Experimental analysis of slip distribution along a fault segment under stick–slip and stable sliding conditions

Tectonophysics 337 (2001) 85±98

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Experimental analysis of slip distribution along a fault segment under stick±slip and stable sliding conditions G. de Joussineau a,*, S. Bouissou a,1, J.P Petit a, M. Barquins b a

Laboratoire de geÂophysique, tectonique et seÂdimentologie (UMR 5573), c.c 060, Universite Montpellier II, Place EugeÁne Bataillon, 34095 Montpellier, cedex 5, France b LPMMH (UMR 7636), ESPCI, 10 rue Vauquelin, 75231Paris, cedex 5, France Received 29 March 2000; accepted 22 February 2001

Abstract This paper presents the results of an experimental study of slip distribution along fault models with different sliding surface morphologies submitted to various orientations of fault and loading direction. Uniaxial compression tests were done on PMMA plates to measure the relative displacement of markers perpendicular to the sliding surfaces. Slip distribution pro®les were then constructed for different stages of shortening of the samples. It was shown that fault morphology determines both the sliding regime and the slip distribution along the sliding surfaces. Along fault models with grounded surfaces (FMGS), stable sliding giving a symmetrical smooth distribution was observed. For fault models with natural fracture surfaces (FMNS), stick±slip was observed. In this case, the slip distribution exhibited signi®cant irregularities in a generally symmetrical pro®le. Analysis of this pro®le enabled four types of local slip behaviour to be identi®ed which may provide a physical basis for the interpretation of coseismic slip distributions. The local slip behaviours were interpreted in terms of interaction of irregular asperities, and helped understand the discrepancy observed between theoretical slip distributions and the displacement measured along the active faults. Analysis of the slip distribution pro®les along the FMGS and the FMNS also showed that they evolved systematically after the appearance of branching. For a large amount of shortening, mode I branching was observed at the tips of the sliding surfaces. After its development, the global amount of slip along the faults increased, and the slip gradient between the centre and the tips of the faults decreased. This effect is consistent with numerical modelling studies. Finally, the observed pro®les were compared with the linear, the tapered, and the elliptic theoretical models, using two different procedures. We conclude that, in the case of the FMNS, where the sliding surface topography is complex, no model could correctly describe the slip distribution, and that, in both cases of FMNS and FMGS, branching caused a change in the shape of the slip distribution pro®les. q 2001 Elsevier Science B.V. All rights reserved. Keywords: slip distribution ; surface morphology; stick±slip ; stable sliding

1. Introduction * Corresponding author. E-mail address: [email protected] (G. de Joussineau). 1 Now at: GeÂosciences Azur, UMR 6526, Universite de NiceSophia Antipolis, 250 Rue Albert Einstein Ð Sophia Antipolis, 06560 Valbonne, France

The modalities of slip distribution along faults is a crucial factor in understanding seismic source behaviour and more generally fault growth mechanisms. Concerning seismicity, the problem of recurrent coseismic displacement is still discussed. Two main

0040-1951/01/$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0040-195 1(01)00058-0

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hypothetical models of seismic recurrence have been established, the ªuniform seismic slipº model (Sieh, 1981), and the ªcharacteristic earthquakesº model (Schwartz and Coppersmith, 1984), but they represent end member cases observed for a few seismic cycles and on few faults. The origin of these behaviours is not explained, although there may be a link with the degree of morphological maturity of the faults (Sieh, 1996; Deng and Liao, 1996). However, the link between morphology and fault displacement behaviour is not clear at present. On short term scale, the main methods of investigating slip distribution along faults, i.e. ®eld studies (Walsh and Watterson, 1987a,b, 1988; Contreras et al., 2000), geodetic trilateration techniques (Murray et al., 1993; Hudnut et al., 1994), and inversions of broadband teleseismic waveforms (Boatwright et al., 1989; Ide et al., 1996) provide useful data but have serious limitations. For example, inversion techniques imply prior knowledge of fault geometry, which is considered as a succession of regular planes for calculations. On long-term scale, there are no means of retracing recurrent coseismic displacement along geological faults, although new paleoseismological techniques are being developed. In addition, modelling is essential to understand such recurrent displacements. Numerical studies have tackled important problems: the reconstitution of temporal and spatial slip distribution for a given event (Wald and Heaton, 1994), the characterisation of the in¯uence of some physical parameters (stress gradient, friction coef®cient, etc) on slip distribution along faults of limited extent (Cowie and Scholz, 1992a,b; BuÈrgmann et al., 1994; Willemse et al., 1995; Cowie and Shipton, 1998), and ®nally the determination of changes in the stress pattern in the medium around a fault during an event (King et al., 1994). However, given the present state of numerical techniques, models of displacement pro®les are not possible in the case of recurrent coseismic displacement. Therefore analogue modelling is essential, for it may provide some physical basis for the hypothetical models of seismic recurrence. In this domain, a few studies devoted to the stick±slip phenomenon and the problem of slip distribution along faults have been carried out (Wu et al., 1972; Brune, 1973; King, 1986, 1991), but to our knowledge, no attempt has been made to link fault morphology

and slip distribution, although it is believed to be a crucial approach (Power et al., 1987; Scholz, 1990). Several fundamental questions remain unsolved. Is there a difference in slip distribution along seismic and non-seismic faults? What about the validity of elastic models commonly used in numerical studies (Willemse et al., 1995)? How does slip distribution evolve with fault propagation? An important case is that of branching at the tips of the fault segments, which may play a role in fault coalescence, or conversely may arrest the dynamic fault growth (Kame and Yamashita, 1999). Studying this issue could also enable discussion of choices made in numerical studies. In this paper, we present the results of a new type of experimental modelling, which aims at analysing stick±slip and stable displacement along fault models. We tested the in¯uence of three parameters on slip distribution: the orientation of the faults to the direction of shortening (b angle, ranging from 158 to 608), the development of branching at the fault tips, and the sliding regime by changing the roughness of the sliding surfaces. We used two types of fault models: the fault model with grounded surface (FMGS) to obtain stable sliding, and the fault model with natural surface (FMNS) to obtain stick±slip regime, which is a classical physical analogue of seismic cycle and recurrence (Brace and Byerlee, 1966). We used PMMA (polymethylmethacrylate) because this material has mechanical properties comparable to those of brittle rocks in the upper crust (Brace and Kholsted, 1980; Kirby and Kronenberg, 1987). Furthermore, it has widely been used as analogue of rocks in rupture mechanics experiments (Nemat-Nasser and Horii, 1982; Petit and Barquins, 1987; Barquins and Petit, 1992) and in friction experiments (Wu et al., 1972; Dieterich and Kilgore, 1994, 1996; Bouissou et al., 1998a,b,c; Bodin et al., 1998). Finally, it becomes birefringent if stressed, which permits photoelastic stress analysis within the samples to be realised. 2. Experimental device and procedure Experiments were carried out on PMMA (polymethylmethacrylate) plates with dimensions of 160 £ 100 £ 6 mm 3, taken from the same plate of

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Fig. 2. Fabrication of FMGS: (a) a PMMA plate has been cut and sanded in order to obtain a well controlled geometry for the future oblique sliding surface; (b) the plate is now reassembled (note the glued joints) and is ready to be tested as a FMGS.

Fig. 1. Experimental device, showing a PMMA sample maintained by vertical tighteners. The oblique fault model wears a net of regular markers, perpendicular to the sliding surface.

Altuglasw. We examined the displacement along 40 mm-length oblique sliding surfaces, at a varying angle (called b) of 158, 30,8, 458 and 608 to the loading direction (Fig. 1). In the tests, uniaxial compression was applied using a strain rate of 4.15 10 26s 21 imposed by an electromechanic testing machine (Davenport 30 kN). Samples were maintained between vertical tighteners to prevent bending effects (Fig. 1). Three stages were needed to prepare the FMGS for stable sliding (Fig. 2). First, the plates were cut with a

0.3 mm blade diameter micro-saw, to impose the geometry of the sliding surfaces (Fig. 2a). Secondly, the oblique surface roughness was imposed using a rotating sandpaper band of 120 grits, which gave reproducible surface morphology. Samples were grounded in such a way that the produced asperities were perpendicular to sliding direction. Thirdly, the two bits of each sample were stuck together using Araldite glue (ref. 3148566/00), along the two segments perpendicular to loading (Fig. 2b). Araldite glue was chosen because it had pure elastic behaviour during the compression tests, and an elastic modulus comparable to that of PMMA (2.5 GPa). The glued surfaces were perpendicular to the direction of movement imposed by the testing machine, in order to avoid shear stress. The FMNS were made in two stages. First, a fracture embryo was made by hitting

Fig. 3. Numerical photograph of a FMGS sliding surface detail: the stripes are regular asperities let by the sander abrasive paper at contact with PMMA. White arrows indicate the slip direction during the tests.

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Fig. 4. Numerical photographs of a FMNS sliding surface: its geometry is more complex than the geometry of a FMGS sliding surface. This is evidenced by the irregular shape and distribution of the stripes (let by the propagation of a ªnaturalº crack trough the PMMA plate).White arrows indicate the slip direction during the tests.

a blade perpendicularly disposed to the sample. Then, the existing fracture was propagated by compressing the plate in a vice, and was stopped at the required dimensions: thus length and orientation of fractures were controlled, but with an irregular distribution of asperities, corresponding to fractographic features, which develop on the wall of mode I fractures (Bahat, 1991). The two surfaces of the same FMNS were complementary, so the asperi-

ties were initially interlocked before the compression test. Figs. 3 and 4 and show a view of a FMGS and a FMNS, respectively, taken after a test. No damage due to wear was observed, apart from slight local striations (Fig. 5). The measurement of the relative shear displacement along the faults was undertaken by tracing a net of regularly spaced orthogonal markers. With a video camera and an image processing

Fig. 5. Detailed view of a FMNS sliding surface after a compression test: the slight stripes are slickensides parallel to slip direction due to friction during the experiment. The global morphology of the surface remains unchanged.

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Fig. 6. Method of slip measurement along a sliding surface, on a numerical photograph: the distance between the two parts of a same marker, due to the slip along the sliding surface, is measured with an appropriate tool.

software (Visiolab 2000w, developed by Biocom), photographs of samples were taken at different stages of experiments, with a shortening increment of 0.5 mm between two successive pictures. The ®rst photograph was taken for 1 or 1.5 mm shortening (below that, displacements were too small to be measured accurately), and the last one after branching development at the tips of the sliding surfaces. The relative displacement of each marker was measured on the numerical images (Fig. 6). The precision was about one per cent. In the case of stick±slip regime experiments, several sliding cycles evidenced by acoustic emissions occurred between two consecutive measurements. 3. Experimental results We ®rst present results obtained with FMGS samples. Fig. 7a shows the slip distribution along an experimental fault, its orientation to the direction of shortening (b) being 158. In this experiment, branching appeared at the extremities of the sliding surface for 2.9 mm shortening, and reached the edges of the sample for 5 mm shortening. Each curve on Fig. 7a represents the slip distribution for a stage of shortening of the sample. The ®rst pro®le from the bottom was built for a shortening of 1.5 mm, and the top one

for a shortening of 5 mm. The three ®rst curves and the ®ve following ones are related to pre- and postbranching steps, respectively (the branching being observed for 2.9 mm shortening). The slip distribution along the FMGS appeared smooth, symmetrical, with a maximum displacement at the middle of the fault, which decreases symmetrically on each side to be minimum at the tip. We observed that this shape does not depend on b value. Normalising displacement and distance data (respectively to the maximum displacement and to the length of the fault model) enabled the in¯uence of branching development on the slip distribution pro®les to be shown. Fig. 7b shows that after branching, the amount of slip along the sliding surfaces was higher than at the beginning of the experiments and that the displacement gradient between the centre and the fault tips was reduced. This was true for the different b values. BuÈrgmann et al. (1994) reached the same conclusions on the effect of branching development on slip distribution along faults by numerical modelling. In the following text, we used dmax for the maximum displacement along the faults for a given axial shortening. Fig. 8a shows average dmax values for increasing stages of shortening for different b values, for FMGS. dmax values are the greatest for each compression stage when b ˆ 308, and decrease for other b values, with a minimum for b ˆ 608. Black

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Fig. 7. FMGS displacement pro®les: (a) slip distributions along a FMGS constructed at 1.5, 2, 2.5, 3, 3.5, 4, 4.5 and 5 mm shortening, (b) same slip distributions normalised as follows: dnorm ˆ d=dmax ; where d is the relative displacement of the considered marker and dmax the maximum displacement observed at the stage and lnorm ˆ l=L; where l is the abscissa of the marker and L the length of the fault.

arrows show the ®rst dmax values after branching for each orientation. It is shown that branching appeared for the lowest axial shortening for b ˆ 308, and did not appear for b ˆ 608, even at the end of the experiment. This shows that, in terms of displacement and occurrence of branching, the preferential sliding orientation on FMGS is 308. We followed the same procedure for experiments performed in the case of the FMNS. Fig. 9 shows a corresponding slip distribution for b ˆ 308. The pro®les represent increasing shortening of the sample from 1 to 3 mm, with branching appearing for 2 mm.

Fig. 9 shows a global slip distribution comparable to that obtained in the case of the FMGS (i.e. symmetrical pro®les, with a maximum displacement at the centre of the faults and minimum at the tips). However strong heterogeneities in the slip distribution are observed (i.e. areas more or less displaced than the neighbouring ones). Four types of behaviour can be distinguished: (1) The ªtemporal lockingº (a on Fig. 9): some zones appear less displaced than their neighbours for a stage of the experiment, but not before and after this stage. (2) The ªpersistent slip perturbationº (b on Fig. 9):

Fig. 8. Average maximum displacements (dmax): (a) Values calculated from measurements along FMGS at different stages of shortening (W: 1 mm, X: 1.5 mm, K: 2 mm, A: 2.5 mm, O: 3 mm, B: 3.5 mm and X: 4 mm). b is the angle between the FMGS and the loading direction. (b) Values calculated from measurements along FMNS (of various orientation b with the loading direction) at different stages of shortening (W: 1 mm, X: 1.5 mm, K: 2 mm, A: 2.5 mm, O: 3 mm, B: 3.5 mm and X: 4 mm).

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Fig. 9. Slip distributions measured along a FMNS at different stages of shortening (1, 1.5, 2, 2.5 and 3 mm). The four typical behaviours observed are underlined: a Ð temporal locking; b Ð persistent slip perturbation; c Ð displacement motif inversion; d Ð displacement motif drift.

some very local zones on the surfaces stay signi®cantly less displaced than the others on both sides during the whole experiment. (3) The ªdisplacement motif inversionº (c on Fig. 9): a `displacement motif' is a part of a pro®le with a characteristic shape. We observe that some motifs are inverted between two successive measurements (i.e. evolve from a v-shape to a ^ -shape). (4) The ªdisplacement motif driftº (d on Fig. 9): some displacement motifs on the pro®les drift laterally between successive measurements.

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We also noted that the in¯uence of branching development on slip distribution was the same for the two types of sliding regimes. Finally, Fig. 8b presents average dmax values measured in the case of FMNS for several orientations. The black arrows indicate the ®rst dmax value after branching. Stick±slip was easily audible when b was 308 or 458, but was more discreet for b ˆ 158 and for b ˆ 608. We also observed that dmax was greatest for b ˆ 308, and minimum for b ˆ 608, for every stage of shortening. Another observation was that branching appeared for increasing axial shortening successively for 308, 458, 158, and did not exist for b ˆ 608. This shows that, in terms of energy release, the most favourable orientation for stick±slip on FMNS is 308. 4. Interpretation We ®rst observed that slip distribution along our experimental faults is strongly in¯uenced by surface morphology. For FMGS, we obtained stable sliding with a smooth and symmetrical slip distribution (Fig. 7a). For FMNS, stick±slip conditions occurred and even if slip distribution pro®les were globally symmetrical, the displacement was not smooth and exhibited four speci®c behaviours.

Fig. 10. Photoelastic views of (a) a FMNS and (b) a FMGS during a compression test. The contact points (i.e. the interactions between asperities) along the sliding surfaces are evidenced by lobes of isochromatic fringes due to differential stress concentration: there are only few ones in the case of FMNS, and they are numerous in the case of FMGS.

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Fig. 11. Proposed scenari for the explanation of the four slip behaviours observed along the FMNS.

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To determine the relations between the sliding surface morphology, the sliding regime, and the measured slip distributions in our experiments, photoelasticity (see HeteÂnyi (1966) and Vishay (1984) for details), an optical method of stress analysis within birefringent materials (i.e. materials like PMMA which have the property of resolving the light which falls on them at normal incidence into two components and transmitting it on planes at right angles) was carried out. Effectively, this technique is particularly useful to evidence zones of high differential stress associated with contact points along sliding surfaces. Fig. 10 shows photoelastic views of a FMGS and a FMNS during an experiment. Observations from this ®gure suggest that the difference in the sliding regime can be related to the difference in the number of contact points along surfaces during the experiments. In fact, for FMGS, the large number of regular asperities implied a large number of contact points (Fig. 10b). The low normal stress which acted on each asperity prevented interlocking and hence led to stable sliding. Conversely, FMNS were composed by irregular asperities in terms of height and spacing, which led to fewer contact points (Fig. 10a). The greater normal stress concentrated at each one implied a better interlocking, and then stick slip appeared (Bouissou et al., 1998a). In the FMGS case, as the interlocking between asperities was low and the contact points were numerous, the displacement was regularised along the fault: the possible perturbations of slip were not perceptible at the observation scale we used. In the FMNS case, the interlocking in some asperities implied that only a few markers were not able to displace freely for each slip event. From the idea that contact points were scarce, the four typical displacement behaviours could be explained as follows: (1) The ªtemporal lockingº (Fig. 11a) could be explained by a temporary contact between local asperities. The marker situated in the interlocking area was not able to displace freely and appeared less displaced than its neighbours during the brief contact. This was only recorded on one slip distribution pro®le, because the resistance of the asperity was quickly overcome. (2) The ªpersistent slip perturbationº (Fig. 11b) could be explained by the following scenario. If one of the two parts of the sliding surface bore a particularly high and local asperity, it was in contact with the

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other side of the sliding surface for longer than the interval between several successive measurements. Thus, the displacement of the marker located near it was signi®cantly braked during all the contact time so that it stayed less displaced than the neighbouring ones throughout the experiment. (3) The local ªdisplacement motif inversionº between two successive slip distribution pro®les (Fig. 11c) could be explained by inversions of contact points in small fault zones. At ®rst, the markers near initial contacts were less displaced than the other ones on the both sides, but when these contacts ceased, new ones appeared in the neighbouring areas. At this time, the slip distribution motif was inverted because of the relative displacement of the contact points. (4) The ªdisplacement motif driftº (Fig. 11d) could be explained by a lateral drift of sets of adjacent asperities, facing the movement of a relatively smooth surface wearing a little hole. Every existing contact point was relaxed when arriving at the hole location, and came back into contact afterwards. The effect of fault obliquity on slip distribution was also observed. Whatever the type of surface used, dmax values were highest when b was 308, and lowest when it was 608 in the experiments (Fig. 8). We also noted that branching appeared for the lowest axial shortening for b ˆ 308. These two facts can be interpreted according to Barquins et al. (1991). The authors analysed experimentally and analytically the path and kinetics of branching for various orientations of open defects in PMMA plates. They showed that, in uniaxial conditions, the maximum tensile stress at the fault tips was obtained for b ˆ 308. In our experiments, the earliest branching at 308 can be related to the maximum tensile stress at fault tips, itself related to the maximum measured displacement. Furthermore, faults were not open but the actual contact area during the compression tests was small in compared to the total fault surfaces. Even if the local stress distribution was different due to the different frictional con®guration, the tensile stress at the tips could have evolved similarly with b. 5. Discussion As one goal of our study is to help understand fault behaviour, our results have to be discussed in the light

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Fig. 12. Surface slip distribution from the three strands of the Superstition Hills fault system (California) after the 1987 seismic event (®rst pro®les) and in 1989, after several creep events (second pro®les) (modi®ed from Rymer, 1989).

of geological data. Evidently, we are aware of the distance between our models and geological cases, but a comparison can be attempted between surface slip distribution along an FMNS during a compression test, and along active faults after some seismic events, which suggests some common physical behaviours. Indeed, the analogue situation could be found at depth in term of high and low normal stress zones associated with fault surface asperities. Such zones correspond respectively to situations of interlocking between asperities facing the movement of the other side of the fault surface, and to situations of separation between the two sides of the fault surface. Fig. 12 shows surface slip measurements from the Superstition Hills fault system after the 1987 event (modi®ed from Rymer, 1989). The ®rst pro®les were constructed along its three segments just after the 1987 event, and the second pro®les were constructed two

Fig. 13. Elliptic, linear and tapered slip distribution models to be compared with slip distributions along FMGS and FMNS.

years later, and take into account after-slip accommodated by more than eighty creep events which can be considered as individual slip events (Bilham, 1989). The four types of behaviours evidenced on Fig. 9 can be identi®ed in the Superstition Hills sequence. In both natural and experimental data sets, temporal locking (a) and displacement motif inversions (c) are observed. Even if there are only two successive pro®les in the Superstition Hills sequence, a persistent slip perturbation (b) example and a lateral displacement motif drift (d) can be evidenced. We propose that, in spite of the distance between such systems, common behaviour of FMNS and seismic faults may have a common origin, i.e. interaction between asperities. Effectively, for active faults, the surface slip distribution may re¯ect the presence of asperities at depth (Scholz, 1990), even if the relations between surface slip heterogeneities and irregular fault morphology are not well known. Furthermore, the fact that the measured slip distributions along FMGS and FMNS showed non-abrupt terminations at the tips of the sliding surfaces can be related to the strength of the surrounding material. Effectively, Cowie and Scholz (1992a) explained that slip distribution models exhibiting such a geometry at the fault tips implied that the surrounding material had an in®nite strength, which is unrealistic. In our experiments, PMMA does not have in®nite strength, and can break (mode I branching), which led to slip decreasing progressively at the tips of the sliding surfaces. Another key result is given by the comparison between the shapes of slip pro®les from FMGS and

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Fig. 14. Displacement area and average displacement values calculated for successive slip pro®les during the compression of a FMNS sample (W before and X after branching) and for successive slip pro®les during the compression of a FMGS (A before and B after branching). The dashed arrows indicate the theoretical values for linear, tapered, and elliptic models.

FMNS, and the shapes of three classical theoretical slip pro®les (linear, tapered and elliptic pro®les shown by Fig. 13) often proposed as models for slip distribution along faults. These theoretical slip distributions have been de®ned by Dawers et al. (1993) using coef®cients Darea and Daverage related to the shape of the pro®les and to the area below them, where: Darea ˆ ‰Sd i Š*Dl; where di is a displacement measurement,

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Dl is the distance between each measurement, and Daverage ˆ Darea =L; where L is the length of the fault. Darea and Daverage characteristic values for the theoretical slip pro®les are the following: linear pro®le: 1 and 0.5, tapered pro®le: 1.07 and 0.52, and elliptic pro®le: 1.57 and 0.77. It was then possible, by calculating Dawers coef®cients in the case of pro®les from FMGS and FMNS, to compare directly such pro®les with linear, tapered, and elliptic pro®les. Fig. 14 shows Darea and Daverage values related to the successive slip pro®les constructed along a FMGS and a FMNS during a compression test, taking into account the appearance of branching at the tips of the sliding surfaces. A ®rst observation was that the experimental slip distributions measured along FMGS and FMNS never exhibited an elliptic shape, although this theoretical slip distribution is commonly used to describe slip distribution along faults. Second, we noted that the shapes of the slip distributions seemed to change with the branching apparition, from a tapered to a linear shape for the FMGS, and from a linear to a tapered shape for the FMNS. However, as the reference values attributed to the three theoretical slip distributions are really close to each other, we think that they do not easily permit to distinguish clearly between the different models, especially when the values of Dawers coef®cients are close to a frontier

Fig. 15. Statistic comparison between slip distributions from our fault models and linear, tapered and elliptic slip distributions: (a) CHI2 values for a comparison between a FMGS slip distribution and an elliptic slip distribution (X), a linear slip distribution (O), and a tapered slip distribution (B). (b) CHI2 values for a comparison between a FMNS slip distribution and an elliptic slip distribution (X), a linear slip distribution (O), and a tapered slip distribution (B). CHI2c is the critical value of the test. If CHI2 value . CHI2c, the studied slip distribution cannot be compared with the considered theoretical slip distribution.

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between two models. For example, the difference in the Daverage value between the linear and the tapered shape is only about four per cent (0.5 for the linear model versus 0.52 for the tapered model), illustrating that the domains of these models are really close to each other. To overcome this dif®culty in de®ning to what model pro®les from FMNS and FMGS can be related, we used the CHI2 statistic test, which enabled us to compare statistically measured slip distributions and theoretical slip distributions. To realise such a statistic test implied three stages. First, displacement data from slip distributions were reduced using the relation: dreduced ˆ d 2 average…d1 ¼di †=Std…d1 ¼di † where …d1 ¼di † represent the whole population of displacement data (and Std is the standard deviation of the population). Second, reduced displacement data were divided up into a few statistic classes, in such a way that each displacement data belonged to a class, each class including at least ®ve displacement data. Third, the CHI2 values were calculated with a significance level a ˆ 5%, using the following relation: CHI2 ˆ S iˆ1¼:n …‰…Oi 2 Ei †Š2 =Ei †; where: ² Oi is the number of displacement data from the measured slip distribution, included in the statistic class i ² Ei is the number of displacement data from the theoretical slip distribution to be compared with the measured slip distribution, included in the statistic class i and ² n is the number of statistic classes. The results from a representative FMGS are given by the Fig. 15a. We see that, before branching, the tapered model, characterised by the lowest value of CHI2, ®ts the data better, and that after branching, the shape of the slip distribution evolves to a linear slip distribution. The results of this last test globally agree with our previous conclusion but permit to link clearly the change between the two slip distributions with branching. In the case of the FMNS, as expected from Dawers coef®cient, the results are more complex (Fig. 15b). There is no clear tendency. This suggests that in the case of complex surface morphology, simple theoretical slip distributions are not adapted. 6. Conclusion Main results are as follows. First, slip distribution

and sliding regime vary depending on fault morphology. In the case of FMGS, we obtained stable sliding and a symmetrical smooth slip distribution, and we observed that the effect of branching development at defect tips was that predicted by numerical studies. In the case of FMNS, where stick±slip appeared, we evidenced and interpreted four original slip behaviours, and showed that they could also be identi®ed on seismic fault displacements. We suggest that they are the results of interaction between irregular asperities along the sliding surfaces. As the surface slip distribution of seismic faults may re¯ect the interactions of deep asperities, the similar behaviour noticed in both the FMNS case and the active faults case suggest that a few types of interplay between asperities could control the displacement pro®les. Secondly, we showed that the fault orientation b in¯uences the branching, the displacement gradient and the maximum displacement along the sliding surfaces whatever the fault morphology. Thirdly, we observed that the slip distributions along the FMGS and the FMNS are not elliptic (as expected from elastic models), and that their pro®le systematically evolves after branching. Further systematic studies on local displacement behaviours both from experimental and ®eld data could help to develop some predictive tools. Acknowledgements This work is ®nancially supported by the CNRS and ANDRA through the GdR FORPRO (Research action number 98 II) and corresponds to the GDR FORPRO contribution number 2000/08 A. The authors acknowledge with thanks Roland BuÈrgmann and the anonymous reviewer for their helpful comments. Thank you to Jean-FrancËois Ritz, Jean CheÂry and AlfreÂdo Taboada for fruitful discussions References Bahat, D, 1991. Tectono-fractography. Springer-Verlag, Germany. Barquins, M., Petit, J.P., 1992. Kinetic instabilities during the propagation of a branch crack: effects of loading conditions and internal pressure. Journal of Structural Geology 14, 893± 903. Barquins, M., Petit, J.P., Maugis, D., Ghalayini, K., 1991. Path and kinetics of branching from defects under uniaxial and biaxial

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