Michel Frémond, University of Roma Tor Vergata

Collisions and fractures

Michel Frémond,University of Roma Tor Vergata,

Laboratorio Lagrangewith

E. Bonetti, F. Caselli, E. Dimnet, F. Freddi

obstacle−U

r −Ur

Positions of the fractures are unknownΓ

−Ur −U

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 planethe plane

We assume collisions are instantaneous

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

Virtual work of the exterior force

Principle of virtual work gives the equation of motion

Constitutive law is needed for the internal percuss ion

Second law of thermodynamics

Experiments give the answer

PT PT

-PN-PN

PT PT

-PN-PN

or the Coulomb’s constitutive law in agreement with experiments

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?

Collisions of three balls on a plane

at rest

incoming

Multiple collisions of rigid bodies

θ

Velocities after

collision

Collisions of three balls on a plane

at rest

incoming

Multiple collisions of rigid bodies

θ

Main Ideas :

• The system is deformable

Collision of three balls on a plane

Multiple collisions of rigid bodies

at rest

incoming

θ

Main Ideas :

• The system is deformable

• At a distance velocity of deformation

Velocities of deformation

Derivative wrt time of d2AB

O1 O2

O3

AB

θθθθ

e1

e2

e3

S1 S2S3

A B

(a) (b)

Collisions of three balls on a plane

Properties

Existence and uniqueness of solution

Easy numerical method to find the solution

Few parameters , identifiable with simple experiments

The predictive theory accounts for the physical properties of multiple collisions

3D Examples

Carreau effect: before collision, ball 1 angular velocity = [0,-10,0] ,linear velocity = [0.5,0,-1]

xy

z

3D Examples

Carreau effect: before collision, ball 1 angular velocity = [0,-10,0] ,linear velocity = [0.5,0,-1]

x

z

Collisions of deformable solids

Velocities of deformation

Virtual work of the interior forces

Equations of Motion

Collisions of solids and liquidsBelly flop of a diver

Skipping stones on the still water of a lake

obstacle−U

r −Ur

Positions of the fractures are unknownΓ

−Ur −U

The velocities are discontinuous:

with respect to time

)()( xUxU −+ −rr

with respect to space

[ ] [ ])()()()()( xUxUxUxUxU −++++ +=−=rrrrr[ ] [ ])()()()()( xUxUxUxUxU lr

−++++ +=−=

Nr

rightleft

Γ

There are closed form solutions for 1-D problems:

A stone is tied to a chandelier.

The impenetrability condition is taken into account by

.0)( ≥+Udivr

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.

The damage after collision

DivU after collision

3.125 /U m s− = −

1.001.001.000.990.990.990.980.980.980.980.9710.90.80.70.60.5

β +

β +

Effect of the velocity

6.25 /U m s− = − 1211109876543210

0.50.40.30.20.10

β

divU +

We have a schematic description of this phenomenon with 7 parameters