Vannucci Anisotropic Elasticity

1. Auflage 2018
ISBN: 978-981-10-5439-6
Verlag: Springer Nature Singapore
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 85, 439 Seiten

Reihe: Lecture Notes in Applied and Computational Mechanics

This book presents a modern and unconventional introduction to anisotropy. The first part presents a general description of Anisotropic Elasticity theories while the second part focuses on the polar formalism: the theoretical bases and results are completely developed along with applications to design problems of laminated anisotropic structures. The book is based on lectures on anisotropy which have been held at Ecole Polytechnique in Paris.

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.

Autoren/Hrsg.

Vannucci, Paolo

Inhaltsverzeichnis

1;Preface;7
2;Contents;10
3;About the Author;16
4;1 Basic Concepts on Anisotropy;17
4.1;1.1 Introduction: What is Anisotropy?;17
4.2;1.2 Mathematical Consequences of Anisotropy;19
4.2.1;1.2.1 Effects on the Algebraic Operators;19
4.2.2;1.2.2 Geometrical Symmetries;20
4.3;1.3 Some Anisotropic Physical Phenomena;22
4.3.1;1.3.1 Paramagnetism and Diamagnetism;23
4.3.2;1.3.2 Dielectric Susceptibility;23
4.3.3;1.3.3 Thermal Conductivity;24
4.3.4;1.3.4 Piezoelectricity;24
4.3.5;1.3.5 Photoelastic and Electro-Optical Effects;25
4.3.6;1.3.6 A General Consideration About Anisotropic Phenomena;26
4.4;1.4 Some Basic Elements About Crystals;27
4.4.1;1.4.1 Lattices and Cells;27
4.4.2;1.4.2 The Symmetries of Crystals;29
4.4.3;1.4.3 Classifications of the Crystals;29
4.4.4;1.4.4 The Neumann's Principle;31
4.5;1.5 Some Fundamental Equations of the Mechanics of Elastic Bodies;31
4.6;References;33
5;2 General Anisotropic Elasticity;34
5.1;2.1 The Hooke's Law for Anisotropic Bodies;34
5.1.1;2.1.1 The Voigt's Notation;37
5.1.2;2.1.2 The Kelvin's Notation;39
5.1.3;2.1.3 The Mechanical Meaning of the Anisotropic Elastic Constants;40
5.2;2.2 Elastic Symmetries;42
5.2.1;2.2.1 Taking into Account for Elastic Symmetries;42
5.2.2;2.2.2 Rotation of Axes;43
5.2.3;2.2.3 A Tensorial Characterization of Elastic Symmetries;46
5.2.4;2.2.4 Triclinic Bodies;46
5.2.5;2.2.5 Monoclinic Bodies;47
5.2.6;2.2.6 Orthotropic Bodies;48
5.2.7;2.2.7 Axially Symmetric Bodies;49
5.2.8;2.2.8 Transversely Isotropic Bodies;51
5.2.9;2.2.9 Isotropic Bodies;52
5.2.10;2.2.10 Some Remarks About Elastic Symmetries;54
5.2.11;2.2.11 Elasticity of Crystals and Elastic Syngonies;54
5.3;2.3 The Technical Constants of Elasticity;56
5.3.1;2.3.1 The Young's Moduli;57
5.3.2;2.3.2 Shear Moduli;57
5.3.3;2.3.3 Poisson's Coefficients;58
5.3.4;2.3.4 Chentsov's Coefficients;59
5.3.5;2.3.5 Coefficients of Mutual Influence of the First Type;59
5.3.6;2.3.6 Coefficients of Mutual Influence of the Second Type;60
5.3.7;2.3.7 Some Remarks About the Technical Constants;61
5.4;2.4 Bounds on the Elastic Constants;63
5.4.1;2.4.1 General Conditions and Results;63
5.4.2;2.4.2 Mathematical Conditions for the Elastic Matrices;63
5.4.3;2.4.3 A Mechanical Approach;66
5.4.4;2.4.4 Bounds on the Technical Constants;67
5.5;2.5 An Observation About the Decomposition of the Strain Energy;69
5.6;2.6 Determination of Symmetry Planes;71
5.6.1;2.6.1 Physical Interpretations;73
5.7;2.7 Curvilinear Anisotropy;74
5.8;2.8 Some Examples of Anisotropic Materials;76
5.9;References;88
6;3 Plane Anisotropic Elasticity;89
6.1;3.1 Introduction;89
6.2;3.2 Stress Functions;90
6.3;3.3 Simplifying the General Relations;94
6.3.1;3.3.1 Rotation of Axes;94
6.3.2;3.3.2 The Tsai and Pagano Parameters;95
6.3.3;3.3.3 Plane and Antiplane States and Tensors;96
6.4;3.4 Plane Strain;98
6.4.1;3.4.1 The Concept of Plane Strain in the Literature;100
6.5;3.5 Plane Stress;102
6.5.1;3.5.1 The Concept of Plane Stress in the Literature;104
6.6;3.6 Generalized Plane Stress;105
6.7;3.7 Mechanical Consistency of Plane States;107
6.8;3.8 Comparison of Plane States;109
6.9;3.9 The Lekhnitskii Theory;111
6.9.1;3.9.1 The General Lekhnitskii Problem;111
6.9.2;3.9.2 The Decomposition of the Displacement Field;112
6.9.3;3.9.3 Strain Field and Compatibility Equations;114
6.9.4;3.9.4 Differential Equations for ? and ?;115
6.9.5;3.9.5 General Solution of the Homogeneous Equations;118
6.9.6;3.9.6 Roots of the Characteristic Equation;120
6.9.7;3.9.7 General Expressions for Stresses and Displacements;123
6.9.8;3.9.8 Boundary Conditions;125
6.9.9;3.9.9 Generalized Plane Strain;126
6.9.10;3.9.10 Plane Deformation;127
6.9.11;3.9.11 Generalized Plane Stress;130
6.9.12;3.9.12 A Final Consideration;130
6.10;3.10 The Stroh Theory;131
6.10.1;3.10.1 The General Stroh Problem;131
6.10.2;3.10.2 The Theory of Eshelby, Read and Shockley;132
6.10.3;3.10.3 The Eigenvalues pj and the Elastic Syngony;135
6.10.4;3.10.4 The Sextic Formalism of Stroh;136
6.10.5;3.10.5 Algebraic Questions;139
6.11;3.11 Plane States: Nomenclature;142
6.12;References;143
7;4 The Polar Formalism;144
7.1;4.1 Introduction: Why the Polar Formalism?;144
7.2;4.2 The Transformation of Verchery;146
7.3;4.3 Tensor Rotation;150
7.4;4.4 Tensor Invariants Under Frame Rotations;153
7.5;4.5 The Polar Components;156
7.5.1;4.5.1 Second-Rank Symmetric Tensors;156
7.5.2;4.5.2 Elasticity Tensors;157
7.6;4.6 Change of Frame;159
7.7;4.7 Harmonic Interpretation of the Polar Formalism;161
7.8;4.8 Polar Parameters of the Inverse Tensor;162
7.9;4.9 Technical Constants and Polar Invariants;163
7.10;4.10 Polar Decomposition of the Strain Energy;165
7.11;4.11 Bounds on the Polar Invariants;167
7.12;4.12 Symmetries;169
7.12.1;4.12.1 Ordinary Orthotropy;173
7.12.2;4.12.2 R0-Orthotropy;179
7.12.3;4.12.3 r0-Orthotropy;184
7.12.4;4.12.4 Square Symmetry;186
7.12.5;4.12.5 Isotropy;188
7.12.6;4.12.6 Final Considerations About Elastic Symmetries in mathbbR2;188
7.13;4.13 The Polar Formulae with the Kelvin's Notation;189
7.14;4.14 Comparison with the Tsai and Pagano Parameters;190
7.15;4.15 Special Plane Elastic Anisotropic Materials;192
7.15.1;4.15.1 Rari-Constant Materials;193
7.15.2;4.15.2 Complex Materials;205
7.16;4.16 Special Topics of the Polar Formalism;207
7.16.1;4.16.1 Polar Projectors;207
7.16.2;4.16.2 Interaction of Geometry and Anisotropy;214
7.16.3;4.16.3 Wrinkling of Anisotropic Membranes;221
7.17;4.17 Applications of the Polar Formalism to Other Fields;233
7.17.1;4.17.1 Plane Piezoelectricity;233
7.17.2;4.17.2 Anisotropic Damage of Isotropic Layers;235
7.17.3;4.17.3 Tensor Strength Criteria for Anisotropic Layers;242
7.18;4.18 Some Examples of Planar Anisotropic Materials;246
7.19;References;255
8;5 Anisotropic Laminates;258
8.1;5.1 Introduction;258
8.2;5.2 Fundamentals of the Classical Laminated Plates Theory;259
8.2.1;5.2.1 The Assumptions of the Classical Model;259
8.2.2;5.2.2 The Kinematical Consequences of the Kirchhoff Hypotheses;259
8.2.3;5.2.3 The Strain and Stress Tensors;260
8.2.4;5.2.4 Internal Actions;263
8.2.5;5.2.5 The Laminates' Fundamental Law;264
8.2.6;5.2.6 Bending-Extension Coupling;265
8.2.7;5.2.7 Heterogeneity of the Elastic Behavior;266
8.2.8;5.2.8 Quasi-homogeneous Laminates;266
8.2.9;5.2.9 Inverting the Fundamental Law of Laminates;267
8.2.10;5.2.10 Laminates Made of Identical Plies;269
8.2.11;5.2.11 Laminates by the Polar Formalism;270
8.2.12;5.2.12 The Case of Identical Layers: The Lamination Parameters;272
8.2.13;5.2.13 Geometrical Bounds;274
8.3;5.3 Laminates with Special General Properties;279
8.3.1;5.3.1 Bending-Extension Uncoupling;279
8.3.2;5.3.2 Quasi-homogeneity;281
8.3.3;5.3.3 Quasi-trivial Solutions;281
8.3.4;5.3.4 Orthotropy;284
8.3.5;5.3.5 Isotropy;293
8.3.6;5.3.6 Sensitivity to Orientation Errors;298
8.4;5.4 Thermal and Hygral Properties;306
8.4.1;5.4.1 The Fundamental Law of Laminates in Thermo-Elasticity;307
8.4.2;5.4.2 The Inverse Fundamental Law of Laminates in Thermo-Elasticity;308
8.4.3;5.4.3 The Polar Formalism for the Thermo-Elastic Tensors;309
8.4.4;5.4.4 Thermally Uncoupled Laminates;310
8.4.5;5.4.5 Thermally Quasi-homogeneous Laminates;311
8.5;5.5 Higher-Order Laminate Theories and the Polar Formalism;312
8.5.1;5.5.1 The First-Order Shear Deformation Theory of Laminated Plates;313
8.5.2;5.5.2 The Third-Order Shear Deformation Theory of Laminated Plates;316
8.6;References;319
9;6 Design Problems and Methods of Anisotropic Structures;322
9.1;6.1 Introduction;322
9.2;6.2 A Basic Problem: The Optimal Orientation of Anisotropy;324
9.2.1;6.2.1 A Short Account of the State of the Art;324
9.2.2;6.2.2 A Polar Approach to the Maximization of the Stiffness;326
9.2.3;6.2.3 A Polar Approach to the Maximization of the Strength;336
9.3;6.3 Design Problems of Anisotropic Structures;339
9.3.1;6.3.1 Different Types of Design Problems;339
9.3.2;6.3.2 Influence of Anisotropy on Optimal Solutions;340
9.4;6.4 Design Problems of the First Type;345
9.4.1;6.4.1 Unified Polar Formulation of the Optimum Problem;346
9.4.2;6.4.2 Identical Layers Laminates: The Lamination Set, Non Uniqueness of the Stacking Sequence;348
9.4.3;6.4.3 Numerical Approaches;352
9.5;6.5 Design Problems of the Second Type;360
9.5.1;6.5.1 Methods for Handling Constraints;360
9.5.2;6.5.2 Some Examples of Design Problems of the Second Type;364
9.6;6.6 Design Problems of the Third Type;369
9.6.1;6.6.1 A Problem Naturally Sequential: The Two-Step Approach;369
9.6.2;6.6.2 Step 1: Structural Anisotropy Optimization Problem;370
9.6.3;6.6.3 Step 2: Constitutive Law Problem;371
9.6.4;6.6.4 Final Commentaries on the Two-Step Approach;372
9.6.5;6.6.5 Some Examples of Design Problems of the Third Type;373
9.7;6.7 Optimization of Anisotropy Fields;382
9.7.1;6.7.1 The Case of Variable Stiffness and Strength;383
9.8;6.8 Optimization of Modular Systems;393
9.8.1;6.8.1 The Code BIANCA;393
9.8.2;6.8.2 Designing Laminates with Minimal Number of Layers;396
9.8.3;6.8.3 An Application: The Design of an Aircraft Wing Box-Girder;400
9.9;6.9 Some Multiphysics Problems of Anisotropic Laminates Design;413
9.9.1;6.9.1 Tailoring the Thermo-Elastic Properties of an Anisotropic Laminate;413
9.9.2;6.9.2 Thermally Stable Laminates;416
9.9.3;6.9.3 Tailoring the Piezo-Electric Properties;428
9.10;References;433