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# Galilean Mechanics and Thermodynamics of Continua

• ID: 4091820
• Book
• 446 Pages
• John Wiley and Sons Ltd
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This title proposes a unified approach to continuum mechanics which is consistent with Galilean relativity.  Based on the notion of affine tensors, a simple generalization of the classical tensors, this approach allows gathering the usual mechanical entities mass, energy, force, moment, stresses, linear and angular momentum in a single tensor.

Starting with the basic subjects, and continuing through to the most advanced topics, the authors′ presentation is progressive, inductive and bottom–up. They begin with the concept of an affine tensor, a natural extension of the classical tensors. The simplest types of affine tensors are the points of an affine space and the affine functions on this space, but there are more complex ones which are relevant for mechanics torsors and momenta. The essential point is to derive the balance equations of a continuum from a unique principle which claims that these tensors are affine–divergence free.

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Foreword  xiii

Introduction  xxi

Part 1. Particles and Rigid Bodies 1

Chapter 1. Galileo s Principle of Relativity  3

1.1. Events and space time  3

1.2. Event coordinates 3

1.2.1. When? 3

1.2.2. Where? 4

1.3. Galilean transformations 6

1.3.1. Uniform straight motion 6

1.3.2. Principle of relativity 9

1.3.3. Space time structure and velocity addition 10

1.3.4. Organizing the calculus  11

1.3.5. About the units of measurement 12

Chapter 2. Statics  15

2.1. Introduction  15

2.2. Statical torsor 16

2.2.1. Two–dimensional model 16

2.2.2. Three–dimensional model 17

2.2.3. Statical torsor and transport law of the moment  18

2.3. Statics equilibrium 20

2.3.1. Resultant torsor  20

2.3.2. Free body diagram and balance equation 20

2.3.3. External and internal forces 23

Chapter 3. Dynamics of Particles 27

3.1. Dynamical torsor 27

3.1.1. Transformation law and invariants  27

3.1.2. Boost method 30

3.2. Rigid body motions  32

3.2.1. Rotations  32

3.2.2. Rigid motions 34

3.3. Galilean gravitation 36

3.3.1. How to model the gravitational forces? 36

3.3.2. Gravitation 38

3.3.3. Galilean gravitation and equation of motion 40

3.3.4. Transformation laws of the gravitation and acceleration 42

3.4. Newtonian gravitation 46

3.5. Other forces 51

3.5.1. General equation of motion  51

3.5.2. Foucault s pendulum 52

3.5.3. Thrust  55

Chapter 4. Statics of Arches, Cables and Beams 57

4.1. Statics of arches  57

4.1.1. Modeling of slender bodies 57

4.1.2. Local equilibrium equations of arches 59

4.1.3. Corotational equilibrium equations of arches 62

4.1.4. Equilibrium equations of arches in Fresnet s moving frame 63

4.2. Statics of cables  67

4.3. Statics of trusses and beams 69

4.3.1. Traction of trusses 69

4.3.2. Bending of beams 71

Chapter 5. Dynamics of Rigid Bodies 75

5.1. Kinetic co–torsor 75

5.1.1. Lagrangian coordinates 75

5.1.2. Eulerian coordinates 76

5.1.3. Co–torsor 76

5.2. Dynamical torsor 80

5.2.1. Total mass and mass–center 80

5.2.2. The rigid body as a particle 81

5.2.3. The moment of inertia matrix 84

5.2.4. Kinetic energy of a body 87

5.3. Generalized equations of motion 88

5.3.1. Resultant torsor of the other forces 88

5.3.2. Transformation laws 89

5.3.3. Equations of motion of a rigid body 91

5.4. Motion of a free rigid body around it 93

5.5. Motion of a rigid body with a contact point (Lagrange s top) 95

Chapter 6. Calculus of Variations 105

6.1. Introduction  105

6.2. Particle subjected to the Galilean gravitation 109

6.2.1. Guessing the Lagrangian expression 109

6.2.2. The potentials of the Galilean gravitation 110

6.2.3. Transformation law of the potentials of the gravitation  113

6.2.4. How to manage holonomic constraints? 116

Chapter 7. Elementary Mathematical Tools 117

7.1. Maps  117

7.2. Matrix calculus 118

7.2.1. Columns 118

7.2.2. Rows 119

7.2.3. Matrices 120

7.2.4. Block matrix 124

7.3. Vector calculus in R3 125

7.4. Linear algebra 127

7.4.1. Linear space  127

7.4.2. Linear form 129

7.4.3. Linear map 130

7.5. Affine geometry  132

7.6. Limit and continuity  135

7.7. Derivative  136

7.8. Partial derivative  136

7.9. Vector analysis 137

7.9.2. Divergence 139

7.9.3. Vector analysis in R3 and curl 139

Part 2. Continuous Media 141

Chapter 8. Statics of 3D Continua 143

8.1. Stresses 143

8.1.1. Stress tensor  143

8.1.2. Local equilibrium equations 148

8.2. Torsors 150

8.2.1. Continuum torsor 150

8.2.2. Cauchy s continuum  153

8.3. Invariants of the stress tensor 155

Chapter 9. Elasticity and Elementary Theory of Beams 157

9.1. Strains  157

9.2. Internal work and power 162

9.3. Linear elasticity  164

9.3.1. Hooke s law 164

9.3.2. Isotropic materials 166

9.3.3. Elasticity problems  170

9.4. Elementary theory of elastic trusses and beams 171

9.4.1. Multiscale analysis: from the beam to the elementary volume 171

9.4.2. Transversely rigid body model  176

9.4.3. Calculating the local fields  179

9.4.4. Multiscale analysis: from the elementary volume to the beam 183

Chapter 10. Dynamics of 3D Continua and Elementary Mechanics of Fluids 187

10.1. Deformation and motion 187

10.2. Flash–back: Galilean tensors 192

10.3. Dynamical torsor of a 3D continuum 196

10.4. The stress mass tensor 198

10.4.1. Transformation law and invariants 198

10.4.2. Boost method 200

10.5. Euler s equations of motion 202

10.6. Constitutive laws in dynamics 206

10.7. Hyperelastic materials and barotropic fluids  210

Chapter 11. Dynamics of Continua of Arbitrary Dimensions 215

11.1. Modeling the motion of one–dimensional (1D) material bodies 215

11.2. Group of the 1D linear Galilean transformations 217

11.3. Torsor of a continuum of arbitrary dimension 219

11.4. Force mass tensor of a 1D material body 220

11.5. Full torsor of a 1D material body  222

11.6. Equations of motion of a continuum of arbitrary dimension  224

11.7. Equation of motion of 1D material bodies 225

11.7.1. First group of equations of motion 226

11.7.2. Multiscale analysis  227

11.7.3. Secong group of equations of motion 231

Chapter 12. More About Calculus of Variations  235

12.1. Calculus of variation and tensors  235

12.2. Action principle for the dynamics of continua  237

12.3. Explicit form of the variational equations 240

12.4. Balance equations of the continuum  244

12.5. Comments for experts . 245

Chapter 13. Thermodynamics of Continua  247

13.1. Introduction  247

13.2. An extra dimension 248

13.3. Temperature vector and friction tensor 251

13.4. Momentum tensors and first principle 253

13.5. Reversible processes and thermodynamical potentials  258

13.6. Dissipative continuum and heat transfer equation 263

13.7. Constitutive laws in thermodynamics 268

13.8. Thermodynamics and Galilean gravitation  272

Chapter 14. Mathematical Tools 281

14.1. Group 281

14.2. Tensor algebra  282

14.2.1. Linear tensors 282

14.2.2. Affine tensors 288

14.2.3. G–tensors and Euclidean tensors  292

14.3. Vector analysis  295

14.3.1. Divergence  295

14.3.2. Laplacian 296

14.3.3. Vector analysis in R3 and curl 296

14.4. Derivative with respect to a matrix 297

14.5. Tensor analysis  297

14.5.1. Differential manifold  297

14.5.2. Covariant differential of linear tensors 300

14.5.3. Covariant differential of affine tensors 303

Chapter 15. Affine Structure on a Manifold 309

15.1. Introduction  309

15.2. Endowing the structure of linear space by transport 310

15.3. Construction of the linear tangent space  311

15.4. Endowing the structure of affine space by transport 313

15.5. Construction of the affine tangent space  316

15.6. Particle derivative and affine functions 319

Chapter 16. Galilean, Bargmannian and Poincarean Structures on a Manifold 321

16.1. Toupinian structure  321

16.2. Normalizer of Galileo s group in the affine group  323

16.3. Momentum tensors  325

16.4. Galilean momentum tensors 328

16.4.1. Coadjoint representation of Galileo s group  328

16.4.2. Galilean momentum transformation law  329

16.4.3. Structure of the orbit of a Galilean momentum torsor  335

16.5. Galilean coordinate systems 338

16.5.1. G–structures 338

16.5.2. Galilean coordinate systems  338

16.6. Galilean curvature  341

16.7. Bargmannian coordinates  346

16.8. Bargmannian torsors 349

16.9. Bargmannian momenta 352

16.10. Poincarean structures  357

16.11. Lie group statistical mechanics  362

Chapter 17. Symplectic Structure on a Manifold  367

17.1. Symplectic form 367

17.2. Symplectic group 370

17.3. Momentum map 371

17.4. Symplectic cohomology 373

17.5. Central extension of a group  375

17.6. Construction of a central extension from the symplectic cocycle 377

17.8. Connections  385

17.9. Factorized symplectic form 387

17.10. Application to classical mechanics  393

17.11. Application to relativity  396

Chapter 18. Advanced Mathematical Tools 399

18.1. Vector fields  399

18.2. Lie group 400

18.3. Foliation  402

18.4. Exterior algebra 402

18.5. Curvature tensor 405

Bibliography 407

Index 411

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