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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?
• 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
Liu and Nagel
JAMMINGPHASEDIAGRAM
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
Liu and Nagel, Nature v 396, 1998
The “J-point”
• Plasticity in “molecular” systems
• Glassy behavior in liquids
• Flow of “multiphase” materials: granular, foams, colloids, pastes, etc..
WHAT ABOUT FOAMS?
Durian, UPENN
FOAM: gas bubbles with liquid wallsSize: microns to millimetersUseful parameter: Liquid fraction or gas fraction
http://www.joiff.com/technical/infoamation.htm
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
Strain: x/r
Strain Rate: d/dt = v/r
Viscosity: = stress/(strain rate)
r
strain
elastic
flowingstress
Shear stress: xy = F/L (two-dimensions)
Stress drop:
( )/
d v rd dt r
dr r
Apparatus
Schematic of Apparatus
Inner radius ri: 3.84 cmOuter radius ro: 7.43 cmArea fraction: 0.95Boundary 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:
(Is there a simple understanding of a broad range of collective behavior?)
What is a solid and a fluid?
4.0 4.5 5.0 5.5 6.0 6.5 7.00.0
0.5
1.0
1.5
2.0
stre
ss (
mN
/m)
radial position (cm)
"flowing"
zero shearrate: "rigid body"
Yield Stress
Sample stress curve
( ) ny
Continuum Facts: Part I
Couette Geometry: average stress, , proportional to 1/r2
shear rate is a continuous function of r.
Effective Viscosity: stress/(strain rate)
-3 -2 -1 01
2
3
4
log
(vis
cosi
ty)
log (strain rate)
1/3 1/3 1/3(0.8 mN/m)( / ) (1.8 mNs /m)( / ) y a d dt d dt
Shear Discontinuity
4.5 5.0 5.5 6.0 6.5 7.00.0
0.2
0.4
0.6
0.8
1.0
v(r)
/(r
)
radial position (cm)
Yield stress fluid
Power law fluid
J. Lauridsen, G. Chanan, M. Dennin, PRL, 2004
“solid”
Another view
4.5 5.0 5.5 6.0 6.54.5x10-4
5.0x10-4
5.5x10-4
6.0x10-4
4 6 8 100
2x10-4
4x10-4
6x10-4
v(r)
/r (
s-1)
radial position (cm)
radial position (cm)
v(r
)/r
(s-1
)
Exponential
Is this a “phase” transition?
THREE DIMENSIONALCoussot, Raynaud, et al., PRL 88, 218301 (2002)
What are the questions?
• Correct description of fluctuations:– Statistical mechanics?
– Chaos theory?
– Spatial fluctuations?
– Something else?
500 1000 15000.0
0.5
1.0
1.5
2.0
2.5
stre
ss (
mN
/m)
time (s)
How can we understand the average velocity behavior?
• Why does it converge so quickly?
• What sets the critical radius?
• What is the role of T1 events? 5 6 7
0.0
0.5
1.0
6 7
0.8
1.0
radial position (cm)
v(r
)/r
v(r
)/r
radial position (cm)
T1Events
# 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 TwardosJohn LauridsenGregory ChananYuhong WangKapil Krishan
Funded by: Department of Energy grant DE-FG02-03ED46071, Sloan Foundation, Petroleum Research Fund, and UCI UROP