Table Of ContentNonlinear Mechanics of Shells and Plates in Composite,
Soft and Biological Materials
Thisbookguidesthereaderintononlinearmodellingofshellstructuresinapplications
where advanced composite and complex biological materials must be described with
great accuracy. To achieve this goal, Marco Amabili guides the readers through non-
linearshelltheories,nonlinearvibrations,buckling,compositeandfunctionallygraded
materials,hyperelasticity,viscoelasticity,nonlineardamping,rubberandsoftbiological
materials.Advancednonlinearshelltheories,notavailableinanyotherbook,arefully
derived in a simple notation and are ready to be implemented in numerical codes.
Ablendofthemostadvancedtheoryandexperimentalresultsisthebook’sotherunique
feature. It is a must-read for graduate students and researchers in applied mathematics
and engineering, especially those interested in mechanics, aeronautics, civil structures,
materials, bioengineering and solid matter at different scales.
Marco Amabili is Canada Research Chair Professor of Mechanical Engineering at
McGill University. He is the author of more than 440 research publications (over 200
journal papers) and the book Nonlinear Vibrations and Stability of Shells and Plates
(Cambridge University Press, 2008). He is Contributing Editor of the International
Journal of Non-linear Mechanics, Associate Editor of the Journal of Fluids and
Structures,JournalofVibrationandAcoustics,andseveralotherjournals.Hisresearch
areas include vibrations of shell structures, nonlinear vibrations, fluid–structure inter-
action, dynamics and stability of shells and plates, cardiovascular biomechanics, vis-
coelasticity and nonlinear damping, and the vibration monitoring of structures and
buildings.
To my parents, Vito and Antonietta
Nonlinear Mechanics of Shells
and Plates in Composite, Soft
and Biological Materials
MARCO AMABILI
McGillUniversity
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107129221
DOI:10.1017/9781316422892
©MarcoAmabili2018
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Firstpublished2018
PrintedintheUnitedStatesofAmericabySheridanBooks,Inc.
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LibraryofCongressCataloging-in-PublicationData
Names:Amabili,M.,author.
Title:NonlinearMechanicsofShellsandPlatesinComposite,SoftandBiological
Materials/MarcoAmabili,McGillUniversity.
Description:Cambridge,UnitedKingdom;NewYork,NY:CambridgeUniversityPress,2018.
Identifiers:LCCN2018013118|ISBN9781107129221(hardback)|ISBN1107129222(hardback)
Subjects:LCSH:Shells(Engineering)|Plates(Engineering)|Compositematerials–Mechanicalproperties.|
Nonlineartheories.
Classification:LCCTA660.S5A3852018|DDC624.1/7762–dc23
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Contents
Preface page xv
Introduction and Constitutive Equations For Linearly ElasticMaterials 1
I.1 ConstitutiveEquationsfor LinearlyElasticMaterials 3
I.1.1 3-D ConstitutiveEquations for a Layerwithina Laminated Shell 4
I.1.2 ConstitutiveEquations inCase of Negligible Transverse
Normal Stress 6
I.1.3 ConstitutiveEquations inCase of Plane Stress (Classical Theories
ofPlatesandShells) 7
References 7
1 Classical Nonlinear Theories ofElasticity ofPlates and Shells 8
1.1 Introduction 8
1.1.1 Literature Review 8
1.2 Large Deflection ofRectangular Plates 10
1.2.1 Green’s and Almansi Strain Tensors for FiniteDeformation 10
1.2.2 Strains for FiniteDeflectionof Rectangular Plates:
Von Kármán Theory 13
1.2.3 Geometric Imperfections 17
1.2.4 Eulerian,Lagrangian and Second Piola-KirchhoffStressTensors 17
1.2.5 Equationsof Motion inLagrangianDescription 21
1.2.6 Elastic Strain Energy 21
1.2.7 Von Kármán Equation ofMotion 22
1.2.8 Von Kármán Equation ofMotionIncluding Geometric
Imperfections 27
1.3 Large Deflection ofCircular Cylindrical Shells 27
1.3.1 EuclideanMetricTensor 27
1.3.2 Green’s Strain Tensor ina Generic Coordinate System 29
1.3.3 Green’s Strain Tensor inCylindrical Coordinates 31
1.3.4 Strains for FiniteDeflectionof CircularCylindrical Shells:
Donnell’sNonlinear Theory 33
1.3.5 Geometric Imperfections inDonnell’s NonlinearShell Theory 36
1.3.6 The Flügge-Lur’e-Byrne Nonlinear Shell Theory 37
1.3.7 The Novozhilov Nonlinear Shell Theory 38
1.3.8 The Sanders-KoiterNonlinear Shell Theory 40
v
vi Contents
1.3.9 ElasticStrain Energy 40
1.3.10 Donnell’s NonlinearShallow-Shell Theory 41
1.3.11 Donnell’s NonlinearShallow-Shell Theory Including Geometric
Imperfections 47
1.4 Large Deflection ofCircular Plates 48
1.4.1 Green’s Strain Tensorfor Circular Plates 48
1.4.2 Strainsfor Finite Deflection ofCircular Plates:
Von Kármán Theory 49
1.4.3 ElasticStrainEnergyfor CircularPlatesand Membranes 50
1.4.4 Von Kármán Equation of Motion for CircularPlates 51
1.5 Large Deflection ofSpherical Caps 52
1.5.1 Green’s Strain Tensorin Spherical Coordinates 52
1.5.2 Strainsfor Finite Deflection ofSpherical Caps: Donnell’s
Nonlinear Theory 53
1.5.3 Donnell’s Equation of Motion for Shallow Spherical Caps 54
1.5.4 The Flügge-Lur’e-ByrneNonlinear Shell Theory 55
1.6 Large Deflection ofSlightly Corrugated Shells 56
References 57
2 Classical NonlinearTheories of Doubly Curved Shells 60
2.1 Introduction 60
2.2 Doubly Curved Shellsof ConstantCurvature 60
2.2.1 ElasticStrainEnergy 64
2.3 General Theory ofDoubly Curved Shells 64
2.3.1 Theoryof Surfaces 64
2.3.2 Green’s Strain Tensorfor aShell inCurvilinear Coordinates 71
2.3.3 Strain–Displacement Relationshipsfor Novozhilov’sNonlinear
Shell Theory 74
2.3.4 Strain–Displacement Relationshipsfor an Improved Version of
the NovozhilovShell Theory 77
2.3.5 Simplified Strain–Displacement Relationships 78
2.3.6 ElasticStrainEnergy 78
2.3.7 KineticEnergy 79
2.4 PressureLoadfor Large Displacements 79
2.4.1 VirtualWork by Displacement Dependent Pressure 79
2.4.2 Pressure Applied to theShell Surface 82
2.4.3 Pressurized Circular Cylindrical Shell:Numerical Solution 85
References 89
3 Composite,Sandwichand Functionally GradedMaterials: Advanced
Nonlinear Shell Theories 90
3.1 Introduction 90
3.2 Composite Materials 90
3.3 Laminate under Plane Stress for Classical Shell Theories 92
3.3.1 Stress–Strain Relations for a Thin Lamina 92
Contents vii
3.3.2 Stress–Strain Relations for a Layer withina Laminate 94
3.3.3 Elastic Strain Energyfor Laminated Shells 94
3.3.4 Elastic Strain Energyfor Orthotropic and Cross-PlyShells 94
3.4 Lamina and Laminate withShear Deformation 96
3.4.1 Elastic Strain Energyfor Laminated Shells 98
3.5 Sandwich Platesand Shells 98
3.6 Functionally Graded Materialsand Thermal Effects 99
3.6.1 Functionally Graded Material inPlane Stress 101
3.6.2 Functionally Graded Material with ShearDeformation 102
3.7 NonlinearShearDeformation Theories for ModeratelyThick,
Laminated and Functionally Graded, DoublyCurved Shells 103
3.8 NonlinearFirst-Order ShearDeformation Theory for Doubly
Curved Shells of ConstantCurvature 103
3.8.1 Elastic Strain Energyfor Laminated Shells 105
3.8.2 KineticEnergywithRotary Inertia for Laminated Shells 106
3.9 NonlinearThird-OrderShear Deformation Theoryfor Laminated,
Doubly Curved Shells 107
3.9.1 Elastic Strain and KineticEnergies for Laminated Shells 111
3.9.2 Elastic Strain Energyfor Heated, Functionally
Graded Shells 112
3.9.3 KineticEnergywithRotary Inertia for Functionally
Graded Shells 113
3.9.4 NonlinearThird-Order ShearDeformation Theory for Circular
Cylindrical Shells 114
3.10 Thermal Effects onPlatesandShells 116
3.10.1 Isotropic Plates and Shells 116
3.10.2 Laminated Platesand Shells 117
3.11 Effect of Initial In-Plane Stress on aVibrating Rectangular Plate 118
3.12 Effect of Initial Axial Sress on a VibratingCircular Cylindrical Shell 119
3.13 Approximated Nonlinear First-Order Shear Deformation Theory
for Conical Shells 120
References 122
4 Nonlinear Shell Theory with Thickness Deformation 124
4.1 Introduction 124
4.2 NonlinearTheorywithThird-Order Thickness andShearDeformation 126
4.3 Elastic Strain Energy for Isotropic and Laminated Shells 134
4.4 Kinetic Energy of theShell 136
4.5 Formulationof theTheoryfor CircularCylindrical Shells 136
4.6 Simply SupportedCircular Cylindrical Shell Loaded by Pressure 139
4.7 NumericalResultsfor aPressure Loaded CircularCylindrical Shell 142
4.7.1 DisplacementIndependent Pressure Load (Distributed
Radial Forces) 142
4.7.2 DisplacementDependent Pressure Load (Actual Pressure) 147
References 148
viii Contents
5 Hyperelasticity of Soft Biological and Rubber Materials 151
5.1 Soft Materials: Rubbers, Foams and Biomaterials 151
5.2 Strain Tensors and Principal Stretches 154
5.3 Invariants ofthe Strain Tensors 159
5.4 Stress Tensors 160
5.5 Strain Energy Density Function 162
5.6 Isotropic Hyperelastic Materials 164
5.6.1 Representation in Terms of Invariants 165
5.6.2 Representation in Terms of Principal Stretches 166
5.7 Incompressible Hyperelastic Materials 167
5.7.1 Incompressible Isotropic Materials 168
5.8 Compressible Hyperelastic Materials 170
5.8.1 CompressibleIsotropic Materials 172
5.8.2 CompressibleIsotropic Materialsin Terms of Invariants 172
5.9 Hyperelastic Modelsof Incompressible Isotropic Materials 173
5.9.1 Neo-Hookean Model 174
5.9.2 Mooney-RivlinModel 179
5.9.3 OgdenModel 183
5.9.4 Gent Model 187
5.9.5 Fung-Demiray Model 191
5.10 Hyperelastic Modelsof Compressible Isotropic Materials 193
5.10.1 Neo-Hookean CompressibleModel 193
5.10.2 Mooney-RivlinCompressibleModel 195
5.10.3 OgdenCompressible Model 196
5.10.4 Linearly Elastic Material 196
5.11 Anisotropic Hyperelastic Materials 197
5.11.1 Transversely Isotropic Materials 197
5.11.2 Transversely Isotropic Incompressible Materials 199
5.11.3 Composite Materials with Two Families ofFibres 199
5.11.4 Materials with Two Orthogonal orEquivalent Families of Fibres 201
5.12 Hyperelastic Modelsof Anisotropic Materials 201
5.12.1 Standard Reinforcing Model for Transversely Isotropic
Incompressible Materials 201
5.12.2 Horgan-Saccomandi Model for Transversely Isotropic
Incompressible Materials 202
5.12.3 Holzapfel–Gasser–Ogden Model for TwoFamilies ofFibres 202
5.12.4 Holzapfel-OgdenModel for Fibre Dispersion 207
5.13 Model of Artery Wall 213
5.13.1 StrainEnergyof theWall 215
5.13.2 Residual Stresses inArterialWall 215
References 223
6 Introduction toNonlinearDynamics 225
6.1 Introduction 225
6.2 PeriodicNonlinear Vibrations: Softening and Hardening Systems 225
