Chapter 2: Rock Friction Jyr-Ching Hu, Dept. Geosciences National Taiwan University
Rock Friction Scholz, Christopher H., 1998. Earthquakes and Friction Laws. Nature. 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
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 In section Plan View Cause of friction: interactions of asperities Asperities: acted as either rigid or elastic springs. A: Apparent or geometric area A r : Real contact area, responsible for friction
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 Bowden and Tabor (1950, 1964) Assumed: yielding occurs at the contacting asperities until the contacting area is just sufficient to support the normal load N. N pa r P: penetration hardness, a measure of the strength of the material.
Summary N pa r 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
Adhesion theory of friction High compressive stress at the contact points: adhesion occurred there, welding the surfaces together at junctions F sa r s: Shear strength of the material Accommodate slip: these junctions would have to be sheared through Frictional force F: sum of the shear strength of the junctions
Coefficient of friction F sa r F N s p N pa r 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 Ar k N 1 2 3 K 1 : combines the elastic constants and a geometrical factor
Elastic contact theory of friction Ar k N 1 2 3 sk N 1 1 3 F N s p Villaggio (1979) Tabor (1964): Friction of a diamond stylus sliding on a diamond surface obeys this Equation
Elastic contact theory of friction Ar k N Archard (1957) 2 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 B Dlog n Greenwood and Williamson (1966) : 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. 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 Brown and Scholz, 1985 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
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. k ( p/ E " ) 2 p 3 P: hardness; : radius of curvature of the tips; E : elastic constant K 3 : 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 10-5 m to 1 m. natural rock surfaces resembled fractional Brownian surfaces over this bandwidth, with a fractal dimension D that was a function of spatial wavelength.
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 D 5 2
Corner frequency 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
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: N 2 r p During sliding: asperity plough a groove of crosssectional area: Ag 2 r 3 3
Ploughing Force necessary to plough this groove: of the order pa g. Necessary to shear the junctions at the surface of the asperity: of the order πr 2 s Shearing term and is the same as in Equation: 3 1 2 2 F sn p2n 3 p 3 2 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. Wang & Scholz, 1994 Force required for this: F sa a s: Shear strength of the asperity A a : its area
Riding Up Surfaces are initially mated at some wavelength & have irregular topography with longwavelength asperities as the natural surfaces F 2 1 N 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
Evolution of friction from initial sliding to steady-state
Wang and Scholz, 1995 Evolution of friction from the initiation of slip until steadystate 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 steadystate 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.
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 和樂石 Vermiculite 蛭石 Illite 伊萊石
Frictional strength for a wide variety of rocks With the exception of several of the clay minerals, friction is independent of lithology Jaeger and Cook, 1967 o m n 50 0.6 n 200 MPa n 0.85 n 200 MPa n
Byerlee s law Friction law: with very few exceptions, independent of lithology n 50 0.6 n 200 MPa n 0.85 200 MPa n 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
Hardness At low and intermediate stresses: a mild effect of hardness on frictional strength This effect becomes negligible at high loads A r increases linearly with N in all cases Data from Logan & Teufel, 1986) Anticipated from either plastic or elastic contact theory
200 MPa 600 MPa 2200 MPa 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
Temperature & ductility 350 o C
Deformed granite gouge: Sense of shear is right lateral (a) Gouge heated to 550 C with Pc=400 MPa and PH 2 O=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 strikeslip 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.
Shear zone 100 m (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. (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 material
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 u K If slip zone is treated as an elliptical crack in an infinite medium (E, ), shear stiffness is given as: K E 1 The instability occurs when the slip zone reaches a critical length 2 L
l c Nucleation Length 21 EL 2 u n n KL u Critical length is referred to the breakdown length, or nucleation length In laboratory experiments: stability transition occurs at a normal stress when l c becomes larger than the test surface Because l c 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 D c (Rabinowicz): Acritical slip distance in order for friction to change from one value to another s d n D c K
Laboratory-derived friction laws Static friction with hold time 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 measurements: Slide- Hold-Slide experiments Static friction & s in slide-hold-slide experiments Loading velocity before and after holding was 3 m/s Healing effect
Dynamic Friction Measurements Dynamic coefficient of friction versus slip velocity Initially bare rock surfaces (solid symbols) & granular fault gouge (open symbols) Data have been offset to d =0.6 at 1 m/s Phenomenon known as velocity weakening
Dynamic Friction Measurements Transient and steady-state effect on friction Change in loading velocity 3-mm thick layer of quartz gouge sheared under nominally dry conditions at 25 MPa normal stress Direct effect
Rate- & State-Variable Friction Law These observations were fit by an empirical constitutive law by Dieterich (1979) Rate & state dependent friction (RSF) formulation by Ruina (1983) V V 0 0 aln bln V 0 Dc V 1 Dc : friction for steady-state slip at velocity V 0 : state variable (Ruina, 1983) a and b: empirical constants Dc: critical slip distance V: frictional slip rate
Rate- and State-Variable Friction Law Dieterich-Ruina or slowness law V V0 V, 0 aln bln V D V 1 Dc 0 c : friction for steady-state slip at velocity V 0 : state variable, average age of contact: D c /V a and b: empirical constants Dc: critical slip distance V: frictional slip rate
Dieterich-Ruina or slowness law V V 0 V V, 0 aln bln, 1 V D D In the static case, = t, 0 c c 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 D c is the sliding distance (at a constant velocity V): a population of contacts is destroyed and replaced by an uncorrelated set
Dieterich-Ruina or slowness law Friction at steady-state velocity V: ss D c V a b V 0 ( )ln V0 if d is defined as ss at velocity V, then d d =a-b d(ln V)
V In the static case, =1- reduces to t, Dc so for long hold time ds d lnt Dieterich-Ruina or slowness law b Friction parameters a & b: always positive quantities of the order 10-2 Upon a sudden jump in the loading point velocity from V 1 to V 2 : velocity of the slider reaches V 2 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 Rate increase, a increase (direct velocity effect): This follow by an evolutionary effect involving a decrease in friction, of magnitude b (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- 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) & D c 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 ( 霍普夫分歧現象 ) between an unstable regime & a conditionally stable one
Frictional stability regimes Bifurcation occurs at a critical value of the effective normal stress c KDc a b 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
Frictional stability regimes Earthquakes can nucleate only within the unstable regime Near the bifurcation, selfsustained oscillatory motion occurs Propagate into the conditionally unstable field by dynamic loading Propagation into a velocitystrengthening region: rapidly terminated by the negative stress drop that so results
Stability transition Response of a springslider 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 K E 2 21 L 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 L c 2 n b a 21 ED c Stable sliding: occur in a nucleation stage until the slipping region grows to L c 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, b 1 & b 2 )
a-b 1 -b 2 : 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 Shimamoto, 1986 Halite experiments Episodic slip Stick-slip 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
Stages in the brittle plastic transition for frictional sliding Experiments on granite Blanpied et al., 1998 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
Upper stability transition Prevent earthquakes from occurring at shallower depths 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 Slider of mass m loaded through a spring of constant K that is extended at a load point velocity mu au F u, u, t, t K( u t) 0 0 1 st term: inertial force 2 nd term: damping including seismic radiation 3 rd term: frictional force during sliding 4 th term: force exerted by the spring
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 N ut () u 1cos t K N ut () u sint Km N ut () a u cost m mu Ku un F t r K m m K
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 stickslip instability in granite Strain records at various points along the length of the sliding surface
Normal stress: 2.33 MPa 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 (L c ) in this experiment was in good agreement with Equation L c 2 n b a 21 ED c
Friction under geological conditions 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 injected injected Rangely oilfield
Friction under geological conditions Critical pressure for triggering: state of stress and frictional strength of the Weber SS
Dieterich-Ruina Law (1979) Dieterich friction law must be coupled with a description of state evolution. V V0 d V 0 aln bln ; 1 V 0 Dc dt Dc V V d aln bln ( 0 )(1 ) 0 V V dt 0
Ruina law (1983) V V 0 0 aln bln V 0 Dc d V V ln( ) dt D D c c