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Friction: • Leonardo Da Vinci • Amonton • Bowden and Tabor • Dieterich From laboratory scale to crustal scale Figure from http://www.servogrid.org/EarthPredict/ Question: Given that all objects shown below are of equal mass and identical shape, in which case the frictional force is greater? Da Vinci law and the paradox Leonardo Da Vinci (1452-1519) showed that the friction force is independent of the geometrical area of contact. Movie from: http://movies.nano-world.org/movies/frictionmodule/en/ The paradox: Intuitively one would expect the friction force to scale proportionally to the surface area. Amontons’ laws Amontons' first law: The frictional force is independent of the geometrical contact area. Amontons' second law: Friction, FS, is proportional to the normal force, FN: FS FN Movie from: http://movies.nano-world.org/movies/frictionmodule/en/ Bowden and Tabor (1950, 1964) A way out of Da Vinci’s paradox has been suggested by Bowden and Tabor, who distinguished between the real contact area and the geometric contact area. The real contact area is only a small fraction of the geometrical contact area. Figure from: Scholz, 1990 FN pAr , where p is the penetration hardness. FS sAr , where s is the shear strength. Thus: FS p . FN s Since both p and s are material constants, so is . The good newsis that this explains Da Vinci and Amontons’ laws. Static versus kinetic friction The force required to start the motion of one object relative to another is greater than the force required to keep that object in motion. static dynamic Ohnaka (2003) static dynamic Slide-hold-slide - Dieterich Dieterich and Kilgore, 1994 Static (or peak) friction increases with hold time. Slide-hold-slide - Dieterich • The increase in static friction is proportional to the logarithm of the hold duration. Dieterich, 1972 Monitoring the real contact area during slip - Dieterich and Kilgore Change in true contact area with hold time - Dieterich and Kilgore Dieterich and Kilgore, 1994 • The dimensions of existing contacts are increasing. • New contacts are formed. Change in true contact area with hold time - Dieterich and Kilgore Dieterich and Kilgore, 1994 • The real contact area, and thus also the static friction increase proportionally to the logarithm of hold time. The effect of normal stress on the true contact area - Dieterich and Kilgore Dieterich and Kilgore, 1994 Upon increasing the normal stress: • The dimensions of existing contacts are increasing. • New contacts are formed. • Real contact area is proportional to the normal stress. The effect of normal stress on the true contact area - Dieterich and Kilgore Indentation yield stress, sy Acrylic 400 MPa Calcite 1,800 MPa SL Glass 5,500 MPa Quartz 12,000 MPa The effect of normal stress - Dieterich and Linker Changes in the normal stresses affect the coefficient of friction in two ways: Linker and Dieterich, 1992 • Instantaneous response, whose trend on a shear stress versus shear strain curve is linear. • Delayed response, some of which is linear and some not. Instantaneous response linear response The law of Coulomb - is that so? Friction is independent of sliding velocity. Movie from: http://movies.nano-world.org/movies/frictionmodule/en/ Velocity stepping - Dieterich Dieterich and Kilgore, 1994 • A sudden increase in the piston's velocity gives rise to a sudden increase in the friction, and vice versa. • The return of friction to steady-state occurs over a characteristic sliding distance. • Steady-state friction is velocity dependent. Velocity stepping and contact area Dieterich and Kilgore, 1994 Summary of experimental result • Static friction increases with the logarithm of hold time. • True contact area increases with the logarithm of hold time. • True contact area increases proportionally to the normal load. • A sudden increase in the piston's velocity gives rise to a sudden increase in the friction, and vice versa. • The return of friction to steady-state occurs over a characteristic sliding distance. • Steady-state friction is velocity dependent. • The coefficient of friction response to changes in the normal stresses is partly instantaneous (linear elastic), and partly delayed (linear followed by non-linear). Recommended reading: • Marone, C., Laboratory-derived friction laws and their applications to seismic faulting, Annu. Rev. Earth Planet. Sci., 26: 643-696, 1998. • Scholz, C. H., The mechanics of earthquakes and faulting, NewYork: Cambridge Univ. Press., 439 p., 1990.