Galilean mechanics and thermodynamics of continua / Géry de Saxcé.

Format:

Book

Author/Creator:

Saxce, Géry de, author.

Vallée, Claude, Author.

Series:

Mechanical engineering and solid mechanics series.

Mechanical Engineering and Solid Mechanics Series

Language:

English

Subjects (All):

Continuum mechanics--Data processing.

Continuum mechanics.

Laws of thermodynamics--Textbooks.

Laws of thermodynamics.

Physical Description:

1 online resource (448 p.)

Edition:

1st ed.

Place of Publication:

Wiley 2016

Summary:

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.-- Source other than Library of Congress.

Contents:

Cover

Title Page

Copyright

Contents

Foreword

Introduction

I.1. A geometrical viewpoint

I.2. Overview

I.2.1. Part 1: particles and rigid bodies

I.2.2. Part 2: continuous media

I.2.3. Part 3: advanced topics

I.3. Historical background and key concepts

PART 1: Particles and Rigid Bodies

Chapter 1: Galileo's Principle of Relativity

1.1. Events and space-time

1.2. Event coordinates

1.2.1. When?

1.2.2. Where?

1.3. Galilean transformations

1.3.1. Uniform straight motion

1.3.2. Principle of relativity

1.3.3. Space-time structure and velocity addition

1.3.4. Organizing the calculus

1.3.5. About the units of measurement

1.4. Comments for experts

Chapter 2: Statics

2.1. Introduction

2.2. Statical torsor

2.2.1. Two-dimensional model

2.2.2. Three-dimensional model

2.2.3. Statical torsor and transport law of the moment

2.3. Statics equilibrium

2.3.1. Resultant torsor

2.3.2. Free body diagram and balance equation

2.3.3. External and internal forces

2.4. Comments for experts

Chapter 3: Dynamics of Particles

3.1. Dynamical torsor

3.1.1. Transformation law and invariants

3.1.2. Boost method

3.2. Rigid body motions

3.2.1. Rotations

3.2.2. Rigid motions

3.3. Galilean gravitation

3.3.1. How to model the gravitational forces?

3.3.2. Gravitation

3.3.3. Galilean gravitation and equation of motion

3.3.4. Transformation laws of the gravitation and acceleration

3.4. Newtonian gravitation

3.5. Other forces

3.5.1. General equation of motion

3.5.2. Foucault's pendulum

3.5.3. Thrust

3.6. Comments for experts

Chapter 4: Statics of Arches, Cables and Beams

4.1. Statics of arches

4.1.1. Modeling of slender bodies

4.1.2. Local equilibrium equations of arches

4.1.3. Corotational equilibrium equations of arches.

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

4.2. Statics of cables

4.3. Statics of trusses and beams

4.3.1. Traction of trusses

4.3.2. Bending of beams

Chapter 5: Dynamics of Rigid Bodies

5.1. Kinetic co-torsor

5.1.1. Lagrangian coordinates

5.1.2. Eulerian coordinates

5.1.3. Co-torsor

5.2. Dynamical torsor

5.2.1. Total mass and mass-center

5.2.2. The rigid body as a particle

5.2.3. The moment of inertia matrix

5.2.4. Kinetic energy of a body

5.3. Generalized equations of motion

5.3.1. Resultant torsor of the other forces

5.3.2. Transformation laws

5.3.3. Equations of motion of a rigid body

5.4. Motion of a free rigid body around it

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

5.6. Comments for experts

Chapter 6: Calculus of Variations

6.1. Introduction

6.2. Particle subjected to the Galilean gravitation

6.2.1. Guessing the Lagrangian expression

6.2.2. The potentials of the Galilean gravitation

6.2.3. Transformation law of the potentials of the gravitation

6.2.4. How to manage holonomic constraints?

Chapter 7: Elementary Mathematical Tools

7.1. Maps

7.2. Matrix calculus

7.2.1. Columns

7.2.2. Rows

7.2.3. Matrices

7.2.4. Block matrix

7.3. Vector calculus in R3

7.4. Linear algebra

7.4.1. Linear space

7.4.2. Linear form

7.4.3. Linear map

7.5. Affine geometry

7.6. Limit and continuity

7.7. Derivative

7.8. Partial derivative

7.9. Vector analysis

7.9.1. Gradient

7.9.2. Divergence

7.9.3. Vector analysis in R3 and curl

PART 2: Continuous Media

Chapter 8: Statics of 3D Continua

8.1. Stresses

8.1.1. Stress tensor

8.1.2. Local equilibrium equations

8.2. Torsors

8.2.1. Continuum torsor

8.2.2. Cauchy's continuum

8.3. Invariants of the stress tensor.

Chapter 9: Elasticity and Elementary Theory of Beams

9.1. Strains

9.2. Internal work and power

9.3. Linear elasticity

9.3.1. Hooke's law

9.3.2. Isotropic materials

9.3.3. Elasticity problems

9.4. Elementary theory of elastic trusses and beams

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

9.4.2. Transversely rigid body model

9.4.3. Calculating the local fields

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

Chapter 10: Dynamics of 3D Continua and Elementary Mechanics of Fluids

10.1. Deformation and motion

10.2. Flash-back: Galilean tensors

10.3. Dynamical torsor of a 3D continuum

10.4. The stress-mass tensor

10.4.1. Transformation law and invariants

10.4.2. Boost method

10.5. Euler's equations of motion

10.6. Constitutive laws in dynamics

10.7. Hyperelastic materials and barotropic fluids

Chapter 11: Dynamics of Continua of Arbitrary Dimensions

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

11.2. Group of the 1D linear Galilean transformations

11.3. Torsor of a continuum of arbitrary dimension

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

11.5. Full torsor of a 1D material body

11.6. Equations of motion of a continuum of arbitrary dimension

11.7. Equation of motion of 1D material bodies

11.7.1. First group of equations of motion

11.7.2. Multiscale analysis

11.7.3. Secong group of equations of motion

Chapter 12: More About Calculus of Variations

12.1. Calculus of variation and tensors

12.2. Action principle for the dynamics of continua

12.3. Explicit form of the variational equations

12.4. Balance equations of the continuum

12.5. Comments for experts

Chapter 13: Thermodynamics of Continua

13.1. Introduction

13.2. An extra dimension.

13.3. Temperature vector and friction tensor

13.4. Momentum tensors and first principle

13.5. Reversible processes and thermodynamical potentials

13.6. Dissipative continuum and heat transfer equation

13.7. Constitutive laws in thermodynamics

13.8. Thermodynamics and Galilean gravitation

13.9. Comments for experts

Chapter 14: Mathematical Tools

14.1. Group

14.2. Tensor algebra

14.2.1. Linear tensors

14.2.2. Affine tensors

14.2.3. G-tensors and Euclidean tensors

14.3. Vector analysis

14.3.1. Divergence

14.3.2. Laplacian

14.3.3. Vector analysis in R3 and curl

14.4. Derivative with respect to a matrix

14.5. Tensor analysis

14.5.1. Differential manifold

14.5.2. Covariant differential of linear tensors

14.5.3. Covariant differential of affine tensors

PART 3: Advanced Topics

Chapter 15: Affine Structure on a Manifold

15.1. Introduction

15.2. Endowing the structure of linear space by transport

15.3. Construction of the linear tangent space

15.4. Endowing the structure of affine space by transport

15.5. Construction of the affine tangent space

15.6. Particle derivative and affine functions

Chapter 16: Galilean, Bargmannian and Poincarean Structures on a Manifold

16.1. Toupinian structure

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

16.3. Momentum tensors

16.4. Galilean momentum tensors

16.4.1. Coadjoint representation of Galileo's group

16.4.2. Galilean momentum transformation law

16.4.3. Structure of the orbit of a Galilean momentum torsor

16.5. Galilean coordinate systems

16.5.1. G-structures

16.5.2. Galilean coordinate systems

16.6. Galilean curvature

16.7. Bargmannian coordinates

16.8. Bargmannian torsors

16.9. Bargmannian momenta

16.10. Poincarean structures

16.11. Lie group statistical mechanics.

Chapter 17: Symplectic Structure on a Manifold

17.1. Symplectic form

17.2. Symplectic group

17.3. Momentum map

17.4. Symplectic cohomology

17.5. Central extension of a group

17.6. Construction of a central extension from the symplectic cocycle

17.7. Coadjoint orbit method

17.8. Connections

17.9. Factorized symplectic form

17.10. Application to classical mechanics

17.11. Application to relativity

Chapter 18: Advanced Mathematical Tools

18.1. Vector fields

18.2. Lie group

18.3. Foliation

18.4. Exterior algebra

18.5. Curvature tensor

Bibliography

Index.

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-119-05809-0

1-119-05803-1

OCLC:

934770229