Antonio Romano Addolorata Marasco Continuum Mechanics using Mathematica Fundamentals, Methods, and Applications Second Edition TECHNISCHE INFORM ATIONSB IBLIOTHEK UNIVERSITATSBtBLIOTHEK HANNOVER 1? Birkhauser
Contents 1 Elements of Linear Algebra 1 1.1 Motivation to Study Linear Algebra 1 1.2 Vector Spaces and Bases 2 Examples 3 1.3 Euclidean Vector Space 5 1.4 Base Changes 10 1.5 Vector Product 12 1.6 Mixed Product 13 1.7 Elements of Tensor Algebra 14 1.8 Eigenvalues and Eigenvectors of a Euclidean Second-Order Tensor 20 1.9 Orthogonal Tensors 25 1.10 Cauchy's Polar Decomposition Theorem 28 1.11 Higher Order Tensors 29 1.12 Euclidean Point Space 31 1.13 Exercises 32 1.14 The Program VectorSys 38 Aim of the Program VectorSys 38 Description of the Problem and Relative Algorithm 38 Command Line of the Program VectorSys 39 Parameter List 39 Worked Examples 40 Exercises 42 1.15 The Program EigenSystemAG 42 Aim of the Program EigenSystemAG 42 Description of the Algorithm 43 Command Line of the Program EigenSystemAG 43 Parameter List 43 ix
' x Contents Worked Examples 43 Exercises 44 2 Vector Analysis 47 2.1 Curvilinear Coordinates : 47 2.2 Examples of Curvilinear Coordinates 50 2.3 Differentiation of Vector Fields 53 2.4 The Stokes and Gauss Theorems 58 2.5 Singular Surfaces 59 2.6 Useful Formulae 63 2.7 Some Curvilinear Coordinates 66 Generalized Polar Coordinates 66 Cylindrical Coordinates 68 Spherical Coordinates 69 Elliptic Coordinates 70 Parabolic Coordinates 70 Bipolar Coordinates 71 Prolate and Oblate Spheroidal Coordinates 72 Paraboloidal Coordinates 72 2.8 Exercises 73 2.9 The Program Operator 76 Aim of the Program 76 Description of the Algorithm 76 Command Line of the Program Operator 77 Parameter List 77 Use Instructions 78 Worked Examples 78 Exercises 82 3 Finite and Infinitesimal Deformations 83 3.1 Deformation Gradient 83 3.2 Stretch Ratio and Angular Distortion 86 3.3 Invariants of C and B 89 3.4 Displacement and Displacement Gradient 90 3.5 Infinitesimal Deformation Theory 92 3.6 Transformation Rules for Deformation Tensors 94 3.7 Some Relevant Formulae 95 3.8 Compatibility Conditions 98 3.9 Curvilinear Coordinates 102 3.10 Exercises 103 3.11 The Program Deformation 107 Aim of the Program 107 Description of the Algorithm and Instructions for Use 107 Command Line of the Program Deformation 108
Contents xl Parameter List 108 Worked Examples 109 Exercises ; 112 4 Kinematics H5 4.1 Velocity and Acceleration 115 4.2 Velocity Gradient 118 4.3 Rigid, Irrotational, and Isochoric Motions 119 4.4 Transformation Rules for a Change of Frame 122 4.5 Singular Moving Surfaces 123 4.6 Time Derivative of a Moving Volume 126 4.7 Exercises 130 4.8 The Program Velocity 133 Aim of the Program, Input and Output 133 Worked Examples 134 Exercises 135 5 Balance Equations 137 5.1 General Formulation of a Balance Equation 137 5.2 Mass Conservation 142 5.3 Momentum Balance Equation 143 5.4 Balance of Angular Momentum 146 5.5 Energy Balance 147 5.6 Entropy Inequality, 149 5.7 Lagrangian Formulation of Balance Equations 152 5.8 The Principle of Virtual Displacements 157 5.9 Exercises 158 6 Constitutive Equations 163 6.1 Constitutive Axioms 163 6.2 Thermoviscoelastic Behavior 167 6.3 Linear Thermoelasticity 173 6.4 Exercises 177 7 Symmetry Groups: Solids and Fluids 179 7.1 Symmetry 179 7.2 Isotropic Solids 182 7.3 Perfect and Viscous Fluids 186 7.4 Anisotropic Solids 189 7.5 Exercises 192 7.6 The Program LinElasticityTensor 193 Aim of the Program 193 Description of the Problem and Relative Algorithm 194 Command Line of the Program LinElasticityTensor 194 Parameter List 195 Worked Examples 195 Exercises 196
xii Contents 8 Wave Propagation 197 8.1 Introduction 197 8.2 Cauchy's Problem for Second-Order PDEs 198 8.3 Characteristics and Classification of PDEs 202 8.4 Examples 204 8.5 Cauchy's Problem for a Quasi-Linear First-Order System 207 8.6 Classification of First-Order Systems 209 8.7 Examples 211 8.8 Second-Order Systems 214 8.9 Ordinary Waves 216 8.10 Linearized Theory and Waves 220 8.11 ShockWaves 224 8.12 Exercises 227 8.13 The Program PdeEqClass 229 Aim of the Program PdeEqClass 229 Description of the Problem and Relative Algorithm 229 Command Line of the Program PdeEqClass 230 Parameter List 230 Use Instructions 230 Worked Examples 231 Exercises 232 8.14 The Program PdeSysClass 234 Aim of the Program PdeSysClass 234 Description of the Problem and Relative Algorithm 234 Command Line of the Program PdeSysClass 235 Parameter List 235 Use Instructions 236 Worked Examples 237 Exercises 238 8.15 The Program WavesI 240 Aim of the Program WavesI 240 Description of the Problem and Relative Algorithm 240 Command Line of the Program WavesI 241 Parameter List 241 Use Instructions 241 Worked Example 242 Exercises 243 8.16 The Program WavesII 246 Aim of the Program WavesII 246 Description of the Problem and Relative Algorithm 246 Command Line of the Program WavesII 248 Parameter List 248 Use Instructions 248 Worked Example 248 Exercises 250
Contents 9 Fluid Mechanics 251 9.1 Perfect Fluid 251 9.2 Stevino's Law and Archimedes' Principle 253 9.3 Fundamental Theorems of Fluid Dynamics 256 9.4 Boundary Value Problems for a Perfect Fluid 261 9.5 2D Steady Flow of a Perfect Fluid 262 9.6 D'Alembert's Paradox and the Kutta-Joukowsky Theorem 270 9.7 Lift and Airfoils 273 9.8 Newtonian Fluids 278 9.9 Applications of the Navier-Stokes Equation 279 9.10 Dimensional Analysis and the Navier-Stokes Equation 281 9.11 Boundary Layer 282 9.12 Motion of a Viscous Liquid Around an Obstacle 287 9.13 Ordinary Waves in Perfect Fluids 295 9.14 Shock Waves in Fluids 297 9.15 Shock Waves in a Perfect Gas 300 9.16 Exercises 304 9.17 The Program Potential 306 Aim of the Program Potential 306 Description of the Problem and Relative Algorithm 307 Command Line of the Program Potential 307 Parameter List 307 Worked Examples 308 Exercises 312 9.18 The Program Wing 313 Aim of the Program Wing 313 Description of the Problem and Relative Algorithm 313 Command Line of the Program Wing 313 Parameter List 313 Worked Examples 314 Exercises 315 9.19 The Program Joukowsky 315 Aim of the Program Joukowsky 315 Description of the Problem and Relative Algorithm 316 Command Line of the Program Joukowsky 316 Parameter List 316 Worked Examples 317 Exercises 318 9.20 The Program JoukowskyMap 318 Aim of the Program JoukowskyMap 318 Description of the Problem and Relative Algorithm 318 Command Line of the Program JoukowskyMap 318 Parameter List 318 Use Instructions 319 Worked Examples 319 Exercises 321
xjv Contents 10 Linear Elasticity 323 10.1 Basic Equations of Linear Elasticity 323 10.2 Uniqueness Theorems 327 10.3 Existence and Uniqueness of Equilibrium Solutions 329 10.4 Examples of Deformations 333 10.5 The Boussinesq-Papkovich-Neuber Solution 335 10.6 Saint-Venant's Conjecture 336 10.7 The Fundamental Saint-Venant Solutions 341 10.8 Ordinary Waves in Elastic Systems 344 10.9 Plane Waves 350 10.10 Reflection of Plane Waves in a Half-Space 356 SH Waves 361 10.11 Rayleigh Waves 362 10.12 Reflection and Refraction of SH Waves 365 10.13 Harmonic Waves in a Layer 368 10.14 Exercises 371 11 Other Approaches to Thermodynamics 373 11.1 Basic Thermodynamics 373 11.2 Extended Thermodynamics 376 11.3 Serrin's Approach 378 11.4 An Application to Viscous Fluids 382 12 Fluid Dynamics and Meteorology 385 12.1 Introduction 385 12.2 Atmosphere as a Continuous System 387 12.3 Atmosphere as a Mixture 390 12.4 Primitive Equations in Spherical Coordinates 394 12.5 Dimensionless Form of the Basic Equations 397 12.6 The Hydrostatic and Tangent Approximations 401 12.7 Bjerknes'Theorem 404 12.8 Vorticity Equation and ErtePs Theorem 406 12.9 Reynolds Turbulence 409 12.10 Ekman's Planetary Boundary Layer 412 12.11 Oberbeck-Boussinesq Equations 416 12.12 Saltzman's Equations 418 12.13 Lorenz's System 421 12.14 Some Properties of Lorenz's System 424 13 Fluid Dynamics and Ship Motion 429 13.1 Introduction 429 13.2 A Ship as a Rigid Body 430 13.3 Kinematical Transformations 434 13.4 Dynamical Equations of Ship Motion 436 13.5 Final Form of Dynamical Equations 439 13.6 About the Forces Acting on a Ship 440
Contents xv 13.7 Linear Equations of Ship Motion 442 13.8 Small Motions in the Presence of Regular Small Waves 444 13.9 The Sea Surface as Free Surface 448 13.10 Linear Approximation of the Free Boundary Value Problem 451 13.11 Simple Waves 454 13.12 Flow of Small Waves 456 13.13 Stationary Waves 459 A A Brief Introduction to Weak Solutions 463 A.1 Weak Derivative and Sobolev Spaces 463 A.2 A Weak Solution of a PDE 467 A.3 The Lax-Milgram Theorem 469 References 471 Index 475