StructuralGeologyAlgorithms VectorsandTensors State-of-the-artanalysisofgeologicalstructureshasbecomeincreasinglyquantitative,buttradition- ally,graphicalmethodsareusedinteachingandintextbooks.Now,thisinnovativelabbookprovides aunifiedmethodologyforproblemsolvinginstructuralgeologyusinglinearalgebraandcomputa- tion. Assuming only limited mathematical training, the book builds from the basics, providing the fundamental background mathematics, and demonstrating the application of geometry and kine- maticsingeosciencewithoutrequiringstudentstotakeasupplementarymathematicscourse. Starting with classic orientation problems that are easily grasped, the authors then progress to more fundamental topics of stress, strain, and error propagation. They introduce linear algebra methodsasthefoundationforunderstandingvectorsandtensors.Connectionswithearliermaterial areemphasizedtoallowstudentstodevelopanintuitiveunderstandingoftheunderlyingmathematics beforeintroducingmoreadvancedconcepts.Allalgorithmsarefullyillustratedwithacomprehensive suiteofonlineMATLAB®functions,whichbuildonandincorporateearlierfunctions,andwhichalso allowuserstomodifythecodetosolvetheirownstructuralproblems. Containing 20 worked examples and over 60 exercises, this is the ideal lab book for advanced undergraduatesorbeginninggraduatestudents.Itwillalsoprovideprofessionalstructuralgeologists withavaluablereferenceandrefresherforcalculations. RICHARD W. ALLMENDINGER is a structural geologist and a professor in the Earth and AtmosphericSciencesDepartmentatCornellUniversity.Heiswidelyknownforhisworkonthrust tectonics and earthquake geology in South America, where much of his work over the past three decadeshasbeenbased,aspartoftheCornellAndesProject.ProfessorAllmendingeristheauthorof morethan100publicationsandnumerouswidelyusedstructuralgeologyprogramsforMacsandPCs. NESTORCARDOZOisastructuralgeologistandanassociateprofessorattheUniversityofStavanger, Norway,whereheteachesundergraduateandgraduatecoursesonstructuralgeologyanditsappli- cationtopetroleumgeosciences.Hehasbeeninvolvedinseveralmultidisciplinaryresearchprojectsto realisticallyincludefaultsandtheirassociateddeformationinreservoirmodels.Heistheauthorof severalwidelyusedstructuralgeologyandbasinanalysisprogramsforMacs. DONALDM.FISHERisastructuralgeologistandprofessoratPennStateUniversity,whereheleads astructuralgeologyandtectonicsresearchgroup.Hisresearchonactivestructures,strainhistories, anddeformationalongconvergentplateboundarieshastakenhimtofieldareasinCentralAmerica, Kodiak Alaska, northern Japan, Taiwan, and offshore Sumatra. He has been teaching structural geologytoundergraduateandgraduatestudentsformorethan20years. STRUCTURAL GEOLOGY ALGORITHMS VECTORS AND TENSORS RICHARD W. ALLMENDINGER CornellUniversity,USA NESTOR CARDOZO UniversityofStavanger,Norway DONALD M. FISHER PennsylvaniaStateUniversity,USA CAMBRIDGE UNIVERSITY PRESS Cambridge,NewYork,Melbourne,Madrid,CapeTown, Singapore,Sa˜oPaulo,Delhi,Tokyo,MexicoCity CambridgeUniversityPress TheEdinburghBuilding,CambridgeCB28RU,UK PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork www.cambridge.org Informationonthistitle:www.cambridge.org/9781107012004 ©RichardW.Allmendinger,NestorCardozoandDonaldM.Fisher2012 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2012 PrintedintheUnitedKingdomattheUniversityPress,Cambridge InternalbooklayoutfollowsadesignbyG.K.Vallis AcatalogrecordforthispublicationisavailablefromtheBritishLibrary LibraryofCongressCataloginginPublicationdata Allmendinger,RichardWaldron. Structuralgeologyalgorithms:vectorsandtensors/RichardW.Allmendinger, NestorCardozo,DonaldM.Fisher. p. cm. ISBN978-1-107-01200-4(hardback)–ISBN978-1-107-40138-9(pbk.) 1. Geology,Structural–Mathematics. 2. Rockdeformation–Mathematicalmodels. I. Cardozo,Nestor. II. Fisher,DonaldM. III. Title. QE601.3.M38A45 2011 551.80105181–dc23 2011030685 ISBN978-1-107-01200-4Hardback ISBN978-1-107-40138-9Paperback Additionalresourcesforthispublicationatwww.cambridge.org/allmendinger CambridgeUniversityPresshasnoresponsibilityforthepersistenceor accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto inthispublication,anddoesnotguaranteethatanycontentonsuch websitesis,orwillremain,accurateorappropriate. Contents Preface pageix 1 Problemsolvinginstructuralgeology 1 1.1 Objectivesofstructuralanalysis 1 1.2 Orthographicprojectionandplanetrigonometry 3 1.3 Solvingproblemsbycomputation 6 1.4 Sphericalprojections 8 1.5 Mapprojections 18 2 Coordinatesystems,scalars,andvectors 23 2.1 Coordinatesystems 23 2.2 Scalars 25 2.3 Vectors 25 2.4 Examplesofstructureproblemsusingvectoroperations 34 2.5 Exercises 43 3 Transformationsofcoordinateaxesandvectors 44 3.1 Whataretransformationsandwhyaretheyimportant? 44 3.2 Transformationofaxes 45 3.3 Transformationofvectors 48 3.4 Examplesoftransformationsinstructuralgeology 50 3.5 Exercises 65 4 Matrixoperationsandindicialnotation 66 4.1 Introduction 66 4.2 Indicialnotation 66 4.3 Matrixnotationandoperations 69 4.4 Transformationsofcoordinatesandvectorsrevisited 77 4.5 Exercises 79 v vi Contents 5 Tensors 81 5.1 Whataretensors? 81 5.2 Tensornotationandthesummationconvention 82 5.3 Tensortransformations 85 5.4 Principalaxesandrotationaxisofatensor 88 5.5 Exampleofeigenvaluesandeigenvectorsinstructuralgeology 91 5.6 Exercises 97 6 Stress 98 6.1 Stress“vectors”andstresstensors 98 6.2 Cauchy’sLaw 99 6.3 Basiccharacteristicsofstress 104 6.4 Thedeviatoricstresstensor 112 6.5 Aprobleminvolvingstress 113 6.6 Exercises 119 7 Introductiontodeformation 120 7.1 Introduction 120 7.2 Deformationanddisplacementgradients 121 7.3 Displacementanddeformationgradientsinthreedimensions 125 7.4 Geologicalapplication:GPStransects 128 7.5 Exercises 132 8 Infinitesimalstrain 135 8.1 Smallerissimpler 135 8.2 Infinitesimalstraininthreedimensions 138 8.3 Tensorshearstrainvs.engineeringshearstrain 140 8.4 Straininvariants 141 8.5 Strainquadricandstrainellipsoid 142 8.6 Mohrcircleforinfinitesimalstrain 143 8.7 Exampleofcalculations 144 8.8 Geologicalapplicationsofinfinitesimalstrain 147 8.9 Exercises 164 9 Finitestrain 165 9.1 Introduction 165 9.2 DerivationoftheLagrangianstraintensor 166 9.3 Eulerianfinitestraintensor 167 9.4 DerivationoftheGreendeformationtensor 167 9.5 Relationsbetweenthefinitestrainanddeformationtensors 168 9.6 Relationstothedeformationgradient,F 169 9.7 Practicalmeasuresofstrain 170 9.8 Therotationandstretchtensors 173 9.9 Multipledeformations 176 9.10 Mohrcircleforfinitestrain 176 9.11 Compatibilityequations 178 9.12 Exercises 180 Contents vii 10 Progressivestrainhistoriesandkinematics 183 10.1 Finiteversusincrementalstrain 183 10.2 Determinationofastrainhistory 199 10.3 Exercises 213 11 Velocitydescriptionofdeformation 217 11.1 Introduction 217 11.2 Thecontinuityequation 218 11.3 Pureandsimpleshearintermsofvelocities 219 11.4 Geologicalapplication:Fault-relatedfolding 220 11.5 Exercises 252 12 Erroranalysis 254 12.1 Introduction 254 12.2 Errorpropagation 255 12.3 Geologicalapplication:Cross-sectionbalancing 256 12.4 Uncertaintiesinstructuraldataandtheirrepresentation 266 12.5 Geologicalapplication:Trishearinversemodeling 270 12.6 Exercises 279 References 281 Index 286 Preface Structuralgeologyhasbeentaught,largelyunchanged,forthelast50yearsormore.Thelecture partofmostcoursesintroducesstudentstoconceptssuchasstressandstrain,aswellasmore descriptivemateriallikefaultandfoldterminology.Thelabpartofthecourseusuallyfocuseson practicalproblemsolving,mostlytraditionalmethodsfordescribingquantitativelythegeometry of structures. While the lecture may introduce advanced concepts such as tensors, the lab commonlytrainsthestudenttouseacombinationofgraphicalmethods,suchasorthographic orsphericalprojection,andavarietyofplanetrigonometrysolutionstovariousproblems.This leads to a disconnect between lecture concepts that require a very precise understanding of coordinatesystems(e.g.,tensors)andlabmethodsthatappeartohavenocommonspatialor mathematicalfoundation.Studentshavenochancetounderstandthat,forexample,seemingly unconnectedconstructionssuchasdown-plungeprojectionsandMohrcirclesshareacommon mathematicalheritage:Theyarebothgraphicalrepresentationsofcoordinatetransformations. In fact, it is literally impossible to understand the concept of tensors without understanding coordinatetransformations.Andyet,wetrytoteachstudentsabouttensorswithoutteaching themaboutthemostbasicoperationsthattheyneedtoknowtounderstandthem. Thebasicmathbehindalloftheseseeminglydiversetopicsconsistsoflinearalgebraand vector operations. Many geology students learn something about vectors in their first two semestersofcollegemath,butareseldomgiventheopportunitytoapply thoseconceptsin theirchosenmajor.Fewerstudentshavelearnedlinearalgebra,asthattopicisoftenreserved forthethirdorfourthsemestermath.Nonetheless,thesebasicconceptsneededforanintro- ductorystructuralgeologycoursecaneasilybemasteredwithoutaformalcourse;weassume nopriorknowledgeofeither.Ononelevel,then,thisbookteachesaconsistentapproachtoa subsetofstructuralgeologyproblemsusinglinearalgebraandvectoroperations.Thissubset ofstructuralgeologyproblemscoincideswiththosethatareusuallytreatedinthelabportion ofastructuralgeologycourse. Thelinearalgebraapproachisideallysuitedtocomputation.Thirtyyearsafterthewide- spread deployment of personal computers, most labs in structural geology teach students increasinglyarcanegraphicalmethodstosolveproblems.Studentsaretaughttheoperations needed to solve orientation problems on a stereonet, but that does not teach them the ix