Mechanics of Earthquakes and Faulting
Lecture 10, 29 Sep. 2015
www.geosc.psu.edu/Courses/Geosc508
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Friction
Base-level friction coefficient
Contact mechanics
Hardness
Amonton’s laws
Hertian contact
Adhesive theory of friction
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Stability of frictional sliding: stable vs. unstable sliding
Stick-slip dynamics
Earthquakes as a stick-slip phenomenon
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Basic observations of time-dependent static friction
Velocity-dependent sliding friction
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Read Rabinowicz, 1951
Read Chapter 2 of Scholz
Read Marone, 1998
Base Friction vs. 2nd order variations
0.7
angular sand (rough)
0.6
0.5
µ
spheres (3-D)
Base Friction, µo
µo
0.4
rods (⊥) (2-D)
0.3
0.2
For metals: µo ~ 1/3
0.1
0
For rocks: µo ~ 2/3
0
5
10
Load Point Displacement (mm)
(Frye and Marone, GRL 2002)
15
base level friction (~ 0.6 for rocks)
Load-up
sequence
0.7
C-SHS Partial unload
SHS
Zero load
SHS
Friction ( µ=τ/σn )
0.6
100s
hold
0.5
0.4
100s
holds
0.3
0.2
Loading rate 10 µm/s
Unloading rate 300 µm/s
0.1
100s
holds
m295
0
0
5
10
15
20
25
Shear Displacement (mm)
Karner & Marone (GRL 1998, JGR 2001)
30
35
Amonton’s Laws (1699)
(Both apply to base friction, µo)
1st: Friction force independent of the size of surface contact dimension A
2nd: Friction force is proportional to normal load
τ
Fn
Fs
σ
Contact area A
Amonton’s Laws (1699)
Friction force is the same for objects small and large as long as is σ ~ equal
µo ~ 1/3 regardless of surface or material for a wide range of metals and technological
materials, excluding lubricated surfaces and modern polymers such as teflon
Why does it hold?
Friction is a contact problem. Therefore base friction is
primarily a surface property and not a material property (but
note that we’ll have to relax this a bit when we talk about
2nd order variations in friction, which are closely connected
with material properties)
Friction ~ independent of surface roughness for low
normal loads and unmated surfaces
Asperities
mated joint
Adhesive Theory of Friction
1: Friction force independent of the size of surface contact dimension A
Why does it hold?
Solution to Amonton’s Problem: Asperities and contact junctions
contact junction of dimension ai
Nominal contact area A
Real area of contact ~ 10% of A for unmated rough surfaces –but this
does not apply for very light loads, mirror-smooth surfaces or lubricated
surfaces
But we still have the problem of
and µo ~ independent of material
Asperities
Why is this a problem?
Friction
Base-level friction
coefficient in terms of
contact mechanics and
hardness
Adhesive Theory of Fricton (Bowden and Tabor)
• Real contact area << nominal area
• Contact junctions at inelastic (plastic) yield
strength
• Contacts grow with “age”
• Add: Rabinowicz’s observations of
static/dynamic friction
• “Static” friction is higher than “Dynamic”
friction because contacts are older (larger)
• -> implies that contact size decreases as
velocity increases
Friction
Base-level friction
coefficient in terms of
contact mechanics and
hardness
Adhesive Theory of Fricton (Bowden and Tabor)
• Real contact area << nominal area
• Contact junctions at inelastic (plastic) yield
strength
• Contacts grow with “age”
• Add: Rabinowicz’s observations of
static/dynamic friction
• “Static” friction is higher than “Dynamic”
friction because contacts are older (larger)
• -> implies that contact size decreases as
velocity increases
Classic theory of friction
Bowden and Tabor [1960]
τ - shear stress
σn - normal stress
Fn - normal force
Fs - shear force
AT - total fault area
Ac- the real area of
contact
S- contact shear
strength
σy - yield strength or
hardness
σn =
σ y Ac
AT
SAc
τ=
AT
τ
S
µ=
=
σn σy
Friction is the ratio of shear
strength to hardness
Modified from Beeler, 2003
This is base level friction
Adhesive Theory of Friction
But we still have the problem of
and µo ~ independent of material
Why is this a problem?
welded contact junction
Consider a hemispherical contact against a flat, under a shear load
(Bowden & Tabor, 1950)
Two assumptions:
1) Yielding at asperities is just sufficient to support normal load
where, p is penetration hardness
2) Slip involves shearing of adhesive contacts and/or asperities
where, s is shear strength
combing these equations shows why µo ~ independent of material
friction is the ratio of two material properties
Adhesive Theory of Friction
(Bowden & Tabor, 1950)
friction is the ratio of two material properties
welded contact junction
Generally see that p ~ 3 σy compressive yield strength and s ~ σy /2
This gives µo = 1/6 --but recall that observation is that µo ~ 1/3.
--difference due to unaccounted effects, such as ploughing, wear and surface
production, interlocking, dilational work, etc.
But we still have the problem of linearity between τo and σ
Hertzian contact predicts
but, this is dealt with by realistic descriptions of surface roughness: asperities have
asperities on them…. Archard (1957), Greenwood and Williamson (1966)
Byerlee’s Law for Rock Friction (Coulomb’s Criterion)
µ = 0.6
Byerlee, 1978
Friction: Observations & Geophysical Experimental Studies
See Scholz Fig 2.5 for common experimental configurations
Rock Mechanics Lab Studies
• Experiments designed to investigate mechanisms and processes, not scale model experiments
• Application of friction/fracture studies to earthquakes/fault behavior
• Scaling problem.
Lab: cm-sized samples,
Field: earthquake source dimensions 10’s to 100’s km
• Friction is scale invariant to 1st order (Amonton) --i.e. µ is a dimensionless constant. But will this
extend to 2nd order characteristics of friction that control slip stability
Byerlee’s Law (Byerlee, 1967, 1978)
Base Friction is:
~ independent of rock type and normal stress
~ the same for bare, ground surfaces and gouge
τ = 0.85 σn for σn < 200 MPa
τ = 50 + 0.6 σn for σn > 200 MPa
This applies (only) to ground surfaces,
primarily Westerly granite
For granular materials, powders, and
fault gouge: τ = 0.6 σn
Note that Byerlee’s law is just Coulomb Failure. It’s simply a statement about
brittle (pressure sensitive) deformation and failure.
Byerlee’s Law (Byerlee, 1967, 1978)
τ = 0.85 σn for σn < 200 MPa
τ = 50 + 0.6 σn for σn > 200 MPa
For granular materials, powders, and
fault gouge: τ = 0.6 σn
Friction of Fault Zones
Penn State Lab, ~ 2000 samples
Now we have an introduction to Base Friction
What about 2nd order variations in friction, for example ‘static’ vs. ‘dynamic’
friction?
0.7
angular sand (rough)
0.6
0.5
µ
Base Friction, µo
spheres (3-D)
0.4
rods (⊥) (2-D)
0.3
Base Friction level, µo
0.2
0.1
0
0
5
10
Load Point Displacement (mm)
(Frye and Marone, GRL 2002)
15
Friction: 2nd order variations, slick-slip and stability of sliding
Rabinowicz 1951, 1956,. 1958
Static vs. dynamic friction & state dependence
Classical view
Rabinowicz recognized that finite slip was
necessary to achieve fully dynamic slip
Static-Dynamic Friction with
critical slip
µs
µ
d
sd
sd is the critical slip distance
Slip
Rabinowicz experiments showed state, memory effects and that µd
varied with slip velocity.
Brittle Friction Mechanics, Stick-slip
• Stick-slip (unstable) versus stable shear
N
N
L
x´
x
K
Fs
f
Why is this a reasonable approach?
1-D fault
zone analog,
Stiffness K
Slip
contours, u
W
Rupture
area, A
Reid’s Hypothesis of
Elastic Rebound
Reid, H.F., The mechanics of the
earthquake, v. 2 of The California
earthquake of April 18, 1906. Report of the
State Earthquake Investigation
Commission, Carnegie Institution of
Washington Publication 87, 1910.
Earth, S. Marshak, W.W. Norton
Elastic strain accumulates during the interseismic period and is released during an earthquake.
The elastic strain causes the earthquake –in the sense that the elastic energy stored around
the fault drives earthquake rupture.
There are three basic stages in Reid’s hypothesis.
1) Stress accumulation (e.g., due to plate tectonic motion --but what about intra-plate
earthquakes?)
2) Stress reaches or exceeds the (frictional) failure strength
3) Failure, seismic energy release (elastic waves), and fault rupture propagation
Stage 1
Stage 2
Reid’s
Hypothesis of
Elastic
Rebound
Stage 1
Stage 2
Brittle Friction Mechanics, Stick-slip
• Stick-slip (unstable) versus stable shear
N
N
L
x´
x
K
Fs
f
1-D fault
zone analog,
Stiffness K
Why is this a reasonable approach?
How do we get at stiffness?
Relation between stress and slip on a
dislocation of radius r. Therefore, the local
stiffness around the slip patch is:
That is, stiffness decreases
as the patch enlarges.
Slip
contours, u
W
Rupture
area, A
Brittle Friction Mechanics, Stick-slip
• Stick-slip (unstable) versus stable shear
N
N
L
x´
x
K
Fs
1-D fault
zone analog,
Stiffness K
f
Slope = -
Force
B
Ν µs
Thi
x
x´
Slip
Rupture
area, A
Frictional stability is determined by the
combination of
K
T
Slip
contours, u
W
f
C
Displacement
1) fault zone frictional properties and
2) elastic properties of the surrounding
material
Brittle Friction Mechanics, Stick-slip
• Stick-slip (unstable) versus stable shear
Stick-slip dynamics
N
x´
x
K
Fs
f
Static-Dynamic Friction
Slope = -
Force
B
K
Ν µs
µs
Thi
T
x
x´
Slip
f
C
Displacement
µd
sd
Slip
Brittle Friction Mechanics, Stick-slip
• Stick-slip (unstable) versus stable shear
Stick-slip dynamics
N
x´
x
K
Fs
f
Static-Dynamic Friction
µs
µd
sd
Slip
slip duration = rise time
Brittle Friction Mechanics, Stick-slip
• Stick-slip (unstable) versus stable shear
N
x´
x
K
Fs
f
Slope = -
Force
B
slip duration = rise time
K
Ν µs
Thi
T
x
x´
Slip
f
C
Displacement
total slip, particle
velocity, and accel. all
depend on friction drop
(stress drop)
Laboratory Studies
Plausible Mechanisms for Instability
Slip Weakening Friction Law
N
µs
x´
x
K
Fs
µ
d
≠
µ
d
(v)
L
f
Slip
Slope = -
Force
B
K
Quasistatic Stability Criterion
Ν µs
Thi
T
x
x´
Slip
f
C
Displacement
K< Kc; Unstable, stick-slip
K > Kc; Stable sliding
Laboratory Studies
N
x´
x
K
Fs
But, there’s a
f
problem…….
Slip Weakening Friction Law
µs
µd≠ µd(v)
L
Slip
Stick-slip stress-drop amplitude
varies with loading rate.
Mair, Frye and Marone, JGR 2002
Duality of time and
displacement dependence of
friction.
“Static” and “dynamic” friction
are just special cases of a more
general behavior called “rate
and state friction”
Sheared layer of quartz
particles. Marone, 1998
Load point
Fault
surface
Time, displacement,
and velocity
dependence of
“static” and
“dynamic” friction
Time (state) dependence of
friction: Healing
Velocity (rate) dependence of
friction.
Duality of time and
displacement dependence of
friction.
“Static” and “dynamic” friction
are just special cases of a more
general behavior called “rate
and state friction”
Friction: 2nd order variations, slick-slip and stability of sliding
Rabinowicz’s work solved a major problem with friction theory: he introduced
a way to deal with the singularity in going from µs to µd
Slip Weakening Friction Law
(for L > x > 0)
µs
µ
d
L
≠
µ
d
(for x > L)
(v)
Palmer and Rice, 1973; Ide, 1972; Rice, 1980
Slip
For solid surfaces in contact (without wear materials), the slip distance L represents the
slip necessary to break down adhesive contact junctions formed during ‘static’ contact.
The slip weakening distance is also known as the critical slip or the breakdown slip
This slip distance helps with the stress singularity at propagating crack tips, because the
stress concentration is smeared out over the region with slip < L.
Friction: 2nd order variations, slick-slip and stability of sliding
Slip Weakening Friction Law
(for L > x > 0)
µs
µ
d
≠
µ
d
(v)
(for x > L)
L
Slip
Adhesive Theory of Friction
Critical friction distance
represents slip necessary
to erase existing contact
For a surface with a
distribution of contact
junction sizes, L, will be
proportional to the average
contact dimension.
Critical friction distance scales with
surface roughness
Time dependent yield strength:
µ=
τ
S
=
σn σy
Dieterich and Kilgore [1994]
Time dependent growth of contact (acyrlic plastic)- true static
contact
Time dependent yield strength:
µ=
Dieterich and Kilgore [1994]
Time dependent growth of contact (acyrlic
plastic)- true static contact
Modified from Beeler, 2003
τ
S
=
σn σy
τ
S
µ=
=
σn σy
σ y = σ o + f (t )
Other measures of changes in ‘static’ friction, contact area, or strength
‘hold’ test
after Dieterich [1972]
time dependent closure (westerly
granite)
- approximately static contact
Modified from Beeler, 2003
Rate dependence of contact shear strength
‘hold’ test
µ=
Rate
dependent
response
Modified from Beeler, 2003
τ
S
=
σn σy
S = So + g( V )
Summary of friction observations:
0. Friction is to first order a constant
1. Time dependent increase in contact area (strengthening)
2. Slip dependent decrease in contact area (weakening); equivalently increase in dilatancy
3. Slip rate dependent increase in shear resistance (non-linear viscous)
Modified classic theory of friction:
µ=
S + g( V )
S
= o
σ y σ o + f ( age)
So + g( V ) #σ o − f ( age) &
%
(
µ=
σ o + f ( age ) %$σ o − f ( age) ('
Discard products of second order terms:
µ=
So g(V ) So f ( age)
+
−
σ o σo
σ o2
Modified from Beeler, 2003
[e.g., Dieterich, 1978, 1979]
S
g(V ) So f ( age)
Summary of friction observations:
µ= o +
−
σ
σ
σ o2
0. Friction is to first order a constant
o
o
1. Time dependent increase in contact area (strengthening)
2. Slip dependent decrease in contact area (weakening); equivalently increase in dilatancy
3. Slip rate dependent increase in shear resistance (non-linear viscous)
1st order term second order terms
Rate and state equations:
µ = µ 0 + a ln
0.
Vθ
V
+ b ln 0
V0
Dc
3.
θ is contact age
1. & 2.
Dieterich [1979]
Rice [1983]
Ruina [1983]
dθ
θV
= 1−
dt
Dc
time dependence
# ∂θ &
dθ # ∂θ &
= % ( +% ( V
dt $ ∂t ' d $ ∂d ' t
Modified from Beeler, 2003
# ∂θ &
% ( =1
$ ∂t ' d
slip dependence
# ∂θ &
θ
% ( =−
$ ∂d ' t
Dc