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?
Da Vinci law and the paradox
Leonardo Da Vinci (1452-1519) showed that the friction force is
independent of the geometrical area of contact.
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Sorenson Video decompressor
are needed to see this picture.
Movie from: http://movies.nano-world.org
The paradox: Intuitively one would expect the friction force to
scale proportionally to the contact 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

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Sorenson Video decompressor
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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 newsis that this explains Da Vinci and Amontons’ laws.
But it does not explain Byerlee law…
Beyrlee law
For  N  200MPa :   0.85
For  N  200MPa :   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.
static
dynamic


Ohnaka (2003)
static  dynamic
The law of Coulomb - is that so?
Friction is independent of sliding velocity.
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Sorenson Video decompressor
are needed to see this picture.
Movie from: http://movies.nano-world.org
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.
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 logarithm of normal stress.
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
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).
The constitutive law of Dieterich and Ruina
* 




V

V
     A ln  *  B ln 

V 

 DC 
and
d
V  d /dt
 1

,
dt
DC
B 
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 :

V  exp 
A
.
• Steady-state friction:

* 



V

V
ss  *  (A  B)ln  ss*  *  (B  A)ln  ss  .
V 
 Dc 
• Static friction following hold-time, thold:

static  (B  A)ln  0  thold  .
The evolution law: Aging-versus-slip
* 




V
V

     Aln  *  Bln 

V 

 DC 
and
d
V
Aging law (Dieterich law):
 1
,
dt
DC
or
Slip law (Ruina law):

d
V V 

ln  
dt
DC DC 
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, NewYork: Cambridge Univ. Press., 439 p., 1990.