Mathematical Modeling in Continuum Mechanics 2nd Edition by Roger Temam,Alain Miranville 9780521617239 0521617235

PART I FUNDAMENTAL CONCEPTS IN CONTINUUM MECHANICS

CHAPTER ONE Describing the motion of a system: geometry and kinematics

1.1. Deformations

Regularity assumption

Displacement

Elementary deformations

a) Rigid deformations

b) Linear compression or elongation

c) Shear

1.2. Motion and its observation (kinematics)

Explicit representation of the motion

Trajectory of a particle

Velocity of a particle

Acceleration of a particle

Stream lines

1.3. Description of the motion of a system: Eulerian and Lagrangian derivatives

Lagrangian description of the motion of a system (description by the trajectories)

Eulerian description of the motion of a system (description by the velocity field)

Eulerian and Lagrangian derivatives

Application to the computation of acceleration in the Eulerian representation

1.4. Velocity field of a rigid body: helicoidal vector fields

Operations on helicoidal vector fields

a) Sum and product by a scalar

b) Topology

c) Differentiation

Structure of a helicoidal vector field

1.5. Differentiation of a volume integral depending on a parameter

CHAPTER TWO The fundamental law of dynamics

2.1. The concept ofmass

Conservation of mass

Consequence of mass conservation: the continuity equation in Eulerian variables

Conservation of mass in Lagrangian variables

Kinetic energy of a system with respect to a frame of reference

Linear and angular momentum and the corresponding HVF (Helicoidal Vector Field)

Quantities of acceleration and the corresponding HVF

2.2. Forces

The corresponding HVF

2.3. The fundamental law of dynamics and its first consequences

2.4. Application to systems ofmaterial points and to rigid bodies

Systems of material points

Analytical expression of the fundamental law

Rigid bodies

Inertia tensor at a point A

2.5. Galilean frames: the fundamental law of dynamics expressed in a non-Galilean frame

Comparison of velocities and accelerations in two different frames

The fundamental law in a non-Galilean frame

Galilean frames

CHAPTER THREE The Cauchy 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 ofmotion

Equilibrium equations

3.4. Symmetry of the stress tensor

Consequences

3.5. The Piola-Kirchhoff tensor

CHAPTER FOUR Real and virtual powers

4.1. Study of a system of material points

The case of a material point

The case of a system of n material points M1,…,Mn

4.2. General material systems: rigidifying velocities

Virtual velocity fields rigidifying a partition

4.3. Virtual power of the cohesion forces: the general case

The virtual power theorem

4.4. Real power: the kinetic energy theorem

CHAPTER FIVE Deformation tensor, deformation rate tensor, constitutive laws

5.1. Further properties of deformations

Distortion of distances

Distortion of angles

5.2. The deformation rate tensor

5.3. Introduction to rheology: the constitutive laws

Some principles of rheology

Main examples in fluid mechanics

Newtonian viscous fluids

Non-Newtonian fluids

Main examples in solid mechanics

Elastic media

a) Linear or classical elasticity

b) Nonlinear elasticity (hyperelasticity)

c) Hypoelasticity

Viscoelastic materials (materials with memory)

Plastic materials

a) Perfectly plastic (or plastic rigid) materials

b) Elastoplastic materials

c) Viscoplastic materials

d) Prandtl–Reuss law

5.4. Appendix.Change of variable in a surface integral

CHAPTER SIX Energy equations and shock equations

6.1. Heat and energy

Heat

The energy conservation principle: the first law of thermodynamics

The energy conservation equation

Particular cases

6.2. Shocks and the Rankine–Hugoniot relations

Principle of the study

Consequence: the shock conditions

Application to conservation laws: the Rankine–Hugoniot relations

a) Conservation of mass

b) Conservation of momentum

Conservation of energy

PART II PHYSICS OF FLUIDS

CHAPTER SEVEN General properties of Newtonian fluids

7.1. General equations of fluids mechanics

The equations

Inviscid fluids

Incompressible fluids

The vorticity equation

Equation of state: barotropic fluids

Boundary conditions

Other cases

7.2. Statics of fluids

Statics of incompressible fluids

a) Contact between two fluids

b) The Venturi device

c) The Pitot tube

Statics of compressible barotropic fluids: a first approximation in meteorology

7.3. Remark on the energy of a fluids

CHAPTER EIGHT Flows of inviscid fluids

8.1. General theorems

The Bernoulli, Kelvin, and Lagrange theorems

8.2. Plane irrotational flows

Velocity potential and stream function

Elementary stationary plane flows

a) Uniform flows

b) Sources and sinks

c) Singular point vortex

d) Doublet

e) Flow in an angle or around an angle

f) Flow with circulation around a circle

Computation of the forces on a wall: Blasius formulas

Airfoil theory: the analytic approach

a) The Kutta–Joukowski condition

b) Numerical methods

8.3. Transsonic flows

8.4. Linear accoustics

“Infinitesimal” motion assumption

CHAPTER NINE Viscous fluids and thermohydraulics

9.1. Equations ofviscous incompressible fluids

9.2. Simple flows of viscous incompressible fluids

Poiseuille flow between two parallel planes

a) The two-dimensional case

b) The Three-Dimensional Case

Poiseuille flow in a cylindrical tube

Flows between two coaxial cylinders (Couette–Taylor flows)

9.3. Thermohydraulics

9.4. Equations in nondimensional form: similarities

9.5. Notions of stability and turbulence

9.6. Notion of boundary layer

A plane wall instantaneously set-in motion

Mathematical model

CHAPTER TEN Magnetohydrodynamics and inertial conflnement of plasmas

10.1. The Maxwell equations and electromagnetism

Conservation of the electric charge

Faraday’s law

Interface laws

Constitutive laws

Electromagnetism in a stable medium

10.2. Magnetohydrodynamics

10.3. The Tokamak machine

CHAPTER ELEVEN Combustion

11.1. Equations for mixtures of fluids

11.2. Equations ofchemical kinetics

11.3. The equations of combustion

Model of premixed laminar frame

11.4. Stefan–Maxwell equations

11.5. A simplifed problem: the two-species model

CHAPTER TWELVE Equations of the atmosphere and of the ocean

12.1. Preliminaries

The fundamental law of dynamics on a frame attached to the earth

Static equations of the atmosphere

The differential operators on the sphere

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

CHAPTER THIRTEEN The general equations of linear elasticity

13.1. Back to the stress–strain law oflinear elasticity: the elasticity coefficients of a material

Other elasticity coefficients: the Young and Poisson moduli

13.2. Boundary value problems in linear elasticity: the linearization principle

Linearization of the equations

Linearization of the boundary conditions

a) Prescribed displacement

b) Prescribed stress

Particular cases

a) Statics

b) Evolutionary case

Boundary value problems

a) Elastostatics

b) Elastodynamics

c) Linearity

13.3. Other equations

Compatibility equations (for the deformations)

The Beltrami equations

Other equations

13.4. The limit of elasticity criteria

The Tresca criterion

The von Mises criterion

CHAPTER FOURTEEN Classical problems of elastostatics

14.1. Longitudinal traction–compression of a cylindrical bar

Comparison with experiment

14.2. Uniform compression of an arbitrary body

14.3. Equilibrium of a spherical container subjected to external and internal pressures

Boundary conditions

Limit of elasticity

14.4. Deformation of a vertical cylindrical body under the action of its weight

14.5. Simple bending of a cylindrical beam

Discussion of the result

14.6. Torsion of cylindrical shafts

Boundary conditions

Elasticity limit (for a cylindrical shaft of radius a)

14.7.The Saint-Venant principle

CHAPTER FIFTEEN Energy theorems, duality, and variational formulations

15.1. Elastic energy of a material

15.2. Duality–generalization

Generalization

15.3. The energy theorems

15.4. Variational formulations

15.5. Virtual power theorem and variational formulations

Reciprocity theorem

CHAPTER SIXTEEN 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

CHAPTER SEVENTEEN Nonlinear elasticity and an application to biomechanics

17.1. The equations of nonlinear elasticity

Constitutive laws

Reference configuration and natural state

Constitutive laws near a reference configuration

Saint Venant-Kirchhoff material

17.2. Boundary conditions–boundary value problems

17.3. Hyperelastic materials

17.4. Hyperelastic materials in biomechanics

Saint Venant-Kirchhoff materials

Hyperelastic materials in biomechanics

PART IV INTRODUCTION TO WAVE PHENOMENA

CHAPTER EIGHTEEN Linear wave equations in mechanics

18.1. Returning to the equations of linear acoustics and oflinear elasticity

Returning to the wave equations of linear acoustics

Returning to the Navier equation of linear elasticity

Particular cases

a) The vibrating cord

b) The vibrating membrane

Flexion of an elastic string

18.2. Solution of the one-dimensional wave equation

18.3 .Normal modes

Cord fixed at endpoints

Sound pipe

Membrane fixed at its boundary

18.4. Solution of the wave equation

General vibrations of a cord fixed at its endpoints

General vibrations of a membrane fixed on its boundary

The three-dimensional case

18.5. Superposition of waves, beats, and packets of waves

CHAPTER NINETEEN The soliton equation: the Korteweg–de Vries equation

19.1. Water-wave equations

Small amplitude waves in shallow water: nondimensional form of the equations

19.2. Simplified form of the water-wave equations

Asymptotic expansions

Simplified equations (based on asymptotic expansions)

19.3. The Korteweg–de Vries equation

19.4. The soliton solutions of the KdV equation

CHAPTER TWENTY The nonlinear Schrödinger equation

20.1. Maxwell equations for polarized media

20.2. Equations of the electric field: the linear case

Linearized equation

20.3. General case

20.4. The nonlinear Schrödinger equation

20.5. Soliton solutions of the NLS equation

APPENDIX The partial differential equations ofmechanics

Hints for the exercises