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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 91, NO. B1, PAGES 521-530, JANUARY 10, 1986

Dynamic Motion of a Single Degree of Freedom SystemFollowing a Rate and State Dependent Friction Law

JAMESR. RICE AND SIMON T. TSE1

Divisionof Applied Sciences, arvard University, Cambridge,MassachusettsSequences f dynamic nstabilitiesare analyzed or a singledegreeof freedom elastic systemwhichslidesalong a surfacehaving frictional resistancedependingon slip rate and slip rate history, in themanner of Dieterich, Ruina and others.The system s represented s a rigid block in contactwith a fixedsurfaceand having a spring attached to it whose oppositeend is forced to move at a uniform slow speed.The resulting"stick-slip"motions are well understood n the classical ase or which there is an abruptdrop from "static" to "sliding" rictional resistance.We analyze them here on the basisof more accuratefrictional constitutivemodels. The problem has two time scales,an inertial scale set by the naturaloscillation period T of the analogous rictionlesssystem as T/2:rr and a state relaxation scale L/Voccurring n evolution, over a characteristicslip distanceL, of frictional stressu towards a "steady state"value uss(v) ssociatedwith slip speed V. We show that u _ uss(v)during motions for which accelerationa satisfies L/V 2

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522 RICE AND TSE: DYNAMIC MOTION WITH RATE AND STATE DEPENDENT FRICTION

I &80-=apIII - r Iunit base area(a)

(b)Fig. 1. (a) Single degree of freedom elastic system. A block ofmass m and unit base area slides distance 5 with frictional stress r.

The load point movesa distance5o at an imposedspeed Vo, stressingthe block through the springwith stiffness . (b) Stress esponse f theblock to a steadily moving load point for the classicalstatic-kineticfriction law. Initially stationary, the block starts to move when thespring orce k5o is equal to the static riction zs,and it then slideswiththe constantkinetic friction r k. The arrows show the directionsof thepaths along which the frictionalstress nd spring orceevolve.

In slip motion at a fixed slip rate and normal stress, he statevariables evolve towards steady state values 0iss satisfyingGi(V, a,, 0 ss,0_, .. , 0, ) = 0 such that the shear strengthevolves towards a steady state value :s(V)corresponding othe fixed speedand normal stress we do not further display a,as a variable).A fairly comprehensivenonlinear analysis of quasi-staticslip motion of a driven spring-blocksystem Figure la) and itspossible nstabilitieshas been given by Gu et al. [1984], Riceand Gu [1983] and Blanpied et al. [1984] for specificone andtwo state variable laws (as special casesof the class (1)). Theone state variable law was proposedby Ruina [1980, 1983] asan approximation to a law proposed by Dieterich [1979a, b,1981] and is cited later here; the two state variable law was ofsimilar form but provided a closer fit to the observed elax-ations in Ruina's experiments.Also, extensivenumerical simu-lations of quasi-staticmotion and its instability are given byDieterich [1980] and G. M. Mavko (unpublishedpaper, 1984).From a linearized stability analysis hat included nertia, Riceand Ruina [1983] showed that d?(V)/dV < 0 from the aboveformulation is a necessary nd sufficientcondition for steadystate sliding to become unstable at sufficiently educed elasticstiffness or surfaces n which a step changeof V, from that ofa previous steady state to a new fixed value, is followed by amontonic approach of : to the new :s(V). Conversely,drs*(V)/dV> 0 on surfaceswith such monotonic approach atfixed V implies stability of steady sliding to small pertur-bations no matter what the elastic stiffness.However, as slipmotion becomes unstable the slip velocity departs from thesteady state value and the linearized analysisbreaks down.

The aim of this paper is to examine the full nonlinear de-scription of motion, including the effect of inertia, and to de-termine when the quasi-static analysis breaks down and howthe inclusion of inertia affects he subsequent lip motion forthe spring-block with imposed load-point motion. We reportnumerical simulations of several cycles of slip motion usingdifferent forms of a one state variable version of the constitu-tive law (1), and discusscharacteristic time scales,dimension-less measures of their importance, and possible approxi-mations in numerical calculation.

We do not suggest hat the singledegree of freedom systemanalyzed can be made to correspondclosely to an actual fault.Rather, the analysis is intended to provide an understandingof stress esponseaccording o the rate and state dependentfriction laws during inertia controlled motions, in the hopethat this understandingwill be in part transferable to morerealsitic fault models involving slip between deformable con-tinua.

RATE AND STATE DEPENDENT LAWSSpecific orms of the general slip rate and slip rate history

dependent aw (1) have been proposedand discussed y Die-terich [1979a, 1980, 1981], Ruina [1980, 1983], Gu eta!.[1984], and Tullis and Weeks [1985]. Based on the a prioriassumption (or approximation) that one state variable 0 in aconstitutive aw as in equations 1) suffices o characterize hesurface state, Rice [1983] recently outlined a procedure forconstructing constitutive relations on the basis of experi-mental features reported by a variety of workers (see refer-encesabove).These are as follows:1. The effect of a suddenly imposed (i.e., at fixed state)increaseor decrease f V is to respectivelyncreaseor decrease:. Thus (&/V)o > 0. We note that when there is a single 0,this derivative can be expressed s a function of : and V, say,as (&/V)o = A/V where A = A0:, V) > 0 is obtained experi-mentally from "velocity ump" tests.A - const is suggestednseveral references cited above.

2. In slip motion at a fixed slip rate V, z evolves towards asteady state value corresponding o that slip rate V, i.e., ?(V).3. The evolution of z towards z(V) at constant V ischaracterized by an approximately exponential decay over acharacteristic slip distance L = L(V) such that &r/dr5=-(z-z)/L in slip 5 at constant V. (It is generally understoodthat a superposition of two or more such decay processes,each with their own L, better fits data and must be included toexplain some phenomena,but here for compatibility with theassumedone state variable form we must assume hat only asingle decay process s active.) Thus, since d: = (&/O)v dO nslip at constant V, we are assuming hatao/at = - [v/(v)][, - eXv)]

As Rice [1983] noted, although the last equation is motivatedas an idealization of approach to steady state at constant V,the expressionhat it gives or dO/dt n terms of V and 0 (: and(&/O)v can be regarded as functions of V and 0) must applyduring general motions with variable V. This is because heone state variable version of (1) requires that dO/dt be deter-mined only by 0 and V and not, e.g., by dV/dt or higherderivatives of V.A procedure for assigningnumerical values to 0 has not yetbeen identified. This can be done with some degree of arbi-trariness, associating0 values with members of the one param-eter family of curves in a , V plane of which each representconstant state response, .e., instantaneous hangesof : and V

[Ruina, 1980, 1983]. However, the exercise s unnecessary.We

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RICE AND TSE: DYNAMIC MOTION WITH RATE AND STATE DEPENDENT FRICTION 523

can rewrite the classof constitutiverelations ust described or,as recognized lso by Ruina [1983], all relations hat involve asinglestate variable)without explicit reference o 0. To do so,note that

('r/?O)v O= du - (u/?V)odV= du - (A/V) dVThus, the constitutive laws discussed n items 1 to 3 above areequivalent o saying hat the first order differentialequation

du A(u, V) dV V= -- [u -- uss(v)] (2a)dt V dt L(V)links historiesof V and u to one another. For example, givenV and u at t = 0 (i.e., given the initial state), the equationassociates stresshistory u(t) with any given velocity historyV(t). The experimentalquantitiesor functions equired are theinstantaneous viscosity parameter A = c9u/c9(ln ) at fixedstate, he steadystate strengthussand the decay slip distanceL. Sometimes he systemmay undergo a motion that is soclose to steady state conditions hat equation (2a) can be re-placedby

= ss(v) (2b)Various forms for A, ussand L can be chosen. The one state

variable law of Ruina [1980, 1983] mentioned above corre-sponds o (again, for fixed an)A = const L = const u's= u, -- (B -- A) In (V/V,)

(3a)whereB is a constant nd V, is an arbitraryreference elocityat which the steadystate strengthwould be u,. The law ismore familiar as Ruina first wrote it,

u = F(V, O)= u, + A In (V/V,) + 0 (3b)dO/dt= G(V, O)= --(V/L)[O + BIn (V/V,)] (3c)

but (2a) and (3a) are fully equivalent to (3b) and (3c). Fittingthis law to Dieterich's [1981] experiments (under constantnormal stress an of 100 bars and at room temperature) onfrictional sliding of intact Westerly granite on a gouge ayer ofthe same material with variable layer thickness,gouge particlesize and degree of surfaceroughness,Gu et al. [1984] foundthat for rough surfacesA/an = 0.006 to 0.008, B/A = 1.12 to1.14, L=40 to 50 #m/s and u,/an=0.6 for V,=I #m/s.When fitted to the other surface roughness the values areslightly different but they are of the same order of magnitude.One can see hat both A and B are small ractions f u, (about1%). This law can therefore be viewed as a small deviationfrom the classicalkinetic friction concept. The state or historydependence ives an effect like "static" friction in certain cir-cumstances e.g., Dieterich, 1980; Ruina, 1983].Experimental observation over a wide range of V suggeststhat the steady state stress uss(v) tends to level out to a re-sidual evel at higher slip speeds above 0.1 mm/s in the resultsof Dieterich [1978] on Westerly granite at room temperature).This was also observed by Scholz and Engelder [1976]. It isfound that uss(v) s, approximately, nverselyproportional tothe logarithmof slip velocity rom about 10 ' mm/s to 10-mm/s. Hence, one can replace he last entry in equations 3a)byuss(v) u, + (B - A) In (V,/V + e n) (4)

and use this in (2a). Here, n is a positive number chosen de-pendingon the choiceof reference peedV,) in sucha waythat uss(v) educes o a residual level for slip velocity higherthan a certainvalue.For example,f V, is equal o 30 mm/yr

(typical latevelocity) nd uss(v) eginso leveloff after 10- xmm/s, n can be chosenas 10.The function uss(v) n equations 3a) and (4) has the proper-ty that duss(v)/dV< 0 for the valuesof B and A quoted. Re-versal of the inequality has been observed n experiments athigh temperatures Steskyet al., 1974; Stesky,1975, 1978] andon surfaces hat have undergone elatively little total slip [Die-terich, 1981]. Based on the stability analysesmentioned in theintroduction, for duss(v)/dV< 0 steadystate sliding s unstablefor small enoughstiffness r large enoughperturbationswhilefor duss(v)/dV> 0 steady state sliding is always stable. Tseand Rice [1984] suggestedhe depth cut-off of crustal earth-quake activity (e.g., between approximately 10 and 15 kmalong the San Andreas fault) can be understood n terms ofthe variation of the friction responsewith depth (increasingtemperature) rom a regime with duss(v)/dV 0 to one withduss(v)/dV>O. Mavko [1980, unpublished paper, 1984]showed hat a change n sign of duss(v)/dV t a certain depthtends o concentrate nstability, n quasi-static trike slip slid-ing between wo elasticplates, nto the shallower egion withduSS/dV O.Rice [1983] also discussed case for which V(cu/cV)oA = 0, with 0= const. In this case integration at fixed 0shows that the variation of u with V is u = W V where theintegration"constant"W is a function of state 0 and, for thatmatter, could be used tself as the variable to be identified as 0.SinceA is typically of order 0.01u,0would have to be of order0.01. In that caseexperimentaldata fitted over several ordersof magnituden V to V(c9u/OV)oA = const= 0.01u,, eadingto the logarithmic direct velocity dependencen (3a), could befitted comparably well to V(&/3V)o = A = czu , 0.01u, eadingto a power form u oc V for the direct velocity dependence,sinceu differsonly modestly rom u, over a wide rangeof V.The latter form remains well defined at V = 0, although suchsimulationsas we report here never nvolve V so close o zerothat the awkwardbehaviorof In (V/V,) very near that pointshowsup.The formulation in equation (2a) can also be applied withthe quotient form

u = anC(O)/f(V)for F(V, O) favored by Dieterich [1979a, 1980, 1981]. In hiscase

C(O)= Cl + c2 loglo (1 + c30)f(V) = 1 +f2 10go 1 +f3/V)

where c, c2, c3, 2, f3 are constantsand 0 is interpretedas aneffective contact lifetime whose steady state value is dc V. Forthis case,A = -uV[df(V)/dV]/f(V) uss(v)= anC(dc/V)/f(V)and L = dc in equation (2a), although the presumed ex-ponential evolution, in the form exp (-i/L), of towardsuss(v) t fixed V, implicit in equation (2a), is different n detailfrom results of the different evolution laws for 0 adopted byDieterich at different times in his work [see Dieterich, 1981].

ELASTIC LOAD SYSTEM NTERACTION, INERTIAL EFFECTS,AND CHARACTERISTIC TIME SCALES

For the spring-block arrangement of Figure la with unitbase area, let k be the spring constant and rn the mass per unitbase area. We write rn = k(T/2rO where T is the vibrationperiod of the analogousreelyslippingsystem.Since he forcesacting on the block are the spring force k(g0- g) and the

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RICE AND TSE: DYNAMIC MOTION WITH RATE AND STATE DEPENDENT FRICTION 525

Po= constant

-2

-4PO constant

I , Isteady state lineTSS(V)T'*-(B-A)nI

o 2

Po>O

6Po:0T:TssV)+&LB

Fig. 2. Trajectoriesf the blockmotionaccordingo quasi-staticnalysisor a stationaryoad point n the normal-izedstress,z - z,)/A, versusogarithm f velocity,n (V/V,), phase iagramor theonestate ariableaw (3) for thecasekL/A = 0.8 and (B- A)/A = 1. The thick solid ine is the steady tate ine. The curvePo = 0 separatesnstable rbits(Po > 0) from stableorbits Po < 0).

However, for a rupture involving slip over a region whichgrowsat the shearwave speed to diameterD and then stopsgrowing, we might approximately associate he period T ofthe oscillator with 2D/. That is, T/2, the portion of acom-

plete cyclewith positivevelocity, s the sum of the time D/2over which rupture grows plus the same time D/2 for arrestphases to spread backwards from the stopped edge of therupture. Thus, for D = 9 km and = 3.5 km/s, T _ 5 s.Stress, (z-%)/A'. 5,1 o/;,

_1o-30 -'(B-A)n -30-0 = (o) -0

-.0-5 0 5 0 5 go g5 30 35 40:st, (-.)/A ].(v/v.) 250 ] ] I ]dynamic,/ 200-10 [ I[ stress/

I dropI 5o-20 100-30

n, 50-0 4 q4(b i0 S0 100 1S0 Z00 ZS0

Slip,

Stress, (z-%)/A/ //

//

//

//

//

/

//

//

///

//

///

//

///

//

cyclee--time,

50' ' ' , ' I i

0 100Slip,

//

//

///

//

///

, i I150

Time, V.t/L

_.[ (d). : : : : : ; : : , : ; : : 0 50 1)0 150Time, V.t/L

Fig. 3. Numerical imulation f slipmotion or the one statevariableaw (3) (z(V)= z, -(B- A) In (V/V,)) withkL/A = 0.8, (B - A)/A -- 1, Vo - V, - 30 mm/yr, L/Vo 2.7 yr and T - 5 s. The blockand load point nitiallyslide nsteadystateat velocityV = Vo= V,. The load point speeds then suddenlyncreasedo 1.5Vo nd subsequentlyeldconstant.Dashed and solid ines are results rom quasi-static nd dynamiccalculations, espectively.a) Plot of dimension-less tressz - z,)/A versusogarithm f velocity,n (V/V,). (b) Plot of dimensionlesstress ersus imensionlesslip,J/L.(c)Plot of dimensionlesstressersus imensionlessime, V,t/L.(d)Plot of dimensionlesslipversus imensionlessime.

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526 RICE AND TSE' DYNAMIC MOTION WITH RATE AND STATE DEPENDENT FRICTION

lO

-lO

-20

-30

lO

-lO

-20

-30

Stress, (-%)/A10

:. 5,1 steadytateine,,%= .t. S(v).t. 0/ __ .2 +{B-A)n{--+'0)_,.... -Z0

(a)::::::::::::::::::::::::::::::::::::::::::::::::::::-10 - 0 10 1 20 2 30 3 40Stress,'-'r.)/A ln(V/V.)

Stress, (-%)/A

-30: :850 Slip, 6/L

i iiii ii/I /I/ !(c)

: : :

ii///!lOO 150

Time, V.t/L

1(8o-8)} I I dynamic' 3' I\\ T stress

dynamicslip(b): : : : I : : : : ', : : : : I , , , , I : : : :o 5o oo 5o zoo 25o

Slip, 5/L

200

150

100

50

..[ - (d): : : : : : : :0 50 100 150

Time, V.t/LFig. 4. Numerical imulation f the slipmotion or the onestatevariableaw (4) with ss= , + (B- A) In (V,/V+ e- o).The parametersnd nitialcondition re denticalo the casen Figure3. Dashed ndsolid ines reresultsromquasi-static nd dynamic analyses, espectively.a) Plot of dimensionlesstress ersusdimensionlesslip, 6/L. (c) Plot ofdimensionlesstress ersus imensionlessime, V,t/L. d)Plot of dimensionlesslipversus imensionlessime.

QUASI-STATIC MOTIONSIn this limit T/2

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RICE AND TSE: DYNAMIC MOTION WITH RATE AND STATE DEPENDENT FRICTION 527

-lO

-20

-30

-lO

-20

-30

Stress,10 10

0 0

steadytateine -10

, . , . , . , . , . , . , . , . , . r. . ,

-10 -5 0 5 10 15 20 25 30 35 40Stress,r-r,)/A In(V/V,) 250, (8o-8) 2oo

II \ J l \ stress150

lOO

I 50b), : : : : ', .... : : : : : : : : ; : : ; : ', 00 50 100 150 200 1250

Slip, 5/L

Stress, (t-t.)/A

o

cycletime

Slip, 6/L

o 50

(c)lOO

' I1,50

Time, V.t/L

(d): : :

loo: : : I

150Time, V.t/L

Fig. 5. Numerical simulation of the slip motion for the one state variable law (3) with limiting speed approximation(see text for details). The parameters and initial condition are identical to the case n Figure 3. For slip speed ower thanlimiting speedVL (i.e., n (VL/V,)= 21.6), he solution s found rom quasi-static nalysis.When the slip speed eacheshelimiting speed, the block is allowed to slide at this speed until the sliding stress s balanced by the spring force. Then,quasi-static nalysiss resumed.a) Plot of dimensionlesstress z - z,)/A versusogarithmof velocity, n (V/V,). (b) Plotof dimensionlesstress ersus imensionlesslip, i/L. (c) Plot of dimensionlesstress ersus imensionlessime, V,t/L. (d)Plot of dimensionless lip versusdimensionlessime.difficult, the full equations (5a) and (2a) must again be used.We find in numerical solutions that the high block speed isslowed down substantially or "arrested" within a short slipdistance. This implies that the "arrest" process is achievedunder a nearly constant or "frozen" state. After the block isarrested, the slip speed becomes low enough so that quasi-static analysis applies again.

A numerical simulation of the slip motion described aboveis performed for the case kL/A = 0.8, (B -- A)/A = 1, imposedload point velocityVo= V, = 30 mm/yr, L/Vo = 2.7 yearsandT- 5 s. The block is initially sliding in a steady state con-dition with constant load point velocity Vo such that V =Vo= V, and rss= r,. The slip motion is then made unstablequasi-staticallyby suddenly ncreasing he load point velocityto a new constant value 1.5Vo.The resultsare shown n Figure3. In the early stage of the slip motion, the block responds othe sudden ncreaseof the load point velocity by a growingoscillation. However, the slip speed s still small so that quasi-static analysis remains valid. Because of this and compu-tational efficiency, the system of equations is solved quasi-statically in the low speedrange and dynamically in the highspeed ange.In the calculations performed here, the quasi-static calcula-tion is switched over to a dynamic calculation when VT/2rLis greater than a prescribedsmall number (5 x 10 4 is usedhere) and vice versa. The dashed and solid lines in Figure 3show results from quasi-staticand dynamic analyses, espec-tively. As shown in the plot of shear stressversusdisplacementin Figure 3b, during the initial stageof the dynamic range, the

excess f spring orce k(Jo -- J) over the sliding riction r accel-erates the block and the sliding stressevolves apidly towardsa steady state value corresponding o the current slip rate V(line 1-2). The block then continues o slide under a nearlysteady state condition until further shortening of spring isprohibited (line 2-3). At this point, the block is "arrested"within an extremely short slip distance (line 3-4). It can beseen from Figure 3a that this arrest occurs at a frozen state.The responseshould be compared to Figure lb for the classi-cal friction law. At the end of the arrest, the spring force andthe sliding friction are equal and the speed s low enough sothat quasi-static calculation is resumed. During this quasi-static stage (line 4-5), the sliding stress estrengthens o a peakvalue at a nominally stationary contact. After the peak, thefriction stressdecreasesapidly so that the block is acceleratedand once again the system enters the dynamic range at highspeed.The result shows that the system exhibits a limit cycle.The stress variations with slip and time during the cycle areshown in Figures 3b and 3c, respectively.Although the cyclesare repeated exactly, the stressbuild up after a slip event is notstrictly linear with time (Figure 3c) and the system is nom-inally stationary (effectively ocked) for most of the cycle time(Figure 3d) with acceleratingslip preceding he next event.Experiments on rocks by Tullis and Weeks [1985] confirmthat arrest after instability occurs at nearly frozen state, repre-sented by a measured r versus n V line analogous to line 3-4in Figures 3 and 4.An identical numerical simulation as described above has

been carried out with the friction law (4). In this case, the

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528 RICE AND TSE' DYNAMIC MOTION WITH RATE AND STATE DEPENDENT FRICTION

-lO

-2o

-3o

-4o

Stress, r- r.) / A0

-10

3 -20feody stote line

/ rss(v)--r.- q (o) -40

-10-5 0 5 10 15 20 25 30 35 40In(V/V.)

Stress,r- %)/A

time0 50 100 150

Time, V. t/L

-10

-2O

-3O

-4O

_ Stress, r- r.)/A

3

(b)

(8o-8) \

d, ,,, 1, ,, , I, , ,, 1,,, , I, ,,,I

25O

2OO

150

100

5O

Slip,8/L

1(d)

0 50 1 O0 150 200 250 0 50 1 O0 150Slip,8/L 'nme,V.t/L

Fig. 6. Numerical simulationof the slip motion for the one statevariable aw (3) with limiting speedapproximationincludingdynamicovershoot see ext for details).The parameters nd initial conditionare identical o the case n Figure3. For slipspeedower han imiting peed . i.e., n (V/V,) = 21.6), hesolutions found romquasi-staticnalysis. henthe slipspeedeacheshe imiting peed,heblock s allowedo slide t thisspeed ntil he spfing'Iorces loweredo thedesiredevel.Then,quasi-staticnalysiss resumed.a) Plot of dimensionlesstress7 r,)/A versusogarithm f velocity,In (V/V,). (b) Plot of dimensionlesstress ersus imensionlesslip,cS/L.c) Plot of dimensionlesstress ersus imension-less ime,V,t/L. (d) Plot of dimensionlesslipversus imensionlessime.

steadystate stress ss(V)decreasesinearly with In (V/V,) atlow speeds nd reduces o residual evel at high speeds here, nin equations 4) is taken to be 10 with V, = 30 mm/yr so thatthe steady state stressbecomes elatively constant for speedshigher than 10- mm/s). The resultsare plotted n Figure 4which show identical featuresas describedabove except thatthe dynamicstress rop, dynamicslip, total strengthdrop andcycle time are expectedlysmaller.As is seen n Figures 3a and 4a, the slidingstress changesrapidly with the slip rate to the steady state value during theaccelerationstage.This seems o take place over a narrow sliprate range (on the logarithmic scale)as compared o the wholeslip rate range calculated. One may ask if this could be ap-proximated by the following procedure:As the slip rate of anunstableslip increases o a limiting speed V., he quasi-staticassumptionbecomesnvalid. The block is then artificially con-strained to slide at this limiting speed such that the stressevolves towards the steady state value. The assumption ofinstability continuesuntil the spring orce s reduced o equalto the sliding stress,after which quasi-static equilibrium isreasserted.This approximation procedure has been adoptedby Dieterich [1981] and G. M. Mavko (unpublishedpaper,1984) in their numerical simulations of experiments and tec-tonic earthquake nstability models respectively. he limitingspeed VL can be estimated from an energy balance conditionsuch that the potential energy released from the unloading

spring is used to increase the kinetic energy of the slidingblock, i.e.,Ar2/2k= mVL2/2 (8a)

or

V = (2r/T)(Ar/k) (80)where T is the period of natural vibration and Ar is the dy-namic stressdrop. This dynamic stressdrop can be estimatedas

AT= q(VO- ss(vo (9)where q(V.)enoteshestressredictedlong hequasi-statictrajectory at speedV..Hence equations 80) and (9) becomeanimplicit equation for V.but are easy to solve teratively sinceAT is a slowly varying function of V. When the case n Figure3 is analyzed n this way (seeFigure 5a for the full quasi-statictrajectories), e find V = 2.40 m/s. Hence n (Vl/ V,)= 21.6and this is very close to the maximum speed attained in thefull dynamiccalculation Figure 3a shows n (VL/V,) = 21.6).Aquasi-staticnumerical calculation for the same elastic systemwith the imitingspeed aken to be such hat In (V/V,) = 21.6has been performed.The resultsare plotted in Figure 5. Com-paring Figure 5 with Figure 3 shows that this procedure ap-proximates the response reasonably well in the acceleration

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RICE AND TSE: DYNAMIC MOTION WITH RATE Ai',tD STATE DEPENDENT FRICTION 529

stage but not in the dynamic overshoot stage. The latter isexpected ecause he procedure mplicitly neglects he dynam-ic overshoot phenomenon.Therefore, in the limiting speedapproximation, he total strengthdrop, dynamicslip and cycletime are generally underestimatedat least for this one degreeof freedomsystem)while the dynamicstressdrop is the same.It is also possible o incorporate dynamic overshooteffectsdirectly into quasi-staticcalculationswith the limiting speedprocedure.Once the quasi-static nalysissuggestshat V hasreached VL, calculatedas above, we suspenduse of equation(5b) and let slip continuear rate V until the spring orce F hasreduced o whatever evel below zss(VL)s desired for example,to make the overshoot rs'(V)- F a certain fraction of the"dynamic" tress rop rq(VD- r"(VD).During his rapid slipat V, r evolves apidly to r's(VL)and the state evolves o thecorrespondingsteady state value. Once the spring force hasbeen lowered to the desired level, we instantaneously reducevelocity (i.e., at constant state, along line 3-4 in Figure 6a), tomake fall from zs'(V) o the spring force, and then resumequasi-staticcalculationsusing equation (5b). Figure 6 showsthe result of this procedure when the overshoot s chosen as90% of the dynamic stressdrop. Comparing Figure 6 andFigure 3, this modification of the quasi-static imiting speedprocedure predicts the dynamic motion remarkably well.Under crustal conditionswhere some energy s lost to seismicradiation, the dynamic overshoot s expected o be smallerthan the virtually completeovershootanalyzed here.

CONCLUSIONS AND DISCUSSIONIn summary, we have shown that two different time scalesexist for single degree of freedom elastic systemssliding ac-cording to rate and state dependent friction. For low slipspeeds, he state relaxation time scale L/V dominates and aquasi-staticanalysis s valid. For higher slip speedsoccurringalong unstable rajectoriesof the quasi-staticanalysis, he timescale characterizedby the period T of natural vibration be-comesessentialand full dynamic effectshave to be considered.Slow quasi-static slip motion that is developing towards insta-bility is characterized by an unrelaxed state which continu-ously diverges from steady state. During unstable slip, thefrictional stressdecreaseswith slip faster than the spring forcedoes, and the excessspring force accelerates he block tohigher speeds,which allows the state to evolve rapidly to-wards a steady state condition. The unstable nertia limitedslip motion occurs n nearly steady state conditions, .e., z -z"(V), until the block overshootsdynamically. Motion is ar-rested apidly such hat final velocity eductionoccursunder anearly frozen state as the friction force reduces o the springforce. Quasi-static analysis s found to be appropriate whenT/2n

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Gu, J.-C., Frictional resistance o accelerating slip, Pure Appl. Geo-phys.,122, 662-679, 1984.Gu, J.-C., J. R. Rice, A. L. Ruina, and S. T. Tse, Slip motion andstability of a singledegree of freedom elastic systemwith rate andstate dependent riction, J. Mech. Phys. Solids,32, 167-196, 1984.Ida, Y., Cohesive orce across he tip of a longitudinal shear crack andGriffith's specific surface energy, J. Geophys.Res., 77, 3796-3805,1972.Kostrov, B. V., Self-similar problems of propagation of shear cracks(in Russian),J. Appl. Math. Mech., 28, 1077-1087, 1964.Li, V. C., and J. R. Rice, Preseismic upture progressionand greatearthquake instabilities at plate boundaries, J. Geophys.Res., 88,4231-4246, 1983.Mavko, G. M., Simulation of creep events and earthquakes on aspatially variable model (abstract),Eos Trans. AGU, 61, 1120, 1980.Palmer, A. C., and J. R. Rice, The growth of slip surfaces n theprogressive failure of overconsolidated clay slopes, Proc. R. Soc.London, Ser. A, 332, 527, 1973.Rice, J. R., Constitutive relations for fault slip and earthquake insta-bilities, Pure Appl. Geophys., 21, 443-475, 1983.Rice, J. R., and J.-C. Gu, Earthquake aftereffectsand triggered seismicphenomena,Pure Appl. Geophys., 21, 187-219, 1983.Rice, J. R., and A. L. Ruina, Stability of steady frictional slipping, J.Appl. Mech., 50, 343-349, 1983.Ruina, A. L., Friction laws and instabilities:A quasistaticanalysisofsome dry frictional behavior, Ph.D. thesis, Brown Univ., Provi-dence, R. I., 1980.Ruina, A. L., Slip instability and state variable friction laws, J. Geo-phys.Res.,88, 10,359-10,370,1983.Ruina, A. L., Constitutive relations for frictional slip, in MechanicsofGeomaterials, dited by Z. P. Bazant, John Wiley, New York, 1984.Scholz,C. H., and J. T. Engelder,The role of asperity ndentation and

ploughing n rock friction, , Asperitycreepand stick-slip,nt. J.Rock Mech. Min. Sci. Geomech.Abstr., 13, 149-154, 1976.Stesky,R. M., The mechanical ehaviorof faulted ock at high tem-perature nd pressure, h.D. thesis,Mass. nst. of Technol.,Cam-bridge, 1975.Stesky,R. M., Mechanisms f high temperaturerictionalsliding nWesterlygranite,Can.J. Earth Sci.,15, 361-375, 1978.Stesky,R. M., W. F. Brace,D. K. Riley,and P.-Y. F. Robin,Frictionin faulted rock at high temperature nd pressure, ectonophysics,23, 177-203, 1974.Stuart, W. D., Strain softening rior to two-dimensionaltrike slipearthquakes,. Geophys. es., 4, 1063-1070,1979a.Stuart,W. D., Strain softeningnstabilitymodel or the San Fernandoearthquake,Science, 03, 907-910, 1979b.Stuart, W. D., and G. M. Mavko, Earthquake nstability on a strikeslip ault,J. Geophys. es., 4, 2153-2160,1979.Tse, S. T., and J. R. Rice, Stick-slipconfinemento upper crust bytemperaturedependent rictional constitutive esponseabstract),Eos Trans. AGU, 65, 993, 1984.Tullis, T. E., and J. D. Weeks,Constitutivebehaviorand stability offrictionalslidingof granite,paper presented t the 5th MauriceEwingSymposium n EarthquakeSourceMechanics, GU, Harri-man, N.Y., May 20-23, 1985.J. R. Rice, Division of Applied Sciences, ierce Hall, Harvard Uni-versity,Cambridge, MA 02138.S. T. Tse, AT&T Bell Laboratories,Murray Hill, NJ 07974.

(Received ebruary19, 1985;revised October 2, 1985;acceptedOctober 3, 1985.)