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Contact: collisions and fractures.
A predictive theoryElena Bonetti, University of Pavia,
Francesco Freddi, University of Parma,Michel Frémond, University of Roma Tor Vergata,
Laboratorio Lagrange.
obstacleU
U
Collision is instantaneous.
There are velocities before collision and velocities after collision
)(xU
)(xU
Fractures result from the collision. Thus velocity is
a discontinuous function of x
)(xU
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
Numerical Simulation
20L
h
1.25h
l
U
1 rigid velocity
undamaged material
Two representative cases have been analyzed:all the parameters are fixed except the density ofthe material thus an heavy material and a light material, having the same resistance, have been considered (a ratio between the material densities equal to 10 has been adopted).
Numerical Simulation: heavy material
3.125 /U m s
1.5625 /U m s 1.000.900.800.700.600.500.400.300.200.100.00
9876543210
1.000.900.800.700.600.500.400.300.200.100.00
131211109876543210
divU
0 1 2 3 4
x
-0 .4
-0 .2
0
0.2
0.4
u x
y= 0 .03 my= 0 .17 m
y
x
Numerical Simulation: heavy material
3.125 /U m s
1.5625 /U m s 1.000.900.800.700.600.500.400.300.200.100.00
1.000.900.800.700.600.500.400.300.200.100.00
U
3.125 /U m s
Numerical simulation: light material 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
U
Numerical SimulationRectangular slab
M. Zineddin, T. Krauthammer, Int. J. Impact Eng. 34, 2007
0U
1
Imposed percussion
We have a schematic description of this phenomenon with 7 parameters
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