Collisions and fractures
Michel Frémond,University of Roma Tor Vergata,
Laboratorio Lagrangewith
E. Bonetti, F. Caselli, E. Dimnet, F. Freddi
obstacleU
U
Positions of the fractures are unknown
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
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 percussion
Second law of thermodynamics
Experiments give the answer
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
θ
Multiple collisions of rigid bodies
Velocities after
collision
Collisions of three balls on a plane
at rest
incoming
θMain Ideas:
• The system is deformable
Multiple collisions of rigid bodies
Collision of three balls on a plane
Multiple collisions of rigid bodies
θMain Ideas:
• The system is deformable
• At a distance velocity of deformation
at rest
incoming
Velocities of deformation
O1 O2
O3
AB
e1
e2
e3
S1 S2 S3
A B
(a) (b)
Derivative wrt time of d2AB
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
3D Examples
Carreau effect: before collision, ball 1 angular velocity = [0,-10,0], linear velocity = [0.5,0,-1]
xy
z
3D Examples
x
z
Carreau effect: before collision, ball 1 angular velocity = [0,-10,0], linear velocity = [0.5,0,-1]
Collisions of deformable solids
Velocities of deformation
Virtual work of the interior forces
Equations of Motion
Collisions of solids and liquids
Belly flop of a diver
Skipping stones on the still water of a lake
obstacleU
U
Positions of the fractures are unknown
The velocities are discontinuous:
with respect to time
)()( xUxU
with respect to space
)()()()()( xUxUxUxUxU lr
N
rightleft
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)( Udiv
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
1.001.001.000.990.990.990.980.980.980.980.97
6.25 /U m s div U
1211109876543210
10.90.80.70.60.50.40.30.20.10
divU
Effect of the velocity
We have a schematic description of this phenomenon with 7 parameters