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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?
Leonardo Da Vinci (1452-1519) 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: http://movies.nano-world.org/movies/frictionmodule/en/
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:
Movie from: http://movies.nano-world.org/movies/frictionmodule/en/
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FS = μFN
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
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FN = pAr ,
where p is the penetration hardness.
where s is the shear strength.
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FS = sAr ,
Thus:
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μ ≡FSFN
=p
s .
Since both p and s are material constants, so is μ.
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.
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μstatic > μdynamicOhnaka (2003)
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μstatic
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μdynamic
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 MPaCalcite 1,800 MPaSL Glass 5,500 MPa
Quartz 12,000 MPa
The effect of normal stress - Dieterich and Linker
Linker and Dieterich, 1992 Instantaneousresponse 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: http://movies.nano-world.org/movies/frictionmodule/en/
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: 643-696, 1998.• Scholz, C. H., The mechanics of earthquakes and faulting, New-York: Cambridge Univ. Press., 439 p., 1990.