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F act, S ir, Not F iction: A F raction of F rictions O bey S trictures of F racture

F act, S ir, Not F iction: A F raction of F rictions O bey S trictures of F racture. Background Talk on Numerical Simulation of A New Theory of Friction Based on Traveling, Self-Healing Cracks. - Philip (Flip) Kromer ∙ 7 Feb 2002 -

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F act, S ir, Not F iction: A F raction of F rictions O bey S trictures of F racture

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  1. Fact, Sir, Not Fiction:A Fraction of FrictionsObey Strictures of Fracture Background Talk on Numerical Simulation of A New Theory of Friction Based on Traveling, Self-Healing Cracks - Philip (Flip) Kromer ∙ 7 Feb 2002 - On A New Theory of Friction By M. Marder and E. Gerde, And on My Numerical Work Using Code By D. Holland. - Center for Nonlinear Dynamics - You may view the slides for this talk athttp://www.mrflip.com/research/talks

  2. Big idea How could Something so Simple Come from Something so Complicated? • Macroscopic: Friction is amazingly simple • Microscopic: Friction is amazingly complicated • F= µN • Where µ: • Depends only on material • Is independent of contact area • Is near 1 • Is independent of sliding velocity

  3. How could Something so SimpleCome from Something so Complicated? • Macroscopic: Friction is amazingly simple • Microscopic: Friction is amazingly complicated • Roughness at every length scale • Plastic and Elastic deformations • Adhesion and Interlocking of Asperities • Oxide Layers, Adsorbed Layers, Hydrocarbon Chains • Phononic and Electronic Drag • Many-body Quantum Mechanical Problem • Aspects of the problem are too big for QM… • A Polished surface: bumps up to ≈ 200nm: too large • … Yet too small for Statistical Mechanics.

  4. A Brief History of Friction Truth is stranger than Friction • Early History • da Vinci 1496 • Amonton’s Laws of Friction 1699 • Coulomb’s Theory of Friction 1781 … • Traditional Theory • Bowden and Tabor Model 1940 • Bowden and Tabor, Friction & Lubrication of Solids • Persson, Sliding Friction … • A New Theory of Friction • Marder and Gerde 2001 • Numerical Investigations 2002-200?

  5. da Vinci is Da Man • Da Vinci made careful, quantitative studies of friction ca. 1490-1496 “Friction produces double the amount of effort if the weight be doubled.” [Codex Forster III 72 r] “The friction made by the same weight will be of equal resistance at the beginning of its movement although the contact may be of different breadth or lengths.” [Codex Forster II, 133r & 133v]

  6. da Vinci is Da Man • Also: Lubricants; Roller Bearings; Inclined Plane with Friction; Abrasion; Roughness

  7. Friction is Simple: Amontons’ Laws • F= µN • µ depends on materials only • Independent of contact area • µk is largely independent of sliding velocity • Coefficient of Friction is rarely outside .02-3 • Very small range for a material property

  8. Coulomb is Cool, Mon (not whole truth) • Friction due to Interlocking Asperities • Lifting over asperities • Bending asperities • Breaking asperities Asperity just means “Bump” • Interlocking: important for metals; not primary • Adhesion is primary friction mechanism • Adhesion should increase with area of contact

  9. The Traditional Theory of Friction STM Scans of Silver Bowden and Tabor, 1940: Solid surfaces are highly irregular True area of contact far less than apparent area Profilometer Scans of Steel Surfaces (note scale)

  10. True Area Increases with Load • Ex: steel cube 10 cm on a side, on steel table • True area δA ≈ 0.1 mm2, 10-5 of apparent area • Junctions have diameter ≈ 10 µm • about 1000 junctions [Persson p47]

  11. Interlocking is not important

  12. We can now produce F = µ N • The true area is proportional to load • This yields the Coulomb equation • Friction is from shearing cold-welded junctions Since τ and σ are usually similar in magnitude, this explains why typically µ ≈ 1

  13. ANon-TraditionalTheory of Friction Science Friction Marder & Gerde, Nature 413, 285-288 (2001)

  14. Analytical Continuum Model

  15. Problem with Continuum Model? Displacement |uy|: Linear Scale • Infinite Oscillations Approaching Crack Tip Open End Crack Tip Distance x: Log Scale

  16. Analytical Lattice Model

  17. Envelope of Solutions Gives µ! • Build a Catalog of Matched Solutions • Lattice: Crack Tips • Continuum: Self-Healing • Only matching solutions accepted • Envelope is a linear threshold of -σ vs. τ: • Static Coefficient of Friction!

  18. Numerical Simulations Or, How to take Simple Physics and make it Difficult, Expensive, and Time-Consuming

  19. Molecular Dynamics Simulations • Boring Details: • Single Verlet – 4th order in dt. • Temperature by kicking/scaling • To prevent O(N2) problem, build Neighbor Lists • Neighbor Lists by cell method with shell • No long-range forces included • See Ph.D. thesis of D. Holland for whole story

  20. Computers are Too Small Space requirements • Consider a sample 0.1 mm x 0.01 mm x 1 µm … • Hardly macroscopic, but still 1015 atoms! • For 100 bytes/atom, need 100,000,000 GB (3D)! Time requirements • Characteristic timescale of chemical interactions: 1 fs • Therefore 1 µs of simulation takes 109 timesteps CPU Speeds • Units are GFlops, Giga Floating point Operations Per Second • Measured 20 min / G atom · timestep · GFlop • 1200 op / atom · timestep • Therefore, 1 µs of simulation takes 20 min/atom·GFlop • We need about a GFlop/atom! Preposterously too slow! • In all, we need 1024 atom·timesteps

  21. My Brand-New Computer: Too Small Dual 1500 MHz, 512MB RAM Needs\ 2·108x more RAM 1.5· 1010 years Tick.ph, an extremely fast workstation

  22. UT’s Supercomputer: Too Small Golden, Here in Texas: #340 in world† 272 Node Cray T3-E, 128 MB/node Needs\ 3·106x more RAM 3.3· 108 years †Rankings from the “World’s Fastest Supercomputer” list, Nov 2001: http://www.top500.org/

  23. Even This is Too Small 8192 Node 1024 MB/node Needs\ 1·105x more RAM 5.3· 106 years ASCI White, Lawrence Livermore: #1 in World† †Rankings from the “World’s Fastest Supercomputer” list, Nov 2001: http://www.top500.org/

  24. Computers are Too Small †Golden data from http://www.tacc.utexas.edu/resources/systems/ ‡ASCI White data from http://www.llnl.gov/asci/platforms/white/

  25. Impatience demands Inexactitude • Approximations • Effective Potential • Effective Potential with Cutoff • Simple Effective Potential with Cutoff • Snapping Hooke Springs • Two Dimensions • Scaling Argument

  26. Impatience demands Cleverness • Scaling Argument • Use Nonlinear Dynamics arguments • Matched Asymptotics • Multi-scale Modeling • MD Simulation of particles for atomistic regime • Adaptive Finite Element Mesh for continuum regime • Tight-Binding region for quantum regime (Not for us, though)

  27. Results Oooh, Pretty Pictures

  28. Spontaneous Pure Shear Crack

  29. Spontaneous Pure Shear Crack

  30. Seeded, Ignited Shear Crack

  31. Does this ModelQualitatively Explain Friction? • Coefficient of Friction • Do we get a threshold -σ vs. τ for traveling cracks? • That is, do we get a Coefficient of Friction µ? • Is robustness or nucleation imposing the threshold? • Is µ Independent of Contact Area? • Is µ a Sensible Physical Value? • Is µk Independent of speed? • What are the size effects?

  32. Future Work OK, Then is any of it True? • Three different issues: • Does it capture qualitative physics of friction? • Does it make quantitatively correct predictions? • If it does succeed in the general sense, why? • That is, how can something so simple replace something so complicated?

  33. How might something so Simplemodel something so Complicated? Right Picture, for Simple Reasons • Carefully prepared atomically flat surfaces • Might be boring: Laboratory Curiosity • Special case: Nanomachines • Earthquakes • Original inspiration for the new theory • Model of the process for individual asperities • Surface behaves as simple aggregate flat interface

  34. How might something so Simplemodel something so Complicated? Right Picture, for Complicated Reasons • This could be a picture of asperities, not atoms • Traveling crack hops from asperity to asperity

  35. How might something so Simplemodel something so Complicated? Right Picture, for Complicated Reasons • Clever Mathematical Mapping • Surface is fractal – contact at every scale • (Polished surfaces: half wavelength ≈ 300nm: large) • Conjecture: A Renormalization argument might imply that the fractal surface ends up being a boring flat surface

  36. Summary For those who fell asleep after the first slide Our research is guided by the Three Virtues of Programming: • Laziness • Discard “Correct” Model for “Simple” Flat Rigid Surfaces • Impatience • Numerical Work requires either way, way too much time, or just the right amount of cleverness • Hubris • Chutzpah to assert this model is correct • Work will give insight to where & how it is correct. Laziness, Impatience, Hubris [Wall 91] You may view the slides for this talk athttp://www.mrflip.com/research/talks

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