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

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 JAMMING PHASE DIAGRAM

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 walls Size: microns to millimeters Useful parameter: Liquid fraction or gas fraction

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r strain elastic flowing stress  Shear stress:  xy = F/ L (two- dimensions) Stress drop: 

Apparatus

Schematic of Apparatus Inner radius r i : 3.84 cm Outer radius r o : 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: (Is there a simple understanding of a broad range of collective behavior?) What is a solid and a fluid?

Yield Stress Sample stress curve Continuum Facts: Part I Couette Geometry: average stress, , proportional to 1/r 2 shear rate is a continuous function of r.

Effective Viscosity: stress/(strain rate)

Shear Discontinuity Yield stress fluid Power law fluid J. Lauridsen, G. Chanan, M. Dennin, PRL, 2004 “solid”

Another view Exponential

Is this a “phase” transition? THREE DIMENSIONAL Coussot, Raynaud, et al., PRL 88, (2002)

What are the questions? Correct description of fluctuations: –Statistical mechanics? –Chaos theory? –Spatial fluctuations? –Something else?

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?

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 DE- FG02-03ED46071, Sloan Foundation, Petroleum Research Fund, and UCI UROP