On the speed of fast and slow rupture fronts along frictional interfaces

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by  Jørgen Kjoshagen Trømborg

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