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Collisions and shape memory alloys
- M. Frémond, M. Marino
Collisions and shape memory alloys M. Frmond, M. Marino University - - PowerPoint PPT Presentation
Collisions and shape memory alloys M. Frmond, M. Marino University - - PowerPoint PPT Presentation
Collisions and shape memory alloys M. Frmond, M. Marino University of Roma Tor Vergata 1 2 T 1 B 21 T 2 B 2 T 1 B 1 2 1 T 2 B 21 Tennis table ball: 1 1 2 3 Soccer ball: 2 Floor: 3 Warm rain droplet
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2 1
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± ± 1 1 B
T
± ± 2 2 B
T
2 1
21 1 B
T
± 21 2 B
T
±
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Soccer ball: 2 Tennis table ball: 1 Floor: 3 1 2 3
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Warm rain droplet Frozen ground
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Water Stone or swimmer Ω Γ contact surface
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- bstacle
−
U r
−
U r Collision is instantaneous. There are velocities before collision and velocities after collision
) (x U − r
) (x U+ r
Fractures result from the collision. Thus velocity is a discontinuous function of x
) ( x U
+
r
Positions of the fractures are unknown
Γ
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The velocities are discontinuous: with respect to time ) ( ) ( x U x U
− +
− r r with respect to space
[ ] [ ]
) ( ) ( ) ( ) ( ) ( x U x U x U x U x U
l r − + + + +
+ = − = r r r r r N r
right left
Γ
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There are closed form solutions for 1-D problems: A stone is tied to a chandelier. Work with Elena Bonetti, Pavia University
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Work with Francesco Freddi, Parma University
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Mathematics with Elisabetta Rocca, Milano University
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Applications
- Impact and post‐collision behavior of a 1D beam structure:
Gp y T−<To , β1
−=β2 −=0.5
The results from the collision SMA predictive theory are the input data for the thermomechanical smooth problem describing SMA evolution (Frémond model).
Computational step size is limited only by the smooth evolution problem (structural natural frequency).
0.01 0.02 0.03 0.04 t [s]
- 0.6
- 0.4
- 0.2
0.2 0.4 0.6 u [m] SMA Linear elastic
〈〉=average along the beam
u
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SMA constitutive modeling: collisions
Impact behavior depends on classical SMAs parameters and on: cc: collision viscosity of phase change; νc: collision viscosity of phase change diffusion; kv: collision viscosity of velocities.
Simplified assumptions: ‐ T−<To , β1
−=β2 −=0.5
‐ Homogenous case: ∇T=0, ∇β=0
- Dissipated power ζ=σp:ε(v) is known
100 200 300 400 500 ζ [MPa] 0.2 0.4 0.6 0.8 1 β [-]
β3 + β1 +=β2 + β1 +=β2 + β3 +
cc cc 100 200 300 400 500 ζ [MPa] 0.9 0.95 1 1.05 1.1 T+/To [-] cc
Lower cc, easier is the phase change
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SMA constitutive modeling: collisions
Impact behavior depends on classical SMAs parameters and on: cc: collision viscosity of phase change; νc: collision viscosity of phase change diffusion; kv: collision viscosity of velocities.
2 4 6 8 10 y [m] 0.2 0.4 0.6 0.8 β3 + β1 +=β2 +
β+ [-]
νc νc
Gp y T−<To , β1
−=β2 −=0.5
Higher νc more uniform is the phase change
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SMA constitutive modeling: collisions
Impact behavior depends on classical SMAs parameters and on: cc: collision viscosity of phase change; νc: collision viscosity of phase change diffusion; kv: collision viscosity of velocities. T−<To , β1
−=β2 −=0.5
Higher kv
2 4 6 8 10 y [m] 0.2 0.4 0.6 0.8 1
β3 + [-]
kv
Smaller the post‐collision velocity Lower is the phase change Gp y