Collisions Engineering: Theory and Applications (eBook)

Preface 6
Contents 7
1 Introduction 13
2 The Theory: Mechanics. An Example: Collision of a Point and a Plane 16
2.1 A System Made of a Point and a Plane 16
2.2 The Velocities 16
2.3 The Velocity of Deformation 17
2.4 The Principle of Virtual Work 18
2.4.1 The Work of the Acceleration Forces 19
2.4.2 The Work of the External Forces 20
2.4.3 The Work of the Internal Forces 21
2.5 The Equations of Motion 22
2.5.1 Properties of the Equations of Motion 24
2.6 The Laws of Thermodynamics 25
2.6.1 The First Law 25
2.6.2 The Second Law 26
2.7 The Constitutive Laws 27
2.7.1 The Free Energy and the Non Dissipative Forces 28
2.7.2 The Dissipative Forces 29
2.8 Examples of Collisions with Internal Forces Defined ƒ 31
2.8.1 First Example 32
2.8.2 Second Example 33
2.8.3 Third Example. Interpenetration Is Possible 36
2.9 Examples of Dissipative Forces Defined with a Function of Dissipation 38
2.9.1 The Coulomb's Friction Law 38
2.9.2 The Coulomb's Collision Law 39
2.9.3 Experimental Results 39
2.9.4 Relationships Between Smooth Friction and Collision Constitutive Laws 41
References 41
3 The Theory: Mechanics and Thermics. An Example: Collision of Two Balls 44
3.1 Introduction 44
3.2 The Velocities and the Velocities of Deformation 44
3.3 The Principle of Virtual Work and the Equations of Motion 45
3.4 The Virtual Works 45
3.4.1 The Theorem of Kinetic Energy 46
3.5 The Equations of Motion 47
3.6 Smooth Evolution of Two Balls with Thermal Effects 48
3.6.1 Laws of Thermodynamics for a Ball 48
3.6.2 Laws of Thermodynamics for the System 49
3.6.3 The Constitutive Laws 51
3.6.4 An Example 51
3.7 Collisions of Two Balls with Thermal Effects 56
3.7.1 First Law of Thermodynamics for a Ball 57
3.7.2 Second Law of Thermodynamics for a Ball 57
3.7.3 A Useful Inequality for a Ball 57
3.7.4 The First Law for the System 58
3.7.5 The Second Law for the System 59
3.7.6 A Useful Inequality for the System 60
3.7.7 The Constitutive Laws 61
3.7.8 An Example of Thermal Effects Due to Collisions 62
3.8 Phase Change and Collisions 65
3.9 Experimental Results 66
References 66
4 Collisions of Rigid Solids: Three Disks in a Plane 68
4.1 Introduction 68
4.2 The Velocities 69
4.3 The Velocities of Deformation 70
4.4 The Work of the Interior Forces 71
4.5 The Work of the Acceleration Forces 71
4.6 The Equations of Motion 72
4.7 The Constitutive Laws 72
4.7.1 Solution of the Equations 73
4.8 Numerical Examples 74
4.8.1 The Mass Moment of Inertia Is Infinite: I=infty 74
4.8.2 The Mass Moment of Inertia Is Finite: I< infty
References 76
5 Collisions of Rigid Solids: Three Balls in a Box 78
5.1 Introduction 78
5.2 Three Balls Evolving on a Plane 78
5.2.1 Numerical Examples 80
5.3 Three Balls Evolving in a Box 84
5.3.1 Numerical Examples 87
References 87
6 Pedestrian Trajectories and Collisions in Crowd Motion 89
6.1 Definitions---Phenomena of Typical Crowd Self-Organization 89
6.2 The Current Methods for Modeling Crowd Movement 95
6.2.1 Macroscopic Models 100
6.2.2 Microscopic Models 101
6.3 The Proposed 2D Discrete Model 104
6.4 Multiple Contacts' Detection 105
6.5 Presentation of Three Approaches to Granular Media 109
6.5.1 Theoretical Aspects of the Three Approaches 110
6.5.2 Numerical Aspects of the Three Approaches 114
6.6 Adaptation of a Granular Approach to the Crowd 120
6.6.1 Introduction of the Desired Velocity of Each Pedestrian into the Particle Movement Approach 120
6.6.2 Definition of the Desired Velocity 120
6.6.3 The Influence of the Relaxation Time Parameter ? 122
6.7 Making the Behaviour of Pedestrians More Realistic 123
6.7.1 The Socio-Psychological Force 123
6.7.2 Subgroups: Pedestrians Holding Hands 124
6.8 Simulations of Crowd Movement 128
6.8.1 Phenomena of Crowd Self-Organization 129
6.8.2 Evacuation Exercises: Comparison Between Numerical and Experimental Results 131
6.8.3 A Predictive Model 140
References 150
7 Collisions of Deformable Solids 155
7.1 Introduction 155
7.2 The Principle of Virtual Work 155
7.3 The First Law of Thermodynamics 158
7.4 The Second Law of Thermodynamics 166
7.5 The Constitutive Laws 167
7.6 Evolution in a Collision 170
7.6.1 The Mechanical Evolution When Decoupled from the Thermal Evolution 172
7.6.2 An Example: Collision of a Bar with a Rigid Support 174
7.6.3 Thermal Evolution When the Mechanical Equations Are Decoupled From the Thermal Equations 175
7.6.4 The Temperature Variation in a Collision 178
References 180
8 Collisions of Rigid Solids and Fluids 181
8.1 Introduction 181
8.2 The Equation of Motion 182
8.2.1 The Principle of Virtual Work 182
8.2.2 The Equations of Motion 185
8.3 The Constitutive Laws 186
8.3.1 The Energy Balance 187
8.3.2 The Second Law of Thermodynamics 187
8.3.3 The Constitutive Laws for an Incompressible Fluid 188
8.4 An Incompressible Fluid in a Tube 191
8.5 The Diver Problem 193
8.5.1 The Equations 194
8.5.2 The Variational Formulation 195
8.5.3 Numerical Results 197
8.6 Skipping Stones 200
8.7 Conclusions 201
References 202
9 Debris Flows and Collisions of Fluids and Deformable Solids 203
9.1 The Solid Liquid Collision 203
9.1.1 The Debris Flows 203
9.1.2 The Principle of Virtual Work 204
9.1.3 The Equations of Motion 206
9.1.4 The Laws of Thermodynamics and Constitutive Laws 206
9.2 An Example 209
9.2.1 The Equations of the Predictive Theory 209
9.2.2 The Numerical Approximation 210
9.3 Properties of the Physical Parameters 210
9.3.1 The Basic Case 211
9.3.2 The Effect of the Density of the Debris Flow ?f 214
9.3.3 The Effect of Percussion Viscosity kf of the Debris Flow 217
9.3.4 The Effect of Percussion Viscosities ks and s of the Wall 220
9.3.5 The Effect of the Friction of the Debris Flow with the Wall 223
9.4 Smooth Predictive Theory Versus Non Smooth Predictive Theory 223
9.5 The Coupled Influence of the Debris Flow Density and Height 226
9.6 The Effect of the Soil Deformation 227
9.7 An Application. The Protection of a Wall by a Damping Sand Layer 228
References 231
10 Shape Memory Alloys and Collisions 234
10.1 Introduction 234
10.1.1 The State Quantities 235
10.1.2 Quantities Which Describe the Evolution 235
10.2 The Principle of Virtual Work and the Equations of Motion 236
10.3 The Mass Balance 237
10.4 The Laws of Thermodynamics 238
10.4.1 The First Law 238
10.4.2 The Second Law 240
10.5 The Free Energy 243
10.6 The Pseudo-potential of Dissipation 245
10.7 The Constitutive Laws 245
10.8 The Equations in a Collision 246
10.8.1 The Energy Balance 246
10.8.2 The Equations of Motion 247
10.8.3 The Mass Balance 247
10.8.4 The Constitutive Laws 247
10.8.5 The Evolution Equations 248
10.9 Mathematics 249
10.10 Closed Form Examples 250
10.10.1 Example 1. The Non Dissipative Case, c=0 250
10.10.2 Example 2. The Non Dissipative Case c=0 and Voids 253
10.10.3 Example 3. The Dissipative Case, c> 0
10.11 Numerical Examples 256
10.11.1 A Percussion Is Applied to a Rod: 1D Application 256
10.11.2 A Surface Percussion Is Applied to a Solid: 2D Application 259
10.11.3 Evolution Following the Collision: Configuration and Alloy Composition Depending on Time 259
10.12 Experimental Results and Other Modeling Approaches 262
References 262
11 Conclusion 265
Appendix A Some Elements of Convex Analysis 266