fault finiteness and initiation of dynamic shear instability

Fault ¢niteness and initiation of dynamic shear instability

Cristian Dascalu a;b;*, Ioan R. Ionescu a, Michel Campillo c

a Laboratoire de Mathematiques, Universite de Savoie, 73376 Le Bourget-du-Lac Cedex, Franceb Institute of Applied Mathematics, Romanian Academy, P.O. Box 122, 70700 Bucharest, Romania

c Laboratoire de Geophysique Interne, Universite Joseph Fourier, P.O. Box 53X, 38041 Grenoble Cedex, France

Received 5 May 1999; received in revised form 28 January 2000; accepted 10 February 2000

Abstract

We study the initiation of slip instabilities of a finite fault in a homogeneous linear elastic space. We consider theantiplane unstable shearing under a slip-dependent friction law with a constant weakening rate. We attack the problemby spectral analysis. We concentrate our attention on the case of long initiation, i.e. small positive eigenvalues. A staticanalysis of stability is presented for the nondimensional problem. Using an integral equation method we determine thefirst (nondimensional) eigenvalue which depends only on the geometry of the problem. In connection with theweakening rate and the fault length this (universal) constant determines the range of instability for the dynamicproblem. We give the exact limiting value of the length of an unstable fault for a given friction law. By means of aspectral expansion we define the `dominant part' of the unstable dynamic solution, characterized by an exponential timegrowth. For the long-term evolution of the initiation phase we reduce the dynamic eigenvalue problem to ahypersingular integral equation to compute the unstable eigenfunctions. We use the expression of the dominant part todeduce an approximate formula for the duration of the initiation phase. Finally, some numerical tests are performed.We give the numerical values for the first eigenfunction. The dependence of the first eigenvalue and the duration of theinitiation on the weakening rate are pointed out. The results are compared with those for the full solution computedwith a finite-differences scheme. These results suggest that a very simple friction law could imply a broad range ofduration of initiation. They show the fundamental role played by the limited extent of the potentially slipping patch inthe triggering of an unstable rupture event. ß 2000 Published by Elsevier Science B.V. All rights reserved.

Keywords: faults; slip rates; eigenvalues; shear; rupture

1. Introduction

We de¢ne the initiation phase as the periodbetween a perturbation of the mechanical condi-tions of a fault system and the onset of rupturepropagation which is associated with wave radia-

tion. We do not discuss here the origin of theperturbation. It may be for example the e¡ect ofa distant earthquake or a local small variation ofstrain due to a rapid change of £uid pressure. Iio[9] and Ellsworth and Beroza [6] observed a weakslow increase in ground motion before the strongseismic pulse associated with the propagating rup-ture front. They interpreted this signal as amarker of the initiation phase. However, the char-acteristic time scale of this observation is typicallyof the order of 1 s. A much longer initiation phase

0012-821X / 00 / $ ^ see front matter ß 2000 Published by Elsevier Science B.V. All rights reserved.PII: S 0 0 1 2 - 8 2 1 X ( 0 0 ) 0 0 0 5 5 - 8

* Corresponding author.E-mail: [email protected]

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can be expected during which the slip acceleratesexponentially. The present-day instruments canonly detect the very last stage associated withlarge enough slip velocity.

On the other hand, one can observe that anearthquake can be triggered by another one withvarious large delays. A typical example is the 3 hdelay between the Landers and Big Bear earth-quakes of 1992, but the delay may be much larger(see Harris [8] for a review). It is also establishedthat the time distribution of seismicity has nocorrelation with the stress changes associatedwith the tides (see Vidale et al. [14]). This suggeststhat the delay between a perturbation of stressand the onset of rupture propagation is variableand may be at least as large as the period of thetides. Do such long initiation times preclude usingsimple elastic models? Do they imply that one hasto consider complex ad-hoc friction laws?

To provide some elements to answer these ques-tions, we studied the range of initiation durationexpected for simple models of fault. The case ofthe single fault which is treated here is a buildingblock for a more developed model of fault. Thecomplete understanding of the physics of its spon-taneous rupture is therefore critical for the set-upof dynamic fault models. We investigated the do-main of evolution between the long-term quasi-static evolution governed by the slow externaldriving (plate) motion and the extremely rapidslip in the rupture propagation stage. This inter-mediate time scale sets speci¢c di¤culties. Thedynamic aspect of the problem cannot be ignoredduring the onset of the instability, that is the pe-riod of continuous acceleration of the slip process.At the same time, the long duration of initiation isnot compatible with the standard numerical tech-niques used to deal with rupture dynamics. Wepropose a spectral resolution of the problemthat takes advantage of the very fundamentalproperties of the solution of the complete problemin the initiation stage.

We consider an elastic body with a fault char-acterized by a slip-dependent friction law. Cam-pillo and Ionescu [3,10] studied a slip weakeninglaw deduced from laboratory experiments (see forinstance [12]). They considered in [3] a piecewiselinear version of the friction law and an in¢nite

fault. They derived the generic form of the unsta-ble behavior in the initiation phase and gave thecharacteristic length lf associated with the frictionlaw. In [10] they studied a nonlinear weakeningand showed the determinant role of the initialweakening for the duration of initiation, whichis only weakly sensitive to the critical slip for agiven the initial weakening rate. They also showednumerically that the ¢niteness of the faultstrongly a¡ects the duration of initiation whenthe fault length 2a is of the same order as a char-acteristic length associated with the friction law lf .However, the numerical simulation does not makeit possible to investigate the behavior of the faultwhen 2a is equal to or less than lf , since theseconditions lead to a very long duration of evolu-tion that cannot be handled with the ¢nite di¡er-ence technique. As a consequence, the minimumfault length 2ac required for an instability to de-velop cannot be evaluated numerically. Its orderof magnitude is known only by a static stabilityanalysis based on the analogy with a block slidersystem. The goal of the present paper was to ¢nda theoretical manner of dealing with the dynamicevolution of the system when a is close to ac, thatis at the limit between the stable and unstablebehaviors.

After the formal statement of the nondimen-sional dynamic problem, we present its spectralexpansion. We emphasize the role of the staticstability analysis to characterize the transition be-tween stable and unstable behaviors. This transi-tion is associated with a (universal) value of a newnondimensional parameter L which emerges fromthe spectral analysis of the problem. We computethe eigenfunctions and give an approximate for-mula for the time of initiation. Finally we presentnumerical tests demonstrating the accuracy of ourapproach.

2. Problem statement

We consider the antiplane shearing on a ¢nitefault y = 0, MxM6 a of length 2a, denoted by yf , ina homogeneous linear elastic space. The contacton the fault is described by a slip-dependent fric-tion law. We assume that the displacement ¢eld is

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0 in directions Ox and Oy and that uz does notdepend on z. The displacement is therefore de-noted simply by w(t,x,y). The elastic mediumhas a shear rigidity G, a density b and a shearvelocity c =

��G=b

p. The nonvanishing shear stress

components are czx = drx +GDxw(t,x,y) and czy =dry +GDyw(t,x,y), and the normal stress on the faultplane is cyy =3S (Ss 0).

The equation of motion is:

D 2wD t2 �t; x; y� � c29 2w�t; x; y� �1�

for ts 0 and (x,y) outside the fault yf . Theboundary conditions on yf are:

c zy�t; x; 0�� c zy�t; x; 03�; MxM6a �2�

c zy�t; x; 0� �

W �N w�t; x��S signD N wD t�t; x�

� �; MxM6a �3�

if DtNw(t,x)g0 and:

Mc zy�t; x; 0�M9W �N w�t; x��S; MxM6a �4�if DtNw(t,x) = 0, where Nw(t,x) = 1/2(w(t,x,0+)3w(t,x,03)) is the half of the relative slip andW(Nw) is the coe¤cient of friction on the fault.

The initial conditions are denoted by w0 andw1, that is:

w�0; x; y� � w0�x; y�; DwD t�0; x; y� � w1�x; y� �5�

Since our intention is to study the evolution of theelastic system near an unstable equilibrium posi-tion, we shall suppose that dry = SWs, whereWs =W(0) is the static value of the friction coe¤-cient on the fault. We remark that taking w as aconstant satis¢es Eqs. 1^4; hence wr0 is a meta-stable equilibrium position, and w0, w1 may beconsidered small perturbations of the equilibrium.We shall suppose that the friction law has theform of a piecewise linear function:

W �N w� � W s3W s3W d

LcN w; N w9Lc �6�

W �N w� � W d; N wsLc �7�

where Ws and Wd (Ws sWd) are the static and dy-namic friction coe¤cients, and Lc is the criticalslip. Let us assume in the following that the slipNw and the slip rate DtNw are non-negative. Bear-ing in mind that we deal with a homogeneousfault plane and with the evolution of one initialpulse, we may put (for symmetry reasons)w(t,x,y) =3w(t,x,3y), hence we consider onlyone half-space ys 0 in Eq. 1 and Eq. 5. Withthese assumptions, Eqs. 2^4 become:

w�t; x; 0� � 0; for MxMva �8�

DwD y�t; x; 0� � 3Kw�t; x; 0�� if

w�t; x; 0�9Lc; for MxM6a �9�

DwD y�t; x; 0� � 3KLc if w�t; x; 0�sLc; for MxM6a

�10�

where K is a parameter which has the dimensionof a wavenumber (m31) and which will play animportant role in our further analysis. The valueK is given by:

K � �W s3W d�SGLc

�11�

It is important to note that K is proportional tothe weakening rate. Note that this parameter wasdenoted by Kc in the case of the in¢nite fault (see[3]) when it represents a characteristic (critical)wave number. It no longer plays such a role inthe present discussion on a ¢nite fault.

3. Nondimensional problem and its spectralexpansion

Since the initial perturbation (w0,w1) of theequilibrium state wr0 is small, we havew(t,x,0+)9Lc for tn[0,Tc] for all x, where Tc isa critical time for which the slip on the faultreaches the critical value Lc at least at one point,that is, supxnRw(Tc,x,0+) = Lc. Hence for a ¢rstperiod [0,Tc], called in what follows the initiation

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C. Dascalu et al. / Earth and Planetary Science Letters 177 (2000) 163^176 165

period, we deal with a linear initial and boundaryvalue problem (Eqs. 1, 5 and 9).

In order to obtain some universal entities (de-pending only on the geometrical con¢guration) tocharacterize the stability of the above problem, wegive here its nondimensional formulation. If weput:

x1oxa; x2o

ya

and we introduce the following nondimensionalconstant:

L � aK � a�W s3W d�S

GLc�12�

then from Eqs. 1, 5 and 9 we deduce:

D 2wD t2 �t; x1; x2� � c2

a292w�t; x1; x2� �13�

w�t; x1; 0� � 0; for MxM1v1 �14�

DwD x2�t; x1; 0� � 3L w�t; x1; 0�; for Mx1M61 �15�

w�0; x1; x2� � w0�x1; x2�; DwD t�0; x1; x2� � w1�x1; x2�

�16�

Let us consider the following eigenvalue prob-lem connected to Eqs. 13^16: ¢nd x : RUR�CRand V2 such that:Z �r3r

Z �r0

x 2�x1; x2�dx1dx2 � 1

and:

9 2x �x1; x2� � V 2x �x1; x2�; for x2s0 �17�

x �x1; 0� � 0; for MxM1v1 �18�

DD x2

x �x1; 0� � 3Lx �x1; 0�; for Mx1M61 �19�

Since we deal with a symmetric operator wehave real-valued eigenvalues V2, i.e. V is real orpurely imaginary. Since the boundary conditions(Eqs. 18 and 19) are included in the de¢nition of

the operator (see Ionescu and Paumier [11] for aprecise functional framework) this symmetryproperty is speci¢c to the slip-dependent frictionlaw used here. Let us denote by (V2

n,xn) the asso-ciated eigenvalues and the eigenfunctions of Eqs.17^19. Bearing in mind the above notations, onecan verify (by simple calculation) that the solutionof Eqs. 13^16 can be generically written (in itsspectral expansion) as:

w�t; x1; x2� �Xrn�0

�cosh �cV nt=a�W 0n�

asinh �cV nt=a�

cV nW 1

n�x n�x1; x2�; for x2s0 �20�

where:

W 0n �

Z �r3r

Z �r0

x n�x1; x2�w0�x1; x2�dx1dx2

�21�

W 1n �Z �r3r

Z �r0

x n�x1; x2�w1�x1; x2�dx1dx2 �22�

are the projections of the initial data on the ei-genfunctions. Let N be such that:

V 20sV 2

1sTsV 2N31s0sV 2

NsT

We remark that the part of the solution asso-ciated with positive eigenvalues V2 will have anexponential growth with time. Hence, after awhile this part will completely dominate the otherpart which has a wave-type evolution. This behav-ior is the expression of the instability caused bythe slip weakening friction law. This is why weput:

w � wd � ww

where wd is the `dominant part' and ww is the`wave part', given by:

wd�t; x1; x2� �XN31

n�0

�cosh�cMV nMt=a�W 0n�

asinh�cMV nMt=a�

cMV nMW 1

n�x n�x1; x2� �23�

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ww�t; x1; x2� �Xrn�N

�cos�cMV nMt=a�W 0n�

asin�cMV nMt=a�

cMV nMW 1

n�x n�x1; x2� �24�

The use of the expression of the dominant partleads to a solution in which the perturbationhas been severely smoothed by the ¢nite wave-number integration. The propagative terms arerapidly negligible and the shape of the slip dis-tribution is almost perfectly described by thedominant part. This was already noticed by Cam-pillo and Ionescu [3] in the case of the in¢nitefault.

4. Stability analysis

One can easily see that wr0 is a stable positionif V2

0 6 0 (i.e. N = 0). In this case the dominantpart wd vanishes and the system has a stable be-havior. Hence it is important to obtain a simplecondition on L which determines the positivenessof the eigenvalues V2. Since L is nondimensionalsuch a condition depends only on the geometry ofthe domain and it completely characterizes thestability. The numerical simulation does not per-mit investigation of the behavior of the fault whenV0 is small and greater than 0, since these con-ditions lead to very long durations of evolutionthat we are not able to handle with the ¢nite-di¡erences technique. As a consequence, the mini-mum fault length required for an instability todevelop cannot be evaluated numerically. Its or-der of magnitude is known only by a static stabil-ity analysis based on the analogy with a blockslider system. In order to obtain a more preciseevaluation a sharper stability analysis has to beconsidered (see Ionescu and Paumier [11] forsome general results in the case of a ¢nite do-main).

In order to perform a stability analysis let usintroduce the eigenvalue problem correspondingto the static case: ¢nd B : RUR�CR and Lsuch that:

Z �r3r

Z �r0

B 2�x1; x2�dx1dx2 � 1

and:

9 2B�x1; x2� � 0; for x2s0 �25�

B�x1; 0� � 0; for Mx1Mv1 �26�

D x2B�x1; 0� � 3L B�x1; 0�; for Mx1M61 �27�

This problem has a sequence of positive eigen-values 06 L0 6 L1 6 T with:

limn!r

L n � �r:

The eigenvalues Lk may be viewed in connectionwith the dynamic eigenvalue problem (Eqs. 17^19). Indeed, they correspond to the intersectionpoints of the increasing curves LCV2

k(L) withthe axis V2 = 0 (i.e. V2

k(Lk) = 0). Such functionsare generically represented in Fig. 1.

The ¢rst eigenvalue L0 has a major signi¢cancein the static stability analysis: if L6L0 thenV2

0 6 0, i.e. :

if a�W s3W d�S

GLc� L6L 0 then wr0 is stable �28�

Fig. 1. The generic representation of the ¢rst three eigenval-ues V2

0 =V20(L), V2

1 =V21(L) and V2

3 = V23(L) versus the nondimen-

sional weakening parameter L. Note the intersection with theaxis V2 = 0 on the eigenvalues L0, L1, L2 of the static prob-lem.

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Moreover, if Ls L0 one obtains (see Fig. 1) thenumber N of positive eigenvalues V2 contained inthe expression of the dominant part, i.e. :

if Ln�L N31; L N� then V 20sV 2

1sTsV 2N31s0sV 2

NsT�29�

The sequence (Lk)kv0 plays an important role inthe stability of the dynamic solution and thestructure of the dominant part. The aim of thissection is to determine the numerical values of Lk.

In Appendix A we have reduced the problem ofEqs. 25^27 to the following hypersingular integralequation for B(x1,0) :

B�x1; 0� � 1L Z

PVZ 1

31

D sB�s; 0�x13s

ds Mx1M61 �30�

where the integral is taken in the Cauchy principalvalue sense. Note that Eq. 30 is similar to thePrandtl equation for the compressible case inaerodynamics (see for instance [4,5]). To solveEq. 30 we seek a solution of the form:

B�x1; 0� �Xrn�1

unsin�n arccos�x1�� 31�

Let us remark that each term of this expansioncontains the right scaling behavior at the faultedges in terms of both the displacement and thesingularity of stress. When this expression is sub-stituted in Eq. 30 we obtain the following gener-alized eigenvalue problem (see the details in Ap-

pendix A) involving the in¢nite matrices A and B :

LXrn�1

Aknun �Xrn�1

Bknun; k � 1; 2; T �32�

where:

Akn � 32kn�1� �31�k�n�

��k3n�231��k � n�231�

�13N n;k31��13N n;k�1�; Bkn � nZ2N k;n �33�

By retaining the ¢rst M terms in the series in Eq.32 we obtain a `generalized eigenproblem' for LM

with the eigenvectors (uMn )n�1;M , i.e. :

L MAMuM � BMuM �34�

where AMij = Akn and BM

kn = Bkn for 19 k, n9M.This problem was numerically solved for M = 20,M = 100 and M = 1000 and the ¢rst 10 eigenvaluesare given in Table 1.

The constant L0 depends only on the geometryof the antiplane problem with a ¢nite fault and itis independent of all physical entities involved inour problem. We remark that the smallest eigen-value was found as:

L 0 � 1:15777388 �35�

This nondimensional parameter gives quantita-tively the limit between the stable (L6L0) and

Table 1The ¢rst 10 eigenvalues of the static nondimensional problem (Eqs. 25^27) computed from the truncated algebraic eigenproblem(Eq. 34) for M = 20, M = 100 and M = 1000

M = 20 M = 100 M = 1000

L0 1.15777389 1.15777388 1.15777388L1 2.75475480 2.75475474 2.75475474L2 4.31680136 4.31680107 4.31680107L3 5.89214801 5.89214747 5.89214747L4 7.46017712 7.46017574 7.46017574L5 9.03285448 9.03285269 9.03285269L6 10.60229691 10.60229310 10.60229310L7 12.17412302 12.17411826 12.17411826L8 13.74413789 13.74410906 13.74410906L9 15.31570893 15.31555500 15.31555500

Note that there is no variation of the eigenvalues at this level of accuracy.

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unstable (Ls L0) behaviors of the fault. For eachrepresentative physical quantity which is includedin the nondimensional parameter L (the stressdrop (Ws3Wd)S, the friction weakening slope(Ws3Wd)S/Lc, the interface sti¡ness G/a, the faulthalf length a, the elastic bulk modulus G, etc.) wecan de¢ne a `critical' value. For instance, accord-ing to Eq. 28 the minimum half length ac of anunstable fault is :

ac � L 0

K� L 0GLc

�W s3W d�S �36�

This critical fault length for unstable behavior is aconcept very di¡erent from that of critical cracklength of a propagating crack as developed byPalmer and Rice [13] or Andrews [1]. The criticalfault length de¢ned here refers to the developmentof instability on a heterogeneous fault surfacewhile the critical crack length is related to theenergy balance of a propagating crack on a ho-mogeneous fault.

5. Spectral analysis of initiation

During the initiation phase the essential behav-ior of the solution w of Eqs. 13^16 is given by itsdominant part wd. In order to compute wd wehave to determine the eigenvalues Vn and the ei-genfunctions xn, for n = 0,N from Eqs. 17^19.This is the objective of this section. These resultswill be used in the next section to compute thelong-term evolution of the slip during the initia-tion phase. Such a ??long initiation period can beobtained only for small values of V. This assump-tion, i.e. :

MV MI1

will be adopted in what follows.

To study the problem in Eqs. 17^19 we employa technique similar to that in the static case. Sincein the expression of the dominant part it appearsMVM for V real-valued we seek only positive V inwhat follows. The details of calculus are presentedin Appendix B. We have deduced the hypersingu-lar integral equation for x(x1,0):

x �x1; 0� �

3VZL

FPZ 1

31x �s; 0�K1�V Ms3x1M�

Ms3x1Mds; Mx1M61

�37�

where K1 is the modi¢ed Bessel function of thesecond kind and the integral is taken in the ¢nite-part sense (see for instance [7]). For small valuesof V (i.e. VI1), we can approximate Eq. 37 by:

ZLx �x1; 0� �

PVZ 1

31

D sx �s; 0�dsx13s

3V 2Z 1

31x �s; 0�ln�V Ms3x1M�ds

�38�

with Mx1M6 1 and the integral is taken in the prin-cipal value sense. We seek a solution for Eq. 38 ofthe form:

x �x1; 0� �Xrn�1

Un sin�n arccos�x1�� 39�

which implies the following generalized eigenvalueproblem involving in¢nite matrices :

LXrn�1

AknUn �Xrn�1

�Bkn � Ckn�V ��Un; k � 1; 2; T

�40�

with Akn, Bkn given by Eq. 33 and:

Ckn�V � � ZV 2 lnV4

N n;1N k;13ZV 2

16

�N k;33�1� 4 ln2�N k;1�N n;13ZV 2

8�n23 1� �13N n;3�

��n31�N k;n�2 � �n� 1�N k;n3232nN k;n� �41�

Our objective is to determine V(L) and Un(L),n = 1,2,T. The form (Eq. 40) of the system suggestsan analysis of the eigenvalue problem with L asthe eigenvalue and V as a parameter. Such a tech-nique will ¢nally provide the values of V(L) andthe corresponding eigenvectors Un(L).

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As in the static analysis, we retain the ¢rst Mterms in the series (Eq. 40). Since for the descrip-tion of the initiation process only the ¢rst value ofV, denoted V0, is relevant, we shall determine nu-merically V0 as a function of L and the corre-sponding eigenvector U0

n(L). Then, by Eq. 39 weobtain x0.

6. Duration of initiation

We can de¢ne two characteristic times in thesystem. The ¢rst one, a/c, is related to the wavevelocity and the other one can be de¢ned throughthe characteristic slip patch K31 by 1/(cK). For thein¢nite system size the latter is the only time scalewe have and it gives the order of the duration ofinitiation Tc on an in¢nite fault (see Campillo andIonescu [3] for an approximative formula of Tc).For a ¢nite system size the duration of initiationdoes not scale with any of these two characteristictimes. Indeed, as follows from the numerical ex-periments of Ionescu and Campillo [10], the du-ration of initiation becomes very large when thefault length 2a approaches the critical value 2ac.In this section we explain this behavior throughthe stability analysis presented above.

Here, we use the expression of the dominantpart wd to ¢nd an approximate formula for Tc,the duration of the initiation phase. Since theevolution of the slip w(t,x,0+) is in essencedescribed by the dominant part, Tc satis¢essupxnRwd(Tc,x,0+) = Lc. Assuming that the initialperturbation is such that the ¢rst point x of thefault for which the slip reaches the critical valueLc is x = 0, we obtain that Tc is the solution of theequation wd(Tc,0,0+) = Lc.

Let us suppose in what follows that:

1:1577T � L 06L � a�W s3W d�S

GLc6L 1 � 2:754T

i.e. we deal with one eigenfunction in the expres-sion of the dominant part wd. This case corre-sponds to `long' initiation periods for which V0

is small.If we note A0 =x0(0,0+) then from Eq. 23 we

obtain:

LcWA0

�cosh�cV 0T c=a�W 0

0 � asinh�cV 0T c=a�

cV 0W 1

0

�

where W00 and W1

0 are the weighted averages ofthe initial perturbation given by Eq. 22. Moreprecisely they are the projection of w0,w1 on the¢rst eigenfunction x0. One can use the aboveequation to deduce the following approximateformula for Tc :

T cWa

cV 0ln

cV 0Lc=A0 ��cV 0Lc=A0�23�V 0cW 0

0�2 � �aW 10�2

qV 0cW 0

0 � aW 10

24 35�42�

We remark that Tc depends on the initial aver-ages W0 and W1 through a natural logarithm,hence the duration of the initiation phase hasonly a weak (logarithmic) dependence on the am-plitude of the initial perturbation.

The duration of the initiation Tc also dependson the nondimensional weakening rate L throughthe functions V0(L) and A0(L). Noticing that forLCL0, with LsL0, we have V0(L)C0 and there-fore TcC+r. That is when approaching the crit-ical value L0 the system has a slow unstable be-havior.

The above formula is valid only for a lineardependence of the friction coe¤cient W on theslip u in the weakening domain (un[0,Lc] inour case). If a nonlinear weakening dependenceW=W(u) is considered, much slower evolution ofthe initiation phase can be expected in the neigh-borhood of the slip u0 for which WP(u0) = 0.

7. Numerical results

The theoretical development in the previoussections indicates some strong and simple proper-ties of the slip during the initiation phase. In par-ticular, the essence of the system evolution is de-scribed by the simple expression in Eq. 23 whichwe refer to as the dominant part. The aim of this

EPSL 5415 4-4-00


section is to compute the leading part. These re-sults are then compared with the fully nonlinearsolution computed with a ¢nite di¡erence scheme.

We have numerically solved the nondimension-al eigenvalue problem (Eqs. 17^19) by truncationof the series in Eq. 40 up to M = 100 terms. As inthe static case, it was observed that such a trun-cation gives a high level of accuracy for the dis-crete solution. In Fig. 2 we plotted the ¢rst eigen-function x0 computed for L= 1.1764 whichcorresponds to V0 = 0.1. It was obtained fromEq. 63 by discretization of the integral througha Gaussian quadrature. We remark that the eigen-function is rather concentrated on the fault. We

have also computed the partial derivative Dx2x0

(see Fig. 3), which gives the variation of the tan-gential stress acting on the fault. To do this wehave employed Eq. 64 with the same numericalapproach as for x. Fig. 3 illustrates the in¢nitelimit of the outside tangential stress at the ends ofthe fault.

In Fig. 4 we have plotted the ¢rst eigenvalue V0

as a function of the nondimensional parameter L.We note a signi¢cant increase of the eigenvalue inthe neighborhood of L0 = 1.1577T, the intersectionof the graphic with the L-axis.

The following numerical results were obtainedusing a ¢nite-di¡erences scheme proposed by Iones-cu and Campillo [10] to approach the nonlinearproblem (Eqs. 1^5). We used a grid of 800U800points in the x,y plane and the following modelparameters: b= 3000 kg/m3, c = 3000 m/s, Ws = 0.8and Wd = 0.72 and the half length of the fault isa = 500 m. The normal stress is assumed to corre-spond to a lithostatic pressure corresponding to adepth of 5 km. The initial condition correspondsto a velocity perturbation w1 while the initial dis-placement perturbation w0 is 0.

The initial velocity perturbation has the follow-ing distribution:

w1�x; y� � A exp�x3x0�2�x3x0�23h2

� �cos�Zy=�2h��;

Mx3x0M6h; MyM6h; w1�x; y� � 0 elsewhere

�43�

Fig. 4. The computed value of the ¢rst eigenvalue V0 =V0(L)versus the nondimensional weakening rate L. The intersectionwith the L-axis is the critical value L0.

Fig. 3. Partial derivative Dx2x0(x1,x2) of the ¢rst eigenfunc-tion computed for L= 1.1764 (which corresponds to V0 = 0.1)as a function of nondimensional variables x1 = x/a,x2 = y/a. Itgives the spatial behavior of the tangential stress cyz. Notethe stress singularities at the ends of the fault.

Fig. 2. First eigenfunction x0(x1,x2) computed for L= 1.1764(which corresponds to V0 = 0.1) as a function of nondimen-sional variables x1 = x/a,x2 = y/a. It gives the spatial behaviorof the dominant part.

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where the half width h is 100 m, the maximumamplitude A is 0.01 m/s.

In order to illustrate the stability analysis, givenin Section 3, we show in Fig. 5 the evolution ofthe slip rate on the fault for L= 1.16L0 (Fig. 5,upper part) and for L= 1.2sL0 (Fig. 5, lowerpart). We note the qualitatively di¡erent behaviorof the solution in the two cases for a small varia-tion of Lc (Lc = 19.8 cm in the ¢rst case andLc = 18.1 cm in the second case). Indeed we seethat after propagation and re£ection on the endpoints of the fault, the initial perturbation van-ishes in the ¢rst case but it has an exponentialgrowth in the second case.

In Fig. 6 we have represented the duration ofthe initiation Tc given by Eq. 42 versus the coef-¢cient L, for a = 500 m and c = 3000 m/s and usingthe computed value of x(0,0+). Note that for L

approaching L0 the duration of initiation, Tc, be-comes in¢nite. In Fig. 6 we have also plotted theduration of the initiation obtained by the ¢nite-di¡erences computation of the dynamic solution.These results on the duration were deduced fordi¡erent values of the critical slip Lc by keepingall the other physical values ¢xed, i.e. for di¡erentvalues of the nondimensional weakening rate L.The values of V0 in Eq. 42 are computed usingthe approach presented in Section 5 which is validfor small values of V, i.e. for great Tc. On theother hand the ¢nite-di¡erences scheme can beapplied on small intervals of time (consequenceof the Courant^Friedrichs^Lewy condition).Nevertheless we note a rather good agreementbetween the numerical results obtained with thetwo methods.

8. Conclusions

To investigate the parameters which control thedelay of triggering, we studied the initiation ofslip instabilities of a ¢nite fault in a homogeneous

Fig. 6. Duration of initiation Tc versus nondimensionalweakening coe¤cient L, for a = 500 m and c = 3000 m/s com-puted from Eq. 42 (continuous line) and obtained after the¢nite-di¡erences computation of the dynamic solution(circles). Note the good agreement between these results. ForL approaching L0 the duration of initiation Tc becomes in-¢nite.

Fig. 5. Slip velocity (DtNw(t,x)) on the fault (y = 0) as a func-tion of space x and time t computed with a ¢nite-di¡erencesmethod. Here the half length of the fault is a = 500 m. In theupper part of the ¢gure L= 1.16 L0 (i.e. Lc = 19.8 cm) and inthe lower part L= 1.2s L0 (i.e. Lc = 18.1 cm). Note the quali-tatively di¡erent behavior with the same initial perturbation.

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linear elastic space. We considered the antiplaneunstable shearing under a slip-dependent frictionlaw with a constant weakening rate. The problemwas attacked by spectral analysis. We concen-trated our attention on the case of long initiation,i.e. small positive eigenvalues. A static analysis ofstability was presented for the nondimensionalproblem. Using an integral equation method the¢rst (nondimensional) eigenvalue, which dependsonly on the geometry of the problem, was deter-mined. In connection with the weakening rate andthe fault length this (universal) constant deter-mines the range of instability for the dynamicproblem. An accurate limiting value of the lengthof an unstable fault for a given friction law wasgiven. By means of a spectral expansion, the`dominant part' of the unstable dynamic solutionwas de¢ned. It characterizes the exponential timegrowth. For the long-term evolution of the initia-tion phase the dynamic eigenvalue problem wasreduced to a hypersingular integral equation. Theexpression of the dominant part was used to de-duce an approximate formula for the duration ofthe initiation phase. The ¢rst eigenfunction wasobtained numerically. The dependence of the ¢rsteigenvalue and the duration of the initiation onthe weakening rate were pointed out. The resultswere compared with those for the full solutioncomputed with a ¢nite-di¡erences scheme. Theseresults suggest that a very simple friction lawcould imply a broad range of duration of initia-tion. They show the fundamental role played bythe limited extent of the potentially slipping patchin the triggering of an unstable rupture event.This point is absent from the numerous discus-sions of the triggering process based on springslider systems or in¢nite fault models, makingtheir conclusions regarding the state of stress onthe faults or the acceptable friction laws highlyquestionable.

Acknowledgements

We thank A. Cochard, J.-P. Vilotte, and ananonymous reviewer for their comments and theirsuggestions to improve this paper. This work was

partially supported by the program MIG of Uni-versite de Savoie and by GDR `FORPRO' (Ac-tion 98 A). C.D. was supported by Rhoªne-AlpesRegion, grant TEMPRA-PECO.[FA]

Appendix A. Static eigenvalue problem

In the following, we give some details of thecalculus for the static eigenvalue problem (Eqs.25^27) of Section 4. The Fourier transform inx1 of Eq. 25 leads to:

D 2 BDx2

2

h 2 B �44�

where B(h,x2) is the Fourier transform in x1 ofB(x1,x2). The ¢nite-energy solutions of Eq. 44,which require that 9B is vanishing at in¢nity,have the form:

B�h ; x2� � A�h �e3Mh Mx2 �45�

The Fourier inverse of Eq. 45 is:

B�x1; x2� � 12Z

Z �r3r

A�h �e3Mh Mx23ih x1 dh �46�

and for x2 = 0 it leads to:

A�h � �Z �r3r

B�s; 0�eih sds �47�

By substitution of A(h) in Eq. 46 and interchangeof the integration order we get:

B�x1; x2� � x2

Z

Z 1

31

B�s; 0�x2

2 � �s3x1�2ds �48

which is a representation formula for the displace-ment ¢eld B(x1,x2). To deduce Eq. 48 we haveused the relationZ �r

0e3h x2 cos�h �s3x1��dh � x2

x22 � �s3x1�2

(see [2], ch. 1.4). By derivation with respect to x2

EPSL 5415 4-4-00


we obtain:

D BD x2�x1; x2� � 1

Z

Z 1

31B�s; 0� �s3x1�23x2

2

�x22 � �s3x1�2�2

ds �49�

which, for x2 = 0, gives:

D BD x2�x1; 0� � 1

ZFPZ 1

31

B�s; 0��s3x1�2ds �50�

where the integral is taken in the ¢nite part sense.For x1n(31,1), from the boundary condition

(Eq. 27), we have:

B�x1; 0� � 31L Z

FPZ 1

31

B�s; 0��s3x1�2ds �51�

which is a hypersingular integral equation forB(x1,0), x1n(31,1). After integration by partsEq. 51 can be written as:

B�x1; 0� � 1L Z

PVZ 1

31

D sB�s; 0�x13s

ds; Mx1M61 �52�

where the integral is taken in the Cauchy principalvalue sense. To solve Eq. 52) we seek a solution ofthe form of Eq. 31. Introducing x1 = cos a,s = cosi and substituting Eq. 31 in Eq. 52, weobtain:

ZLXrn�1

un sin�na � �

Xrn�1

nun

Z Z

0

cos�ni �dicosi3cosa

; an�0; Z� �53�

The Glauert integral formula:Z Z

0

cos�ni �dicosi3cosa

� Zsin �na �

sin a�54�

reduces Eq. 53 to:

LXrn�1

un sin�na � �Xrn�1

nunsin�na �

sina�55�

for an(0,Z). Multiplication of this equation with

sin(ka)sin a and integration on (0,Z) yields:

LXrn�1

Aknun �Xrn�1

Bknun; k � 1; 2; T �56�

where:

Akn �Z Z

0sin�na �sin�ka �sina da �

32kn�1� �31�k�n�

��k3n�231��k � n�231��13N n;k31��13N n;k�1�

�57�

Bkn � nZ Z

0sin�na �sin�ka �da � nZ

2N k;n �58�

Appendix B. Dynamic eigenvalue problem

In this Appendix we develop the mathematicalapproach of the dynamic eigenvalue problem(Eqs. 17^19) presented in Section 3. By Fouriertransform in x1 of Eq. 17 we get:

D 2xDx2

2

� �h 2 � V 2�x �59�

For x2 s 0 the ¢nite-energy solution of Eq. 59 is:

x �h ; x2� � A�h �e3��h 2�V 2p

x2 �60�

Its Fourier inverse is :

x �x1; x2� � 12Z

Z �r3r

A�h �e3��h 2�V 2p

x23ih x1 dh

�61�

which, for x2 = 0, yields :

A�h � �Z 1

31x �t; 0�eih tdt �62�

When this expression of A(h) is substituted in Eq.61 and using the formula (see [2]) :

EPSL 5415 4-4-00


Z r

0e3x2

��V 2�h 2p

cos�h �s3x1��dh �

MV Mx2��x2

2 � �s3x1�2q K1 MV M

��x2

2 � �s3x1�2q� �

we ¢nd:

x �x1; x2� 12Z

Z 1

31x �s; 0�

Z �r3r

e3��h 2�V 2p

x23ih �x13s�dh ds �

MV Mx2

Z

Z 1

31x �s; 0�

K1�MV M��x2

2 � �s3x1�2q

��x2

2 � �s3x1�2q ds

�63�

with K1 being the modi¢ed Bessel function ofthe second kind. This formula is the representa-tion of the displacements in the upper half plane.By taking the derivative with respect to x2 wehave:

DxD x2�x1; x2� � MV M

Z

Z 1

31x �s; 0�1 �s; x1; x2�ds �64�

where:

1 �s;x1;x2� ��s3x1�23x2

2�K1 �MV M��x2

2 � �s3x1�2q

��s3x1�2 � x2

2��x2

2 � �s3x1�2q 3

MV Mx22K0�MV M

��x2

2 � �s3x1�2q

��s3x1�2 � x2

2

�65�

For x2C0 from Eqs. 64 and 65 we ¢nd:

DxD x2�x1; 0� � MV M

ZFPZ 1

31x �s; 0�K1�MV �s3x1�M�

Ms3x1Mds

�66�

The boundary condition (Eq. 19) yields:

x �x1; 0� �

3MV MZL

FPZ 1

31x �s; 0�K1�MV �s3x1�M�

Ms3x1Mds; Mx1M61

�67�

which is a hypersingular integral equation forx(x1,0) on (31,1). Integrating by parts and usingthe relations:

K 00�x1� � 3K1�x1�; K 01�x1� � 31x1

K1�x1�3K0�x1��68�

with the prime meaning the derivative, we obtainfrom Eq. 67:

ZLx �x1; 0� �

PVZ 1

31D sx �s; 0�MV �x13s�MK1�MV �s3x1�M�

x13sds�

V 2Z 1

31x �s; 0�K0�MV �x13s�M�ds �69�

For small values of MVM (i.e. MVMI1), we can ap-proximate:

K0�x1�W3ln�x1�; K1�x1�W 1x1

so that Eq. 69 becomes:

ZLx �x1; 0� �

PVZ 1

31

D sx �s; 0�x13s

ds3V 2Z 1

31x �s; 0� ln�MV �s3x1�M�ds

�70�

for Mx1M6 1. To solve Eq. 70 we use the sametechnique as in Appendix A. We seek a solutionfor Eq. 70 of the form of Eq. 39. In this case the

EPSL 5415 4-4-00


in¢nite system corresponding to Eq. 56 is:

LXrn�1

AknUn �Xrn�1

�Bkn � Ckn�V ��Un; k � 1; 2; T

�71�

with Akn,Bkn given by Eqs. 57 and 58 and:

Ckn�V � � 3V 2

Z

Z Z

0

Z Z

0sin�ka � sin�nB� sina sinB ln

�V Mcosa3cosBM�dBda � ZV 2 lniV4

N n;1N k;1

3ZV 2

16�N k;33�1� 4 ln2�N k;1�N n;1

3ZV 2

8�n231��13N n;3�

��n31�N k;n�2 � �n� 1�N k;n3232nN k;n� �72�

References

[1] D.J. Andrews, Rupture propagation with ¢nite stress inantiplane strain, J. Geophys. Res. 81 (1976) 3575^3582.

[2] A. Erdelyi, W. Magnus, F. Oberhettinger, F.G. Tricomi,Tables of Integral Transforms, Bateman ManuscriptProject, MacGraw-Hill, New York, 1954.

[3] M. Campillo, I.R. Ionescu, Initiation of antiplane shear

instability under slip dependent friction, J. Geophys. Res.102 (B9) (1997) 20363^20371.

[4] L. Dragos, Method of fundamental solutions. A noveltheory of lifting surface in a subsonic £ow, Arch. Mech.35 (1983) 575.

[5] L. Dragos, Integration of Prandtl's equation with the aidof quadrature formulae of Gauss type, Q. Appl. Math. 52(1994) 23^29.

[6] W.L. Ellsworth, G.C. Beroza, Seismic evidence for anearthquake nucleation phase, Science 268 (1995) 851^855.

[7] Ch. Fox, A generalisation of the Cauchy principal value,Can. J. Math. 9 (1957) 110.

[8] R.A. Harris, Introduction to special section: Stress trig-gers, stress shadows, and implications for seismic hazard,J. Geophys. Res. 103 (1998) 24347^24358.

[9] Y. Iio, Slow initial phase of the P-wave velocity pulsegenerated by microearthquakes, Geophys. Res. Lett. 19(1992) 477^480.

[10] I.R. Ionescu, M. Campillo, In£uence of the shape of thefriction law and fault ¢niteness on the duration of initia-tion, J. Geophys. Res. 104 (B2) (1999) 3013^3024.

[11] I.R. Ionescu, J.-C. Paumier, On the contact problem withslip dependent friction in elastostatics, Int. J. Eng. Sci. 34(1996) 471^491.

[12] M. Ohnaka, Nonuniformity of the constitutive law pa-rameters for shear rupture and quasistatic nucleation todynamic rupture: A physical model of earthquake gener-ation model, Paper Presented at Earthquake Prediction:The Scienti¢c Challenge, US Acad. Sci., Irvine, CA, 1996.

[13] A.C. Palmer, J.R. Rice, The growth of slip surfaces in theprogressive failure of over consolidated clay, Proc. R. Soc.London A 332 (1973) 527^548.

[14] J.E. Vidale, D. Carr Agnew, M.J.S. Johnston, D.H. Op-penheimer, Absence of earthquake correlation with earthtides: an indication of high preseismic fault stress rate,J. Geophys. Res. 103 (1998) 24567^24572.

EPSL 5415 4-4-00


fault finiteness and initiation of dynamic shear instability

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