Galilean Mechanics and Thermodynamics of Continua

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.

Sommario

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
1.4. Comments for experts 14
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
2.4. Comments for experts 25
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
3.6. Comments for experts 56
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
5.6. Comments for experts 103
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
7.9.1. Gradient  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
13.9. Comments for experts  279
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
Part 3. Advanced Topics  307
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
17.7. Coadjoint orbit method 383
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

Info autore

Géry de Saxcé is Professor at Lille 1 University - Science and Technology, France.

Riassunto

This title proposes a unified approach to continuum mechanics which is consistent with Galilean relativity.

Dettagli sul prodotto

• Autori: Gery De Saxce, Géry De Saxcé, Claude Vallee
• Editore: Wiley
• Lingue: Inglese
• Formato: Copertina rigida
• Pubblicazione: 31.03.2016
• EAN: 9781848216426
• ISBN: 978-1-84821-642-6
• Pagine: 448
• Dimensioni: 160 mm x 239 mm x 30 mm
• Peso: 820 g
• Categorie: Scienze naturali, medicina, informatica, tecnica > Matematica > Analisi; Maschinenbau, Thermodynamik, Mechanik, Mechanical Engineering, Computational / Numerical Methods, Rechnergestützte / Numerische Verfahren im Maschinenbau