4.06 Dynamic Shear Rupture in Frictional Interfaces ... · insight into some aspects of earthquake mechanics, such models are more appropriate for providing fric-tional constitutive

4.06 Dynamic Shear Rupture in Frictional Interfaces:Speeds, Directionality, and ModesA. J. Rosakis, California Institute of Technology, Pasadena, CA, USA

K. Xia, University of Toronto, Toronto, ON, Canada

G. Lykotrafitis, Massachusetts Institute of Technology, Cambridge, MA, USA

H. Kanamori, Seismological Laboratory, Caltech, Pasadena, CA, USA

ª 2007 Elsevier B.V. All rights reserved.

4.06.1 Introduction 153

4.06.2 Experimental Techniques for Creating Earthquakes in the Laboratory 154

4.06.2.1 Early Experimental Studies 154

4.06.2.2 Experimental Design Philosophy 156

4.06.2.2.1 Simulating tectonic loading 156

4.06.2.2.2 Simulating the nucleation process 157

4.06.2.2.3 Choosing appropriate model materials and diagnostics 158

4.06.3 Supershear and Sub-Rayleigh to Supershear Transition in Homogeneous

Fault Systems 160

4.06.3.1 Purely Sub-Rayleigh and Purely Supershear Earthquake Ruptures 161

4.06.3.2 The Experimental Visualization of the Sub-Rayleigh to Supershear Earthquake

Rupture Transition 163

4.06.3.3 A Theoretical Model for the Sub-Rayleigh to Supershear Transition 164

4.06.3.4 Discussion 167

4.06.4 Directionality of Ruptures along Faults Separating Weakly Dissimilar Materials:

Supershear and Generalized Rayleigh Wave Speed Ruptures 167

4.06.4.1 Two Types of Ruptures along Inhomogeneous Fault Systems 168

4.06.4.2 Experimental Setup 170

4.06.4.3 Experimental Results 171

4.06.4.4 Comparison of the Experimental Results to Early Numerical and Theoretical Studies 173

4.06.4.5 The Parkfield Earthquake Discussion in the Context of Experiments and of

Recent Numerical Studies 174

4.06.4.6 Discussion of the Historic, North Anatolian Fault Earthquake Sequence

in View of the Experimental Results 175

4.06.5 Observing Crack-Like, Pulse-Like, Wrinkle-Like and Mixed Rupture Modes in the

Laboratory 177

4.06.5.1 Experimental Investigation of Dynamic Rupture Modes 179

4.06.5.1.1 Specimen configuration and loading 180

4.06.5.1.2 Using particle velocimetry to measure slip and opening histories 181

4.06.5.2 Visualizing Pulse-Like and Crack-Like Ruptures in the Laboratory 181

4.06.5.3 Wrinkle-Like Opening Pulses along Interfaces in Homogeneous Systems 185

4.06.5.4 Discussion 187

References 189

4.06.1 Introduction

The dramatic progress of research on earthquake

rupture in the last decades along with advances in

the various branches of geophysics has decisively

improved our understanding of seismic source

events. A tremendous scientific activity including

theoretical seismomechanics studies, field data pro-

cessing, and laboratory investigations have answered

many outstanding questions on earthquake dynamics.

153

Other problems yet remain to be solved whereas newquestions have recently emerged. One of these ques-tions is related to the ability of laboratoryexperiments to faithfully ‘mimic’ natural earthquakerupture events. In particular, this question is con-cerned with the ‘scalability’ of the laboratory resultsand observations. Can conclusions regarding the natureof dynamic rupture (e.g., modes, speeds, directionalityetc.), obtained from such experiments, be accuratelytransferred from the laboratory scale (meters) to thenatural earthquake rupture scale (hundred ofkilometers)? Another question relates to similar issuesin modeling. Do mechanics models and numericalsimulations of rupture also suffer from scaling issuesand do they need to first be validated by direct com-parison to experiments before their results areextrapolated to the scale of rupture processes in theEarth’s crust? These are some of the questions which,although only partially addressed here, have motivatedmuch of the work reviewed in the present chapter.

In this chapter, we first review past work onexperimental simulations of earthquake rupture inthe laboratory. We describe the basic traditionalexperimental techniques used in the last few decadesand we then focus on a new experimental designphilosophy introduced by the authors. This newapproach to laboratory earthquakes has resulted inan experimental configuration particularly suited tothe study of fast rupture events.

Following the description of the design philoso-phy, the chapter also concentrates on experimentalresults which demonstrate the attainability of inter-sonic rupture speeds (i.e., speeds in the intervalbetween the shear and the dilatational wave speeds)and elucidate the sub-Rayleigh to supershear transi-tion phenomenon in homogeneous fault systems. Thistransition is discussed in the light of well-establishedtheoretical and numerical models based on the prin-ciples of fracture mechanics. In addition, recent fieldevidence of intersonic rupture that has occured duringnatural earthquakes is presented and compared to theexperimental observations.

The issue of the directionality of ruptures alongfaults separating weakly dissimilar materials is cur-rently in the center of a very active debate betweenseismologists. In the present chapter, theoretical ana-lyses together with field evidence and recentexperimental and numerical results are utilized toprovide a complete picture of the state of contem-porary knowledge on this problem.

Finally, the various rupture modes (crack-like,pulse-like, wrinkle-like) which have been proposed

on the basis of field evidence and theoretical analysesare critically discussed. Experimental results, whichshow that the activation of these modes in the labora-tory scale is possible, are presented.

Here, we have made the conscious decision not toinclude an extensive general introduction to thisarticle. Instead, we have chosen to strengthen theintroductory comments of individual sections and toblend reference to previous and current work withineach section.

4.06.2 Experimental Techniques forCreating Earthquakes in theLaboratory

4.06.2.1 Early Experimental Studies

The first efforts of simulating earthquakes in thelaboratory have utilized the classic block-slider con-figuration. Based on valuable observations gatheredthrough this configuration, Brace and Byerlee (1966)have proposed that stick-slip may be an importantmechanism for shallow earthquakes occurring alongpre-existing faults. The basic assumption of theblock-slider model is that the sliding rock blocksare rigid so that edge effects, leading to an initiallynonuniform stress distribution along the interface,can be ignored. Under the rigidity assumption, fric-tion along the interface can be assumed uniform andas a result only the displacement history of the sliderneeds to be measured. This assumption is clearly avery strong one since the Earth’s crust is far frombeing rigid during earthquake faulting. Although theblock-slider model experiments do provide usefulinsight into some aspects of earthquake mechanics,such models are more appropriate for providing fric-tional constitutive information instead of modelingthe detailed mechanics of earthquake rupture.Indeed, based on the pioneering experimental resultsof Dieterich (1979), the famous state- and rate-dependent friction law was formulated by Ruina(1983) and Rice (1983).

There are two configurations suitable for produ-cing laboratory earthquakes, namely those of directshear and of biaxial compression (Brune, 1973;Johnson and Scholz, 1976; Wu et al., 1972). In orderto extract meaningful rupture dynamics informationfrom these configurations, the assumption of rigidityshould be abandoned. Indeed, if strain gauges areattached along the fault in a two-dimensional (2-D)domain, the dynamic strain history resulting from therupture can be recorded at various points. With

154 Dynamic Shear Rupture in Frictional Interfaces: Speeds, Directionality, and Modes

this modification, rupture velocities can also be esti-mated (Brune, 1973; Johnson and Scholz, 1976; Wuet al., 1972) in addition to the dynamic friction-sliprelation (Dieterich and Kilgore, 1994; Ohnaka andShen, 1999; Okubo and Dieterich, 1984). However,strain gauges have approximate sizes of few milli-meters, and this limits the spatial resolution of thesediscrete measurements. As a result, the data are point-wise, averaged, and are arguably difficult to interpret.For example, although supershear ruptures were sug-gested by this type of experiment (Johnson and Scholz,1976; Okubo and Dieterich, 1984; Wu et al., 1972),this evidence was considered far from being conclu-sive due to the ‘averaged’ quality of the data.Furthermore, since the diagnostics system (theWheatstone bridge and oscilloscope) was triggeredby one of the strain gauge signals and there was noinformation on the exact time and location of thenucleating point of earthquake (hypocenter), the sam-pling rate was necessarily set too low to capture thewhole process. This has limited the time resolution ofthe measurements. Despite their shortcomings, theseearly studies have produced very valuable informationconcerning the nature and modes of frictional sliding.Indeed, by using this type of setup, the two importantphenomena of frictional healing and velocity weaken-ing were first observed by Scholz et al. (1972) and byDieterich (1972).

Recently, two important modifications of the ori-ginal block-slider model were introduced. The firstone involved introducing a granular layer into thesliding interface to simulate the fault core (Gu andWong, 1994). The second one involved using softmaterials (e.g., foam rubber) and multipoint measure-ments performed by burying accelerometers insidemeter-sized blocks (Brune, 1973; Brune et al., 1993).The first type of test usually ignored the dynamicfeatures of the earthquake, and is thus beyond thescope of this chapter. As to the second type of test,namely, the ‘foam model’, the usage of slow wavespeed material has some distinct advantages overusing rocks in that: (1) the whole process can becaptured without the need of high sampling rates orhigh-speed photography and (2) 3-D effects can bemore effectively addressed, since it is relatively easy tobury gauges inside the model. However, some con-cerns exist regarding the model material itself. Theseinclude its nonlinear and viscous nature and the spe-cial frictional properties of the foam rubber (thecoefficient of friction can be larger than unity).Regarding their constitutive behavior, foam rubberscan sustain large deformations and are not well

described by the theory of linear elasticity. The pre-sence of material nonlinearities (both kinematic andconstitutive) raises concerns regarding the scalabilityof the results from the laboratory scale to naturallyoccurring earthquake ruptures. This together withtheir known viscoelastic response makes foam modelsunlikely candidates to simulate earthquakes, whichoccur in much stiffer and more brittle materials (i.e.,rocks) and also involve sliding processes extendingover lengths that are four to five orders of magnitudelonger than in the laboratory. Furthermore, like allother direct shear-type experiments, the ‘foammodel’ has also suffered from unavoidable edge effectsas first pointed out by Scholz et al. (1972). Despite thefew shortcomings described above, it should beemphasized that the foam rubber model has producedvery interesting observations whose scientific valuehas been to elucidate some of the basic physicsgoverning the sliding of frictional interfaces(Anooshehpoor and Brune, 1994, 1999; Brune andAnooshehpoor, 1997; Brune et al., 1993).

At least three important simultaneously acting pre-requisites are needed in order for an earthquake tooccur either in the laboratory or in the Earth’s crust.These are: proper tectonic loading, a pre-existing fault,and the occurrence of a certain type of triggeringmechanism. The first two prerequisites were verywell simulated and controlled in the previously men-tioned block slider and foam rubber models. However,the triggering of the rupture was mainly left uncon-trolled and was dominated by the configuration edgeeffects discussed above. The triggering mechanism isimportant for both the earthquake process and its sub-sequent evolution (Lu et al ., 2005) . In e xp er iments,control of the triggering mechanism is crucial since iteffects the synchronization of high-speed diagnosticsand also serves as an important boundary/initial con-ditions that may affect the repeatability and theinterpretation of the results. Except for one particularstudy specifically targeting the problem of the dynamictriggering of earthquakes by incoming Rayleigh waves(Uenishi et al., 1999), this aspect has largely beenignored by the previous literature. In this importantinvestigation, two photoelastic polymer plates of dif-ferent sizes were held together at their convex sides toform a fault (line of frictional contact) with an extendedfree surface. A Rayleigh wave pulse was subsequentlygenerated on the free surface by a point explosion andthis wave propagated along the free surface toward thecontact part (simulated fault). The incoming Rayleighpulse was used to trigger dynamic slip along the fault ina controlled and repeatable fashion. In addition to

Dynamic Shear Rupture in Frictional Interfaces: Speeds, Directionality, and Modes 155

controlling rupture initiation, these experiments werealso the first ones to feature the use of full-field opticaldiagnostics to study the dynamic rupture process in‘incoherent’ (frictional) interfaces.

In addition to the experimental studies describedabove, there also exists a class of dynamic shearfracture experiments originating from the engineer-ing community which are closely related to thephenomenon of geological fault rupture. As reviewedby Rosakis (2002), fracture mechanicians are recentlypaying more and more attention to the dynamicshear rupture propagating along material interfaceswhich are becoming common in modern engineeringstructures and composite materials. These interfacesusually feature finite strength and toughness and canresist both opening and shear loading. They are oftenreferred to as ‘coherent’ interfaces. Since they are ingeneral weaker than the constituent matrix material,these interfaces often trap dynamic ruptures, of boththe opening and shear types and serve as sites ofcatastrophic failure. On a much larger length scale,crustal faults provide natural weak frictional inter-faces where shear-dominated earthquake rupturesoccur under the superimposed action of tectonicpressure. Experimental methods for dynamic fracturemechanics, such as photoelasticity and coherent gra-dient sensing (CGS) combined with high-speedphotography, have been applied successfully in theexperimental study of shear ruptures propagatingdynamically along coherent interfaces in engineeringcomposites of various types (Coker and Rosakis,2001; Coker et al., 2003; Lambros and Rosakis, 1995;Rosakis et al., 1998, 1999; 2000). These two diagnostictechniques, which are 2-D in nature, are able toprovide very valuable full-field and real-time stressinformation. Specifically, the photoelastic method,which measures maximum in-plane shear stress con-tours, is an attractive candidate for the study of shear-dominated processes such as earthquake ruptures.

Further discussion of experimental methodologiesand results pertaining to the investigation of variousmodes of ru pture can be found in Secti on 4.06.5.1 . Inthe following section, we present the principles thathave guided the design of simple experiments invol-ving incoherent (frictional) interfaces whose purposehas been to simulate earthquake rupture in a highlycontrolled laboratory setting.

4.06.2.2 Experimental Design Philosophy

One goal in designing dynamic frictional experimentssimulating earthquake rupture has been to create a

testing environment or platform which could repro-duce some of the basic physics governing the rupturedynamics of crustal earthquakes while preservingenough simplicity so that clear conclusions can beobtained by pure observation. Another goal hasbeen to ensure that this platform would be versatileenough to accommodate increasing levels of modelingcomplexity. We have concentrated in creating experi-ments that are well instrumented and featurediagnostics of high temporal and spatial resolution.In order to keep the design as simple as possible, wehave decided to first concentrate on experimentsinvolving essentially 2-D processes which may bethought of as ‘experimental’ equivalents of the multi-plicity of 2-D numerical models of rupture which existin the open literature. Because of its predominant twodimensionality this configuration is perhaps bettersuited to mimic processes corresponding to largerearthquake events which have already saturated theseismogenic zone and as such, involve primarily 2-D,mode-II rupture growth. The design of the genericexperimental configuration, the earthquake triggeringmechanism, and the choice of model materials anddiagnostics are guided by the following principles:

4.06.2.2.1 Simulating tectonic loading

As shown in Figure 1, the crust is simulated by a‘large’ photoelastic plate with a thickness of 9.5 mmand 150 mm� 150 mm in 2-D plane dimensions. Theplate is large enough so that during the time windowof observation there are no reflected wave arrivals.The plate is cut into two identical quadrilaterals,which are then put together, and the frictional inter-face is used to simulate a fault. The angle of the faultline to the horizontal is denoted by � while theuniaxial pressure acting at the top and the bottomends of the sample is denoted by P. Both P and � aresystem variables. Following these definitions, theresolved shear traction � and the resolved normaltraction � along the fault can be expressed in termsof angle � and pressure P as

� ¼ P sin� cos� and � ¼ P cos2 � ½1�

To make the connection with commonly used geo-physics terminology, a dimensionless factor s, which isoften used by seismologists to describe the loadingalong faults with respect to the strength of the fault(Scholz, 2002), is introduced. By invoking the slip-weakening frictional model (Palmer and Rice, 1973)which is shown schematically in Figure 1(c), and bydenoting the maximum strength of the fault by � y and


the f in al stre ngth of th e fau lt b y � f, t hi s load ing fact or

(or dimensionless driving force) s is defined by s ¼(�

y�� )/(�� f). According to the linea r slip-weakening

fault strength model, � y is the static frictional strength

wh ile a nd � f is the dynamic frictional strength of the

interface. By denoting the static and dynamic coeffi-cients of frictions by �

s an d �d, respectively, and byus in g eqn [1], t he l oad in g factor s corresponding to thegeometry of the experiment of Figure 1 becomes

s ¼ ð�scos� – sin�Þ=ðsin� –�d cos�Þ ½2�

As is obvious fro m eqn [2] , s , and throug h it theearthquake rupturing process, can be controlled byvarying the fault angle � for given frictional proper-ties of the fault in the experiments. The magnitude ofthe uniaxial pressure P does not appear in s.However, it controls the total amount of deformationand the total slip.

4.06.2.2.2 Simulating the nucleation process

Researchers have tried to understand the nucleationprocess of earthquakes (triggering earthquake rup-ture) and indeed, they have shown that nucleation

can be understood by invoking proper friction rela-tions (Dieterich, 1992; Lapusta and Rice, 2003;Uenishi and Rice, 2003). Unfortunately, it is imprac-tical to explore the detailed physics of the nucleationevent experimentally and instead, we have decidedto mimic the way that triggering of rupture isachieved in numerical simulations. The triggeringof natural earthquakes can be achieved either by anincrease of the shear loading or by a decrease of thefault strength at a specific location. In the first casetriggering due to a loading increase can occur dyna-mically if the requisite jump of shear stress isprovided by stress waves generated by a nearbyearthquake source. It can also occur quasi-staticallydue to the presence of inhomogeneities or stressconcentrations along a fault (e.g., kinks or forks)which, as slip accumulates, may result in a criticalincrement of the local resolved shear stress on theinterface. In the second case, triggering occursthrough a local decrease of fault strength whichmay result, among other things, from the flow ofpore fluid into the fault interstice generating a localrelease of fault pressure. Both mechanisms (pre-stress

150

mm

α

(a)

150 mm

(b)

GC

t f

t y

t

Slip

She

ar s

tres

s

d0

(c)

Figure 1 Laboratory fault model (a) and the loading fixture inside a hydraulic press (b). Slip-weakening frictional law (c).


and pressure release) have been applied in numericalsimulations of earthquake rupture dynamics(Aagaard et al., 2001; Andrews, 1976; Andrews andBen-Zion, 1997; Cochard and Rice, 2000; Fukuyamaand Madariaga, 1998).

In the laboratory earthquake models described inthis chapter, the triggering mechanism is a localpressure release, which is achieved by the explodingwire technique shown in Figure 2. A capacitor ischarged by a high-voltage power supply. The char-ging time is determined by the resistance of thecharging resistor and the capacity of the capacitor.Upon closing the switch, the electric energy stored inthe capacitor causes a high current in a thin metalwire (buried inside a hole of 0.1 mm in diameter inthe interface) for a short duration. The high currentturns the metal wire into high-pressure, high-tem-perature plasma in approximately 10 ms. Theexpansion of the high-temperature, high-pressureplasma causes a local pressure release. The adjustablepower supply can provide electric potential in a widerange and different intensities and time durations ofexplosion can be obtained easily.

Before the explosion, the shear traction along thefault is less than the maximum static frictionalstrength. At the simulated hypocenter, after theexplosion, the local normal traction on the fault isreduced and so is the static frictional strength. As aresult, the applied shear traction, which is initiallysmaller than the static frictional strength and unaf-fected by the isotropic explosion, can be momentarilylarger than the reduced frictional strength. Theresulting net driving force, defined by the differencebetween the shear traction and the frictional resis-tance, drives the slip along the interface.Furthermore, the slip or the slip velocity also reducesthe coefficient of friction as described by either a slip-weakening, or a velocity-weakening friction law.In other words, the frictional strength changes from

static to dynamic. If the original shear traction is

larger than the dynamic friction, the slip continues

to propagate away from the explosion site (corre-

sponding to the hypocenter of an earthquake)where normal traction reduction due to the explosion

is not important any more. In this way a spontaneous

rupture or a laboratory earthquake is triggered.

4.06.2.2.3 Choosing appropriate model

materials and diagnostics

The diagnostic method of choice is dynamic photo-elasticity. This technique is a classical method which

measures the maximum shear stress in transparent,

birefringent plates (Dally and Riley, 1991). As such

the technique is very well suited for the study of

shear-dominated processes. Two photoelastic solids,

namely Homalite-100 and Polycarbonate, are used in

all of the experiments described in the various sec-

tions of this chapter. The two materials have been

chosen because of their enhanced birefringence rela-

tive to other transparent polymers such aspolymethyl methacrylate (PMMA). Relevant prop-

erties of several photoelastic materials are listed in

Table 1.

Switch

Capacitor

Exploding wire

Powersupply

Loading resistor

Figure 2 Schematic drawing of the exploding wire system coupled with a photoelastic fault model. Isochromatic fringes

due to the explosion are visible.

Table 1 Summary of optical and mechanical propertiesof Homalite-100 and Polycarbonate

Material propertyHomalite100 Polycarbonate

Young’s Modulus E (MPa) 3860 2480

Poisson’s Ratio v 0.35 0.38

Stress fringe value f�(kN m�1)

23.6 7.0

Yielding Stress �Y (MPa) 48.3 34.5

P Wave Speed CP (km s�1) 2.498 2.182

S Wave Speed CS (km s�1) 1.200 0.960Density � (kg m�3) 1230 1129


The stress fringe values quoted in the table are forgreen light at a wave length 525 nm. The static elasticproperties listed in Table 1 are from the literature(Dally and Riley, 1991), while the dynamic elasticproperties (wave speeds) were measured using5 MHz ultrasonic transducers.

A typical of dynamic photoelasticity setup isshown in Figure 3. A polarized laser provides ahigh-intensity beam continuously at a power levelof a few watts. The beam is then expanded by acollimator to a size of 100 mm or 130 mm in diameter.The large beam is transmitted through a combinationof circular polarizers and the transparent photoelasticspecimen, and it is arrang ed so tha t an isochrom aticfringe pattern is obtained and focused into the cam-era. The isochromatic fringe pattern obtained fromthe two-dimensional photoelastic model gives fringesalong which the in-plane princi pal stres s diffe rence�1�� 2 is equal to a con stant. When the frin ge ord erN is known, the in-plane principal stress differencecan be computed as follows:

�1 – �2 ¼ Nf�=h ½3�

where h is the plate thickness of the model material.The high-speed camera (Cordin 220) used can takepictures at spe eds up to 10 8 frames/s econd. In mos texperiments an interframe time of 2–4 ms was used.

Another and perhaps the most important reason ofusing brittle polymers in our experiments is the smallrupture nucleation length, LC, resulting when suchmaterials are frictionally held together by loads onthe order of 10–2 0 MPa. As one can see by us ing eqn

[1] in Secti on 4.0 6.3.3 , these lengths are on the order

of 5 mm and unlike rocks allow for relatively small

laboratory specimen sizes. If rock samples were to be

used instead of stiff polymers, the resulting rupture

nucleation lengths would be approximately 30 times

larger under similar far-field loading and surface

finish. Finally, both Homalite and Polycarbonate

are very brittle and are known to deform elastically

at strain rates exceeding 102 s�1.An example of a dynamic photoelastic pattern

corresponding to a ‘coherent’ (glued) and horizontal

(�¼ 0) interface subjected to far-field tension is

shown in Figure 4. This configuration which is of

relevance to dynamic self-similar fracture of the

mode-I (opening) type has been discussed in the

recent work by Xia et al. (2006).

Polarized laserbeam

Pressure

Laser

Collimator

Circular polarizer I

Leads

Explosion boxHigh-speed camera

Exploding wire

Circular polarizer II

Focusing lens

a

Figure 3 The setup of dynamic photoelasticity combined with high-speed photography for laboratory earthquake studies.

Figure 4 Specimen configuration and photoelastic fringe

pattern for self-similar crack growth of a mode-I (opening)

crack along a coherent interface. White arrows indicatecrack tips.


4.06.3 Supershear and Sub-Rayleighto Supershear Transition inHomogeneous Fault Systems

The question of whether natural earthquake ruptures

can ever propagate at supershear speeds was until

recently a subject of active debate within the seismo-

logical community. This is because of the often

insufficient field data, as well as the limited resolu-

tion and nonuniqueness of the inversion process.

However, a recent experimenta l study (Xia et al .,

2004) has almos t settled this ques tion by demo nstrat-

ing that under controlled laboratory conditions

supershear frictional rupture can occur under condi-

tions similar to those present in some natural settings.

A widespread view in seismology speaks of crustal

earthquake ruptures mainly propagating at sub-

Rayleigh speeds between 0.75 and 0.95 CR

(Kanamori, 1994). However, the multiplicity of inde-

pendently collected evidence warrants further

investigations of the mechanics of supershear rupture

(velocity faster than the shear wave speed of the rock,

CS) propagation. Whether and how supershear rup-

ture occurs during earthquakes has an important

implication for seismic hazard because the rupture

velocity has a profound influence on the character of

near-field ground motions (Aagaard and Heaton,

2004).In the 8 August 2003 issue of Science, Bouchon and

Vallee (2003) took advantage of the unusual length of

the rupture event and used both seismic waves and

geologically observed total slip distribution to infer

the rupture velocity history of the great Ms 8.1 (Mw

7.8) central Kunlunshan earthquake that occured in

Tibet on 14 November 2001. This is an extraordinary

event from the point of view of both earthquake

dynamics and dynamic rupture mechanics. The rup-

ture occurred over a very long, near-vertical, strike-

slip fault segment of the active Kunlunshan fault and

featured an exceptionally long (400 km) surface rup-

ture zone and large surface slip (Lin et al., 2002).

Although it may not be unique, their modeling sug-

gests speeds that are close to the Rayleigh wave

speed of the crust, CR, for the first 100 km of rupture

growth, and transitioning to a supershear speed for

the remaining 300 km of propagation. Their results

were later corroborated by Das (private communica-

tion, 2006).Recently, several other seismological reports also

point to the possibility of supershear ruptures. These

reports concentrate on events such as the 1979

Imperial Valley earthquake (Archuleta, 1984;Spudich and Cranswick, 1984), the 1992 Landersearthquake (Olsen et al., 1997), and most recentlythe 2002 Denali earthquake in Alaska (Ellsworthet al., 2004). The 1999 Izmit earthquake in Turkey(Bouchon et al., 2001) is yet another event featuring avery long segment of supershear rupture growth. Itshould be noted here that for most of those examplesmentioned above (Izmit and Denali excluded), thesupershear ruptures occurred only on short patchesalong the whole rupture length and the results arenot conclusive. Bouchon and Vallee’s (2003) papertogether with Das’s work on Kunlunshan, as well asthe work of Ellsworth et al. (2004) on Denali are themost recent of a series of investigations reportingsupershear rupture growth occurring during ‘large’earthquake events. Moreover, they present the firstseismological evidence for transition from sub-Rayleigh to supershear.

Classical dynamic fracture theories of growingshear cracks have many similarities to the earthquakerupture processes (Broberg, 1999; Freund, 1990;Rosakis, 2002). Such theories treat the rupture frontas a distinct point (sharp tip crack) of stress singular-ity. These conditions are close to reality in cases thatfeature strong ‘coherent’ interfaces of finite intrinsicstrength and toughness. The singular approach ulti-mately predicts that dynamic shear fracture isallowed to propagate either at a sub-Rayleigh wavespeed or at only one supershear speed, which is

ffiffiffi2p

times the shear wave speed. As a result, it excludesthe possibility of a smooth transition of a steady-staterupture from sub-Rayleigh to supershear speed for asteady-state rupture. The introduction of a distribu-ted rupture process zone has allowed fracturemechanics to better approximate the conditions thatexist during real earthquake events (Ida, 1972;Palmer and Rice, 1973). Based on this so-called cohe-sive zone fracture mode, there is a forbidden speedrange between CR and CS (Burridge et al., 1979;Samudrala et al., 2002a, 2002b). In the sub-Rayleighspeed range all speeds are admissible, but only theRayleigh wave speed is a stable speed; in the super-shear speed range all speeds are admissible, but onlyspeeds larger than

ffiffiffi2p

CS are stable (Samudrala et al.,2002b). Ruptures with unstable speeds will accelerateto a stable speed as determined by the fault strengthand the loading conditions. The theoretical results ofthe cohesive zone rupture model ultimately predictthat earthquake ruptures can propagate either atRayleigh wave speed or at supershear speeds lyingwithin the interval between

ffiffiffi2p

CS and CP.


Early theoretical results by Burridge and co-workers (Burridge, 1973; Burridge et al., 1979), alongwith num erical results by And rews (1976) and Dasand Aki (1977) have predicted the possibility ofsupershear rupture and have alluded to a mechanism(Rosakis, 2002) for transition from the sub-Rayleighto the supershear rupture velocity regime. This isoften referred to as the ‘Burridge–Andrews’ mechan-ism. According to the 2-D Burridge–Andrewsmechanism, a shear rupture accelerates to a speedvery close to CR soon after its initiation. A peak inshear stress is found sitting at the shear wave frontand is observed to increase its magnitude as the mainrupture velocity approaches CR. At that point, theshear stress peak may become strong enough to pro-mote the nucleation of a secondary microrupturewhose leading edge propagates at a supershearspeed. Shortly thereafter, the two ruptures join upand the combination propagates at a speed close toCP, the longitudinal wave speed of the solids. It isinteresting that this transition was also clearly visua-lized by recent 2-D, atomistic calculations (Abrahamand Gao, 2000; Gao et al., 2001) of shear rupture inthe microscale, which provided an impressivedemonstration of the length scale persistence (14orders of magnitude difference between the nanos-cale and the scale of natural earthquake rupture) ofthis sub-Rayleigh to supershear rupture transitionmechanism. The Burridge–Andrews mechanism isalso known as the mother–daughter mechanism inmechanics literature.

For mixed-mode (tensile and shear) ruptures, adifferent transition model has also been suggested(Geubelle and Kubair, 2001 ; Kubair et al ., 2002,2003). Based on num erical simu lation, they sugges tthat a mix-mode rupture can speed up and cross theforbidden speed range between CR and CS continu-ously. Finally, recent numerical investigations offrictional rupture have identified alternate, asperitybased, mechanisms that provide a 3-D rationalizationof such a transiti on ( D ay, 1982 ; Dunh am et al ., 2003 ;Madariaga and Olsen, 2000). In this case, 3-D effectsplay an important role in the transition. The rupturefront focusing effect provides extra driving force tospeed up the spontaneous rupture in the supershearrange.

The experimental confirmation of the possibility ofsupershear (intersonic) fracture followed many yearsafter the first theoretical predictions. Indeed, the longseries of experiments summarized in a recent review byRosakis (2002) demonstrate that intersonic crackgrowth in constitutively homogenous systems featuring

coherent interfaces (interfaces with inherent strength)is possible and may also occur in various combinationsof bimaterial systems. However, in all of thevarious cases discussed by Rosakis (2002), the crackswere nucleated directly into the intersonic regime andthere was no observation of a transition from sub-Rayleigh to supershear speeds. This was due to thenature of the impact induced stress wave loadingwhich was imparted in the absence of pre-existingstatic loading and the nature of the relatively strongcoherence of the interface (provided by adhesive). Themajor differences between the conditions during earth-quake rupture and those early fracture experimentshave left questions regarding the plausibility of spon-taneously generated intersonic rupture in frictionallyheld, incoherent interfaces unanswered. The recentwork by Xia et al. ( 2004) on friction ally held interfaceshas addressed this issue in detail. In the followingsections, w e w ill summari ze th eir most imp or tant find -ings and we will attempt to relate these findings to fieldmeasurements related to recent natural earthquakeevents.

4.06.3.1 Purely Sub-Rayleigh and PurelySupershear Earthquake Ruptures

In this section, experimental parameter ranges leadingto either exclusively sub-Rayleigh or exclusivelysupershear ruptures will be discussed. The configura-tion of interest is the one described in Section 4.06.2.2.The physics governing possible transitions betweenthese two speed regimes within a single ruptureevent will be examined later. Figure 5 shows twodifferent experiments featuring sub-Rayleigh speedruptures. In all the photographs, we can clearly seethe circular shear wave front emitted from the simu-lated hypocenter. Rupture tips, characterized by thestress concentration (fringe concentrations in thephotographs), are identified just behind the shearwave front. The P-wave is long gone out of the fieldof view (100 mm). The ruptures as shown in thephotograph feature right lateral slip while the rupturetips are equidistant from the hypocenter (symmetricbilateral slip). From the rupture length history, it ispossible to estimate the rupture velocity. In these twocases, the rupture velocities are very close to theRayleigh wave speed of the material.

All experiments featuring lower inclination angles� and lower magnitude of uniaxial compression pres-sure P involve exactly the same features and rupturesspeeds. This observation is consistent with manyobservations of earthquake rupture velocities in the


field. As we have discussed earlier, in the sub-

Rayleigh speed range the only stable rupture velocityis the Rayleigh wave speed according to various

cohesive zone models of rupture. The experimentsof Xia et al. (2004) described in this section confirm

this observation.In Figure 6, the inclination angle was kept at 25�

while the pressure was increased to 15 MPa. For

comparison purposes, the same time instants (28and 38 ms after nucleation) are displayed.

In this case, the circular traces of the shear waveare also visible and are at the same correspondinglocations as in Figure 6. However, in front of thiscircle supershear disturbances (propagating to theleft, marked in the photograph as the rupture tipand featuring a clearly visible Mach cone) areshown. The formation of the Mach cone is due tothe fact that the rupture front (tip) is propagatingfaster than the shear wave speed of the material.For this case, the sequence of images, other than

(a)

Rupture tipRupture tip

(b)

Figure 6 Experimental results for the purely supershear case. (a) and (b) both correspond to an experiment with the

pressure of P¼ 15 MPa and angle �¼ 25� at the two time instants of 28 and 38ms, respectively.

(a)

(c) (d)

(b)

Figure 5 Experimental results of purely sub-Rayleigh cases. (a) and (b) are from one experiment with a pressure of

P¼ 13 MPa and angle �¼20� at time instants of 28 and 38ms, respectively. (c) and (d) are from an experiment with a pressure

of P¼7 MPa and angle �¼25� at the time instants of 28 and 38ms, respectively. For (a) and (b), we can also identify twomode-I cracked in the lower half of the sample caused by the explosion itself. We expect that the effect of these cracks is

localized.


those at 28 and 38 ms, have a very similar form and

reveal a disturbance that was nucleated as supershear.

The speed history v(t) is determined independently

by either differentiating the rupture length history

record or measuring the inclination angle, �, of the

shear shocks with respect to the fault plane, and using

the relation v¼CS/sin �. The speed was found to be

almost constant and very close to the plane stress

P-wave speed CP of the material. These experiments

provide the first evidence of supershear growth of a

spontaneous shear rupture propagating along a fric-

tional interface. The supershear rupture initiated

right after the triggering of the earthquake rupture.

This is determined from the fact that the Mach cones

are nearly tangential to the shear wave front which

was itself created at the time of rupture nucleation.In previous experiments involving strong, coher-

ent interfaces and impact induced stress wave

loading, stable rupture velocities slightly exceedingffiffiffi2p

CS were observed (Rosakis et al., 1999). This

apparent discrepancy can be explained by referring

to the rupture velocity dependence on the available

energy per unit shear rupture advance within the

supershear regime (Samudrala et al., 2002b). This

dependence was obtained by means of an analytical

model of a dynamic steady state, shear rupture fea-

turing a shear cohesive zone of the velocity

weakening type. The initial, static, strength of the

interface was represented by the parameter �0. For a

given normalized far-field loading level, �D12=�0, this

energy, shown here in normalized form (Figure 7) as

a function of rupture speed, attains a maximum value

at speeds closer toffiffiffi2p

CS for interfaces that are

‘strong’. This is consistent with the purely singular

prediction which corresponds to idealized interfacesof infinite strength. To this idealized case, (�0!1),the only value of rupture velocity corresponding tofinite G is exactly

ffiffiffi2p

CS. For weaker interfaces (orhigher driving stresses), this maximum moves towardCP. In the situation described in this section, theinterface is ‘weak’ and the driving force (resolvedshear minus dynamic frictional force) is relativelylarge and constant. Hence, a rupture velocity closeto CP is expected.

4.06.3.2 The Experimental Visualization ofthe Sub-Rayleigh to Supershear EarthquakeRupture Transition

As discussed earlier, investigating how supershearruptures are nucleated experimentally is of greatinterest. For the experimental cases described in theabove section, the supershear rupture event wasnucleated immediately after triggering of the ruptureevent . Since the rupture velocity is controlled byboth the inclination angle � and the magnitude ofuniaxial compression P, it is possible to vary both ofthem carefully to suppress or perhaps delay theappearance of supershear rupture. Specifically, theinclination angle � was fixed at 25� while P wasdecreased in order to induce and capture the nuclea-tion process of a supershear rupture.

Figures 8(a)–(c) corresponds to a case with anintermediate far-field pressure compared to theones displayed in Figures 5(c) and 5(d) andFigure 6. Here, the angle is kept the same (25�) andthe pressure is decreased to 9 MPa in an attempt tovisualize a transition within our field of view(100 mm). Three different time instances of thesame rupture event are displayed. In Figure 8(a),the circular traces of both P- and S-waves are visible,followed by a rupture propagating at CR. InFigure 8(b), a small secondary rupture appears infront of the main rupture and propagates slightlyahead of the S-wave front. In Figure 8(c), the tworuptures coalesce and the leading edge of the result-ing rupture grows at a speed of 1970 m s�1 which isvery close to CP. A well-defined ‘shear’ Mach cone isvisible in Figure 8(c). Unlike Figure 6, the Machlines are not tangent to the shear wave circle sincethe supershear rupture was not generated at initia-tion. Figure 8(d) displays the length versus time ofthe two ruptures, in which the length is directly readfrom the figure. The figure also compared the slopesto the characteristic wave speeds of the materialbefore and after their coalescence. A magnified view

10–1

β∗ = –0.4

10–3

10–6

0.0∗

1.8

1.5

1.2

0.9

0.6

0.3

01 1.1 1.2 1.3 1.4 1.5 1.6 1.7

σ D /τα12

Plane stressv

=

0.34

2 β α

G / G

0

↑

υ / β →

√

Figure 7 Normalized energy release rate as a function of

intersonic rupture speed. The figure illustrates that the energy

flux available to the tip has a maximum at or above � ¼ffiffiffi2p

CS.


of the secondary rupture as it nucleates in front of themain rupture is shown in the inset. Both ruptures areindicated by arrows. The transition length L for thiscase is approximately 20 mm, as is visually obviousby Figure 8(b).

In cases where the transition length cannot bemeasured directly if it is too small, an indirectmethod is developed. As shown in Figure 9, at timeT0, the supershear rupture is nucleated at the inter-section of the shear wave front and the fault line(point A). If the transition length L ¼ OA is verysmall, spatial resolution may not be good enough tomeasure it accurately. Alternatively, a photographmay not have been taken at that instant. Assuming

that at all times before transition the rupture speed isvery close to CS and that the shear wave position (A)and supershear rupture-tip position (B) at a later timeinstance T1 can be measured, the transition lengthcan be inferred by pure geometry. To do so weobserve that relations OA ¼ CST0; OB ¼ CST1, and

AC ¼ V ðT1 –T0Þ hold provided that the supershearrupture tip also grows at a constant speed V > CS (asconfirmed by Figure 8).

A simple manipulation gives AC=OB ¼ð1 –T0=T1ÞV=CS, which leads to T0 ¼½1 – ðCS=V ÞðAC=OBÞ�T1. Multiplication of bothsides of the last relation by CS gives L ¼ OA ¼CST0 ¼ ½OB –OC � CS=V �. This relation allows forthe estimation of the supershear rupture transitionlength, L, even if no photographic records have beenobtained at or before the time instance of the transi-tion. Several results listed in this chapter were eitherobtained or verified by using this method.

4.06.3.3 A Theoretical Model for the Sub-Rayleigh to Supershear Transition

The above physical picture is comparable with theBurridge–Andrews mechanisms already described inthe introduction of this chapter. Andrews (1976,

P wave

10 mm

(a)

Secondary

S wave

(b)

10 mm

Main rupture

rupture

(c)

Secondary

S wave 10 mm

rupture

(d)

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.215 20 25 30

Time (ms)35 40 45

Rup

ture

-tip

pos

ition

P

SR

Figure 8 Visualization of the sub-Rayleigh to supershear rupture transition.

T1

T0

O A B C

Figure 9 Method of transition length L estimation.


1985) quantifie d this transiti on in a parameter space

spanned by a nor malized supers hear transition length

L/LC and the nondi mensio nal driving stres s par a-

meter s [s ¼ (� y – � )/( � – �

f)]. T he parameter s � ,�

y, and � f are the resolve d shear stress on the fault,

the static fricti onal strength, and the dynam ic

strength of the fault, respect ively; they describe the

linear slip-we akenin g fricti onal law ( Ida, 1972 ;

Palmer and Rice, 1973 ) us ed in Andr ews’

computa tions.The Andrews result can be symbolically written as

L ¼ LCf (s). The function f (s) h as be en gi ve n n um er i-

cally by Andrews as an increasing function of s, and

can be well approximated by the equation f (s)

¼ 9.8(1.77 – s)� 3. The normalizing length LC is the

critical length for unstable rupture nucleation and is

proportional to the rigidity G an d to d0, which is

defined as the critical or breakdown slip of the slip-

weakening model. LC can then be expressed as

LC ¼1þ v

ð�y – � f Þð� – � f Þ2

Gd0 ½4�

By adopting the above relation to the geometryof the experim ental configur ation and by using eqn

[1] , the transiti on length L is fou nd to be inversely

proportional to the applied uniaxial pressure P and

to be governed by the following general functional

form:

L ¼ f ðsÞ 1þ v

ð�s –�dÞðtan� –�dÞ2

Gd0

P½5�

Figure 10 displays the dependence of the transi-tion length L on pressure from a set of experiments

corresponding to the same inclination angle of 25�

(s¼ 0.5) and identical surface finish (roughness isabout 17 mm). The static frictional coefficient wasmeasured to be �s¼ 0.6 using the traditional inclinedplane method. In this method, one block was placedon top of an inclined plane and the inclination angleuntil the block slides. By this way the critical inclina-tion angle �C was measured. The static coefficient offriction was then determined from the relation �s¼tan�1 (�C). To estimate the average, steady state,dynamic frictional coefficient, � was graduallyincreased from 10� to a critical angle �c at whichslip was initiated under the action of far-field loadsand the dynamic triggering. It was assumed that theshear traction was approximately equal to thedynamic frictional strength at this critical angle �c.This angle was found to be between 10� and 15�,from which the average coefficient of dynamic fric-tion was estimated to be �d¼ 0.2. This low value ofdynamic friction coefficient is very consistent withrecent experimental measurements conducted byTullis and his research group at slip velocitiesapproaching seismic slip rates (Di Toro et al., 2004;Tullis and Goldsby, 2003). Using �s¼ 0.6 and�d¼ 0.2, the experiments can be compared to thepredi ctions of eqn [5] of Andrews ’ theor y as show nin Figure 10. Although the theory qualitatively cap-tures the decreasing trends of the experiments, thedata exhibit a dependence on pressure that is visiblystronger than P�1.

A natural way to modify Andrews’ results and tointroduce some scaling into the pressure dependenceof L is to consider the effect of pressure on the criticalbreakdown slip d0. As pointed out by Ohnaka, on thebasis of frictional experiments on rocks, and as alsosupported by dimensional analysis, there exists alinear relation between a characteristic surface length(half-distance between contacting asperities, denotedas D in this case) and the critical slip distance d0 as(Ohnaka, 2003)

d0 ¼ c½ð�y – � f Þ=� f �MD ½6�

where c and M are experimentally determined con-stants. In addition, D depends on the normal stress, �,applied on the fault (which is �¼ P cos2� in thiscase). As shown in Figure 11, a classical plastic con-tact model may be used to establish this dependence.In this model the average radius of n contactingasperities is a0 (assumed as a constant in thismodel). As the pressure over a macroscopic contactarea A (¼nD2) is increased, the number of contacts,

36

Experimental data

Andrews’ theoryModified theory

32

28

24

20

16

12

8

6 8 10Uniaxial pressure, P (MPa)

Tran

sitio

n le

ngth

, L (

mm

)

12 14 16 18 20

Figure 10 Transition length as a function of the far-field load.


n, as well as the re al contac t area Ar ¼ n a 20� �

, alsoincreases .

By defining the hardness H as the rati o of the totalnormal force N to the real contac t area Ar (Bowden

and Tabor, 1986 ), N can be expr essed as

N ¼ HAr ¼ Hna 20 ¼ �A ¼ AP cos 2 � ½ 7�

Substitu tion of A and Ar in term s of D and a0 gives

D ¼ffiffiffiffiHp

a0 cos � P – 1 = 2 . Usin g the linear relationbetween D and d0 , the breakdow n slip is furth erfound to depe nd on the pressu re as d0 � P� 1/2 . Bysubstitut ing the above relations into the expres sionrelating L and d0 / P, discussed abov e, a modifiedexpressi on of transiti on length to pressu re th at fea-tures a stronger depe ndence (L � P� 3/2 ) on pressu reemerges. For the loadi ng conditi ons and geom etry ofthe experim ental con figurati on pr esented inFigure 1 , this ex pression becomes

L ¼ f ðs Þ 1 þ v

G

� s –�

d

ðsin � –� d cos �Þ2 2 c�

s –� d

� s

� �M

�ffiffiffiffiHp

a0 P – 3= 2 cos – 1 � ½ 8�

As show n in Figure 10 , this modifi ed relation agreesvery well with the experim ental data presented inthis paper for appr opriate cho ices of material andgeometrical par ameters of the microm echani cs con-tact mode l. In plottin g Fig ure 10 , the hardness andthe elastic constants of Homa lite wer e used while thefriction coe fficients were taken to be �

s ¼ 0.6 and�d ¼ 0.2 as esti mated above. The values for the con-stants c and � were from the experiments of Ohnak a(2003) . The aspe rity radius a0 w as taken to be equalto 1.78 mm, for a best fit to the experim ental dat a.This value was found to be consiste nt to averagedmeasuremen ts of asperit y radii obtained throug hmicrosco pic observ ations of the slidi ng surfaces .Relation [8 ] can also easily be generali zed to biaxialloading con ditions. Indeed, if P1 ¼ P and P2 ¼ bP

denote the horiz ontal and ve rtical compon ents of

the applied loads su ch a situa tion, the transitionlength can be expr essed as

L ¼ f ½ s ð�Þ�1 þ v

G

2c ð� s –�

d Þ1 þ M ðcos2 �þ b sin 2 �Þ1 :5

½ð 1 – b Þsin � cos �–� d ðcos2 �þ b sin 2 �� 2

� ð� s Þ– M

ffiffiffiffiHp

a0 P – 1 :5 ½ 9�

In additi on to tectonic loading magnitude and direc-tion relativ e to fault inclinati on, eqn [9] involves thematerial pr operties G, v , and H, th e fricti onal proper-ties �s and �

d, as well as fault related length scale a0 .Because of this, it can perhaps be used as a simpl escaling relation connecti ng laboratory observ ationsto re ports of supershear transiti on in ‘larg e’ nat uralearthq uake rupture events su ch as these related to2001 Kunl ushan event in Tibet.

For making order of magnitu de level cont act withseismo logical applications , we rewrite the genera lform of eqn [ 5] in terms o f the effec tiv e s tress �

e ¼� – �

f , a co mm onl y us e d a n d e s tim at ed s e is mo lo-gica l parameter. Equation [5 ] can be rewritten as

L ¼ f ðs Þ 1 þ v

ð1 þ s ÞGd0

� e ½10 �

Appli cation of this equation to both seismic faultin gand lab oratory data allows us to scale the transitio nlength L from labo ratory to seism ological conditi ons.The effective stress �

e in our experim ent is chosen tobe of the sam e order of mag nitude as th at meas uredin seism ology. The ratio of rigi dity of the Earth’ scrust to Hom alite is about 25. From the experim entdescrib ed in Figu re 8 (P ¼ 9 MPa and �¼ 25 � ) thetransiti on length is esti mated to be L ¼ 20 mm fromwhich d0 ¼ 10 mm is obtained using eqn [4] . Thevalues of d0 for large earth quak es are often esti matedas 50 cm to 1 m (Ide and Takeo, 1997). If s is approxi-mately the same under laboratory and crustalconditions, the transition length for earthquakes canbe estimated to be in the range between 25 and 50 km.Because s can be different and the estimate of d0 forearthquakes is very uncertain at present, this value

Side viewTop view

2a 0 2D 2a0

2D

2D

Figure 11 Schematic drawing of the microcontact based frictional model. The top figure is the side view of the contact andthe bottom figure is the top view. As the normal force increases, the number of contacts, n, increase.


should only be taken as an order of mag nitude esti-mate. Nev ertheless, it is of th e same order as thatinferred for Kunlun shan (100 km) and for D enali(75 km). T he large trans ition length required forsupershear is perhaps one of the reasons tha t rela-tively few earthquak e events hav e be en obser ved tofeature su ch high rupture velociti es and that all ofthem corres pond to lar ge mag nitude earthq uakes. Ifthe tectonic stres s is well below the stat ic faultstrength (i.e., large s ), the trans ition length be comestoo large for many earthquak e ruptures to attainsupershear . The observa tion that during severallarge earthquak es the rupture velocity became ve ryfast, possib ly supers hear, suggests that the tectonicstress is fairly clo se to th e stat ic fault strength (i.e.,small s ), which has importan t impli cations for theevolution of rupt ure in lar ge earth quakes.

4.06.3.4 Dis cussion

A final word of caution is in ord er here. It hasrecently be en show n in rece nt pr eliminar y numericalsimulations of the homog eneous experimentsdescribed in this section (Lu et al ., 2005) that thedetailed variati on of the transiti on length on theload P may also str ongly depend on the nat ure ofthe ru pture nucleation pro cess. The severi ty of thiseffect is still unknow n. If this depe ndence pr oves tobe signi ficant, an experim ental investig ation invol-ving variou s pr essure release histo ries (inten sities,rise time s, and durat ions) will be necess ary in ord erto complete the experim ental inves tigation of super-shear transitio n presented above. A related issu eregarding the influ ence of nucl eation conditi ons ina bimat erial setti ng (see Section 4.06.4 ) wa s recentlydiscussed by Shi and Ben-Zion (2006) .

4.06.4 Directionality of Rupturesalong Faults Separating WeaklyDissimilar Materials: Supershear andGeneralized Rayleigh Wave SpeedRuptures

In mos t mature fault system s, the elastic pro pertiesvary acros s the fault plane ( Magistral e and Sand ers,1995 ; Pel tzer et al ., 1999 ) and also the shear wa vespeeds may vary by as mu ch as 30% ( Ben-Zion andHuang, 2002 ; Cochard and Rice, 2000 ). Rec ently,Rubin and Gillard (2000 ) studied several thous andsof pairs of con secutive earthq uakes that occurre d on asegment of the central San Andreas fault, south of the

Loma Pr ieta rupture. Amo ng the secon d events ofeach pair, th ey fou nd that ove r 70% more occurre dto the nor thwest than to the sou thwest. They inter-pret this asymme try as being a result of the contrastin material properties acros s th e faul t. Indeed, at thislocation of the San And reas Fault, the rock body ismore compl iant nor theast of the faul t than it is south-west (Eberhart-P hillip s and Michael , 19 98 ).

Early theoretical and n umerical stu dies of rupturet hat e mploy va rio u s f rict io nal l aw s wi th a co nsta ntcoefficient o f friction which would be ind epen dent ofslip or slip rate ( Ad ams, 1995 ; Andrew s a nd Ben- Zion,1997; Cochard a nd Rice, 2000 ; Harris and D ay, 1997 ;Heaton, 1990 ; Ranjith and Rice, 1999 ; Ric e et al ., 2001;Weertman , 1980 ) imply that if rupture occurs o n th eboun dary be tw een tw o frictionally he ld solids, havingdifferent elastic p ro perties and wave spe eds, such arupture may preferen tially propagate in a particu lardirection. T his is the same direction as the direction ofslip in the lower w ave speed solid. T his direction isoften referred to as the ‘preferred’ direction (Ben- Zion2001). Since such implied directionality of fault rup-ture may have a profound influence on the distributionof damage caused by earthquake ground motion, itwould be extremely useful if this behavior could beconfirmed under controlled laboratory conditions.While many of the physical aspects of dynamic rupture(including supershear) are recently becoming progres-sively clearer in relation to homogeneous faults (Ben-Zio n, 2 001 ; Rice, 2001; Rosakis, 2002; Xia et al ., 2004) ,the behavior of spontaneously nucleated ruptures ininhomogeneous faults, separating materials with differ-ent wave speeds, has until recently (Xia et al., 2005)remained experimentally unexplored. In this sectionwe elaborate on the results of this experimental inves-tigation and we discuss their implications in relation tothe theoretical concept of a ‘preferred’ rupture direc-tion in the presence of bimaterial contrast.

The recent large earthquakes (1999 Izmit andDuzce) and the seismic migration history along theNorth Anatolian Fault may represent a unique fieldexample of the effect of the material contrast acrossthe fault. The 1999 Izmit and Duzce events featuredboth supershear and sub-Rayleigh rupture branches(Bouchon et al., 2001). Most significantly, they are thelast of a series of large (M � 6.8) earthquakes thathave occurred since 1934 in the North AnatolianFault. These earthquakes have occurred on a ratherlong and allegedly inhomogeneous fault system (Zoret al., 2006; Le Pichon et al., 2003) that has hosted tensof major migrating earthquakes in the past century.Following the work of Stein et al. (1997) and


of Parsons et al. (2000), tens of large (M� 6.8) earth-quakes occurred over 1000 km along the NorthAnatolian Fault between the 1939 earthquake atErcinzan and the 1999 Izmit and Duzce earthquakes.Such a long series of earthquakes are believed to betextbook examples of how the transfer of stress from arecent nearby event can trigger the next major eventin due time. This presumably happens by adding ortransferring stress to the fault segment, which is adja-cent to the tips of a segment that has last failed. Thestress distribution is highly nonuniform and itoccurs in addition to the long-term stress renewaland to the pre-existing stress inhomogeneities.However, as much as this model seems to be com-plete and convincing, a few questions remain thatneed to be resolved and are of relevance to thework described here.

4.06.4.1 Two Types of Ruptures alongInhomogeneous Fault Systems

Inhomogeneous faults separate materials with differ-ent wave speeds. When such faults experiencespontaneous rupture, the equibilateral symmetry,expected in the homogeneous case, is broken. Thisleads to various forms and degrees of rupture direc-tionality. Dynamic rupture along bimaterialinterfaces is known to involve substantial couplingbetween slip and normal stress (Ben-Zion, 2001; Rice,2001; Weertman, 1980). As a consequence, the rela-tive ease or difficulty for a rupture to propagate in aspecific direction along a bimaterial interface shouldbe closely related to the degree of mismatch in wavespeeds. This phenomenon is also related to the fault’sfrictional characteristics which play a dominant rolein facilitating the phenomenon of normal shear cou-pling. For bimaterial contrast with approximately lessthan 35% difference in shear wave speeds (as in thecase of most natural faults), generalized Rayleighwaves can be sustained. These waves are waves offrictionless contact propagating at a speed, CGR,called the generalized Rayleigh wave speed(Rice, 2001).

The 1980 rupture solution by Weertman (1980)involves a dislocation like sliding pulse propagatingsubsonically with a velocity equal to CGR along aninterface governed by Amonton–Coulomb friction.However, the classical Amonton–Coulomb’s descrip-tion has been shown to be inadequate for addressingfundamental issues of sliding (Ranjith and Rice,2001), since sliding becomes unstable to periodicperturbations. Instability, in the above sense, implies

that periodic perturbations to steady sliding growunbounded for a wide range of frictional coefficientand bimaterial properties (Adams, 1995; Renardy,1992). The growth rate is proportional to the wavenumber. This issue has been discussed in relation ofboth homogeneous and bimaterial systems (Rice,2001). For bimaterial systems and in particular,when generalized Rayleigh waves exist, Ranjith andRice (2001) demonstrate that unstable periodicmodes of sliding appear for all values of the frictioncoefficient. Mathematically, instability to periodicperturbations renders the response of a materialinterface to be ill-posed (no solution exists to theproblem of growth of generic, rather than periodic,self-sustained perturbations to steady sliding). Theproblem is regularized by utilizing an experimentallybased frictional law (Prakash and Clifton, 1993), inwhich shear strength in response to an abrupt changein normal stress evolves continuously with time(Cochard and Rice, 2000; Ranjith and Rice, 2001).In such a case, the problem becomes well-posed andgeneric self-sustained pulse solutions exist whilenumerical convergence through grid size reductionis achieved (Cochard and Rice, 2000; Coker et al.,2005). However, despite the fact that this specialfrictional law provides regularization, self-sustainedslip pulses may still grow in magnitude with time.This is a phenomenon that has been demonstratednumerically by Ben-Zion and Huang (2002).Moreover, self-sustained pulses were found to existand to propagate at discrete steady velocities and atspecific directions along the inhomogeneous inter-face by analytical (Ranjith and Rice, 2001) andnumerical means (Andrews and Ben-Zion, 1997;Cochard and Rice, 2000).

Two types of such steady, self-sustained pulseswere discovered by Ranjith and Rice (2001) theore-tically. Consistent with Weertman’s analysis(Weertman, 1980), the first type corresponds to rup-ture growth in the direction of sliding of the lowerwave speed material of the system. This direction isreferred to in the literature (Ben-Zion, 2001; Rice,2001) as the ‘positive’ direction and sometimes as the‘preferred’ direction (Ben-Zion, 2001). The rupturepulses belonging to this type are subshear and alwayspropagate with a steady velocity V¼þCGR, wherethe plus denotes growth in the ‘positive’ direction.Thus, in this work these rupture pulses will bereferred to as ‘positive’ generalized Rayleigh rup-tures and will be abbreviated as ‘þGR’ ruptures.The second type of self-sustained rupture corre-sponds to growth in the direction opposite to that of


sliding in the lower wave speed material of the bima-terial system. This direction is often referred to as the‘negative’ direction or ‘opposite’ direction (Cochardand Rice, 2000). Such ruptures are supershear andthey always propagate with a steady velocity that isslightly lower than the P-wave speed of the materialwith the lesser wave speed V ¼ –C2

P

� �. Such rup-

tures are generated for sufficiently high values of thecoefficient of friction (Ranjith and Rice, 2001) andare less unstable than the ‘þGR’ ruptures describedabove (Cochard and Rice, 2000). In the present papersuch ruptures will be abbreviated as ‘�PSLOW’ rup-tures. This second type of rupture was later studiedby Adams (2001), who showed that the leading edgesof these supershear (intersonic) ruptures are weaklysingular, a result which is consistent with numericalanalysis (Cochard and Rice, 2000).

From the point of view of numerics, the earlywork of Andrews and Ben-Zion (1997) has broughtto light the persistence and interesting properties ofrupture pulses of the ‘þGR’ type. This was possibleeven in the ill-posed context of sliding governed byAmontons–Coulomb friction before much of the the-oretical concepts were at hand. In their work, thesliding ‘þGR’ pulses were triggered by a local releaseof interfacial pressure spread out over a finite regionat the interface and over finite time. No pulses of thesecond type (‘�PSLOW’ pulses) were excited in theseearly simulations despite the fact that the coefficientof friction was high enough to have permitted theirexistence as suggested by the modal analysis ofRanjith and Rice (Cochard and Rice, 2000). Perhapsthis is not altogether surprising if one considers thefact that in these early simulations, nucleation wasachieved by a ‘biased’ localized stress drop having afavored propagation direction. In particular, the‘positive’ direction of growth was artificially seededin the nucleation process (Ben-Zion, 2006). The sub-sequent numerical simulations of Cochard and Rice(2000), which utilized the modified Prakash andClifton Law, still featuring a constant coefficient offriction, were able to sequentially excite regularizedself-sustained pulses of both types. This was achievedby introducing small changes in the parameters of thefriction law and in the geometry of the nucleationzone. At the same time, no simultaneous excitation ofboth modes was reported. Moreover, the ‘�PSLOW’pulses were found to be slightly more difficult toexcite than the ‘þGR’ pulses. However, the degreeof relative difficulty was not examined in detail. Inpartial agreement to the above 2-D numerical stu-dies, Harris and Day (1997) demonstrated the

simultaneous existence of both types of slidingmodes, propagating in opposite directions duringthe same rupture event. They considered variousbimaterial and trilayered configurations featuringmodest wave speed mismatch and a slip-weakeningfrictional law. As first pointed out by Xia et al. (2005),the inconsistency between the various numerical stu-dies is most probably due to the different frictionlaws utilized. Indeed, up to very recently, all studiesexcept this of Harris and Day (1997) have assumed aconstant coefficient of friction. The need for experi-mental analysis becomes clear at this point since onlyexperiments can be used to judge the physical rele-vance of various assumed friction laws and to validatevarious proposed numerical methodologies. Asemphasized by Xia et al. (2005), the goal of some ofthe early theoretical and numerical studies (Adams,1995, 1998; Andrews and Ben-Zion, 1997; Cochardand Rice, 2000; Ranjith and Rice, 2001; Weertman,1980) was to investigate what kind of unstable slipwould develop on a surface which, as judged fromconventional friction notions, was superficiallystable, in the sense that its friction coefficient, �, didnot decrease, or vary otherwise, with slip and/or sliprate. For most brittle solids, however, ample evidenceexists that, � does decrease with increase of slip and/or slip rate (or, more fundamentally, � varies withslip rate and contact state). As a result, a propermodel for natural faulting along a bimaterial interfaceshould include both a weakening of � and the slip-normal stress coupling effects of the bimaterial situa-tion. Indeed, various models for � are expected tostrongly influence the effectiveness of the bimaterialcontrast in either enhancing or retarding rupturegrowth. Such a weakening model was first includedby Harris and Day (1997) in a bimaterial context.Given the above, Xia et al. (2005) first pointed outthat it would be an invalid interpretation of theresults of the earlier set of papers (Adams, 1995,1998; Andrews and Ben-Zion, 1997; Cochard andRice, 2000; Ranjith and Rice, 2001; Weertman,1980) to conclude that the rupture scenarios (includ-ing preference for specific rupture mode) which theypredict constitute the full set of possibilities availableto a real earthquake, of which � decreases withincreasing slip and/or slip rate. Indeed, according toXia et al. (2005) the consistently bilateral nature ofrupture predicted by Harris and Day (1997) is anindication of the effect of including a strongly slip-weakening frictional law in their calculations. Inaddition, recent refined calculations by Harris andDay (2005) have reconfirmed their conclusions


within a fully 3-D setting. T hese con clusions, howe ve r,have re cen tly been partiall y c hallenged by Shi andBen-Zion (2006) on the b asis of a n exte ns ive p ar a m ete rstudy. A more complete analysis of this on-goingdiscussion w ill be presen ted in Se ct ion 4 .06.4. 5.

In the immediately following section, we elabo-rate on recent experiments (Xia et al., 2005) designedto investigate some of the issues discussed above. Wealso briefly summarize the recent debate on the exis-tence or nonexistence of a preferable rupturedirection which has been triggered by theseexperiments.

4.06.4.2 Experimental Setup

The experiments described in this section mimicnatural earthquake rupture processes in fault systems,where bimaterial contrast between intact rock massesseldom feature more than a 30% difference in shearwave speeds (Rice, 2001).

The laboratory fault model is similar to the onedescribed in Figure 1(a) and is shown in Figure 12.The figure shows a Homalite-100 plate (material 1,top) and a polycarbonate plate (material 2, bottom)that are held together by far-field load, P. The higherwave speed material at the top (Homalite-100) has a

shear wave speed CS1¼ 1200 m s�1 and a longitudinal

wave speed CP1 ¼ 2498 m s�1. The lower wave speed

material at the bottom (Polycarbonate) has a shearwave speed CS

2¼ 960 m s�1 and a longitudinal wavespeed CP

2 ¼ 2182 m s�1. The fault is simulated by africtionally held contact interface forming an angle tothe applied load that is varied to mimic a wide range

of tectonic load conditions. Spontaneous rupture istriggered at the hypocenter through the explodingwire mechanism described in Chapter 1.02. The sta-tic compressive load P is applied through a hydraulicpress. By arbitrary convention, the fault line runs inthe east–west direction with the lower wave speedsolid located at the south side. This choice was moti-vated in order to eventually facilitate comparisonswith the 1999 Izmit earthquake in Turkey. Accordingto the work of Le Pichon et al. (2003), the material tothe south of the North Anatolian Fault (at its westernend near the sea of Marmara) is the lower wave speed

solid. As viewed from the camera, a rupture willproduce right lateral slip.

The ratio of shear wave speeds, C1S=C2

S ¼ 1:25,was chosen to be within the naturally occurring

bimaterial range so that the interfacial phenomenacan be applied to some field observations. In parti-cular, the bimaterial difference is big enough to allowfor a high-enough growth rate of sliding instabilities(Rice, 2001) and to permit the clear distinctionbetween various wave speeds. Within roughly thesame range generalized Rayleigh waves exist aswell. The shear wave speeds were directly measuredfor each material by following the shear wave frontsthrough high-speed photography and photoelasticity.The listed P wave speeds were calculated by using

measured values of Poisson’s ratios (v1¼ 0.35,v2¼ 0.38) and by using the listed shear wave speeds.An independent measurement of the P-wave speedsin the plates using ultrasonic transducers confirmedthese listed values to within 5%. The value of CGR

can be determined (Rice, 2001) from the equation:

f ðV Þ ¼ 1 – b21

� �a1G2D2 þ 1 – b2

2

� �a2G1D1 ¼ 0

where an ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 –V 2= Cn

Pð Þ2;

qbn ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 –V 2= Cn

S

� �2q

;

Dn ¼ 4anbn – 1 – b2n

� �2, V is the rupture speed, Gn are

rigidities of the two materials, and n¼ 1,2.Substituting the material constants for Homalite-100 and Polycarbonate into the equation, the gener-alized Rayleigh wave speed is calculated asCGR¼ 950 km s�1. This is a value that is extremelyclose to the shear wave speed of Polycarbonate.

Pressure, P

Material 1(Fast)

Hypocenter

Material 2(Slow)

150 mm × 150 mm

a

Fault

North

Figure 12 Laboratory earthquake fault model composed

of two photoelastic plates of the same geometry.


4.06.4.3 Experimental Results

More than 30 experiments featuring different angles,

� (20�, 22.5�, and 25�), and far-field loading, P (10–

18 MPa), were performed and the repeatability of

rupture events was confirmed. The higher level of

angles was limited by the static frictional characteris-

tics of the interface. Depending on P and �, three

distinct and repeatable rupture behaviors were

observed. In all cases, the two separate, semicircular

traces of the shear waves in the two materials were

clearly visible as discontinuities in the maximum shear

stress field. The ruptures were always bilateral and

became progressively asymmetric with time, within

the time window of all experiments. As shown in

Figure 13, two distinct rupture tips, one moving to

the west and the other moving to the east, with velo-

cities VE and VW, respectively, were identified by a

distinct concentration of fringe lines. For this case

(case-1), both tips propagate at subshear velocities

VE < VW < C2S < C1

S. Differentiation of the rupture

length–time histories, obtained from a series of high-

speed images, allows for the estimation of the rupture

velocity histories. On the one hand, the rupture mov-

ing to the west is the one propagating in the direction

of sliding of the lower wave speed material (positive

direction). Within experimental error this rupture

was found to grow at a constant velocity equal to

the speed of the generalized Rayleigh waves

(VW¼ 950 m s�1�þCGR). The rupture moving to

the east, on the other hand, was the one propagating

in the direction opposite to that of sliding in the lowerwave speed material (opposite direction). This rupturegrew at an almost constant sub-Rayleigh velocity ofVE¼�900 m s�1, which is clearly slower than theRayleigh wave speed, C2

R, of the slower wave speedmaterial. The observations were very similar for smal-ler angles, �, and compressive loads, P, as well. Therupture to the west (positive direction) always propa-gated with þCGR. The rupture to the east remainedsub-Rayleigh VE < C2

R < C1R

� �. However, its velocity

varied across experiments with different load levelsand angles. In particular, smaller angles of � (or smal-ler values of the s factor described earlier) and lower P

resulted in VE being lower fractions of C2R. Judging

from the number of near-tip fringes per unit area, theeastward moving rupture resulted in a visibly smallerlevel of stress drop than the one moving to the west.This observation is consistent with predictions byCochard and Rice (2000) and Adams (2001) who alsopredict a weaker singularity for ruptures moving inthe negative direction.

A very distinct but equally repeatable rupture case(case-2) was observed for higher values of � and P.These conditions correspond to much higher valuesof driving stress or to conditions closer to incipientuniform sliding of the entire interface (smaller valuesof s). A typical example corresponding to �¼ 25� andP¼ 17 MPa is shown in Figure 14. In this case therupture is still bilateral with a westward tip trailing

Figure 13 The photoelastic patterns for an experimentwith �¼22.5, P¼ 17 MPa, and smooth surface. Both

ruptures to the east and the west are subshear (case-1).

Figure 14 For �¼25�, P¼ 17 MPa, and smooth

surface finish the bilateral rupture features two distinct

tips. The one moving to the west (positive direction) has

a velocity VW�þCGR, while the one moving to the east(opposite direction) is supershear. (case-2).


behind both shear wave traces. This tip moves at aconstant velocity VW�þCGR along the ‘positive’direction. This observation is identical to the situa-tion described above in relation to lower values � and

P. The eastward moving tip however is clearly dif-ferent from the previously described case. Its tip ismoving with a velocity faster than both the shear

wave speeds. Moreover, its structure, shown in detailin the upper insert of Figure 14, is distinctly differentto the structure of the sub-Rayleigh, westward mov-ing rupture shown in the lower inset.

As conclusive proof of its supershear velocity, twodistinct shear shock waves are clearly visible. Themagnitude of the velocity of the eastward rupture VE

is 1920 m s�1, which is approximately 12% less than theP-wave speed, C2

P, of the lower wave speed material. VE

is also equal to 1.6 C1S, or is slightly higher than

ffiffiffi2p

C1S.

The upper inset in Figure 14 shows two clear lines of

discontinuity in the maximum shear contours of photo-elasticity. Each of these lines (shear shock waves) islocated at two different angles 1¼ 41� and 2¼ 30�,to the north and to the south of the fault respectively.The two angles n (n¼ 1, 2) are related to the shearwave speeds Cn

S and to the rupture velocity VE, by

n ¼ sin – 1 VE=CnS

� �. This relation provides indepen-

dent means of estimating VE from each individualframe of the high-speed camera record without relianceon the less accurate rupture length history. Both meth-ods yield consistent values of VE¼�1920 m s�1.

Both cases described above feature westward mov-ing ruptures that are of the ‘þGR’ type. Irrespective ofthe values of � and P, these ruptures have a constant

speed VW�þCGR, and they propagate in the ‘positive’direction. However, those two cases also feature east-ward ruptures that are distinctly different in nature. For

sufficiently low P and � (or large s), the eastward

ruptures, which propagate in the opposite direction,

are purely sub-Rayleigh within the time window of

our experiments. For large-enough P and � however,

eastward ruptures propagate in the opposite direction

with a constant supershear velocity, which is slightly

less than C2P and are thus of the ‘�PSLOW’ type.

To visualize an intermediate situation and a con-trolled transition from one case to the other within

the field of view, P was reduced to 13 MPa. For this

case (case-3), Figure 15 shows a smooth transition

from case-1 to case-2 within the same experiment.

While the westward rupture remains of the ‘þGR’

type throughout the experiment, the eastward rup-

ture jumps from a constant sub-Rayleigh velocity

(�910 m s�1) to a constant supershear velocity

(�1920 m s�1), and thus transitions to the ‘�PSLOW’

type. The rupture length plot of Figure 16 also

Figure 15 Experimental results for �¼25�, P¼13 MPa, and rough surface showing transition of the eastward moving

rupture to supershear. The westward rupture retains a constant velocity VW�þCGR (case-3).

55

50

45

West East

East ruptureWest rupture

40

35

30

25

2040 20 0 20

Distance (mm)

Tim

e

40 60 80

950 m s–1 ≈CGR

910 m s–1 ≤ C

2 ≤ C1

S S

1920 m s–1≈C

2P

Figure 16 Rupture length plot of an experiment for

�¼ 25�, P¼13 MPa, and rough surface finish.


shows the abrup t transiti on of the eastward rupturefrom a sub-Rayl eigh ve locity to a veloc ity slightlyless than C 2P . This happe ns at a transitio n lengt h, L,which is appro ximately equal to 25 mm . Howe ver,the westwa rd rupture retains its con stant þ CGR velo-city throu ghout the ex periment. The eastwardtransition beha vior of case-3 is qua litatively simila rto the one w e hav e discussed in relation to homo-geneous interf aces (X ia et al ., 2004), while thetransition length, L, is also a decreas ing fu nction of� and P . Mos t imp ortant to the disc ussion of thepresent paper is the obser vation that the rupturesthat pro pagate to the opposite directi on re quire acertain minim um rupture length be fore they becomesupershear . T his obser vation provides a clear intui-tive link between super shea r growth in th e ‘oppos ite’direction and large earth quakes. In con trast, no suchtransition was observed for ‘positive ly’ growing‘þ GR’ ruptur es irrespect ive of � , P , and rupturelength. As a re sult, the experim ents do not providean obvious link between ‘pos itively’ growin g rupt uresand large ea rthquakes.

In other words, a certain minim um length of rup-ture growth in the negat ive directi on is requiredbefore such a transition can be observed. This sug-gests that in smaller earthquake events, such as the2004 Parkfield rupture such a transition may nothave had the chance to happen before the negativemoving rupture was arrested. (for more on Parkfield,see Secti on 4.06.4. 5).

In Section 4.06.3 .3, we hav e dis cussed the depen-dence of the transition length L on the uniaxialpressure P. In the homogeneous case there is a verywell-defined point for transition, while in the inho-mogeneous case the transition point is not always soclear. This difference again is due to the presence of amaterial contrast. In the homogeneous case, there isan energetically forbidden velocity zone between CR

and CS (Rosakis, 2002). As a result, the secondarycrack is initiated exactly at the shear wave front. Incontrast for inhomogeneous fault systems, the forbid-den zone no longer exists (Rosakis, 2002) and thesubshear crack in the opposite direction acceleratesto the supershear speed in a smoother way(Figure 16). Nevertheless, we can still define thetransition length where a rapid speed change occurs.The plot of transition length L is in Figure 17. L has aweaker dependence on P (L � P�0.4) than the homo-geneous case (L � P�1.5). This is expected becausefor the inhomogeneous case, and for subshear rupturespeeds, the coupling between the slip and the normaltraction causes a local rupture-tip compression. This

compression increases the resistance to slip for rup-ture growing in the negative direction. On the otherhand, the shear traction driving the rupture is con-stant and hence it takes a longer slip distance for arupture in bimaterial interface to reach the super-shear velocity.

4.06.4.4 Comparison of the ExperimentalResults to Early Numerical and TheoreticalStudies

The experiments by Xia et al. (2004, 2005) describedabove provide the first full-field and real-time visua-lization of dynamic frictional rupture eventsoccurring along inhomogeneous interfaces, whichfeature low-wave-speed mismatch such that the gen-eralized Rayleigh wave speed can be defined. Whileit is very difficult to access whether the ruptures arepulse-like, crack-like, or a mixture of the two, theobservations confirm the existence of two distinctself-sustained and constant speed rupture modes.These bare strong similarities to the ones that havebeen theoretically and numerically predicted overthe recent years (Andrews and Ben-Zion, 1997;Ben-Zion, 2001; Ben-Zion and Huang, 2002;Cochard and Rice, 2000; Rice, 2001; Weertman,1980). In particular, a ‘þGR’ type of rupture modeis always excited instantaneously along the ‘positive’direction of sliding. Besides the ‘þGR’ rupture mode,a ‘–PSLOW’ mode is observed as long as the rupturepropagating in the ‘opposite’ direction is allowed togrow to sufficiently long distances from the hypo-center. The triggering of the ‘–PSLOW’ mode isalways preceded by a purely sub-Rayleigh, crack-like

36

32

HomogeneousInhomogeneous

L~P–1.5

L~P–0.4

28

24

20

16

12

8

8 10 12 14Uniaxial pressure, P (MPa)

Tran

sitio

n le

ngth

, L (

mm

)

16 18 20

Figure 17 Transition length as a function of pressure P for

experiments with �¼ 25�. The data and fitting curves for thehomogeneous case are included for comparison.


rupture whose velocity depends on loading, on geo-metry, and on the bimaterial characteristics.Therefore, the existence of this preliminary andapparently transient stage is one of the main differ-ences with the early numerical predictions (Andrewsand Ben-Zion, 1997; Cochard and Rice, 2000).However, its existence does not contradict early the-oretical studies (Adams, 2001; Ranjith and Rice,2001), which can only predict stable rupture eventswhose constant velocities relate to the wave speeds ofthe bimaterial system.

A far more striking difference to some of the earlynumerical predictions (Andrews and Ben-Zion, 1997;Ben-Zion and Huang, 2002; Cochard and Rice, 2000) isthe consistent experimental observation of bilateral slip.In contrast to the experiments, the above numericalpredictions seem capable only of exciting one or theother of the two self-sustained rupture modes (Cochardand Rice, 2000), giving rise to purely unilateral ruptureevents. They also seem to primarily favor the triggeringof the ‘þGR’ mode in low-wave-speed mismatch bima-terial systems (Andrews and Ben-Zion, 1997). This kindof preference has led to the labeling of the ‘positive’direction as the ‘favored’ rupture direction and ofþCGR as the ‘favored’ rupture velocity. These numer-ical results indirectly support the closely related notionof ruptured directionality (McGuire et al., 2002). Anotable exception to this rule is provided by the earlynumerical analysis by Harris and Day (1997), as well astheir subsequent work (Harris and Day, 2005; Andrewsand Harris, 2005) which consistently reports asym-metric bilateral rupture growth in a variety of low-speed contrast, inhomogeneous fault systems. Theseresults are qualitatively very similar to the experimen-tal observations of cases 1 and 2. In particular, the latter3-D results by Harris and Day (2005) also clearly reporton sub-Rayleigh to supershear transition for the rup-ture propagating along the negative direction. Asbriefly discussed by Cochard and Rice (2000) and Xiaet al. (2005), the excitation of various modes or theircombinations should be related to the details of thenumerically or experimentally implemented triggeringmechanisms. In an attempt to further reconcile theobserved differences between various models and theexperiments, Xia et al. (2005) have noted that unstableslip rupture propagation has also been observed (Xiaet al., 2004) on homogeneous Homalite/Homalite andPolycarbonate/Polycarbonate interfaces. Such unstablerupture growth would be possible only if there was asubstantial reduction of the friction coefficient with slipand/or slip rate, and hence such reduction must be aproperty of both these model materials when sliding

against each other. It is then plausible to assume that asimilar reduction of friction coefficient occurs along theHomalite/Polycarbonate interface, and to thus inferthat its rupture behavior should not be expected tofully correspond to the idealized models of a dissimilarmaterial interface with constant coefficient of friction(Adams, 1995, 1998; Andrews and Ben-Zion, 1997;Cochard and Rice, 2000; Ranjith and Rice, 2001;Weertman, 1980). Indeed, the stronger the weakeningbecomes, the reduced frictional resistance is expectedto further neutralize the effect of bimaterial contrast onnormal shear coupling thus rendering the ‘preferable’rupture direction less preferable than originallythought.

The present experiments neither support exclu-sivity nor show a strong preference for rupturedirection. Although they support the idea that fric-tional ruptures that grow in the positive directionwill always do so at a specific constant velocity(V¼þCGR), they still allow for a significant possibi-lity of self-sustained supershear ruptures growing inthe opposite direction. This possibility becomes sig-nificant, provided that their transient, sub-Rayleighprecursors grow for a large-enough length and arenot arrested prior to transitioning to supershear. Therequirement of a critical transition length along the‘opposite’ direction provides a link between largeearthquakes and the occurrence of self-sustainedsupershear rupture in the ‘opposite’ direction. Oneperhaps can contemplate the existence of a weakstatistical preference for positively growing ruptures,since this link to large earthquakes is absent for‘þGR’ ruptures.

4.06.4.5 The Parkfield EarthquakeDiscussion in the Context of Experimentsand of Recent Numerical Studies

The Parkfield earthquake sequence presents a veryinteresting case in the context of bimaterial ruptureand the issue of the existence, or lack of, a preferablerupture direction. The slip on the San Andreas Faultis right-lateral, and the crust, near Parkfield, on thewest side of the fault features faster wave speeds thanthe east side (Thurber et al., 2003). The three mostrecent major Parkfield earthquakes ruptured thesame section of the San Andreas Fault. The rupturedirections of the 1934 and the 1966 events weresoutheastward (positive direction), whereas the 2004earthquake ruptured in the opposite direction (nega-tive direction). Following the discussion above,results from the early constant friction coefficient


studies in bimateri als would imply that the positive(southeas tward) directi on of the 1934 and the 1966events is the prefe rable directi on. Accordi ng to thisnotion, the negativ e (northwes tward) directi on of the2004 earth quake wo uld not be favored. The experi-ments describ ed above (Xia et al ., 2005 ), however,have clearly demonstrat ed that a rupture in the nega-tive direction can indeed occur at least in modelmaterials. Such a rupture, dependi ng on tectonicloading conditio ns and on available rupture length,may grow at either su b-Rayleigh or at supers hearrupture spe eds. Motivated by the mos t recent 2004Parkfield event an d suppor ted by the Xia et al. (2005) ,Harris and Day (2005) extende d their ea rlier 1997work to three dimens ions and discusse d their resu ltsin relation to exte nsive obs ervational evidence col-lected from the pa st 70 year s of ea rthquakes at thevicinity of Par kfield. They con cluded that naturalearthquak e rupture propagati on directi on is unlikel yto be predictabl e on the basis of bimateri al contrast .

T he scientific disc ussion on the notion of rupturedirectiona lity is however far from being over. In arecent paper, Shi and Ben-Zion (2006 ) have pre-sented an extens ive parameter stud y for variousbimateri al contrasts which clearly shows the effectof ass umed frictiona l pr operties and nucl eation con-ditions in pro moting either unilatera l (pr eferred) orbilateral (nonpre ferred) rupture in the presence ofbimateri al contras t. T heir study utilized a classicallinear slip-weak ening mode l of fricti on of the sametype used by Har ris and Day (1997 , 2005) . It showsthat for sm all differences �

s – �d � 0.1 be tween thestatic and dynam ic coeffici ent of friction and fornonbiased nucl eation events , the bimat erial effectindeed induces ruptures which prim arily grow inthe posit ive directi on at þ CGR . These cases are con-sistent with the constant coeffici ent friction mode ls.However, when the difference be tween the two coef-ficients of fricti on is taken to be lar ger than or equalto 0.5 an d for 20% shear wave speed contrast , bilat-eral rupture in both directi ons be comes possibl e withspeeds eq ual to þ CGR in the posi tive directi on, anddepending on the nucl eation conditio n, with eithe rsubshear or supers hear (– PSLOW ) speeds in the nega-tive directi on. These cases are consi stent with theexperiments and perhaps with the Parkfield observa-tions. Their results also agree with the above-discussed explanation, originally offered by Xia et al.(2005), regarding the root cause of the differencesbetween the experimental results and the predictionof a preferable direction in the early numerical stu-dies. Indeed the difference of friction coefficient in

the experiments is expected to be in the range of0.4–0.5 (see Secti on 4.0 6.2.2.3 for the homogene ouscase) which, within the slip weakening assumption,would promote bilateral ruptures of various types.Moreover, this recent study does not conclusivelysettle the issue of rupture direction preference innatural earthquakes where frictional characteristicsare generally unknown. Nevertheless it clearlyshows that the main arbitrator in whether bimaterialcontrast is capable of inducing rupture directionality,through normal to shear coupling, is the frictionallaw. Clearly, much more work is needed to investi-gate this phenomenon. The inclusion of realisticfrictional laws including strong velocity weakeningor enhanced rate and state frictional descriptionswhich regularize the rupture problem could be afirst step. Another step would necessitate the numer-ical and experimental study of a variety of bimaterialcontrasts and of more naturally based nucleationconditi ons. Nuclea tion conditi ons are show n by Luet al. (2005) (homog eneous case) and by Shi and Ben-Zion (2006) and Rubin and Ampuero (2006) (inho-mogene ous case) to have a strong influ ence onsubsequent rupture growth. We believe that transi-tion lengths for example may be heavily dependenton nucleation conditions. The recent papers byAndr ews and Harr is (2005) an d Ben-Zion (2006) , aswell as the work of Rubin and Ampuero (2006) arealso very important recent references reflecting theon-going discussion on this subject.

4.06.4.6 Discussion of the Historic, NorthAnatolian Fault Earthquake Sequencein View of the Experimental Results

The 1999, M7.4, Izmit earthquake in Turkey is per-haps a prime example of a recent large earthquakeevent for which both modes of self-sustained rupturemay have been simultaneously present, as is the casein our experiments. The event featured right-lateralslip and bilateral rupture of a rather straight strike-slip segment of the North Anatolian Fault shown inFigure 18. As reported by Bouchon et al. (2001), thewestward propagating side of the rupture grew with avelocity close to the Rayleigh wave speed, while theeastward moving rupture grew at a supershear velo-city that was slightly above the

ffiffiffi2p

CS times the shearwave speed of crustal rock. The visible difference inthe nature of the seismographs from stations ARCand SKR situated almost equidistantly to the hypo-center (star) support their conclusion (see bottom ofFigure 18). Since the laboratory ruptures of the


current paper are intentionally oriented similarly tothe Izmit event (lower-wave-speed material to thesouth as indicated by Le Pichon et al., 2003), a directcomparison with the case described in Figure 14becomes possible and it reveals some striking simila-rities between Bouchon et al., (2001) interpretation ofthe earthquake and the experiment. In addition tofeaturing right lateral slip and asymmetric bilateralrupture, this experimental case (case-2) featured asubshear westward rupture propagating at þCGR.To the east however, the rupture propagated at avelocity slightly lower than C2

P, which also happensto be equal to 1.6 C1

S for the particular bimaterialcontrast of the experiments. If one interprets theIzmit event as occurring in an inhomogeneous faultwith the lower-wave-speed material being situated atthe southern side of the fault (as is in the experiment),the field observations and the experimental measure-ments of both rupture directions and velocities arevery consistent. Moreover, when the bimaterial con-trast is low enough, the differences between CGR andthe average of the two Rayleigh wave speeds,

C1R þ C2

R

� �=2, as well as the difference between an

inferred rupture speed of 1.6 C1S and

ffiffiffi2p

C1S þ C2

S

� �=2

would be small enough not to be discriminated by theinversion process, even if the fault geology was com-pletely known. In this respect, the agreement withexperiment is as good as it can ever be expected. In

addition, viewing the fault as inhomogeneous can

explain the choice of direction for both the sub-

Rayleigh and the supershear branches, respectively.

This choice of rupture direction is consistent with

both the present experiments and with the theories

reviewed in the introduction.The 1999 Duzce earthquake can also be inter-

preted through a similar line of argument used for

Izmit. The Duzce rupture also featured right lateral

slip (as did all events that occurred in the North

Anatolian Fault between 1939 and 1999) and it

extended the Izmit rupture zone 40 km eastward

(negative direction) through asymmetric bilateral

slip (Bouchon et al., 2001). Thus, similar to the Izmit

earthquake, numerical modeling by Bouchon et al.

(2001) indicates both sub-Rayleigh westward and

supershear eastward rupture fronts. As a result the

direct comparison with case-2 described in Figure 14

provides an explanation for the two rupture direc-

tions and respective velocities, similar to the one

given for Izmit. This explanation is of course plau-

sible only if one assumes, once again, that the

material to the south of the North Anatolian Fault

(at its western end) is the lower-wave-speed solid.

Strong evidence supporting this assumption has

recently been presented by Le Pichon et al. (2003).By using similar arguments to the ones used for

Izmit and Duzce, one can perhaps attempt to provide

ARCSKR

29° 30°

29° 30°

(a)

(b)ARC SKR

0.21 g

0.41 g

0 5 10 15 20 25 30 0 5 10 15 20 25 30

PPP-waves

S-wavesE-W

Time (s)Ground acceleration

Time (s)

30° 00°

30° 00°

30° 30°

30° 30°

31° 00°

31° 00°

Figure 18 (a) Map view indicating the hypocenter of 1999 Izmit earthquake and two seismological stations ARC and SKR;

(b) Seismograms obtained at ARC and SKR. Reproduced from Bouchon M, Bouin MP, Karabulut H, Toksoz MN, Dietrich M,and Rosakis AJ (2001) How fast is rupture during an earthquake? New insights from the 1999 Turkey earthquakes.

Geophysical Research Letters 28(14); 2723–2726.


a unified rati onalizati on of the seeming ly randomrupture directi ons and rupture ve locities of the inter-related seri es of earthquak es that occu rred since 1939along the North Anatolian Fault and ende d in 1999with the Izmi t and Du zce events. T he foll owingargument requires the assum ption tha t, in averageand along its entire length, the North Anatol ianFault featu res the sam e type of bimat erial in-h omo-geneity as the one th at has been su mmarize d forIzmit and Du zce. Howeve r, rathe r limited ev idencesupportin g such an assumption is currently available(Zor et al ., 20 06 ). If in some average sense this is tru e,one wo uld expect that the slight maj ority (60%) ofthe large ( M � 6.8) earth quake events (i.e., (1939-M7.9), (1942-M 6.9), (1944-M7 .5), (1951-M 6.8),(1957-M6 .8), (1967-M 7.0)) which featu red westw ardgrowing ruptures, were pro bably of the ‘ þGR’ type.In other words, this assum ption impli es that theywere classical su b-Rayleigh ruptures that movedwith ve locity equal to þ CGR in the ‘positive ’ direc-tion. The remaining ruptures of the seri es were‘special’ in the sense that they featured domi nanteastward growth. As previous ly detailed, out of theremaining fou r ru ptures of the seri es, the Izmi t andDu zce ev ents w ere bilatera l with a wester n branch ofthe ‘þ GR’ type (c onsistent with the oth ers) and aneastward, su pershear branch of the ‘ � PSLOW ’ type.The 1943 and 1949 rupt ures were purely unidirec-tional and eastward moving; howev er, their rupturevelocities are not known. If these ruptures are to beconsistent with the remaining events in the sequencethen they could also have developed as the ‘�PSLOW’type. This possibility is more likely for the 1943event that featured over 250 km of growth length.As estimated in Secti on 4.06.2 .2.3 , this length ismuch larger than the critical length required fortransition to supershear. By observing that the 1943and the two 1999 (Izmit and Duzce) events were of ahigher magnitude than most of the other events of thecomplete series, as reported by Stein, et al. (1997),further supports the assertion that at least three out offour ‘special’ events featured partial or total super-shear growth along the ‘opposite’ (eastward)direction. The basic support for this assertion is pro-vided by the experimentally established link betweenlarge earthquakes and supershear ruptures growingin the ‘opposite’ direction, and is consistent with thedirect evidence of supershear from the two mostrecent ‘special’ events of 1999.

Finally, it should be noted that if earthquakes oflesser magnitude (in the range between M6.4–6.8) arealso included in the discussion, the North Anatolian

Fault series will feature a weak preference for wes-tern propagation. This is not very surprising giventhe above discussed link between large earthquakesand self-sustained supershear along the oppositedirection, a link that does not exist for ‘positive’(westward growing) ‘þGR’ ruptures. Indeed, in addi-tion to the actual number of ruptures that grew to theeast or west, what is of importance here is the actualgrowth lengths. The results reported by Stein et al.(1997) show that the total length of westward growthis only slightly higher than that of eastward growth.This is again consistent with experiments whichshow that the self-sustained ‘þGR’ mode is alwaysand instantaneously present after nucleation. In con-trast, the self-sustained ‘�PSLOW’ mode is oftenpreceded by an unstable subshear phase. For smallerearthquakes this unstable phase may never transitionto supershear and instead it may be arrested. This inturn would result in a total eastward rupture length,which is slightly shorter than the total western rup-ture length of the earthquake series.

4.06.5 Observing Crack-Like, Pulse-Like, Wrinkle-Like and Mixed RuptureModes in the Laboratory

The duration of slip at a point on an interface (orfault) in comparison to the duration of the rupture ofthe entire fault is a central issue to the modeling ofearthquake rupture. There are two widely acceptedapproaches to the description of dynamic sliding(Rice, 2001). The most classic approach uses elasto-dynamic shear rupture (or crack) models in whichthe surfaces behind the leading edge of the rupture(rupture tip) continuously slide and interact throughcontact and friction. More recently, pulse-like mod-els in which sliding occurs over a relatively smallregion have been introduced (Heaton, 1990). Inthese models, the sliding is confined to a finite lengthwhich is propagating at the rupture speed and isfollowed by interfacial locking.

In faults separating identical materials (homoge-neous systems) the crack-like mode of rupture hasoften (but not exclusively) been generated in manynumerical simulations of spontaneous rupture whena rate-independent friction law was implemented(Madariaga, 1976; Andrews, 1976, 1985; Das andAki, 1977; Day, 1982; Ruppert and Yomogida, 1992;Harris and Day, 1993). It has been pointed out, how-ever, that inversions of seismic data for slip histories,from well-recorded events, indicate that the duration


of slip at a point on the fault is one order of magni-tude shorter than the total event duration. An attemptto explain this phenomenon has given rise tothe concept of a pulse-like rupture mode (Heaton,1982, 1990; Hartzell and Heaton, 1983; Liu andHelmberger, 1983; Mendoza and Hartzell, 1988,1989). Some eyewitness accounts have also reportedshort slip durations during some earthquakes(Wallace, 1983; Pelton et al., 1984).

The concept of a pulse-like rupture went against awidely accepted view of how seismic rupture occurs.Its introduction was followed by various efforts toilluminate the physics leading to this process throughanalytical and numerical investigations. Differentmechanisms for ‘self-healing’ pulse generation alongfaults in homogeneous systems have been proposed(Heaton, 1990). One postulation is that if the faultstrength is low immediately behind the rupture frontand is increased rapidly at a finite distance, then slipmight be restricted to a short, narrow propagatingarea (Brune, 1976). Recent theoretical and numericalinvestigations show that a strong velocity-weakeningfriction law model could indeed allow for pulse-likebehavior of rupture under certain conditions (Zhengand Rice, 1998). However, simulations utilizing velo-city weakening have sometimes resulted in crack-likeruptures and sometimes in ‘self-healing’ pulses (e.g.,Cochard and Madariaga, 1994, 1996; Perrin et al.,1995; Beeler and Tullis, 1996; Ben-Zion and Rice,1997; Lapusta et al., 2000; Cochard and Rice, 2000;Nielsen et al., 2000; Coker et al., 2005). Friction lawsalong interfaces between two identical elastic solidshave to include laboratory-based rate and state evo-lution features and they must not exhibit illposednessor paradoxical features (Cochard and Rice, 2000;Ranjith and Rice, 2001). It has been proved thatgeneralized rate and state friction laws are appropri-ate candidates for homogeneous fault systems(Cochard and Rice, 2000; Ranjith and Rice, 2001;Rice et al., 2001; Zheng and Rice, 1998). Within theframe of rate and state friction laws, the followingthree requirements have to be fulfilled for rupture tooccur as a ‘self-healing’ pulse (Zheng and Rice, 1998;Rice, 2001). One requirement is that the friction lawmust include strengthening with time on slippedportions of the interface that are momentarily instationary contact (Perrin et al., 1995). Another isthat the velocity weakening at high slip rates mustbe much greater than that associated with the weaklogarithmic dependence observed in the laboratoryduring low-velocity sliding experiments. This isoften termed ‘enhanced’ velocity weakening. Lastly,

the third requirement is that the overall driving stresshas to be lower than a certain value, but high enoughto allow for self-sustained pulse propagation (Zhengand Rice, 1998).

While strong velocity weakening is one mechan-ism which explains the onset of short duration slip-pulses along interfaces (faults) which separate similarmaterials, it is important to note that other mech-anisms exist as well. One involves geometricconfinement of the rupture domain by unbreakableregions (barrier model). That is, sliding consists of anumber of crack-like ruptures of short duration on asmall rupture area that are separated by lockedregions (Aki, 1979; Papageorgiou and Aki, 1983a,1983b). In one implementation of this scenario, apulse-like rupture behavior was found in a 3-D geo-metry when the rupture process was confined withina long but narrow region by unbreakable barriers(Day, 1982). It was observed that the rupture startsin a classic crack-like mode and it propagates in alldirections. After some time, arresting waves arrivefrom the boundaries and they effectively relock thefault behind the rupture front, resulting in two slippulses. Alternatively, a rupture nucleates and propa-gates bilaterally, but may arrest suddenly at a strongbarrier at one end. Following its arrest, the reflectedwaves from the barrier spread back and heal therupture surface. The combination of the still-propa-gating end of the rupture with the healing reflectedwave forms a moving pulse-like configuration(Johnson, 1990).

In general, narrow slip pulses can be generatedduring dynamic sliding along interfaces by stronglyvelocity-weakening friction on a homogeneous sys-tem, by strong fault zone heterogeneities (Rice, 2001)or by variations in normal stress along the ruptureinterface. All of the above conditions (velocity weak-ening, heterogeneities, and local normal stressvariation) can produce slip pulses with low dynamicstress at the active part of the slip. An extensivediscussion of the subject is presented by Nielsenand Madariaga (2003). Most relevant to the discus-sion here is a set of recent finite element calculationswhich have been carried out by Coker et al. (2005) fora configuration which is very similar to the experi-mental setup described in the next section. Thesesimulations have shown that it is possible to generatea great variety of dynamic rupture modes (cracks,pulses, and pulse trains) propagating along an inter-face characterized by a rate-enhanced, rate- andstate-dependent frictional law. The choice of rupturespeed and rupture mode clearly depends on the load


intensity and rate but it is also very sensi tive to thetype of con stitutive law emp loyed, as well as thevalues of its parameters .

4.06.5.1 Experi mental Inve stigati on ofDynamic Rup ture Mode s

As discus sed in Secti on 4.06.2.1 , the maj ority of theexisting experim ental studies on sliding concentr ateon proce sses occu rring over lar ge time scales (lar gecompared to wa ve transit within the spe cimen) andare con cerned with developin g relationshi ps betweentime-aver aged fricti on data an d variou s sys tem par a-meters. Many dynam ic friction laws mot ivated byusing various exper imental configu rations and appa-rati lack the reproduci bility of friction da ta (surveyby Ibrah im, 1994 ) to be of defini te value to thetheorists. The results of these experim ents are multi-branched friction ve rsus slip velocity curves, whicheven for the same material and the sam e experimen-tal configur ation depend not only on the pro pertiesof the fricti onal interf ace but also on the dynam icproperties of the appar atus, suc h as mass, stiff ness,and damp ing. T his sugges ts that the fricti on dat aobtained in the course of stick-s lip motion s do notpurely reflec t the intrinsic properties of the surfac esin con tact but are also greatly affected by several ofthe dynam ic variabl es involved in each par ticula rexperimental setu p. Anoth er shortc oming of theexperiments described above, in relation to theirrelevance to re al earthq uakes, is the achievablerange of slidi ng speeds. T ypically, natural earth -quakes involve sliding velociti es on the ord er of1–10 m s� 1 while these exper iments involve slidi ngin the range of 1 mm s� 1 to 1 mm s � 1 . An exc eption tothis rule are the experim ents of Pra kash and Clifton(1993) and of Prakash (1998) , who employed a plate-impact pressu re-shear fricti on con figuratio n to inves-tigate the dynam ic sliding respon se of an incohe rentinterface in the micros econd timescal e (slidingspeeds on the order of 10 m s� 1 ). The ex perimentalresults, deduc ed from the response to step chang esimposed on the normal pressu re at the fricti onalinterface, reinfor ce the importan ce of incl uding fric-tional memor y in the devel opme nt of the rate-dependent state variable friction models. However,this setup is essentially 1-D. As such it is not able toprovide information about the detailed stress fielddeveloped along the length of the interf ace nor toshed light on the nature (e.g., uniformity or lack of ) ofthe rupture process during dynamic sliding. As inmany cases described in Section 4.06.2.1.1 , the

implicit assumption crucial to the interpretation ofthe data is that both stress and sliding processesremain spatially uniform. Another exception is thevery recent work by Tullis and his co-workers (DiToro et al., 2004; Tullis and Goldsby, 2003) who haveapproached seismic slip rates in experiments invol-ving crustal and quartz rock. They have investigatedthe phenomenon of flash heating as the mainmechanism resulting in severe velocity weakening.They have shown that the phenomenon of flashheating (Rice, 1999) can be experimentally linkedto the observed behavior of drastic velocity weaken-ing observed in their experiments. Finally, thescarcity of fast sliding velocity data in the openliterature has to be emphasized. We believe thatthis is an essential data set which could be provedas the ultimate arbitrator for the validation of variousdynamic sliding models and friction laws.

Rubinstein et al. (2004) conducted experiments toinvestigate the onset of dynamic sliding. They con-sidered two blocks of PMMA separated by a roughinterface and subjected to vertical static compressionand to a gradually increased horizontal driving force.During the sliding initiation, they recorded the rela-tive change of the net local contact area as a functionof time by measuring the changes of the light inten-sity transmitted across the interface. The transmittedlight was imaged by a camera. Because of the lowframing rates of their recording apparatus, theirexperimental configuration was particularly suitedto the study of ‘slow’ phenomena, with respect tothe characteristic wave speeds of the material. Theydiscovered slow waves propagating at speedsbetween 40 m s�1 and 80 m s�1, which is one orderof magnitude slower than the Rayleigh wave speed ofPMMA. These detachment waves, which are perhapsrelated to Schalamach waves, because of their lowpropagation speeds (Schallamach, 1971), are veryimportant since they give rise to a significant amountof slip. There are some doubts, however, to whetherthe characteristics of these waves are determinedonly by the properties of the surfaces in contact andby the loading. Because of their low rupture speeds, itis also likely that the dynamic properties of the entirespecimen may play an important role in the forma-tion of these waves. If the above is true, then the slowdetachment waves, though very important to slidingof finite bodies, may be less relevant to natural earth-quakes. Another very interesting question, which alsoremains to be answered, is whether these ‘detach-ment’ waves generate opening at the interface, thusjustifying their name.


As it is clear from the experiments presented bothin Section 4.06.2.1.1 and above, most of the experi-mental methods employed up to this point lack thespatial or/and the temporal resolution to address someimportant questions on dynamic sliding related to thesliding initiation process, to the duration of sliding (orsliding mode), and to the changes in the stress fielddistribution along the interface. This is particularlytrue in cases where sliding speeds on the order of1 m s�1, or above, are involved. To remedy this situa-tion, the physical plausibility of generating pulse-likeand crack-like rupture modes along ‘incoherent’ (fric-tional) interfaces in homogeneous systems has beeninvestigated by Lykotrafitis et al. (2006a) in a numberof well-controlled experiments. The sliding speedswere on the order of 1 m s�1. This sliding speed is ofthe same order of magnitude as the one expected inmost ‘large’ natural earthquake rupture events.

Dynamic photoelasticity combined with a new laserinterferometry-based technique has provided directphysical evidence of the rupture mode type, theexact point of rupture initiation, the sliding velocityhistory at a point on the interface, and the rupturepropagation speed. A summary of these experimentsand a discussion of the conditions leading to variousmodes of ruptures is given in the following sections.

4.06.5.1.1 Specimen configuration and

loading

Two Homalite-100 plates, subjected to a uniformcompressive stress, were frictionally held along theinterface. The top plate was also subjected to anasymmetric impact shear loading (Figure 19). Eachof the plates was 139.7 mm long, 76.2 mm wide, and9.525 mm thick. The average roughness of the sur-faces in contact was approximately Ra¼ 400 nm. All

Homalite

Ste

el

d

(a)

Laser beams

V

M1

M2Interface

Vibrometerheads

Homalite

Homalite plate

surface

Homalite plate

surface

Reflective

membrane

Reflective

membrane

(b) (d)

Reflective

membrane

d

A

B

Homalitest

eel V

Homalite

Interface

Homalite plate surface

Homalite plate surface

Reflective

membrane

(c)

Figure 19 (a) Schematic illustration of the experimental configuration for the sliding velocity measurement. The area inside the

dotted circle is shown magnified. Points M1 and M2 were at the same horizontal distance d from the impact side of the Homaliteplate. (b) Photography of the actual setup displaying the arrangement of the velocimeters’ heads for the measurement of the

sliding velocity. (c) Schematic illustration of the experimental configuration for the relative vertical velocity measurement. The

area inside the dotted line is shown magnified. Points M1 and M2 are at the same horizontal distance from the impact side of the

Homalite plate. (d) Photography of the actual setup displaying the arrangement of the velocimeters’ heads.


the experim ents were executed at the same externalconfining stres s of P ¼ 10 MPa, appli ed by a cali-brated press. The asym metric impa ct loading wasimposed via a cy lindrical stee l projecti le of diam eter25.4 mm and length 50. 8 mm, fire d using a gas gun.The imp act speeds ran ged from 9 m s �1 to 72 m s� 1 .A steel buff er 73 mm high, 25.4 mm long , and9.525 mm thick wa s attache d to the impac t side ofthe upp er plate to prevent shatteri ng by direct impa ctand to ind uce a more or less planar loadi ng wave.

4.06.5.1.2 Using particle velocimetry to

mea su re s l ip a nd op en ing historie s

The time evolutio n of the dynamic stres s field in thebulk was recorded by high-speed digital photographyin conjunction with dynamic photoelasticity (seeSection 4.06.2. 1). In additi on, a new techniqu e basedon laser interferometry was adapted to locally mea-sure the horizontal and vertical components of thein-plane particle velocity at various points above andbelow the sliding interface, thus allowing for themeasurement of slip and opening velocity histories.The combination of the full-field technique of photo-elasticity with the local technique of velocimetry isproved to be a very powerful tool in the study ofdynamic sliding.

The configuration employed in the measurementof the slip velocity is as follows. A pair of indepen-dent fiber-optic velocimeters was used to measurethe horizontal particle velocities at two adjacentpoints M1 and M2 across the interface. A schematicillustration of the experimental setup is shown inFigure 19(a), whereas a photograph of the setup isshown in Figure 19(b). The vertical distance of eachpoint from the interface was less than 250 mm. Bothpoints had the same horizontal distance from theimpact side of the Homalite plates. By subtractingthe velocity of the point below the interface (M2)from that of the point above the interface (M1), therelative horizontal velocity history was obtained. Thevelocimeter consists of a modified Mach–Zehnderinterferometer and a velocity decoder. The decoderwas set to a full range scale of 10 m s�1 with amaximum frequency of 1.5 MHz and a maximumacceleration of 107 g. The beam spot size wasapproximately 70 mm, whereas the error of the velo-city measurements was 1%. The technique and thecorresponding experimental setup are presented indetail in Lykotrafitis et al. (2006b).

Following the same strategy, a pair of indepen-dent velocimeters was employed to measure thevertical in-plane components of the velocities of

two adjacent points M1 and M2, located across theinterface. The relative vertical velocity was obtainedby adding algebraically the corresponding velocities.The arrangement of the velocimeters is shown sche-matically in Figure 19(c), whereas a photograph ofthe actual setup is shown in Figure 19(d). These twotypes of measurements enabled them to record var-ious modes of rupture propagating along thefrictional interface.

4.06.5.2 Visualizing Pulse-Like and Crack-Like Ruptures in the Laboratory

The specimen was subjected to a uniform confiningpressure of 10 MPa, whereas the impact speed wasV¼ 19 m s�1. An instantaneous isochromatic fringepattern is shown in Figure 20(a). An eye-like fringestructure is observed traveling, from right to left,behind the longitudinal wave front. The rupture tipA followed this fringe structure at a supershear speedof 1.36CS. Consequently, two Mach lines forming ashear Mach cone emanated from the rupture tip. Thetip can be located by tracing the Mach lines to theinterface. The rupture-tip speed was found to beconstant.

The time evolution of the horizontal relativevelocity of two adjacent points M2 and M1, belongingto the upper and lower plate respectively at a hor-izontal distance of 70 mm from the impact side of theHomalite plate, is displayed in Figure 20(b). If theentire time history of photoelastic frames is takeninto consideration in relation to the velocimetertime record, then this record can be superimposedon the photographs (see red line trace inFigure 20(a)) by converting time to spatial distance.

When the longitudinal wave front arrived at themeasurement positions M1 and M2 of Figure 20(a),where the pair of the interferometers was pointed,the velocities of both points started to increase.However, the relative horizontal velocity was zerofor the next few microseconds and it remained verylow for a time interval of approximately 13 ms. Anumerical integration of the relative velocity withrespect to time from 0 to 13 ms resulted in an accu-mulated relative horizontal displacement of 2mmbetween points M1 and M2, which can easily beidentified to elastic shear deformation prior tosliding. Indeed, the actual sliding started at approxi-mately 13 ms when the rupture tip (point A asidentified by the photoelastic image) arrived at themeasurement position and the relative velocityincreased sharply. The correlation between the two


measurements has allowed us to thus establish anestimate of elastic displacement (�2 mm) precedingthe sliding event for a confining pressure of 10 MPa.After it reached its maximum value of approximately6 m s�1, the relative velocity decreased and then itfluctuated, but never dropped below 4 m s�1 duringthe recording time. As is evident from Figure 20(b),the sliding was continuous and thus we can safely saythat rupture occurred in a classic crack-like mode.

As was already noted, the speed of the rupture tipwas substantially higher than the shear wave speed ofHomalite, and therefore the situation is similar tointersonic shear rupture propagation observed tooccur along both ‘coheren t’ ( Rosak is et al., 1999 ,2000; Needleman, 1999; Coker and Rosakis, 2001;Samudrala et al ., 2002a , 2002b ) and ‘in coherent’ (Xiaet al ., 2004) interfa ces sepa rating iden tical monolithi csolids. In the former experiments, the ‘coherent’interfaces were bonded and they featured intrinsicstrength and toughness in the absence of confiningpressure. Unlike the present study, the resultingmodes were always crack-like and the rupture speedswere not constant throughout the event.

In contrast to these early shear crack growthexperiments, the present work involves incoherentor frictional interfaces and static far-field compres-sive loading. Here, the frictional resistance to slidingdepends on the normal stress through the frictionlaw. The normal stress, however, is a superpositionof the static externally imposed pressure and adynamic (inertial) compression generated by the

impact loading as follows. The P-wave produced bythe impact loading creates a primarily horizontalcompressive stress in the upper plate close to theinterface, and due to the Poisson effect it also createscompression in the direction vertical to the interface.As the sliding proceeds, the vertical stress to therupture interface changes, and thus the frictionchanges as well. The change in friction, however,affects the evolution of sliding. Thus, we infer thatsliding is dependent on impact loading not onlythrough the horizontal compression, which is thedriving force for sliding, but also through the verticalcompression which affects the resistance to sliding.Because of that dependence, essential changes in therupture process are to be expected as the impactspeed is decreased.

In order to investigate the above line of reasoning,the impact speed was reduced to 15 m s�1.Figure 21(a) shows an instantaneous isochromaticfringe pattern obtained at the above impact speed.The corresponding relative horizontal velocity his-tory, measured at 30 mm from the impact side of theHomalite plate, is shown in Figure 21(b). By com-bining the horizontal relative velocity measurementwith the recorded photoelastic frames, we can iden-tify two rupture tips A1 and A2, which are fringeconcentration points and are propagating along therupture interface at speeds of 1.09CS and 0.98CS,respectively. The deformation at the velocitymeasurement position remains elastic until approxi-mately 18 ms when the rupture tip A1 arrived there

Rel

ativ

e ve

loci

ty (

m s

–1)

(b)

7

6

5

4

3

2

1

00 5 10 15 20 25 30 35 40 45 50

A

Time (μs)

Crack-like

(a)

M1

10 mm

M2

A

A

θ

θ

10

6

4

2

0

Figure 20 (a) Isochromatic fringe pattern generated during an experiment for which the impact speed was 19 m s�1 and the

external compression was 10 MPa. The rupture tip is at the fringe concentration point A. The insets highlight the location of the

Mach lines emanating from the rupture tip and the specimen configuration. The slip profile is superimposed (red line).(b) Relative velocity history of points M1 and M2 located at a distance of 70 mm from the impact side of the Homalite plates.

The rupture commenced when the rupture tip A reached the velocity measurement position.


and sliding commenced. As in the previous case, thecommencement of slip corresponds to an accumulatedrelative horizontal displacement of 2mm.Subsequently, the horizontal relative velocityincreased rapidly from 0.6 m s�1 to a local maximumof 2.5 m s�1. After 5ms, the relative velocity decreasedabruptly back to 0.6 m s�1 at point A2 (Figure 21(b)).The slip ceased since the relative velocity was verylow, allowing surface asperities to re-establish contactand be deformed elastically. The above observationsshow that the stable fringe structure (A1A2) representsa ‘self-healing’ slip pulse of approximately 7ms induration. Directly after the pulse, the relative velocityincreased rapidly again to 6.4 m s�1 and retained itslarge value of approximately 4 m s�1 for a long periodof time, of about 40ms. This suggests that the initialrupture of the pulse-like mode was immediately fol-lowed by a second rupture of the crack-like mode.Thus, as it has been anticipated, the experimentalresults presented up to this point indicate that therupture process is very sensitive to impact speed.Indeed, as the projectile speed was decreased whilekeeping the external confining stress constant, therupture-tip speed was also decreased. Finally, therupture mode changed from a crack-like mode to amixed mode where a single ‘self-healing’ slip pulsewas followed by a crack.

By further reducing the impact speed to 10 m s�1,the rupture mode became purely pulse-like. Thehorizontal relative velocity was measured at a

distance of 70 mm from the impact side of the

Homalite plate and its evolution over time is shown

in Figure 22. The rupture started at A1 and propa-

gated at sub-Rayleigh speed of 0.76CS, whereas after

15 ms the sliding ceased at A2. The duration of sliding

was very short compared to the approximately 100 ms

duration of the impact event, and thus we infer that

an isolated ‘self-healing’ pulse was formed. Such a

case clearly indicates that a pulse-like mode of rup-

ture can definitely occur under appropriate

conditions. Indeed this is the first time that such a

dynamic pulse has been clearly seen in a controlled

laboratory setting. The fact that ‘self-healing’ pulses

(b)

8

7

6

5

4

3

2

1

0–1 10 20 30 40 50

Time (μs)60 70 800

Rel

ativ

e ve

loci

ty (

m s

–1)

Crack

A1

A2

Pulse

(a)

M2

M1

10 mm

A1 A2

8

6

4

2

–2

0


external compression was 10 MPa. The fringe concentration points A1 and A2 are the rupture tip and the rear edgerespectively of the pulse-like rupture mode. The crack-like mode initiated at A2 right after the pulse. The slip profile is

superimposed (red line). (b) Relative velocity history of points M1 and M2 located at a distance of 30 mm from the impact side

of the Homalite plates. The rupture commenced when the rupture tip A1 reached the velocity measurement position and a

pulse A1A2 was formed. The crack-like rupture mode initiated at A2 right behind the second pulse.

Isolated pulse

Rel

ativ

e ve

loci

ty (

m s

–1)

A1 A2

0 10 20 30 40 50 60 70 80 90 100Time (μs)

3

2.5

2

1.5

1

0.5

0

–0.5

Figure 22 Relative velocity history of points M1 and M2

located at a distance of 70 mm from the impact side of the

Homalite plates. The rupture commenced when the rupture

tip A1 reached the velocity measurement position and anisolated pulse A1A2 was formed.


were obtained as the impact spe ed wa s decrea sedbelow certai n point is con sistent with the theoreticalresults of Zhen g and Rice (1998) . Accordi ng to theirpredictions , an d as discus sed in Secti on 4.06.3 .1, oneof the necess ary requirem ents of self-heali ng pul seformatio n is that the ove rall driving stress could belower than a threshold value.

T he combine d use of classic dynam ic phot oelas-ticity with ve locimetr y allowed us to fullycharacte rize the frictiona l sliding process. Howe ver,the max imum particle velocity which can be mea-sured by the velocim eter is 10 m s �1 . In order tocomply with the above limit, low impac t spe edswere only used up to this point. For higher impa ctspeeds, only dynam ic photoel asticit y in conju ctionwith high-spee d photog raphy was used as diagnos tictools.

In Figu re 23 , an instanta neous isochrom aticfringe pattern is show n at a specific time for thesame compres sive load of 10 MP a as in the pr eviousexperiments and at much higher imp act speed of42 m s �1 . The rupture pr opagated interso nically atapproxim ately 1948 m s� 1 ¼ 1.5 6CS ¼ 0.75 C P givingrise to a shear Mach cone ori ginating from the rup-ture tip B1 . In addition, a second Mach lin e whichoriginated from point B2 and was nonpara llel to thefirst one was obser ved behind the rupturepoint (Figu re 23 (a )). The Mach line was at a shal-lower slope correspond ing to a supers hear(approxim ately sonic) propagati on speed of2514 m s �1 ¼ 2.01CS ¼ 0.9 7C P . A more detail ed viewof the isochromat ic fringe pattern in the neighb or ofthe rupture tip is show n in the inset of Figu re 23(a ).

The two shock wave s are highl ighted by dot ted lines.Nonpar allel shock lines imply a highly transie nt andunstab le rupture process. Ind eed, study of the entireset of the captured photoel astic picture s show s thatthe tip B1 of the secon d Mach line gradu allyapproa ched the end B2 of the first Mach line.Finally , thes e two points merg ed as the secondpoint caugh t up with the first point. The slidi ngcontinu ed at the lower speed and thus only oneMach line was observed in th e next re corded frame s(not shown here). Fi gure 23 (b ) show s the posi tionhistori es of the first and secon d sliding tips for thecase above. It is evide nt that the secon d slidi ng tipmoved faster than the first slidi ng tip and at approx i-matel y 50 ms th e pair of points coalesc ed. T heexistenc e of two Mach lines mean s that behin d theonset of slidi ng (poin t B1 ), there was a second pointon the interfac e (poin t B2), where the sliding spe edagain cha nged rapidly. Then, one could conjectur ethat the initial slidi ng which star ted at point B1

stopped after a while and new sliding started atpoint B2. In this way, an unstable sliding pulse wasformed between the points B1 and B2 followed by acrack-like sliding which started at point B2 in asimilar manner to that shown in Figure 21 thoughat higher sliding speeds. We finally note that behindthe second Mach line a ‘wrinkle’ like pulse appearedat point C propagating at a speed of 0.92CS which isclose to the Rayleigh wave speed of Homalite. Weextensively elaborate on the nature of this distur-bance in Secti on 4.0 6.5.3 .

An important comment on the frictional slidingexperiments discussed above is that the rupture-tip

(a) 10 mm

B1B2

C

44 μs

(b)

80

70

60

50

40

30

20

10

00 10 20 30 40 50 60

Time (μs)

Dis

tanc

e (m

m)

Sliding rear edge2.01 Cs

Sliding leading edge1.56 Cp

Impact speed: 42 m s–1

Sliding tips speedsB1: 1.56 Cs, B2: 2.01Cs

Confining pressure: 10 Mpa

Figure 23 (a) Isochromatic fringes of sliding propagation at different time instances. Two sliding tips (B1 and B2) were

propagating at different intersonic speeds. In the insets the dual and single Mach lines are marked. (b) Positions of the sliding

leading edge and the second sliding tip as functions of time.


speeds wer e constants during the entire observ ation

time. T hat is a genera l result and it holds true for all

the experiments perform ed. T his is a strong charac-teristic of fricti onal sliding and it agrees with

theoretical results (Adams , 19 98 ; Ric e et al ., 2001 ;

Ranjith and Rice, 2001 ) which predi ct constant and

discrete pr opagati on speeds for all the different dis-

turbances and singular ities along homog eneous

interfaces subjected to uniform prestres s.In the last pa rt of this sectio n, the influ ence of the

impact spe ed on the propa gation speeds of the rup-ture tip is explored . Figu re 24 show s the variation of

the pr opagati on speed of the slidi ng front with the

impact speed at a constant unifor m con fining stres s of

10 MPa. The slowes t achieved impac t speed was

9 m s� 1 . In this case and in other cases with impa ct

speeds close to 10 m s �1 the rupture- tip speed was

sub-Rayleig h. At higher impact speeds the rupture-tip speed became supershear. It was observed that for

impact speeds in the range of 20 to 40 m s�1, the

sliding speed was slightly aboveffiffiffi2p

CS. This is a

special rupture speed (als o discusse d in Sections

4.06.3.1, 4.06.4.3, and 4.06.4.5) and it has been shown

that it separates regions of unstable and stable inter-

sonic shear crack growth (Samudrala et al., 2002a,

2002b). When the impact speed increased, the rup-ture-tip speed increased toward the plane stress P-

wave speed. It is worth mentioning that no sliding

speed was observed in the interval between the

Rayleigh wave speed and the shear wave speed of

Homalite-100. This experimental observation agrees

with theoretical predictions on steady-state shear

crack propagation which exclude this speed intervalbased on energetic arguments (Freund, 1990;

Broberg, 1999; Rosakis, 2002). It should also be

noted that the experimental rupture-tip speed mea-

surements presented here feature, as is proven below,

high-enough resolution to obtain propagation speedsin the interval between CR and CS, if such speedsexisted. It is finally noted that experiments per-formed at lower compression show that the speed ofthe rupture tip was influenced by the change in thestatic confining stress and it was supershear even atthe lowest achieved impact speed.

4.06.5.3 Wrinkle-Like Opening Pulsesalong Interfaces in Homogeneous Systems

The possibility of generating wrinkle-like slidingpulses in incoherent frictionless contact between twodissimilar solids, when separation does not occur, wasfirst investigated by Achenbach and Epstein (1967).These ‘smooth contact Stonely waves’ (also known asslip waves or generalized Rayleigh waves) are qualita-tively similar to those of bonded contact (Stonelywaves) and occur for a wider range of material combi-nations. Comninou and Dundurs (1977) found thatself-sustained slip waves with interface separation(detachment waves or wrinkle-like opening slippulses) can propagate along the interface of two simi-lar or dissimilar solids which are pressed together. Theconstant propagation speed of these waves was foundto be between the Rayleigh wave speed and the shearwave speed of the slowest wave speed material. Forinterfaces separating identical solids the propagationspeed was between CR and CS. Weertman (1980)obtained a 2-D self-sustained wrinkle-like slip pulsepropagating at the generalized Rayleigh wave speedalong a bimaterial interface governed by Coulombfriction when the remote shear stress was less thanthe frictional strength of the interface. Finite-differ-ence calculations of Andrews and Ben-Zion (1997)show the propagation of wrinkle-like opening pulsesalong a bimaterial interface governed by Coulombfriction. Particle displacement in a direction perpen-dicular to the fault is much greater in the slowermaterial than in the faster material, resulting in aseparation of the interface during the passage of theslip pulse. Anooshehpoor and Brune (1999) discoveredsuch waves in rubber sliding experiments (using abimaterial system consisting of two rubber blockswith different wave speeds). The above-mentioneddetachment waves are radically different from theSchallamach waves (Schallamach, 1971) which propa-gate very slowly compared to the wave speeds of thesolid.

Lykotrafitis and Rosakis (2006b) observed wrin-kle-like opening pulses propagating along theinterfaces of Homalite-steel bimaterial structures

2500

2000

1500

1000

500

00 10 20 30 40 50 60

Impact speeds (m s–1)

Slid

ing

tip s

peed

s (m

s–1

)

CP

2½ Cs

Cs

CR

8070

Figure 24 Variation of the sliding tip speed with theimpact speed. The confining stress was 10 MPa.


subjected to impact shear loading. The propagationspeeds of these wrinkle-like pulses of finite openingwere constants and their values were always betweenthe Rayleigh wave speed and the shear wave speed ofHomalite, in accordance with the theoretical predic-tion of Comninou and Dundurs (1977). The wrinkle-like pulse in the bimaterial case generated a charac-teristic fringe pattern. Similar photoelastic fringestructures propagating behind the rupture tip alonginterfaces in homogeneous systems of Homalite werealso observed by the same researchers (Lykotrafitisand Rosakis, 2006a). Prompted by this similarity,they decided to investigate whether wrinkle-likepulses were once more responsible for the generationof these fringe structures, in interfaces separatingidentical solids. Their results are summarized inthis section. It is known that a system consisting oftwo identical half-planes and subjected to compres-sion and to far-field shear loading cannot sustain awrinkle-like pulse propagating along the interface. Inthe setup used in these experiments, however, theloading was not strictly shear and there is not anyphysical reason to exclude the possibility of generat-ing wrinkle-like pulses.

In this section, photoelasticity and velocimetry isused to investigate the physical attainability of suchwrinkle-like pulses along frictional interfaces separat-ing identical solids. The equivalent issue for the caseof Homalite-steel bimaterial system was investigatedby Lykotrafitis and Rosakis (2006b) and Lykotrafitiset al. (2006b). Figure 25 is used to illustrate the

methodology used to identify the wrinkle pulses.The figure features an instantaneous isochromaticfringe pattern obtained at confining stress of 10 MPaand at an impact speed of 28 m s�1. Point B is ofparticular interest. The rupture tip A is shown topropagate at a constant supershear speed of 1.49 CS.A shear Mach cone is clearly visible in the photoelasticimage. A fringe structure located at point B, inFigure 25(a), propagates at a speed of 0.96 CS. Thetime evolution of the relative vertical displacement ofpoints M1 and M2, which were located 70 mm fromthe impact side of the Homalite plate, is displayed inFigure 25(b). A simple 1-D calculation shows that theinitial static compression of 10 MPa caused a negativerelative displacement of approximately 1.3mm.Negative relative displacement means that the twopoints approached each other under compression.The P-wave front arrives at about 2.5ms after thetriggering of the oscilloscope. Because of the Poissoneffect, the horizontal dynamic (inertial) compressiongenerated a vertical dynamic compression in additionto the static compression. A negative relative displace-ment was caused by this dynamic compression. Theshear rupture point crossed the velocity measurementposition at approximately 10ms (marked A inFigure 25(b)) and it did not cause any visible changeto the vertical components of the velocities of pointsM1 and M2. The relative vertical displacementbecomes positive at approximately 31ms. The photo-elastic picture, captured at 30ms, clearly shows thatthe fringe structure at B, which is the point under

B

105

0

0 10 20 30 40 50 60 70 80 90 100–5–10–15–20–25–30–35

–45–40

Decompression

CompressionCompression

Time (μs)

Rel

ativ

e ve

rtic

al d

ispl

acem

ent

(μm

)

A

(b)

(a)

M2

M1A B

5

0

–5

–10

–15

–20

–25

–30

–35

10

10 mm


external compression was 10 MPa. The rupture tip is at the fringe concentration point A and the wrinkle-like pulse is at point B.

The relative vertical displacement profile is superimposed (red line). (b) Relative vertical displacement history of points M1 andM2, located at a distance of 70 mm from the impact side of the Homalite plate. The rupture tip and the wrinkle-like disturbance

crossed the velocity measurement position at approximately 10ms and 32ms after triggering, respectively.


investigation, is very close to the measurement posi-tion. The study of the entire set of the 16 recordedphotoelastic images in combination with the relativevertical displacement history sheds light on the dis-tribution of the dynamic compression along theinterface during sliding. The interface was locallyunder dynamic compression immediately after thearrival of the P-wave front until the arrival of thementioned fringe formation at B. The entire areafrom the rupture point A to the location of the fringestructure at B was sliding under compression. Thelength of the area AB was estimated to be approxi-mately 40 mm, whereas the entire length of theinterface was 139.7 mm. The fringe structure at Bcau se d a l oca l o pe ni ng ( � 5 mm) at the interface. Atap pr ox im at el y 4 2 ms the relative vertical displacementbecame again negative and the compression increasedabruptly. During the rest of the recording time theinterface at the velocity measurement position wassliding under compression. The fringe structure at Bcan thus be clearly related to a wrinkle-like pulsepropagating along the interface. As was stated earlier,its propagation speed was very close to 0.96 CS whichlies in the interval between the shear and the Rayleighwave speeds of the material in agreement with theprediction of Comninou and Dundurs (1977).

Re viewing the resu lts from the entire spect rum ofexperiments , it is ve rified that , at a confinin g stress of10 MPa, the lowes t impac t speed which can gen eratea wrinkle -like pulse in the pr esent setu p is appr oxi-mately 17 m s � 1. Fig ure 26(a ) displays anisochromat ic fringe pat tern obtai ned at a confiningstress of 10 MPa and at an impac t spe ed of 9 m s � 1. It

is clear that no fringe structu re related to a wrinkle-like pulse appear s. That is reflect ed by the re lativevertic al displace ment his tory shown in Figu re 26(b ),where the maximum po sitive value of the displace-ment was only about 0.3 m m. A compar ison of thephotoel astic image shown in Figu re 26 (a ) withphotoel astic imag es obtained at the same static com-pression of 10 MPa an d at simil ar impact speedsduring experiments where the horiz ontal par ticlevelociti es measurem ents were avai lable, show s thatthe rupture star ted at point A.

The propagati on speeds of wrink le-like pulse s atdiffere nt imp act speeds and at th e same confinin gstress of 10 MPa are show n in Figu re 27. The slidingspeed was measured to be between the Raylei ghwave and the shear wave spe ed of Hom alite- 100.This re sult, in combinati on with the experim entalresults obtained via ve locimetry , favor s our conjec-ture that the interfa ce dist urbance was actually awrinkle -like pulse . As has been already mentione d,the available theoretical and numerical analysis onthe subject predicts the same speed range with thisidentified in Figure 27. It is also noted that both theprestress and the impact speed do not affect thepropagation speed of the wrinkle-like pulses pro-vided that such pulses could be generated.

4.06.5.4 Discussion

Finite element calculations of dynamic frictionalsliding between deformable bodies have alwaysbeen a very challenging task for the numerical ana-lysts. As it was recently recogniz ed (see Section

(a)

M1

M2

A

10 mm

(b)

Rel

ativ

e ve

rtic

al d

ispl

acem

ent

(μm

)

00

0.5A

Time (μs)–0.5

–1

–1.5

–2

–2.5

20 40 80 100 120 140 16060


external compression was 10 MPa. The rupture tip is at the fringe concentration point A. No wrinkle-like pulse appeared.

(b) Relative vertical displacement history of points M1 and M2 located at a distance of 100 mm from the impact side of the

Homalite plate.


4.06.4.1 ), the main source of difficult y was the ill-posedne ss of the correspond ing boundar y and initialvalue problem , if a rich-en ough fricti on law was notimplemen ted. Cok er et al. (2005) took adv antage ofthe lates t advances in the theory of th e fricti onalsliding (Rice, 2001 ) an d by using a rate- enhanc ed,rate and state fricti on law of Prakash–C lifton ty pe,they wer e abl e to perform stab le (grid -size indepen -dent) finite elemen t calcul ations for a configu rationvery simil ar to the experiments desc ribed above. Theuse of a rate-enhan ced Prakash–C lifton type of lawwas an essential element for the su ccessful com ple-tion of the simulations becaus e of the presenc e of fastchanges of the local compressi on at the interface,caused by wave propagati on, durin g dynam ic sliding .As it is ex plained in Section 4.0 6.4.1 , the Prakash–Clifton law is currently consi dered the only fricti onlaw which correctly describ es the effect of fastchanges in compres sion on the re sulting fricti onal

resistance . We note that for various imp act speeds

the numerical simu lations exhibit a ric her behaviorthan the ex perimental re sults and gen erate not only

crack-li ke, pulse -like and mixed mode ruptures but

also trains of pulse s and pulses follo wing a crack-lik erupture. A repr esentative result is shown in

Figu re 28 (a), where the distrib utions of the sliding

speed ( � _uslip ) and the shear traction T s along the

interfa ce at three times (t ¼ 32, 38, 44 ms) are illu-strated, for the case of 40 MPa stat ic com pression

and of a 2 m s� 1 imp act speed. Contou rs of max imum

shear stress, which corres pond to isoch romat ic frin ges

in photoel asticit y, at t ¼ 38 m s are shown inFigu re 28 (b). In this numerical ex periment, a mixe d

mode of rupture, w here a pulse was followed by a

crack-li ke ru pture, was obtai ned. This is reminisc entof the experimental case shown in Figure 21 . We

believe that the similarity between the numerical and

the experimental results can be further improved by

binding and eventually fine tuning the parameters ofthe friction law.

The ex perimental re sults presented in Section4.06.5 elucidate the sliding process and provide con-

clusive evidence of the occurrence of various slidingrupture modes (crack-like, pulse-like, or mixed) pro-

pagating dynamically along incoherent interfaces. Of

particular interest here is the experimental evidence

of the formation of both supershear and sub-Rayleighsliding pulses of the ‘self-healing’ type, leading the

direct validation of predictions made on the basis of

theoretical and numerical models of dynamic shear

rupture. These pulses were found to propagate in theabsence of interfacial opening. The experiments also

provide hints of the dominant physical mechanisms

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 10 20 30 40 50 60 70 80Impact speeds (m s–1)

Wrin

kle-

like

puls

e sp

eeds

(m

s–1

)

CSCR

Figure 27 The wrinkle-like pulse speed remained

between the Rayleigh wave speed and the shear wavespeed of Homalite-100, independent of the impact speed.

(b)

x1 (mm)

x 2 (m

m)

6

4

2

0

–2

–4

20 22 24 26 28 30 32 34 36 38 40

(a)

600

500

400

300

200 32 μs

38 μs

44 μs

100

00 10 20 30

x1 (mm)

Ts

(MP

a)

Δùsl

ip (m

/s)

40 50 60

25

20

15

10

5

0

–5

–10

–15

–20

–25

Figure 28 (a) Distributions of the sliding speed (� _uslip) at t¼ 32, 38, and 44ms, and of the shear traction (TS) at t¼ 44ms

along the interface for external compression of 40 MPa and impact speed of 2 m s�1. (b) Contours of maximum shear stress

(isochromatic fringe pattern) at t¼38 ms.


governing the choice of various rupture modes andtheir evolution. It is finally noted that ‘wrinkle-like’pulses which feature finite opening in addition tosliding were discovered propagating along the inter-face at speeds between CR and CS for various loadingconditions.

Acknowledgments

The authors gratefully acknowledge the support ofNSF (Grant EAR 0207873), the US Department ofEnergy (Grant DE-FG52-06NA 26209) and the con-sistent support of the Office of Naval Researchthrough (Grant N00014-03-1-0435) and MURI(Grant N000140610730) Dr. Y.D.S. Rajapakse,Program Manager.

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4.06 Dynamic Shear Rupture in Frictional Interfaces ... · insight into some aspects of earthquake mechanics, such models are more appropriate for providing fric-tional constitutive