At a time when mathematical modeling is pervading many areas of science and master's degree programs in industrial mathematics are being initiated in many universities, this book is intended as an introduction to continuum mechanics and mathematical modeling.One of the aims of the book is to reduce the gap slightly between mathematics and this area of natural science - a gap that is usually due to the language barrier and to the differences in thinking andreasoning.This book is written in a style suitable for mathematicians and adapted to their training.We have tried to remain very close to physics and to mathematics at the same time by making,in particular, a clear separation between what is assumed and what is proved.

Introduction

A Few Words About Notations

PART ONE.FUNDAMENTAL CONCEPTS IN CONTINUUM MECHANICS

1 Describing the Motion of a System: Geometry and Kinematics

 1.1 Deformations

 1.2 Motion and Its Observation (Kinematics)

 1.3 Description of the Motion of a System: Eulerian and Lagrangian Derivatives

 1.4 Velocity Field of a Rigid Body: Helicoidal Vector Fields

 1.5 Differentiation of a Volume Integral Depending on a Parameter

2 The Fundamental Law of Dynamics

 2.1 The Concept of Mass

 2.2 Forces

 2.3 The Fundamental Law of Dynamics and Its First Consequences

 2.4 Application to Systems of Material Points and to Rigid Bodies

 2.5 Galilean Frames: The Fundamental Law of Dynamics Expressed in a Non-Galilean Frame

3 The Cauchy Stress Tensor - Applications

 3.1 Hypotheses on the Cohesion Forces

 3.2 The Cauchy Stress Tensor

 3.3 General Equations of Motion

 3.4 Symmetry of the Stress Tensor

4 Real and Virtual Powers

 4.1 Study of a System of Material Points

 4.2 General Material Systems: Rigidifying Velocities

 4.3 Virtual Power of the Cohesion Forces: The General Case

 4.4 Real Power: The Kinetic Energy Theorem

5 Deformation Tensor, Deformation Rate Tensor,Constitutive Laws

 5.1 Further Properties of Deformations

 5.2 The Deformation Rate Tensor

 5.3 Introduction to Rheology: The Constitutive Laws

6 Energy Equations and Shock Equations

 6.1 Heat and Energy

 6.2 Shocks and the Rankine-Hugoniot Relations

PART TWO.PHYSICS OF FLUIDS

7 General Properties of Newtonian Fluids

 7.1 General Equations of Fluid Mechanics

 7.2 Statics of Fluids

 7.3 Remark on the Energy of a Fluid

8 Flows of Inviscid Fluids

 8.1 GeneralTheorems

 8.2 Plane Irrotational Flows

 8.3 Transsonic Flows

 8.4 Linear Acoustics

9 Viscous Fluids and Thermohydraulics

 9.1 Equations of Viscous Incompressible Fluids

 9.2 Simple Flows of Viscous Incompressible Fluids

 9.3 Thermohydraulics

 9.4 Equations in Nondimensional Form: Similarities

 9.5 Notions of Stability and Turbulence

 9.6 Notion of Boundary Layer

10 Magnetohydrodynamics and Inertial Confinement of Plasmas

 10.1 The Maxwell Equations and Electromagnetism

 10.2 Magnetohydrodynamics

 10.3 The Tokamak Machine

11 Combustion

 11.1 Equations for Mixtures of Fluids

 11.2 Equations of Chemical Kinetics

 11.3 The Equations of Combustion

 11.4 Stefan-Maxwell Equations

 11.5 A Simplified Problem: The Two-Species Model

12 Equations of the Atmosphere and of the Ocean

 12.1 Preliminaries

 12.2 Primitive Equations of the Atmosphere

 12.3 Primitive Equations of the Ocean

 12.4 Chemistry of the Atmosphere and the Ocean

 Appendix: The Differential Operators in Spherical Coordinates

PART THREE.SOLID MECHANICS

13 The General Equations of Linear Elasticity

 13.1 Back to the Stress-Strain Law of Linear Elasticity:The Elasticity Coefficients of a Material

 13.2 Boundary Value Problems in Linear Elasticity:The Linearization Principle

 13.3 Other Equations

 13.4 The Limit of Elasticity Criteria

14 Classical Problems of Elastostatics

 14.1 Longitudinal Traction-Compression of a Cylindrical Bar

 14.2 Uniform Compression of an Arbitrary Body

 14.3 Equilibrium of a Spherical Container Subjected to External and Internal Pressures

 14.4 Deformation of a Vertical Cylindrical Body Under the Action of Its Weight

 14.5 Simple Bending of a Cylindrical Beam

 14.6 Torsion of Cylindrical Shafts

 14.7 The Saint-Venant Principle

15 Energy Theorems - Duality: Variational Formulations

 15.1 Elastic Energy of a Material

 15.2 Duality - Generalization

 15.3 The Energy Theorems

 15.4 Variational Formulations

 15.5 Virtual Power Theorem and Variational Formulations

16 Introduction to Nonlinear Constitutive Laws and to Homogenization

 16.1 Nonlinear Constitutive Laws (Nonlinear Elasticity)

 16.2 Nonlinear Elasticity with a Threshold(Henky's Elastoplastic Model)

 16.3 Nonconvex Energy Functions

 16.4 Composite Materials: The Problem of Homogenization

PART FOUR.INTRODUCTION TO WAVE PHENOMENA

17 Linear Wave Equations in Mechanics

 17.1 Returning to the Equations of Linear Acoustics and of Linear Elasticity

 17.2 Solution of the One-Dimensional Wave Equation

 17.3 Normal Modes

 17.4 Solution of the Wave Equation

 17.5 Superposition of Waves, Beats, and Packets of Waves

18 The Soliton Equation: The Korteweg-de Vries Equation

 18.1 Water-Wave Equations

 18.2 Simplified Form of the Water-Wave Equations

 18.3 The Korteweg--de Vries Equation

 18.4 The Soliton Solutions of the KdV Equation

19 The Nonlinear Schrodinger Equation

 19.1 Maxwell Equations for Polarized Media

 19.2 Equations of the Electric Field: The Linear Case

 19.3 General Case

 19.4 The Nonlinear Schrodinger Equation

 19.5 Soliton Solutions of the NLS Equation

Appendix The Partial Differential Equations of Mechanics

References

Index