Grain size evolution and the rheology of ductile shear zones:from laboratory experiments to postseismic creep

Laurent G.J. Monte¤si �, Greg HirthWoods Hole Oceanographic Institution, Department of Geology and Geophysics, Woods Hole, MA 02543, USA

Received 15 November 2002; received in revised form 27 March 2003; accepted 2 April 2003

Abstract

Geologic, experimental, and geodetic studies indicate that ductile shear zones do not deform in steady state. Wepresent a theoretical model for an elastically loaded ductile shear zone deforming by combined dislocation anddiffusion creep, in which grain size is allowed to evolve toward its equilibrium (recrystallized) value. Several grain sizeevolution laws are considered. Simulations of laboratory experiments show that grain size evolution can lead tochanges in the dominant deformation mechanism that are accompanied by transient strain-weakening or strain-strengthening episodes reminiscent of laboratory observations. Ductile shear zones can also produce postseismic creepas observed in geodetic data following large earthquakes. Non-steady-state ductile creep is difficult to differentiatefrom frictional sliding with velocity-strengthening friction. Hence, ductile creep may play a role in the transition fromvelocity-weakening to velocity-strengthening friction with increasing temperature.3 2003 Elsevier Science B.V. All rights reserved.

Keywords: ductile creep; grain size evolution; transient creep; postseismic creep; friction laws

1. Introduction

At depths larger than a few kilometers, rocksare hot enough to £ow by ductile creep, accom-modated by motion of dislocations or di¡usion ofvacancies [1,2]. By de¢nition, ductile behavior in-dicates that deformation remains distributed, incontrast to brittle behavior, for which deforma-tion localizes on shear bands [3,4]. In the ductileregime, rocks are strain rate-hardening, which sta-

bilizes deformation and impedes localization.However, localized shear zones are common innatural rocks that deformed by dislocation or dif-fusion creep [5]. Localization may be promoted byfeedback loops involving temperature or grainsize that lead to strain weakening [6^8]. Localiza-tion in the ductile regime is therefore transient,associated with a change in temperature or micro-structure. Geological evidence supports the hy-pothesis that ductile shear zones are transient phe-nomena [9,10].Strain weakening resulting from shear heating

might cause strain localization in the ductile re-gime [9,11,12]. However, thermal e¡ects alone areinsu⁄cient to cause strain localization unless de-formation is already localized [8,13,14]. Indeed,

0012-821X / 03 / $ ^ see front matter 3 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0012-821X(03)00196-1

* Corresponding author. Tel. : +1-508-289-3708;Fax: +1-508-457-2187.E-mail addresses: montesi@whoi.edu (L.G.J. Monte¤si),

ghirth@whoi.edu (G. Hirth).

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there is only scant evidence for a thermal anomalyassociated with ductile shear zones [15].By contrast, grain size reduction is observed in

natural ductile shear zones [16^23]. Recrystalliza-tion can weaken the rock by producing smallgrains, which enhances di¡usion creep, and byreducing the dislocation density. Laboratory sam-ples deformed to large strain document recrystal-lization-induced weakening [24^31]. These experi-ments may be direct analogues of the rheologicalevolution of a ductile shear zone. We present amodel for ductile shear zones that includes theevolution of grain size, and investigate its rheo-logical response to sudden changes of loading ve-locity.In nature, there are several reasons why the

grain size of a rock evolves. At low temperature,the grain size is reduced through brittle processes.At higher temperature, the competing processes ofstatic grain growth and dynamic recrystallizationoperate. Small grains grow to reduce surface en-ergy, while subgrain rotation and grain boundarybulging produce new small grains (dynamic re-crystallization). Hence, the average grain sizeevolves towards an equilibrium value, D, forwhich the two phenomena balance one another.If a sudden change of loading velocity, like the

start of a laboratory experiment, induces micro-structural and rheological evolution, it is possiblethat earthquakes, viewed also as sudden changesof loading velocity, produce a short-lived defor-mation episode on nearby ductile shear zones.Geodetic data show that large (Mwx7.0) earth-quakes are followed by fast aseismic creep on aplanar zone surrounding the rupture area [32^36].The creeping plane has been interpreted as a faultplane with velocity-strengthening friction [37,38].Alternatively, we propose that the creeping planeis a ductile shear zone undergoing a transient epi-sode of enhanced creep triggered by the earth-quake. We use our ductile shear zone model toshow that an episode of enhanced creep can in-deed follow an earthquake.In this paper, we use several descriptions of

grain size evolution to explore the transient be-havior of ductile shear zones. Two types of sim-ulations are presented that illustrate the rheolog-ical consequences of grain size evolution. The ¢rst

simulations aim at reproducing laboratory experi-ments and produce strain^stress curves that dis-play transient strain weakening or hardening. Thesecond simulations investigate the response of aductile shear to earthquake cycles and comparethe behavior of the ductile shear zone modelwith velocity-strengthening friction.

2. Ductile shear zone model

2.1. Rheology

At su⁄ciently high temperature, rocks deformby the motion of intracrystalline defects. The mo-tion of linear defects (dislocations) results macro-scopically in dislocation creep, in which the strainrate, _OOD, and the applied stress, c, are related by apower law:

_OOD ¼ ADc nD ð1Þ

where nDW3^5 and AD is a function of temper-ature, water content, oxygen fugacity, etc. Asthese quantities do not evolve in the simulationspresented here, AD is treated as a constant.Deformation can also be accommodated by the

di¡usion of point defects, such as vacancies,through grain interiors or boundaries, resultingmacroscopically in di¡usion creep. The strainrate accommodated by di¡usion creep, _OOG, de-pends on the stress, but also depends on the grainsize, d :

_OOG ¼ AGd3mc

nG ð2Þ

with nG =1, mW2^3, and AG a constant.These deformation mechanisms are indepen-

dent [1]. Thus, the total strain rate of a materialdeforming by dislocation and di¡usion creep is:

_OO ¼ _OOD þ _OOG ¼ ADc nD þ AGd3mc

nG ð3Þ

For a given grain size d, the combined creep isdominated by dislocation creep at high stress anddi¡usion creep at low stress (Fig. 1). This behav-ior can be idealized as a transition from di¡usionto dislocation creep at a critical stress [17,20,39].ter Heege et al. [40] show that when a continuousdistribution of grain size is taken into account, thecoe⁄cient AG depends on the width of the grain

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size distribution. Di¡usion creep is enhanced ifthe grain size distribution is broader, which in-creases the critical stress for the transition fromdi¡usion- to dislocation-dominated regimes.Additional deformation mechanisms, such as

low temperature plasticity, grain boundary slid-ing, or pressure solution, may also be includedin Eq. 3. For simplicity, we restrict this study tothe combined e¡ect of dislocation and di¡usioncreep.As already mentioned, the average grain size of

a rock tends to evolve toward an equilibrium val-ue, D. The equilibrium grain size depends on thestress, c, as:

D ¼ D0c3r ð4Þ

where D0 and r are empirically determined con-stants [1]. If the grain size is at its equilibriumvalue (Eq. 4), the combined creep law becomes[8,41^43] :

_OO ¼ ADc nD þ AGD3m0 c

nGþmr ð5Þ

The values of nD =3.5, m=3, and r=1.3, pub-lished for dry olivine [39,44^46], result innD6 nG+mr. In this case, the combined creep atsteady state is dominated by dislocation creep atlow stress, where the equilibrium grain size is

large and di¡usion creep ine⁄cient, and by di¡u-sion creep at high stress, where the equilibriumgrain size is small (Fig. 1). The relation betweendislocation- and di¡usion-dominated regimes isinverted compared to the case for a constant grainsize d (Eq. 3). Hence, transient e¡ects can arise asthe dominant deformation mechanism changeswhen the grain size evolves from an initial valuedi to the equilibrium value D.If the grain size obeys Eq. 4, the ratio of dis-

location and di¡usion creep depends on the ap-plied stress. Alternatively, de Bresser et al. [47,48]reasoned that the equilibrium grain size is suchthat the ratio of dislocation- to di¡usion-accom-modated strain rate is a critical value, R, of orderunity. They obtain:

D ¼ ðRAGA31D Þ1=mc3ðnD3nGÞ=m ð6Þ

Substitution of Eq. 6 into Eq. 5 gives:

_OO ¼ ð1þ 1=RÞADc nD ð7Þ

2.2. Grain size evolution law

To fully characterize the deformation of a duc-tile rock, we need to know both the rheology for agiven grain size (Eq. 3), and an equation thatdescribes how the grain size evolves with time.In this study, we consider ¢ve grain size evolutionlaws. The ¢rst three laws are based on the com-petition between static grain growth and recrystal-lization, whereas the last two are ad hoc expres-sions for the evolution of grain size towards theequilibrium value D.

2.2.1. Field boundary lawStatic grain growth is usually presented as a

power law of time, dp3dp0 =G(t3t0), where Gand p are constants and d0 is the grain size attime t0 [49]. This law can be recast as:

_dd ¼ Gp31d13p ð8ÞDynamic recrystallization may be treated as a

statistical phenomenon, where the probability ofproducing a new grain is higher for larger grainsthan small grains [43]. That probability also in-creases with the mobility of dislocations, approxi-mated by the strain rate accommodated by dislo-

Fig. 1. Strain rate^stress curves for combined creep law for¢xed grain size (thin solid lines), grain size at equilibrium(thick dashed line), and for a case with only dislocationcreep (thick solid line). The scaling stress c0, strain rate _OO 0,and grain size d0 are de¢ned by Eqs. 20^22.

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cation creep, _OOD [8,43]. Hence, the rate of de-crease of the average grain size by dynamic re-crystallization is proportional to _OOD and to thegrain size, d [43] :_dd ¼ 3 _OODd=O c ð9Þ

where the proportionality constant is expressed asOc, the critical strain for microstructural evolution.Experimental studies suggest that Oc is of orderunity [29,41,42].de Bresser et al. [47,48] argue that dynamic re-

crystallization can be neglected in the di¡usion-dominated regime, and that static grain growthdoes not operate in the dislocation-dominated re-gime. They propose that the boundary betweenthese regimes occurs at a critical ratio of disloca-tion to di¡usion creep, R. In this theory, calledthe ¢eld boundary theory of grain size evolution(FB), the grain size evolves following Eq. 8 if _OOD/_OOG6R and following Eq. 9 if _OOD/ _OOGsR :

FB : _dd ¼Gp31d13p; if _OOD= _OOG6R

3 _OODd=O c; if _OOD= _OOGsR

(ð10Þ

In steady state, _OOD/ _OOG =R, which results in Eqs. 6and 7.In the ¢eld boundary theory, it is assumed that

the critical ratio, R, is a constant. As a result, theequilibrium grain size is di¡erent from Eq. 4.However, it is equally possible that R depend onstress. In particular, if R ¼ ðADA31

G Dm0 Þc nD3nG3mr,

the steady-state grain size relation follows Eq. 4.In this case, the boundary between dislocation-and di¡usion-dominated regimes di¡ers fromthat advocated by de Bresser et al. [47,48]. Ex-pressing this boundary as a function of the grainsize rather than the ratio of dislocation- and dif-fusion-induced strain rates, we obtain a modi¢ed¢eld boundary theory (mFB):

mFB : _dd ¼Gp31d13p; if d6D

3 _OODd=O c; if dsD

(ð11Þ

2.2.2. Continuous recrystallization lawThe grain growth and recrystallization process-

es may not vanish when the critical ratio of dis-location to di¡usion creep is reached. Hall andParmentier [43] proposed to simply use the sum

of these two processes in their continuous recrys-tallization grain size evolution law (CRX):

CRX : _dd ¼Gp31d13p3 _OODd=O c

Gp31d13p3ADc nDd=O c

(ð12Þ

In this theory, the equilibrium grain size is de¢nedby the condition that grain growth and recrystal-lization balance one another:

D ¼ ðA31D O cGp31Þ1=pc3nD=p ð13Þ

2.2.3. Grain size relaxation lawsMathematically simpler grain size evolution

laws leading to the steady-state grain size Dhave been proposed [8,41]. In these laws, therate of grain size evolution is assumed to be pro-portional to the di¡erence between the currentgrain size d and the equilibrium grain size Dand to the strain rate. Kameyama et al. [41] andBraun et al. [42] use the total strain rate, whereasMonte¤si and Zuber [8] include only the strain rateaccommodated by dislocation creep, reasoningthat recrystallization mechanisms involve the mo-tion of dislocations. These assumptions lead re-spectively to the total rate relaxation (TRR) anddislocation rate relaxation (DRR) laws:

TRR : _dd ¼ _OO ðD3dÞ=O cðADc nD þ AGd3m

cnGÞUðD0c3r3dÞ=O c

�ð14Þ

DRR : _dd ¼_OODðD3dÞ=O cADc nD ðD0c3r3dÞ=O c

(ð15Þ

In the DRR model, an increase in grain size oc-curs only associated with dynamic recrystalliza-tion. Therefore, when _OODW0, there is no graingrowth. This aspect of the DRR model is unreal-istic for single-phase aggregates, where graingrowth is driven by a reduction in surface areaand occurs even if the aggregate is not deforming.However, we suggest that the DRR model may beapplicable for polyphase rocks, where static graingrowth is impeded by secondary minerals [49].

2.2.4. Characteristics of the grain size evolutionlaws

Each of the ¢ve evolution laws (Eqs. 10, 11, 12,14, and 15) predicts a speci¢c relation between d;

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and d, for a given strain rate (Fig. 2). The equi-librium grain size is de¢ned as the grain size forwhich d; =0. In the mFB, TRR, and DRR laws,the equilibrium grain size is speci¢ed a priori withEq. 4. For the FB theory to give Eq. 4 at equilib-rium, the critical ratio R must depend on stress,which gives the mFB theory. For the CRX theoryto give Eq. 4 at equilibrium, the following rela-tions must hold:

p ¼ nD=r ð16Þ

G ¼ Dp0ADp=O c ð17Þ

For olivine, this implies pW2.7 which is compat-ible with experimental results.For a large grain size, all of the grain size evo-

lution laws result in an exponential decay towardsD. These laws are equivalent if dED (Fig. 2).When dID, the FB, mFB, and CRX modelspredict rapid static grain growth. Eqs. 16 and 17show that in the CRX theory, the rate of staticgrain growth is tied to the equilibrium grain sizerelation. By contrast, for the ¢eld boundary the-ories, the rate of static grain growth can be ad-justed independently, without in£uencing theequilibrium grain size relation. The TRR law re-sults in a rapid increase of grain size at small d,driven by the larger equilibrium value. While thisresembles static grain growth, grain growth in theTRR law requires strain whereas static graingrowth requires time. In the DRR law, grain

growth requires dislocation creep and a smallgrain size. Hence, when the stress is low, as isthe case when the grain size is small, the grainsdo not grow (Fig. 2). Therefore, the DRR law canrepresent conditions where static grain growth isinhibited by a secondary phase or £uids. Whilethese conditions are not met in laboratory experi-ments on monomineralic rocks, they may be rep-resentative of natural conditions [50].

2.3. Loading system and shear zone simulations

For the simulations, we consider that a ductileshear zone is loaded elastically at a load pointvelocity V that can vary with time (Fig. 3). Thestress changes at a rate proportional to the di¡er-ence between the velocity across the shear zone,VS, and the load point velocity, V :

_cc ¼ KðV3VSÞ ð18Þ

where K is the sti¡ness of the elastic couplingsystem.If the shear zone has thickness H and deforms

at strain rate _OO , VS = _OOH. Then, using Eq. 3, weobtain an equation linking the rate of change ofthe stress to the current grain size and appliedstress:

_cc ¼ K ½V3HðADc nD þ AGd3mc

nGÞ� ð19Þ

The shear zone model is complete when anequation for the evolution of the grain size as afunction of the applied stress and current grainsize is chosen among Eqs. 10, 11, 12, 14, or 15.The resulting system of two equations is solved by

Fig. 2. Absolute value of the rate of grain size evolution as afunction of current grain size at a strain rate of _OO_OO =1, forwhich the FB and mFB models are identical. We use GΠ=1.The grain size evolution rate is positive if d6D and negativeif dsD. Scaling described in Section 2.4.

VK

H

Fig. 3. Schematic representation of the shear zone model. Ashear zone of thickness H and obeying the combined creeplaw and one of the grain size evolution law is loaded at ve-locity V through an elastic system of sti¡ness K.

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a numerical Runge^Kutta method [51] for a givenhistory of loading velocity V(t) and initial condi-tions di and ci.

2.4. Scaling

All the simulations are conducted with the fol-lowing non-dimensional quantities.The ¢rst scale is the stress for which _OOD = _OOG, if

the grain size is at equilibrium:

c 0 ¼ ðADA31G Dm

0 Þ1=ðnGþmr3nDÞ ð20Þ

From the stress scale, we de¢ne the strain ratescale as the total strain rate in steady state at c0 :

_OO ¼ ADcnD0 þ AGD3m

0 cnGþmr0 ð21Þ

and the grain size scale as the equilibrium grainsize at c0 :

d0 ¼ D0c3r0 ð22Þ

A natural time scale is that needed for evolu-tion of the grain size by recrystallization at _OO 0 :

t0 ¼ 2O c= _OO 0 ð23Þ

All non-dimensional quantities are denotedwith a circum£ex. With this scaling, the combinedcreep law becomes:

_OO_OO ¼ ðcc nD þ dd3mcc

nGÞ=2 ð24Þ

the equilibrium grain size is:

DD ¼ cc3r ð25Þ

and the various grain size evolution laws are:

TRR : _dd_dd ¼ ðcc nD þ dd3mcc

nGÞðcc3r3ddÞ ð26Þ

DRR : _dd_dd ¼ ccnDðcc3r3ddÞ ð27Þ

CRX : _dd_dd ¼ dd13nD=r3ccnD dd ð28Þ

FB : _dd_dd ¼GGd13nD=r; if ddm

ccnD3nG6R

3ccnD dd; if ddm

ccnD3nGsR

(ð29Þ

mFB : _dd_dd ¼GGd13nD=r; if dd6DD

3ccnD dd; if ddsDD

(ð30Þ

where GΠ=Grt0/nDdp0 is a non-dimensional param-

eter describing the importance of static graingrowth over dynamic recrystallization. For theCRX law, GŒ is ¢xed to 1 because the equilibriumgrain size is de¢ned by a balance between recrys-tallization and static grain growth.The stress evolution equation, which is derived

from the elastic loading, becomes:

_cc_cc ¼ KK ½VV3ðcc nD þ dd3mcc

nGÞ=2� ð31Þ

where VŒ =V/H _OO 0 and KŒ =2KHOc/c0 are the non-dimensional load point velocity and sti¡ness ofthe loading system, respectively.For all applications, we use KŒ =1 and the pa-

rameters for olivine aggregates: nD =3.5, nG =1,m=3, and r=1.3. We veri¢ed that changing theseparameters does not a¡ect the general behaviorpresented below.

3. Simulation of laboratory experiments

In a simple laboratory experiment, a sample isloaded at constant velocity VŒ , with an initial grainsize d“ i and no initial stress. We conduct similarsimulations, in which the initial stress is cc i = 1/100V0 and the initial grain size is not necessarilythe steady-state value for the loading velocity. We¢rst consider the response of a shear zone to sim-ple shear loading and then compare it with labo-ratory experiments.

3.1. Simple shear experiments

Simple shear simulations are presented in Fig.4, with di¡erent initial grain sizes, the same load-ing velocity VŒ =1, and two di¡erent grain sizeevolution laws, DRR (Eq. 27) and CRX (Eq.28. At steady state, we expect equal contributionsfrom dislocation and di¡usion creep, cc =1, andd“ =1.If the initial grain size is large, the two grain

size evolution laws result in similar stress^straincurves. Initially, the stress increases linearly withtime as the shear zone is loaded elastically. Athigh enough stress, the shear zone deforms at arate similar to the loading velocity. Creep is inquasi-steady state, but is dominated by disloca-tion creep, because the grain size is still large.

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Hence, the stress is higher than the steady-statevalue. During this period, the grain size decreasestowards the equilibrium value. When the grainsize is small enough for di¡usion creep to be im-portant, the shear zone displays strain weakening.The shear zone accelerates and deforms fasterthan it is loaded, relieving the stress. Finally, theshear zone reaches steady state.If the initial grain size is small, the shear zone

starts deforming at low stress in the di¡usion-dominated regime. It then undergoes a period ofstrain hardening as the grain size increases to-wards the equilibrium value. There is no quasi-steady state before the strain-hardening period.However, if static grain growth is inhibited (i.e.,using the law DRR) and the grain size is verysmall, the applied stress can be very small andgrain size evolution extremely slow. If static graingrowth is included, the initially small grain sizeincreases quickly, even during the nominally elas-tic loading period (Fig. 4d). When the stress issu⁄cient for creep to occur, the grain size canbe close to or even exceed the equilibrium value.

In this case, there is no di¡erence between theloading curves for any initial grain size smallerthan the equilibrium value.In Fig. 5, we present the stress^strain curves for

di¡erent simulations with the same loading veloc-ity VŒ =1, the same initial grain size d“ i = 0.5, butdi¡erent evolution laws. For the TRR and CRXlaws, which have fast grain growth at small grainsize (Fig. 2), the initial hardening is easily mistak-

0 2 4 6 8 100

0.5

1

1.5

Str

ess

DRRTRRCRXmFB FB

Strain

FB and mFB

DRR

TRR

and

CR

X

Fig. 5. Evolution of stress for shear zones obeying di¡erentgrain size evolution laws with an initial grain size of d“ =0.5,and loading velocity VŒ =1. GŒ =0.1 for the simulation usingthe mFB and FB laws.

0 2 4 6 8 100

0.5

1

1.5

Str

ess

0 2 4 6 8 100

0.5

1

1.5

Str

ain

rate

0 2 4 6 8 1010

-1

100

101

102

Gra

in s

ize

Strain

Total strain rate

Dislocation creep strain rate

a)

c)

e)

10010 0.80.5

initial grain size

0 2 4 6 8 100

0.5

1

1.5

Str

ess

0 2 4 6 8 100

0.5

1

1.5

Str

ain

rate

0 2 4 6 8 1010

-1

100

101

102

Gra

in s

ize

Strain

Total strain rate

Dislocation creep strain rate

b)

d)

f)

10010 10.1

initial grain size

Fig. 4. Evolution of stress (a,), strain rate (c,d; thicker lines: total strain rate; thinner lines: strain rate by dislocation creeponly), and grain size (e,f) for simulations of laboratory experiments with di¡erent grain size. The DRR grain size evolution lawis used for a,c,e, and the CRX grain size evolution law is used for b,d,f.

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en for elastic loading. For the DRR law, whichdoes not include static grain growth, there is amuch slower hardening rate. If GŒ =1 with theFB and mFB laws, static grain growth is fastand the simulation results are similar to theCRX case. With the FB and mFB laws, we canreduce the rate of static grain growth indepen-dently of the equilibrium grain size. By setting GŒ

at 0.1, we observe a behavior intermediate be-tween the CRX (GΠ=1) and DRR (GΠ=0) simula-tions (Fig. 5).In summary, the ductile shear zone model ex-

hibits two types of responses to loading at a con-stant velocity: one with transient weakening, theother with transient hardening. The transition be-tween these two types of responses is presented asa function of initial grain size in Fig. 4. The sametransition occurs with changes in loading velocity(Fig. 6). If the loading velocity is high, dislocationcreep is favored. The resulting stress^strain curvedisplays transient weakening, similar to the casewith large initial grain size. If the loading velocityis low, which initially favors di¡usion creep, weobserve strain hardening as the grain size in-creases towards the steady-state value and di¡u-sion creep becomes less important. As before,static grain growth mitigates this e¡ect by produc-ing large grains quickly.

3.2. Comparison with laboratory experiments

Several recent experiments on microstructuralevolution display stress^strain curves similar toour simulations. As we use parameters relevantfor dry olivine, our simulations are most readilycompared with the experiments of Bystricky et al.[28]. In these experiments, the recrystallized grainsize is about a factor of 10 smaller than the initialgrain size. For these conditions, we predict aninitial yield point about 20% higher than the ¢nalsteady-state stress and no stress plateau as thesample recrystallizes, in agreement with experi-mental results.With an initial grain size 10 times larger than

the ¢nal grain size, as observed in Bystricky etal.’s experiments [28], the di¡erence between sim-ulations using alternative £ow laws is minimal. Toaddress which grain size evolution law is mostapplicable to these experiments, it will be impor-tant to consider the e¡ect of velocity steps inwhich the loading velocity is suddenly reducedor increased. The grain size evolution laws alsogive very di¡erent results in situations where thegrain size increases. Karato et al. [39] report ontriaxial experiments on olivine with a variety ofinitial grain size, including some with small initialgrain size. These experiments display hardeningassociated with grain growth, as in our simula-tions. An experiment in which deformation wasinterrupted showed time-dependent hardening,which is interpreted as evidence for static graingrowth. This implies that laws like CRX or FBare more relevant in the laboratory for single-phase aggregates. The alternative laws, such asTRR and DRR, link grain growth to deformationof the sample, not time, and cannot replicate Ka-rato et al.’s observations. However, in nature,static grain growth may be impeded by a pinningsecond phase [49] and a law for which smallgrains do not grow, such as DRR, may be appli-cable.One di¡erence between our simulations and the

experiments is the shear zone geometry. Whereaswe assume a simple shear geometry, the experi-ments are conducted in a rotary shear apparatus,in which the angular velocity of the shear zone isimposed, and the total torque on the sample is

0 2 4 6 8 100

1

2

3

4

Str

ess

0 2 4 6 8 10

1

Gra

in s

ize

Strain

a)

b)2

0.3

0.5

100 10 10.1

Loading velocity

Fig. 6. Evolution of stress (a) and grain size (b) for a shearzone obeying the DRR grain size evolution law, an initialgrain size of d“ =1 and varying loading velocity.

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measured. The torque is an integration of thestress over the entire sample. At each location,the stress is a function of the local strain andstrain rate, which depend on radius. We adaptedour shear zone model for a rotary shear geometry.The resulting time^torque curves are similar tothe simple shear results presented in Figs. 4^6.

4. Response to earthquakes

An earthquake may be seen as a short period oftime during which shear displacement on a fault ismuch more rapid than during the intervening pe-riod. If a ductile shear zone is present in the vi-cinity of a seismogenic fault, for instance as anextension of the fault plane at depth, it will beloaded during an earthquake at a faster ratethan during interseismic periods. We present sim-ulations of the response of ductile shear zones toan earthquake-like loading.We consider that earthquakes, of duration TΠE,

occur periodically with a recurrence intervalTΠRETΠE. The loading velocity is VΠis during the

interseismic period and VΠcs during the coseismicinterval. In the simulations presented herein, weuse TΠR = 1000, VΠis = 1036, and VΠcs = 103. The ve-locity jump of nine orders of magnitude betweencoseismic and interseismic displacement rates issimilar to that experienced in nature. We varyTΠE from 1036 to 1033. The simulation is con-ducted for 20 earthquake cycles in order to reacha limit cycle, independent of initial conditions. Wepresent only the last of these cycles in Fig. 7. Wenote that in most simulations, the limit cycle isreached at the second or third cycle.If the earthquake lasts long enough, steady

state is attained for the fast coseismic velocity(Fig. 7a). The grain size at the end of the earth-quake is very small and tends to increase in theinterseismic interval. However, in the absence ofstatic grain growth (DRR law), the grain size isvirtually unchanged during the interseismic periodbecause of the low residual stress (Fig. 7c,e).Hence, the shear zone is able to deform at thecoseismic rate during the next earthquake withoutrequiring a large microstructural adjustment.However, by including grain growth (CRX law),

Fig. 7. Evolution of strain rate (a,b), stress (c,d), and grain size (e,f) for simulations of shear zone response to earthquake loadingfor di¡erent earthquake durations and using the DRR grain size evolution law (a,c,e) or the CRX grain size evolution law(b,d,f). The arrows indicate the end of the coseismic period for simulations with matching line style.

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the grain size reaches the steady-state value at theinterseismic velocity (Fig. 7f). The strength of theshear zone, when loaded at coseismic velocities, isvery high. As a result, a slip de¢cit builds upduring the earthquake, leading to prolonged creepafter the earthquake ends. The rate of postseismiccreep decreases approximately as a power law intime (Fig. 7b).With short-duration earthquakes, steady state

is not attained in the coseismic period, regardlessof the grain size evolution law (Fig. 7). The grainsize remains close to the large equilibrium valuefor interseismic velocities and the shear zone de-forms for a signi¢cant portion of the interseismicinterval at a rate intermediate between the coseis-mic and interseismic rates.The periods of high strain rate that follow

earthquakes in these simulations may be observedgeodetically after large earthquakes. Such earth-quakes are followed by a prolonged period ofaseismic creep occurring mainly on the extensionof the fault plane at depth [32^35]. Because of thehigher temperatures at depth, it is possible thatthe extensions of the fault plane are ductile shearzones [52]. Hence, postseismic creep may bedue to ductile creep triggered by the earthquake.More realistic simulations, including depth depen-dence of the shear zone parameters, elastic cou-pling, and slip complexity on the fault surface,must be conducted before this hypothesis can betested.

5. Comparison with rate- and state-dependentfriction

Postseismic creep may also be due to velocity-strengthening friction [37,38]. Indeed, frictionalsliding becomes velocity-strengthening above acritical temperature, leading to a stably slidingdeep fault plane [53^58]. One prediction of ourmodel is a power law decay of the creep rate,which contrasts with the exponential decay pre-dicted by velocity-strengthening friction. It is pos-sible that geodetic data can di¡erentiate betweenthese models although recent numerical simula-tions of postseismic creep following the 1999 Iz-mit earthquake show that even very di¡erent as-

sumptions about the deep fault behavior canproduce similar creep patterns [38]. Next, weshow how ductile creep can result in a behaviorsimilar to that predicted by velocity-strengtheningfriction in typical laboratory experiments.Rock friction can be described in the frame-

work of rate- and state-dependent friction(RSDF) theory, whereby the coe⁄cient of fric-tion, W, depends on the sliding velocity, V, andon a state variable a, which is related to the mi-crostructure of the sliding surface [59]. The statevariable may evolve with time or slip towards asteady value that depends only on the currentsliding velocity [60,61]. In RSDF theory, the shearstress on the fault is :

d ¼ c nW ¼ d 0 þ A ln ðV=V 0Þ þ B ln ðaV 0=DCÞ ð32Þ

where cn is the normal stress on the fault, d0 andV0 are reference values of the shear stress and thesliding velocity, and A, B, and DC quantify theresponse of W to a change in V at constant a, theevolution of a, and the critical distance for evolu-tion of a, respectively. At steady state, a=DC/V,and the shear stress is:

d ¼ d 0 þ ðA3BÞ ln ðV=V 0Þ ð33Þ

If A3B6 0, the steady-state friction decreaseswith increasing velocity. The situation is unstableand may lead to earthquakes [62,63]. By contrast,when A3Bs 0, sliding is stable. In the laborato-ry, the transition to velocity strengthening maycorrespond to the onset of ductility in the rock[52,54]. The transition corresponds to the appear-ance of a second state variable with strongly pos-itive A3B and large DC, which dominates thesteady-state regime without eliminating the e¡ectsof the ¢rst state variable [57,58].The second state variable may be a manifesta-

tion of ductile behavior in the fault gouge. Theconstitutive equations de¢ned in Section 2 are atype of rate- and state-dependent rheology, forwhich the state variable is explicitly the grainsize. The coe⁄cients A and A3B of the RSDFlaws are most simply de¢ned from the change offrictional resistance to an in¢nitesimal change ofV at constant a (i.e., constant d) and in steady

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state (i.e., d=D), respectively. Velocity step ex-periments conducted on a ductile shear zonewith evolving grain size give apparent values ofA and A3B :

A ¼

D c

DV

����d constant

c 01þ cc

nGþmr3nD

nD þ nGcc nGþmr3nD

8>><>>: ð34Þ

A3B ¼

dcdV

����d¼D

c 01þ cc

nGþmr3nD

nD þ ðnG þmrÞcc nGþmr3nD

8>>><>>>:

ð35Þ

where cc is the stress immediately before the ve-locity step. These relations are illustrated in Fig.8. Note that A3B is always positive, indicatingthat deformation is always stable in this model.Hence, ductile creep is inconsistent with RSDF atlow temperature, where A3B6 0.The parameters A, B, and DC (Eq. 32) are de-

termined experimentally from the response of afault to sudden changes in loading velocity. Wesimulate such experiments with our shear zonemodel and analyze them within the frameworkof RSDF laws (Fig. 9). In these simulations, theinitial velocity is VΠ=1 and the shear zone is ini-tially in steady state. At time zero, the velocity isincreased (Fig. 9a) or decreased (Fig. 9b) by afactor of 10. The simulation is continued until

the shear zone is at steady state for the ¢nal load-ing velocity. Then, we ¢t the stress^state curveswith the RSDF formulation. In addition to Eq.32, we need to de¢ne an evolution law for thestate variable a. We use the evolution law pro-posed by Dieterich [60] :

da =dt ¼ 13Va =DC ð36Þ

Alternative state evolution laws give comparableresults.In our ¢tting algorithm, the parameters A and

A3B are determined a priori from Eqs. 34 and 35modi¢ed for a ¢nite change in loading velocity.The parameter DC, however, cannot be speci¢ed apriori. Hence, we explore a large range of candi-date values, and retain the value of DC that pro-duces the closest match to the ductile shear zonesimulation.In all cases, an almost perfect match can be

found (Fig. 9). It would be impossible to di¡er-entiate experimentally between RSDF and ductilecreep if only one of these stress^strain curves wasavailable. However, during ductile creep, the val-ues of A, B, and DC inferred from the stress^strain curves change as a function of the magni-

Fig. 9. Thick dashed lines: response of a shear zone deform-ing by combined di¡usion and dislocation creep to (a) an in-crease or (b) a decrease of the loading velocity by a factor of10. The curves are labelled by the grain size evolution law.Thin solid lines: Best-¢tting simulation with rate- and state-dependent friction. In each case, the ¢t is virtually perfect.

10-2 10-1 100 101 1020

0.2

0.4

0.6

0.8

1

A/σ

0, (

A-B

)/σ 0

Stress

A-B

A

Fig. 8. Apparent rate- and state-dependent friction parame-ters A3B (solid line) and A (dashed line) for a shear zonedeforming by combined dislocation and di¡usion creep withnD =3.5, nG =1, m=3, and r=1.3.

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tude and direction of the velocity step, with anincrease of velocity yielding smaller A and A3Band larger B and DC values than a decrease ofloading velocity. It may be possible to di¡erenti-ate between the combined creep rheology andRSDF using relaxation experiments because thestrain rate of the ductile shear zone cannot belower than the dislocation creep rate, which de-pends on the applied stress only. By contrast, thefrictional theory allows arbitrarily small velocityat a given stress.

6. Conclusions

We have presented a formulation for combineddislocation and di¡usion creep that considers theevolution of grain size with time and/or deforma-tion. Several grain size evolution laws were em-ployed to explore the e¡ects of static grain growthand dynamic recrystallization. The grain size evo-lution laws were included in a shear zone modelto simulate the response of a ductile shear zone tosudden changes of loading velocity. The transi-tional e¡ects observed in the simulations comparefavorably with laboratory experiments on the ef-fect of microstructural evolution on rock rheol-ogy. We also found that the ductile shear zoneresponse resembles rate- and state-dependent fric-tion. Simulations of the shear zone response toearthquake cycles predict episodes of enhancedcreep following earthquakes, which are possibleanalogues of postseismic creep observed geodeti-cally following major Ms 7.0 earthquakes.

Acknowledgements

We thank Scott King and Norm Sleep for theircomments on the manuscript. We are grateful toChad Hall for sharing with us his insight on grainsize evolution and pre-publication manuscripts.This work was supported by the PostdoctoralScholar Program at the Woods Hole Oceano-graphic Institution, with funding provided bythe USGS, by NSF Grant OCE-9907244 and agrant from the Deep Ocean Exploration Instituteat WHOI.[SK]

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