Friction: Leonardo Da Vinci Amonton Bowden and Tabor Dieterich

From laboratory scale to crustal scale Figure from

Question: Given that all objects shown below are of equal mass and identical shape, in which case the frictional force is greater?

Leonardo Da Vinci ( ) showed that the friction force is independent of the geometrical area of contact. Da Vinci law and the paradox The paradox: Intuitively one would expect the friction force to scale proportionally to the surface area. Movie from:

Amontons’ laws Amontons' first law: The frictional force is independent of the geometrical contact area. Amontons' second law: Friction, F S, is proportional to the normal force, F N : Movie from:

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. Bowden and Tabor (1950, 1964) Figure from: Scholz, 1990

where p is the penetration hardness. where s is the shear strength. Thus: Since both p and s are material constants, so is . The good news is 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. Ohnaka (2003)

Slide-hold-slide - Dieterich Static (or peak) friction increases with hold time. Dieterich and Kilgore, 1994

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 The dimensions of existing contacts are increasing. New contacts are formed. Dieterich and Kilgore, 1994

Change in true contact area with hold time - Dieterich and Kilgore The real contact area, and thus also the static friction increase proportionally to the logarithm of hold time. 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 Dieterich and Kilgore, 1994

The effect of normal stress on the true contact area - Dieterich and Kilgore Indentation yield stress,  y Acrylic 400 MPa Calcite 1,800 MPa SL Glass 5,500 MPa Quartz 12,000 MPa

The effect of normal stress - Dieterich and Linker Linker and Dieterich, 1992 Instantaneous response linear response Changes in the normal stresses affect the coefficient of friction in two ways: Instantaneous response, whose trend on a shear stress versus shear strain curve is linear. Delayed response, some of which is linear and some not.

The law of Coulomb - is that so? Friction is independent of sliding velocity. Movie from:

Velocity stepping - Dieterich 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. Dieterich and Kilgore, 1994

Velocity stepping and contact area Dieterich and Kilgore, 1994

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). Summary of experimental result

Recommended reading: Marone, C., Laboratory-derived friction laws and their applications to seismic faulting, Annu. Rev. Earth Planet. Sci., 26: , Scholz, C. H., The mechanics of earthquakes and faulting, New- York: Cambridge Univ. Press., 439 p., 1990.