Structural Geology Algorithms

313 Pages·2013·9.43 MB·English

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
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