On the influence of fault bends on the growth of sub ...mcise.uri.edu/rousseau/CER_AJR_JGR03.pdfcontains numerous weak interfaces, or faults, and therefore from a fracture standpoint

$Page 1: On the influence of fault bends on the growth of sub ...mcise.uri.edu/rousseau/CER_AJR_JGR03.pdfcontains numerous weak interfaces, or faults, and therefore from a fracture standpoint$
On the influence of fault bends on the growth of sub-Rayleigh

and intersonic dynamic shear ruptures

Carl-Ernst Rousseau1

Department of Mechanical Engineering, University of Rhode Island, Kingston, Rhode Island, USA

Ares J. RosakisGraduate Aeronautical Laboratories, California Institute of Technology, Pasadena, California, USA

Received 15 November 2002; revised 27 March 2003; accepted 19 June 2003; published 4 September 2003.

[1] Earthquake ruptures are modeled as dynamically propagating shear cracks with theaim of gaining insight into the physical mechanisms governing their arrest or, otherwise,the often-observed variations in rupture speeds. Fault bends have been proposed as beingthe main cause for these variations. Following this line of reasoning, the existence ofdeviations from fault planarity is chosen as the main focus of this study. Asymmetricimpact is used to generate shear loading and to propagate dynamic mode-II cracks alongthe bonded interfaces of two otherwise identical homogeneous constituents. Secondarypaths inclined at various angles are also introduced to represent fault bends or kinks. Theexperiments show that certain fault bend inclinations are favored as alternate paths forrupture continuation, whereas others suppress further motion of the incoming rupture. Theasymptotic elastodynamic stress fields at the tip of the growing rupture are used to developtwo criteria (one energetic and one stress based) for rupture propagation or arrest atthe kinked interfaces. These criteria correlate very well with the experimental results.Since most field evidence suggests that the average rupture speeds during crustalearthquakes are sub-Rayleigh, this work first focuses on incoming rupture speeds that arejust below the Rayleigh wave speed. Reports of intersonic crustal fault rupture speedshaving surfaced recently, experiments and analyses are also performed within that speedregime. INDEX TERMS: 7209 Seismology: Earthquake dynamics and mechanics; 7203 Seismology:

Body wave propagation; 7260 Seismology: Theory and modeling; 8010 Structural Geology: Fractures and

faults; 8020 Structural Geology: Mechanics; KEYWORDS: shear crack, wave propagation, dynamic fracture,

fault bend, energy release rate, fault propagation criteria

Citation: Rousseau, C.-E., and A. J. Rosakis, On the influence of fault bends on the growth of sub-Rayleigh and intersonic dynamic

shear ruptures, J. Geophys. Res., 108(B9), 2411, doi:10.1029/2002JB002310, 2003.

1. Introduction

[2] Earthquakes generally result from a sudden rupture inthe Earth’s crust, which occurs under the influence ofslowly increasing compressive and shear tectonic stresses.Nucleation and propagation of such events take place alongpreexisting faults in the Earth’s crust. However, these weakpaths are rarely planar but rather exhibit kinks, discontinu-ities, jogs, and varying levels of strength, which assume agreat importance in seismology. For example, branches andsteps are believed to control the extent to which earthquakeruptures are able to propagate, thereby potentially restrictingtheir magnitude. King and Nabelek [1985], in particular,have noted the apparent confinement of earthquakes toregions between fault bends, and in a few cases, have

isolated their initiation to the immediate vicinity of bendsor jogs. Arrest at those locations, however, is not aninevitable outcome. Indeed, earthquakes have in some casesbeen observed to bypass geometrical discontinuities. Forexample, the 1966 Parkfield earthquake jumped across astep over in the San Andreas fault [Segall and Du, 1993].[3] The impact of fault kinks on overall earthquake

progression, though acknowledged, has rarely undergonerigorous scrutiny to the extent that jogs have. This arisesfrom the difficulty that exists in quantifying their role. Thusfar, the most advanced analysis on the subject is due toPoliakov et al. [2002] who have approached the problemfrom two different angles. Successively, the propagatingearthquake is likened to a steadily propagating and singularelastodynamic shear crack and is then represented by a slip-weakening dynamic rupture model. Off-fault maximumshear stresses in the vicinity of the propagating tip areevaluated and compared to the frictional resistance withinthe Earth’s crust. When the former exceeds the latter ina particular direction, that direction becomes a plane ofpotential failure. In cases where extensive damage is

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. B9, 2411, doi:10.1029/2002JB002310, 2003

1Also at Graduate Aeronautical Laboratories, California Institute ofTechnology, Pasadena, California, USA.

Copyright 2003 by the American Geophysical Union.0148-0227/03/2002JB002310$09.00

ESE 4 - 1

present, this results in secondary faulting. They also findthat were an intersecting weak plane aligned to thatdirection, this plane would then promptly rupture. In addi-tion to the nature of the local field of the incoming rupture,their results are highly dependent upon the level of tectonicstress. Further, Aochi et al. [2002] and Kame et al. [2003]have studied the effect of these stresses on symmetric andasymmetric forked fault geometries with the aid of numer-ical analysis. Depending on tectonic stress level, orientation,and incoming rupture speed, the rupture may favor themore compressional or the more tensional of the tectonicbranches.[4] In this work, a close analogy between earthquakes

and cracks will be embraced. The traditional engineeringapproach to fracture mechanics has been primarily guidedby opening (mode-I) cracks, largely neglecting shear(mode-II) cracks. This is indeed very practical as crackpropagation in constitutively homogeneous, monolithicmaterials is strictly limited to the opening mode, even iffar-field conditions are asymmetric. As a crack extendsunder the influence of external tractions, it will continuallykink and assume a local mode-I field, even if the externaltractions are such as to generate mixed-mode stress states[Cotterell and Rice, 1980]. Seismologists, on the otherhand, face different concerns. The Earth’s crust inherentlycontains numerous weak interfaces, or faults, and thereforefrom a fracture standpoint can no longer be construed asmonolithic. In addition, active faulting brings into frictionalcontact rocks that are compositionally dissimilar, therebyinducing material inhomogeneity. In some cases, elasticmaterial properties across several faults have been found tovary substantially, and shear wave speeds have beenmeasured to differ by as much as 30% [Das, 1985; Rice,2001]. The latter case simply constitutes a bimaterial with aweaker interface fracture toughness than either of its twoconstituents. Bimaterial cracks have a natural affinitytoward acquiring a mixed-mode stress state, and evenpurely tensile external tractions generate a substantialmode-II component at the crack tip, due to the influenceof material inhomogeneity. Also, the fault represents aready-made path along which the rupture propagation isconstrained. Thus the running rupture is prevented fromkinking and thus is disallowed the opportunity of assumingpure mode I.[5] The different crack-tip running modes and the pres-

ence or absence of weak paths also translate in differences inattainable crack speeds. Dynamically propagating tensilecracks are theoretically limited to cR, the Rayleigh wavespeed of the material [Freund, 1990]. However, in brittlemonolithic materials, mode-I cracks can propagate smoothlyonly up to nearly 40% of cR, at which point they becomeunstable, generating microbranches and engaging intotortuous paths [Ramulu and Kobayashi, 1985]. However,the introduction of a weak path in an otherwise homoge-neous solid does set the environment for propagation ofmode-I crack at its theoretical speed limit. For dynamic,mode-II, shear ruptures, the situation is very different. Herethe existence of a weak plane or fault is a prerequisite for theexistence of dynamic shear rupture of any speed. In contrastto their mode-I counterparts, mode-II cracks (or shearruptures) propagating along faults are not only allowed toreach cR theoretically but also are capable of jumping that

barrier to reach the intersonic region between the shear wavespeed (cs) and the longitudinal wave speed (cl) [Burridge etal., 1979]. Field observations seem to confine the rupturespeed of most earthquakes to the range between 0.7cs(0.76cR) and 0.9cs (0.98cR) [Kanamori, 1994]. In onlyvery few cases have earthquake rupture speeds beeninferred to have become intersonic over a limited portionof the fault [Archuleta, 1984; Olsen et al., 1997; Bouchonet al., 2001].[6] Of particular interest to the present investigation is the

rupture event that occurred along the North Anatolian faultduring the 1999 Izmit earthquake. In this event, Bouchon etal. [2001] present compelling evidence that the east boundrupture propagated intersonically with a speed just aboveffiffiffi2

pcs, while the west bound rupture remained sub-Rayleigh.

Partially confirming these results are the more recentnumerical evaluations of R. A. Harris (personal communi-cation, 2002) that require the existence of supershear speedsas a necessary condition for complete rupture of the NorthAnatolian fault. The intersonic east bound rupture encoun-tered a 22.5�-retaining bend in the fault (compression side)that was unable to stop the rupture (see analysis by Harris etal. [2002]). Situations mimicking such an encounter of anintersonic rupture with a compression bend will be dis-cussed in the second part of this study.[7] Despite the caution in claiming the existence of

intersonic earthquake ruptures, strong evidence of intersoniccrack propagation has surfaced recently in laboratory set-tings. Lambros and Rosakis [1995] have witnessed anumber of cases in which intersonic speeds were attainedalong the bonded interface between two highly dissimilarmaterials. This was further discussed in the review paper bySingh et al. [1997]. Even more notable is the discovery ofbonded identical materials sustaining propagating crackspeeds in the intersonic regime [Rosakis et al., 1999], whichcertainly adds credence to the possibility of intersonicearthquake rupture.[8] The above discussion has raised several issues perti-

nent to dynamic rupture propagation in the Earth’s crust. Itwas shown that the reasons for the initiation and the arrestof ruptures remain evasive. Using experimental techniquesof dynamic fracture mechanics, this paper aims at partiallyanswering the second question. Indeed, relations are devel-oped that correlate the presence of fault bends to variationsin the speed of the propagating rupture. However, certainlimitations are associated with the present methodology. Forinstance, earthquake ruptures are resisted mainly by friction,which in this case is simulated by an interfacial bondstrength. Also, in the present work, rupture is triggered byimpact, whereas in earthquake ruptures they are excited byslow crustal motion and the resulting tectonic loading aswell as the sudden release of local pressure. Above all, thiswork presents some valid conceptual answers which shoulduniversally apply to the rupture of a weak bond by apropagating crack as well as the overcoming of fault frictionby a propagating earthquake rupture.[9] In the following, specimens and experimental setup

are described. Results of the experiments are then presented,and the analytical basis of this work is developed. Finally,the experimental results are discussed in view of twodifferent failure criteria. The outline hereby laid out forthe case of sub-Rayleigh cracks is then repeated for the

ESE 4 - 2 ROUSSEAU AND ROSAKIS: INFLUENCE OF FAULTS BENDS ON SHEAR RUPTURES

intersonic case, as it is believed here that earthquakeruptures have the potential of running intersonically.

2. Specimen Description and ExperimentalApparatus

[10] The study investigates the progression, transmission,or arrest behavior of shear-dominated sub-Rayleigh andintersonic cracks growing along weak paths, as theseruptures encounter fault bends or kinks along their paths.The optical method of photoelasticity is used as a means ofvisualizing stress fields in real time. The method lends itselfparticularly well to revealing the essential features ofintersonic cracks, such as shear shock waves, and willtherefore be used in both cases. In addition, this full-fieldtechnique also relates stress field information around prop-agating cracks. The events are recorded using high-speedphotography.[11] The specimens were composed of Homalite-100, a

birefringent polymer with longitudinal, shear, and Rayleighwave speeds of cl = 2295 m s�1, cs = 1310 m s�1,and cR = 1205 m s�1, respectively. At room temperatureand at strain rates prevailing during the experiments(103 s�1), the material is extremely brittle and linear elastic.[12] The specimens were 5 mm thick, 175 mm high, and

200 mm long. The narrowness of the specimens ensured theprevalence of two-dimensional, generalized plane stressconditions. Preferred paths were generated by machiningthem, beginning halfway through the specimen height onthe left side, and following the prescribed route. The twohalves were then bonded with a solution having weakenedfracture properties with respect to the bulk material butsimilar elastic properties and density, as measured bySamudrala et al. [2002a]. The weaker adhesive joint trapsthe moving crack to the prescribed interface, compelling itto retain its originally imposed mode-II state. Although notmonolithic, the resulting specimens can be considered to beconstitutively homogeneous. Indeed, the existence of apath of lower fracture toughness makes these specimensfracture-wise inhomogeneous but does not, in any sense,affect their continuum mechanics description. Along theinterface, a 15-mm-long and 1-mm-wide starter notch was

machined. The presence of the notch prevented the imme-diate transmission of the incoming impact stress waves(applied at the top) to the bottom half. This in turn guaran-teed a relatively clean mode-II initiation loading of the notchtip by preventing notch face contact. The kinks in the weakpath were placed 75 mm from the left edge. Placement ofthese alternate paths away from the loading (left edge) of theplate allowed the incoming crack to establish a steady statespeed before reaching the bend site, thereby ensuring uni-form conditions at the intersection of the two paths. Also, thelocation of the bend was even farther away from the distantright edge. The specimens were thus designed so thatreflected longitudinal waves returning from that edge wouldnot reach the location of the bend until the crack would havemoved through the major portion of the alternate path, wellbeyond the experimental field of observation.[13] The photoelastic setup is sketched in Figure 1. It

consists of a collimated beam generated by an argon-ioncontinuous laser that is transmitted through a circular polar-izer, the specimen, and a second circular polarizer. Theinformation is directed to the iris of a digital high-speedCordin camera capable of recording 16 frames up to a ratenearing 100 million frames per second. This time resolutionis fortuitous, as very high crack speeds are present in somecases encountered here.

3. Experimental Observations in the SubsonicRegime

[14] Loading was generated by impact of a steel bufferbonded to the top half of the specimen, near the notch, by ahardened steel projectile. Following impact, an initiallyplanar compressive wave is transmitted to the specimenvia the steel buffer. Over the length of the notch, only the topportion of the specimen is loaded. Beyond that location, thecompressive wave is gradually transmitted to the bottom. Asa result, loading to the latter always lags that of the top. Astate of stress thus exists along the interface, wherein alarger amount of shear and compression continually prevailsabove the adhesive joint. The condition is therefore propi-tious to crack initiation at the notch tip in a combination ofcompression and mode II, followed by propagation of a

Figure 1. The dynamic photoelastic setup and the high-speed camera.

ROUSSEAU AND ROSAKIS: INFLUENCE OF FAULTS BENDS ON SHEAR RUPTURES ESE 4 - 3

shearing crack, in a manner similar to the stress statepresented in the sketch of Figure 2. In addition, since thespecimens are impacted on the top half, from left to right,the current experiments are akin to cases where earthquakeruptures feature right-lateral slip.

[15] Further, a coordinate system (h1, h2), moving withthe crack tip is defined in Figure 2. The origin of thestationary prenotch is located at (x, y) and is related to themoving crack-tip location by h1 = x �

R0tv(t0)dt0 and h2 = y.

In addition, angular conventions are presented, with positiveangles situated above the fault line (counterclockwise) andnegative angles situated below the fault line (clockwise).This angular convention is carried over to the specimengeometry, as secondary bend paths are placed at variousangles on either side of the main horizontal fault line.Recorded experimental images were centered at the inter-section of the primary and secondary planes and encom-passed a circular area having a radius of 50 mm.[16] Figure 3 shows selected frames of the isochromatic

fringe patterns recorded for the case of a fault bent �35�toward the extensional side of the specimen. Each frame isidentified by the time elapsed following impact and theangle of the bend. These are inscribed on the top left and onthe top right corners of the frames, respectively. As in all theexperiments conducted in this study, the specimen is loadedat the left edge of the top half. Compressive waves travel

Figure 2. Schematic defining coordinate system conven-tion and the initial state of stress ahead of the growingrupture.

Figure 3. Isochromatic fringe pattern around a mode-II crack propagating at subsonic speed along aweak plane in Homalite-100. The crack continues propagating along a secondary weak path, inclined�35� toward the extensional side of the specimen. Times after impact are (a) 80 ms, (b) 90 ms, (c) 130 ms,and (d) 170 ms. The field of view, of 100 mm, is centered at the intersection of the two weak paths.


through the specimen predominantly in the top half and aretrailed by disturbances corresponding to the shear waves.Finally, the shear crack follows, since its initiation isdelayed and it propagates more slowly along the weakplane. Figure 3a shows the crack as it has just reached theintersection of the two paths. The crack tip can be identifiedusing several indicators. First, its location is usually accom-panied by a miniscule shadow spot, which is caused by thevery high level of deformation within the immediate vicin-ity of the crack tip. Second, due to high stresses and thepresence of discontinuity, the crack tip is always a rallyingpoint for a very large number of fringes. Also present in theframe are fringes, corresponding to the initial stress wavepropagation, that are still making their way through the rightportion of the field of view as they are preloading thespecimen in a combination of shear and compression.

[17] In the next frame, the crack has veered down tofollow the bend and in the process has lost some of itsenergy. Note that this change in direction is also accompa-nied by a change in stress state. No longer do predominantlymode-II conditions prevail at the crack tip, but rather, thecrack assumes a mixed-mode state. This is not apparentfrom the photograph. However, it will be shown in a latersection that this is the case.[18] At 130 ms, the crack has almost reached the edge of

the field of view, with its speed tapered substantially, whilestill following the weakened interface. Striking in that frameis the fact that the bottom section is almost devoid of fringesand is therefore devoid of stresses. This is caused by theinterfacial separation brought about by the moving crackthat prevents further transfer of energy between the twosections. The opening component of this decohesion is clear

Figure 4. Isochromatic fringe pattern showing a subsonically propagating mode-II crack encountering aweakened plane inclined �100� toward the extensional side of the specimen. Times after impact are(a) 110 ms and (b) 160 ms.


in the next frame, as a narrow white space along theinterface testifies to the complete severance between thetwo sections.[19] The next two images, presented in Figures 4a and 4b,

are of great interest as they depict the behavior of a subsoniccrack upon reaching a backward orientated path (a =�100�).However, prior to exploring the behavior pertaining to thispeculiar geometry, focus will be directed to a characterwhich seems to be universal to all the specimens tested. Onthe first photograph, a series of shadow spots are presenton the extensional side, along the main fault. They corre-spond to the tips of opening cracks generated as the mainshear crack travels along the weak plane. These secondarycracks are periodic and are all inclined at an angle a* ��79� from the direction of crack propagation. The pres-ence of the secondary tensile cracks was explained bySamudrala et al. [2002b] by introducing a finite slip-rate-weakening shear cohesive zone behind the shear crack tip.Contact and friction are conjectured to be present along thecrack face. Within this model, the biaxial stress state thatexists in the cohesive region behind the tip results inmaximum principal tensile stresses oriented at an angle�a* from the interface. This angle is a function of thecohesive strength of the interface, the crack-tip speed, andthe level of velocity weakening. Their possible relation tothe field observations of periodic tensile microcrackingperpendicular to fault planes in rocks has been discussedby Rosakis [2002].[20] The results presented in the next frame (Figure 4b)

are not a priori intuitive since they essentially show thecrack returning toward the direction fromwhich it originated.The crack has indeed turned the corner and proceededalong the provided �100� path, though at a constantlydiminishing speed. Again, secondary tensile, periodic cracksaccompany the main crack along the weakened bent path,at an angle of a* � �71�. This time, the secondary crackshave grown from the right face of the fault, which has turnedextensional with respect to its left side, indicating a changefrom right-lateral to left-lateral slip as the rupture extendsfrom left to right.

[21] Figure 5a is representative of a specimen having thesecondary fault on its compressional side, for a bend anglea = 35�. In this frame, the crack has already veered aroundthe corner and is halfway through its progression along thebend, a process accompanied by a slight deceleration.Finally, a case is presented in Figure 5b, for which themoving crack failed to follow the bend but instead pene-trated the bulk of the material. In this frame, the secondaryweakened path is bent 71� toward the compressional side ofthe specimen. Here and in all the subsonic cases where thecrack failed in its attempt to follow the specimen interfacepast the bend, the penetration angle into the Homalite wasabout �40� on the extensional side. Progression into thematerial is achieved through a dominant mode-I crack, asevident from the symmetric shadow spots surrounding thegrowing crack tip. This crack propagates steadily at a speedcorresponding to 55% of the Rayleigh wave speed, beforeprevalence of extensive branching in the form of a fan oftensile branches. In addition to this mechanism, failure alsoproceeds by the prolonged extension of the last of theperiodic microcracks (generated just before the bend),which is inclined at �79� to the horizontal.

4. An Energy Approach to Subsonic CrackDeflection

[22] Consider an incipient crack initiating its growth as aresult of an applied far-field pure shear loading. The crackwill very promptly try to kink or curve relative to its initialdirection in an attempt to assume a pure mode-I symmetricstate, unless the latter is suppressed by application ofsuitably oriented compressive stresses or unless the crackis confined to its own plane in a manner similar to thatespoused in the current experiments. If the imposed restric-tions come to an abrupt end, the crack may breach into thematerial under mode I, provided the bulk material fracturetoughness is low enough relative to the fault plane. If, onthe other hand, another weakened path is presented, thecrack may elect to follow it, provided motion along thatpath would generate energy release rate values surpassing

Figure 5. Isochromatic fringe patterns showing subsonically propagating mode-II cracks encounteringweakened planes inclined toward the compressional side of the specimens (a) at an angle of 35� and time200 ms and (b) at an angle of 71� and time 170 ms.


those offered by other alternatives. The said propagationwould occur under mixed-mode conditions and is preciselythe type of crack extension observed experimentally. Itremains to be found whether a study of the energiesexpanded prior to and potentially after the bend can beused to predict continuation around the bend, and if so,forecast postbend crack speed and mode. A similar meth-odology was used by Xu et al. [2003], who succeeded inpredicting the crack speed of a transmitted crack afterdeflection at a bent. In their case, the approaching crackwas purely mode I while the deflected crack propagatedunder mixed-mode conditions.[23] Consider a shear crack propagating along a weak

bond in an elastic solid. The crack will advance if thematerial fracture toughness KIIc of the weak bond, lyingahead of it, is attained by the mode-II stress intensity factorKII or if the shear crack growth energy release rate, GII,exceeds the critical shear fracture energy, �, associated withKIIc. If however, the crack were to propagate along aninfinitesimal kink making an angle q with the initial crackdirection, mixed-mode conditions will in general exist at itstip. Static or dynamic stress intensity factors associated withthe infinitesimal kinked crack can both be obtained in termsof the static stress intensity factor, KII, of the prekinked pathas follows [e.g., Broberg, 1999]:

KdI ¼ � 3

4sin

q2

� �þ sin

3q2

� �� KII kI ;

KdII ¼

1

4cos

q2

� �þ 3 cos

3q2

� �� KII kII ;

ð1Þ

where kI ¼ 1� v=cRð Þ½ =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� v=clð Þ

pand kII ¼ 1� v=cRð Þ½ =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� v=csð Þp

.[24] Inserted in these two relations are the universal

functions of crack-tip speed, kI and kII [Freund, 1990],which account for the kinetic state of the rupture. For thequasi-static case (v ! 0) both functions reduce to a valueof 1. After crack kinking, equation (1) will be evaluatedwith v = v2, while prior to kinking, v will be taken to be v1and q = 0�. A dynamic failure criterion based on energy cannow be formulated for growth both before and after kinking:

Gq ¼1� u2ð Þ

E

n2

1� uð Þc2sDal K

dI

� 2þas KdII

� 2h i� �; ð2Þ

where

D ¼ 4alas � 1� a2s

� 2; ð3aÞ

al ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� v2=c2l

� q; ð3bÞ

and

as ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� v2=c2s

� q: ð3cÞ

[25] Inserting equation (1) into equation (2) and taking theratio of the resulting inequality for values of q = a (v = v2) andq = 0� (v = v1), respectively, one can conclude that the crack

would be able to proceed in the kinked direction only if thisratio reaches or exceeds unity. In the expression to follow,subscript ‘‘1’’ is associated with prebend speed and subscript‘‘2’’ refers to crack speed after the kink. The requisiteinequality, which is valid for both plane stress and planestrain, can be written as

Gq¼a;2

Gq¼0;1¼ v22D1

v21D2

9al;2 kI ;2

� 2sin

q2

� �þ sin

3q2

� �� 2þas;2 kII ;2

� 2cos

q2

� �þ 3 cos

3q2

� �� 216as;1 kII ;1

� 2 � 1:

ð4Þ

[26] Speeds v2 exceeding zero indicate a propensity forthe crack to follow the secondary path, provided the left sideof inequality (4) exceeds unity. The expression is plotted inFigure 6 at the limit v2 ! 0+, for several values of v1registered during the experiments. The angles for which theenergy ratio is greater than one have therefore the potentialfor kinked growth. The graphical data of Figure 6 indicatethe possibility for crack growth around a bend for all anglesinferior to 153�. Note that energy does not differentiatebetween angular orientation and gives equal probabilisticweight to crack motion along angles q and �q. However, themethodology cannot account for the highly compressivestate of stress that suppresses the possibility of crackadvancement in the major portion of the positive angulardomain by producing extensive frictional dissipation asso-ciated with crack-face contact. This shortcoming is clearlyassociated with the assumption that the transmitted crackfaces remain traction and dissipation free even for positivekink angles. Consequently, it is felt that this criterion canonly be accurate for the extensional side of the specimens.Finally, inequality (4) can also be used as a predictive toolfor the speed v2. By using the equation as an equality, the

Figure 6. Ratio of the energy release rates of thesecondary path to that of the primary path at the limitv2 ! 0. The analysis is valid only for mode-II crackpropagating at subsonic speeds along the primary plane.Crack deflection toward the incline is allowed only in theregion above the line (�2/�1), which represents thetoughness ratio of the incline to that of the main plane.


speed along the kink can be retrieved, providing motionthrough the kink angle is allowed as per Figure 6.

5. The Subsonic Singular Crack-Tip Stress Field

[27] The inability of the previous method in segregatingpositive and negative angles of crack orientation will beremedied by alternatively scrutinizing the dynamic, asymp-totic stress fields in the immediate vicinity of a shear crackpropagating dynamically in the sub-Rayleigh regime, alonga prescribed straight line path representing the ‘‘incoming’’branch of a kinked weak plane or fault. Let such a crackassume a pure mode-II stress state as it grows along thehorizontal direction. Further, let the polar coordinates atthe instantaneous crack tip be defined as in Figure 2. Theasymptotic form of the stress and deformation fields forsuch a case has been provided by Freund [1990] and isrepeated here:

s11 ¼� KdIIffiffiffiffiffiffiffiffi2pr

p 2as

D

1þ 2a2l � a2

s

� sin

ql2

� �=

ffiffiffiffigl

p � 1þ a2s

� sin

qs2

� �=

ffiffiffiffiffigs

p �

;

ð5Þ

s22 ¼KdIIffiffiffiffiffiffiffiffi2pr

p2as 1þ a2

s

� D

sinql2

� �=

ffiffiffiffigl

p � sinqs2

� �=

ffiffiffiffiffigs

p� �

; ð6Þ

s12 ¼� KdIIffiffiffiffiffiffiffiffi2pr

p 1

D

4alas cosql2

� �=

ffiffiffiffigl

p � 1þ a2s

� 2cos

qs2

� �=

ffiffiffiffiffigs

p �

:

ð7Þ

[28] All variables pertaining to these equations have beendefined earlier, except for the following:

gl ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� v sin qð Þ=clð Þ2

q; ql ¼ tan�1 al tan qð Þð Þ;

gs ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� v sin qð Þ=csð Þ2

q; qs ¼ tan�1 as tan qð Þð Þ:

ð8Þ

[29] These stresses govern deformation only in theimmediate vicinity of the crack tip. Further away from thecrack tip, geometric and far-field effects can make asignificant contribution. These, however, are disregarded,as only the singular stresses are conjectured to contribute toinitiation of local bifurcation off the fault path, especially inthe absence of static far-field loading as in the experimentsdescribed earlier. When substantial levels of far-fieldloading are present, the above assumption is expected tobe less accurate, as rationalized by Kame et al. [2003], andin such cases the entire stress state is expected to play animportant role. Along every possible kink angle, a combi-nation of the shear stress (srq) and the opening hoop stress(sqq) along that direction will act in concert to control thebend direction and overcome local toughness. Thus thesestresses, derived from equations (5)–(7), are plotted in

Figure 7, normalized with respect to the factor KII=ffiffiffiffiffiffiffiffi2pr

p,

which is common to all stress components. The plots aregenerated for incoming speeds along the main rupture pathat levels of v1 = 0.80cR and v1 = 0.95cR, which encompassesthe speed range recorded during the experiments. Thisapproach is indeed the one suggested by Poliakov et al.[2002] for the analysis of the more general case of astatically preloaded kinked fault hosting a subsonicallygrowing rupture. As noted by these authors and as is evidentfrom the figure, the dependence of the stresses on incomingspeed is striking as can also be surmised from the structureof the above equations. First, within that narrow speedrange, the stresses experience an increase in magnitude,with speed sometimes exceeding a factor of two. Second,

Figure 7. Angular variation of the dynamic crack-tipstress fields for subsonic mode-II rupture occurring atspeeds of (a) 0.80 cR and (b) 0.95 cR.


for lower speeds, whereas three shear extrema are present at0� and ±136�, as the incoming speed approaches cR, the 0�maximum is transposed into two maxima appearing at ±71�.Note also that the shear component is symmetric withrespect to the main fault line, whereas the hoop stress isantisymmetric about the same plane. Since the sign of thehoop stresses is positive on the extensional side, thelikelihood of activating a branch that lies on that side isfar greater than achieving the same on the compressionalside, where the hoop stress is negative. This would tend tocounteract secondary growth along an upward bend. Acriterion based on critical levels of stress can now beformulated.[30] Generalizing the type of reasoning presented above,

one can attempt to formulate a stress-based criterion ofkinking that would perhaps describe both opening andshear-dominated crack initiation along a kinked path. Fora coherent interface and in the absence of frictional strength,one may follow the work of Camacho and Ortiz [1996] andintroduce a particular combination of the traction compo-nents sqq and srq acting on the interface, due to anapproaching crack tip, as the driving force responsible fordecohesion. This driving force should reach or exceed thestrength of the interface for cracks to be initiated along it.This criterion is usually stated as follows:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2qq þ b2s2rq

q� s0; ð9Þ

where s0 is the opening strength of the interface and b is aweight factor allowing for different levels of opening versusshear strength. In fact, inspection of inequality (9) impliesthat b = s0/t0, where t0 is the shear strength of the interface.Relation (9) can now be generalized to include possiblecontributions to the shear strength of the interface arisingfrom frictional resistance to sliding in the presence ofcompressive components of traction. A simple way to do sois to require that the generalized traction should reach orexceed a level of strength which also depends on the levelof the compressive normal tractions, if present. The kinkingcriterion will now be expressed as

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisqqh i2þb2s2rq

q� b t0 þ m �sqqh i½ : ð10Þ

[31] In the above inequality, any function enclosed withinthe bracket is defined as:

mh i ¼0; m < 0

m; m � 0

8<: :

[32] The combination (to + mh�sqqi) represents the shearstrength of the interface, augmented by a Coulombic con-tribution which is activated only if the normal hoop stress iscompressive. Indeed, when this normal stress sqq is com-pressive, the criterion becomes

srq þ msqq ¼ srq � m sqqj j � t0; ð11Þ

which is exactly the criterion used by Poliakov et al. [2002]in their discussion of kinking and branching.

[33] When, on the other hand, sqq is tensile or zero, thenthe criterion reduces to the one presented in inequality (9). Itshould also be noted at this point that the above-postulatedcriterion has been expressed with respect to the tractioncomponents sqq(r, q) and srq(r, q) acting on the kinked faultline due to the presence of a dynamically growing crackwhose position is at the kink point. The traction componentsare expressed with respect to distance r measured from theinstantaneous location of this crack tip which is movingwith speed, v, and can either be sub-Rayleigh or intersonic.In both cases, the general form of the asymptotic stresscomponents is separable in r and q and can be expressed as

srq ¼ A=rq vð Þ� �

srq q; vð Þ; sqq ¼ A=rq vð Þ� �

sqq q; vð Þ: ð12Þ

[34] In the above expression, srq q; vð Þ and sqq q; vð Þ are theangular distributions of stress and q is measured counter-clockwise from the direction of propagation of the incomingcrack. The dependence on the radial distance r is governedby a function of speed q(v) called the singularity exponent,while A is an amplitude factor common to all stresscomponents. In the sub-Rayleigh regime, A = KII

d, thedynamic mode-II stress intensity factor, and q(v) = 1/2 forevery v between zero and cR. For intersonic crack growth, Ais denoted by KII

*d while q(v) is a continuous and smoothfunction of crack-tip speed that varies between zero at v = csto 1/2 at v ¼

ffiffiffi2

pcs down to zero at v = cl. In both speed

regimes (with the exception of the exact speed v = cs andv = cl) the crack-tip field is singular according to the linearelastic model and the stresses become unbounded as r ! 0.Indeed, if one substitutes relation (12) into the full-fledgedcriterion expressed in relation (10), one has

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisqqh i2þb2s2rq

q� b torq vð Þ þ m �sqqh i

� �: ð13Þ

[35] As r! 0, the first term of the right-hand side becomesmuch smaller than all other terms and can be neglected (this istrue only for v 6¼ cs, cl where q(cs) = q(cl) = 0). By collectingall contributions in the left-hand side of inequality (13), thecriterion can be expressed with respect to a normalized stress,s q; vð Þ, involving the angular functions of crack-tip speed asfollows:

s q; vð Þ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisqqh i=bð Þ2þs2rq

q� m �sqqh i

� �� 0: ð14Þ

[36] The expression no longer involves the shear bondstrength t0. Inequality (14) suggests that to first order and inthe absence of substantial static far-field stress, theapproaching crack tip has to overcome only the frictionalstrength m �sqqh i caused by its own asymptotic and singularfield through compression and the resulting frictionalresistance. This analysis is valid only for a strictly asymp-totic interpretation of the rupture behavior. If, on the otherhand, a cohesive model were to be implemented, neglect-ing the fault strength would be inappropriate, and theargument would be subject to modification. The variableb � s0/t0 was determined experimentally to be 1.875. Thefriction coefficient m in this crude model should be iden-tified as the dynamic friction coefficient and lies much


below the static value measured to be 0.84. In the firstseries of discussions, m will be set to tan (30�) or 0.577.This is a value very close to the one assumed by Harris etal. [2002] in their recent study of intraearthquake triggeringin relation to the 1999 Izmit earthquake. A parametricinvestigation that follows the initial analysis will allow m todrop all the way down to 0.1.[37] The dependence of the stresses on the speed of the

incoming crack again becomes obvious upon inspection ofthe angular variation of the stresses plotted in Figure 7, as isthe speed dependence of s q; vð Þ plotted in Figure 8. Note, inparticular, that as the incoming crack speed approaches cR,the angular stress variation deviates substantially from thosecorresponding to lower values of velocities, all of whichcongregate within a relatively limited range. This isexpected, as Figures 6 and 7 do forecast a unique separationnear cR.[38] Finally, it should be noted that, given a fixed speed,

v1, of the incoming crack, for those kink angles a < 0 suchthat s a; v ¼ v1ð Þ < 0, the criterion would preclude anycrack growth along the weak kinked path. On the otherhand, for angles such that s a; v1ð Þ � 0, crack growth alonga kink would be favored. For the latter case, the kinkedcrack would propagate with a speed v2 6¼ v1. The magnitudeof s a; v1ð Þ represents the level of crack driving forceavailable to an outgoing kinked crack at each angle a dueto the arrival of an incoming crack whose speed was v1. Onewould expect the speed of the outgoing crack to varyproportionally to this level of driving force. This sugges-tion will be investigated experimentally in the followingsections.

6. Discussion of the Subsonic Results

[39] In this section, the experimental, results will berevisited and compared to the two criteria developed in

the previous sections. Raw speeds were first obtained byrecording the location of the crack tip from the high-speedcamera records and by using a central difference scheme todifferentiate the crack length time history. These speedswere then normalized by cR and plotted for each experi-ment, with respect to instantaneous crack-tip location.Figure 9 shows such a speed history for the case of asecondary path bent 35� toward the compressional side ofthe specimen. The dashed vertical line dividing the plot,with an abscissa value of zero, represents the intersectionpoint between the main horizontal and the incline planes. Inother words, it marks the location of the kink. Thereforedata points left of the vertical line refer to progression alongthe main path, and to the right of the vertical line are datapoints corresponding to motion along the incline. The crackcruises along the primary fault within a narrow range, arrivingat the corner of the bend with a speed of v1 = 0.88cR.Following kinking around the bend, a drop in speed tov2 = 0.82cR is recorded. Only very few data points areavailable along the secondary path. These, however, do notshow much variation, which would indicate that a steadystate speed is reached rather quickly in this case. Figure 10includes several plots of the same general geometrycorresponding to secondary faults situated on the extension-al side of the specimen. In the first plot (Figure 10a),measured speeds at kink angles of �35� and �71� arecombined. Kink angles of �100� and �103.5� are com-bined in Figure 10b. Common to these four cases is thedecrease in crack-tip speed following the intersection of thetwo paths. This speed drop becomes more pronounced asthe angles become more obtuse. Also, the general trend isfor the speed along the secondary fault to diminish furtherbefore attaining a steady state level.[40] Some level of uniformity in the interpretation of

these results is accomplished by computing the ratio of

Figure 8. Variation of the normalized driving stressmeasure s q; vð Þ as a function of incoming rupture speed, v1.

Figure 9. Speed history for a crack propagating in mode IIalong a weak plane, in the subsonic regime, and encounter-ing an incline at 35� toward the compressional side of thespecimen.


the initial speed of the moving crack recorded just afterkinking to the crack speed along the primary plane justbefore kinking for the various specimens tested. The speedratio was then plotted with respect to the kink angle of thesecondary fault, as presented in Figure 11. The referencepoint is at an angle of 0�, at the vertical line separating thepositive and negative portions of the abscissa, for whichcase, v2/v1 = 1. On the extensional side, there is a gradualdecrease in the speed ratio with increasing angle, up to�100�, after which point the drop in speed ratio becomesdrastic. At �136� and beyond, the crack can no longercontinue on the secondary path, but instead branches intothe bulk material. At an angle of 35�, the initial branching

crack speed is slightly lower than that of its tensile coun-terpart. Finally, at 71� and beyond, further crack extensionis not possible. Note that no data were available between�103.5� and �136� or between 35� and 71�. Thus the exactangles for which secondary fault triggering is unattainable isnot known.[41] The experimental results are highly dependent upon

the incoming crack-tip speed, confirming the original ob-servation of Poliakov et al. [2002]. Because v1 variesbetween experiments, the results are better understood ifexperiments and theoretical projections (left-hand side ofinequality (4)), which includes the same speed dependence,are viewed simultaneously. The theoretical predictionsbased on the energetic arguments are superimposed on theplot for the extensional side. Only the discrete theoreticalpoints are computed by using, in each case, the exactincoming crack speed measured in each individual experi-ment. There is excellent agreement between the two from 0�to �100�, where theoretical and experimental data partiallyconceal each other. At �136�, the theory predicts a sharpdrop in v2, not drastic enough, however, as to suppresssecondary branching, as observed experimentally. Recallthat Figure 6 would forecast crack arrest at the end of theprimary plane only for secondary plane angles more obtusethan a = �153�. Nevertheless, this difference constitutesbut a mere shift in the data, whereas overall behavior andtrends remain identical. Thus it can be concluded that theenergy criterion may be used with great confidence instudying extensional fault branching.[42] Next, the focus must be placed on the theoretical

considerations based on the singular crack-tip stress fieldand the stress criterion of inequality (14). To that aim, theexperimental data are plotted again in Figure 12, accompa-nied by visual representation of inequality (14) at locationscorresponding to the experiments. Theoretical points arenormalized to unity for the case of a straight, unkinked fault(a = 0�). It is suspected here that there exists a correlationbetween predicted driving stress and normalized kink speed.Consequently, the two normalized quantities are plotted on

Figure 10. Speed history for a crack propagating in modeII along a weak plane, in the subsonic regime, andencountering a secondary weak plane. All fault bends aretoward the extensional side of the specimens at angles of(a) �35� and �71� and (b) �100� and �103.5�.

Figure 11. Experimentally measured and theoreticallypredicted ratio of outgoing to incoming crack speeds asthe cracks are engaged along fault bends of variousinclinations. Theoretical predictions v2/v1 are based on theenergy criterion. Incoming mode-II cracks are subsonic.


the same graph to verify this suspicion. The first fewtheoretical points, up to �100�, overestimate the experi-mental ones only by a small percentage. The speed ratios at�103.5� are not as accurate but still well predicted by thetheory. At �136�, on the other hand, the theory highlyoverestimates the speed ratio. Based on the crack-tip stressfield, branching would also be expected in the remainingangles corresponding to the extensional side of the fault,though with a greatly reduced probability. On the compres-sional side, the normalized branching speed is overesti-mated at 71� but differs only slightly at 35�. Though notfitting impeccably for each angle studied, the relativenormalized driving stress levels from inequality (14) offerthe same overall trends as do speeds obtained experimen-tally. The predictive abilities of inequality (14) are compa-rable to those of inequality (4). Furthermore, the formerpresents the additional benefit of predicting the level ofpropagation possible along a kink situated on the compres-sional side. Predictions of inequality (14) over the pertinentrange of data, for a single average speed v1, are alsopresented in Figure 12. Overall agreement with the exper-imental data is present. However, the plot is burdened byoscillations that make it depart from full accord with theexperimental data. It must be reemphasized, however, thatthe experimental data correspond to different values ofincoming crack-tip speed, while the continuous s q; vð Þcurve was calculated assuming a constant incoming speedvalue taken to be the average of the experimentally mea-sured incoming crack-tip speeds. Therefore this apparentdissension does not detract from the excellent predictivecapabilities of the crack-tip stress field approach, which isvery apparent in the comparison between discrete solid oropen symbols.

7. Experimental Observations in the IntersonicRegime

[43] The remainder of this paper is devoted to examiningthe behavior of incoming intersonic cracks as they reach

the alternate kinked path. In the intersonic regime, crackinitiation is followed by sudden acceleration to levels belowcl, and finally by deceleration to steady state values nearffiffiffi2

pcs. It must be noted that the recent work by Samudrala et

al. [2002a] has identified the region between cs and acharacteristic rupture speed close to but higher than

ffiffiffi2

pcs

as being unstable. Speeds betweenffiffiffi2

pcs and cl were

identified as stable. The exact value of this characteristicspeed, delineating the border between the stable and theunstable regimes, was found to depend on the nature ofthe bond strength and on other parameters of the slip-weakening cohesive model used in their analytical study.The minimum value of this characteristic speed correspondsto high bond strength and is equal to

ffiffiffi2

pcs. This observation

is consistent with the early stability results obtained by theslip-weakening models of Burridge et al. [1979] and laterobtained numerically by Needleman [1999] and Abrahamand Gao [2000] on the basis of cohesive element andatomistic calculations, respectively. It is also noteworthythat at an entirely different length scale, Bouchon et al.[2001] reported stable intersonic rupture speeds slightlyhigher than

ffiffiffi2

pcs occurring over the segment of the North

Anatolian fault, east of the hypocenter of the 1999 Izmitearthquake.[44] Figure 13 illustrates several cases for which crack

propagation along the primary plane reached steady stateintersonic speed prior to arriving at the intersection with thesecondary plane. An important feature identifying theintersonic nature of the crack propagation is revealed inFigure 13a. In that frame, the presence of a shear stressMach cone is clear. It is characterized by a wedge-shapestructure, separated from the remainder of the fringes by aline of discontinuity along which there is a jump in stresses.The extremity of the Mach cone coincides with the instan-taneous location of the crack tip. As an alternative todifferentiating consecutive crack-tip locations with respectto time, the crack speed, v, can also be obtained from theMach angle, y = sin�1(cs/v). Because of its superioraccuracy, evaluation of crack-tip speed by means of mea-suring the inclination of the Mach cone is used wheneveravailable.[45] A sequence showing progression of the crack as it

approaches and follows a secondary plane bent at an anglea = �35� is presented in Figures 13a and 13b. As the crackturns onto the bend, its state of the motion is no longer oneof primary shear, as a normal component is now associatedwith it. Figure 13b shows the crack as it nears the edge ofthe field of view, along the secondary plane. The presenceof the Mach cone clearly indicates that the crack hasretained its intersonic nature beyond the bend.[46] Next is a frame from a specimen with its secondary

plane bent at angle of �56� (Figure 13c), recorded afterkinking. Along the secondary plane, the contrast betweenthe behavior of this specimen and the previous one isstriking. First, the Mach structure is no longer present, anindication of a sharp drop in speed of the crack from itsprekinked, intersonic state. Instead, vague remnants of theMach cone still drift on the top and bottom sections and arefully dissociated from the crack tip, which is visible on thesecondary plane. Upon reaching the kink site, the tip ofthe cone detaches itself from either of the two paths, and inthe process, becomes blunt. The entire structure eventually

Figure 12. Experimentally measured and theoreticallypredicted ratio of outgoing to incoming crack speeds asthe cracks are engaged along fault bends of variousinclinations. Theoretical predictions of the normalizeddriving stress measure s q; v1ð Þ are based on the asymptoticcrack-tip stress field. Incoming mode-II cracks are subsonic.


acquires a semihemispherical shape before finally disap-pearing. In addition to the crack speeds along the inclinepath having fallen to within the subsonic range, it is notedthat microcracks are absent from the secondary plane,

whereas they were still prominent on the primary plane.Postmortem examination of four specimens confirms thatthe kinked portion of the interface does not feature anymicrocracks. Thus it may be concluded that crack-tip stress

Figure 13. Isochromatic fringe pattern showing an intersonically propagating mode-II crackencountering secondary fault. Angular inclinations are (a) �35� at 50 ms, (b) �35� at 80 ms, (c) �56�at 120 ms, (d) �80� at 96 ms, (e) �80� and 124 ms, and (f ) 45� at 99 ms after impact. The fault bends ittoward the extensional side in the first five frames and toward the compressional side in the last frame.


states have rotated in such a manner as to align the principaltensile stresses perpendicularly to the kinked interface,opening a path to the crack in primarily mode I, therebyminimizing any mode-II component. A primarily tensilestress state would minimize crack-face contact as well asany mode-II components, conditions necessary for theformation of secondary tensile microcracks.[47] The sequence presented in Figure 13d displays yet

another trend. The figure shows a specimen having itssecondary path rotated �80� toward the extensional side.The frame shows the crack after it has traveled two thirds ofthe way along the incline. Some similarity exists withthe previous case having the weakened kinked path ata = �56�. Indeed, upon reaching the kink, the Mach coneovershoots past the intersection of the two paths, and itsacute angle becomes blunt. However, unlike the previouscase, the bottom edge of the discontinuous enveloperemains attached to the moving crack, whereas the edgesituated above the primary plane has barely shifted. Thisresults in a very pronounced asymmetry of what wasformerly the Mach cone enclosure. Following the progressbetween several frames, the apparent behavior is that of anextremely blunted and obtuse angle being rotated counter-clockwise about a fixed point along the extended primaryplane, such as to allow the bottom section to retain itsassociation to the crack tip. Nevertheless, crack propagationalong the incline does resume its advance in the intersonicregime. Though not very distinct, a new Mach cone hasregenerated, moving downward along the secondary plane.The narrow inclination of the secondary shock wave con-firms the large acceleration experienced by the propagatingcrack along that path.[48] In Figure 13e, secondary tensile cracks developing

from the alternate weakened path are also noticeable. Thephotograph is specifically meant to put these microcracks inevidence, the frame having been recorded after full propa-gation of the main crack. The location of these tensile cracksis indicated by the large shadow spots present at their tips.Most important here is that they are located to the rightrather than to the left of the incline, indicating an abruptshift from right- to left-lateral slip after the fault kink. Thisestablishes a change in the sign of shear after kinking.[49] The last photoelastic fringe pattern (Figure 13f)

represents a marked departure from all the previous inter-sonic cases, as the angle of the secondary path is relativelylarge (45�) and is situated on the compressional side.Narrower compressional kink angles were surmounted bythe cracks almost as effortlessly as their extensional coun-terparts. In the present frame, the crack is only a fewmillimeters past the intersection of the two paths andprogresses slowly and laboriously within the subsonicregime. Concurrently, a branch is initiated into the bulk ofthe material, at an angle of �60� on the extensional side,and extends at approximately half the speed of the second-ary kinked crack or 55% of the Rayleigh wave speed. It willbe shown later that �60� is a principal stress direction. Thusfor the present configuration, despite the higher toughnessof the bulk material, the crack finds it as equally effortless tobranch into the bulk material in mode I as it is to follow theincline, weakened path. Further increasing the angle that thealternate path makes with the primary plane prohibits entryonto the incline, leaving only a slow-moving mode-I crack

to proceed into the bulk material, away from the incline.Also, note that for slightly lower secondary plane inclina-tion (a = 42�) in which a dual path is still present, entry intothe homalite occurs less forcefully, with a speed of 0.25cR,while the concurrent continuing compression/shear crackpropagation along the secondary plane remains intersonic.

8. The Intersonic Singular Crack-Tip Stress Field

[50] Consider a crack extending in its own plane underpure shear in a homogeneous, isotropic material. Thespeed at which the crack is moving need not be constantas long as it is continuous and varies smoothly. As in theprevious sections, let there be a moving Cartesian coordi-nate (h1, h2) attached to the crack tip, with related polarcoordinates (r, q), as shown in Figure 2. Then the near-tipsingular stress field has been determined by Freund [1979]and is expressed as

s11 ¼KII*

d

2al

ffiffiffiffiffiffi2p

p� 1þ 2a2

l þ a2s

� rql

sin qqlð Þ"

þ1� a2

s

� sin sgn h2ð Þqp½

�h1 � as h2j jð Þq H �h1 � as h2j jð Þ#; ð15Þ

s22 ¼KII*

d

2al

ffiffiffiffiffiffi2p

p1� a2

s

� rql

sin qqlÞð"

�1� a2

s

� sin sgn h2ð Þqp½

�h1 � as h2j jð Þq H �h1 � as h2j jð Þ#; ð16Þ

s12 ¼KII*

dffiffiffiffiffiffi2p

p 1

rql

cos qqlð Þ�

� cos sgn h2ð Þqp½ �h1 � as h2j jð Þq H �h1 � as h2j jð Þ

�;

ð17Þ

where

rl ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih21 þ a2

l h22

q; ql ¼ tan�1 alh2=h1ð Þ;

as ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2=c2s�

� 1q

; q ¼ 1=pð Þ tan�1 4alas= 1� a2s

� 2h i:

ð18Þ

Functions ‘‘sgn’’ and ‘‘H’’ are the sign and the Heavisidefunctions, respectively. Equations (15)–(17) are valid onlyif the crack-tip-dominant singular terms are more significantthan the remainder of the expansion.[51] The Heaviside step function indicates a discontinuity

in the stresses that translates into the formation of the twoedges of the Mach cone. This solution to the intersonicproblem features another major difference from the subsoniccase in that singularity exists not only at the crack tip butalso along the Mach front and governs the intensity of thestress jump across the Mach cone. In addition, when v isprecisely equal to

ffiffiffi2

pcs, the step functions vanish, thereby

pointing to the disappearance of the Mach cone at thatpeculiar speed and to the return of singularity solely to thecrack tip. Special conditions resulting in the disappearance


of the Mach wave are often referred to as radiation free [Gaoet al., 1999]. Finally, for intersonic cracks, the strength of thesingularity is 1/2, as in the subsonic case, only for v ¼

ffiffiffi2

pcs.

On either side of that special speed, q monotonicallydecreases to be zero at v = cs and v = cl.[52] To further understand the behavior of a mode-II

crack extending intersonically, polar component of stressesare obtained from equations (15) to (17) and plotted inFigure 14 for v = 1.5 cs. This particular speed of 1.5 cs ischosen based on experimental evidence that in most casesthe running crack hovers slightly above

ffiffiffi2

pcs while follow-

ing a straight path. Also recall theoretical predictions thatthe segment between

ffiffiffi2

pcs and 1.5 cs requires the minimum

amount of critical load to sustain intersonic crack propaga-tion and is therefore an obvious region for the crack speedto try and achieve steady state. The relevant stresses, sqq,srq, are normalized through multiplication by (rl

q/KII*d) The

resulting angular variations sqq and srq are plotted as afunction of angular location, in the immediate vicinity of thecrack tip. Radial stresses act in the direction of crack growthand are therefore not pertinent to the current problem.[53] Since the crack is advancing due to shear loading, the

effect of srq will be prevalent. The main features of thisstress component are reviewed. As expected, the maximumshear occurs at q = 0�, along the plane of the crack, and issymmetric about that plane. On either side of the crackplane, the shear stress decreases to zero at q � ±56�. Thisangle is therefore a principal stress direction. Thereafter, thesign of shear changes to attain an extrema at ±121� beforereturning to zero at ±180�. This latter portion of the curve ispunctuated by a strong discontinuity at ±138�, which againcorresponds to the Mach cone. The location of this discon-tinuity gradually shifts toward larger angles with increasingcrack speed and toward smaller angles as speed decreases.The exact position is given by q = (p � sin�1[cs/v]).

[54] The opening stress sqq is asymmetric with respect tothe crack plane and is zero directly ahead of the crack.Therefore the solution correctly recognizes the existence ofa pure shear condition at that location. The profile of thehoop stress as it varies with orientation is akin to that of aninverted sine function, with its extrema shifted to ±94�.[55] From the above, the shear stress distribution offers

equal probability for kinking toward either side of theprimary crack plane. On the other hand, the sign of thehoop stress clearly supports growth toward the extensionalside. A criterion for determining whether a propagatingcrack can leap from the main path onto an incline can beappropriated from the subsonic case. Inequality (10), whichwas used with much success in the subsonic regime, willagain be adopted here to establish the likelihood of deco-hesion of the weak bond along an incline as it is beingapproached by an intersonic crack along the primary plane.In the present case, normalization of relation (10) will beimplemented with respect to (KII

*d/rlq) and conditions as

rl ! 0 will be considered. This results in inequality (14)involve only the angular variations sqq and srq, which arealso strong functions of rupture speed.

9. Discussion of the Intersonic Results

[56] In this section, the experimental results will bereviewed in the light of the aforementioned analytical solu-tion. First, speed histories of the moving cracks within thefield of view are presented. Figure 15a shows crack speeds oftwo specimens having weakened secondary planes inclinedat an angle a = �10� on the extensional side. Speeds arenormalized with respect to cs. Most of the speeds recordedare based on the Mach cone angle. At locations where thisinformation was ambiguous or unavailable, speeds werebased on a three-point regression of the crack length history.The figure shows the prekink speed to be mostly confined tothe range between

ffiffiffi2

pcs and cl. The crack speed retains its

prekink value over the entire length of the incline.[57] The next set of experiments, for a = �35�, shows a

slight decline in speed right after the crack engages alongthe secondary plane (Figure 15b). Again, in this case thecrack speed along the incline remains nearly constant nearits initial level. At a = �42� (Figure 15c), the initialpostkink speed suffers slight postkink speed deteriorationwhen compared to shallower angles. Once more, in thiscase, the speed remains fairly constant along the incline.The same figure includes data for a = �53� and features aninitial postkink decrease in speed followed by a slightincrease and a plateau. Very drastic is the loss of speedsuffered by the crack as it veers toward a secondary planebent at �56�. Indeed, for both tests presented (Figure 15d),the crack speed has fallen either to cs or to within thesubsonic range in a very short time but then recovers itsspeed back into the intersonic range and oscillates. A lesserdrop in speed is observed for a = �60� (Figure 15e).However, very large speed variations follow as the crackspeed continues its decline to below cR. Figure 15f revealsthe beginning of a transition. For secondary path inclina-tions a = �65� and a = �75.5�, the postkink crack speedremains constant and increases, respectively, compared toits individual prekink levels. The next two angles (a = �80�and �90�) capture further drastic gains in speed following

Figure 14. Dynamic crack-tip stress fields for intersonicmode-II rupture occurring at a speed of 1.50 cs. Thenormalized stresses are plotted with respect to angularinclination.


Figure 15. Speed history for a crack propagating in mode II along a weak plane, in the intersonicregime, and encountering a secondary fault path. The following inclines are toward the extensional sideof the specimens: (a) �10�, (b) �35�, (c) �42� and �53�, (d) �56�, (e) �60�, (f ) �65� and �75.5�, (g)�80�, (h) �90�, and (i) �100�. Inclines oriented toward the compressional side of the specimens are ( j)10� and 35�, (k) 42� and 45�, and (l) 53�.


Figure 15. (continued)


the turn, reaching levels hovering in the vicinity of cl (seeFigures 15g and 15h). However, whereas the formerremains at a relatively constant speed, the latter sustainsirregular oscillations. Finally, the specimen with the back-ward incline angle of �100� experiences an initial crackspeed jump similar to those of the two specimens just citedbut is unable to maintain high speed levels, dropping tobelow cR (Figure 15i).[58] On the compressional side, the behavior of cracks

along shallow angles (a = 10�, 35�) is almost indistinguish-able from that of their tensile counterparts (see Figure 15j).The inability of the 22.5� Karadere bend of the NorthAnatolian fault to substantially slow down the intersonicallymoving East bound rupture of the 1999 Izmit earthquake[Harris et al., 2002] is consistent with the above observa-tion. Larger incline angles of 42�, 45� (Figure 15k), and 53�(Figure 15l) show even more severe drops in speed withincreasing angles to very low subsonic values. Beyond 53�,the crack is incapable of further motion along the incline.Also, starting with the specimen having the incline angle of42�, upon reaching the end of the primary plane, the crackelects to follow the incline while simultaneously branchinginto the bulk material. The speed of the mode-I branchedcrack seems to increase with increasing angle of theweakened secondary path, gradually changing from 0.25 cRto 0.5 cR.[59] The most relevant information from these numerous

tests consists in the speeds immediately following the crackturn onto the secondary plane. This data is captured andsummarized in Figure 16a, normalized with respect to cs.For secondary paths inclined toward the extensional side,there is a steady decline from 0� to �56�, from the rangeffiffiffi2

pcs < v < cl to a level between cs < v <

ffiffiffi2

pcs, after

which point an increase takes the crack speed to levelsnearing cl at �90�. On the compressional side, the crackspeed remains fairly constant up to 35�, then drops drasti-cally to zero near 60�.[60] Referring back to Figure 14, a local maximum value

of shear stress occurs for q = 0�, with no associated normalstress present. It is expected, therefore, that this straight pathwould be preferential for crack propagation under thepresent loading condition when compared to possibleadjoining paths. As the crack encounters alternate weakplanes of large angular orientation, conditions prevailing atthe crack tip, corresponding to the direction of the providedincline, will come into play. In addition to the effect of srq,the circumferential stress, sqq, would open a path to theincoming crack, if tensile, thereby facilitating and perhapseven expediting a jump onto the secondary plane. Ifcompressive, the effect of sqq would be that of sealing thesecondary plane further. The practical effect of this is one ofa virtually strengthened bond (frictional strengthening)offering even more resistance to crack penetration andcontinuing motion. In light of this discussion, particularattention should be paid to three distinct angular segments.First, notice the region of �121� < q < �94�, where thevalue of shear attains another local maximum that is evenstronger than at q = 0�. Coupled with the fact that this is alsoa region of maximum tensile hoop stress, a prediction ofenhanced crack continuation along secondary paths withinthe said zone can be advanced. Second, near q = �56�, noshear is manifest. Therefore conditions here can be surmised

as being ominous to secondary crack motion. Finally, for allpositive values of q, compressive hoop stresses abound, andthis should also discourage secondary crack propagation.[61] Figure 16b attempts to capture the ease of penetration

into a secondary plane. This can be achieved effectively interms of the already visited postkink speeds. The effect ofvarying incoming crack speed is annulled by normalizationof the postkink speeds with respect to that entity. In addition,it provides the requisite comparison of post- to prekinkspeed, thereby providing a ratio of secondary to primaryenergy expended into the crack. At shallow angles, say�35� < a < 35�, the relative shear stress has dropped slightlywith respect to its value at 0�, which is unity. Concurrently, asmall amount of positive circumferential stress is present fornegative kink angles, whereas negative hoop stresses of thesame magnitude exist for positive kink angles. However, dueto the relatively limited size of these stresses and the fact thatthe shear stress is still near a maximum level, little differencein behavior can be perceived between these opposite cases.Also, based on their speed history, neither can they bedifferentiated from an intersonic shear crack running selfsimilarly and continuously along its original plane.[62] As the negative kink angles of the secondary plane

become more obtuse, the amount of shear available topotentially initiate a crack along that path decreases. At thesame time, increasing circumferential stress allows for moremode-I opening, leading to mixed-mode conditions. Atangle a = �56�, with no more shear present, advance of acrack along that path is theoretically limited to cR, thus thelarge drops in crack speed recorded experimentally in thevicinity of that angle. Increasing the kink angle furtherchanges the sign of shear, resulting in an abrupt reversal indisplacement direction between the two sections of thespecimen. As the angle draws near �90�, the most favorableconditions for crack initiation along secondary paths are met.Indeed, the highest tensile, opening stress of any angle isattained, considerably weakening the effectiveness of thebond. Simultaneously, the shear stress reaches a value whosemagnitude approaches the absolute maximum shear stressobtainable at any kink angle q. This is reflected by thecontinuous escalation in crack speed, with increasing anglebeyond a =�56�, to levels exceeding unity. This culminatesin the very high speed ratios obtained at �80�, �90�, and�100�. Propagation along the latter kink angle seemssomewhat counterintuitive since upon reaching the intersec-tion of the two paths, the crack must make a turn and moveback in the direction from which it originally came, changinglocal slip direction from right-lateral to left-lateral slip.However, the solution does guarantee this outcome basedon high-shear and high-tensile hoop stresses. Nevertheless, aslight drop is observed in the figure, when the data point ata = �100� is compared to those of the two previous kinkangles. Though the crack speed drop appears miniscule, it isactually of considerable importance as it is indicative of atrend, since tests performed at a =�125� and higher resultedinstead in crack penetration into the bulk of the material.Analysis of the crack-tip stress field does not exhibit anyimpediment to initiation along these obtuse paths, except forthe constantly diminishing levels of both shear and tensilestress components to eventually zero at �180�.[63] On the compressional side, the conditions for shal-

low angles are similar to those of negative angles of


Figure 16. Crack speed along fault bends of various inclinations for incoming intersonic shear cracks.(a) Raw data. (b) The crack speed along the secondary fault normalized by the terminal crack speed of theprimary plane. Theoretical predictions of the crack speed based on the intersonic crack-tip stress field arealso plotted (broken line) as a function of fault inclination.


equivalent magnitude. Therefore ease to propagation isexpected, as evidenced by the high-speed ratios observedin Figure 16b at a = 10� and 35�. As angular inclination isincreased, shear stress drops. At the same time, compressivehoop stress increases and gradually frictionally enhances theeffective bond strength, impeding crack motion. This iscorroborated by the continuously decreasing crack speedrecorded experimentally as kink angle increases. Near 56�,the shear stress declines to zero, while hoop stress is stillhighly compressive. Incline paths near that point cantherefore no longer be triggered by the incoming crack.This is observed experimentally up to a = 90�. Althoughnear that point the shear stress has increased substantially,so has the compressive hoop stress, to the extent ofpreventing further motion along the incline.[64] Thus far, qualitative analyses based on the crack-tip

stress fields (equations (15)– (17)) have been used tounderstand the behavior of cracks as they enter an incline.However, quantitative measurements are necessary to betterpredict the relative incoming speeds onto the inclines.Inequality (14) is used for that purpose, in combinationwith the intersonic crack-tip stress fields. The expression iscomposed of two elements, one in which cohesion prevailsand is valid only on the extensional side, and another whichtakes friction into account and is applicable only to thecompressional side. Inequality (14) is plotted as a continu-ous curve with respect to angular inclination of the second-ary plane and is superimposed on the experimental datapoints of Figure 16b. Note that the angular variationsinvolved in the expression are automatically normalized insuch a way as to predict a value of unity at q = 0�. Thequantity s a; v1ð Þ represents a compound measure of crackdriving force and fault resistance along q = a and isexpected to correlate with postkink speed v2. Indeed, withinthe experimental window, the angular trend between speedratio and s a; v1ð Þ is remarkable. On the extensional side,the theoretical s a; v1ð Þ curve registers a gradual declinewith increasing angle, up to q = �45�, followed by a risethat ends at q = �100�, at a level comparable to that of theactual experimentally measured crack speed of the sameangular inclination. Larger values of the shear stress factor,b (say a 20% increase), ameliorate the match between thecurve and the measured data points, as the level of the curveis lowered slightly to almost coincide with the experimentalsymbols. Such an increase in the value of b would not beunreasonable since it is well within the range of errorassociated with the experimental evaluation of s0 and t0.Furthermore, the minimum is shifted to the left, towardq = �56�, where the minimum is also registered experi-mentally. On the compressional side, where normalizedstresses have been scaled to fit the range between zeroand one, the theory bypasses the experimental plateaupreceding the drop in crack speed with increasing angle.Instead, the decrease is immediate, ending with a value ofzero at q = 56�, precisely where crack propagation along theinclines is no longer observed experimentally.[65] Inequality (14) can also be used as a simple predictor

of the ability of the crack to engage into an incline. Thepredicted value of s a; v1ð Þ is largely influenced by thevalue of the free parameters b and m. On the compressionalside, as the friction coefficient increases, the value ofs a; v1ð Þ decreases because of the contribution of the term

accounting for the frictional strengthening of the bond. As aresult, the curves shift down and to the left, restricting therange of angles within which crack motion is allowed(Figure 17a). For the friction coefficient assumed in thisstudy, 0.5 < m < 0.6, incline angles beyond a = 35� wouldnot permit crack growth along their paths. Crack motionalong the bend, within the range 35� < a < 60�, does occurbut is further complicated by the fact that the availableenergy must be shared with a simultaneously generated

Figure 17. Variation of the normalized driving stressmeasure s q; v1ð Þ (a) as a function of friction coefficient m,(b) as a function of shear stress factor b, and (c) as afunction of incoming rupture speed v1.


tertiary mode-I crack which proceeds laboriously within thebulk of the material. This dual and intricate phenomenoncannot be captured by inequality (14), which within itsdefined parameters still delivers an excellent performance.[66] On the extensional side, s q; vð Þ is heavily influenced

by the value of b. The left side of inequality (14) isparameterized and plotted for several values of 1/b. As seenin Figure 17b, failure along the bend is forecasted every-where within the experimental range for all values of theshear stress factor. However, as 1/b decreases, its minimumvalue shifts to the left, toward the experimentally registeredincline angle that corresponds to the minimum crack speed.[67] Finally, inequality (14) is plotted as a function of

incline angles, for several values of the incoming speed v1(Figure 17c). The speed was varied at equal intervals,between 1.10 cs and 1.60 cs. Variation of s q; vð Þ withincoming crack speed was not tested experimentally. Nev-ertheless, from the excellent correlation between inequality(14) and experimental data, it can be surmised that for anyincline situated on the extensional side, as is observed in thefigure, greater incoming speed will also result in greateroutgoing speed along the secondary path. On the compres-sional side, the opposite happens, as the increase in crack-tip shear stress is simultaneously accompanied by evenlarger increase compressive stresses along the secondarypath. However, in this case, the dependence on incomingspeed is not so pronounced as on the extensional side.Nevertheless, increasing the incoming crack speed along themain path would result in smaller crack speed along thesecondary path and would also decrease the range ofsecondary path angles allowing rupture continuation.

10. Conclusion

[68] Several experiments were conducted with the aim ofduplicating conditions that may be present during crustalearthquake ruptures. Differences do exist, however, betweenthe actual events and these laboratory earthquake simula-tions. In particular, static precompression is deliberatelyoverlooked as these experiments are meant to uniquelyisolate the influence of secondary fault bends on a runningrupture. Both subsonic and intersonic incoming ruptures areinvestigated as earthquake ruptures are believed to have thecapability of propagating in both regimes. In the subsoniccase, criteria based on energy and on the asymptoticelastodynamic crack-tip stress field are effectively used toexplain experimental observations. Only the latter approachis valid in the intersonic regime and is used successfully topredict the behavior of ruptures as they reach the intersec-tion of two planar faults. Finally, it is worth noting that thesecondary tensile microcracks that appear on the extensionalside of the shear rupture faces are thought to be common toboth events, suggesting that a dynamic fracture mechanicsapproach and bonded interfaces adequately mimic condi-tions present along earthquake faults.

[69] Acknowledgments. The authors would like to acknowledge thesupport of the Office of Naval Research (Ship Structures Program), grantN00014-95-1-0453, as well as the support of the National Science Foun-dation (Geophysical Sciences Directorate) through grant EAR-0207873.Many helpful discussions with James Rice and Renata Dmowska ofHarvard University, Hiroo Kanamori of Caltech, and Ruth Harris of theU.S. Geological Survey are also acknowledged.

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��A. J. Rosakis, Graduate Aeronautical Laboratories, California Institute of

Technology, Pasadena, CA 91125, USA. ([email protected])C.-E. Rousseau, Department of Mechanical Engineering, University of

Rhode Island, 222B Wales Hall, Kingston, RI 02881, USA. ([email protected])


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On the influence of fault bends on the growth of sub ...mcise.uri.edu/rousseau/CER_AJR_JGR03.pdfcontains numerous weak interfaces, or faults, and therefore from a fracture standpoint