Discrete Mechanics

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

2.3. Postulates 51

2.4. Modeling of the pressure forces 52

2.5. Modeling of the viscous forces 57

2.5.1. Modeling of the viscous effects of volume 57

2.5.2. Modeling of the viscous surface effects 59

2.5.3. Stress state 62

2.6. Objectivity 64

2.7. Discrete motion balance equation 67

2.7.1. Fundamental law of dynamics 67

2.7.2. Eulerian step 73

2.7.3. Mechanical equilibrium 74

2.8. Formulation in terms of density and temperature 78

2.9. Similitude parameters 81

2.9.1. Impact on the surface of a liquid 85

2.10. Hypercompressible media 88

CHAPTER 3. CONSERVATION OF HEAT FLUX AND ENERGY 91

3.1. Introduction 91

3.2. Conservation of flux 92

3.3. Conservation of energy 95

3.3.1. Conservation of total energy 95

3.3.2. Conservation of kinetic energy 97

3.3.3. Conservation of the internal energy 98

3.4. Discrete equations for the flux and the energy 99

3.5. A simple heat-conduction problem 100

3.5.1. Case of anisotropic materials 102

CHAPTER 4. PROPERTIES OF DISCRETE EQUATIONS 105

4.1. A system of equations and potentials 105

4.2. Physics represented 107

4.2.1. Poiseuille flow and potentials 110

4.2.2. Celerity and maximum velocity 112

4.2.3. Remarks about turbulence 113

4.3. Boundary conditions 114

4.3.1. Contact surface 114

4.3.2. Shockwaves 117

4.3.3. Edge conditions 118

4.3.4. Slip condition 119

4.3.5. Capillary effects 120

4.3.6. Thermal boundary conditions 124

4.4. Penalization of the potentials 125

4.5. Continua and discrete mediums 129

4.5.1. Differences with the Navier–Stokes equation 129

4.5.2. Dissipation 133

4.5.3. Case of rigidifying motions 135

4.5.4. An example of the dissipation of energy 137

4.6. Hodge–Helmholtz decomposition 139

4.7. Approximations 141

4.7.1. Bernoulli’s law 141

4.7.2. Irrotational flow 143

4.7.3. Inviscid fluid 144

4.7.4. Incompressible flow 145

4.8. Gravitational waves 147

4.9. Linear visco-elasticity 150

4.9.1. Viscous dissipation in a visco-elastic medium 153

4.9.2. Dissipation of longitudinal waves in a visco-elastic medium 155

4.9.3. Consistency with Continuum Mechanics 156

4.9.4. Pure compression 159

4.9.5. Pure shear stress 160

4.9.6. Bingham fluid 162

CHAPTER 5. MULTIPHYSICS 165

5.1. Extensions to other branches of physics 165

5.1.1. Coupling between a fluid and a porous medium 167

5.2. Flow around a cylinder in an infinite medium 169

5.2.1. Darcian model 170

5.2.2. Stokes model 174

5.2.3. Model of an ideal fluid 175

5.2.4. Brinkman model 176

5.3. Fluid statics 178

5.3.1. Perfect gas in isothermal evolution 179

5.3.2. Perfect gas in adiabatic evolution 181

5.4. Injection of a gas into a cavity 183

5.4.1. Isothermal injection 184

5.4.2. Adiabatic injection 185

5.5. Nonlinear wave propagation 188

5.5.1. Sod shock tube 190

5.6. Thermo-acoustics 192

5.6.1. Heating of a cavity filled with air 193

5.7. Natural convection in an enclosed cavity 198

5.8. Multi-component transport 200

5.9. Modeling of phase change 203

5.10. Critical opalescence 207

5.11. Conclusions regarding the multiphysics approach 209

APPENDIX 211

BIBLIOGRAPHY 215

INDEX 219

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