Table Of ContentNONLINEAR SOLID MECHANICS
Thisbookcoverssolidmechanicsfornonlinearelasticandelastoplas-
tic materials, describing the behavior of ductile materials subjected
to extreme mechanical loading and their eventual failure. The book
highlights constitutive features to describe the behavior of frictional
materialssuchasgeologicalmedia.Onthebasisofthistheory,includ-
inglargestrainandinelasticbehaviors,bifurcationandinstabilityare
developed with a special focus on the modeling of the emergence of
localinstabilitiessuchasshearbandformationandflutterofacontin-
uum. The former is regarded as a precursor of fracture, whereas the
latteristypicalofgranularmaterials.Thetreatmentiscomplemented
withqualitativeexperiments,illustrationsfromeverydaylifeandsimple
examplestakenfromstructuralmechanics.
Davide Bigoni is a professor in the faculty of engineering at the
University of Trento, where he has been head of the Department
of Mechanical and Structural Engineering. He was honored as a
EuromechFellowoftheEuropeanMechanicsSociety.Heisco-editor
oftheJournalofMechanicsofMaterialsandStructures(aninternational
journalfoundedbyC.R.Steele)andisassociateeditorofMechanics
ResearchCommunications.
Nonlinear Solid Mechanics
BIFURCATION THEORY AND
MATERIAL INSTABILITY
Davide Bigoni
UniversityofTrento
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,SãoPaulo,Delhi,MexicoCity
CambridgeUniversityPress
32AvenueoftheAmericas,NewYork,NY10013-2473,USA
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107025417
©DavideBigoni2012
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2012
PrintedintheUnitedStatesofAmerica
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloginginPublicationData
Bigoni,Davide,1959–
Nonlinearsolidmechanics:bifurcationtheoryandmaterialinstability/DavideBigoni.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-1-107-02541-7
1. Nonlinearmechanics. 2. Materials–Mechanicalproperties. 3. Elasticanalysis
(Engineering) 4. Bifurcationtheory. I. Title.
TA405.B4983 2012
(cid:2)
620.11292–dc23 2012013657
ISBN978-1-107-02541-7Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofURLs
forexternalorthird-partyInternetWebsitesreferredtointhispublicationanddoesnot
guaranteethatanycontentonsuchWebsitesis,orwillremain,accurateorappropriate.
Contents
Preface page xiii
ForewordbyGiulioMaier xv
1 Introduction 1
1.1 Bifurcationandinstabilitytoexplainpatternformation 2
1.2 Bifurcationsinelasticity:Theelasticcylinder 6
1.3 Bifurcationsinelastoplasticity:TheShanleymodel 8
1.4 Shearbandsandstrainlocalization 12
1.5 Bifurcation,softeningandsizeeffectastheresponseofastructure 17
1.6 Chainswithsofteningelements 22
1.7 Shearbandsaturationandmultipleshearbanding 31
1.8 Brittleandquasi-brittlematerials 33
1.9 Coulombfrictionandnon-associativeplasticity 37
1.10 Non-associativeflowrulepromotesmaterialinstabilities 41
1.11 Aperturbativeapproachtomaterialinstability 42
1.12 Asummary 48
1.13 Exercises,detailsandcuriosities 52
1.13.1 Exercise:TheEulerelasticaandthedoublesupportedbeam
subjecttocompressiveload 52
1.13.2 Exercise:Bifurcationofastructuresubjecttotensile
deadload 69
1.13.3 Exercise:Degreesoffreedomandnumberofcriticalloads
ofelasticstructures 70
1.13.4 Exercise:Astructurewithatrivialconfigurationunstable
atacertainload,returningstableathigherload 73
1.13.5 Exercise:Flutteranddivergenceinstabilityinanelastic
structureinducedbyCoulombfriction 80
2 Elementsoftensoralgebraandanalysis 91
2.1 Componentsontoanorthonormalbasis 92
2.2 Dyads 93
2.3 Second-ordertensors 95
2.4 Rotationtensors 98
vii
viii Contents
2.5 Positivedefinitesecond-ordertensors,eigenvalues
andeigenvectors 99
2.6 Reciprocalbases:Covariantandcontravariantcomponents 101
2.7 Spectralrepresentationtheorem 102
2.8 Squarerootofatensor 103
2.9 Polardecompositiontheorem 104
2.10 Oncoaxialitybetweensecond-ordertensors 104
2.11 Fourth-ordertensors 105
2.12 Onthemetricinducedbysemi–positivedefinitetensors 106
2.13 TheMacaulaybracketoperator 107
2.14 Differentialcalculusfortensors 107
2.15 Gradient 108
2.16 Divergence 110
2.17 Cylindricalcoordinates 111
2.18 Divergencetheorem 113
2.19 Convexityandquasi-convexity 114
2.20 Examplesanddetails 116
2.20.1 Example:Jordannormalformofadefectivetensorwitha
doubleeigenvalue 116
2.20.2 Example:Jordannormalformofadefectivetensorwitha
tripleeigenvalue 117
2.20.3 Example:Inverseoftheacoustictensorofisotropic
elasticity 117
2.20.4 Example:Inverseoftheacoustictensorforaparticular
classofanisotropicelasticity 118
2.20.5 Example:Arepresentationforthesquarerootofatensor 118
2.20.6 Proofofapropertyofthescalarproductbetweentwo
symmetrictensors 119
2.20.7 Example:Inverseandpositivedefinitenessofthe
fourth-ordertensordefininglinearisotropicelasticity 120
2.20.8 Example:Inverseandpositivedefinitenessofa
fourth-ordertensordefiningaspecialanisotropiclinear
elasticity 121
2.20.9 Example:Inverseoftheelastoplasticfourth-ordertangent
tensor 121
2.20.10Example:Spectralrepresentationoftheelastoplastic
fourth-ordertangenttensor 122
2.20.11Example:Strictconvexityofthestrainenergydefining
linearisotropicelasticity 124
3 Solidmechanicsatfinitestrains 125
3.1 Kinematics 125
3.1.1 Transformationoforientedlineelements 127
3.1.2 Transformationoforientedareaelements 129
3.1.3 Transformationofvolumeelements 129
Contents ix
3.1.4 Angularchanges 130
3.1.5 Measuresofstrain 131
3.2 Onmaterialandspatialstrainmeasures 135
3.2.1 Rigid-bodyrotationofthereferenceconfiguration 135
3.2.2 Rigid-bodyrotationofthecurrentconfiguration 136
3.3 Motionofadeformablebody 137
3.4 Massconservation 141
3.5 Stress,dynamicforces 142
3.6 Powerexpendedandwork-conjugatestress/strainmeasures 146
3.7 Changesoffieldsforasuperimposedrigid-bodymotion 150
4 Isotropicnon-linearhyperelasticity 152
4.1 Isotropiccompressiblehyperelasticmaterial 153
4.1.1 Kirchhoff–SaintVenantmaterial 154
4.2 Incompressibleisotropicelasticity 155
4.2.1 Mooney-Rivlinelasticity 156
4.2.2 Neo-Hookeanelasticity 158
4.2.3 J -Deformationtheoryofplasticity 158
2
4.2.4 TheGBGmodel 159
5 Solutionsofsimpleproblemsinfinitelydeformednon-linear
elasticsolids 162
5.1 Uniaxialplanestraintensionandcompressionofan
incompressibleelasticblock 162
5.2 UniaxialplanestraintensionandcompressionofKirchhoff–Saint
Venantmaterial 168
5.3 Uniaxialtensionandcompressionofanincompressible
elasticcylinder 170
5.4 Simpleshearofanelasticblock 173
5.5 Finitebendingofanincompressibleelasticblock 179
6 Constitutiveequationsandanisotropicelasticity 188
6.1 Constitutiveequations:Generalconcepts 188
6.1.1 Changeinobserverandrelatedprincipleofinvarianceof
materialresponse 189
6.1.2 Indifferencewithrespecttorigid-bodyrotationofthe
referenceconfiguration 192
6.1.3 Materialsymmetries 195
6.1.4 Cauchyelasticity 198
6.1.5 Greenelasticorhyperelasticmaterials 201
6.1.6 Incompressiblehyperelasticityandconstrainedmaterials 203
6.2 Rateandincrementalelasticconstitutiveequations 207
6.2.1 Elasticlawsinincrementalandrateform 207
6.2.2 RelativeLagrangeandescription 210
6.2.3 Hypoelasticity 220
