Continuum Mechanics using Mathematica®: Fundamentals, Applications and Scientific Computing

This book examines mathematical tools, principles, and fundamental applications of continuum mechanics, providing a solid basis for a deeper study of more challenging problems in elasticity, fluid mechanics, plasticity, piezoelectricity, ferroelectricity, magneto-fluid mechanics, and state changes. The work is suitable for advanced undergraduates, graduate students, and researchers in applied mathematics, mathematical physics, and engineering. Table of Contents Cover Continuum Mechanics using Mathematica - Fundamentals, Applications and Scientific Computing ISBN-10 0817632409 eISBN 081764458X ISBN-13 9780817632403 Contents Preface Chapter 1 Elements of Linear Algebra 1.1 Motivation to Study Linear Algebra 1.2 Vector Spaces and Bases 1.3 Euclidean Vector Space 1.4 Base Changes 1.5 Vector Product 1.6 Mixed Product 1.7 Elements of Tensor Algebra 1.8 Eigenvalues and Eigenvectors of a Euclidean Second-Order Tensor 1.9 Orthogonal Tensors 1.10 Cauchy's Polar Decomposition Theorem 1.11 Higher Order Tensors 1.12 Euclidean Point Space 1.13 Exercises 1.14 The Program VectorSys 1.15 The Program EigenSystemAG Chapter 2 Vector Analysis 2.1 Curvilinear Coordinates 2.2 Examples of Curvilinear Coordinates 2.3 Di.erentiation of Vector Fields 2.4 The Stokes and Gauss Theorems 2.5 Singular Surfaces 2.6 Useful Formulae 2.7 Some Curvilinear Coordinates 2.7.1 Generalized Polar Coordinates 2.7.2 Cylindrical Coordinates 2.7.3 Spherical Coordinates 2.7.4 Elliptic Coordinates 2.7.5 Parabolic Coordinates 2.7.6 Bipolar Coordinates 2.7.7 Prolate and Oblate Spheroidal Coordinates 2.7.8 Paraboloidal Coordinates 2.8 Exercises 2.9 The Program Operator Chapter 3 Finite and In.nitesimal Deformations 3.1 Deformation Gradient 3.2 Stretch Ratio and Angular Distortion 3.3 Invariants of C and B 3.4 Displacement and Displacement Gradient 3.5 In.nitesimal Deformation Theory 3.6 Transformation Rules for Deformation Tensors 3.7 Some Relevant Formulae 3.8 Compatibility Conditions 3.9 Curvilinear Coordinates 3.10 Exercises 3.11 The Program Deformation Chapter 4 Kinematics 4.1 Velocity and Acceleration 4.2 Velocity Gradient 4.3 Rigid, Irrotational, and Isochoric Motions 4.4 Transformation Rules for a Change of Frame 4.5 Singular Moving Surfaces 4.6 Time Derivative of a Moving Volume 4.7 Worked Exercises 4.8 The Program Velocity Chapter 5 Balance Equations 5.1 General Formulation of a Balance Equation 5.2 Mass Conservation 5.3 Momentum Balance Equation 5.4 Balance of Angular Momentum 5.5 Energy Balance 5.6 Entropy Inequality 5.7 Lagrangian Formulation of Balance Equations 5.8 The Principle of Virtual Displacements 5.9 Exercises Chapter 6 Constitutive Equations 6.1 Constitutive Axioms 6.2 Thermoviscoelastic Behavior 6.3 Linear Thermoelasticity 6.4 Exercises Chapter 7 Symmetry Groups: Solids and Fluids 7.1 Symmetry 7.2 Isotropic Solids 7.3 Perfect and Viscous Fluids 7.4 Anisotropic Solids 7.5 Exercises 7.6 The Program LinElasticityTensor Chapter 8 Wave Propagation 8.1 Introduction 8.2 Cauchy's Problem for Second-Order PDEs 8.3 Characteristics and Classi.cation of PDEs 8.4 Examples 8.5 Cauchy's Problem for a Quasi-Linear First-Order System 8.6 Classi.cation of First-Order Systems 8.7 Examples 8.8 Second-Order Systems 8.9 Ordinary Waves 8.10 Linearized Theory and Waves 8.11 Shock Waves 8.12 Exercises 8.13 The Program PdeEqClass 8.14 The Program PdeSysClass 8.15 The Program WavesI 8.16 The Program WavesII Chapter 9 Fluid Mechanics 9.1 Perfect Fluid 9.2 Stevino's Law and Archimedes' Principle 9.3 Fundamental Theorems of Fluid Dynamics 9.4 Boundary Value Problems for a Perfect Fluid 9.5 2D Steady Flow of a Perfect Fluid 9.6 D'Alembert's Paradox and the Kutta-Joukowsky Theorem 9.7 Lift and Airfoils 9.8 Newtonian Fluids 9.9 Applications of the Navier-Stokes Equation 9.10 Dimensional Analysis and the Navier-Stokes Equation 9.11 Boundary Layer 9.12 Motion of a Viscous Liquid around an Obstacle 9.13 Ordinary Waves in Perfect Fluids 9.14 Shock Waves in Fluids 9.15 Shock Waves in a Perfect Gas 9.16 Exercises 9.17 The Program Potential 9.18 The Program Wing 9.19 The Program Joukowsky 9.20 The Program JoukowskyMap Chapter 10 Linear Elasticity 10.1 Basic Equations of Linear Elasticity 10.2 Uniqueness Theorems 10.3 Existence and Uniqueness of Equilibrium Solutions 10.4 Examples of Deformations 10.5 The Boussinesq-Papkovich-Neuber Solution 10.6 Saint-Venant's Conjecture 10.7 The Fundamental Saint-Venant Solutions 10.8 Ordinary Waves in Elastic Systems 10.9 Plane Waves 10.10 Re.ection of Plane Waves in a Half-Space 10.11 Rayleigh Waves 10.12 Re.ection and Refraction of SH Waves 10.13 Harmonic Waves in a Layer 10.14 Exercises Chapter 11 Other Approaches to Thermodynamics 11.1 Basic Thermodynamics 11.2 Extended Thermodynamics 11.3 Serrin's Approach 11.4 An Application to Viscous Fluids References Index