目錄
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 Cauehy stress tensor and the Piola-Kirchhoff
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
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 theology: 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 irrotational 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 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 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 biomechanies
17.1 The equations of nonlinear elasticity
17.2 Boundary conditions - boundary value problems
17.3 Hyperelastic materials
17.4 Hvoerelastic 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 Sehrodinger 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
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