Table Of ContentSolid Mechanics and Its Applications
David J. Steigmann
Mircea Bîrsan
Milad Shirani
Lecture Notes
on the Theory
of Plates and
Shells
Classical and Modern Developments
Solid Mechanics and Its Applications
FoundingEditor
G.M.L.Gladwell
Volume 274
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AndersKlarbring,MechanicalEngineering,LinköpingUniversity,Linköping,
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· ·
David J. Steigmann Mircea Bîrsan Milad Shirani
Lecture Notes on the Theory
of Plates and Shells
Classical and Modern Developments
DavidJ.Steigmann MirceaBîrsan
DepartmentofMechanicalEngineering FacultyofMathematics
UniversityofCalifornia UniversityDuisburg-Essen
Berkeley,CA,USA Essen,Germany
DepartmentofMathematics
MiladShirani
DepartmentofMechanicalEngineering University“AlexandruIoanCuza”ofIas,i
UniversityofCalifornia Ias,i,Romania
Berkeley,CA,USA
ISSN 0925-0042 ISSN 2214-7764 (electronic)
SolidMechanicsandItsApplications
ISBN 978-3-031-25673-8 ISBN 978-3-031-25674-5 (eBook)
https://doi.org/10.1007/978-3-031-25674-5
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Preface
The theory of plates and shells originated in the need on the part of engineers for
tractablemodelstoanalyzeandpredicttheresponseofthinstructurestovariouskinds
ofloading.Thebasicideaistoexploitthethinnessofthestructuretorepresentthe
mechanicsoftheactualthinthree-dimensionalbodyunderconsiderationbyamore
tractable two-dimensional theory associated with an interior surface. In this way,
therelativelycomplexthree-dimensionalcontinuummechanicsofthethinbodyis
replacedbyafarmoretractabletwo-dimensionaltheory.Toensurethattheresulting
model is predictive, it is necessary to compensate for this ‘dimension reduction’
byassigningadditionalkinematicalanddynamicaldescriptorstothesurfacewhose
deformations are modeled by the simpler two-dimensional theory. The manner in
whichthisisachieved,onthebasisofthethree-dimensionalparenttheory,constitutes
mostofthecontentsofthisbook.
FewsubjectsexhibitsuchanintimateandbeautifulinterplaybetweenMechanics
andGeometryasShellTheory.However,preciselybecauseofitsconnectiontogeom-
etry, learning the theory is in many ways more challenging than learning standard
continuummechanicsintheflatthree-dimensionalEuclideanspaceofourcommon
experience. Accordingly, in this book, we begin with a fairly extensive exposition
ofthedifferentialgeometryofsurfaces.Thisinturnrequiresafacilitywithtensor
analysis in curvilinear coordinates, which is also covered in detail. As the book is
concernedwiththetheoryofelasticshells,furtherbackgroundonthree-dimensional
elasticitytheory,bothlinearandnonlinear,isprovidedasapreludetothediscussion
ofthedimensionreductionprocedure.
Withtheseprerequisitesinhand,wegoontodeveloplinearandnonlinearplateand
shellmodels,includingclassicalplatebucklingtheory,systematically.Ourmethod
isbasedontheexpansion ofthepotentialenergy ofthethinbody inpowers ofits
thickness. This is achieved by combining the Leibniz Rule, for differentiating an
integralwithrespecttoitsintegrationlimits,withaTaylorexpansion.Theexpansion
isterminatedattheorderofthecubeofthicknesstocapturemembraneandleading-
order bending effects in a two-dimensional energy density defined over the shell
midsurface.Though itisnotstrictlynecessarytobasethetwo-dimensionaltheory
v
vi Preface
on the midsurface, it proves advantageous, from the viewpoint of the theoretical
development,todoso.
Ourprocedureisreminiscentoftherecentapplication,prevalentinthemathemat-
icalliterature,ofgammaconvergencemethodstothinbodies.Infact,ourprocedure
delivers precisely the same two-dimensional models as those obtained by gamma
convergenceincircumstanceswherethelattermaybeapplied.However,ourproce-
dure is far more accessible to those not trained in the sophisticated mathematics
underpinninggammaconvergence,andextendstocircumstanceswherethelatter—
concerned exclusively with the limiting variational problem as thickness tends to
zero—cannotbeapplied.Thatis,whilegammaconvergencehasdeliveredrigorous
limit models for membrane theory and pure bending theory separately, in contrast
to our procedure it has not yielded models for combined bending and membrane
behaviorthatareofprincipalinteresttopractitionersofshelltheory.
Thus,wesystematicallyderiveexistingclassicallinearandnonlineartheoriesof
platesandshellsinaconceptuallyclearmanner.Importantly,weavoidthevarious
adhocassumptionsmadeinthehistoricaldevelopmentofthesubject,mostnotably
theclassicalKirchhoff-Lovehypothesisrequiringthatmateriallinesinitiallynormal
to the shell surface remain so after deformation. Instead, such conditions, when
appropriate, are here derived rather than postulated. We show how they emerge
naturallyinthecourseofourdimension-reductionprocedureundercertainconditions
onthesymmetriespossessedbytheunderlyingthree-dimensionalmaterial.Afurther
importantexampleofaconditionthatwaspostulatedhistoricallybutwhichwehere
deriveisthatofplanestressattheshellmidsurface.
Thoughthebookisprimarilyconcernedwiththedevelopmentoftheory,anumber
ofsimpleexplicitsolutionsareincludedwiththeaimofexhibitingtherelativeease
withwhichthetheorycanbeusedtomakepredictions.Thiscontrastssharplywith
the formidable obstacles one faces when using three-dimensional elasticity. Many
problemsareposedforthestudent.Theseareintendedtopromoteengagementwith
thematerialandtoreinforceunderstanding.However,asourintentionistousethe
book in our future teaching of shell theory we do not include solutions to these
exercises.
Berkeley,CA,USA DavidJ.Steigmann
September2022 MirceaBîrsan
MiladShirani
Acknowledgements The second author (M.B.) acknowledges support from the Deutsche
Forschungsgemeinschaft(DFG,GermanResearchFoundation)—Projectno.415894848.
Contents
1 TensorAnalysisinEuclideanSpaceUsingCurvilinear
Coordinates ................................................... 1
1.1 CartesianandCurvilinearCoordinates ........................ 1
1.2 RepresentationsofVectorsandTensors ....................... 6
1.2.1 Vectors ............................................ 6
1.2.2 RepresentationinCartesianCoordinates ................ 6
1.2.3 Tensors ............................................ 7
1.2.4 CoordinateTransformations .......................... 9
1.3 DifferentialOperatorsinCurvilinearCoordinates .............. 10
1.3.1 Gradients .......................................... 10
1.3.2 Divergence ......................................... 12
1.3.3 ProductRuleforCovariantDerivatives ................. 17
1.4 OrthogonalCoordinates .................................... 18
1.5 Exercises ................................................. 19
References ..................................................... 22
2 LocalGeometryofDeformation ................................. 23
2.1 DescriptionoftheDeformation .............................. 23
2.1.1 DeformationTensor ................................. 24
2.1.2 VolumesandAreas .................................. 25
2.2 Displacement,Strain,andStress ............................. 27
2.3 LocalGeometryofSurfaces ................................ 32
2.4 DifferentiationonSurfaces.Curvature ........................ 38
2.4.1 CurvatureofaCurveontheSurface ................... 41
2.4.2 MeanCurvatureandGaussianCurvature
oftheSurface ...................................... 44
2.5 CompatibilityConditionsfortheFundamentalTensors ......... 46
2.6 Green-StokesFormula ..................................... 49
2.7 SurfaceDifferentialOperators ............................... 52
2.8 Exercises ................................................. 55
References ..................................................... 59
vii
viii Contents
3 HyperelasticSolids:PurelyMechanicalTheory ................... 61
3.1 ConstitutiveRelations ...................................... 61
3.2 SmallStrains ............................................. 64
3.2.1 SpecialCase:Isotropy ............................... 65
3.3 PotentialEnergy .......................................... 68
3.4 Legendre-HadamardConditionforStability ................... 72
References ..................................................... 76
4 LinearlyElasticPlates .......................................... 77
4.1 Three-dimensionalTheory .................................. 77
4.2 DerivationofthePlateModel ............................... 78
4.2.1 IntegrationovertheThickness ........................ 80
4.2.2 EdgeLoads ........................................ 81
4.2.3 OptimalExpressions ................................ 83
4.2.4 LateralLoads ...................................... 91
4.3 EquilibriumEquations ..................................... 92
4.4 IsotropicPlates ........................................... 94
4.4.1 DisplacementGradients .............................. 95
4.4.2 StrainEnergy ....................................... 97
4.4.3 DecoupledEquilibriumEquations ..................... 99
4.5 EnergyMinimizers ........................................ 102
4.6 Exercises ................................................. 107
References ..................................................... 108
5 LinearShellTheory ............................................ 109
5.1 GeometryoftheCurvedShell ............................... 109
5.2 Kinematics ............................................... 113
5.3 DerivationoftheShellModel ............................... 115
5.4 OptimalExpressionforPotentialEnergy ...................... 119
5.4.1 MembraneEnergy .................................. 122
5.4.2 PureBending ....................................... 124
5.4.3 EstimateoftheCouplingTermintheEnergy ............ 127
5.5 MeasuresofDistortion ..................................... 131
5.6 Isotropy .................................................. 138
5.7 EquilibriumEquationsandBoundaryConditions .............. 141
5.8 ClassicalMembraneTheory ................................ 145
5.8.1 EquilibriumUnderNormalPressure ................... 147
5.9 BendingTheory ........................................... 151
5.9.1 CircularCylindricalShells ........................... 153
5.10 Exercises ................................................. 166
References ..................................................... 167
Contents ix
6 NonlinearEquationsforPlatesandShells ........................ 169
6.1 AsymptoticDerivationofNonlinearPlateModels .............. 170
6.1.1 BackgroundfromThree-DimensionalNonlinear
ElasticityTheory ................................... 170
6.1.2 Small-ThicknessEstimateoftheEnergy ................ 173
6.1.3 MembraneLimit .................................... 177
6.1.4 PureBending ....................................... 179
6.1.5 AsymptoticModelforCombinedBending
andStretching ...................................... 181
6.1.6 ReflectionSymmetryandIll-Posedness ................ 185
6.1.7 Koiter’sModel ..................................... 186
6.2 Koiter’sShellTheory ...................................... 187
6.2.1 GeometricandKinematicFormulae ................... 188
6.2.2 ExpansionoftheThree-DimensionalEnergy ............ 190
6.2.3 TheOptimumOrder—h3Energy ...................... 191
6.2.4 Koiter’sEnergyastheLeading-OrderModel ............ 195
6.2.5 TheExplicitEnergyforIsotropicMaterials ............. 198
6.3 EquilibriumEquations ..................................... 199
6.4 Exercises ................................................. 204
References ..................................................... 204
7 BucklingofElasticPlates ....................................... 207
7.1 NonlinearElasticityandStability ............................ 208
7.2 LinearIncrementalElasticityandBifurcationofEquilibria ...... 211
7.3 Two–dimensionalModelforPlate-Buckling ................... 214
7.3.1 DescriptionofPlatesasThinPrismaticBodies .......... 214
7.3.2 Thickness-WisePowerExpansionoftheSecond
Variation ........................................... 216
7.3.3 TheRefinedModel .................................. 217
7.3.4 ConditionsPertainingtoReflectionSymmetry
andthePre-stress ................................... 220
7.3.5 DerivationoftheClassicalModel ..................... 225
7.4 Exercises ................................................. 228
References ..................................................... 228
8 Saint-VenantProblemforGeneralCylindricalShells .............. 231
8.1 RelaxedSaint-Venant’sProblemforCylindricalShells .......... 231
8.2 Closed–formExactSolutions ............................... 237
8.2.1 Extension-Bending-TorsionProblem ................... 239
8.2.2 FlexureProblem .................................... 243
8.3 SimplifiedSolution ........................................ 247
8.4 CircularCylindricalTubes .................................. 250
References ..................................................... 251
