 3 t a u 0 0 . 3 3 3 e6 gdot 0 0 . 0 0 1 n 20 h0  1 e6 t a u s a t 0 . 6 6 7 e6 w0 2 . 2 5 a t o l _ r e s i s t a n c e 1     < m i c r o s t r u c t u r e > [ b a s e p l a t e ] c r y s t a l l i t e 1 ( c o n s t i t u e n t ) phase  2 t e x t u r e   1 f r a c t i o n 1 . 0 [ a i r l a y e r ] c r y s t a l l i t e 1 ( c o n s t i t u e n t ) phase  3 t e x t u r e   1 f r a c t i o n 1 . 0 [ Grain 01 ] c r y s t a l l i t e 1 ( c o n s t i t u e n t ) phase  1  t e x t u r e 1 f r a c t i o n 1 . 0 [ Grain 02 ] c r y s t a l l i t e 1 ( c o n s t i t u e n t ) phase  1  t e x t u r e 2 f r a c t i o n 1 . 0 95  [ Grain 03 ]  c r y s t a l l i t e 1 ( c o n s t i t u e n t ) phase 1 t e x t u r e 3 f r a c t i o n 1 . 0 [ Grain 04 ]       c r y s t a l l i t e 1       ( c o n s t i t u e n t ) phase 1 t e x t u r e 4 f r a c t i o n 1 . 0 [ Grain 05 ]       c r y s t a l l i t e 1       ( c o n s t i t u e n t ) phase 1 t e x t u r e 5 f r a c t i o n 1 . 0 [ Grain 06 ]       c r y s t a l l i t e 1       ( c o n s t i t u e n t ) phase 1 t e x t u r e 6 f r a c t i o n 1 . 0 [ Grain 07 ]       c r y s t a l l i t e 1       ( c o n s t i t u e n t ) phase 1 t e x t u r e 7 f r a c t i o n 1 . 0 [ Grain 08 ]       c r y s t a l l i t e 1       ( c o n s t i t u e n t ) phase 1 t e x t u r e 8 f r a c t i o n 1 . 0 [ Grain 09 ]       c r y s t a l l i t e 1       ( c o n s t i t u e n t ) phase 1 t e x t u r e 9 f r a c t i o n 1 . 0 [ Grain 10 ]       c r y s t a l l i t e 1       ( c o n s t i t u e n t ) phase 1 t e x t u r e 10 f r a c t i o n 1 . 0 [ Grain 11 ]       c r y s t a l l i t e 1       96  ( c o n s t i t u e n t ) phase 1 t e x t u r e 11 f r a c t i o n 1 . 0 [ Grain 12 ]       c r y s t a l l i t e 1       ( c o n s t i t u e n t ) phase 1 t e x t u r e 12 f r a c t i o n 1 . 0 [ Grain 13 ]       c r y s t a l l i t e 1       ( c o n s t i t u e n t ) phase 1 t e x t u r e 13 f r a c t i o n 1 . 0 [ Grain 14 ]       c r y s t a l l i t e 1       ( c o n s t i t u e n t ) phase 1 t e x t u r e 14 f r a c t i o n 1 . 0 [ Grain 15 ]       c r y s t a l l i t e 1       ( c o n s t i t u e n t ) phase 1 t e x t u r e 15 f r a c t i o n 1 . 0 [ Grain 16 ]       c r y s t a l l i t e 1       ( c o n s t i t u e n t ) phase 1 t e x t u r e 16 f r a c t i o n 1 . 0 [ Grain 17 ]       c r y s t a l l i t e 1       ( c o n s t i t u e n t ) phase 1 t e x t u r e 17 f r a c t i o n 1 . 0 [ Grain 18 ]       c r y s t a l l i t e 1       ( c o n s t i t u e n t ) phase 1 t e x t u r e 18 f r a c t i o n 1 . 0 < t e x t u r e >       [ Grain 01 ]         ( g a u s s ) ph i 1 93 . 2191 Phi  44 . 9554 ph i 2 4 3 . 3 0 5 6 \ 97  s c a t t e r 0 . 0 f r a c t i o n 1 . 0 [ Grain 02 ]  ( g a u s s ) ph i 1 74.389 Phi 28 . 602 p h i 2 4 4 . 2 0 1 4 \ s c a t t e r 0 . 0  f r a c t i o n  1 . 0 [ Grain 03 ]   ( g a u s s ) ph i 1 87.5264 Phi  25 . 9108 phi 2 4 7 . 6 2 7 3 \ s c a t t e r 0 . 0  f r a c t i o n  1 . 0 [ Grain 04 ]   ( g a u s s ) ph i 1 65.882 Phi  40 . 6057 p h i 2 4 7 . 5 0 6 8 \ s c a t t e r 0 . 0  f r a c t i o n  1 . 0 [ Grain 05 ]   ( g a u s s ) ph i 1 106.452 Phi  35 . 1883 phi 2 4 8 . 8 9 0 1 \ s c a t t e r 0 . 0  f r a c t i o n  1 . 0 [ Grain 06 ]   ( g a u s s ) ph i 1 108 . 138 Phi 44 . 982 ph i 2 4 4 . 1 0 2 1 \ s c a t t e r 0 . 0  f r a c t i o n  1 . 0 [ Grain 07 ]   ( g a u s s ) ph i 1 61.7808 Phi  29 . 7165 phi 2 3 0 . 0 2 1 5 \ s c a t t e r 0 . 0  f r a c t i o n  1 . 0 [ Grain 08 ] 98    ( g a u s s ) ph i 1 125 . 367 Phi  48 . 5804 p h i 2 3 7 . 1 6 8 8 \ s c a t t e r 0 . 0  f r a c t i o n  1 . 0 [ Grain 09 ]   ( g a u s s ) ph i 1 50 . 9184 Phi  43 . 8141 ph i 2 4 7 . 3 9 9 7 \ s c a t t e r 0 . 0  f r a c t i o n  1 . 0 [ Grain 10 ]   ( g a u s s ) ph i 1 45 . 0427 Phi  47 . 8092 ph i 2 4 8 . 6 5 8 2 \ s c a t t e r 0 . 0  f r a c t i o n  1 . 0 [ Grain 11 ]   ( g a u s s ) ph i 1 88 . 9443 Phi  44 . 4786 ph i 2 4 2 . 6 4 3 4 \ s c a t t e r 0 . 0  f r a c t i o n  1 . 0 [ Grain 12 ]   ( g a u s s ) ph i 1 17.1677 Phi  51 . 8046 phi 2 4 3 . 7 2 6 \ s c a t t e r 0 . 0  f r a c t i o n  1 . 0 [ Grain 13 ]   ( g a u s s ) ph i 1 105 . 945 Phi 50 . 722 ph i 2 5 5 . 3 7 1 5 \ s c a t t e r 0 . 0  f r a c t i o n  1 . 0 [ Grain 14 ]   ( g a u s s ) ph i 1 43.5652 Phi  51 . 5874 phi 2 3 9 . 9 6 4 \ 99  s c a t t e r 0 . 0 f r a c t i o n 1 . 0 [ Grain 15 ]  ( g a u s s ) ph i 1 113 . 782 Phi  52 . 8362 ph i 2 4 4 . 1 1 1 7 \ s c a t t e r 0 . 0  f r a c t i o n  1 . 0 [ Grain 16 ]   ( g a u s s ) ph i 1 62.2733 Phi  45 . 8902 phi 2 4 3 . 0 5 7 1 \ s c a t t e r 0 . 0  f r a c t i o n  1 . 0 [ Grain 17 ]   ( g a u s s ) ph i 1 82.2969 Phi  47 . 3403 phi 2 4 7 . 1 1 0 5 \ s c a t t e r 0 . 0  f r a c t i o n  1 . 0 [ Grain 18 ]   ( g a u s s ) ph i 1 48.625 Phi  51 . 2483 p h i 2 3 9 . 8 3 5 3 \ s c a t t e r 0 . 0  f r a c t i o n  1 . 0 100  APPENDIX C NUMERICS CONFIGURATION FILE    ###  $ Id$  ### ###  numerical parameters  ###    r e l e v a n t S t r a i n d e f g r a d T o l e r a n c e i J a c o S t i f f n e s s 1 . 0 e7 1 . 0 e7 1 i J a c o L p r e s i d u u m 1 p e r t _ F g 1 . 0 e7 p e r t _ m e t h o d 1 i n t e g r a t o r 1 i n t e g r a t o r S t i f f n e s s 1 a n a l y t i c J a c o 1 u n i t l e n g t h 1 useping pong 1   ## c r y s t a l l i t e numerical parameters  ## n C r y s t 20 s u b S t e p M i n C r y s t 1 . 0 e3 s u b S t e p S i z e C r y s t 0 . 2 5 s t e p I n c r e a s e C r y s t 1 . 5 n S t a t e 10 n S t r e s s 40 r T o l _ c r y s t a l l i t e S t a t e 1 . 0 e6 101  r T o l _ c r y s t a l l i t e S t r e s s 1 . 0 e6 a T o l _ c r y s t a l l i t e S t r e s s 1 . 0 e8 r T o l _ c r y s t a l l i t e T e m p e r a t u r e 1 . 0 e6   ##  homogenization numerical parameters  ## nHomog 20 subStepMinHomog 1 . 0 e3 sub Step Size Homog 0 . 2 5 s t e p I n c r e a s e H o m o g 1 . 5 n MPstate 10   ##  RGC  scheme  numerical parameters  ## aTol_RGC 1 . 0 e+4 rTol_RGC 1 . 0 e3 aMax_RGC 1 . 0 e +10 rMax_RGC 1 . 0 e+2 perturb Penalty_ RGC 1 . 0 e7 max Relaxation_ RGC 1 . 0 e+0   relevant Mismatch_ RGC 1 . 0 e5   viscosity Power_ RGC 1 . 0 e+0 viscosity Modulus_ RGC 0 . 0 e+0   r e f R e l a x a t i o n R a t e _ R G C 1 . 0 e3 102  maxVolDiscrepancy_RGC 1 . 0 e5 volDiscrepancy Mod_RGC 1 . 0 e +12 discrepancy Power_ RGC 5 . 0  f i x e d _ s e e d 0   ## s p e c t r a l parameters  ## e r r _ d i v _ t o l A b s 1 . 0 e3 e r r _ d i v _ t o l R e l 5 . 0 e4 e r r _ c u r l _ t o l A b s 1 . 0 e 12 e r r _ c u r l _ t o l R e l 5 . 0 e4 e r r _ s t r e s s _ t o l A b s 1 . 0 e3 e r r _ s t r e s s _ t o l R e l 0 . 0 1 f f t w _ t i m e l i m i t 1.0   r o t a t i o n _ t o l 1 . 0 e 12 f f tw_ plan_ mode FFTW_PATIENT i tmax 250 i t m i n 2 maxCutBack 3 m e m o r y _ e f f i c i e n t 1 update_gamma 0 d i v e r g e n c e _ c o r r e c t i o n 2 s p e c t r a l s o l v e r basic PETSc s p e c t r a l f i l t e r none p e t s c _ o p t i o n s s n e s _ t y p e ngmres 103  s n e s _ n g m r e s _ a n d e r s o n  regrid Mode 0 p o l a r A l p h a 1 . 0 p o l a r B e t a 1 . 0 104  APPENDIX D  ALTERATIONS TO NUMERICS CONNFIGURATION FILE     e r r _ d i v _ t o l R e l 5 . 0 e3 s p e c t r a l s o l v e r p o l a r i s a t i o n 105  APPENDIX E LOADCASE FILE    F  1 . 0 7 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 t   700  i n c s   70 r e s t a r t 10   f r e q  10    F is the deformation gradient tensor. 106  APPENDIX F QUEUE SUBMISSION FILE    # ! / b i n / bash l o g i n      #PBS  l w a l l t i m e = 24 : 00 : 00   ###  I e s t i m a t e d   1days w a l l t i m e   #PBS  l f e a t u r e= i n t e l 1 4    #PBS  l nodes =1: ppn=4     #PBS l  mem=5gb      #PBS  N   Siddharth_MSE_CMM_75x21x16 107      module  l o a d  I n t e l / 1 3 . 0 . 1 . 1 1 7 \ MKL/ 1 1 . 1 p o w e r t o o l s / 1 . 2 \ CMake / 3 . 1 . 0 Abaqus / 6 . 11 2  e x p o r t  DAMASK_NUM_THREADS=4     cd / mnt / r e s e a r c h /CMM/ P r o j e c t s / S i d d h a r t h / cd t e s t i n g / d a t a / 5 _ d e g r e e s    DAMASK_spectral l o a d t e n s i o n . l o a d \ geom 75 x21x16 . geom >  75 x21x16 . o u t 108  APPENDIX G POST PROCESSING   To get the ASCII table with output from the .spectralOut file:  >  p o s t R e s u l t s  r a n g e  0  70  70 s p l i t \  s e p a r a t i o n  x , y , z  c r fe , f , p , o r i e n t a t i o n , \ g r a i n r o t a t i o n  i n c r e m e n t s \ 74 x 2 0 x 1 0 _ t e n s i o n . s p e c t r a l O u t  This gives us the results in an easily manipulated ASCII table. The next steps (generic) are outlined below: >   add Cauchy f f   p   p   ASCIIfilename . t x t >  a d d S t r a i n T e n s o r s  l e f t   l o g a r i t h m i c  f f \ ASCIIfilename . t x t >   add Mises e (V)  s   Cauchy \ ASCIIfilename . t x t >   a d d C a l c u l a t i o n  l r o t a t i o n A n g l e  f \  damask . Q u a t e r n i o n (# g r a i n r o t a t i o n # ) . as Angle Axis ( d e g r e e s = True ) [ 0 ] ASCIIfilename . t x t >  geom_ check  < c o r r e s p o n d i n g G e o m e t r y f i l e . geom  > v t r f i l e . v t r > v t k _ a d d R e c t i l i n e a r G r i d D a t a   v t k v t r f i l e . v t r   i n p l a c e \ mode  c e l l  s  r o t a t i o n A n g l e  ,  Mises ( l n (V) )  ASCIIfilename . t x t  This step gives us a .vtr file that can be rendered in Paraview to view the strain maps and grain rotations geom_unpack  o n e d i m e n s i o n a l \ < c o r r e s p o n d i n g G e o m e t r y f i l e . geom  >ID . t x t 109  This step gives the geometry file, which is a two dimensional array, in the form of a one dimen- sional listing of material point texture IDs. We then assign a label to the singl >  add Table t a b l e  ID . t x t ASCIIfilename . t x t > f i l t e r T a b l e  c  ID# >2  w    i p i n i t i a l c o o r d   , \ ID Mises ( l n (V)  ID . t x t \ | a d d C a l c u l a t i o n  l t e x t u r e  f  ID# | \ r e L a b e l  l  i p i n i t i a l c o o r d  s  pos >   a d d E u c l i d e a n D i s t a n c e  t   boundary p   pos i   t e x t u r e   ID . t x t # adds  g r a i n   boundary  d i s t a n c e  the  strains are then normalized by dividing the strains at each material point by the average strain of the grain it belongs to. The normalized strains and the Euclidean distances are then sent into binXY which gives each material point a probability density, and thereafter it is rendered as a nice plot by gri. 110  APPENDIX H  EULER ANGLES FROM GEOMETRY FILE HEADER     # ! / usr / b i n / env python import  s y s import  r e    t r y : i n f i l e n a m e   =  s y s . a rg v [ 1 ] ; o u t f i l e n a m e   =  s y s . a rg v [ 2 ]  e x c e p t : p r i n t   " Usage : " , s y s . a r g v [ 0 ] , " i n f i l e  o u t f i l e " ; s y s . e x i t ( 1 )     o u t p u t   =   [ ]   wi t h open ( i n f i l e n a m e ,   ) as i n f i l e : i   =  0 f o r l i n e in i n f i l e : r e g e x p  =  r e . compile  ^ \ ( g a u s s \ ) \ s+ ph i 1 \ s +([+  . 0123456789 ]+)   \ s+ Phi \ s +([+  . 0123456789 ]+)\ s+ phi 2 \ s +([+  . 0123456789 ]+)  ) m  =  r e g e x p . match ( l i n e ) i f  m: i   +=  1 s t r ( i ) ,m. group ( 1 ) ,m. group ( 2 ) , \ m. group ( 3 ) ] ) ) 111  wi t h open ( o u t f i l e n a m e ,  ) as o u t f i l e : o u t f i l e . w r i t e (  \ n  . j o i n ( [  1   head  , \ \  112                          BIBLIOGRAPHY 113  BIBLIOGRAPHY    Lim, H., Carroll, J. 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