Microsoft PowerPoint - MEF2013_Ch02.ppt []Rock Friction
1. Constitutive law of rock friction 2. Friction stability regimes and seismogenesis 3. Seismic coupling and seismic styles 4. Stages in the seismic cycle 5. Earthquake insensitivity to transients 6. Outstanding problems in earthquake mechanics
Scholz, Christopher H., 1998. Earthquakes and Friction Laws. Nature.
Rock Friction
Once a fault has been formed: further motion is controlled by friction, which is a contact property rather than a bulk property
Schizosphere: Micromechanics of friction involve brittle fracture, but frictional behavior is fundamentally different from bulk brittle fracture
Stability of friction: Determines if fault motion is seismic or aseismic.
Two main laws of friction: Amontons (1699)
Amontons’s first law: Frictional force is independent of the size of the surfaces in contact
Amontons’s second law: Friction is proportional to the normal load
Asperities: Protrusions on surfaces
Cause of friction: interactions of asperities
Asperities: acted as either rigid or elastic springs.
A: Apparent or geometric area
Difference between static & kinetic friction
Coulomb noticed, for wooden surfaces: initial friction increased with the time the surfaces were left in stationary contact
Surfaces were covered with protuberances like bristles on a brush
When brought together the bristles interlocked, and this process became more pronounced the longer the surfaces were in contact
Coulomb used this mechanism to explain the general observation that static friction is higher than kinetic friction
Weaknesses of early theories of friction
They failed to account for the energy dissipation characteristic of friction and for frictional wear
Both of these point to asperity shearing as an important mechanism, but to establish that required two developments
A model for asperity shearing: still compatible with Amontons’s first law
Microscopes: allow examination of the surface damage produced during friction
The adhesion theory of friction
Assumed: yielding occurs at the contacting asperities until the contacting area is just sufficient to support the normal load N.
Bowden and Tabor (1950, 1964)
rN pA P: penetration hardness, a measure of the strength of the material.
Summary
Real area of contact: controlled by the deformation of asperities in response to the normal load
It explains Amontons’s first law
Equation implicitly satisfies: Second law as well, as long as the equation itself is linear in N
rN pA
Adhesion theory of friction
High compressive stress at the contact points: adhesion occurred there, welding the surfaces together at “junctions”
Accommodate slip: these junctions would have to be sheared through
Frictional force F: sum of the shear strength of the junctionss: Shear strength of the
material
To first order : independent of material, temperature, & sliding velocity
s & p: dependent on those parameters, differ between themselves by only a geometric constant
Elastic contact theory of friction
Harder materials: such as the silicates, we might expect contact to be largely elastic.
If this were true, the contact area for an asperity would obey Hertz’s solution for contact of an elastic sphere on an elastic substrate
2 3
Elastic contact theory of friction
Tabor (1964): Friction of a diamond stylus sliding on a diamond surface obeys this Equation
2 3
1 3
1sk N
Villaggio (1979)
Study the contact of surfaces: surface comprised a large number of spherically tipped elastic indenters, each of which was covered by smaller spheres, and so on
Although each sphere obeyed Hertz’s equation: an asymptotic solution in the limit of a large number of hierarchies of sphere sizes that produced a linear law
Elastic contact theory of friction
2rA k N Archard (1957)
Elastic contact theory of friction
: The closure of the surfaces under the action of a normal stress;
n: normal stress B and D: constants that scale with the elastic
constants and are otherwise determined by the topography of the surfaces.
log nB D Greenwood and Williamson (1966)
Contact of a rough surface with a flat surface in which the rough surface was described with a random distribution of asperity heights.
Closure of two elastic surfaces in contact under the action of a normal load
Certain amount of permanent closure, due to brittle fracture or plastic flow of asperity tips.
After several cycles the closure: though hysteretic, becomes completely recoverable
Contact is indeed elastic
Brown and Scholz, 1985
Closure of two elastic surfaces in contact under the action of a normal load
In more ductile rocks, such as marble, the permanent closure becomes more pronounced and flattening of asperities by plastic flow can be observed.
" 2 3 ( / )p k p E
P: hardness; : radius of curvature of the tips; E”: elastic constant K3: geometrical factor
Nature of surface topography
Spectrum from 1 m to 10 mm of a natural joint surface
Brown and Scholz (1985) made a general study of the topography of rock surfaces over the spatial bandwidth 10-
5m to 1 m.
Nature of surface topography
Spectrum from 1 m to 10 mm of a natural joint surface
A fractal surface has a power spectrum that falls off as:
: spatial frequency : between 2 and 3

Nature of surface topography
spectra (1 cm to 10 mm) of ground surfaces of various roughnesses
The corner in the spectra: grinding and occurs approximately at the dimension of the grit size of the grinding wheel
Corner frequency
Elastic spheres in contact subject to normal & shear loads
A small shear load results in slipping over an annulus surrounding a nonslipping region.
Initial frictional behavior: small displacements for roughnesses & loads
(Boitnott et al., 1992)
Extended surface by applying Mindlin’s solution to the full contact
Obtained from the measured topography of the surfaces using the method of Brown and Scholz (1985)
Slip begins at the onset of application of the shear stress
Friction curve is well fit by assuming a microscopic friction coefficient of = 0.33
Model fit the data for only the first few micrometers of slip
After that point the majority of asperity contacts are fully sliding and the initial conditions assumed in the model are no longer applicable
Other frictional interactions
(a) ploughing: (b) riding up: (c) interlocking
Ploughing A hard asperity penetrates into a softer surface: ploughs through during sliding
If p is the hardness of the softer material, a spherically tipped asperity with radius of curvature will penetrate until the radius of contact r is given by:
2N r p During sliding: asperity plough a groove of cross-
sectional area: 32 3gA r
Ploughing
Force necessary to plough this groove: of the order pAg.
Necessary to shear the junctions at the surface of the asperity: of the order πr2s
3 1 2 23
22 3F s N p N p Shearing term and is the same as in Equation:
F N s p ploughing term
Interlocking Asperities: occur on all scales and asperity interlocking is likely to be common
If the interlocking distance is greater than a critical value: Sliding occur by shearing through the interlocked asperities.
Force required for this:
aF sA s: Shear strength of the asperity Aa: its area
Wang & Scholz, 1994
Riding Up
Surfaces are initially mated at some wavelength & have irregular topography with long- wavelength asperities as the natural surfaces
Sliding commences: asperities ride up on one another,
Sliding occurs at a small angle to the direction of the applied force F & the mean plane of the surfaces
“Joint dilatancy” during sliding: since a component of the motion is normal to the mean sliding surface
21F N
Evolution of friction from initial sliding to steady-state
Evolution of friction from the initiation of slip until steady- state friction was achieved: for their relatively finely ground surfaces, required about a millimeter of slip.
Once full sliding occurs: asperity interlocking results in an increase in friction
Followed by a period of slip hardening associated with a steady increase in real contact area as the surfaces wear into each other
Wang and Scholz, 1995
Steady-state friction: achieved after sliding a characteristic distance, which depends on the initial topography of the surfaces
Characteristic distance also corresponds to the change from transient to steady state wear
Wang and Scholz, 1995
Evolution of friction from the initiation of slip until steady- state friction was achieved: for their relatively finely ground surfaces, required about a millimeter of slip.
Once full sliding occurs: asperity interlocking results in an increase in friction
Followed by a period of slip hardening associated with a steady increase in real contact area as the surfaces wear into each other.
Wang and Scholz, 1995
Experimental configurations used in friction studies
(a) triaxial compression; (b) direct shear; (c) biaxial loading: (d) rotary.
Frictional strength for a wide variety of rocks
Halloysite
Vermiculite
Illite
m o n
With the exception of several of the clay minerals, friction is independent of lithology
Jaeger and Cook, 1967
n MPa
Byerlee’s law Friction law: with very few exceptions, independent of lithology
To first order: independent of sliding velocity & roughness; for silicates, up to temperatures of 350°C
Because of its universality: use it to estimate the strength of natural faults
It holds over a very wide range of hardness & ductility, from carbonates to silicates
50 0.6 200
Data from Logan & Teufel, 1986)
At low and intermediate stresses: a mild effect of hardness on frictional strength
This effect becomes negligible at high loads
Ar increases linearly with N in all cases
Anticipated from either plastic or elastic contact theory
Data from Logan & Teufel, 1986)
Growth of Ar: accomplished in the SS/SS case by a rapid increase in the number of contact spots with normal stress
SS/LS & LS/LS: Spots did not increase greatly
Spots grew in size, more like the observations of plastic blunting of asperities in the contact of marble
600 MPa
2200 MPa
200 MPa
Temperature & ductility
is right lateral
(a) Gouge heated to 550 °C with Pc=400 MPa and PH2O=100 MPa but not sheared
100 m
(b) Sample deformed wet at 150 °C shows grain size reduction and formation of R1 Riedel shears and C surfaces.
A laboratory model of strike- slip development
R and R’: Riedel shears
P: P fractures
A later stage of deformation, in which Riedel shears have been linked by P fractures.
Characteristic geometry of C-S and C- C’ structures in a dextral shear zone
1. Most mylonites show at least one well-developed foliations at low angle to the boundary of shear zone. 2. S-foliation: S comes from French word for foliation, “schistosité”. 3. C-foliation: C comes from French word for shear, “cisaillement”.
(c) Sample deformed dry at 702 °C:
1. Gouge is pervasively sheared.
2. Grains of biotite & magnetite are flattened and cut and offset by many closely spaced R1 shears.
100 m
(d) Sample deformed wet at 600 °C:
1. Riedel shears cut the gouge at a low angle to the layer.
2. Particle size remains similar to the starting materialShear zone
Strength of sandstone surfaces sliding on a sandwich of halite at three sliding velocities
Ratio of gouge thickness to slip (T/D) versus normal stress
Slipped 30 cm
Stick-slip Behavior Stable sliding: At low confining pressure, frictional
sliding occurs as smooth, continuous motion Stick-slip: Increasing confining pressure, the motion
changes to stick-slip behavior, characterized by ”stick” intervals of no motion
Stick-slip Behavior
Variation of frictional resistance during sliding: dynamic instability occurs; sudden slip with an associated stress drop
Instability is followed by a period of no motion during which the stress is recharged, followed by another instability
All sliding occurs during the instabilities
Frictional behavior is called regular stick slip
During the latter stage the spring unloads following a line of slope -K.
Stick-slip Behavior
Tangent point B is reached: F will decrease faster with u than K, an instability occurs: (force imbalance produce an acceleration of the slider)
Beyond point C: F becomes greater than the force in the spring & the slider decelerates, coming to rest at point D (in the absence of other dissipation)
Area between the curves between B and C is just equal to that between C and D
Stick-slip phenomenon: Earthquakes are recurring slip instabilities on preexisting faults which remain stationary between earthquakes
Frictional Instabilities Condition for instabilities (slip weakening): F K
u

If slip zone is treated as an elliptical crack in an infinite medium (E, ), shear stiffness is given as:
The instability occurs when the slip zone reaches a critical length
Nucleation Length


Critical length is referred to the breakdown length, or nucleation length
In laboratory experiments: stability transition occurs at a normal stress when lc becomes larger than the test surface
Because lc varies inversely with normal stress it will become large at shallow depths, which will tend to inhibit earthquake nucleation there
Frictional Instabilities
Static friction coefficient s must be exceeded for slip to commence, during which slip is resisted by a dynamic friction d.
If s > d, unstable slip will occur Healing mechanism: For regular stick slip, there
must be a mechanism for friction to regain its stable value following the unstable motion
Critical slip distance Dc (Rabinowicz): A critical slip distance in order for friction to change from one value to another
s d n
s
Initially bare rock surfaces (solid symbols) & granular fault gouge (open symbols)
Data have been offset to s = 0.6 at 1 s and represent relative changes in static friction
Rate effects on friction: rate and state variable friction laws
Static friction increases logarithmically with hold time
Static friction with hold time
Static friction measurements: Slide- Hold-Slide experiments
Healing effect
Loading velocity before and after holding was 3 m/s
Dynamic Friction Measurements
Phenomenon known as velocity weakening
Initially bare rock surfaces (solid symbols) & granular fault gouge (open symbols)
Data have been offset to d=0.6 at 1 m/s
Dynamic coefficient of friction versus slip velocity
Dynamic Friction Measurements
Change in loading velocity
3-mm–thick layer of quartz gouge sheared under nominally dry conditions at 25 MPa normal stress
Rate- & State-Variable Friction Law
: friction for steady-state slip at velocity V0 : state variable (Ruina, 1983) a and b: empirical constants Dc: critical slip distance V: frictional slip rate
1 c
V D


These observations were fit by an empirical constitutive law by Dieterich (1979)
Rate & state dependent friction (RSF) formulation by Ruina (1983)
0 0


0 0
V D






: friction for steady-state slip at velocity V0 : state variable, average age of contact: Dc/V a and b: empirical constants Dc: critical slip distance V: frictional slip rate
Dieterich-Ruina or slowness law
In the static case, = t, Dieterich (1979a) suggested that can be
interpreted as the average age of contacts i.e., the average elapsed time since the contacts
existing at a given time were first formed Dc is the sliding distance (at a constant velocity V):
a population of contacts is destroyed and replaced by an uncorrelated set
0 0
V V VV a b V D D



cD V
=a-b (ln )
d ss
so for long hold time d
d lnt
Dieterich-Ruina or slowness law
Friction parameters a & b: always positive quantities of the order 10-2
Upon a sudden jump in the loading point velocity from V1 to V2: velocity of the slider reaches V2 at the peak of the response spike
How static and dynamic friction
measurements could be related?
Static friction: depends on the history of sliding surface increase slowly as log t
If the surfaces are in static contact under load for time t
Dynamic friction: depends on the sliding velocity increases/decreases as log V, depends on the rock type and certain other parameters such as temperature
Rate and state friction resolved the problems come from other friction laws
Rate- and State-Variable Friction Law
(a-b) > 0, velocity-strengthening behavior (stable) 1. No earthquake can nucleate in this field 2. Any earthquake propagating into this field produce there a negative stress drop 3. Rapidly terminate propagation (a-b) < 0, velocity weakening behavior (unstable)
Rate increase, a increase (direct velocity effect): This follow by an evolutionary effect involving a decrease in friction, of magnitude b
Rate- and State- Variable Friction
Law
Material property and Temperature Taking granite for example, low temperature: (a-b) < 0
high temperature: (a-b) > 0
For faults in granite (representative rock of the continental crust): not expect earthquakes to occur below a depth at which the temperature is 300.
Rate- and State- Variable Friction
Law
Faults are not simply frictional contacts of bare rock surfaces: lined with wear detritus (cataclasite of fault gouge)
(a-b) is positive when the material is poorly consolidated (a-b) decreases at elevated pressure and temperature as
the material becomes lithified Faults may have a stable region near the surface: owning
to the presence of loosely consolidated materials
Frictional stability regimes
Stability of system: effective normal stress, , K, friction parameters (a-b) & Dc
Independent of the base friction 0
Velocity-strengthening (a-b ≥ 0): system is intrinsically stable
Velocity weakening (a-b < 0): there is a Hopf bifurcation ( ) between an unstable regime & a conditionally stable one
Frictional stability regimes
Bifurcation occurs at a critical value of the effective normal stress
Higher values of normal stress: system is unstable under quasi-static loading
Normal stress less than the critical value: system is stable under quasi-static loading
Unstable under dynamic loading if subjected to a velocity jump higher than V
c
within the unstable regime
Propagate into the conditionally unstable field by dynamic loading
Propagation into a velocity- strengthening region: rapidly terminated by the negative stress drop that so results
Stability transition
Response of a spring- slider system being driven at constant load point velocity
As normal load is decreased: Transition between stick-slip & stable sliding is crossed, with a narrow region of oscillatory motion at the stability boundary
Stability transition For a 2- or 3-dimensional
case: Stiffness is inversely proportional to a length scale
If slipping region is treated as an elliptical crack
22 1 EK

E: Young’s modulus : Poisson’s ratio L: length of the slipping region
Stability transition Stability transition takes
place at a critical value of L
22 1 c

Stable sliding: occur in a nucleation stage until the slipping region grows to Lc and the instability occurs
Nucleation process is of considerable interest in earthquake prediction theory (Section 7.3.1)
Rate–state friction parameter : function of temperature
Granite & quartz gouge measured under hydrothermal conditions: elaboration of the RSF law with two state Variables (b, b1 & b2)
a-b1-b2: nearly zero or slightly positive at low T, becomes negative above 90°C & strongly positive above 350°C
Strongly positive temperature dependence of the direct velocity-dependent parameter a
High T: a regime of solution–precipitation creep
Velocity dependence for RSF
At the lowest normal stresses: velocity weakening, & hence stick slip
At higher normal stress: semibrittle regime, velocity strengthening at all rates & the sliding is stable
Stick-slip Episodic slip
Stages in the brittle–plastic transition for frictional sliding
Stability transition at 350°C: temperature of the onset of quartz plasticity
Change from abrasive to adhesive wear to occur at the lower transition boundary
Lower stability transition: of profound significance in the mechanics of faulting
Blanpied et al., 1998 Experiments on granite
Upper stability transition
Change from consolidated to unconsolidated fault gouge at shallow depth
Change in the sign of (a-b) from negative to positive and thus a stability transition at shallow depth
Upper stability transition
Granular materials: dilatancy during shear that has a positive rate dependence that overwhelms the negative rate dependence intrinsic to the grain-to-grain contact friction
This effect may be reduced considerably if shear is localized in narrow shear bands
Synoptic model of a shear zone Section 3.4
Earthquake afterslip: Frictional properties & seismic behavior of a mature crustal fault zone
Dynamics of stick slip
0, , , ( ) 0mu au F u u t t K u t
1st term: inertial force 2nd term: damping including seismic radiation 3rd term: frictional force during sliding 4th term: force exerted by the spring
Slider of mass m loaded through a spring of constant K that is extended at a load point velocity
Dynamics of stick slip 1. Assumption: No damping & friction to drop from an initial
static value s to a lower dynamic value d upon sliding 2. At the onset of sliding (t = 0): spring has been extended an
amount 0 just sufficient to overcome the static friction 3. Load point velocity: negligible compared with the average
velocity of the slider mu Ku uN F
( ) 1 cos
Nu t a u t m








Dynamics of stick slip
Slip duration: given after which static friction is reestablished and the loading cycle begins anew
Rise time (slip duration): depends only on the stiffness and mass and is independent of and N
Total slip u=2(N/K) and the particle velocities and acceleration are directly proportional to the friction drop
Corresponding force drop is F =2N and the stress drop is = 2(s-d)n
Dynamics of stick slip & precursory
stage
Stable slip must occur during nucleation prior to the instability
Nucleation of the stick- slip instability in granite
Strain records at various points along the length of the sliding surface
Precursory stage
Nucleation begins: quasi-static slip becomes fast enough to reduce the stress at point A
Propagates along the fault at a velocity much smaller than sonic
Dynamic slip begins at point B & propagates backdown the fault with a rupture velocity slightly lower than the shear velocity
Normal stress: 2.33 MPa
Precursory stage
Cessation of slip (healing) similarly propagates at near the shear velocity
Nucleation length (Lc) in this experiment was in good agreement with Equation
22 1 c
Rangely oilfield
Intensive microearthquake activity: on a fault within an oil field in the vicinity of Rangely
Activity induced by overpressurization of the field during secondary recovery operations
injected
Friction under geological conditions
Critical pressure for triggering: state of stress and frictional strength of the Weber SS
Dieterich-Ruina Law (1979)
VV d Va b V D dt D



Dieterich friction law must be coupled with a description of state evolution.
Ruina law (1983)