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