Discrete Mechanics [Hardcover]

This book presents the fundamental principles of mechanics to re-establish the equations of Discrete Mechanics. It introduces physics and thermodynamics associated to the physical modeling.  The development and the complementarity of sciences lead to review today the old concepts that were the basis for the development of continuum mechanics. The differential geometry is used to review the conservation laws of mechanics. For instance, this formalism requires a different location of vector and scalar quantities in space. The equations of Discrete Mechanics form a system of equations where the Helmholtz-Hodge decomposition plays an important role.

PREFACE ix

LIST OF SYMBOLS xv

INTRODUCTION xxi

CHAPTER 1. FRAMEWORK OF DISCRETE MECHANICS 1

1.1. Frames of reference and uniform motions 1

1.2. Concept of a Discrete Medium 4

1.2.1. Vectors and components 6

1.2.2. Physical meaning of the differential operators 8

1.2.3. Use of the theorems of differential geometry 10

1.2.4. Two essential properties 12

1.2.5. Tensorial values 17

1.2.6. The scalar and vectorial potentials 19

1.3. The physical characteristics 20

1.4. Equilibrium stress state 22

1.4.1. Two examples of mechanical equilibrium 25

1.5. Thermodynamic non-equilibrium 26

1.5.1. Forces and fluxes 29

1.6. Conservation of mass 30

CHAPTER 2. MOMENTUM CONSERVATION 33

2.1. Classification of forces 33

2.2. Three fundamental experiments 35

2.2.1. Equilibrium in a glass of water 35

2.2.2. Couette flow 44

2.2.3. Poiseuille flow 47