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
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