Downloaded by guest on September 30, 2021 omo aeadsaefito stesons a ie by given accepted law widely slowness the most is The friction 7). rate-and-state 6, of (3, form studies laboratory explain to or (4). direction molecules averaged well constituent the as the locally field, of describes velocity orientation that a field by director orien- described a of is as amount phase certain This a order. possess tational does have but not order, par- does positional any phase any in crystal remain liquid nematic they the do orientation; nor ticular neither position molecules average the specific ori- phase, a and liquid occupy the position strong in has molecular whereas material phase fixed, the a entation solid keep of The that phases. phase forces liquid a intermolecular and is solid crystal the liquid between A fluids. a non-Newtonian in flow shear (3). a fluid model a to widely as practice is material standard and granular geologic gouge is fault a It as of (2). known case recognized is the material In granular surfaces. this of the fault, development between sliding the media two in granular between a resulting friction granulation, Dry also generates friction. can surfaces model sliding fluid for a used justified however, is be oil surfaces; of sliding layer two a lubricate when to example, to for applicable occurs, directly which introduces is friction, that model wet crystal fluid-based liquid A a frictional process. be the relaxation to be a fluid to the take take we We which resistance. stress, shear resulting the mine rnecefiin ffito trfrnevelocity reference at friction of coefficient erence where F in variable faulting state the to analogous role friction. a rate-and-state plays field channel. director the within The resis- localized of in is Strain increase sliding relaxation. rapid by to a followed undergoes response tance fluid In the plane, system. moving top function the the viscosity of of behavior choice the specific determines the the orientation; of function field a as director changes that viscosity nonlinear a field introduce we director field velocity a the of the with order acting orientational the by charac- terized fluids, non-Newtonian and studied anisotropic well of are examples they sys- interesting states; in between phenomena switch physical channel rapidly Nematic of that a sliding. variety tems wide frictional in a model crystal model crystals to liquid liquid planes nematic parallel a two use between We of behavior surface. the frictional to approach a empirical an is friction Rate-and-state of Departments Cheng A. H. C. friction for model liquid-crystal A 7930–7935 hcns ewe w oi lcs h lcssieps each past velocity slide slip blocks prescribed prescribed The a of blocks. at layer solid fluid other a two consider orienta- between We the in field. thickness with director friction varies the fluid for of the model tion of properties liquid-crystal physical nematic the a which present we paper otiue yD .Trot,Nvme 7 2007 27, November Turcotte, L. D. by Contributed eea miia aeadsaefito ashv enwritten been have laws friction rate-and-state empirical Several anisotropic of examples studied well are crystals liquid Nematic ern n epyisbtaepol nesod() nthis In (1). understood poorly are engi- but in applications geophysics of and range neering wide a have rheologies rictional µ u = rheology ssi velocity, slip is µ 0 PNAS + ∗ ahmtc and Mathematics a ∗ ln .H Kellogg H. L. , ue1 2008 , 10 June u u 0 µ + scefiin ffriction, of coefficient is b ln † elg,Uiest fClfri,Dvs A95616 CA Davis, California, of University Geology, u †‡ ( u x .Shkoller S. , L o.105 vol. 0 u , θ t 0 omdlfitoa sliding, frictional model To ). h betv st deter- to is objective The . , d dt θ o 23 no. = ∗ n .L Turcotte L. D. and , 1 − µ d u 0 θ L ( 0 u x steref- the is , , , θ t ) sthe is inter- [1] tpices ntesi eoiyfrom velocity slip the in increase step a parameters. h nl ftedrco field director the of angle the ofceticessisatnosyt h value the to instantaneously increases coefficient director simulate. function we that the of choice The tt variable, state hre nraei rcincag safnto fvlct,but velocity, of function a as when change traction in increase sharper a coefficient relaxation the Reducing friction. state director The channel. the of center field the near localized is strain and fol- director resistance, The in relaxation. a increase by rapid lowed a undergoes fluid the increased, osraino oetmrqie that requires momentum of Conservation to back relaxation resulting state. the equilibrium and the velocity in changes with ated function the of choice the restricts but behavior, of iiu viscosity minimum 08b h ainlAaeyo cecso h USA the of Sciences of Academy National The by 2008 © [email protected]. h nlvlegvnby given value final the in hra h aiu viscosity maximum the whereas tion, h uhr elr ocnito interest. paper. ‡ of the conflict wrote D.L.T. no and declare S.S., authors L.H.K., The C.H.A.C., L.H.K., and C.H.A.C., research; research; performed D.L.T. designed D.L.T. and and S.S., S.S., L.H.K., C.H.A.C., contributions: Author If euigtemnmmviscosity minimum the Reducing xdlwrpaeand plate lower a fixed by characterized is field, crystal directional liquid The (4). rapidly states that between systems a switch including to channel systems, 1). applications engineered (Fig. have horizontal of variety and sliding wide a studied frictional extensively in model are crystals to flowing Liquid plates, fluid, parallel two liquid-crystal between a use We Model Liquid-Crystal stick–slip the to faults. leads with and associated weakening behavior velocity is This velocity. ing †‡ owo orsodnemyb drse.Emi:kloggooyudvseu or [email protected]. E-mail: addressed. be may correspondence whom To h hrceitcbhvo fteeeutosi lutae by illustrated is equations these of behavior characteristic The hr parameter third A h oe sdsrbdb h olwn qain fmotion. of equations following the by described is model The eaaino h rcincefiin au hntkspaeto place takes then value coefficient friction the of relaxation A h icst sgvnby given is viscosity The b α (θ > d  γ ; ,i un otosbt h ieo h rcinjm associ- jump traction the of size the both controls turn, in ), ly oeaaoost h tt aibei rate-and- in variable state the to analogous role a plays svr ml,nmrclisaiiycnetrtesimulation. the enter can instability numerical small, very is a u  d  t h nlcefiin ffito erae ihincreas- with decreases friction of coefficient final the hntevlct fteuprplate upper the of velocity the When . + ( u  L ·∇ d  sacaatrsi lplnt,and length, slip characteristic a is ) (y, ν µ u  0 = = cuswhen occurs t t µ γ ,where ), stime. is i µ ν div( eoe h eaaincefiin fthe of coefficient relaxation the denotes = w.nsog/ci/di/1.03/pnas.0710990105 / 10.1073 / doi / cgi / www.pnas.org 0 − µ ν α 0 = ∇ (b (θ + d  u)  y α − eemnstetp ffriction of type the determines ) a ihrsett h etcl The vertical. the to respect with stevria itnefo the from distance vertical the is (θ − ν ln d  a)ln d  0 )ν onsi h oiotldirec- horizontal the in points ρ sdflce rmtevertical, the from deflected is 1 o0poue stick–slip-like produces 0 to 1 ν ∇ u u 1 + 0 1 p cuswhen occurs u u (1 0 1 u . n(0, in − 0 . to α (θ u α T γ ))ν 1 (θ ) h friction The . u of × ;techoice the ); 0 d a  where , ssuddenly is d  , svertical. is and produces b θ are [4] [2] [3] is

GEOPHYSICS Downloaded by guest on September 30, 2021 ae ntebhvo flqi rsas()btmdfidt yield to modified but (8) crystals behavior. liquid friction-like a of behavior the on based udvlct.W hoearltosi between relationship a choose We velocity. angle fluid the on only depends viscosity. on influence its field field velocity the Eq. in viscosity the h vlto fti system. this of evolution the and set, open p h aeto base the eed ntevria coordinate vertical the on depends where odtosaetknt be to taken are conditions cos where h icst ntefli sdtrie yteoinaino the velocity: the of to orientation analogous the field, by determined director is vector fluid the in viscosity The Where 1. Fig. w.nsog/ci/di/1.03/pnas.0710990105 / 10.1073 / doi / cgi / www.pnas.org velocity horizontal The liquid-crystal. a with filled in,w sueta h etclcmoeto h eoiyfield velocity the of that component so vertical vanishes the that assume we tions, ie w-iesoa nnt hne othat so channel infinite two-dimensional a sider initial and boundary specified denote ic conditions. and BC subscript where oet h oiotlcmoeti eoa h onaisadicessto center. increases the and boundaries at the maximum at a zero is component horizontal the ponent; (d o sdie yteipsdvlct fteuprbudr of boundary upper the the that of so velocity zero imposed channel be the the to by gradient driven is pressure flow horizontal the take we (0, spressure, is 1 h icst ftefluid the of viscosity The easm httefli sicmrsil,s that so incompressible, is fluid the that assume We oueti oe ocaatrz h eairo al,w con- we fault, a of behavior the characterize to model this use To elet We ihcodnts(x coordinates with L) (x , d  y), h ietrfil ed akt h eoiyfil through field velocity the to back feeds field director the ; γ u lutaino u al oe,ahrzna hne fthickness of channel horizontal a model, fault our of Illustration  d sarlxto aaee htpasarl nlgu to analogous role a plays that parameter relaxation a is stevelocity, the is 2 θ u d  u(x ν  (x = t = u.  = ,  +∇ [0, , ) rceigwt h sa hne o assump- flow channel usual the with Proceeding y)). u u y)  eoe ihrato rtredmninlsmooth three-dimensional or two- a either denotes α ttetp h ietrfil a osatvria com- vertical constant a has field director The top. the at × T u (θ hc hne h retto ftedirector the of orientation the changes which u, d   2 ] d  = )ν · eoe h nevlo time of interval the denotes /( d d u u =     div oeta hr sa neato between interaction an is there that Note 4. u  1 = = = = d  −∇ ,adta h oiotlcmoetonly component horizontal the that and 0, (u + h ietrfil eae namanner a in behaves field director The . u  u  u d u d     ρ 1 (1 , BC ic BC ic = (x .Tecanli le ihafli and fluid a with filled is channel The y). u  sdensity, is − , d  θ 0in · y), .Terqie nta n boundary and initial required The ). d  ν α ewe h ietrfil n the and field director the between on on on on = (θ u eed nafunction a on depends 2 {t {t [0, [0, ))ν (x γ [0, , = = eset We y. T T 0 )adsimilarly and y)) T ν d  n(0, in , ) ) 0 0 ) sknmtcvsoiy and viscosity, kinematic is }×, }×, × × × n(0, in ∂ ∂ . u , , nrae from increases u( T t T nwihw study we which on ) )  y): × × ν = , , = and (−∞, u d  1 α α (x ( which , u .For y). , htis that ∞) y) = [10] [11] 0at [7] [6] [9] [5] [8] × = L o otct,teknmtcviscosity kinematic the vorticity, for number director The h w rcse.If processes. two the field, director the for ytmo qain ipie sfollows: as simplifies equations of system eoiyi h nta rsrbdvlct fteuprboundary upper the of velocity prescribed initial the width is channel the velocity is length reference Our ables. hc oen h icu eair n h ietrnumber, director the and behavior, viscous the governs which odmninlprmtr r h enlsnumber: Reynolds the are parameters nondimensional h oe silsrtdi i.1. Fig. in illustrated is model The ν 0;terfrnevsoiyi h icst tteboundaries the at viscosity the is viscosity ν reference the u(0); enwtake now we h hne emty eset we geometry, channel the rcs iefravsolsi material. to viscoelastic time a relaxation for a time of process ratio observa- a the for number, the timescale Weissenberg of the the to ratio or material tion, the viscous is equiv- a which not of (9), is time number but relaxation Deborah resembles, the it either Because times, to, field. relaxation alent velocity two the of ratio than the time is shorter much a If in field. relaxes director the than time shorter a component tal where aiu ttso icst nEq. in viscosity of states maximum 1 → osmlf u nlss eitouenniesoa vari- nondimensional introduce we analysis, our simplify To ihteeasmtos n sn h notation the using and assumptions, these With Let edfieasot rniinfunction transition smooth a define We where ν u 1 0 steiiilsiigvlct ftetpplate. top the of velocity sliding initial the is u(0) ν → etecagso aibe;te Eqs. then variables; of changes the be ν ν α θ = (θ = u(0) u(0, = u(0, d cos cos ) d α α (0, d (0, ,adterfrnetm is time reference the and 0, = d (θ (θ u(y,0) t u; d u t 1 t θ θ ) = )ν ) t t ⎩ ⎨ ⎧ t )ν u d d t u( nydpnso h etclcodnt othat so coordinate vertical the on depends only ) u ) ======t (y,0) ( 1 1 = DRe PNAS = → = y,0) y,0) γ + e + 0, γ (1 D (1 0, 1 Re 0cos 10 d = ti lotertoo h eaaintm for time relaxation the of ratio the also is It . 1 d D d (ν e (1 u(L, (1 (L, u(0) u(1, + >> (1, +  9 = (ν d u = u(0) = stertoo h ifso coefficient diffusion the of ratio the is f0.9 if 1 + − d −  − Re d  θ d L ) u t t ( 00 00 2 + y ) ) 2  u; ue1 2008 , 10 June − u 1 t t  α h eoiyfil eae namuch a in relaxes field velocity the 1 y): α D ) ) ) ) ) d = =  −1/2 =  −1/2 (θ y (θ u y 1 = = 2 =  = 0 0 → ))ν ))ν = u(0) ut ut ν γ d f0 if ν n sueta h horizon- the that assume and 1 0 1 0 echoose We 16. Ly 1 1 ν n(0, in , n(0, in , . ( L 1 y) ; otedfuincoefficient diffusion the to , , t D n(0, in (0, in (0, in n(0, in (0, in (0, in ≤ → << ≤ α cos 0 o.105 vol. 0 Lt rmtemnmmto minimum the from cos L/ ≤ ≤ t ≤ ≤ T T T T t T T T T / ,tedrco field director the 1, θ ≥ ≥ ≥ ≥ ) ) ) ) ) ) ) ) u(0), y y 0.Tederived The u(0). y y 13 θ ≤ × × × × × × × × 0 0 ≤ ≤ 0 0 u reference our L, ≤ ≤ ≤ u 0.9, (0, (0, (0, (0, 0 1) (0, 1) (0, 1) (0, 0 1) (0, to  L L 1 1 1 = d o 23 no. 20 L) L) L) L) → ∂ ∂ u y become h full the , d 0 d [17] [27] [16] [26] [19] [20] [29] [14] [24] [15] [13] [25] [23] [22] [21] [12] [18] [28] 7931 and D

GEOPHYSICS cases we have velocity strengthening for small Re (small u) and velocity weakening for large Re (large u). The velocity weakening behavior is similar to that of rate and state friction but without the singular behavior at u = 0. The maximum of the director d indicates the largest angle the director is deflected by the fluid. In both cases, it increases linearly initially, while starting to increase more rapidly when it is bigger than 0.5. Note that when d > 0.5, cos θ = (1 + d2)−1/2 is smaller than 0.9, in which case α starts to drop dramatically. When this occurs, τ, the traction at the top, decreases accordingly. The laboratory experiments used to derive rate-and-state fric- tion laws use a sudden increase of the sliding velocity from u(0) to u(0) + δu and then a sudden decrease back to u(0) (5–7, 10). We next study the transient response of the liquid-crystal layer to a rapid change in the top-plate sliding velocity u. In the following simulations, a tanh(tan)-type of transition is considered: the top plate sliding velocity is defined as

Fig. 2. Dependence of the transition function α for the viscosity as defined ⎧ ⎪ 1ift ∈[0, 14.9], in Eq. 16 on the angle of the director field θ. ⎪ ⎪ δu ⎨⎪u + tanh[tan(5(t − 15)π)] if t ∈[14.9, 15.1], 2 = + ∈[ ] u ⎪ 1 δu if t 15.1, 19.9 , [30] ⎪ δu and this dependence is illustrated in Fig. 2. The model is now fully ⎪u − tanh[tan(5(t − 20)π)] if t ∈[19.9, 20.1], ⎩⎪ 2 prescribed. 1ift ∈[20.1, 25],

Simulations where u = 1 + δu/2. The change of the top-plate sliding veloc- We first give the results in the steady state. The shear stress ity is shown in Fig. 4. This dependence is a close approximation (traction) τ is a constant across the channel. Using our nondimen- to a step increase followed by a step decrease in the upper plate → sionalization, we have τ ν1u(0)τ/L. We give the dependence velocity. of the nondimensional traction τ and the maximum value of the We give numerical solutions for two examples: horizontal component of the director field dmax on the Reynolds = number in Fig. 3 for two values of the director number D 30 Case 1: Re = 0.04129, D = 300 and δu = 0.0145. and D = 300. The dependence on the Reynolds number is equiv- alent to a dependence on the velocity of the upper plate u. In both Case 2: Re = 0.02443, D = 30 and δu = 0.0914.

Fig. 3. Steady-state dependence of the nondimensional traction τ and the maximum horizontal value of the traction field d on the Reynolds number Re. Two values of the director number are considered: D = 300 and D = 30. Also shown are the Reynolds numbers of the two transient solutions that we present.

7932 www.pnas.org / cgi / doi / 10.1073 / pnas.0710990105 Cheng et al. Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 n ein h nlsed-tt tutr fCs sillustrated is 1 Case of strengthen- structure velocity steady-state a final in The is region. 2 ing Case and region, weakening velocity 4. Fig. w.nsog/ci/di/1.03/pnas.0710990105 / 10.1073 / doi / cgi / www.pnas.org h odmninltime nondimensional the h taysaebhvo silsrtdi i.3 ae1i na in is 1 Case 3. Fig. in illustrated is behavior steady-state The eedneo h odmninltppaesiigvelocity sliding plate top nondimensional the of Dependence i.5. Fig. rfie fvelocity of Profiles t o u rnin simulations. transient our for u ietrfield director , d viscosity , u ν on n ietrangle director and , nta nraes httesed tt rcinfrea h higher the at force the traction than state less velocity steady is the dur- decrease that force the so traction However, increase the 2. initial in Case as decrease in case a transient also the this is ing for There 3. behavior Fig. weakening in shown velocity the to corresponds aei i.3 irriaebhvo sse hntevelocity the when seen value is initial its behavior to image returned mirror is this A for 3. given Fig. behavior in case strengthening velocity the to corresponds betv st opr h eairo hsmdlwt the with model Eqs. this The in of illustrated friction. behavior law for the slowness model rate-and-state compare empirical liquid-crystal to is a objective introduced have We Discussion h etro h hne,where channel, ν the of center the for 5, Fig. in hne Fg ) h ietrfil a o eae Fg )so transient. of 7) increase velocity in (Fig. transient the a relaxed during exhibit not constant cases nearly Both has is field viscosity director the that The 6). (Fig. changed h nta nraes httesed tt rcinfrea the at force traction than state greater steady is the force velocity traction that higher the so in increase decrease initial the the 1, Case In nel. n angle and o ohcases both For θ n eraei h viscosity the in decrease a and u = θ θ costecanlfrCs ttime at 1 Case for channel the across t u r minima. are = 2 u 5 h udseri togycnetae near concentrated strongly is shear fluid The 25. sgetrta h nta au at value initial the than greater is = τ PNAS u 2 nrae ail nwe h eoiyis velocity the when in rapidly increases sls hnteiiilvleat value initial the than less is ue1 2008 , 10 June u = d 1. ν smxmmadteviscosity the and maximum is ertecne ftechan- the of center the near d o.105 vol. edn oadecrease a to leading , t = 25. o 23 no. u u = = .This 1. .This 1. 1 7933 and

GEOPHYSICS Fig. 6. Dependence of the nondimensional traction force τ on time t for Case 1 (Left) and Case 2 (Right).

Fig. 7. Dependence of the maximum horizontal value of the director field d on time t for Case 1 (Left) and Case 2 (Right).

3. The steady-state behavior of the liquid-crystal model is illus- the velocity field through the viscosity. An increase in fluid shear trated in Fig. 3. At low Reynolds numbers (low velocities) the results in an alignment of the director field with the flow and a traction increases with an increase in Reynolds number (veloc- reduction in the viscosity. This reduction leads to the reduction of ity). This is velocity strengthening and solutions are stable. At the traction and velocity weakening. high Reynolds number (high velocities), the traction decreases The alignment of the director field is a transient process obeying with an increase in Reynolds number (velocity). This is veloc- the diffusion equation as can be seen from Eq. 6. The weaken- ity weakening, and solutions are unstable, leading to stick–slip ing associated with rate-and-state friction has a similar relaxation behavior. from velocity strengthening at short times to velocity weakening The transient behavior of the liquid-crystal model is illus- at long times. trated with two cases. Case 1 is the unstable velocity-weakening Friction is attributed to the roughness of surfaces and/or the region and Case 2 is the stable velocity-strengthening region. interactions of the granular material between the surfaces. The The velocity-weakening Case 1 is very similar to the transient interacting surfaces are generally referred to as asperities. We behavior of rate and state friction when a > b. The velocity- believe that it is appropriate to associate the reorientation of the strengthening Case 2 is very similar to rate and state friction director field to the horizontal direction at high velocities with when b > a. the weakening of asperity contacts in faults at higher slip veloc- The velocity-weakening friction is directly responsible for the ities. The validity of this association remains to be explored in stick–slip behavior of faults and the occurrence of earthquakes. detail. The empirical rate-and-state friction laws based on laboratory experiments reproduce the velocity weakening friction but the basic physics of this behavior is poorly understood. The physics ACKNOWLEDGMENTS. We thank Nadia Lapusta and Chun Liu for helpful of the velocity weakening in our liquid-crystal model is clear. It reviews of the manuscript. This work was supported by National Science is a direct result of the coupling between the director field and Foundation Grant EAR 0327799.

7934 www.pnas.org / cgi / doi / 10.1073 / pnas.0710990105 Cheng et al. Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 .DeeihJ 17)Mdln frc rcin .Eprmna eut n constitutive and results Experimental 1. friction. rock of Modeling (1979) JH Dieterich 5. .D ensP,PotJ(1993) J Prost PG, Gennes De 4. (1992) RM Nedderman 3. (2002) CH Schulz (1996) 2. PJ Blau 1. w.nsog/ci/di/1.03/pnas.0710990105 / 10.1073 / doi / cgi / www.pnas.org equations. Ed. abig,UK). Cambridge, Ed. 2nd UK), Cambridge, epy Res Geophys J rcinSineadTechnology and Science Friction h ehnc fErhuksadFaulting and Earthquakes of Mechanics The ttc n ieaiso rnlrFlows Granular of Kinematics and Statics 84:2161–2168. h hsc fLqi Crystals Liquid of Physics The Dke,NwYork). New (Dekker, Caedn xod,2nd Oxford), (Clarendon, CmrdeUi Press, Univ (Cambridge CmrdeUi Press, Univ (Cambridge 0 itrc H igr D(94 ietosraino rcinlcnat:Nwinsights New contacts: frictional of observation Direct (1994) BD Kilgore JH, Dieterich 10. .Mrn 19)Lbrtr-eie rcinlw n hi plcto oseismic to application their laws. and friction laws friction variable Laboratory-derived state (1998) and C instability Marone Slip 7. (1983) AL Ruina 6. .CuadD holrS(01 elpsdeso h ulEike-elemdlof model Ericksen-Leslie full number. the Deborah The of (1964) M Well-posedness Reiner (2001) 9. S Shkoller D, Coutand 8. o tt-eedn properties. state-dependent for faulting. 88:359–370. eai liquid-crystals. nematic n e at lntSci Planet Earth Rev Ann cdSiPrsSrIMath I Ser Paris Sci Acad R C PNAS ueadApidGeophys Applied and Pure 26:643–696. ue1 2008 , 10 June hsc Today Physics 333:919–924. 17:62. o.105 vol. 143:283–302. o 23 no. epy Res Geophys J 7935

GEOPHYSICS