Anisotropic Elasticity (eBook)

Paolo Vannucci is Professor of Mechanics at the LMV - Laboratoire de Mathématiques deVersailles, University of Versailles and Saint-Quentin. His main research activities concern planeanisotropic elasticity and multiphysics problems, optimization methods for anisotropic structures,metaheuristics for structural optimization, mechanics of no-tension materials applied to the study ofmonumental structures.

Preface 7
Contents 10
About the Author 16
1 Basic Concepts on Anisotropy 17
1.1 Introduction: What is Anisotropy? 17
1.2 Mathematical Consequences of Anisotropy 19
1.2.1 Effects on the Algebraic Operators 19
1.2.2 Geometrical Symmetries 20
1.3 Some Anisotropic Physical Phenomena 22
1.3.1 Paramagnetism and Diamagnetism 23
1.3.2 Dielectric Susceptibility 23
1.3.3 Thermal Conductivity 24
1.3.4 Piezoelectricity 24
1.3.5 Photoelastic and Electro-Optical Effects 25
1.3.6 A General Consideration About Anisotropic Phenomena 26
1.4 Some Basic Elements About Crystals 27
1.4.1 Lattices and Cells 27
1.4.2 The Symmetries of Crystals 29
1.4.3 Classifications of the Crystals 29
1.4.4 The Neumann's Principle 31
1.5 Some Fundamental Equations of the Mechanics of Elastic Bodies 31
References 33
2 General Anisotropic Elasticity 34
2.1 The Hooke's Law for Anisotropic Bodies 34
2.1.1 The Voigt's Notation 37
2.1.2 The Kelvin's Notation 39
2.1.3 The Mechanical Meaning of the Anisotropic Elastic Constants 40
2.2 Elastic Symmetries 42
2.2.1 Taking into Account for Elastic Symmetries 42
2.2.2 Rotation of Axes 43
2.2.3 A Tensorial Characterization of Elastic Symmetries 46
2.2.4 Triclinic Bodies 46
2.2.5 Monoclinic Bodies 47
2.2.6 Orthotropic Bodies 48
2.2.7 Axially Symmetric Bodies 49
2.2.8 Transversely Isotropic Bodies 51
2.2.9 Isotropic Bodies 52
2.2.10 Some Remarks About Elastic Symmetries 54
2.2.11 Elasticity of Crystals and Elastic Syngonies 54
2.3 The Technical Constants of Elasticity 56
2.3.1 The Young's Moduli 57
2.3.2 Shear Moduli 57
2.3.3 Poisson's Coefficients 58
2.3.4 Chentsov's Coefficients 59
2.3.5 Coefficients of Mutual Influence of the First Type 59
2.3.6 Coefficients of Mutual Influence of the Second Type 60
2.3.7 Some Remarks About the Technical Constants 61
2.4 Bounds on the Elastic Constants 63
2.4.1 General Conditions and Results 63
2.4.2 Mathematical Conditions for the Elastic Matrices 63
2.4.3 A Mechanical Approach 66
2.4.4 Bounds on the Technical Constants 67
2.5 An Observation About the Decomposition of the Strain Energy 69
2.6 Determination of Symmetry Planes 71
2.6.1 Physical Interpretations 73
2.7 Curvilinear Anisotropy 74
2.8 Some Examples of Anisotropic Materials 76
References 88
3 Plane Anisotropic Elasticity 89
3.1 Introduction 89
3.2 Stress Functions 90
3.3 Simplifying the General Relations 94
3.3.1 Rotation of Axes 94
3.3.2 The Tsai and Pagano Parameters 95
3.3.3 Plane and Antiplane States and Tensors 96
3.4 Plane Strain 98
3.4.1 The Concept of Plane Strain in the Literature 100
3.5 Plane Stress 102
3.5.1 The Concept of Plane Stress in the Literature 104
3.6 Generalized Plane Stress 105
3.7 Mechanical Consistency of Plane States 107
3.8 Comparison of Plane States 109
3.9 The Lekhnitskii Theory 111
3.9.1 The General Lekhnitskii Problem 111
3.9.2 The Decomposition of the Displacement Field 112
3.9.3 Strain Field and Compatibility Equations 114
3.9.4 Differential Equations for ? and ? 115
3.9.5 General Solution of the Homogeneous Equations 118
3.9.6 Roots of the Characteristic Equation 120
3.9.7 General Expressions for Stresses and Displacements 123
3.9.8 Boundary Conditions 125
3.9.9 Generalized Plane Strain 126
3.9.10 Plane Deformation 127
3.9.11 Generalized Plane Stress 130
3.9.12 A Final Consideration 130
3.10 The Stroh Theory 131
3.10.1 The General Stroh Problem 131
3.10.2 The Theory of Eshelby, Read and Shockley 132
3.10.3 The Eigenvalues pj and the Elastic Syngony 135
3.10.4 The Sextic Formalism of Stroh 136
3.10.5 Algebraic Questions 139
3.11 Plane States: Nomenclature 142
References 143
4 The Polar Formalism 144
4.1 Introduction: Why the Polar Formalism? 144
4.2 The Transformation of Verchery 146
4.3 Tensor Rotation 150
4.4 Tensor Invariants Under Frame Rotations 153
4.5 The Polar Components 156
4.5.1 Second-Rank Symmetric Tensors 156
4.5.2 Elasticity Tensors 157
4.6 Change of Frame 159
4.7 Harmonic Interpretation of the Polar Formalism 161
4.8 Polar Parameters of the Inverse Tensor 162
4.9 Technical Constants and Polar Invariants 163
4.10 Polar Decomposition of the Strain Energy 165
4.11 Bounds on the Polar Invariants 167
4.12 Symmetries 169
4.12.1 Ordinary Orthotropy 173
4.12.2 R0-Orthotropy 179
4.12.3 r0-Orthotropy 184
4.12.4 Square Symmetry 186
4.12.5 Isotropy 188
4.12.6 Final Considerations About Elastic Symmetries in mathbbR2 188
4.13 The Polar Formulae with the Kelvin's Notation 189
4.14 Comparison with the Tsai and Pagano Parameters 190
4.15 Special Plane Elastic Anisotropic Materials 192
4.15.1 Rari-Constant Materials 193
4.15.2 Complex Materials 205
4.16 Special Topics of the Polar Formalism 207
4.16.1 Polar Projectors 207
4.16.2 Interaction of Geometry and Anisotropy 214
4.16.3 Wrinkling of Anisotropic Membranes 221
4.17 Applications of the Polar Formalism to Other Fields 233
4.17.1 Plane Piezoelectricity 233
4.17.2 Anisotropic Damage of Isotropic Layers 235
4.17.3 Tensor Strength Criteria for Anisotropic Layers 242
4.18 Some Examples of Planar Anisotropic Materials 246
References 255
5 Anisotropic Laminates 258
5.1 Introduction 258
5.2 Fundamentals of the Classical Laminated Plates Theory 259
5.2.1 The Assumptions of the Classical Model 259
5.2.2 The Kinematical Consequences of the Kirchhoff Hypotheses 259
5.2.3 The Strain and Stress Tensors 260
5.2.4 Internal Actions 263
5.2.5 The Laminates' Fundamental Law 264
5.2.6 Bending-Extension Coupling 265
5.2.7 Heterogeneity of the Elastic Behavior 266
5.2.8 Quasi-homogeneous Laminates 266
5.2.9 Inverting the Fundamental Law of Laminates 267
5.2.10 Laminates Made of Identical Plies 269
5.2.11 Laminates by the Polar Formalism 270
5.2.12 The Case of Identical Layers: The Lamination Parameters 272
5.2.13 Geometrical Bounds 274
5.3 Laminates with Special General Properties 279
5.3.1 Bending-Extension Uncoupling 279
5.3.2 Quasi-homogeneity 281
5.3.3 Quasi-trivial Solutions 281
5.3.4 Orthotropy 284
5.3.5 Isotropy 293
5.3.6 Sensitivity to Orientation Errors 298
5.4 Thermal and Hygral Properties 306
5.4.1 The Fundamental Law of Laminates in Thermo-Elasticity 307
5.4.2 The Inverse Fundamental Law of Laminates in Thermo-Elasticity 308
5.4.3 The Polar Formalism for the Thermo-Elastic Tensors 309
5.4.4 Thermally Uncoupled Laminates 310
5.4.5 Thermally Quasi-homogeneous Laminates 311
5.5 Higher-Order Laminate Theories and the Polar Formalism 312
5.5.1 The First-Order Shear Deformation Theory of Laminated Plates 313
5.5.2 The Third-Order Shear Deformation Theory of Laminated Plates 316
References 319
6 Design Problems and Methods of Anisotropic Structures 322
6.1 Introduction 322
6.2 A Basic Problem: The Optimal Orientation of Anisotropy 324
6.2.1 A Short Account of the State of the Art 324
6.2.2 A Polar Approach to the Maximization of the Stiffness 326
6.2.3 A Polar Approach to the Maximization of the Strength 336
6.3 Design Problems of Anisotropic Structures 339
6.3.1 Different Types of Design Problems 339
6.3.2 Influence of Anisotropy on Optimal Solutions 340
6.4 Design Problems of the First Type 345
6.4.1 Unified Polar Formulation of the Optimum Problem 346
6.4.2 Identical Layers Laminates: The Lamination Set, Non Uniqueness of the Stacking Sequence 348
6.4.3 Numerical Approaches 352
6.5 Design Problems of the Second Type 360
6.5.1 Methods for Handling Constraints 360
6.5.2 Some Examples of Design Problems of the Second Type 364
6.6 Design Problems of the Third Type 369
6.6.1 A Problem Naturally Sequential: The Two-Step Approach 369
6.6.2 Step 1: Structural Anisotropy Optimization Problem 370
6.6.3 Step 2: Constitutive Law Problem 371
6.6.4 Final Commentaries on the Two-Step Approach 372
6.6.5 Some Examples of Design Problems of the Third Type 373
6.7 Optimization of Anisotropy Fields 382
6.7.1 The Case of Variable Stiffness and Strength 383
6.8 Optimization of Modular Systems 393
6.8.1 The Code BIANCA 393
6.8.2 Designing Laminates with Minimal Number of Layers 396
6.8.3 An Application: The Design of an Aircraft Wing Box-Girder 400
6.9 Some Multiphysics Problems of Anisotropic Laminates Design 413
6.9.1 Tailoring the Thermo-Elastic Properties of an Anisotropic Laminate 413
6.9.2 Thermally Stable Laminates 416
6.9.3 Tailoring the Piezo-Electric Properties 428
References 433