Galilean Mechanics and Thermodynamics of Continua

Inbunden, Engelska, 2016

2 379 kr

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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.

Produktinformation

Utgivningsdatum2016-01-08
Mått165 x 241 x 28 mm
Vikt794 g
FormatInbunden
SpråkEngelska
Antal sidor448
FörlagISTE Ltd and John Wiley & Sons Inc
ISBN9781848216426

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Foreword xiiiIntroduction xxiPart 1. Particles and Rigid Bodies 1Chapter 1. Galileo’s Principle of Relativity 31.1. Events and space–time 31.2. Event coordinates 31.2.1. When? 31.2.2. Where? 41.3. Galilean transformations 61.3.1. Uniform straight motion 61.3.2. Principle of relativity 91.3.3. Space–time structure and velocity addition 101.3.4. Organizing the calculus 111.3.5. About the units of measurement 121.4. Comments for experts 14Chapter 2. Statics 152.1. Introduction 152.2. Statical torsor 162.2.1. Two-dimensional model 162.2.2. Three-dimensional model 172.2.3. Statical torsor and transport law of the moment 182.3. Statics equilibrium 202.3.1. Resultant torsor 202.3.2. Free body diagram and balance equation 202.3.3. External and internal forces 232.4. Comments for experts 25Chapter 3. Dynamics of Particles 273.1. Dynamical torsor 273.1.1. Transformation law and invariants 273.1.2. Boost method 303.2. Rigid body motions 323.2.1. Rotations 323.2.2. Rigid motions 343.3. Galilean gravitation 363.3.1. How to model the gravitational forces? 363.3.2. Gravitation 383.3.3. Galilean gravitation and equation of motion 403.3.4. Transformation laws of the gravitation and acceleration 423.4. Newtonian gravitation 463.5. Other forces 513.5.1. General equation of motion 513.5.2. Foucault’s pendulum 523.5.3. Thrust 553.6. Comments for experts 56Chapter 4. Statics of Arches, Cables and Beams 574.1. Statics of arches 574.1.1. Modeling of slender bodies 574.1.2. Local equilibrium equations of arches 594.1.3. Corotational equilibrium equations of arches 624.1.4. Equilibrium equations of arches in Fresnet’s moving frame 634.2. Statics of cables 674.3. Statics of trusses and beams 694.3.1. Traction of trusses 694.3.2. Bending of beams 71Chapter 5. Dynamics of Rigid Bodies 755.1. Kinetic co-torsor 755.1.1. Lagrangian coordinates 755.1.2. Eulerian coordinates 765.1.3. Co-torsor 765.2. Dynamical torsor 805.2.1. Total mass and mass-center 805.2.2. The rigid body as a particle 815.2.3. The moment of inertia matrix 845.2.4. Kinetic energy of a body 875.3. Generalized equations of motion 885.3.1. Resultant torsor of the other forces 885.3.2. Transformation laws 895.3.3. Equations of motion of a rigid body 915.4. Motion of a free rigid body around it 935.5. Motion of a rigid body with a contact point (Lagrange’s top) 955.6. Comments for experts 103Chapter 6. Calculus of Variations 1056.1. Introduction 1056.2. Particle subjected to the Galilean gravitation 1096.2.1. Guessing the Lagrangian expression 1096.2.2. The potentials of the Galilean gravitation 1106.2.3. Transformation law of the potentials of the gravitation 1136.2.4. How to manage holonomic constraints? 116Chapter 7. Elementary Mathematical Tools 1177.1. Maps 1177.2. Matrix calculus 1187.2.1. Columns 1187.2.2. Rows 1197.2.3. Matrices 1207.2.4. Block matrix 1247.3. Vector calculus in R3 1257.4. Linear algebra 1277.4.1. Linear space 1277.4.2. Linear form 1297.4.3. Linear map 1307.5. Affine geometry 1327.6. Limit and continuity 1357.7. Derivative 1367.8. Partial derivative 1367.9. Vector analysis 1377.9.1. Gradient 1377.9.2. Divergence 1397.9.3. Vector analysis in R3 and curl 139Part 2. Continuous Media 141Chapter 8. Statics of 3D Continua 1438.1. Stresses 1438.1.1. Stress tensor 1438.1.2. Local equilibrium equations 1488.2. Torsors 1508.2.1. Continuum torsor 1508.2.2. Cauchy’s continuum 1538.3. Invariants of the stress tensor 155Chapter 9. Elasticity and Elementary Theory of Beams 1579.1. Strains 1579.2. Internal work and power 1629.3. Linear elasticity 1649.3.1. Hooke’s law 1649.3.2. Isotropic materials 1669.3.3. Elasticity problems 1709.4. Elementary theory of elastic trusses and beams 1719.4.1. Multiscale analysis: from the beam to the elementary volume 1719.4.2. Transversely rigid body model 1769.4.3. Calculating the local fields 1799.4.4. Multiscale analysis: from the elementary volume to the beam 183Chapter 10. Dynamics of 3D Continua and Elementary Mechanics of Fluids 18710.1. Deformation and motion 18710.2. Flash-back: Galilean tensors 19210.3. Dynamical torsor of a 3D continuum 19610.4. The stress–mass tensor 19810.4.1. Transformation law and invariants 19810.4.2. Boost method 20010.5. Euler’s equations of motion 20210.6. Constitutive laws in dynamics 20610.7. Hyperelastic materials and barotropic fluids 210Chapter 11. Dynamics of Continua of Arbitrary Dimensions 21511.1. Modeling the motion of one-dimensional (1D) material bodies 21511.2. Group of the 1D linear Galilean transformations 21711.3. Torsor of a continuum of arbitrary dimension 21911.4. Force–mass tensor of a 1D material body 22011.5. Full torsor of a 1D material body 22211.6. Equations of motion of a continuum of arbitrary dimension 22411.7. Equation of motion of 1D material bodies 22511.7.1. First group of equations of motion 22611.7.2. Multiscale analysis 22711.7.3. Secong group of equations of motion 231Chapter 12. More About Calculus of Variations 23512.1. Calculus of variation and tensors 23512.2. Action principle for the dynamics of continua 23712.3. Explicit form of the variational equations 24012.4. Balance equations of the continuum 24412.5. Comments for experts . 245Chapter 13. Thermodynamics of Continua 24713.1. Introduction 24713.2. An extra dimension 24813.3. Temperature vector and friction tensor 25113.4. Momentum tensors and first principle 25313.5. Reversible processes and thermodynamical potentials 25813.6. Dissipative continuum and heat transfer equation 26313.7. Constitutive laws in thermodynamics 26813.8. Thermodynamics and Galilean gravitation 27213.9. Comments for experts 279Chapter 14. Mathematical Tools 28114.1. Group 28114.2. Tensor algebra 28214.2.1. Linear tensors 28214.2.2. Affine tensors 28814.2.3. G-tensors and Euclidean tensors 29214.3. Vector analysis 29514.3.1. Divergence 29514.3.2. Laplacian 29614.3.3. Vector analysis in R3 and curl 29614.4. Derivative with respect to a matrix 29714.5. Tensor analysis 29714.5.1. Differential manifold 29714.5.2. Covariant differential of linear tensors 30014.5.3. Covariant differential of affine tensors 303Part 3. Advanced Topics 307Chapter 15. Affine Structure on a Manifold 30915.1. Introduction 30915.2. Endowing the structure of linear space by transport 31015.3. Construction of the linear tangent space 31115.4. Endowing the structure of affine space by transport 31315.5. Construction of the affine tangent space 31615.6. Particle derivative and affine functions 319Chapter 16. Galilean, Bargmannian and Poincarean Structures on a Manifold 32116.1. Toupinian structure 32116.2. Normalizer of Galileo’s group in the affine group 32316.3. Momentum tensors 32516.4. Galilean momentum tensors 32816.4.1. Coadjoint representation of Galileo’s group 32816.4.2. Galilean momentum transformation law 32916.4.3. Structure of the orbit of a Galilean momentum torsor 33516.5. Galilean coordinate systems 33816.5.1. G-structures 33816.5.2. Galilean coordinate systems 33816.6. Galilean curvature 34116.7. Bargmannian coordinates 34616.8. Bargmannian torsors 34916.9. Bargmannian momenta 35216.10. Poincarean structures 35716.11. Lie group statistical mechanics 362Chapter 17. Symplectic Structure on a Manifold 36717.1. Symplectic form 36717.2. Symplectic group 37017.3. Momentum map 37117.4. Symplectic cohomology 37317.5. Central extension of a group 37517.6. Construction of a central extension from the symplectic cocycle 37717.7. Coadjoint orbit method 38317.8. Connections 38517.9. Factorized symplectic form 38717.10. Application to classical mechanics 39317.11. Application to relativity 396Chapter 18. Advanced Mathematical Tools 39918.1. Vector fields 39918.2. Lie group 40018.3. Foliation 40218.4. Exterior algebra 40218.5. Curvature tensor 405Bibliography 407Index 411