�

Mechanics and Interpretations of Fault Slip

Stephen J. Martel and Christin Shacat

Department of Geology and Geophysics, University of Hawaii, Honolulu, Hawaii, USA

Profiles of fault slip reflect the redistribution of strain energy associated with faulting. Real slip profiles typically are irregular and commonly are drawn as piece-wise linear, although the data permit other interpretations. In a linear elastic body, stress singularities would occur at the ends of profile segments along which slip varies linearly, and strain incompatibilities would occur where the third derivative of the slip is discontinuous: concentrated fracturing (energy sinks) along a fault should arise there. Elastic solutions for interacting faults with uniform strengths yield slip profiles consistent with the first-order shapes of many observed slip profiles, including those with near-linear tapers, which are difficult to explain by on-fault effects. Slip data collected and analyzed in conjunction with information on both along-fault fracturing and neighboring faults can lead to improved insight into how faults slip, propagate, and dissipate energy.

�. IntRoduCtIon

Slip, the relative displacement of the walls of a fault, defines faulting, provides a quantitative measure of a fault’s mechani-cal behavior, and manifests the redistribution of strain energy in an earthquake [e.g., Scholz, 2002]. Slip typically is irregu-lar along fault traces both for single earthquakes and for cumulative fault slip [e.g., dawers et al., �993; nicol et al., �996 and references therein; Shipton and Cowie, 200�; Aydin and Kalafat, 2002; d’Alessio and Martel, 2005]. Slip profiles commonly are interpreted as piecewise linear, especially near fault trace tips, with line segments connecting, passing through, or enveloping discrete slip data [e.g., Peacock and Sanderson, �994; Manighetti et al., 200�]. In addition, slip distributions commonly are presented or analyzed without accounting for fault interaction, even though the distance between faults commonly is much less than the fault trace length and fault interaction could be expected [Segall and Pollard, �980]. We compare observations of slip to theoretical predictions involving one, two, or three faults to assess what slip profiles might reveal about fault mechanics, fracturing

accompanying slip (energy sinks), and fault interaction. In particular, we address near-linear tapers in slip along interact-ing faults, and profiles with flat midsections (“quasi-elliptical profiles”), topics not addressed explicitly by other mechanics studies [Willemse et al., �996; Willemse, �997; Crider and Pollard, �998; Maerten et al., �999; 2002].

2. obSERvAtIonS of SlIP

We consider the extensive data of Manighetti et al. [200�, 2004], who compiled cumulative slip profiles from 255 nor-mal faults along active rifts of East Africa (fig. �). they clas-sified the profiles into eight sets (fig. 2) and emphasized their piecewise linear character: 72% have long, roughly linear portions, ~60% taper nearly linearly towards a fault trace end, and ~�5% have roughly flat tops. the spacing of the subparal-lel fault traces is generally small relative to the trace length. these profiles reflect slip accumulation over millions of years but resemble slip distributions during single earthquakes [see also Schwartz and Coppersmith, �984].

3. MEChAnICAl ModElS

We model slip (D) along faults idealized as two-dimen-sional shear fractures in infinite, uniform, isotropic, isother-

Fig. 1Fig. 1

Fig. 2Fig. 2

titleGeophysical Monograph SeriesCopyright 2006 by the American Geophysical union##.####/###GM##

2 Mechanics and interpretations of fault slip

mal, linear elastic bodies [e.g., Pollard and Segall, �987]. A fault of half-length a parallels the x-axis, is normal to the y-axis, and extends an infinite distance in the z-direction. for an isolated fault, x = 0 is midway between the fault tips. We treat the remote shear stress parallel to the fault (τl) as uniform. the shear stress acting on the walls of a fault is τ2. the two-dimensional treatments avoid many of the mechani-cal and geometric complications of three-dimensional mod-els and permit a more direct focus on the mechanics of fault interaction. We use both analytical and numerical solutions, the latter being for antiplane strain, where displacements parallel the z-axis [e.g., Gupta and Scholz, 2000], but note that slip profiles for plane and antiplane strain commonly have identical shapes [e.g., tada et al., 2000].

3.1 Slip Along Isolated Faults

3.1.1 Elliptical and bell-shaped reference slip distribu-tions. the slip profile for an isolated fault with a constant driving stress (∆τ = τl-τ2) has an elliptical form, with a stress singularity at the fault tip [tada et al., 2000]. the simplest slip distribution where the stress singularity is eliminated is bell-shaped; it is produced by a driving stress distribu-tion in the form of a step function [bürgmann et al., �994]. Although elliptical and bell-shaped profiles can pass through some of the data in fig. 2, neither can account for the range of observations in fig. 2.

3.1.2 Piecewise linear slip distributions. Manighetti et al. [200�, 2004] emphasized piecewise linear interpretations of their slip profiles (fig. 2b, c). using hilbert transformations [Weertman, �996], we show in the appendix (see supporting material for this volume) that for the simplest piecewise linear profile, a triangular slip distribution with the peak slip (dmax) at x = c, the normalized driving stress (∆τ*) is non-linear:

∆∆

ττ

π

*'

ln lnmax

= =+

+−

+−

−−

G D

a

ac a

x ax c

aa c

x ax c

2

, (�)

where G' is the shear modulus (G) for antiplane strain, and G/(�-ν) for plane strain (ν = Poisson’s ratio). our results con-firm the assertion of Manighetti et al. [2004] that a logarithmic stress singularity arises at the tip of a fault (x = ±a) if the slip distribution tapers in a perfectly linear fashion to the end of the fault. this singularity is weaker than the one at the end of a fault with an elliptical slip distribution. Additionally, how-ever, a logarithmic stress singularity also arises where the slip peaks (x = c). It is at least twice as strong as the singularity at the fault end, becoming stronger as the profile becomes more asymmetric (fig. 3). Moreover, the term in the denominator of the integrand in equation (A3) ensures that stress singulari-ties will exist anywhere piecewise linear segments of a slip profile meet.

Fig. 3Fig. 3

Figure 1. Map of normal faults from the Afar rift, east Africa [modified from Manighetti et al., 200�], showing possible locations of fault triplets like those of fig. 7.

MARtEl And ShACAt 3

3.1.3 Composite slip profiles with f lat tops. We now explore slip distributions that arise from a random distribu-tion of strength losses along an isolated fault. figure 4 shows 20 profiles, calculated using a boundary element method [e.g., Crouch and Starfield, �983] for antiplane strain. Each fault is divided into 40 constant-slip elements, with the shear strength of each element selected at random from a uni-form distribution from 0 to τl. Individual profiles in fig. 4, such as the one shown by the heavy line, generally have an irregular skewed elliptical form and do not match composite profiles of fig. 2 particularly well. Collectively, however, the normalized profiles yield a composite “quasi-elliptical” profile like that of fig. 2h. the resemblance occurs because the peak normalized slip for each profile (a) equals �, and (b) generally develops over the range |x/a| < 0.5. the flat top of the composite profile of fig. 4 is an artifact of normal-izing and superposing individual profiles; it does not reflect piecewise linear slip along the central portions of individual faults traces.

3.2 Slip Along Interacting Parallel Faults

3.2.1 Periodic faults. tada et al. [2000] provide the slip distributions along two periodic sets of parallel faults of equal length. for coplanar faults with a center-to-center spacing of w (w > 2a),

DG

w

xwa

w

=

−∆τπ

π

πcoshcos

cos

1 . (2)

Fig. 4Fig. 4

Figure 2. normalized composite slip profiles along the traces of faults from the Afar rift, east Africa [modified from Manighetti et al., 2004]. of the 255 faults studied, 82% fall into one of the eight groups here. the percentage of faults in each group is taken from Manighetti et al. [200�]; the percentage of faults represented by fig. 2d is undetermined.

Figure 3. normalized triangular slip profiles (D* = D/Dmax) and corresponding normalized driving stress profiles (∆τ* = 2πa∆τ/[G’Dmax]) for (a) a symmetric slip distribution (c/a = 0), and (b) an asymmetric slip distribution (c/a = 0.9).

4 Mechanics and interpretations of fault slip

If w decreases while a is fixed, the peak of the slip profiles becomes flatter (fig. 5a), with slip increasing everywhere

along each fault in the array. the departure from an ellipti-cal form is not pronounced even for w/a = 2.�, but fig. 5a suggests that a close spacing of coplanar faults provides a second explanation for the “quasi-elliptical” profiles of fig. 2h. for a periodic set of “stacked” parallel faults of equal length arranged like rungs on a ladder, with a center-to-center spacing of H,

DG

H

xHa

H

=

−∆τπ

π

πcoscosh

cosh

1 . (3)

If H decreases while a is fixed, the slip profiles again become flatter (fig. 5b); in this case slip decreases everywhere along each fault in the array. A close spacing of stacked faults thus provides a third explanation for the composite profile of fig. 2h. Manighetti et al. [2004] argued that a flattened slip profile can reflect highly fractured compliant material near fault trace ends. thus, at least four explanations exist for how “quasi-elliptical” slip profiles could develop, and none of them involve true piecewise linear slip along an isolated fault.

Fig. 5Fig. 5

Figure 4. normalized slip profiles (D* = D/Dmax) for 20 faults with random strength distributions. Slip is plotted at the center of each element. line segments connect these points.

Figure 5. Slip profiles for infinite periodic arrays of faults, with slip normalized by the peak slip on each fault (D/Dmax). (a) Coplanar faults with various normalized fault center spacings (w/a). (b) Stacked faults for various normalized fault center spacings (H/a).

MARtEl And ShACAt 5

3.2.2 Two slightly overlapped faults of equal length. As the spacing (s) shrinks between two slightly overlapped faults of equal length, our boundary element solution (fig. 6a) shows that the faults share slip such that they yield a superposed slip profile like that of an equivalent single fault. the peak slip on each fault shifts from the fault center towards the region of overlap, akin to the three-dimensional results of Willemse et al. [�996]. the skewed individual profiles also taper in nearly linear fashion in the region of overlap, a point not previously recognized. two

key effects contribute. first, the region of overlap is near the center of an equivalent single fault, where slip is a maximum and is nearly constant (fig. 6a). Within that region, as the slip at the overlapped tip of one fault drops to zero, slip at neighboring points on the adjoining fault approaches a maximum. Second, the slip profiles must be mirror images because of symmetry. for widely separated faults the slip profiles are convex (ellip-tical), but for closely spaced overlapped faults the two cited conditions require slip to approach a linear taper in the region

Fig. 6Fig. 6

Figure 6. (a) normalized slip profiles (D* = D/Dmax) for two sets of overlapping faults with an overlap of 0.5a. the normalized slip on each fault peaks at a value of �. the bumps in the profile in the region of overlap show the superposed slip of both faults. note the near-linear taper of slip in the region of overlap for the closely spaced faults (s/a = 0.05). (b) two overlapped aplite dikes from the Sierra nevada. the stick of lip balm in the photograph is 6.7 cm long. note the near-linear tapering of dike aperture in the region of overlap.

6 Mechanics and interpretations of fault slip

of overlap. this conclusion is based primarily on symmetry arguments and is likely to hold even in materials that are not perfectly elastic. It also is consistent with observations of natural faults [Willemse, �997, Manighetti et al., 200�; Crider and Pol-lard, �998; Cartwright and Mansfield, �998; Contreras et al., 2000]. Closely spaced dikes commonly exhibit a mechanically analogous near-linear taper in aperture where they overlap (fig. 6b). near-linear relative displacement profiles characterize one way that closely-spaced, slightly overlapped fractures of similar size interact.

3.2.3 Three faults. to further investigate fault interaction we calculate slip profiles with the boundary element method for various fault triplets (fig. 7, right column) and compare them to the composite profiles of fig. 2 (fig. 7, left column). Above each set of profiles in the right column of fig. 7, the associated fault triplet configuration appears, with the bold and dotted profiles corresponding to the bold and dotted faults, respec-tively. the bold profiles are reproduced over the data of fig. 2 in the left column, where they provide a good visual match to the composite profiles. fault triplets like those in fig. 7 occur in fig. �, supporting our inference that fault interaction strongly influences the first-order composite profiles.

4. dISCuSSIon

4.1 Continuous Slip Profiles Constructed From Discrete Data

Constructing continuous slip profiles from discrete data requires an interpolation scheme. linear interpolation com-monly is the method of choice. for piecewise linear profiles in a linear elastic medium, however, stress singularities arise where the first derivative along a slip profile is discontinuous. Profiles produced with cubic splines are continuous through the second derivative [Press et al., �990], and hence can avoid strain and stress singularities, but their third derivative is discontinuous, violating the beltrami-Michell compatibility equations [Malvern, �969] and introducing strain incompat-ibilities. Even though a bell-shaped slip profile produced by a driving stress with a step function can avoid a fault tip stress singularity, strain incompatibilities arise at the inflection points, where the second derivative of the driving stress and the third derivative of the slip are discontinuous. higher-order polynomial interpolation methods can avoid introducing strain incompatibilities, but they can introduce artificial oscilla-tions between data points [Press et al., �990]. We provide no satisfactory solution to this dilemma here, but note that data and interpolation methods are important to identify and keep distinct since interpolation schemes can introduce, alter, or eliminate mechanically significant aspects in slip profiles.

4.2 Slip Profiles Constructed From Effectively Continuous Data

Recent surveying methods (e.g., laser altimetry and satel-lite interferometry) effectively allow accurate, high-resolu-tion, spatially continuous slip profiles to be prepared for faults. fractures can be anticipated where such a slip profile is sufficiently steep [Cooke, �997] or where any of its first three derivatives are discontinuous. the locations of these possible energy sinks are unlikely to be spatially uniform and might be expected where a fault is non-planar.

4.3 Development of “Near-linear” Tapers in Slip

Several theoretical explanations exist for why near-linear tapers in slip appear to be so common. these include varia-tions in fault strength [e.g., Cooke, �997] or driving stress [bürgmann et al., �994]; off-fault deformation concentrated near the fault tip [e.g., Manighetti et al. 200�; Scholz, 2002; d’Alessio and Martel, 2004]; and the lack of a “barrier” to impede fault propagation [Manighetti et al. 200�]. our analyses indicate that sharing of slip between closely spaced, slightly overlapped faults provides a robust explanation for a near-linear slip taper that is supported by several data sets [e.g., dawers et al., �993; Willemse, �997; Cartwright and Mansfield, �998; Contreras et al., 2000; Manighetti et al., 200�; Shipton and Cowie, 200�; d’Alessio and Martel, 2004]. We conclude that a near-linear slip taper by itself is not diagnostic of a particular cause. Slip data can be col-lected and analyzed in conjunction with information on both along-fault fracturing and neighboring faults to test whether the possibilities cited here provide viable explanations for near-linear slip tapers on a case by case basis.

4.4 “Barriers” to Fault Propagation

our findings bear on how faults stop propagating. Willemse and Pollard [2000] showed how mechanical inter-action can impede the propagation of closely spaced faults of similar size if their tips overlap, and our results show that this should yield near-linear slip tapers. In contrast, Manighetti et al. [200�] concluded that fault tips encountering barriers to propagation would develop an elliptical variation in slip. faults thus might respond differently to different barriers. by collecting and analyzing fault slip data in conjunction with information on near-fault fracturing, rock type, and slip dis-tributions on nearby faults [e.g., d’Alessio and Martel, 2004] the effects of different types of barriers might be clarified.

the history of fault propagation and slip could depend on how faults nucleate. one possibility involves widely spaced nucleation points. In this scenario, when the faults are small

Fig. 7Fig. 7

MARtEl And ShACAt 7

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8 Mechanics and interpretations of fault slip

and interact little, the slip profiles might be roughly elliptical or skewed elliptical, depending on the fault strength distribu-tion and regional stresses. As the faults grow and interact, near-linear tapers could evolve where closely spaced faults eventually overlap. A second end member involves faults initially clustered along a pre-existing flaw. In this scenario, faults could interact and develop slip profiles with near-linear tapers early in their growth history. these two possibilities are not mutually exclusive, and other scenarios exist as well.

5. ConCluSIonS

Slip profiles that are truly piecewise linear require singular driving stress distributions, and profiles without continuous third derivatives imply strain incompatibilities. Such profiles, if appropriate, suggest localized fracturing (energy sinks) along a fault. In many cases the profiles are interpretations based on discrete data, and alternative interpretations without these physical implications are permitted. fault interaction provides a simple explanation for the general form of many observed slip distributions, especially near-linear slip tapers, which are difficult to explain by on-fault effects. Many faults are closely spaced and can be expected to interact. Slip data collected and analyzed in conjunction with information on both along-fault fracturing and neighboring faults can lead to improved insight into how faults slip, propagate, and dis-sipate energy.

Acknowledgements. We also thank Isabelle Manighetti and her colleagues for their research, Michele Cooke and Geoffrey King for helpful reviews, and Garrett Ito and david Pollard for helpful discussions.

APPEndIx

hilbert transforms relate the driving stress distribution (∆τ), the slip distribution (D), and dislocation density (b) along a two-dimensional fault [Weertmann, �996]. the slip distribution can be found from the following two equa-tions:

D x B x dxa

x

( ) = ( )−∫ ' ' , (A�)

where

B xG

xx x

dx( ) =−−∞

+∞

∫2

πτ

'( ')

''∆, (A2)

Alternatively, the driving stress distribution can be found from the slip:

∆τπ

x G B xx x

dx( ) =−−∞

+∞

∫' ( ')

''

2, (A3)

where

B x dDdx

( ) = , (A4)

for a triangular slip distribution with the peak slip Dmax at x = c

D x D x ac a

( ') 'max=

++

for –a ≤ x'≤ c (A5)

D x D a xa c

( ') 'max=

−−

for c ≤ x'≤ a. (A6)

Substituting (A5) and (A6) into (A4) yields

B x Dc a

( ') max=+

1 for –a ≤ x'≤ c, (A7)

B x Da c

( ') max= −−

1 for c ≤ x'≤ a . (A8)

the non-zero values of B occur for -a ≤ x'≤ a, so

∆τπ π

x G B xx x

dx G B xx x

dxa

a

( ) = ′−

= ′−−∞

+∞

−

+

∫ ∫2 2( ')

'' ( ')

'' . (A9)

Substituting (A7) and (A8) into (A9) yields

. (A�0)

the integrals in (A�0) are elementary and lead to the follow-ing solution for the driving stress:

∆τπ

x G Dc a

x ax c a c

x ax c

( ) = ′+

+−

+−

−−

2

1 1max ln ln . (A��)

So a piecewise linear slip distribution is associated with a non-linear logarithmic driving stress distribution possessing stress singularities. As equations (A2) and (A3) have the same form, piecewise linear driving stress distributions will produce non-linear slip distributions.

MARtEl And ShACAt 9

REfEREnCES

Aydin, A., and d. Kalafat (2002), Surface rupture of the �7 August Izmit and �2 november duzce earthquakes in nW Anatolia, turkey, their tectonic and kinematic significance and the associated damage, Bull. Seis. Soc. Am., 92, 95–�06.

bürgmann, R., d.d., Pollard, and S.J. Martel (�994), Slip distributions on faults: effects of stress gradients, inelastic deformation, heteroge-neous host-rock stiffness, and fault interaction, J. Struct. Geol. 16, �675–�680.

Cartwright, J., and C.S. Mansfield (�998), lateral displacement variation and lateral tip geometry of normal faults in the Canyonlands national Park, utah, J. Struct. Geol., 20, 3–�9.

Contreras, J., M.h. Anders, and C.h. Scholz (2000), Growth of a normal fault system: observations from the lake Malawi basin of the east African rift, J. Struct. Geol., 22, �59–�68.

Cooke, M.l. (�997), fracture localization along faults with spatially vary-ing friction, J. Geophys. Res., 102, 22,425–22,434.

Crider, J. G., and d. d. Pollard (�998), fault linkage: three-dimensional mechanical interaction between echelon normal faults, J. Geophys. Res., 103, 24373–24392.

Crouch, S.l., and A.M. Starfield (�983), Boundary element methods in solid mechanics, 322 pp., Allen and unwin, london.

d’Alessio, M. A., and S.J. Martel (2004), fault terminations and barriers to fault growth, J. Struct. Geol., 26, �885–�896.

d’Alessio, M. A., and S.J. Martel (2005), development of strike-slip faults from dikes, Sequoia national Park, California, J. Struct. Geol., 27, 35–49.

dawers, n.h., M.h. Anders, and C.h. Scholz (�993), Growth of normal faults: displacement-length scaling, Geology, 2�, ��07–���0.

Gupta, A., and C.h. Scholz (2000), A mode of normal fault interaction based on observations and theory, J. Struct. Geol., 22, 865–879.

Maerten, l., E.J.M. Willemse, d.d., Pollard, and K. Rawnsley (�999), Slip distributions on intersecting normal faults, J. Struct. Geol., 21, 259–27�.

Maerten, l., P. Gillespie, d.d. Pollard (2002), Effects of local stress pertur-bation on secondary fault development, J. Struct. Geol., 24, �45–�53.

Malvern. l.E. (�969), Introduction to the Mechanics of a Continuous Medium, 7�3 pp., Prentice-hall, Englewood Cliff, new Jersey.

Manighetti I., G. King, Y. Gaudemer, C. Scholz, and C. doubre (200�), Slip accumulation and lateral propagation of active normal faults in Afar, J. Geophys. Res., 106, �3,667–�3,696.

Manighetti, I., G. King, and C.G. Sammis (2004), the role of off-fault damage in the evolution of normal faults, Earth Planet. Sci. Lett., 217, 399–408.

nicol, A., J. Watterson, J.J. Walsh, and C. Childs (�996), the shapes, major axis orientations and displacement patterns of fault surfaces, J. Struct. Geol., 18, 235–248.

Peacock, d.C., and d.J. Sanderson (�994), Geometry and development of relay ramps in normal fault systems, Bull. Am. Ass. Petrol. Geol., 78, �47–�65.

Pollard, d.d., and P. Segall (�987), theoretical displacements and stresses near fractures in rock: With applications to faults, joints, veins, dikes, and solution surfaces, in Fracture Mechanics of Rock, edited by b.K. Atkinson, pp. 277–348, Academic Press, london.

Press, W.h., b.P. flannery, S.A. teukolsky, and W.t. vetterling (�990), Numerical Recipes, 2nd ed., 702 pp., Cambridge university Press, new York.

Scholz, C.h. (2002), The Mechanics of Earthquakes and Faulting, 2nd ed., 47� pp., Cambridge university Press, new York.

Schwartz, d.P., and K.J. Coppersmith (�984), fault behavior and charac-teristic earthquakes: Examples from the Wasatch and San Andreas fault zones, J. Geophys. Res., 89, 568�–5698.

Segall, P., and d.d. Pollard (�980), Mechanics of discontinuous faults, J. Geophys. Res., 75, 4337–4350.

Shipton, Z.K., and P.A. Cowie (200�), damage zone and slip-surface evolu-tion over µm to km scales in high-porosity navajo sandstone, utah, J. Struct. Geol., 23, �825–�844.

tada, h., P.C. Paris, and G.R. Irwin (2000), The Stress Analysis of Cracks Handbook, 677 pp., American Society of Mechanical Engineers, new York.

Weertman, J. (�996), Dislocation Based Fracture Mechanics, 524 pp., World Scientific, Singapore.

Willemse E.J.M. (�997), Segmented normal faults: correspondence between three-dimensional mechanical models and field data, J. Geo-phys. Res., 102, 675–692.

Willemse, E.J.M., d.d., Pollard, and A. Aydin (�996), three-dimensional analyses of slip distributions on normal fault arrays with consequences for fault scaling, J. Struct. Geol., 18, 295–309.

Willemse, E.J.M., and d.d. Pollard (2000), normal fault growth: evolution of tipline shapes and slip distribution, in Aspects of Tectonic Faulting, edited by f.K. lehner and J.l. urai, pp. �93–226, Springer -verlag, berlin.