Preface A few words about notations PART I 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 Canchy stress tensor and the Piola—Kirchhoff tensor.Applications 3.1 Hypotheses on the cohesion forces 3.2 The Canchy stress tensor 3.3 General equations of motion 3.4 Symmetry of the stress tensor 3.5 The Piola—Kirchhoff 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 5.4 Appendix.Change of variable in a surface integral 6 Energy equations and shock equations 6.1 Heat and energy 6.2 Shocks and the Rankine——Hugoniot relations PART Ⅱ 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 General theorems 8.2 Plane h'rotational flows 8.3 Transsonic flows 8.4 Linear accoustics 9 Viscous fluids and thermohydraulics 9.1 Equations of viscous incompressible fluids 9.2 Simple flows of viscous incompressible fluids 9.3 Thermohydranlics 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 Ⅲ 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， and 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 17 Nonlinear elasticity and an application to biomechanics 17.1 The equations of nonlinear elasticity 17.2 Boundary conditions—boundary value problems 17.3 Hyperelastic materials 17.4 Hyperelastic materials in biomechanics PART Ⅳ INTRODUCTION TO WAVE PHENOMENA 18 Linear wave equations in mechanics 18.1 Returning to the equations of linear acoustics and of linear elasticity 18.2 Solution of the one—dimensional wave equation 18.3 Normal modes 18.4 Solution of the wave equation 18.5 Superposition of waves， beats， and packets of waves 19 The soliton equation： the Korteweg—de Vries equation 19.1 Water—wave equations 19.2 Simplified form of the water—wave equations 19.3 The Korteweg—de Vries equation 19.4 The soliton solutions of the KdV equation 20 The nonlinear Schrodinger equation 20.1 Maxwell equations for polarized media 20.2 Equations of the electric field： the linear case 20.3 General case 20.4 The nonlinear Schrodinger equation 20.5 Soliton solutions of the NLS equation Appendix.The partial differential equations of mechanics Hints for the exercises References Index
