Mathematical Modeling in Continuum Mechanics

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ISBN: 9780521617239 | 0521617235
Cover: Paperback
Copyright: 6/20/2005

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Temam and Miranville present core topics within the general themes of fluid and solid mechanics. The brisk style allows the text to cover a wide range of topics including viscous flow, magnetohydrodynamics, atmospheric flows, shock equations, turbulence, nonlinear solid mechanics, solitons, and the nonlinear Schr_dinger equation. This second edition will be a unique resource for those studying continuum mechanics at the advanced undergraduate and beginning graduate level whether in engineering, mathematics, physics or the applied sciences. Exercises and hints for solutions have been added to the majority of chapters, and the final part on solid mechanics has been substantially expanded. These additions have now made it appropriate for use as a textbook, but it also remains an ideal reference book for students and anyone interested in continuum mechanics.

Preface

A few words about notations

PART I FUNDAMENTAL CONCEPTS IN CONTINUUM MECHANICS

1 Describing the motion of a system: geometry and kinematics

(21)

1.1 Deformations

(3)

1.2 Motion and its observation (kinematics)

(4)

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

(3)

1.4 Velocity field of a rigid body: helicoidal vector fields

(5)

1.5 Differentiation of a volume integral depending on a parameter

(6)

2 The fundamental law of dynamics

(18)

2.1 The concept of mass

(6)

2.2 Forces

(2)

2.3 The fundamental law of dynamics and its first consequences

(2)

2.4 Application to systems of material points and to rigid bodies

(4)

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

(4)

3 The Cauchy stress tensor and the Piola-Kirchhoff tensor. Applications

(15)

3.1 Hypotheses on the cohesion forces

(3)

3.2 The Cauchy stress tensor

(3)

3.3 General equations of motion

(2)

3.4 Symmetry of the stress tensor

(2)

3.5 The Piola-Kirchhoff tensor

(5)

4 Real and virtual powers

(13)

4.1 Study of a system of material points

(4)

4.2 General material systems: rigidifying velocities

(2)

4.3 Virtual power of the cohesion forces: the general case

(4)

4.4 Real power: the kinetic energy theorem

(3)

5 Deformation tensor, deformation rate tensor, constitutive laws

(20)

5.1 Further properties of deformations

(5)

5.2 The deformation rate tensor

(2)

5.3 Introduction to rheology: the constitutive laws

(10)

5.4 Appendix. Change of variable in a surface integral

(3)

6 Energy equations and shock equations

(13)

6.1 Heat and energy

(5)

6.2 Shocks and the Rankine-Hugoniot relations

(8)

PART II PHYSICS OF FLUIDS

7 General properties of Newtonian fluids

103

(13)

7.1 General equations of fluid mechanics

103

(6)

7.2 Statics of fluids

109

(5)

7.3 Remark on the energy of a fluid

114

(2)

8 Flows of inviscid fluids

116

(21)

8.1 General theorems

116

(4)

8.2 Plane irrotational flows

120

(10)

8.3 Transsonic flows

130

(4)

8.4 Linear accoustics

134

(3)

9 Viscous fluids and thermohydraulics

137

(21)

9.1 Equations of viscous incompressible fluids

137

(1)

9.2 Simple flows of viscous incompressible fluids

138

(6)

9.3 Thermohydraulics

144

(2)

9.4 Equations in nondimensional form: similarities

146

(2)

9.5 Notions of stability and turbulence

148

(4)

9.6 Notion of boundary layer

152

(6)

10 Magnetohydrodynamics and inertial confinement of plasmas

158

(14)

10.1 The Maxwell equations and electromagnetism

159

(4)

10.2 Magnetohydrodynamics

163

(2)

10.3 The Tokamak machine

165

(7)

11 Combustion

172

(13)

11.1 Equations for mixtures of fluids

172

(2)

11.2 Equations of chemical kinetics

174

(2)

11.3 The equations of combustion

176

(2)

11.4 Stefan-Maxwell equations

178

(3)

11.5 A simplified problem: the two-species model

181

(4)

12 Equations of the atmosphere and of the ocean

185

(16)

12.1 Preliminaries

186

(2)

12.2 Primitive equations of the atmosphere

188

(4)

12.3 Primitive equations of the ocean

192

(1)

12.4 Chemistry of the atmosphere and the ocean

193

(2)

Appendix. The differential operators in spherical coordinates

195

(6)

PART III SOLID MECHANICS

13 The general equations of linear elasticity

201

(14)

13.1 Back to the stress-strain law of linear elasticity: the elasticity coefficients of a material

201

(2)

13.2 Boundary value problems in linear elasticity: the linearization principle

203

(5)

13.3 Other equations

208

(3)

13.4 The limit of elasticity criteria

211

(4)

14 Classical problems of elastostatics

215

(20)

14.1 Longitudinal traction-compression of a cylindrical bar

215

(3)

14.2 Uniform compression of an arbitrary body

218

(1)

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

219

(4)

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

223

(2)

14.5 Simple bending of a cylindrical beam

225

(4)

14.6 Torsion of cylindrical shafts

229

(4)

14.7 The Saint-Venant principle

233

(2)

15 Energy theorems, duality, and variational formulations

235

(13)

15.1 Elastic energy of a material

235

(2)

15.2 Duality - generalization

237

(3)

15.3 The energy theorems

240

(3)

15.4 Variational formulations

243

(3)

15.5 Virtual power theorem and variational formulations

246

(2)

16 Introduction to nonlinear constitutive laws and to homogenization

248

(11)

16.1 Nonlinear constitutive laws (nonlinear elasticity)

249

(2)

16.2 Nonlinear elasticity with a threshold (Henky's elastoplastic model)

251

(2)

16.3 Nonconvex energy functions

253

(2)

16.4 Composite materials: the problem of homogenization

255

(4)

17 Nonlinear elasticity and an application to biomechanics

259

(12)

17.1 The equations of nonlinear elasticity

259

(3)

17.2 Boundary conditions - boundary value problems

262

(2)

17.3 Hyperelastic materials

264

(2)

17.4 Hyperelastic materials in biomechanics

266

(5)

PART IV INTRODUCTION TO WAVE PHENOMENA

18 Linear wave equations in mechanics

271

(18)

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

271

(4)

18.2 Solution of the one-dimensional wave equation

275

(1)

18.3 Normal modes

276

(5)

18.4 Solution of the wave equation

281

(4)

18.5 Superposition of waves, beats, and packets of waves

285

(4)

19 The soliton equation: the Korteweg-de Vries equation

289

(14)

19.1 Water-wave equations

290

(2)

19.2 Simplified form of the water-wave equations

292

(3)

19.3 The Korteweg-de Vries equation

295

(4)

19.4 The soliton solutions of the KdV equation

299

(4)

20 The nonlinear Schrödinger equation

303

(16)

20.1 Maxwell equations for polarized media

304

(2)

20.2 Equations of the electric field: the linear case

306

(3)

20.3 General case

309

(4)

20.4 The nonlinear Schrödinger equation

313

(3)

20.5 Soliton solutions of the NLS equation

316

(3)

Appendix. The partial differential equations of mechanics

319

(2)

Hints for the exercises

321

(11)

References

332

(5)

Index

337

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Mathematical Modeling in Continuum Mechanics

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