Tensor Analysis (eBook)

The author is a Professor Emeritus at the Norwegian University of Science and Technology (NTNU). His major field of interest is continuum mechanics. He obtained an M.Sc. degree in Theoretical and Applied Mechanics from Iowa State University in 1960, and completed a degree in Civil Engineering, majoring in Structural Engineering at NTNU in (1960) He started his academic carrier as an Assistant and Associate Professor in Mechanics at NTNU in 1963 and was appointed Professor in Solid Mechanics at NTNU in 2003. He has been a Visiting Professor at the University of Connecticut and University of Colorado. He has been involved in a variety of projects in which fluids and fluid-like materials have been modeled as non-Newtonian fluids, such as avalanching snow, granular materials in landslides, extrusion of aluminium, modeling of biomaterials as bone, and viscoplastic modeling of paper to simulate the calendaring process. He is author or coauthor of 26 papers and 11 conference presentations on mechanics and biomechanics. He has published the following textbooks in the Norwegian language: Statics, 7 editions; Strength of Materials, 7 editions; Dynamics, 4 editions; Continuum Mechanics; and Tensor Analysis. In 2008 his book Continuum Mechanics was published by Springer Verlag. In 2014 his book Rheology and Non-Newtonian Fluids was published by Springer Verlag. At NTNU he has lectured on elementary and advanced topics in the entire field of theoretical and applied mechanics: engineering mechanic, dynamics, fluid mechanics, biomechanics, rheology, continuum mechanics and tensor analysis. For NTNU he has developed new courses in Continuum Mechanics, Tensor Analysis, Rheology and Non-Newtonian Fluids, and Biomechanics.  

Preface 5
A Short Presentation of the Contents of This Book 7
References 9
Contents 10
Symbols 15
1 Mathematical Foundation 20
1.1 Matrices and Determinants 20
1.2 Cartesian Coordinate Systems. Scalars and Vectors 25
1.2.1 Displacement Vectors 27
1.2.2 Vector Algebra 32
1.3 Cartesian Coordinate Transformations 37
1.4 Curves in Space 40
1.5 Dynamics of a Mass Particle 45
1.6 Scalar Fields and Vector Fields 47
2 Dynamics. The Cauchy Stress Tensor 53
2.1 Kinematics 53
2.1.1 Lagrangian Coordinates and Eulerian Coordinates 53
2.1.2 Material Derivative of Intensive Quantities 56
2.1.3 Material Derivative of Extensive Quantities 58
2.2 Equations of Motion 59
2.2.1 Euler’s Axioms 59
2.2.2 Newton’s Third Law of Action and Reaction 64
2.2.3 Coordinate Stresses 65
2.2.4 Cauchy’s Stress Theorem and Cauchy’s Stress Tensor 68
2.2.5 Cauchy’s Equations of Motion 72
2.3 Stress Analysis 76
2.3.1 Principal Stresses and Principal Stress Directions 76
2.3.2 Stress Deviator and Stress Isotrop 82
2.3.3 Extreme Values of Normal Stress 84
2.3.4 Maximum Shear Stress 85
2.3.5 State of Plane Stress 87
2.3.6 Mohr-Diagram for State of Plane Stress 89
Reference 92
3 Tensors 93
3.1 Definition of Tensors 93
3.2 Tensor Algebra 99
3.2.1 Isotropic Tensors of Fourth Order 106
3.2.2 Tensors as Polyadics 107
3.3 Tensors of Second Order 109
3.3.1 Symmetric Tensors of Second Order 111
3.3.2 Alternative Invariants of Second Order Tensors 115
3.3.3 Deviator and Isotrop of Second Order Tensors 116
3.4 Tensor Fields 116
3.4.1 Gradient, Divergence, and Rotation of Tensor Fields 117
3.4.2 Del-Operators 119
3.4.3 Directional Derivative of Tensor Fields 121
3.4.4 Material Derivative of Tensor Fields 121
3.5 Rigid-Body Dynamics. Kinematics 122
3.5.1 Pure Rotation About a Fixed Axis 122
3.5.2 Pure Rotation About a Fixed Point 124
3.5.3 Kinematics of General Rigid-Body Motion 129
3.6 Rigid-Body Dynamics. Kinetics 133
3.6.1 Rotation About a Fixed Point. The Inertia Tensor 133
3.6.2 General Rigid-Body Motion 138
3.7 Q-Rotation of Vectors and Tensors of Second Order 141
3.8 Polar Decomposition 142
3.9 Isotropic Functions of Tensors 144
References 148
4 Deformation Analysis 149
4.1 Strain Measures 149
4.2 Green’s Strain Tensor 150
4.3 Small Strains and Small Deformations 156
4.3.1 Small Strains 157
4.3.2 Small Deformations 158
4.3.3 Principal Strains and Principal Directions for Small Deformations 160
4.3.4 Strain Deviator and Strain Isotrop for Small Deformations 161
4.3.5 Rotation Tensor for Small Deformations 162
4.3.6 Small Deformations in a Material Surface 163
4.3.7 Mohr–Diagram for Small Deformations in a Surface 165
4.4 Rates of Deformation, Strain, and Rotation 166
4.5 Large Deformations 172
5 Constitutive Equations 181
5.1 Introduction 181
5.2 Linearly Elastic Materials 181
5.2.1 Generalized Hooke’s Law 182
5.2.2 Some Basic Equations in Linear Elasticity. Navier’s Equations 187
5.2.3 Stress Waves in Elastic Materials 189
5.3 Linearly Viscous Fluids 192
5.3.1 Definition of Fluids 192
5.3.2 The Continuity Equation 195
5.3.3 Constitutive Equations for Linearly Viscous Fluids 195
5.3.4 The Navier–Stokes Equations 197
5.3.5 Film Flow 197
References 200
6 General Coordinates in Euclidean Space E3 201
6.1 Introduction 201
6.2 General Coordinates. Base Vectors 202
6.2.1 Covariant and Contravariant Transformations 206
6.2.2 Fundamental Parameters of a General Coordinate System 208
6.2.3 Orthogonal Coordinates 210
6.3 Vector Fields 212
6.4 Tensor Fields 216
6.4.1 Tensor Components. Tensor Algebra 216
6.4.2 Symmetric Tensors of Second Order 220
6.4.3 Tensors as Polyadics 221
6.5 Differentiation of Tensor Fields 222
6.5.1 Christoffel Symbols 222
6.5.2 Absolute and Covariant Derivatives of Vector Components 223
6.5.3 The Frenet-Serret Formulas for Space Curves 228
6.5.4 Divergence and Rotation of Vector Fields 230
6.5.5 Orthogonal Coordinates 230
6.5.6 Absolute and Covariant Derivatives of Tensor Components 234
6.6 Two-Point Tensor Components 243
6.7 Relative Tensors 246
Reference 247
7 Elements of Continuum Mechanics in General Coordinates 248
7.1 Introduction 248
7.2 Kinematics 248
7.2.1 Material Derivative of Intensive Quantities 250
7.3 Deformation Analysis 252
7.3.1 Strain Measures 254
7.3.2 Small Strains and Small Deformations 255
7.3.3 Rates of Deformation, Strain, and Rotation 258
7.4 General Analysis of Large Deformations 261
7.5 Convected Coordinates 263
7.6 Convected Derivatives of Tensors 267
7.7 Cauchy’s Stress Tensor. Equations of Motion 271
7.7.1 Physical Stress Components 273
7.7.2 Cauchy’s Equations of Motion 275
7.8 Basic Equations in Linear Elasticity 276
7.9 Basic Equations for Linearly Viscous Fluids 278
7.9.1 Basic Equations in Orthogonal Coordinates 280
References 286
8 Surface Geometry. Tensors in Riemannian Space R2 287
8.1 Surface Coordinates. Base Vectors. Fundamental Parameters 287
8.2 Surface Vectors 293
8.2.1 Scalar Product and Vector Product of Surface Vectors 295
8.3 Coordinate Transformations 296
8.3.1 Geodesic Coordinate System on a Surface 298
8.4 Surface Tensors 299
8.4.1 Symmetric Surface Tensors of Second Order 300
8.4.2 Space-Surface Tensors 302
8.5 Differentiation of Surface Tensors 303
8.6 Intrinsic Surface Geometry 310
8.6.1 The Metric of a Surface Imbedded in the Euclidean Space E3 310
8.6.2 Surface Curves 311
8.7 Curvatures on a Surface 317
8.7.1 The Codazzi Equations and the Gauss Equation 323
Reference 324
9 Integral Theorems 325
9.1 Integration Along a Space Curve 325
9.2 Integral Theorems in a Plane 327
9.2.1 Integration Over a Plane Region in Curvilinear Coordinates 331
9.3 Integral Theorems in Space 336
9.3.1 Integration Over a Surface Imbedded in the Euclidean Space E3 337
9.3.2 Integration Over a Volume in the Euclidean Space E3 338
Appendix 355
Index 392