Transcript Nonlinear Dynamics Experiments with soft materials

Continuum Mechanics: Research
Questions for the Classroom
Michael Dennin
U. C. Irvine
Department of Physics and
Astronomy
“One of the oddities of contemporary
physics education is the nearly complete
absence of continuum mechanics in the
typical undergraduate or graduate
curriculum.”
Jerry Gollub, Reference Frame, Physics
Today, Dec. 2003.
What do we teach?
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Single particle classical
Rigid body classical
EM
Quantum
Waves (strings)
Relativity
WHY DO WE TEACH THESE TOPICS?
How does it help understand …
FLOW
VERSUS
JAMMING
PHASE
DIAGRAM
Liu and Nagel
JAMMING
What happened to continuum
mechanics?
Two Big Questions in Physics:
1) Transition from quantum to classical.
2) Transition from single particle to
continuum.
Educational Benefits
• Physically accessible tensors: stress/strain.
• Practice with differential equations (ODE AND
PDE).
• Exposure to CLASSICAL FIELD THEORY.
• Fun Demonstrations!!
• Relevance for undergrads moving into
engineering positions
• CRITICAL BACKGROUND FOR CURRENT
RESEARCH AREAS!!!
Jamming Phase Diagram
• Plasticity in
“molecular” systems
• Glassy behavior in
liquids
The “J-point”
Liu and Nagel, Nature v 396, 1998
• Flow of “multiphase”
materials: granular,
foams, colloids,
pastes, etc..
WHAT ABOUT FOAMS?
http://www.joiff.com/technical/infoamation.htm
FOAM: gas bubbles with liquid walls
Size: microns to millimeters
Useful parameter: Liquid fraction or gas fraction
Durian, UPENN
Main Features of Sheared foam
• Initial elastic response (yield stress)
• Flowing regimes:
– Slow shear: “irregular” stress response
– Fast shear: “smooth” flow
BUBBLES PLAYS CENTRAL ROLE
Definition of Terms: Part I
T1 event:
Neighbor switching
Definition of Terms: Part II
Outer barrier moves with V
flowing
Ds
stress
Dr
elastic
Strain: g = Dx/Dr
Strain Rate: dg/dt = v/Dr
d  v(r ) 
dg / dt = r 

dr  r 
Viscosity: h = stress/(strain rate)
strain
Shear stress: sxy = F/L (twodimensions)
Stress drop: Ds
Apparatus
Schematic of Apparatus
Inner radius ri: 3.84 cm
Outer radius ro: 7.43 cm
Area fraction: 0.95
Boundary conditions: no slip at both walls, but
inner cylinder is free to move.
Basic measurements
• Stress on inner cylinder
• Individual bubble motions
– Automatic tracking gives average
properties and topological
rearrangements
Bubble Motions
One problem in continuum mechanics:
What is a solid and a fluid?
(Is there a simple understanding of a broad
range of collective behavior?)
Continuum Facts: Part I
Couette Geometry: average stress, s, proportional to 1/r2
Sample stress curve
2.0
Yield Stress
"flowing"
stress (mN/m)
1.5
zero shear
rate: "rigid body"
s (g ) = s y  g
1.0
n
0.5
shear rate is a continuous
function of r.
0.0
4.0
4.5
5.0
5.5
6.0
radial position (cm)
6.5
7.0
Effective Viscosity: stress/(strain rate)
log (viscosity)
4
3
2
1
-3
-2
-1
0
log (strain rate)
s = s y  a(dg / dt ) = (0.8 mN/m)  (1.8 mNs /m)(dg / dt )
1/3
1/3
1/3
Shear Discontinuity
1.0
Yield stress fluid
v(r)/(r)
0.8
“solid”
0.6
Power law fluid
0.4
0.2
0.0
4.5
5.0
5.5
6.0
6.5
radial position (cm)
J. Lauridsen, G. Chanan, M. Dennin, PRL, 2004
7.0
Another view
-4
6x10
6.0x10
-4
5.5x10
-4
5.0x10
-4
-1
v(r)/r (s )
-1
v(r)/r (s )
-4
4x10
-4
2x10
Exponential
4.5x10
-4
4.5
0
4
6
5.0
5.5
6.0
radial position (cm)
8
radial position (cm)
6.5
10
Is this a “phase” transition?
THREE DIMENSIONAL
Coussot, Raynaud, et al.,
PRL 88, 218301 (2002)
What are the questions?
• Correct description of fluctuations:
stress (mN/m)
– Statistical mechanics?
– Chaos theory?
2.5
– Spatial fluctuations? 2.0
– Something else?
1.5
1.0
0.5
0.0
500
1000
time (s)
1500
How can we understand the average
velocity behavior?
1.0
1.0
0.5
v(r)/r
v(r)/r
• Why does it
converge so
quickly?
• What sets the
critical radius?
• What is the role of
T1 events?
0.8
6
7
radial position (cm)
0.0
5
6
radial position (cm)
7
T1
Events
# of neighbors
• distribution of neighbors
• changes in distribution
• size separation?
• ordering/disorder?
Conclusions
• Even continuum mechanics has
interesting physics questions left.
• We need to inspire our students with
exciting, challenging QUESTIONS, not
just elegant past solutions.
• One such question – Can we describe
collective behavior based on simple
principles?
Thanks to …
Michael Twardos
John Lauridsen
Gregory Chanan
Yuhong Wang
Kapil Krishan
Funded by: Department of Energy grant DEFG02-03ED46071, Sloan Foundation,
Petroleum Research Fund, and UCI UROP