Multiscale Structural Mechanics

List of Figures xv
List of Tables xxi
Foreword xxiii
Preface xxv
Acknowledgments xxxi
Acronyms xxxiii

1 Introduction 1
1.1 Continuum Hypothesis 1
1.2 Isotropic vs. Anisotropic Materials 4
1.3 Homogeneous vs. Heterogeneous Materials 4
1.4 Materials vs. Structures 4
1.5 3D Structures, Plates, Shells, and Beams 6
1.6 Structures, Models, and Assumptions 8
1.7 Composite Materials 20
1.8 Benefits of Using Composites 24
1.9 Mechanics of Composite Materials 25
1.10 Challenges for Modeling Composites 26
1.11 Multiscale Modeling* 28

2 Mathematical Preliminaries 43
2.1 Scalars, Vectors, and Tensors 43
2.2 Cartesian Coordinate System 44
2.3 Index Notation 44
2.4 Vectors 48
2.5 Transformation Between Different Coordinate Systems 52
2.6 Second-order Tensors 55
2.7 Quotient Rule and Higher-order Tensors 60
2.8 Special Tensors 61
2.9 Isotropic Tensors 62
2.10 Tensor Calculus 62
2.11 General Coordinate Systems* 63
2.12 Calculus of Variations* 73

3 Theory of Anisotropic Elasticity 103
3.1 Kinematics 103
3.2 Kinetics 109
3.3 Constitutive Relations 114
3.4 Theory of Linear Elasticity 147
3.5 Boundary Conditions and Continuity Conditions 149
3.6 A Few Anisotropic Elasticity Problems 151
3.7 Variational Principles for Anisotropic Elasticity* 173

4 Micromechanics 187
4.1 Introduction 187
4.2 Microstructures and Their Idealizations 188
4.3 Volume Average 192
4.4 Effective Stiffness and Compliance 192
4.5 Voigt and Reuss Rules of Mixtures 193
4.6 Hybrid Rules of Mixtures 202
4.7 Macro and Micro Coordinates* 209
4.8 Average Stress Theorem* 210
4.9 Average Strain Theorem* 211
4.10 Hill-Mandel Macro-homogeneity Condition* 213
4.11 Computational Homogenization* 216

5 Composite Plate Models 233
5.1 Introduction 233
5.2 Composite Laminates 234
5.3 Why Composite Plate Theories? 236
5.4 Kirchhoff-Love Model Derived Using the Newtonian Method 238
5.5 Reissner-Mindlin Model* 288
5.6 MSG-based Composite Plate Models* 292

6 Composite Beam Models* 301
6.1 Introduction 301
6.2 Ad Hoc Methods 303
6.3 Beam Models Derived Using MSG 319
6.4 A Few Composite Beam Problems 328

7 Mechanics of Structure Genome* 337
7.1 Introduction 338
7.2 Kinematics 339
7.3 Variational Statement for SG 349
7.4 MSG Illustrated 353
7.5 Numerical Examples 359

8 Failure of Composite Materials 371
8.1 Introduction 371
8.2 Failure Criteria for Isotropic Materials 374
8.3 Failure Criteria for Orthotropic Materials 382
8.4 Strength Ratio 399
8.5 Failure Envelope 403
8.6 Progressive Failure Analysis 406
8.7 Nonlocal Approach for Computing Strength 408

Problems 409
References 413
Index 421