Nonlinear Dynamics in Granular Systems

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Nonlinear Dynamics in Granular Systems
Surajit Sen, Physics, SUNY-Buffalo

• Introduction
• Granular contacts, Solitary waves, Shock
  dispersion
• Quasi-equilibrium
• Driven chains granular breathing
• Ultrashallow depth geophysics
• Summary

• M. Manciu, F. Manciu, J. Pfannes, R. Doney, L.
  Gilcrist, D. Sun, R.P. Simion
• E. Avalos, T. Krishna Mohan
• S. Swaminathan
• G. Baker, M. Nakagawa, J. Agui, C. Daraio, J.
  Julien, D. Visco, K. Shenai

sen_at_nsm.buffalo.edu
Support Army, NSF
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1. Introduction
3
1. Introduction
4
2. Granular contacts, Solitary waves Shock
dispersion
?12 d r12

• Hertz contact

V a dn, n5/2 for elastic spheres
d
1
2
a (2/5D)R1R2/(R1 R 2)0.5 D (1-s2)/Y s
Poissons ratio Y Youngs modulus
n
r12
Magnitude of n (ngt2) depends on the nature of the
interface!
H. Hertz, J. reine u Angew. Math. 92, 156 (1881)
5
2. Granular contacts, Solitary waves Shock
dispersion Squishy Hard makes for hurried
energy transmission

• One-sided potential no interaction when grains
  lose contact
• Rapid stiffening of repulsion leads to
  ballistic-like energy transport

hard
squishy
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2. Granular contacts, Solitary waves Shock
dispersion
2R - d
d
R
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2. Granular contacts, Solitary waves Shock
dispersion
(Solid of revolution)
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2. Granular contacts, Solitary waves Shock
dispersion
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2. Granular contacts, Solitary waves Shock
dispersion Restitution
restitution factor ?

• exponential attenuation of amplitude
• no dispersion

simulation
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2. Granular contacts, Solitary waves Shock
dispersion The Chain
v0 (impulse on edge grain to start system)
1
i-1
i
i1
N
i
i1
i-1
N
1
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2. Granular contacts, Solitary waves Shock
dispersion Eq. of motion
?loading, ?0? no loading, ngt2
?
Non-perturbative case, no sound waves possible
solitary wave solutions
Nesterenko (1983), Sen Manciu, (1999, 2001)
Solutions reviewed in Sen et al., Phys Repts
(2008)
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2. Granular contacts, Solitary waves Shock
dispersion Solitary wave width
Numerically and analytically obtained (Sun Sen,
preprint)
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2. Granular contacts, Solitary waves Shock
dispersion How SWs form
In a chain of spheres, solitary wave formation
happens across a distance of 10 grain diameters
A. Sokolow, E. Bittle and SS, Europhys. Lett. 77,
24002 (2007)
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2. Granular contacts, Solitary waves Shock
dispersion Interactions between SWs
M. Manciu, S. Sen A.J. Hurd, Phys Rev E 63,
016616 (2001)
Grain 250 suffers no motion during collision
event ? m250 ?
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2. Granular contacts, Solitary waves Shock
dispersion Interactions between SWs
Collision problem is symmetric about grain 250
M. Manciu, S. Sen A.J. Hurd, Phys Rev E 63,
016616 (2001) F. Manciu S. Sen, Phys Rev E 66,
016614 (2002)
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2. Granular contacts, Solitary waves Shock
dispersion Interactions between SWs
Solitary wave collision in chain with even
number of grains S. Sen, J. Hong, H. Bang, E.
Avalos and R. Doney, Phys. Rept., 462, 21 (2008)
E. Avalos and SS, PRE (to be pub, 2009)
Possible implications of breaking of solitary
waves?
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2. Granular contacts, Solitary waves Shock
dispersion Experimental Observation
Hard end-wall
Softer end wall showing secondary solitary wave
formation
Job, Melo, Sokolow, Sen , Phys Rev Lett 94,
178002 (2005)
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2. Granular contacts, Solitary waves Shock
dispersion Interactions between SWs
Force (N)
Time (ms)
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2. Granular contacts, Solitary waves Shock
dispersion Interactions between SWs
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2. Granular contacts, Solitary waves Shock
dispersion KEs of the Central Grains in Big and
Small SWs
Kinetic energy
In a relative sense, the SMALL wave GROWS and the
BIG wave SHRINKS
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2. Granular contacts, Solitary waves Shock
dispersion Dynamics of Tapered Chains
Scalable system down to nanoscales
Simple TC
Decorated TC
Smaller grains will suffer many collisions gt
large restitutive losses
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2. Granular contacts, Solitary waves Shock
dispersion A simple experiment
Expts by Nakagawa et al. at NASA Glenn, Gran Matt
4, 167 (2003)
Paper is dented - nobody escapes!
Paper is broken and a sphere escapes
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2. Granular contacts, Solitary waves Shock
dispersion Max energy is dissipated at the center
Energy Dissipation within the Tapered Chain, A.
Sokolow, J. Pfannes, et al., Appl Phys Lett 87,
254104, (2005)
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2. Granular contacts, Solitary waves Shock
dispersion Effect of Restitution
KE phase diagram
Force phase diagram
Effects of precompression/loading
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2. Granular contacts, Solitary waves Shock
dispersion Phase diagrams of the Decorated TC
R. Doney S. Sen, PRL 97, 155502 (2006)
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2. Granular contacts, Solitary waves Shock
dispersion Phase diagrams of the Decorated TC
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2. Granular contacts, Solitary waves Shock
dispersion Experimental Setups
Close-up view
Interstitial Nitinol alloy spheres
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2. Granular contacts, Solitary waves Shock
dispersion Experimental Result
5 tapering
R. Doney, J. Agui, S. Sen, submitted
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2. Granular contacts, Solitary waves Shock
dispersion Shock absorbing armor
Typically using 55-nitinol or 60-nitinol alloys
R. Doney S. Sen, Proc of 22nd Intl Ballistics
Symp. Banff (2005) ARL-Tech Rept 3612 (2005)
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3. Quasi-equilibrium
?
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3. Quasi-equilibrium
Solitary Wave has Kinetic Potential Energy
?Velocity or Position Perturbation alone cant
make a Solitary Wave at t0
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3. Quasi-equilibrium

Solitary wave is destroyed by boundaries and
reconstructed imperfectly
q0
Long-time regime
Particle Number
qgt0
t (?s)
Organized wave propagation
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3. Quasi-equilibrium
Gaussian distribution of velocities in
monodisperse chain
N20,q0
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3. Quasi-equilibrium
Average kinetic energy per grain at t1000µs
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3. Quasi-equilibrium
KE(t)
q0
Measures fluctuation against gray scale
q0.1
t

• No memory of initial conditions
• Gaussian velocity distribution
• Equipartition is (likely) not respected

(?)
Time independent
Harmonic systems equilibrate as
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3. Quasi-equilibrium
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3. Quasi-equilibrium
Poincare Recurrence?
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• 3. Quasi-equilibrium Extension to Nonlinear
  Chains and the Fermi-Pasta-Ulam Problem

Note The harmonic term was not killed off in the
FPU problem. The dynamics revealed recurrent
system oscillations.
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• 3. Quasi-equilibrium Extension to Nonlinear
  Chains and the Fermi-Pasta-Ulam Problem

Due to repeated collisions between solitary waves
and with boundaries, a quasi-equilibrium phase
emerges in a quartic chain. Physica A 342, 336
(2004) Phys Rev E (to appear)
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4. Driven chains and Granular Breathing
Promising Possibilities in Noise Filtration?
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4. Driven chains and Granular Breathing
Promising Possibilities in Noise Filtration?
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4. Driven chains and Granular Breathing
Promising Possibilities in Noise Filtration?
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4. Driven chains and Granular Breathing
Promising Possibilities in Noise Filtration?
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4. Driven chains and Granular Breathing
Promising Possibilities in Harnessing Wave Energy?
Typical periods associated with waves range
between 3s and 25s with a dominant frequency of
about 0.1 Hz, R. Boud (UK Dept of Trade and
Industry) estimates a resource base of 1-10
Trillion Watts of energy in ocean waves. This is
an interesting number given that the worlds
energy needs are now estimated at lt 10 Trillion
Watts
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4. Driven chains and Granular Breathing
Promising Possibilities in Harnessing Wave Energy?
Wave dissipation on the beach few seconds
Tapered chain shock absorber assembly (Physica A
2001, Appl Phys Lett 2005)
Half-bridge power harvester with a leakage
resistance (Elvin et al, Smart Mater Struc.
2001)
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4. Driven chains and Granular Breathing Time
Dependent Driving
Driven system
F F0 sin(ft) F0 200 N
N 15, q 0
F
F
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4. Driven chains and Granular Breathing Time
Dependent Driving
N 15 q 0 F0 200 N frequency 120
N 15 q 0 F0 200 N frequency 150
Plots of the total energy of the system
N 15 q 0 F0 200 N frequency 180
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4. Driven chains and Granular Breathing Time
Dependent Driving
Density plot of particles kinetic energies
N20 q0 F0 200 N frequency90
N20 frequency90
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N 20 frequency 125
N 20 frequency 129.5
N 20 frequency 129
N 20 frequency 130
N 20 frequency 131
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Ti6 Al4 V D 0.01206 (mm2 /kN) ? 4.42
(mg/mm3) q 0 R 5.0 mm w0 F100N
N15
N16
Kinetic energies of all particles
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5. Ultrashallow Depth Geophysics Impulse
Acoustics
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5. Ultrashallow Depth Geophysics Backscattering
from Empty Beds
S. Swaminathan, D. Visco and S. Sen, Phys. Rev. E
70, 051306 (2004)
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5. Ultrashallow Depth Geophysics Backscattering
from Buried Inclusions
S. Swaminathan, D. Visco and S. Sen, Appl. Phys.
Lett. 90, 154107 (2007)
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5. Ultrashallow Depth Geophysics Overall Picture
S. Swaminathan, D. Visco and S. Sen, Appl. Phys.
Lett. 90, 154107 (2007)
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6. Summary

• Solitary waves in particulate media are unique
  creatures we can see them forming,
  interacting pointing to the possible existence
  of a new kind of equilibrium-like phase
• Granular Alignments promise new possibilities for
  Shock Absorption
• Granular Breathing promises new possibilities for
  harnessing wave energy, noise cleaning,
• Ultrashallow bed geophysics promises new ways of
  detecting shallow buried objects