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Friction
Why friction? Because slip on faults is resisted by frictional forces.
In the coming weeks we shall discuss the implications of the friction law to:
• Earthquake cycles,
• Earthquake depth distribution,
• Earthquake nucleation,
• The mechanics of aftershocks,
• and more...
Question: Given that all objects shown below are of equal mass and identical shape, in which case the frictional force is greater?
Question: Who sketched this figure?
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 contact area.
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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:
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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.
But it does not explain Byerlee law…
Beyrlee law
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For σ N < 200MPa : μ = 0.85
For σ N > 200MPa : μ = 0.60
Byerlee, 1978
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
The law of Coulomb - is that so?
Friction is independent of sliding velocity.
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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
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 logarithm of normal stress.
The effect of normal stress on the true contact area - Dieterich and Kilgore
Dieterich and Kilgore, 1994
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.
• 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
The constitutive law of Dieterich and Ruina
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τσ=μ =μ∗+ A ln
V
V *
⎛
⎝ ⎜
⎞
⎠ ⎟+ B ln
θV *
DC
⎛
⎝ ⎜
⎞
⎠ ⎟
and
dθ
dt=1−
θV
DC−αθ
B
dσ /dt
σ ,
were:• V and are sliding speed and contact state, respectively.• A, B and are non-dimensional empirical parameters.• Dc is a characteristic sliding distance.• The * stands for a reference value.
The set of constitutive equations is non-linear. Simultaneous solution of non-linear set of equations may be obtained numerically (but not analytically). Yet, analytical expressions may be derived for some special cases.
• The change in sliding speed, V, due to a stress step of τ:
• Steady-state friction:
• Static friction following hold-time, thold:
€
V = exp ΔτAσ( ).
€
μss = μ* + (A − B)lnVssV *
⎛
⎝ ⎜
⎞
⎠ ⎟= μ* + (B − A)ln
θ ssV*
Dc
⎛
⎝ ⎜
⎞
⎠ ⎟ .
€
μstatic ∝ (B − A)ln θ0 + Δthold( ) .
€
τσ=μ =μ∗+ A ln
V
V *
⎛
⎝ ⎜
⎞
⎠ ⎟+ B ln
θV *
DC
⎛
⎝ ⎜
⎞
⎠ ⎟
and
dθ
dt=1−
θV
DC ,
or
dθ
dt= −θV
DClnθV
DC
⎛
⎝ ⎜
⎞
⎠ ⎟
The evolution law: Aging-versus-slip
Aging law (Dieterich law):
Slip law (Ruina law):
Slip law fits velocity-stepping better than aging law
Linker and Dieterich, 1992
Unpublished data by Marone and Rubin
Aging law fits slide-hold-slide better than slip law
Beeler et al., 1994
In the coming weeks we shall discuss the implications of the friction law to:
• Earthquake nucleation,
• Earthquake depth distribution,
• Earthquake cycles,
• The mechanics of aftershocks, and more.
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.
