On the speed of fast and slow rupture fronts along frictional interfaces Jørgen Kjoshagen Trømborg,1,2, Henrik Andersen Sveinsson,1 Kjetil ∗ Thøgersen,1 Julien Scheibert,2, and Anders Malthe-Sørenssen1 † 1Department of Physics University of Oslo Sem Sælands vei 24, NO-0316, Oslo, Norway 2Laboratoire de Tribologie et Dynamique des Syst`emes, CNRS, Ecole Centrale de Lyon 36, Avenue Guy de Collongue, 69134 Ecully cedex, France (Dated: January 14, 2015) 5 1 The transition from stick to slip at a dry frictional interface occurs through the breaking of 0 the junctions between the two contacting surfaces. Typically, interactions between the junctions 2 throughthebulkleadtorupturefrontspropagatingfromweakand/orhighlystressedregions,whose junctions break first. Experiments find rupture fronts ranging from quasi-static fronts with speeds n proportional to external loading rates, via fronts much slower than the Rayleigh wave speed, and a frontsthatpropagateneartheRayleighwavespeed,tofrontsthattravelfasterthantheshearwave J speed. Themechanismsbehindandselectionbetweenthesefrontsarestillimperfectlyunderstood. 3 Here we perform simulations in an elastic 2D spring–block model where the frictional interaction 1 between each interfacial block and the substrate arises from a set of junctions modeled explicitly. We find that a proportionality between material slip speed and rupture front speed, previously ] h reportedforslowfronts,actuallyholdsacrossthefullrangeoffrontspeedsweobserve. Werevisita p mechanismforslowslipinthemodelanddemonstratethatfastslipandfastfrontshaveadifferent, - inertialorigin. Wehighlight thelongtransients infrontspeed evenin homogeneousinterfaces, and s we study how both the local shear to normal stress ratio and the local strength are involved in the s a selectionoffronttypeandfrontspeed. Lastly,weintroduceanexperimentallyaccessibleintegrated l measure of block slip history, the Gini coefficient, and demonstrate that in the model it is a good c predictorofthehistory-dependentlocalstaticfrictioncoefficientoftheinterface. Theseresultswill . s contribute both to building a physically-based classification of the various types of fronts and to c identifying the important mechanisms involved in the selection of their propagation speed. i s y PACSnumbers: 81.40.Pq,46.55.+d,62.20.Qp,46.50.+a h p [ I. INTRODUCTION [1,3,16–18],oratspeedsoneortwoordersofmagnitude slower than the Rayleigh speed (slow) [3, 16, 19, 20]. In 1 v this context, the present paper is mainly devoted to the 0 The onset of sliding at a frictional interface occurs study of the mechanisms responsible for front speed se- 1 through the breaking of the many contacts that were lection. 1 preventing the relative motion of the surfaces. When 3 asinglecontactbreaks, thestressitboreisredistributed Experiments to shed light on the nature of the rup- 0 ture fronts have been performed. The authors of [16, 21] to its neighbors, bringing them closer to or past their . placed arrays of sensors close to the interface and used 1 load-bearing capacity. In extended frictional interfaces 0 (i.e. largerthanacharacteristiccorrelationlengthscale), continuum theory to infer the properties of the elastic 5 this process can lead to propagating ruptures – rupture fields at the interface. Svetlizky and Fineberg [22] re- 1 cently showed that the dynamic fields associated with fronts. The recent direct observation of rupture fronts v: in laboratory friction experiments (see e.g. [1–5]) have sub-Rayleigh fronts are consistent with the ones pre- i provided new insights and opened new questions. It was dicted by linear elastic fracture mechanics [23]. The au- X thors of [5, 20, 24] used microtextured surfaces where found,forinstance,thatnotallfrontsspantheentirein- r each contact can be tracked individually to follow the a terface [6, 7]. The selection of the propagation length of rupture fronts directly at the individual micro-contact the fronts has been intensely investigated [8–14]. It was level. Common to all the experiments mentioned till also found that the fronts can propagate at a variety of now is that they seek to increase both the spatial and speeds, either quasi-statically [4, 5, 15], at speeds close thetemporalresolutionofthebehavioroftheveryinter- to that of surface waves (sub-Rayleigh) [3, 16, 17], at face. These experiments can be usefully complemented speeds faster than the shear wave speed c (supershear) s by computer simulations, which by their nature pro- vide complete information of all quantities in the mod- els they implement, thus filling in the information gaps ∗Electronicaddress: [email protected] that remain despite the experimental progress; for ex- †Electronicaddress: [email protected] ample, simulations can provide simultaneous access to 2 shear stress, normal stress, local contact strength and for the onset of sliding became available in laboratory front propagation, a combination which remains hard to frictionexperiments,the1Dspring–blockmodelwasalso access experimentally. appliedinthefrictionliterature[7,10,12,25,26,30,55]. To represent the propagation of the front separating a The main limitations of the 1D spring–block model are stuck part oftheinterfacefromaslipping one, modelsof itsinabilitytoaccuratelyreproducethestressfieldsthat thetransitiontoslidingneedtoincludeatleastonelevel ariseintheexperimentsandthelackofaphysicallength ofdiscretizationofthemacroscopicinterface. Depending scale [56, 57] unless such a length scale is introduced in on the model, this so-called mesoscopic scale may itself the friction law (e.g. [10]). hostapopulationofsmallerscaledmicro-junctions, such To better reproduce the experimental loading condi- as the micro-contacts forming the multi-contact between tions and how they translate to heterogeneous shear and rough surfaces (see e.g. [25–27]). However, the emerg- normal stress fields at the frictional interface, we used, ingcollectivebehaviorofthemicro-junctionscanalsobe in [9], a 2D spring–block model, which can be shown lumpedintoalocalfrictionlaw,ofteninspiredbylawsde- to reproduce 2D linear elasticity [58]. With Amontons– veloped empirically from macroscale experiments, acting Coulomb friction this model agreed well with experi- directly at the meso-scale (see e.g. [7–9, 28–35]). The ments for static measures related to the onset of sliding, existing models also differ in how they treat the stress such as for the length of precursors and the evolution transfer in the bulk. The options range from 1D sys- of the normal stress along the interface, but the model tems [7, 8, 10, 12, 25, 26, 28–30, 34, 35] via 2D models did not capture the full dynamics of the rupture fronts. [9, 11, 27, 31–33] to a full 3D visco-elasto-plastic dis- Radiguetetal.[11]studiedthememoryofthestressstate cretization. In principle, the most comprehensive fric- throughthepassageofmultiplerupturesinavisco-elastic tion models could be combined with the most compre- 2D finite element model with a slip-weakening friction hensive bulk models, but the dynamics that arise from law. They too focused on successive precursors rather this approach tend to be nearly as complicated and dif- than the dynamics of each rupture event. Kammer et al. ficult to disentangle as in the experiments themselves, [32] studied the properties of fast rupture front propaga- and instead, authors have focused on one or a few model tion in a 2D finite element model with a static+velocity properties. weakening dynamic friction law. Otsuki and Matsukawa Continuum models of friction, from the Amontons– [33] studied how the normal force and the size of the in- Coulomb description to more sophisticated rate-and- terface influence the effective macroscopic static friction state friction laws [36–40], are successful when they re- coefficient, using a 2D finite element model with a lo- produce a robust average behavior of the myriad micro- cal velocity-weakening Amontons–Coulomb friction law. junctionsthatmakeupeachmesoscopicregionofthefric- While the above-mentioned models were able to model tional interface. However, by their nature, these models various aspects of the properties of rupture fronts, they donotexplainhowtheindividualmicro-junctionsevolve do not provide a framework that within the same model and interact to produce the overall friction behavior. To is able to account for the full richness of front dynam- approach this question, numerical [41–43] and analytical ics that was observed experimentally. In particular, the [44–47] models have been made that explicitly include a self-selection of front type leading to sub-Rayleigh, slow set of junctions, each representing one or a few micro- and supershear fronts, as well as the transitions between scopic contacts. The main missing ingredient in these them, was missing. models is a solid foundation for the junction evolution In [27] we combined an asperity model of the friction laws. In principle, this could be addressed by molecular at the interface with a 2D elastic solver in order to ac- dynamics simulations [48], but these simulations enable curately reproduce the experimental loading conditions tooshorttimescalesandtoosmalllengthscalesforasys- used in [3]. In the asperity model we included a time tematic study of the onset of macroscopic sliding. Even scale inspired by the time scale identified in the same models in which junctions represent thermally activated experimental system [59] that in the model controls the single molecular bonds in a simplified way [49, 50] would healing of the interface back to a fully pinned state after require a very large number of junctions and prohibitive slipping. Thiscombinedmodelproducedspatio-temporal calculation times in order to resolve the spatiotemporal featuresoftherupturedynamicsverysimilartothoseob- dynamicsofthetransitiontoslidingobservedexperimen- served in the experiment. In particular, we reproduced tally. theabrupttransitionsbetweenfastandslowfrontpropa- Thesimplestbulkmodelthatincludesspatialandtem- gation,whichcanbeunderstoodfromtheunderlyingfast poralstructureinthetransitiontoslidingisprobablythe and slow slip mechanisms. Here, we use this very same 1D spring–block model. Friction has been of interest to model to gain significant additional insights into the re- the earthquake community at least since Brace and By- lation between the micro-scale junction dynamics, the erlee[51]suggestedthatshallowearthquakesaretheslip meso-scale slow and fast slip dynamics, and the macro- events in a stick–slip cycle, and since the spring–block scale friction dynamics and front propagation. Our re- experimentandsimulationbyBurridgeandKnopoff[52] sultsshowthatwhilefastslipandfastfrontsareinertially the1Dspring–blockmodelhasbeenpopularintheearth- controlled, slow slip and slow fronts are non-inertial and quakeliterature,seee.g. [53,54]. Asspatiotemporaldata instead depend on properties of the friction model. Nev- 3 ertheless, the same proportionality relates front speed II. MODEL DESCRIPTION and slip speed in both velocity regimes. This suggests that the key to understanding complicated experimen- The frictional stability of a system made of two solids tal front dynamics lies in understanding the underlying in contact locally depends on the level of normal and slip dynamics. Further, we show that the spatial extent shear stresses at the contact interface. These interfa- of transients in front speed is comparable to the size of cialstressesresultfromtheexternalforcesappliedatthe our system (we take the system size from experiments). boundaries of the solids, transmitted through the bulk. This suggests that, although front speed is influenced Slip motion will in general be triggered when the local strongly by local quantities like the stress state and the interfacial shear stress reaches a threshold, the level of local strength [27], front speed selection is intrinsically a which crucially depends on the interface behavior law at non-local quantity which depends on the rupture history themicroscale. Themodelweemployhereisthesameas ofthewholeinterface. Finally,weshowthatbyapplying in [27]. In this section we begin by describing the physi- theGinicoefficient,whichisanintegratedmeasureofin- calaspectsthatunderliethemodelassumptions;wethen equality commonly used in demography [60, 61, p. 186], describe the model and its parameters in detail; we end we can characterize the complicated slip dynamics of a by discussing briefly the relationship to other models. rupture event in the simulation and predict the subse- quent(slip-history-dependent)localfrictionalstrengthof the interface. A. Physical aspects The paper is structured as follows. In Section II we introduce the model. We first describe the motivating The net contact between two solids generically con- physical picture, and then make the model explicit by sists of a large number of stress bearing micro-junctions. defining its equations and parameters. Then, we high- The properties of these junctions depend on the type of lightsomesimilaritiesanddifferenceswithotherpopular interface. For rough solids, each micro-junction corre- frictionmodels. Nextfollowfourresultssections. InSec- sponds to a micro-contact between asperities on the op- tion III we show that the friction law introduced on the posing surfaces, whereas for smoother surfaces the junc- junction-level leads to stick–slip on the system level. We tions can be solidified patches of an adsorbate layer [43]. also revisit the slow slip mechanism identified in [27]. In We include in our model the following three physical as- Section IV we present simulations with both fast–only pects of the junction behavior. 1) A micro-junction in and fast–slow–fast rupture events. We identify a possi- its pinned state behaves elastically and can bear a shear ble signature of slow fronts in the macroscopic loading force f , provided f remains smaller than a threshold T T curve, and demonstrate the inertial nature of fast slip f . When f is reached, a local fracture-like event thres thres and fast fronts. We show how changes to parameters in occurs, and the junction enters a slipping state. 2) In themodelorthestateoftheinterfacebeforerupturecan the slipping state, the micro-junction moves with the turn fast–slow–fast fronts into fast–only fronts and vice slider’s surface. The physical picture can be e.g. the versa. We map out the initial conditions that lead to micro-slipping of micro-asperities in contact or the flu- fast–only and to slow fronts. In Section V we study the idization of an adsorbate layer. During slip, the micro- range of front speeds within each event and within each junctionsustainssomeresidualforcef =f ,withf T slip slip type of front (fast or slow). We show that in the model, smaller than f . 3) Slipping micro-junctions have a thres thespatialextentoffrontspeedtransientsiscomparable certain probability to disappear or relax. For example, a to the current system size. We show that the influence micro-contact disappears when an asperity moves away of local stress and strength on front speed within each fromitsantagonistasperitybyatypicaldistanceequalto type of front is similar to their role in selecting the front themeansizeofmicro-contacts, asclassicallyconsidered type. We show that fast front speed is proportional to for slow frictional sliding, e.g. in rate-and-state friction fastslipspeedandthattheconstantofproportionalityis laws. However, another picture may arise from the sud- the same as for slow fronts vs slow slip, providing a uni- den release of energy when pinned junctions break. This versalrelationshipacrossthevelocityscalesinthemodel. energy will dissipate as heat in the region around the In Section VI we come full circle by resolving how a rup- micro-junction [59]. The rise in temperature will signif- ture event sets up the state of the interface, which in icantly increase the rate of a thermally activated relax- turndeterminesthepropertiesofthenextruptureevent. ation of the slipping micro-junction during the time it This links back to our results in [47]. Section VII is a takes for the interface to cool down [43]. The effects of brief discussion. The appendices provide additional de- such temperature rises on friction have recently received tails we deemed important to the understanding of the renewed attention (see e.g. [62]), but remain poorly un- model,butthatwouldhampertheflowofthearguments derstood. In an attempt to include such thermal pro- weretheytoappearwithinthemaintext. Theseinclude cesses in our model, we recognize that they will lead to the stencil used for finding the front speed numerically, time- rather than distance-controlled relaxations, which a data collapse demonstrating the utility of some of our will distribute the shear force drop in time. In order for dimensionlessparameters, andadetailedaccountofhow the interface to continue bearing the normal forces ap- we apply initial and boundary conditions. pliedtoit,themicro-junctionsthatrelaxarereplacedby 4 new, pinned junctions that bear a small tangential force (f =0)andanewcyclestarts. HereweuseT(t )asa new R f . simplifiedwayofmodelingthedistributionoftimesafter new which micro-junctions relax. Due to the variety and the complexity of the underlying thermal processes, we did not try to derive T(t ) for a specific situation. Rather, R B. Technical aspects we chose to model T(t ) in the simplest way, as a Gaus- R sian with average time t and width δt . The shape R R (cid:104) (cid:105) The physical aspects described above have been mod- of T(t ) is not crucial: we obtain qualitatively similar R eledinasimplewayusingthefollowingassumptions. We results with an exponential distribution. The width of considertheroughfrictionalinterfacebetweenahorizon- T(t )istheonlysourceofrandomnessinourmodeland R tal track and a thin linear elastic slider (Fig. 1a). The causestheinterfacespringsofablocktoevolvedifferently slider has mass M and sizes L and H in the horizontal from each other. (x) and vertical (z) directions, respectively. We present The2N equationsofmotionaresolvedsimultaneously the values of all parameters in Table I in Appendix C. using a leapfrog / velocity Verlet integrator [76] on a The bulk elastodynamics of the slider are solved using uniform temporal grid of resolution ∆t. a square lattice of blocks connected by internal springs (Fig. 1b) [9, 58]. The slider is divided into a square lat- tice of N =N N blocks of mass m=M/N. Blocks are x z coupled to their four nearest neighbours and their four next-nearestneighboursbyspringsofequilibriumlengths l = L/(N 1) = H/(N 1) and √2l and stiffnesses C. Relationship to other models x z − − k and k/2, respectively, giving an isotropic elastic model with Poisson’s ratio 1/3. The force exerted on block i 1. Bulk modeling by block j is thus k (r l )∆xij when blocks are con- ij ij− ij rij nected, 0 otherwise, where x = (x,z), ∆x = x x , ij j − i A spring–block discretisation of the bulk elasticity is r = ∆x and k and l are the stiffness and equilib- ij | ij| ij ij particularly convenient for models where the friction is riumlengthofthespringconnectingblocksiandj. Block describedasanensembleofmicro-junctionsratherthana oscillations are damped by introducing a viscous force continuum law, because the blocks provide natural units η(x˙ x˙ )ontherelativemotionofconnectedblocks. We j− i on which to couple the frictional and the bulk elastic chose the coefficient η = √0.1km so that blocks are un- behaviour. derdamped and event-triggered oscillations die out well Let us note that like finite element (FEM) and finite before the next event. The non-frictional boundary con- difference (FDM) methods, the spring–block discretiza- ditions are the same as setup 2 in [9], which was also tion satisfies the equations of linear elasticity. In par- used in [27]. The top blocks are submitted to uniformly distributed, time-independent vertical forces FN. The ticular, longitudinal (P) and shear (S) waves in the bulk Nx propagatewiththecorrectspeedsandtherightreflection bottom blocks lie on an elastic foundation of modulus and refraction properties [58]. To verify our implemen- k =k/2, i.e. each block is submitted to a vertical force f tationwecheckedthatthecodereproducestheexpected of amplitude p = k z if z < 0 or 0 otherwise, where i f i i | | bulk wave speeds. z is the vertical displacement of block i. Both verti- i cal boundaries are free, except for a horizontal driving Thechoiceofthespatialresolution,thesizeofasingle force FT = K(Vt xh) applied on the left-side block block, is made according to the following physical argu- − situated at height h above the interface, where xh is the ments. First,asdiscussedforexamplebyPersson[44],by x-displacement of this block. This models a pushing de- CaroliandNozi`eres[63]andbyBraunetal.[64],belowa vice of stiffness K driven at a small constant velocity V. characteristic length scale λ, called the elastic screening The multi-contact nature of the interface is modeled length, the interface behaves rigidly. λ is thus the max- through an array of N tangential springs represent- imum block size allowing for a correct representation of s ing individual micro-junctions, attached in parallel to the elasticity of the interface. For a linear elastic rough each interfacial block (Fig. 1c) [25, 26]. The individ- interface, λ d2/a, with a the typical lateral size of mi- ∼ ual spring behavior is as follows (Fig. 1f, [27, 47]). A crocontactsanddthetypicaldistancebetweenthem. For spring pinned to the track stretches linearly elastically micrometer-rangedroughnesses,weexpecta 1µmand ∼ as the block moves, acting with a tangential force f d 10 100µm, yielding λ 0.1 10mm. Second, T ∼ − ∼ − on the block. When the force reaches the static fric- the frictional behaviour of each block is then a statisti- tion threshold f (we neglect aging, so that f is cal average over the many micro-junctions connected to thres thres time independent), the micro-junction ruptures and the it. This statistical approach is increasingly relevant for spring becomes a slipping spring acting with a dynamic larger blocks involving more junctions. One thus looks friction force f = f . After a random time t drawn for the largest possible block size. Combining both re- T slip R from a distribution T(t ), the slipping spring relaxes. quirements, λ appears as the natural block size for such R It is replaced immediately by a pinned, unloaded spring spring-block models. 5 2. Interface modeling (d) (a) FN 0.2 FT/FN 0.15 I III II The rate-and-state picture of friction, which includes 0.1 V displacement-controlleddisappearanceofmicro-contacts, z 0.05 x has proved to be adequate for slow (typically up to Time (s) 100µm/s range) sliding in a variety of materials. Such 0 0 0.5 1 slow velocities imply a negligible temperature rise of the (b) (e)0.4 τ/p interface and thus a slipping state the duration of which 0.3 is of purely geometrical origin, i.e. it is controlled by a length scale of the order of the mean micro-contact size. 0.2 0.1 Here, we focus on a drastically different situation, in Time (s) which the transition from static to kinetic friction is ex- 0 0.58 0.6 0.62 0.64 0.66 tfarestmselliyps(h1o0r0tm(mmil/lsisercaonngde)r.anAgse)reacnodgnisizaedccoinmrpeacneinetdebxy- (c) (f)0.4 -fT/fN fthres/fN periments by Ben-David et al. [59, 65], the sudden rup- 0.3 d e tureoftheinterfaceanditssubsequentslipwillgenerate n 0.2 n slipping a significant heating of the interface, sufficient to melt pi (durationt )fslip/fN R 0.1 the broken micro-asperities. In these severe conditions, Block displacement (µm) precise knowledge about the micro-contact behavior at 0 the millisecond time scale is currently lacking. 0 1 2 3 FIG. 1: Color) Sketch and behaviour of the multiscale In our description, we acknowledge the fact that the model. (a) Slider and external loading conditions. (b) onset of sliding is far from the slow steady sliding sit- Spring–block network modeling elastodynamics. (c) uation. During the time in which the interface is sig- Surface springs modeling friction on a block. (d) nificantly heated, thermal activations for the transitions Macroscopic loading curve, the ratio F /F of driving from slipping to pinned states are highly probable. Such T N shear force to total normal force. (e) Mesoscopic loading activation is classically described by time-rates, rather curve, the ratio τ/p of shear to normal stress on a block. than by displacement-related quantities. It is therefore (f) Microscopic friction model for the spring loading natural to propose an alternative picture that incorpo- curve, the ratio f /f of friction to normal force for ratesthepossibilitythattransitionsbetweentheslipping T N one spring (f =p/N ). Figure adapted from [27]. and the pinned states can be controlled by a time. N s We emphasize that experimental data did show that III. LOADING CURVES AND SLIP DYNAMICS the dynamics at the onset of sliding involves a transi- tion from fast to slow slip which occurs after a constant time, rather than a constant displacement [see 59]. This A. Loading curves timescale controls the fast dynamics with which the in- terface comes back to a fully pinned state after slipping. The evolution in time of the driving force F is called T Itthusdrasticallydiffersfromtheclassicaltimescalefor theloading curve. Itis readilymeasured inexperiments, aging, also found in [59], and which controls the slow and is used to characterize the motion as smooth slid- strength recovery of the interface when it is at rest. ing, or regular or chaotic stick–slip. The loading curve for our chosen set of parameters is shown in Fig. 1d. It Note, however, that we would not be surprised if a starts with an initial buildup from zero load. This ini- length scale would also be relevant to the dynamics of tial buildup is linear, as we apply the load through a the onset of sliding in the same experimental system. linearly elastic spring, one end of which is pushing the However, the study of a complete model involving both sample while its other end is being driven with a con- a time scale and a length scale is far beyond the scope of stant speed. From about 0.2s the linear increase is in- the present work. terrupted by small drops in the driving force. These are associated with precursors: rupture events confined to In terms of modeling approach, let us also stress the only part of the interface [see e.g. 6–10, 12, 14, 25, 67]. fact that a number of reference models from the litera- The first event where the entire interface breaks and the tureconsidered,beforeus,time-controlledtransitionsbe- slider moves macroscopically occurs at about 0.45s and tween micro-junction states [see e.g. 25, 26, 43, 47, 66]. is seen in the loading curve as a larger drop in F whose T As shown in Thøgersen et al. [47], our friction law is duration is not resolved on this figure. Then the system actually one particular case of a more general family of entersregularstick–slip,withalternatingeventsshowing models. partialandfullbreakingoftheinterface(smallandlarge 6 drops in FT). In the model, as in experiments, the be- (a)6 Partialslipevent Fullsliding haviouronlargerscalesemergesfromtheinteractionson Fullslidingevent the scales below. Figure 1f illustrates the single junction m)4 Arrest law described in Section II. µ junEcatciohnsm(ehseorbelwocekusiseNcou=ple1d00t)o. Athneextaramcpklewoifththemeavnoy- Slip(2 Slow slip s Fastslip lutionofthenetforceinthejunctions, whichapartfrom 0 asmallinertialtermisequivalenttothesumofforceson 1000 2000 3000 4000 Timeaftert (µs) the block from its neighbours, is shown in Fig. 1e. The 0 dataisfromthestick–slipphase. Thetemporalstructure (b) relaxation of slip increment is rich in detail, but can overall be described as periods one slipping spring of very slow increase (the driving is slow compared to t0- t0+ t0 + dt the internal dynamics) separated by faster increase and decrease when rupture fronts pass. ... ...... ...... ... Fext=Σ fT Fext>Σ fT Fext=Σ fT The force maxima attained on the mesoblock level are smaller than the sum of the individual junction thresh- olds, which would give τ/p = 0.4. The explanation is simple: due to the disorder in the individual stretching ps s s ps s p p s s p states, some junctions will break before others, and so they will not all contribute their maximum force simul- FIG. 2: Color) (a) Example slip profiles from a partial taneously (see detailed discussion in [27, 47]). Similarly, slip and a full sliding event. Line without markers the maxima in the macroscopic loading curve of Fig. 1d (black): slip profile for block at x=0.16L with are smaller than those on the mesoblock level, as the t =0.7872s, i.e. the partial slip event preceding event 0 mesoblocks do not reach their individual maxima at the III. Line with markers (grey/multicolored): slip profile same time. This difference between macroscopic and lo- for block at x=0.34L with t =1.0571s, i.e. event II 0 cal friction coefficients has been discussed in e.g. [7– (more details of this event in Fig. 7). In both cases we 9, 26, 33, 68]. That the overall strength in a system is chose a block located near the middle of the region usually smaller than the sum of the individual strengths where the rupture front speed was fast. For blocks of the constituents is familiar from other fields, for ex- closer to the fast–slow transition or to the front arrest ample fracture mechanics [see e.g. 69] and fiber bundle point, the amplitude of fast slip is smaller. (b) Sketch theory [see e.g. 70]. showing how the relaxation of force associated with the junctions’ relaxation from the slipping (s) to the pinned state (p) can lead to a slow slip motion of the block. B. Block slip dynamics and a slow slip mechanism Figure 2a shows the slip dynamics of a block in a par- different physical origin that was explained in [27, 47]. tial slip event and a block in a full sliding event. We This slow slip mechanism is illustrated in Fig. 2b. When find both fast and slow slip regimes of the motion, in ex- f < f the net friction force on a block is reduced new slip cellent agreement with the experiments reported in [59]. slightlywheneveraspringleavestheslippingstate,yield- The initial fast slip regime begins after the passage of a ing a small positive acceleration as the net friction drops fast rupture front. In both these cases the fast slip is below the net external force from neighboring blocks. followed by the block coming nearly to rest (no visible Thefrictionreductionissoonbalancedbythechangesin increaseinnetslipbetween450and550µs)andthenby thepinnedjunctionsandtheexternalforcesontheblock a slow slip regime with roughly linear increase of slip vs as it slowly moves. This slow slip mechanism is present time, i.e. constant slip speed. The slow slip regime can as long as some junctions are going from the slipping to end in two ways. In full sliding events, slow slip changes thepinnedstateandf <f ,butitismaskedbythe new slip back to fast slip when the slider enters the full sliding fastslipwhilethefastsliplasts. Thedependenceofslow regime. For arresting events, the slow slip regime ends slip speed on model parameters was discussed in detail when the block comes to rest. in [27]. The initial fast slip regime corresponds to an inertial motion of the block when a large number of junctions break in a short time interval as the rupture front passes by. The net friction force is rapidly reduced, bringing IV. FRONT TYPE RESULTS the block out of mechanical equilibrium. The result is a large positive acceleration (in the direction of the net Inthissectionwepresentourresultsonthequalitative forceduetotheneighboringblocks). Theinertialnature features of rupture fronts. That is, we discuss the condi- of this motion is demonstrated in Fig. 6. tions under which we observe fast rupture, slow rupture, The subsequent slow slip observed in the model has a and the transitions between these regimes. 7 0 Fractionofpinnedsprings 100% level to mean that a certain fraction of the block’s junc- tions are in the slipping state. This criterion is robust 0.672 (a) Event I to the choice of threshold fraction, because as is seen in Fig. 3 and other figures (7ab, 8, 21), it is typical for a 0.67 ) block to go from having nearly all its junctions pinned s ( 0.668 to having nearly all of them broken in a time short com- e Fast Tim Slo w pared to the other time scales in the simulation. For 0.666 the events in Fig. 3 the time to go from 80% of springs Fast pinnedto20%ofspringspinnedisapproximately0.03ms in the fast part of the fronts and 0.3ms in the slow part 0.832 (b) Event III of the fronts. We have used a threshold value of 30%, 0.83 that is, the start time of block slipping is taken as the ) instant when the fraction of pinned junctions dropped s ( e 0.828 below 30%. m iT 0.826 Figure3ashowsarupturefrontwhosespeedvcchanges from fast (v c /3) to slow (v c /100) and back to c s c s Fast fast again. Th∼e left-travelling fron∼t that starts when the primaryfrontisreflectedfromtheleadingedgere-breaks (c) 103 cs the junctions that had healed behind the front tip. Fig- ) s /m 2 ure3bshowsarupturefrontthatisfastacrosstheentire (10 interface. By defining the transition to block sliding as d ee above, the location of the front tip in time can be mea- p 1 s10 sured. The local front speed is then the ratio of the tn Fast-slow-fast distance travelled by the front to the time for that prop- orF100 Fast agation. Because of disorder in the junction state along 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 the interface remaining from earlier events, the propa- Positionalong interface,x (m) gation time from one block to the next can vary signifi- cantly. We have found that using only the end-points in FIG. 3: Color) Two interface-sized events. (a) A a 5 blocks wide moving stencil (Appendix A) gives the fast–slow–fast event (I in Fig. 1d). Spatiotemporal plot best balance between robustness and spatial resolution of the fraction of pinned springs. (b) A fast–only event in the calculated front speed. The results are shown in (III in Fig. 1d) shown as in (a). (c) Rupture front speed Fig. 3c. v vs. front location for both events. Block rupture is c defined to occur when 70% of interface junctions have broken (white dashed line in the colorbar). Front speed B. The influence of front type on the loading curve is measured as the inverse slope of the rupture line (indicated by arrows in (a) and (b)) using the endpoints in a five-point-wide moving stencil. Figure adapted The two events in Fig. 3 are different. One is a fast– from [27]. slow–fast front, the other is fast across the entire inter- face. Both events are marked in Fig. 1d, and the associ- ateddropsinF areseentohaveapproximatelythesame T amplitude. This indicates that the details of the front A. Rupture front characteristics propagation do not influence the loading curve strongly, atleastnotfortheseevents. Nevertheless,theslowfront With the driving force applied on the trailing edge of propagation does have a signature in the loading curve the slider, the blocks near the trailing edge are the first that appears when we zoom in on a few events as in toreachtheireffectivestaticfrictionthresholds. Ablock Fig. 4. Namely, the drop in the loading force has a sig- that slips increases the load on its neighbors, which can nificant change in slope while the slow front lasts, which start to slip in turn. The rupture front tip, i.e. the distinguishes it visually from the force drop associated boundary between a region of stuck blocks and a region with a fast–only event. of slipping blocks, then propagates away from the nucle- Let us consider the evolution of F in more detail. T ation point. If the rupture arrests before reaching the As long as the slider remains pinned, F increases with T leadingedgeoftheslidertheeventiscalledapartialslip the motion of the driving stage (the driving stage moves event; if the leading edge is reached, we use the name the end of the driving spring that is not attached to full/globalslidingevent;precursorsarepartialslipevents the slider). F decreases only when the point on the T that occur before the first global sliding event. sliderwherethedrivingspringattaches,thetrailingedge, In the interaction law the distinction between pinned moves away from the driving stage. This occurs in two and slipping states is made on the junction rather than distinct ways. First, the trailing edge moves away from theblocklevel. Wethereforedefineslippingontheblock thedrivingstagewhenthesliderdeformsincompression 8 during the passage of a rupture front, which happens Fast-only both for partial slip and full sliding events. Second, the 0.19 Fast-slow-fast trailing edge moves with the rest of the slider when the entire interface is slipping in a full sliding event. For full 0.185 sliding events we define the boundary between rupture front passage and full sliding as the moment the rupture front reaches the leading edge, seen e.g. in Fig. 3. The 0.18 relative amplitude of the FT reductions associated with Partialslip each of these two motions of the trailing edge depends N on the relative stiffnesses of the slider and the driving F 0.175 / spring. When the slider is stiff compared to the driving FT spring, so that little motion of the trailing edge can oc- 0.17 cur unless the entire slider moves, the drop in F is only T appreciable during the sliding part of full sliding events. 0.165 With a softer slider, the deformation occurring in par- tial slip events and during the rupture front passage in a full sliding event accounts for a larger fraction of the 0.16 net reduction in F . As seen by the relative amplitudes T of the force drops in Fig. 4, with the present parameters 0.155 the (trailing edge) slip associated with the slider defor- 0.96 0.98 1 1.02 1.04 1.06 1.08 mation accounts for about a fifth of the net slip of full Time (s) sliding events. Further quantification of this feature is presented in FIG. 4: A zoom on the macroscopic loading curve Fig. 5, where the force drops associated with either the F /F . The drop in a full sliding event is comprised of T N entire event or with the rupture front passage only are two parts: first, the drop associated with deformation of presented for all events during the developed stick–slip the slider during rupture front passage (dashed arrow); regime. We find that the loading force drop occurring second, the drop associated with motion of the entire during the rupture front passage, regardless of the type slider (full arrow minus dashed arrow). The amplitude of event, is always much smaller than the one associated of the first drop is approximately the same as the with sliding. The net force drop in full sliding events amplitude of the drop associated with a partial slip has comparable amplitudes for fast–slow–fast and fast– event (see Fig. 5). only events. This is because the block motion during full sliding accounts for the largest part of the net block motion in an event, and after the rupture front reaches neighboring blocks located within the part of the inter- the leading edge, the distinction between fast–slow–fast face where the front speed was high as the front passed, events and fast–only events is unimportant. foreachofthefoursimulations. Thefigureclearlyshows that the fast slip speed was modified by the change of density. Figure6bdemonstratesthataρ 1/2 scalingcol- − C. Fast slip and fast front speeds are inertial lapses the data. The reference time t for each block rup is the time of block rupture as defined in Fig. 3, and the block slip is measured with respect to the block position In Fig. 2a we showed example block slip profiles and at t . the regimes of fast slip, slow slip and full sliding. In sec- rup tionIIIBwefocusedontheslowslippart,anddiscussed Figure 6c shows the front speed as a function of po- howthejunctionevolutionlawleadstoaslowslipmech- sition along the interface for the four simulations, while anism on the block level. In Fig. 3 we showed that the Fig.6dshowsthesamedatawiththefrontspeedrescaled model exhibits both fast–only and fast–slow–fast fronts, by ρ−1/2. The fast front speeds are collapsed onto each and explained how we measure the front speed. In this other by this rescaling. The slow front speeds, in con- section we demonstrate that both the fast slip part of trast, remain the same in all four simulations (Fig. 6c), the block slip evolution and the fast front speed are of and are split from each other by the scaling. This indi- inertial origin. That is, like the bulk wave speeds, these cates a non-inertial origin of the slow front speed, which speeds scale as ρ 1/2, where ρ = M/(LBH) is the mass is the topic of the next section. − density of the system. To isolate the effect of inertia from the stress and fric- tional state at the interface we have performed four sim- D. From slow slip to slow fronts ulations starting from the same state (Appendix C), but with ρ decreased to 1, 1/2, 1/4 and 1/8 of its value in In the model the propagation of fast rupture and slow Table I; we changed the mass and kept the system size rupture is governed by different mechanisms. To demon- constant. Figure6ashowsthefastslipdynamicsforfour strate this we start from the fast–slow–fast event in 9 0.04 −6 −6 x 10 x 10 )m 2 )m 2 ( ( ) ) p 1.5 p 1.5 u u tr tr 0.03 x( 1 x( 1 − − ) ) t t /FN fast-only,fulldrop xp,( 0.5 (a) x 10−4 xp,( 0.5 (b) x 10−6 FT0.02 ffaasstt--oslnolwy,-fdasrto,pfudlulrdinrogprupture Sli 00 0.5 1 1.5Sli 00 2 √ 4 6 n Time,t−t (s) (t−t )/ ρ (arb. units) i fast-slow-fast,dropduringrupture rup rup p 4 6 o partialslip 10 10 Dr m/s) nits) 0.01 ( u 104 vd,c102 (arb. e 2 e ρ 10 p √ s nt (c) × (d) 0 Fro 100 vc100 0.8 1 1.2 1.4 1.6 0 0.05 0.1 0 0.05 0.1 Position,x (m) Position,x (m) Time event occurred (s) FIG. 6: Color) The fast slip and the fast front speeds FIG. 5: Color) The force drops in the loading curve scale with inertia. (a) Block slip motion for four (Fig. 1d) grouped according to event type. We include neighboring blocks within the fast front region, for four partial slip events and full sliding events occurring simulations of the same fast–slow–fast event. In each between t=0.63s and t=1.50s (the developed simulation the initial state is the same, but the mass stick–slip regime). The drops occurring during the density ρ is different between the simulations. The passage of the rupture front have comparable amplitude block slip is measured from the rupture time t of rup between the partial slip events, fast–slow–fast events each block as defined in Fig. 3. Lines with the same and fast–only events. Also, the net force drop in full color and marker are from the same simulation. Lines sliding events have comparable amplitudes for with the same line style represent the same block in fast–slow–fast and fast–only events. different simulations. (b) Rescaling the time of slip with ρ 1/2 collapses the data in (a). (c) The rupture front − speed as a function of position for the same simulations Fig. 7a. We then modify the junction law by increas- as the data in (a) (corresponding colors and markers). ingf tof ,sothatthereisnochangeinthefriction The change of density modified the fast front speed, new slip force when a slipping junction relaxes and is replaced by while the slow front speed remained nearly unchanged. a pinned junction. This does not affect the initial force (d) Rescaling the front speed data in (c) by ρ 1/2 − relaxationofpinnedjunctionsreachingf ,butitdoes collapses the fast front speed measurements, but splits thres affectthesecondrelaxationofslippingjunctionsrelaxing the slow front measurements. back to the pinned state. In practice, this change turns off the slow slip mechanism that was discussed in Sec- tionIIIB.Theresultofrestartingfromthesamestateas thus simply a matter of the origin of the underlying slip inFig.7aandwithonlythismodificationtothejunction motion, which is inertial for fast fronts (see Fig. 6), and law is seen in Fig. 7b. The first, fast part of the rupture isrelatedtotheintrinsicarrestdynamicsoftheinterface isunaffected,buttheslowruptureissuppressed,andthe for slow fronts (see [27, 47]). front stops where the fast-slow transition used to occur. Our understanding of the transition from fast to slow Figure7cillustrateshowslowslipintheregionbehind frontsisthatthefastpropagationstopsatthesamepoint the front tip can result in additional front propagation: in both Fig. 7a and b. In Fig. 7a, slow slip becomes as the blocks at and behind the front tip move towards important when the fast slip ends, and results in a slow the stuck region, the external forces on the block just frontpropagating. InFig.7b,theslowslipmechanismis ahead of the front tip increase. Two outcomes are possi- turned off, and the event is over once the initial fast slip ble. Eithertheblockremainsstuckandthefrontarrests, ends. or the forces on this block eventually overcome its effec- We emphasize that the physical origin of the slip on tivestaticfrictionthreshold, andtheblockstartstoslip, theblocklevelisunimportanttothewayslipbehindthe moving the front tip one block ahead. This last part of front tip leads to rupture front propagation. We showed the explanation would be the same for fast slip and fast in[27]thatamechanismforslowslipcompletelydifferent fronts. The difference between fast and slow fronts is fromtheoneinthepresentmodelwouldalsoleadtoslow 10 (a) (b) mind, the strategy for turning fast–slow–fast events into 1.063 Event II restarted Event II restarted fast–only fronts and the other way around was to mod- 1.062 SSllooww sslliipp tauctrinveed of SSllooww sslliipp ttuurrnneedd oofff ify the width of the initial force distribution of selected s)1.061 blocks along the interface. ( me 1.06 Slowslipbecomesimportanttofrontpropagationonly Ti1.059 after a fast front stops. We can keep the initial fast part 1.058 of the event in the re-simulation equal to the original event by leaving the stress state unchanged. By increas- 0 0.05 0.1 0 0.05 0.1 Position,x (m) Position,x (m) ing the width of the junction force distribution we make the interface weaker, which may enable fast propagation (c) front tip across the whole interface. An example of this is shown Vslowslip Vslowslip in Fig. 8b. Conversely, in Fig. 8d, by significantly decreasing the ... ... width of the initial junction force distribution, we made theinterfacestrongeralongapartoftheinterface,sothat blocks undergoing slow slip motion block stuck the original fast–only propagation was stopped. This al- slowly loaded to its threshold lowedtheslowslipmechanismtobecomeimportant,and the front was turned into a fast–slow–fast one. FIG. 7: Color) (a) and (b) Event II restarted at 1.0571s We note that in both cases, the stress state in the with driving speed V =0, shown as in Fig. 3. Slow slip original and modified simulations are the same, and the is either (a) active or (b) turned off. (c) A sketch only change is in the width of φ(f ) in the region where T showing how slip behind the front tip leads to front tip it is modified. To keep the initiation of the events the propagation. Panels (a) and (b) adapted from [27]. same in the modified and original simulations, we did not modify φ near the trailing/left edge of the system. front propagation. F. Front type phase diagram and its predictive power E. Junction force distribution affects front type selection Wehaveseenthatincreasing(decreasing)thewidthof thejunctionforcedistributionmakestheinterfaceweaker We argued in the previous section that even when the (stronger)andthatthisfavoursfast(slow)frontpropaga- slow slip mechanism is active, its effect on the rupture tion. It also makes intuitive sense that higher prestress, frontspeedcanbemaskedbyfastslipandtheassociated hence smaller distance to the breaking threshold, would fastfrontpropagation. Toillustratethis(Fig.8),wewill favorfastfrontpropagation(wereturntothisinmorede- here re-simulate a fast–slow–fast event with the initial tail in Section V). We have performed simulations where statemodifiedsothattheeventbecomesfast–only. Simi- these two parameters are varied systematically (see Sec- larly,wewillre-simulateafast–onlyeventwiththeinitial tion VB and Appendix C), and the observed front types state modified so that the event becomes fast–slow–fast. arepresentedinFig.9. Inthearrestingregion,thefronts To understand these results, recall the connection be- are partial slip events, that is, they stop before reaching tween the distribution φ(f ) of forces among the junc- the leading edge (some of them stop early, some almost T tions attached to one block and the corresponding ef- reachtheleadingedge). Theregionlabelledslowincludes fective static friction coefficient of this same block. As all those events that have a slow front part, even if the shown numerically in e.g. [27] and theoretically in e.g. event is fast along most of the interface. They share the [42,47]andAppendixB,forfriction,andmoregenerally characteristicthattheeventswouldarrestintheabsence fortheruptureofheterogeneoussystemsinwhichanum- oftheslowslipmechanism. Inthefastregion,eventsare berofjunctionsareloadedinparallel[70],themaximum fast across the entire interface. load that an interface can bear is related to the width In Section V we discuss the transient behavior of the of the load distribution and/or threshold distribution of rupture fronts. For now, let us stress that while the ar- the various junctions. Homogeneous systems (vanishing resting, slow and fast regions of the rupture fronts are distribution width) have the maximum possible macro- robust in their relative positions (the fast front region scopic rupture threshold because all junctions will con- is found at higher values of normalized prestress τ¯ and 0 tributewiththeirmaximumforcewhencollectiverupture junction distribution width σ than the slow front region, is reached. In contrast, in heterogeneous systems (finite which is itself found at higher values than the arrest re- width) the weaker and/or initially more highly loaded gion), the precise locations of the boundaries between junctions will break first, so that when macroscopic rup- them depend also on how the events are triggered. For ture occurs, only a fraction of the initial population of example, it is possible to prepare two simulations with junctions will contribute to the total force. With this in the same stresses and distributions in the propagation