JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 105, NO. B10, PAGES 23,431-23,442, OCTOBER
10, 2000
Factors controlling normal fault offset
in an ideal brittle layer
LucL. Lavier• andW. RogerBuck
Lamont-Doherty
EarthObservatory
ofColumbia
University,
Palisades,
NewYork
Alexei N. B. Poliakov
Laboratoirede G6ophysique
et Tectonique,CNRS UMR 5573, Universit6de MontpellierII, Montpellier,France
Abstract. We studythe physicalprocesses
controllingthe developmentandevolutionof normal
faultsby analyzingnumericalexperiments
of extensionof an idealtwo-dimensional
elastic-plastic
(brittle) layer floatingon an inviscidfluid. The yield stressof the layer is the sumof the layer
cohesionand its frictionalstress.Faultsare initiatedby a smallplasticflaw in the layer. We get
finite fault offsetwhen we makefault cohesiondecreasewith strain. Even in thishighly idealized
systemwe vary six physicalparameters:
the initial cohesionof the layer,the thicknessof the
layer,therateof cohesionreductionwith plasticstrain,the frictioncoefficient,the flaw sizeand
the fault width. We obtaintwo main typesof faulting behavior: (1) multiple major faultswith
smalloffsetand (2) singlemajorfault thatcandevelopvery largeoffset.We showthatonly two
parameters
controlthesedifferenttypesof faultingpatterns:(1) the brittlelayerthicknessfor a
givencohesionand (2) the rateof cohesionreductionwith strain. For a largebrittlelayer
thickness(> 22 km with 44 MPa of cohesion),extensionalwaysleadsto multiplefaults
distributedover the width of the layer. For a smallerbrittlelayer thicknessthe fault patternis
dependenton therate of fault weakening:a very slowrateof weakeningleadsto a very largeoffset
fault anda fastrateof weakeningleadsto an asymmetricgrabenandeventuallyto a very large
offsetfault. When the offsetis very large,the modelproducesmajorfeaturesof the patternof
topographyandfaultingseenin somemetamorphiccorecomplexes.
1. Introduction
Karsonet al., 1987; Axen andBartley, 1997; Cannet al.,
1997; Tucholkeet al., 1998] are major featuresof Earth's
Thenaturalfaultsthat either boundor cut an asymmetricor
symmetricgrabenin rifts showa widerangeof offsets [e.g.,
Robertset al., 1991; Morley, 1995]. The offset varies from
microscopic
slip on incipientfaultsto ~5 km slip on rift basin
boundingfaults [e.g., VeningMeinesz, 1950; Ebingeret al.,
1987] up to severaltens of kilometers on metamorphiccore
complexesstructures
[e.g., Coney, 1980; Karsonet al., 1987;
topography. Some authors [e.g., Davis and Lister, 1988;
Davis and Lister, 1988; Wernicke, 1985; Lister et al., 1986,
1991; Cannet al., 1997; Tucholke et al., 1998]. Fault with
Wernicke,1995] believethatthesefaultsformedandslippedat
a low dip. However, this mechanism violates Andersonian
faulting theory [Anderson,1951; Wills andBuck, 1997] which
statesthat normalfaultsformandslip at a high dip angle. An
alternativeto this modelis that normalfaultsoriginateat high
angleand,asfaultoffsetincreases,arerotatedflexurallyto an
inactive low-angle configuration[Hamilton, 1988; Wernicke
andAxen, 1988; Buck, 1988].
Field observations in favor of
offsetssmallerthanthethicknessof the brittlelayer generally or againsteitherof thesemodelsarenumerous(seereviewfrom
dip at high-angle (> 45ø) [Anderson,1951; Jackson, 1987].
Axen and Bartley [1997]) so that it is often difficult to
These faults are the most common features encountered in rift
Suezandthe East African Rift [Hamilton, 1988; Patton et al.,
determine
whetherthe faultrotatedto its presentconfiguration
afterlargeoffsetor if it initiatedat low angle.
We wouldlike (1) to understand
the physicscontrollingthe
differentkindsof normalfaultingobtainedin a set of idealized
numericalexperimentsand(2) to definethe possiblerangeof
behaviorsthat are relevantto geologic environments. It is
1994, Morley at al., 1992, BosworthandMorley, 1994] or at
mid-oceanicridges [e.g., MacDonaldand Luyendyk, 1977,
likely that many processesnot treatedhere, such as thermal
advection,magmatism,andcrustalthinning affect the pattern
Karson et al., 1987].
of faultingin a brittle layer. However,our goal is to startto
understandone of the simplest systemsthat leads to the
settings [Jackson, 1987]. They usually define a series of
grabensstructured
by major or secondaryhigh-angle normal
faults. Such features are consistently observed in most
extensionalsettingssuchas the BasinandRange,the Gulf of
Recent
observations
in
both
continental
and
oceanic
settingsshowthat largeoffsetnormalfaultswith dip < 30 ø or
evenwith negativedips [e.g., Coney, 1980; Wernicke, 1985;
•NowatGeoForschungsZentrum
Potsdam,
Potsdam,
Germany.
Copyright2000 by theAmericanGeophysicalUnion.
formation
2. Previous
2.1.
We therefore use a
Work
Modeling
of Normal
Faults
A great deal of work has been done to understandthe
Papernumber2000JB900108.
0148-0227/00/2000JB 900108509.00
and offset of normal faults.
dynamicmodelof faultingto studythe evolutionof fault offset
duringthe extensionof a uniformthicknessbrittle layer.
mechanics
of normalfaultingusingdifferentapproaches.The
23,431
23,432
LAVIER ET AL.: FACTORS CONTROLLING
main differencebetween these methods is the simplifying
assumptionmade in order to study the evolution of normal
faulting. A first approachconsidersthe initiation of normal
faults and ignores the long-term developmentof faulting.
Anderson[1951] showsthat in a uniformbrittle layer subject
to side-drivenextension,faultsareproducedwith dips ranging
from about~70ødownto 45 ø, dependingon the coefficientof
friction. On the basisof laboratory estimatesof rock friction
[e.g., Byerlee, 1978] normalfaultsshouldform with dipsclose
NORMAL
FAULT OFFSET
only the initiation of fault patterns when the stressfield is
slightly aboveyield. Thusthesepredictionsare only useful
for small deformations,and give erroneousresultsfor large
deformation
2.2.
encountered
Modeling
in natural faults.
of Normal
Faults With Finite
Offset
Forsyth [ 1992] andBuck [ 1993] have attemptedto define
the parameterscontrolling finite offset on normal faults by
to 60 ø.
usingvery simplified modelsof lithosphericbending. Forsyth
Most continuum models realistically approximate the [ 1992] treatedthe brittle layer as a thin perfectly elastic beam,
overall deformationof the lithosphere during rifting but andBuck [1993] choseto treat it as a thin elastic plate having
generally do not examine the evolution of primary versus a finite yield strength (elastic-plastic). Both assumedthe
secondaryfaults. They either do not treat the localization of lower crust to be an inviscid fluid and prescribedan initial
deformationassociatedwith faultsor must explicitly specify cohesionlessfault in the model domain. Both studiesattempt
the initial fault properties. Also, a systematicanalysis of the to show that a normal fault will continueslip as long as the
influenceof eachphysical parameterson the faulting behavior material aroundit is stronger than the fault itself. A weak
is lacking in most of thesestudies.
frictional fault surrounded
by a strongerfrictional andcohesive
Severalauthorsmodeledtopographycausedby normal fault materialfits that criterion. To slip on this fault, thereneedsto
offset by treating the lithosphereas an elastic plate [e.g., be a forceincreaseto compensatefor the resistanceof the layer
Vening Meinesz, 1950; Kusznir et al., 1987; Weissel and to bending and the resistanceof gravity to the buildup of
Karner, 1989]. They assumedslip on one or more high-angle topography. If the force increase is sufficient to cause the
normal faults and look at the topographyresulting from the materialaroundthe fault to reachyield, a new fault will form.
bendingof the lithospherearoundthe fault. Vening Meinesz If the forceincreaseis too small to causea significant change
[ 1950] proposedthat grabenswereproduced
by flexureof the in stress around the fault, the fault may slip indefinitely.
hanging wall of a major normal fault boundinga rift when Therefore both of these authors studied the effect of this
secondary
faultsantitheticto the first fault form in the part of "gravitational" force increaseon the strength of the layer
the hangingwall that undergoes
the mostbending.
surroundingthe fault.
In orderto study the potential formation of antithetic or
Using the simpler approximation,Forsyth [1992] inferred
syntheticfaults in the vicinity of a high-angle normal fault, that the maximumfault offset is controlledby the dip angle of
Melosh and Williams [1989] useda finite element method to
the fault andthe layer cohesion. A very large offset fault can
modelthe lithosphereasa thick elastic plate. They inserteda only developon a fault with an initial low dip angle, lower
normal fault in the lithosphere and predict the initiation of than that predictedfor the formation of a normal fault in a
new faults by assuminga Mohr-Coulomb criterion for brittle homogenouslayer [Anderson,1951; Wills and Buck, 1997].
failure in the elastic plate. However, they could not model By using a more complete descriptionof the strength of the
largefaultoffsets. In a furtherimprovementof thesetypes of layer, Buck [ 1993] foundthat for a given cohesion,fault offset
studies,Hassani and Chery [ 1996] were able to simulate the is controlledby the thicknessof the elastic-plastic layer. For
localization
and formation of new faults around an initial
a brittle layer thickness >10 km and for a reasonably low
prescribedweakfault by assumingplastic behavior in a thick cohesionvaluethe fault can build up only a few kilometers of
elastic-plastic plate to allow for the localization of offsetbeforebeing replacedby a newone. For a thinnerbrittle
deformation in shear zones.
layer the fault offset may be unlimited.
Stein et al. [1988] showedthat, to accuratelymodel the
It can be seen (Figure 1), that in a frictional brittle layer
observedpattern of deformationaround high-angle normal with a reasonablylow value of cohesion (20 to 40 MPa [e.g.,
faults assumingan elastic lithosphere, the effective elastic Handin, 1966]), the remaining strength on an active fault
thicknessof the lithosphere must be much smaller than the (proportional to the area in light shadedarea on the yield
seismogenicthickness. By treating the lithosphereas a thin stressprofiles) representsa much smaller proportion of the
elastic-plasticplate, Buck [1988] confirmedthat result by total strength(proportionalto the areain dark shadedareaon
showingthat finite offset on a normal fault could greatly the yield stressprofiles) in a thin layer than in a thick layer.
reducethe wavelengthof the flexural responsein the area Moreover, Buck [1993] showedthat the gravitational force
surroundingthe fault.
increaseis proportional the squareof the layer thickness.
Several authorshave studiedloading of elastic layers in Thereforea fault will remain relatively much weakerthan the
differentwaysthat might causeinitiation of normal faults with
surroundingmaterialin a thin layer (Figure la) than in a thick
low-angledip (<30ø) [¾in, 1989; Parsonsand Thompson, layer (Figure lb), and the gravitational force increasewill be
1993].
However, these studies looked only at stress smallerin a thin layer than in a thick layer. For these reasons,
orientationandnot at magnitudeof stressesneededfor slip on normalfaultscanaccumulatelarge offset in a thin layer. In a
faults. Wills andBuck [ 1997] looked at •tress magnitudeand thick layer, fault offset should be small and multiple faults
showedthat the loading describedby Yin and Parsons and should form.
Thompsonwouldnot leadto slip on low-anglefaults cutting
an entire layer. Indeed,they show that in the areaswherethe 3. Model Formulation
stressorientationssuggestlow-anglefault initiation the stress
differences(andso shearstresses)approachzero,in contrastto
Forsyth's [1992] and Buck's [1993] studies are limited
adjacentareaswherethe stressorientationsand magnitudes becausethey assumed
thin plate behaviorfor the lithosphere
allow for high-angle fault initiation. Further, Gerbaultet al. and that faults are set as preexisting surfaces in the
[1998] showedthat elastic solutionsfor faulting can predict lithosphere. The thin plate approximation does not render
LAVIERET AL.:FACTORSCONTROLLING
NORMALFAULTOFFSET
23,433
Figure 1. Strengthdifferencesbetween (a) a thin and(b) a thick layer. The strength is the integration of the
yieldstressoverthethickness
of thelayer.Yieldstressoyis proportionalto the shearstressassuming
principal stresses
arehorizontalor vertical. Thedifferentstrengthsarethereforeequalto the shaded
areas. The layer
is frictional andcohesive. Therefore,when a fault has formed, its strengthis proportional to the remaining
frictional strengthof the layer. The strengthof the materialsurrounding
the fault is proportional to the cohesive and frictional strengthof the layer. One can seethat in a thin layer the strength on the fault is much
smallerthan that of the surroundingmaterial. In a thick layer the fault is still strongcomparedto the surrounding layer.
accuratelythe dynamic state of strain and stressin an elastic-
of the brittle layer andthe inviscid substrateequalto 2700 kg
plasticlayer. More importantly,the assumption
that faultsare
preexisting surfacesin the layer obliterates the processesof
m-3andtheacceleration
of gravityg equalto 10 m s-2. At the
weakeningleadingto the formation of the fault.
In the following experimentswe usea numericalmodel that
allows for the calculationof the dynamic state of strain and
stressin an idealbrittle (elastic-plastic)layer in which plastic
strainis given by a non-associated
plastic flow rule [Poliakov
and Buck, 1998]. Non associatedplasticity allows for the
determination
of
the
..................
y•u
uppersurface,shearandnormal stressesareassumed
to be zero.
The right and left side of the box are pulled steadily apart
(Figure2a). At the bottom we apply normal stressequalto the
lithostatic pressurein the brittle layer and zero shear stress
(Winklet foundation).
The cohesionis reduce•with increasingstrain after yielding
(Figure 2b). The shear stress at yield is given by MohrCo,,•,•mh theory:
Coulombyield condition andfor localization in shearzones or
"faults." Moreover, we can parametrize the weakening
q:-- ILl.On
+ C(œp•),
(1)
processes
thatleadtotheformation
of faultby assuming
that, ' wherex is the shearstressat yield,g is the coefficient
of
whenin the layer the yield stressis reached,the layer locally
loses its cohesion with plastic strain [Buck and Poliakov,
1998; Poliakov and Buck, 1998]. This approachis a major
advance
sinceit provides
a tool that allowsus accurately
to
friction, o, is the normalstress,andC is the cohesion,which
is defined
to depend
onthetotal plasticstrainœpsTheplastic
strain is the non recoverable strain accumulatedby plastic
deformationwhenthe stressin the layer is locally greaterthan
takeinto accountthe effectof the gravitationalforceincrease the yield stress. Up to the point wheremateriallosesall
andtheeffectof thedecrease
in forcecaused
by the weakening cohesion, the reduction of cohesion with strain is linear
processes occurring in the fault zone. Because of the
complexity of the numericalmethodand to reducethe number
of inputparameters
wechoose
thefollowingsetup(Figure2).
3.1. Model Setup
The width of the model domain is taken as 10-15 times the
(Figure2b):
C(œps)
= C(0)[1-(œp,/œb],
(2)
whereC(0)is theinitial cohesion
of thelayer. We defineœc
as
a characteristic
valueof plasticstrain.Whenthe plasticstrain
reachesœc,the fault is cohesionless. However,as seenin all
modelsallowingfor the localizationof deformation
[e.g.,
Cundall,1989], thewidthof the faultAw is consistently
2 to .
inviscid substrateand has a constant thickness. Each time we
4 timesthegridsize. Forthisreason,fora givendisplacement
remesh,new materialis addedto the bottomof the grid, AL the strainis dependent
on the grid size. In orderto scale
keeping the interface between brittle and inviscid materials the characteristic
strainandthe rateof cohesionweakening
flat, assumingthat materialaddedto the bottom has the same betweentwo models with different grid sizes we use
properties
asthe materialin the layer. Wetakethe density9 characteristic
offset,Ax,= œ,Aw,ratherthan%asa measure
of
layerthicknessH. The brittle layeris assumed
to overlie an
23,434
LAV1ER ET AL.' FACTORS CONTROLLING NORMAL FAULT OFFSET
(a)
Yield stress
0
Brittlelayer
x
EXTENSION
0 COMPRESSION
H
z
r
r.
Z'
Inviscid substrate
-
(b)
Plastic
strain,
œp
Figure 2. (a) Modelsetup
forextension
ofa brittlelayeroverlying
aninviscidfluid.Thelayerhasthefinite
yieldstress
of a Mohr-Coulomb
typematerial.Yieldstress
oyis proportional
to the shearstressassuming
principal
stresses
arehorizontal
orvertical.(b)Thereduction
of cohesion
C is linearwithplasticstrain.The
maximum
cohesion
lossis givenby C(O),andtherateof strength
reduction
is givenby C(0)/œc,
œc
beingthe
characteristic
plasticstrainfor whicha faulthaslost all its cohesion.
the amountof deformationneededto form a cohesionlessfault.
summed to determine
the new out-of-balance
forces.
This
A small perturbation,
one to three elementslarge, is dynamicresponseis then dampedto approacha quasi-static
initiallyplaced
atthecenter
ofthemodeldomain
to induce
the equilibrium.
formation of the first fault.. It is set to have a plastic strain
FLAC is a very powerfultechniquefor simulatingnon linear
theological behavior at very high resolution becausethe
explicit time-marching scheme does not require storage of
parameters:
(1) thethickness
of the brittlelayerH, (2) the largematriceswhich areneededfor implicit methods. The time
frictioncoefficient,
g whichwechoseto fix at a givenvalueto stepof the calculationscaleswith the elastic-plasticproperty
limit ouranalysis
to theeffectof cohesion
on layerstrength, of ourmodel. If the problem is purely elastic the time step of
(3)theinitialcohesion
C(0)ofthelayer,(4) thecharacteristicthe dynamicresponsescaleswith the velocity of the elastic
plasticstrainϥat whichthe faulthasbecome
cohesionlesswavepropagatingthrough the elements. This time step is of
equalto ϥin orderto becohesionless.
Even in this very simplified setting we input six
andAC(O)/e•
setstherateof cohesion
reduction
withstrain,(5)
the order of a few hundredths
the sizeof theinitial perturbation,and(6) the grid sizeof the
resolution
numericalmodel which setsthe characteristicoffset, Axe.
our numerical experiments would yield very long running
times. In orderto decrease
the CPUtime neededto performthe
numericalexperimentswe increasethe speedof calculationby
settingthe boundarydisplacementasa fractionof grid spacing
per time step. To set the boundarydisplacement,we chosea
3.2.
Numerical
Method
of the domains
of a second.
Therefore
studied and the timescale
the
needed for
The numericalmethodusedfor the experimentsis basedon
velocityto soundvelocityof 5x10-s. We
an explicit finite element method similar to fast lagrangian ratioof boundary
analysisof continua(FLAC) technique[Cundall,1989]. It has find that this ratio allowsfor fast enoughrunsand at the same
been usedto simulatelocalized deformation(approximating time minimizes the error on the strain calculation.
When the layer cohesion reacheszero at the surfaceof the
faults) in elastic-plasticmaterials in a variety of problems
[Hobbs and Ord, 1989; Poliakov et al., 1993; Poliakov and
model, the material at the surface behaves as a cohesionless
t-Ier•nn,
1994; Hassani and Chery; 1996; Poliakov and
Buck, 1998]. For each numerical time step the modeling
involves directsolutionof Newton'ssecondlaw for every grid
point. In order to approximate quasi-staticprocessesthe
effectsof inertia mustbe dampedin a way akin to oscillations
in a dampedoscillator. Starting from a non equilibriumstate,
the forces present at each grid point are summed. The
corresponding"out-of-balance"forcesandthe massat the grid
point give rise to acceleration. The accelerations are
integrated to calculate the velocities which are used to
determinethe incrementof strainat eachgridpoint. By using
the constitutivelaw for elastic and plastic theologies(Hook's
law for elasticity and Mohr-Coulombcriterion for plasticity),
the correspondingstress increments are determined,and the
forceswhich they produceon the surroundinggrid points are
pile of sand flowing on the slope of the topography. The
resolution of the model does not resolve such flow,
and
thereforenumericalinstabilities arise from that phenomenon.
To solve that problem, we set the cohesionof the layer to a
negligible value of 4 MPa in all experiments. As a result,
whenthe processof cohesionreductionhastaken place,4 MPa
of cohesion
remain on the fault.
The initial mesh of the model is made of quadrilaterals
subdividedinto two pairs of superimposedconstant-strain
triangularzones The use of triangular zones eliminates the
problem of "hourglassing" deformation sometimes
experienced
in finite differences
[Cundall,1989]. Sincethis
methodis Lagrangian(i.e., the numericalgrid follows the
deformations),the simulation of very large deformations
(locallymorethat 50%) involvesremeshingto overcomethe
LAVIER El' AL.: FACq'ORSCONTROLLINGNORMAL FAULT OFFSET
problemof degradationof numericalprecisionwhen elements
aredistorted. We trigger remeshingwhen one of the triangles
in the grid elementsis distortedenough that one of its angles
becomessmaller than a given value. Every time remeshing
occurs,strainsat each grid point are interpolatedbetweenthe
old deformed mesh and the new undeformedmesh using a
nearest-neighboralgorithm. The new state of strain is then
used with the rheological laws to calculate the stress and
resulting out-of-balance forces to start the time step cycle
again. Also, every time we remesh,errors in the interpolation
of the strains result in an increase in the out-of-balance forces,
and artificial
accelerations
and oscillations
occur.
For this
reason, the solution may not be in equilibrium immediately
after remeshing. This results in transient variations in
stressesof order 10% from the averagevaluesas we will see in
section4. After abouta hundrednumericaltime steps, damping
brings the solution back to equilibrium. We have tested
different criteria to trigger remeshing in order to reducethe
time to obtain a reproducible result and chose to use a
minimumangle beforeremeshingof 10ø.
4. Results
We ran 44 models (Table 1), varying the layer thickness,
the cohesionof the layer, the characteristicplastic strain, the
grid size, andthe flaw size and location. The model resultsin
the spontaneousdevelopmentof an evolving system of faults.
For all modelsa fault or faults start wherethe plastic flaw is
inserted(Table 1). In all casesthe faultsinitially form at a dip,
0 of 60 ø in agreementwith the internal friction coefficient (g =
0.6) assumedin the brittle layer.
We find two fundamentally different regimes of faulting
with some additional regimes included within each domain
(Plate 1). In the "large offset" or "unlimited mode"the initial
fault accommodatesnearly all the extension for as long as the
numerical experiment continues (Plate l a). For the "small
offset" mode, two or more faults form and accommodate
significant offset (Plates lb and lc)and no fault is offset by a
large amount (i.e., with an off'set greater than the layer
thickness).
Within
Table
l e). We believe that this is due to a local reduction of the
bending moment in the elastic-plastic layer [Buck, 1997].
When a beamis bent sufficiently, it can break in a distributed
mannerover a broadareaor it can break in one place (the way a
cracker snaps). Buck [1997] showedthat an elastic-plastic
layer shouldbreakin a distributedmanneror "crunch"when its
bending moment increaseswith increasing bending of the
plate. It should snap when the bending moment decreases
locally with increasingbending.We believethat hanging wall
snappingand footwall snappingaroundthe major fault occur
whenthe bendingof the plate is largeenoughthat the bending
moment starts to decrease. This qualitatively explains why
this behavior dependson the rate of strain weakening. Note
that snapping generatessecondaryfaults for the unlimited
Parameters
Used
for
Each
of
the
Numerical
Experiments
œc
KXc,km
Perturbation
Size,
M_JH
elts
Faulting
Mode
H = 20 km,Aw = 3000 m, C(O) = 24 MPa
0.01
0.1
0.4
0.6
1
1.2
1.5
0.01
0.4
0.03
0.3
1.2
1.8
3
3.6
4.5
3 bottom
3 bottom
3 bottom
3 bottom
3 bottom
3 bottom
3 bottom
0.01
0.01
0.01
0.01
0.01
0.01
0.01
H = 20 km, Aw = 3000 m, C(O) = 44 MPa
0.03
3 bottom
0.01
multiple
multiple
multiple
multiple
multiple
multiple
multiple
1.2
3 top
0.25
hanging
footwall
0.8
1.
2.4
3
4 bottom
3 bottom
0.62
0.75
footwall
footwall
1.2
1.5
3.6
4.5
4 middle
4 bottom
0.01
0.01
multiple
multiple
2.
3.
6
9
3 top
3 top
0.01
0.01
multiple
multiple
0.5
1.2
H = 20 km, Aw = 1500 m, C(O) = 44 MPa
0.75
4 middle
1.
1.8
3 middle
o•
footwall
unlimited
0.01
H = 10 km, Aw = 150Ore, C(O) = 44 MPa
0.015
3 bottom
0.01
0.3
0.5
0.45
0.75
0.8
1.
1.5
2.
3.
0.01
hanging
4 middle
4 bottom
.9.
1.33
footwall
footwall
1.2
4 top
1.46
footwall
1.5
2.25
4 middle
4 middle
o•
o•
unlimited
unlimited
3
4.5
3 bottom
2 bottom
0.01
0.01
multiple
multiple
H = 5 kin, Aw = 900 m, C(O) = 44 MPa
0.009
3 bottom
0.01
hanging
0.1
0.3
0.5
0.09
0.27
0.45
2 bottom
2 bottom
2 bottom
0.71
1.
1.2
0.8
1.
0.72
0.9
3 top
3 top
2.2
o•
1.5
2.
3.
1.5
1.8
2.7
3 bottom
2 bottom
2 bottom
o•
o•
o•
4.
3.6
3 bottom
0.01
0.015
2 middle
0.1
0.3
0.4
0.8
1
1.5
2
0.15
0.45
0.6
1.2
1.5
2.25
3
2
2
2
2
2
2
2
middle
middle
middle
middle
middle
middle
middle
0.5
0.8
o•
o•
o•
o•
o•
3
4.5
2 middle
0.01
the unlimited mode we find that in some cases,
secondaryfaults canbreakthe layer in the hanging wall and/or
in the footwall of the initial fault. For "hanging wall
snapping",both the hangingwall andthe footwall of the fault
breakafter a very small offset (i.e., a few hundredsof meters)
on the initial fault (Plate l d). For "footwall snapping" the
initial fault develops a limited amount of offset (a few
kilometers)beforea secondaryfault breaksthe footwall (Plate
1.
23,435
footwall
footwall
footwall
footwall
unlimited
unlimited
unlimited
unlimited
0.01
multiple
H = 3 km, Aw = 1500 m, C(O) = 44 MPa
0.01
hanging
footwall
footwall
unlimited
unlimited
unlimited
unlimited
unlimited
multiple
Becauserunninga numericalexperimentwith the adequateresolutiontakesbetween3 CPUdaysto3 CPU weeks,the investigation
of the
parameterspacewaslimited.H is thethi-ckness
of thelayer;C(0) is the
initialcohesion
of the layer or the maximumcohesion
loss;Aw is approximately
thewidthof thefaultin the models;e½is the characteristic
strainfor maximum
strength
reduction;
&re the characteristic
offsetfor
maximumstrengthreduction,A/. is the maximumfault offset before a
new fault forms. The offset is taken as the maximum offset before a
second
faultforms.We estimate
theoffsetby measuring
the dip of the
fault and the horizontal and vertical offsets of the fault.
When the fault
offsetis largeandthe fault surfacecurved,we measureeachstraight
portionof thefaultfootwallandaddthemup. The perturbation
sizeindicatesin numberandposition
of elements.The perturbation
is a plastic
strainflaw of strainequaltothecharacteristic
straine½of everyindividualmodels.The perturbation
altersthe initialstrength
of the layer and
allowsfor theinitiation
of faultingin themiddleof thelayer.
23,436
LAVIER.El' AL.: FACTORSCONTROLLINGNORMAL FAULT OFFSET
(A)UNLIMITED,
H = 10kin,C(0)= 44 MPa, •
(B)
= 1.Skm
MULTIPLE,
H = lO km,C(O)= 44 MPa, Axe = 3km
P_._
-1500 •
lO
•
[:)•a•
• (kin) 70
•o
ao I:)ista•nce
(kin) •o
VE 1.5:1
VE
l.S:t1
010
O10:
Plastic strain
Plastic strain
OO
4.5
9.0
o.o
o 6•
1.7
1500
10
30
50
70
90
Z
Z
LU
I-X
LU
Plast• •aln
•
0.0
Pta•c
45
9.0
strain
oo
Z
033
1500•
o
Z
0:
2'
4
8
0
.50O
-1000
-1500
10
3o
50
713
90
0
ß
4'
,
lO
Pb•Uc
Plastic strain
stv•m
0.0
16
o.o
3,1
(D)
(C) MULTIPLE,
H = 2okm,C(O)= 44 MPa Axc= 4.5km
o 12
024
UNUMITED
HANGING
WALL
SNAPPING,
H = •o re,C(0)= 44 MPa.•xc= 0.015km
•;•.lOOO
•o
Distance (km)
•
DistanCe
(kin) •o
•o
•o
"5
ß : 10
8
n10
o.o
2.0
40
OO
0.69
14
00
047
o_q4
Z
Z
w
Iw
m8
n10
Plasticstrain
o.o
1.6
3.1
uJ
-1500
10
3o
5o
7o
go
'•'0
•8
OlO
Pb•s•c strain
_Plastic strain
0.0
0.9
I 8
0.0
023,
046
LAVIER ET AL.' FACTORSCONTROLLING NORMAL FAULT OFFSET
(E)
dueto the resistanceof the layer to bendingand the resistance
of gravity to the buildupof topographyand(2) the reductionin
force dueto the weakeningprocesseson the fault. We know
from our parametrization(equation(2)) that the reductionin
forcehFwdueto cohesionloss is linear with fault offset hx and
UNEMITED
FOOTWALL
SNAPPING,
H = 10krn,C(0)= 44 MPa, Axc= 0.75km
10
3o
can be described as
70
5o
23,437
Dlsta,nce (kin)
m8
AF,•= - C(O)H(7 + (1- ?)Ax / Ax)
Ax < Ax•
AF,•= C(O)H (• - 1)
Ax _<Ax,, (3)
• 10
Plastic•'ain il
'
,,,
where? is the size of the initial perturbationdivided by the
layerthickness,the subscriptw denotesweakening.
The form of the increase in bending related force as a
functionof fault offset is not specified;therefore,we ran a set
of numericalexperimentsthat allows us to estimate it. We
performedsix numericalexperimentswherea fault zone with
negligible cohesion(4 MPa) and zero friction is prescribedin
the middle of 5-, 10-, and 20-km-thick layers with two
I
00
2.8
5.7
1000-'•
10
30
50
70
9O
different values of cohesion, C(0) = 4 MPa, C(0) = 24 MPa and
•lastiC
strain
00
10
•
2.0
30
50
[t =0.6 (Figure3). The numberof grid elementsis 100 in the x
direction and 20 in the z direction for each experiments, the
20-km-thick layer having the lowest resolution (1000 m) and
the 5-km-thick layer having the highest resolution (250 m).
The result of each of these experiments is the bending
componentof the forcechangeAF, asa function of horizontal
offset (Figure3).
As mentionedpreviously, for each remeshing,the out-ofbalanceforcesin the modelarereinterpolatedfar awayfrom the
static equilibrium. This results in a discontinuity in the
calculationof forces that occursabout every 300 to 500 m of
horizontal offset for a grid size of 250 m and every 700 to
1500 m for a grid size of 1000 m. The forcesreacha maximum
4.1
70
90
strain
oo
i.o
z1
Plate 1. (continued)
modebut that the primaryfault still can potentially develop
very large offset.
for each case after an amount of horizontal
offset
that
scales
approximatelywith layer thickness(700 to 800 m for H = 5
kin, 1700 to 2100 m for H= 10 km and 6000 to 6400 m for H =
5. Analysis of Results
We expectthat two processes
will affectthe developmentof
fault offset in the layer: (1) the increasein gravitational force
7.1011
H - 20 kin, C(O)-•0 M
6.10"
Plate 1. Exampleof the two different regimes of faulting.
The modeledtopographyandplastic strainareplottedfor three
stepsin the evolutionof eachrun with H = 10 km andH = 20
km (Table 1). Faults are in warmcolors in the plastic strain
plot, vertical exaggerationanddirectionfor the evolution of
extension are the same for each plots. (a) Very large offset
modefor Axc= 1.5 kin. For the last step, the modeledfault
has 27 km of offset.
The footwall
of the fault rotates
as the
fault offsetincreases.Elasticandplasticdeformationoccursin
the footwall
of the fault.
The accumulation
of deformation
producesa double peakedtopographic profile. Locally, the
fault hasa negative dip. (b) and (c) Multiple faults with small
offset modefor Axe= 3 and 4.5 km and H= 10 km and 20 kin,
respectively. Initially, one or two faults form in a V or invertedV shapeand are offset alternatively. (d) Hanging wall
snappingfor Ax, = 15 m. Aftera small amountof offset in the
initial fault, the hanging wall and the footwall have bent
enough to break in secondaryfaulting. The topography
evolves into an asymmetricgraben. (e) Footwall snapping
modefor Ax,= 750 m. After 3 km of offset the hanging wall
of the initial fault stops bending. The footwall keep on
bendingasthe fault offset increases. After a bendingequalto
0
2000
4000
6000
8000
Horizontal offset (m)
Figure 3. Plotof the bendingforceneeded
to stretchlayers
(5, 10 and 20 km thick) of increasingstrengthagainstthe
horizontaloffset. The layershavea friction coefficientt.t=
0.6 anddifferent initial cohesions C(0) = 4 MPa and 24 MPa.
Thereis no lossof strengthwith plasticstrainin the layer. A
strengthless
faultis initially setin the layer. The forcesin-
creaseuntil they reach their yield point. The thicker layer,
10-4m
-•of curvature,
thelayersnapson its footwall'
anda new thelargertheforceneedsto be. Afterreachingthe yield point
fault forms.
The initial
fault has a maximum
offset of 5.5 km.
the forces decrease to a constant value.
23,438
LAVIER ET AL.: FACTORS CONTROLLING NOR•MAL FAULT OFFSET
.-
>
I
8888
i
• ß
i0
• ø
o.oõõ•õ
o
,-
(W•l}tlldeo
LAVIER ET AL.: FACTORS CONTROLLING NORMAL FAULT OFFSET
23,439
bendingcomponentof the force changeAFbby an exponential
(a)
function:
Thin Layer
Bend
AFt,= AH2 [ 1 -exp(-BAx/H)],
(4)
whereA definesthe maximumforce change and B controls the
rate of initial increasein force change with horizontal offset.
A and B are obtainedfrom our numericalexperiments (Figure
ent ofAF
3). A is estimated
at 1200Pam-• andB is estimated
to be equal
,-,AH
to 50.
Fault offset, Ax
I AF
,-,C(O)H
Weakening Component ofA F
When added, the two components give the total force
change necessaryto extend the layer. In Figure 4 we sketch
eachcomponentof the averageforce changeandtheir sumfor a
thin and a thick brittle layer. This suggests that two
mechanismsmay control whetherextensionof the brittle layer
leadsto the large offset or the small offset mode. The first is
related to the maximum possible changes in AF• and AF•
while the secondis relatedto the ratesof changeof these force
components.
Pertinentto the first mechanism,the maximumin bending
forceAF• is proportionalto the square
of the thickness/_/e
of
(b)
ThickLayer
Ben,
Com
of A F
2
AF b
the layer and the maximum decrease in force AFw is
proportional to the thickness H of the layer. Thus, the
maximumof AF• will exceedthe maximum of the AFwonly for
a thick enough layer (Figure 4b). In this case the bending
changes are larger than the weakening and a second fault
forms, limiting the offset on the initial fault (small offset
mode) (Plate lb). For a thin layer (Figure 4a) the bending
componentis small enough and the weakening component is
large enoughthat the weakening processesdominatethe total
force changeneededto offset the fault. In that case the initial
fault can accumulatevery large offset (Plate l a) (large offset
mode).
Total AF
For either a thick or a thin layer the rateof changeof forces
couldcontrol whetherthe forcechangeis positive andso a new
fault wouldform. When the rate of cohesionreductionis very
slow, the reduction in force is so small that the initial increase
Fault offset, Ax
,.,C(0)H
in bending force dominatesthe force changein the layer. In
our experimentsafter a few ten's of metersof offset on the fault
another fault forms (Plate lc) (small offset mode). If the rate of
cohesionreductionis moderateto fast, for a thin enoughlayer
the weakening processesdominatethe force change. In our
experiments we were able to obtain large offset faults (Plates
la, ldand le).
•
Axc
Weakening Componentof/t F
Figure 4. Schematicrepresentationof the two components
of the regionalforceneeded
to extendandfaultthe brittle layer
for (a) a thin and (b) a thick layer. Initially, the transition
betweenvery large offset faults and small offset faults is
controlledby the competitionbetweenthe rate of increasein
bendingforceandthe rateof decrease
in forcedueto the weakening on the fault. In the long term the transition is
controlledby the thicknessof the layer H.
20 km). From there, the regional force neededto pull on the
layerdecreases,
finally reachinga steadystatevalue(Figure3).
Ourresultsconfirmthe analysisof Buck [1993] suggesting
that the maximumincreasein force is proportional to the
squareof the layer thickness/½.
There is also some
dependence
on the layer cohesion. On the basis of the results
of the present experiments we choose to approximate the
As we have seen, the modes of faulting in our numerical
experiments are controlled by the thickness of the layer H
relating to the amountof weakeningpossible on a fault andthe
maximumincreasein bendingforce and the characteristicfault
offset Axe relating to the rate of weakening on a fault. We
plotted ourmodelcasesusingaxesof Hand Axe (Figure 5a) (as
definedin Table 1) for a cohesion C(0) = 44 MPa. Using this
parameterspaceto plot the resultsof our model experiments
yields two well-defineddomainsof fault behavior confirming
the theory developed previously: (1) a large offset mode
domainand(2) a small offset mode domain. Also confirming
ourtheory, we find two main transitions betweenlarge offset
andsmall offset domains:(1) Oneis transition that dependson
the maximumchangesin cohesionand bending relatedforces,
which hereoccurat aboutH = 22 km. Below 22 km the layer is
thin enough for the given cohesion to allow for large or
unlimitedoffset. Above 22 km the layer is so thick that the
bendingforceneededto offset the fault is suchthat the stresses
aroundthe fault exceedyield and another fault forms. (2) A
secondtransition correspondsto the point at which the
23,440
LAVIEREl' AL.:FACTORSCONTROLLINGNORMALFAULTOFFSET
lower characteristic
(a)
small offset
I
Multiplefaults
ß
Unlimited
_
Hangingwallsnapping
6. Discussion
ßAAAee• ©
....
I ....
I ....
I ....
I ....
I
I ....
2
I ....
I ....
I ....
3
I
4
5
Oba•ct•,cfaua
o•t, •c (km)
(b)
MORE
I
BRITTLE
ß
LESS BRITTLE
ß
ß
ß
•40
ß
ß
I Multiple
faults
H
•
•3o
•'
and Conclusion
We have developeda simple theory for understandingthe
processesandthe factorscontrolling normal fault offset in an
idealbrittle layer. We find that not only the thickness of the
layer [Buck, 1993] but also the rate of reductionin cohesion in
the fault controls whethera fault can develop large offset or
small offset. In this idealizedsystem we were able to show
that for a given spaceof geologically reasonablevalues of
layer thickness, cohesion loss, and rate of cohesion loss a
broad range of behaviors is possible. In order to test the
validity of our model we compare our results to some
geologicalfeaturesobservedgeologicalfaults.
large offset
0
This decreases the size of the
domainwherelarge offset can occur (for example Figure 5b
also showsthe transition for a 1% perturbation). However, in
natural systems, preexisting fault structures and thermal
structurecould representlarge perturbationsin our domain.
These types of perturbation may increase the size of the
domainin which largeoffsetis possible.
ß Footwall
snapping
$
fault offset.
C
6.1. Core Complex
We predictthat a primary fault can slip by an unlimited
:
amountonly whena layer with 40 MPa of cohesion is thinner
ß hanging
wallsnapping
than 22 km and when the fault forms for a moderate rate of
• 20
•e
ß
-e
PER= 1%
el0
ß
:
ß
ER10
ß
ß
cohesionreductioncorrespondingto a characteristicfault
offset (Ax•- 1 to 4 km dependingon the layer thickness)
ß
;
I
• 0 , .... , .... •.... , .... , .... , .... , .... , .... , .... •....
0
I
2
3
4
5
Characteristic
fault
offset,
•c (krn)
Figure •,
Domain plot for snappingand faulting by re-
giona]stretching.(a) Thicknessof the layeras a functionof
(Figure 5b). In that case, the inactive footwall of the fault
rotatesto becomenearly horizontal. Theseare the first selfconsistent numerical models that producethis kind of
behavior. It is possible,thoughhighly controversial,that
such fault rotation may occur in some continental core
complexesand along someparts of slow spreadingridges
[Laviere! al., 1999]. We compareour unlimitedoffset model
results(Platela andPlate 2) to someof the main geologic
features
of knowncorecomplexes.Theseare(1) a largemassif
the characteristicfault offset •.
The plot showsthe two domainsof very largeoffset and small. The very large offset located at the inside comer of the intersection between the
faults domainis separatedinto an unlimited and a snapping Mid-AtlanticRidgeandthe Atlantistransformfault and (2) the
domain where seconda• faults form in the footwall and Whipple Mountains (Plate 2). Plate 2a is the sea-floor
hangingwall of the initial fault.(b) Sameplot as in Figure5a topography in the vicinity of the Atlantis transform fault
but highlightedin dark and light shadingthe domainsfor [Tucholke e! al., 1998; Blackman e! al., 1998]. Plate 2b
small offset and large offset predictedby theory for a layer showsa satellite image of the area of the Whipple Mountains
with a very smallinitial perturbation(PER= 1%). The bestfit[e.g., Lister andDavis, 1989].
ting theoreticalcurveto ourexperiments
is the one for a layer
with a largerperturbation(PER: 10%).
cohesionreductionis so slow that the systembreaksanother
fault ratherthan continueto slip on the initial fault.
Using equations (3) and (4) we can further analyze the
transition in faulting modethat dependson the rate of fault
weakening. For given values of H and Axc we computedthe
maximum of AFw+ AFbas a functionof fault offset Ax. When
that maximumequalszero, we predicta transition from small
to largeoffset fault. For a given valueof layer thicknessH the
largeoffset fault occursfor smallervaluesof Axesincethe rate
of fault weakeningis inversely proportional to Ax• (Figure
5b). We find that this transition dependson the initial
perturbationsize. The theoretical curvethat best fits the data
is for an initial perturbationof 10% of the thickness of the
layer (Figure5b). However, in an ideal systemwhereonly
infinitely small perturbationsexist, the transition occursfor
The Whipples and other terrestrial core complexes have
been extensively mappedand sampled. However, the severe
erosion and sedimentationaffecting these continental areas
and the fact that they have been inactive for several million
yearscomplicatethe observationsandtheir interpretation. In
fact, no real consensus exists as to whether some normal faults
observedin corecomplexesformedat high angle or originated
with a low dip angle. The detachmentfault of the Mineral
Mountains of southwesternUtah is thought to be a clear
exampleof a normal fault that initially dipped at 60 ø in the
brittle crust [Coleman and Walker, 1994] and then rotated to a
low angle at the surface. In contrast, most workers conclude
that the Whipple detachmentmay have had an original dip of
<35 ø [e.g., Davis and Lister, 1988; Axen andBartley, 1997].
However, even the Whipples remain a controversial area as
someauthorscontendthat the rolling-hinge model explains
some of the major structural features [Hamilton and Howard,
1991] whereas other authors believe that it cannot [Beratan
and Nielson, 1996].
LAVIER El' AL.' FACTORS CONTROLLING
Work over the past 10-20 years on large-offset low-angle
normal faults along the slow spreadingMid-Atlantic Ridge
shedsnew light on this problem. These structuresmay be
currently active and are far less affected by erosion and
sedimentation.As a result,observationsand interpretation of
the first-order topographic features are easier than in the
continental domain. However, detailed mapping is more
difficult than for terrestrial core complexes: it is undertaken
with a combination of sonar imagery, near-bottom
photography,samplingfrom submersibles,anddrilling.
In the oceanic case illustrated in Plate la the dip of the
exposed detachment can vary from horizontal to --30ø or
greaternearthe contactbetweenthe footwall and the hanging
NORMAL
Extension
FAULT OFFSET
at rifts results in subsidence within
23,441
the rift
and
uplift of the rift flanks. The factors that control these basic
componentsof rift expression are poorly understoodat
present.We believe that the partitioning of strain and the
pattern of faulting in a rift dependon a small number of
parametersthat control either localizing or delocalizing
processes.Heat advectionoccurringduring necking of the
lithospherereduces
the strengthof the lithosphereandtendsto
narrowthe width of the necking area. The corresponding
diffusionof heat cools and strengthens
the lithosphereand
tendsto delocalizethe deformationin the upper crust. The
delocalizing
effectof buoyancy
forcesgenerated
by thedensity
differencesfrom surfacetopography and Moho relief is
counteredby lower crustal flow which decreasesrelief and
massifis roughly 30 km [Tucholkeet al., 1998; Blackman et buoyancy forces. The localizing process of cohesion
al., 1998]. Thus, we show our model results for about 30 km of
reduction
of faults[BuckandPoliakov, 1998] is countered
by
horizontalextension(Plate 2c). The shapeand the amplitude viscousstrengtheningin the lower crust, dueto its strain-rate
(2600 m) (Plate 2a) of the topographyis readilycomparableto dependenttheology. Since our model does not take into
the modeledtopography (2100 m) (Plate 2c). As in the model, accountall of theseprocesses,
wedonot comparethe resultsof
inferredhigh-densitylower-crustalmaterialwasuplifted during our modelto currentdatafrom rifts. However, we believe that
exposureof the inactive footwall of the fault [Blackman et al.,
the smalloffsetbehaviorof ourmodel(Plateslb andlc) is the
wall of these faults.
The fault offset inferred for this oceanic
closest to the behavior observed in continental
1998] (Plate 2c).
rifts.
What
The topographicfeaturesat the Whipple Mountains are at
first orderdifferentfrom the modeledtopography. However,
the modeledinactive-footwall dip and offset and the position
of the mylonitic front (Plate 2c) (15 km away from the
controlsthe maximumoffsetin this regimewhenthe effectof
the viscouslowercrustandthe effectof crustalthinning are
includedis still an openquestion.Ebingeret al. [ 1987, 1991]
andScholzandContreras[1998] have shownthat in places
detachment) are all similar to what is observed on the real
suchas the EastAfricanRift, thereis a relationshipbetween
topographicprofiles(Plate2b). Also, filling the topographic the maximumpossibleoffseton a majornormalfault andthe
low with sedimentswouldmake the amplitudeof the modeled effectiveelasticthicknessof the lithosphere. They find that
relief similar to the topographictrendsof the Whippies.
6.2.
Implications
for Material
Parameters
the effectiveelasticthicknessof the lithosphereis larger
wheretheobserved
offsetis the largest. In the future,we will
explorethe effect of viscousstresses,heat transfer, crustal
thinning and erosion and sedimentationin orderto determine
To model the evolution of low-angle normal faults, an the natural condition that should lead to the formation of
important requirementbesidehaving a thin layer, is that the
multiplefaults. Also we will attemptto definethe factors
rateof cohesionreductionwith plastic strain be moderate(Axe
controllingnormalfaultoffsetin settingsuchas narrowand
: 1 to 4 km). Thereis some geologic and laboratory evidence wide rifts.
for rapidlossof cohesion
with strain (i.e., that Ec - 2%
[Scholz, 1990]. For such behavior our results suggestthat
while a large or unlimited offset fault develops and slips,
secondaryfaults form in the hanging wall of the major fault
andaccumulate
slip in a way similar to an asymmetricgraben.
We believethat in ourmodelsucha rapidloss of cohesionwith
strain does not scale with the weakening processesfor the
formationof naturalfaults. If we considerthe simple scaling
relationship betweenthe offset Ax and the length AL of a
natural fault: Ax - 3x10-2 AL [Cowie and Scholz, 1992;
Dawers, 1996], an offset of 1 to 4 km (i.e., a moderaterate of
weakeningin ourmodel)corresponds
to a fault length of 30 to
120 km. Thesevaluesarereasonableandshowthat the scaling
of the weakening processin our model scales is consistent
whatis observedon faults. This may meanthat the formation
of a weaknaturalfaultsoccursonly aftera considerableamount
of damagehas taken place over its width and length. This
complexproblemclearly requiresfurtherstudy. However,we
believe that in our model a moderate rate of cohesion
is the one that scales best with natural faults.
6.3.
Continental
Rifts
reduction
Acknowledgments.
We thankChrisScholz,
MikeSteckler,
Denny
Hayes,BrianTucholke,
andRitskeHuismans
for theirsupport
and
insightful
comments.
We wouldliketo thankCindyEbinger,
John
Hopper,and DavisWatthamfor theirvery usefuland constructive
reviewsof thepaper.Thisworkwassupported
by the U.S. National
ScienceFoundation
grantEAR-9812345anda NSF-CNRScollaborative
grant.LDEOcontribution
6089.AlexeiPoliakov
thanks
theprogramme
INSU "Dynamique
dela fracturation
dela lithosphere."
References
Anderson,
E. M., TheDynamics
of FaultingandDykeFormation,
With
Applications
toBritain,183pp.,OliverandBoyd,WhitePlains,
N.Y.,
1951.
Axen,G. J., andJ. M. Bartley,Fieldtestsof rollinghinges:
Existence,
mechanicaltypes,and implications
for extensional
tectonics,J.
Geophys.Res.,102, 20,515-20,537,1997.
Axen,G. J., andJ. Selverstone,
Stressstateand fluid-pressure
level
alongtheWhippledetachment
fault,California,
Geology,
22, 835838, 1994.
Beratan,
K. K.,andJ.E.Nielson,
Tests
ofdetachment
faultmodels
using
Miocenesyntectonic
strata,ColoradoRiver extensional
corridor,
southeastern
California
andwest-central
Arizona,
inReconstructing
theHistory
ofBasinandRangeExtension
UsingSedimentology
and
Stratigraphy,
editedby K. K., Beratan,Spec.Pap.Geol.Soc.Am.,
303, 171-182, 1996.
Studiesof rifts and continental margins have shown the
D. K.,R.C.Johnson,
B. Janssen,
andD. K. Smith,
Originof
importanceof half grabensas a basic unit in accommodating Blackman,
extensional
corecomplexes:
Evidence
fromtheMid-Atlantic
Ridge
extension. In addition, the asymmetryof faulting in both
at AtlantisFractureZone,J. Geophys.
Res.,103, 21,315-21,333,
1998.
continentalrifts and passive margins has been documented.
23,442
LAVIF__•ET AL.: FACTORSCONTROLLINGNORMAL FAULT OFFSET
Bosworth,W., and C. K. Morley, Structuraland stratigraphicevolution
of the Anza rift, Kenya, Tectonophysics,
236, 93-115, 1994.
Buck, W. R., Flexural rotation of normal faults, Tectonics, 7, 959-973,
1988.
Buck,W. R., Effect of lithosphericthicknesson the formationof highandlow-anglenormalfaults,Geology,21,933-936, 1993.
Buck,W. R., Bendingthin lithospherecauseslocalized "snapping"and
not distributed"crunching":Implicationsfor abyssalhill formation,
Geophys.Res.Letr, 24, 2531-2534, 1997.
Buck,W. R., andA. N. B. Poliakov,Abyssalhills formed by stretching
oceaniclithosphere,Nature,392, 272-275, 1998.
Byeflee, J. D., Frictionof rock, Pure Appl. Geophys.,116, 515-526,
1978.
Cann, J. R., D. K. Blackman, D. K. Smith, E. McAllister, B. Janssen, S.
Mello, E. Avgerinos,A. R. Pascoe,and J. Escartin, Corrugatedslip
surfacesformedat ridge-transform
intersections
on the Mid-Atlantic
Ridge,Nature,385, 329-332, 1997.
Coleman,D. S., and J. D. Walker, Modes of tiltingduring extensional
core complexdevelopment,Science,263, 215-218, 1994.
Coney,P. J., CordilleranmetamorphicCore Complexes:An overview,
Mem. Geol. Soc. Am., 153, 7-134, 1980.
Cowie, P. A., and C.H. Scholz, Physical explanation for the
displacement-length
scalingrelation for faults using a post-yield
fracture mechanics model, J. Struct. Geol., 14, 1133-1148, 1992.
Cundall, P. A., Numerical experimentson localization in frictional
materials,Ing. Arch., 58, 148-159, 1989.
Davis, G. A. and G. A. Lister, Detachment faulting in continental
extension:Perspectivefrom the southwesternU.S. Cordillera, in
Processesin continentallithosphericdeformation,edited by S. P.
Clark,B.C. Burchfiel,andJ. Suppe,Spec.Pap. Geol. Soc.Am., 218,
133-159, 1988.
Dawers, N.H., Observationsof fault growth scaling relations,Ph.D.
thesis,113pp.,ColumbiaUniv., New York, 1996.
Ebinger,C. J.,B. R. Rosendahl,
andD. J. Reynolds,
Tectonicmodelof
theMalawi Rift, Africa, Tectonophysics,
141,215-235, 1987.
Ebinger,C. J.,G. D. Karner,andJ. K.Weissel, Mechanicalstrength
of
extendedcontinentallithosphere:
Constraints
from the westernrift
system,EastAfrica, Tectonics,
10, 1239-1256,1991.
Forsyth,
D. W., Finiteextension
andlow-anglenormalfaulting,Geology,
20, 27-30, 1992.
Gerbault,F., A. N. B. Poliakov,andM. Daigni•res,Predictionof faulting
from the theoriesof elasticityand plasticity:What are the limits?,J.
Struct. Geol., 20, 301-320, 1998.
Lister,O. A., M. A. Etheridge,and P. A. Symonds,
Detachmentmodels
for theformationof passivecontinental
margins,Tectonics,
10, 10381064, 1991.
MacDonald,K. C., andB. Luyendyk,Deep-Towstudies
of the structure
of the Mid-AtlanticRidgenear 37øN (FAMOUS), Geol .Soc.Am.
Bull., 88, 621-636, 1977.
Melosh,H. J., andC. A. WilliamsJr.,Mechanicsof grabenformationin
crustalrocks:A finite elementanalysis,J. Geophys.Res.,94, 13,96113,973, 1989.
Morley,C. K., Developments
in the structural
geologyof riftsover the
last decade and their impact on hydrocarbonexploration,in
Hydrocarbon
HabitatinRiffBasins,
editedbyJ.J.Lambiase,
J. Geol.
Soc.LondonSpec.Publ., 80, 65-77, 1995.
Morley, C. K., S. M. Cunningham,
R. M. Harper, and W.A.Wescott,
Geologyand geophysics
of the Rukwarift, EastAfrica, Tectonics,
11, 69-81, 1992.
Parsons,
T., andG. A. Thompson,
Doesmagrnatism
influencelow-angle
normalfaulting?,Geology,21,247-250, 1993.
Patton,T. L., A. R. Moustafa, R. A. Nelson and A. S. Abdine, Tectonic
evolution
andstructuralsettingof theSuezrift, in InteriorRiff Basins,
editedby S.M. Landon,AAPGMem., 58, 9-55, 1994.
PoliakovA. N. B., and W. R. Buck, Mechanics of stretchingelasticplastic-viscous
layers: Applicationsto slow-spreading
mid-ocean
ridges,in Faultingand Magmatism
at Mid-oceanRidges,Geophys.
Monogr.Ser.,vol. 106, editedby W. R. Buck et al., pp. 305-325,
AGU, Washington,D.C., 1998.
Poliakov,A. N. B., and H. J. Herrmann, Self-organizedcriticality in
plasticshearbands,Geophys.
Res.Lett.,21, 2143-2146,1994.
Poliakov, A. N. B., Y. Podladchikov, and C. Talbot, Initiation of salt
diapirs with frictional overburdens-numericalexperiments,
Tectonophysics,
228, 199-210, 1993.
Roberts,A.M., G. Yielding,andB. Feeman(Eds.), TheGeometryof
NormalFaults,Geol.Soc.Spec.Publ.,56, 61-78, 1991.
Scholz,C. H., The Mechanicsof Earthquakes
and Faulting,439 pp.,
CambridgeUniv. Press,New York, 1990.
Scholz, C. H., and J. C. Contreras, Mechanics of continental rift
architecture,Geology,26, 967-970, 1998.
Stein,R. S., G. C. P. King, andJ. B. Rundle,The growthof geological
structures
by repeatedearthquakes,
2, Fieldexamplesof continental
dip-slipfaults,J. Geophys.Res.,93, 13,319-13,331,1988.
Tucholke,B. E., J.Lin, andM. C. Kleinrock,Megamullions
and mullion
structure
definingoceanicmetamorphic
corecomplexes
on the MidAtlanticRidge,J. Geophys.Res.,103,9857-9866,1998.
VeningMeinesz,F. A, Les grabensafricainsresultants
de compression
oude tensiondansla crofiteterrestre?,
Inst.R. Colon.BelgeBull.,21,
Hamilton,W., Extensional
faultingin the deathvalley region(abstract),
Geol. Soc.Am. Abstr.Programs,20, 165-166, 1988.
Hamilton, W. G., and K. A. Howard, Detachment faults-Steepactive
539-552, 1950.
segments
are boundedby migratinghingesat bothtopsand bottoms Weissel,J. K., and G. D. Karner, Flexuralupliftof rift flanks due to
(abstract),Geol. Soc.Am. Abstr.Programs,23, 189-190, 1991.
tectonic
denudation
of the lithosphere
duringextension,
J. Geophys.
Handin,J., Strengthand ductility,in Handbookof PhysicalConstants,
Res., 94, 13,919-13,950, 1989.
editedby S.P. ClarkJr.,Mem. Geol.Soc.Am.,97, 223-290, 1966.
Wernicke, B., Uniform-sensenormal simpleshear of the continental
Hassani,R., and J. Chery,Anelasticityexplainstopographyassociated
lithosphere,Can.J. Earth Sci.,22, 108-125, 1985.
with Basin and Range normal faulting, Geology, 24, 1095-1098, Wernicke,B., Low-anglenormalfaultseismicity:
A review,J. Geophys.
1996.
Hobbs,B. E., and A. Ord, Numerical simulationof shearband formation
in a frictional-dilatational
materials,Ing. Arch., 59, 209-220, 1989.
Jackson,J. A., Active normal faulting and crustal extension, in
ContinentalExtensionalTectonics,edited by M.P. Coward, J. F.
Dewey, andP. L. Hancock,Geol.Soc.Spec.Publ.,28, 3-17, 1987.
Karson,J. A., et al., Along-axisvariationsin seafloorspreadingin the
MARK area, Nature, 328, 681-685, 1987.
Kusznir,N.J., G. D. Karner and S.S. Egan, Geometric,thermal and
isostaticconsequences
of detachmentsin continentallithosphere
extensionand basin formation.in SedimentaryBasins and Basin
FormingMechanisms,
editedby C. Beaumontand A. J. Tankard,
Mem. Can. Soc. Pet. Geol., 12, 185-203, 1987.
Lavier, L. L.,W. R. Buck, and A. N. B. Poliakov, A self consistent
rolling-hingemodel for large-offset low-angle normal faults,
Res., 100, 20,159-20,174, 1995.
Wernicke,B., andG. J.Axen,On theroleof isostasy
in the evolutionof
normalfault systems,
Geology,16, 848-851, 1988.
Wills, S., and W. R. Buck, Stress-field rotation and rooted detachment
faults:A Coulombfailureanalysis,
J. Geophys.Res., 102, 20,50320,514, 1997.
Yin, A., Origin of regional, rooted low-angle normal faults: A
mechanicalmodeland its tectonicimplications,
Tectonics,8, 469482, 1989.
A. N. B. Poliakov,Laboratoirede C&ophysiqueet T ectonique,
CNRS UMR 5573, Universit• de Montpellier II, F-34095 Montpellier
Cedex05, France.(aljosha@dstu.univ-montp2.
fr)
L. L. Lavier, GFZ Potsdam, Albert-Einstein-Strasse, Haus C426,
Geology, 24, 561-564, 1999.
Telegrafenberg,D- 14473Potsdam,
Germany.(lucl@gfz-potsdam.de)
Lister, G. A., and G. A. Davis, The origin of metamorphiccore
W. R. Buck, Lamont-DohertyEarth Observatoryof Columbia
complexes
anddetachment
faultsformedduringTertiary continental
NY, 10964.(buck@ldeø'cølumbia'edu)
extensionin the northernColoradoRiver region, U.S.A., J. Struct. University,Palisades,
Geol., 11, 65-94, 1989.
Lister,G. A., M. A. Etheridge,andP. A. Symonds,
Detachmentfaulting
andtheevolutionof passivecontinentalmargins,Geology,14, 246250, 1986.
(ReceivedAugust18, 1999;revisedFebruary21, 2000;
acceptedMarch 29, 2000)