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	75
	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	254
	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