Mathematica. 1? Birkhauser. Continuum Mechanics using. Fundamentals, Methods, and Applications. Antonio Romano Addolorata Marasco.

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1 Antonio Romano Addolorata Marasco Continuum Mechanics using Mathematica Fundamentals, Methods, and Applications Second Edition TECHNISCHE INFORM ATIONSB IBLIOTHEK UNIVERSITATSBtBLIOTHEK HANNOVER 1? Birkhauser

2 Contents 1 Elements of Linear Algebra Motivation to Study Linear Algebra Vector Spaces and Bases 2 Examples Euclidean Vector Space Base Changes Vector Product Mixed Product Elements of Tensor Algebra Eigenvalues and Eigenvectors of a Euclidean Second-Order Tensor Orthogonal Tensors Cauchy's Polar Decomposition Theorem Higher Order Tensors Euclidean Point Space Exercises 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 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

3 ' x Contents Worked Examples 43 Exercises 44 2 Vector Analysis Curvilinear Coordinates : Examples of Curvilinear Coordinates Differentiation of Vector Fields The Stokes and Gauss Theorems Singular Surfaces Useful Formulae 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 Exercises 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 Deformation Gradient Stretch Ratio and Angular Distortion Invariants of C and B Displacement and Displacement Gradient Infinitesimal Deformation Theory Transformation Rules for Deformation Tensors Some Relevant Formulae Compatibility Conditions Curvilinear Coordinates Exercises 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

4 Contents xl Parameter List 108 Worked Examples 109 Exercises ; Kinematics H5 4.1 Velocity and Acceleration Velocity Gradient Rigid, Irrotational, and Isochoric Motions Transformation Rules for a Change of Frame Singular Moving Surfaces Time Derivative of a Moving Volume Exercises The Program Velocity 133 Aim of the Program, Input and Output 133 Worked Examples 134 Exercises Balance Equations General Formulation of a Balance Equation Mass Conservation Momentum Balance Equation Balance of Angular Momentum Energy Balance Entropy Inequality, Lagrangian Formulation of Balance Equations The Principle of Virtual Displacements Exercises Constitutive Equations Constitutive Axioms Thermoviscoelastic Behavior Linear Thermoelasticity Exercises Symmetry Groups: Solids and Fluids Symmetry Isotropic Solids Perfect and Viscous Fluids Anisotropic Solids Exercises 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

5 xii Contents 8 Wave Propagation Introduction Cauchy's Problem for Second-Order PDEs Characteristics and Classification of PDEs Examples Cauchy's Problem for a Quasi-Linear First-Order System Classification of First-Order Systems Examples Second-Order Systems Ordinary Waves Linearized Theory and Waves ShockWaves Exercises 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 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 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 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

6 Contents 9 Fluid Mechanics Perfect Fluid Stevino's Law and Archimedes' Principle Fundamental Theorems of Fluid Dynamics Boundary Value Problems for a Perfect Fluid D Steady Flow of a Perfect Fluid D'Alembert's Paradox and the Kutta-Joukowsky Theorem Lift and Airfoils Newtonian Fluids Applications of the Navier-Stokes Equation Dimensional Analysis and the Navier-Stokes Equation Boundary Layer Motion of a Viscous Liquid Around an Obstacle Ordinary Waves in Perfect Fluids Shock Waves in Fluids Shock Waves in a Perfect Gas Exercises 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 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 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 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

7 xjv Contents 10 Linear Elasticity Basic Equations of Linear Elasticity Uniqueness Theorems Existence and Uniqueness of Equilibrium Solutions Examples of Deformations The Boussinesq-Papkovich-Neuber Solution Saint-Venant's Conjecture The Fundamental Saint-Venant Solutions Ordinary Waves in Elastic Systems Plane Waves Reflection of Plane Waves in a Half-Space 356 SH Waves Rayleigh Waves Reflection and Refraction of SH Waves Harmonic Waves in a Layer Exercises Other Approaches to Thermodynamics Basic Thermodynamics Extended Thermodynamics Serrin's Approach An Application to Viscous Fluids Fluid Dynamics and Meteorology Introduction Atmosphere as a Continuous System Atmosphere as a Mixture Primitive Equations in Spherical Coordinates Dimensionless Form of the Basic Equations The Hydrostatic and Tangent Approximations Bjerknes'Theorem Vorticity Equation and ErtePs Theorem Reynolds Turbulence Ekman's Planetary Boundary Layer Oberbeck-Boussinesq Equations Saltzman's Equations Lorenz's System Some Properties of Lorenz's System Fluid Dynamics and Ship Motion Introduction A Ship as a Rigid Body Kinematical Transformations Dynamical Equations of Ship Motion Final Form of Dynamical Equations About the Forces Acting on a Ship 440

8 Contents xv 13.7 Linear Equations of Ship Motion Small Motions in the Presence of Regular Small Waves The Sea Surface as Free Surface Linear Approximation of the Free Boundary Value Problem Simple Waves Flow of Small Waves 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

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