Continuum Mechanics using Mathematica®: Fundamentals, Methods, and Applications

Preface
Contents
1 Elements of Linear Algebra
	1.1 Motivation to Study Linear Algebra
	1.2 Vector Spaces and Bases
		Examples
	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
		Aim of the Program VectorSys
		Description of the Problem and Relative Algorithm
		Command Line of the Program VectorSys
		Parameter List
			Input Data
			Output Data
		Worked Examples
		Exercises
	1.15 The Program EigenSystemAG
		Aim of the Program EigenSystemAG
		Description of the Algorithm
		Command Line of the Program EigenSystemAG
		Parameter List
			Input Data
			Output Data
		Worked Examples
		Exercises
2 Vector Analysis
	2.1 Curvilinear Coordinates
	2.2 Examples of Curvilinear Coordinates
	2.3 Differentiation of Vector Fields
	2.4 The Stokes and Gauss Theorems
	2.5 Singular Surfaces
	2.6 Useful Formulae
	2.7 Some Curvilinear Coordinates
		Generalized Polar Coordinates
		Cylindrical Coordinates
		Spherical Coordinates
		Elliptic Coordinates
		Parabolic Coordinates
		Bipolar Coordinates
		Prolate and Oblate Spheroidal Coordinates
		Paraboloidal Coordinates
	2.8 Exercises
	2.9 The Program Operator
		Aim of the Program
		Description of the Algorithm
		Command Line of the Program Operator
		Parameter List
			Input Data
			Output Data
		Use Instructions
		Worked Examples
		Exercises
3 Finite and Infinitesimal 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 Infinitesimal 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
		Aim of the Program
		Description of the Algorithm and Instructions for Use
		Command Line of the Program Deformation
		Parameter List
		Worked Examples
		Exercises
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 Exercises
	4.8 The Program Velocity
		Aim of the Program, Input and Output
		Worked Examples
		Exercises
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
6 Constitutive Equations
	6.1 Constitutive Axioms
	6.2 Thermoviscoelastic Behavior
	6.3 Linear Thermoelasticity
	6.4 Exercises
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
		Aim of the Program
		Description of the Problem and Relative Algorithm
		Command Line of the Program LinElasticityTensor
		Parameter List
		Worked Examples
		Exercises
8 Wave Propagation
	8.1 Introduction
	8.2 Cauchy's Problem for Second-Order PDEs
	8.3 Characteristics and Classification of PDEs
	8.4 Examples
	8.5 Cauchy's Problem for a Quasi-Linear First-Order System
	8.6 Classification 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
		Aim of the Program PdeEqClass
		Description of the Problem and Relative Algorithm
		Command Line of the Program PdeEqClass
		Parameter List
		Use Instructions
		Worked Examples
		Exercises
	8.14 The Program PdeSysClass
		Aim of the Program PdeSysClass
		Description of the Problem and Relative Algorithm
		Command Line of the Program PdeSysClass
		Parameter List
		Use Instructions
		Worked Examples
		Exercises
	8.15 The Program WavesI
		Aim of the Program WavesI
		Description of the Problem and Relative Algorithm
		Command Line of the Program WavesI
		Parameter List
		Use Instructions
		Worked Example
		Exercises
	8.16 The Program WavesII
		Aim of the Program WavesII
		Description of the Problem and Relative Algorithm
		Command Line of the Program WavesII
		Parameter List
		Use Instructions
		Worked Example
		Exercises
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
		Aim of the Program Potential
		Description of the Problem and Relative Algorithm
		Command Line of the Program Potential
		Parameter List
		Worked Examples
		Exercises
	9.18 The Program Wing
		Aim of the Program Wing
		Description of the Problem and Relative Algorithm
		Command Line of the Program Wing
		Parameter List
		Worked Examples
		Exercises
	9.19 The Program Joukowsky
		Aim of the Program Joukowsky
		Description of the Problem and Relative Algorithm
		Command Line of the Program Joukowsky
		Parameter List
		Worked Examples
		Exercises
	9.20 The Program JoukowskyMap
		Aim of the Program JoukowskyMap
		Description of the Problem and Relative Algorithm
		Command Line of the Program JoukowskyMap
		Parameter List
		Use Instructions
		Worked Examples
		Exercises
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 Reflection of Plane Waves in a Half-Space
		P and SV Waves
		SH Waves
	10.11 Rayleigh Waves
	10.12 Reflection and Refraction of SH Waves
	10.13 Harmonic Waves in a Layer
	10.14 Exercises
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
12 Fluid Dynamics and Meteorology
	12.1 Introduction
	12.2 Atmosphere as a Continuous System
	12.3 Atmosphere as a Mixture
	12.4 Primitive Equations in Spherical Coordinates
	12.5 Dimensionless Form of the Basic Equations
	12.6 The Hydrostatic and Tangent Approximations
	12.7 Bjerknes' Theorem
	12.8 Vorticity Equation and Ertel's Theorem
	12.9 Reynolds Turbulence
	12.10 Ekman's Planetary Boundary Layer
	12.11 Oberbeck–Boussinesq Equations
	12.12 Saltzman's Equations
	12.13 Lorenz's System
	12.14 Some Properties of Lorenz's System
13 Fluid Dynamics and Ship Motion
	13.1 Introduction
	13.2 A Ship as a Rigid Body
	13.3 Kinematical Transformations
	13.4 Dynamical Equations of Ship Motion
	13.5 Final Form of Dynamical Equations
	13.6 About the Forces Acting on a Ship
	13.7 Linear Equations of Ship Motion
	13.8 Small Motions in the Presence of Regular Small Waves
	13.9 The Sea Surface as Free Surface
	13.10 Linear Approximation of the Free Boundary Value Problem
	13.11 Simple Waves
	13.12 Flow of Small Waves
	13.13 Stationary Waves
Appendix A A Brief Introduction to Weak Solutions
	A.1 Weak Derivative and Sobolev Spaces
	A.2 A Weak Solution of a PDE
	A.3 The Lax–Milgram Theorem
References
Index