1. Collisions and fractures Michel Frémond, University of Roma Tor Vergata, Laboratorio Lagrange with E. Bonetti, F. Caselli, E. Dimnet, F. Freddi

2. obstacle Positions of the fractures are unknown

3. Collision of a point and a fixed plane The system {Point U Plane} is deformable Velocity of defomation: The relative velocity of the point with respect to the plane We assume collisions are instantaneous

4. Virtual work of the acceleration force Actual work The internal force is defined by its virtual work: A linear function of the velocity of deformation

5. Virtual work of the exterior force Principle of virtual work gives the equation of motion

6. Constitutive law is needed for the internal percussion Second law of thermodynamics Experiments give the answer

10. or the Coulomb’s constitutive law in agreement with experiments

11. The first law of thermodynamics? The temperature is discontinuous The theory answers the question, Does a warm rain droplet turns into ice when falling on a deeply frozen soil?

12. Collisions of three balls on a plane at rest incoming θ Multiple collisions of rigid bodies Velocities after collision

13. Collisions of three balls on a plane at rest incoming θ Main Ideas: The system is deformable Multiple collisions of rigid bodies

14. Collision of three balls on a plane Multiple collisions of rigid bodies θ Main Ideas: The system is deformable At a distance velocity of deformation at rest incoming

15. Velocities of deformation O1 O1 O2 O2 O3 O3 A B  e1 e1 e2 e2 e3 e3 S1S1 S2S2 S3S3 AB (a)(b) Derivative wrt time of d 2 AB

16. Collisions of three balls on a plane Properties Existence and uniqueness of solution Easy numerical method to find the solution The predictive theory accounts for the physical properties of multiple collisions Few parameters, identifiable with simple experiments

17. 3D Examples Carreau effect: before collision, ball 1 angular velocity = [0,-10,0], linear velocity = [0.5,0,-1] xy z

18. 3D Examples x z Carreau effect: before collision, ball 1 angular velocity = [0,-10,0], linear velocity = [0.5,0,-1]

19. Collisions of deformable solids Velocities of deformation

20. Virtual work of the interior forces Equations of Motion

24. Collisions of solids and liquids Belly flop of a diver Skipping stones on the still water of a lake

30. obstacle Positions of the fractures are unknown

31. The velocities are discontinuous: with respect to time with respect to space right left

35. There are closed form solutions for 1-D problems: A stone is tied to a chandelier.

37. The impenetrability condition is taken into account by This is an old idea of Jean Jacques Moreau. CRAS, 259, 1965, p. 3948-3950, Sur la naissance de la cavitation dans une conduite. Journal de Mécanique, 5, 1966, p. 439-470, Principes extrémaux pour le problème de la naissance de la cavitation.

41. The damage after collision DivU after collision

42. Effect of the velocity

44. We have a schematic description of this phenomenon with 7 parameters