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