Mechanics of extension and inversion in the ... - Wiley Online Library

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. Bll, PAGES 26,655-26,670,NOVEMBER 10, 2001

Mechanics of extensionand inversion in the hanging walls of listric

normal

faults

S.GreggErickson, • LutherM. Strayer, 2 andJohnSuppe Princeton3D StructureProject,Departmentof Geosciences, PrincetonUniversity,Princeton,New Jersey,USA

Abstract. Fault shape,materialpropertiesandbeddinganisotropydeterminethe

styleof deformation in thehangingwallsof listricnormalfaults.We usenumerical modelsto studythisdeformationin bothextensionandinversionduringdisplacementon a varietyof masterfault shapes.Elastic-plastic materialpropertiesin the

modelsallowthedevelopment of shearbands,whichsimulate secondary faults withinthe hangingwall. If the masterfault is composed of a planarrampandplanar flat separated by a sharpfault bend,a seriesof antitheticnormalshearbandsdevelopsin the hangingwall, propagating up from the fault bend. Eachshearbandis progressively abandoned asdisplacement on the masterfaultmovesit awayfromthe fault bend. Displacements on the antitheticfaultsproducethe limb of a hangingwall monoclineandboundonesideof a graben,the othersideof which is boundedby the masterfault. Theantitheticshearbandsarenotrotated,andlayeringwithinthegraben remains subhorizontal. On the other hand, if the fault bend is curved rather than

sharp,symmetricalnestedgrabendevelopin the upperpartof the hangingwall abovethe baseof the ramp.Displacementon the masterfault movestheseshear bandsaway from the fault bend,after which they are abandonedin favor of new shearbands. Early formedsyntheticshearbandsbecomeshallowerand concave upwardbecauseof the foldingandrotationof the hangingwall. With increasingradius of curvatureof the masterfault, localization into shearbandsdecreasesand,

with a largeradiusof curvature,shearbandsdo not develop. If weak beddinglayers are includedwithin the hangingwall, they becomesitesof bedding-parallelshear bandsthat accommodate flexuralslip foldingandreplacethe syntheticshearbands thatdevelopin homogeneous models. If extensionis followedby shorteningandinversion,normalshearbandsthat developedduringextensionarereactivatedas reverseshearbandsbut are alsocrosscutby new reverseshearbands. The models produceresultsthat are similarto bothnaturalstructures andanaloguemodelsand provideexplanations for manyobservations of deformationin seismicprofiles throughextensionalterranes. 1. Introduction

ure la). Folding of the hangingwall by rollover leadsto distinctivesequences of growth strata,determinedby fault shapeas well as the relative rates of fault displacement and sedimentation[Xiao and Suppe, 1992]. Kinematic modelsof extensionalterranes incorporate assumptionsthat are meant to correspondto appropriatedeformationmechanisms. The assumption of bulk simpleshear,alongvertical planes[Gibbs,1984]or alonginclinedplanes[14/hite et al., 1986; Rowan and Kligfield, 1989; Dula, 1991], approximatesantithetic or syntheticfaults orientedparallel to shearplanes.In these models, domainsof bulk simpleshearcan be separatedby fault discontinuities [Nunns, 1991; Schultz-Ela, 1992]. 14/altham [1989, 1990] hasusedthe finite differencetechniqueto modelvelocitiesin the hanging wall of normalfaults for given displacement directions. In contrastto inclined or vertical shear, the assumptionof constantbed length [Davison, 1986; Xiao and Suppe,1992] impliesthat slip alongbedding surfaces is the dominant deformation mechanism.Seismic profiles through extensional

Normal faultsin extensionalterraneshave planar, listric, or ramp-flat trajectories,and the shapesof thesefaultshelpto determinethe natureof deformation in their hanging walls [Bally et al., 1981; Shelton1984; 14/illiamsand Vann, 1987]. Physical modelsof listric normal faults [Ellis and McClay, 1988; 14/ithjack et al., 1995;McClay, 1996] displaya distinctivegeometryand sequenceof hangingwall deformation,includingan early formed crestalcollapse graben and normal faults with orientations bothantitheticandsyntheticto the masterfault (Fig-

•Nowat Science andMath,SullivanCountyCommu-

nity 2College, Loch Sheldrake, New York, USA. Now at Departmentof Geology,CaliforniaStateUniversity,Hayward,California,USA. Copyright2001 by AmericanGeophysicalUnion Papernumber2001JB000245. 0148-0227/01/2001JB000245

$09.00

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terranes may exhibit a combination of antithetic faults, syntheticfaults, or bedding slip (Figures l c and l d). In some seismicprofiles, antitheticfaults have beeninterpreted[Tearpockand Bischke,1991] to merge into beddingsurfacesalong the axial surface of the rollover anticline(Figure l d), indicating that bedding slip is an important mechanismin steeplydippingbedsand replacessyntheticfaulting if beddinganisotropyis strong. Shorteningcommonly follows extension,giving rise to inversion.During inversion,normalfaultsare reactivated as reverse faults, and new reverse faults

develop[Wanget al., 1995; Hill et al., 1995]. Only some of the normal faults in inverted basins are re-

activated, which may result from preferred fault orientation,fault zone weakening,or heterogeneous fluid pressure[Sibson, 1995]. Physical modeling indicatesthat reversefaults propagatefrom the upper tips of preexistingnormal faults (Figure lb) [Buchananand McClay, 1991; McClay, 1995] or crosscutnormal faults insteadof reactivatingthem [Eisenstadtand Withjack,1995]. Weak beddinglayers influence the structuralstyle of inversionby acting as detachments [Bishop and Buchanan, 1995]. In this paper, we presentnumericalmechanical modelsthat simulatethis full range of deformation mechanismsabovelistric faultsduringextensionand subsequentinversion. The models use an elasticplasticmaterial for the hangingwall, which allows the localization

of deformation

into shear bands and

simulatesfaulting. We investigatethe influenceof fault shape,specificallyradius of curvatureof the ramp, on deformationin the hangingwall. We also investigatethe role of stratigraphiclayering in determiningthe styleof hangingwall deformationand follow the sequenceof deformationduringextension and subsequent shorteningand inversion. The models explain many features that are observed on seismic profiles through extensionalterranes, including nested graben, progressivelyolder graben toward the basin, and antithetic faults whose dis-

placementis transferredto beddingslip surfacesat the axial surface of the rollover anticline.

2. Method

We use Fast LagrangianAnalysisof Continua (FLAC) [Cundall and Board, 1988; Coetzeeet al., 1995], a two-dimensional,explicit finite difference

codethathasbeenusedto modelgeologicstructures [Riley, 1996;Strayerand Hudleston,1997;Erickson et al., 2001]. The explicit finite differencemethod differs from finite elementmethodsin that grid points are numericallyassociatedonly with their immediateneighbors;there is no global stiffness matrix to invert. The finite difference

method is

computationally fasterthanothermethods,although time stepsmustbe smallenoughto ensurenumerical convergence[Cundall and Board, 1988]. Coordinate positionsare updated using displacements calculated fromthe previoustime step,andthe grid is displacedalong with the materialit represents.

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Becausethe dynamic equationsof motion are included in the formulation,the numerical schemeis stable,evenif unstablephysicalprocesses are modeled. The simulations are quasistatic; accelerations

from unbalancedforces are dissipatedthrough damping[CundallandBoard,1988]. The mechanical behavior of the material is elas-

tic-plastic. For the constitutiverule we use Mohr-

Coulombplasticity,whichconsists of a yield function anda plasticflow rule. The yield function f, whichgovernsthe onsetof plasticbehavior,is f= • + osin• - Ccos•

(1)

where • is the maximumshearstress,cris the mean stress,• is the internalfrictionangle,and C is the

cohesion.The plasticpotentialfunctiong, which governsplasticflow, is

g = • + osin•'- Ccos•'

(2)

where•' is the dilationangle.Tensionis positivein (1) and(2). The dilationangleis theratioof plastic volumechangeover the plasticshearstrain[Vetmeet and de Borst, 1984] and is positive for a volumeincrease with increasing plasticshearstrain. The valuesof cohesion, internalfrictionangle,and dilationanglemay hardenor soften(increaseor decrease)as a functionof increasingplastic shear strain.Unequalfrictionanddilationanglesresultin nonassociated plasticity,in which the plasticflow rule (equation(2)) is not associated with the yield function(equation(1)) [Vetmeet and de Borst, 1984]. For upper crustalgeologicmaterials,in whichplasticbehavioris frictionalanddilatant,valuesof thedilationanglegenerallyarelessthanthose of theinternalfrictionangle[Cundall,1990]andare between 10øand20ø[Vermeer anddeBorst,1984].

Withnonassociated plasticity, localization of deformation is possible[Rice,1976;Cundall,1989; Hobbsand Ord, 1989]. Strainmay be accommodatedeitherby distributed deformation or by initiationof shearbands.Localization mayoccur evenwithouta strain-softening constitutive relation, providedthatthe dilationangle• is lessthanthe internalfrictionangle• [Rudnicki andRice,1975; Vardoulakis, 1980]. Localization is griddependent; the shearbandthickness depends on the meshsize [BazantandBelytschko, 1987;Needleman, 1988]. Therefore, in order to observedetails of shearband

development a fine grid is needed.Localizationof

deformation intoshearbands simulates uppercrustal faulting.However,thedevelopment of a shearband doesnot introduce any discontinuity into the numericalmodel; all initially continuous regions remaincontinuous throughout a simulation.In ad-

ditiontothelocalized shear bands thatmaydevelop withinthe continuum, discreteinterfaces, along which slip occurs,can be built into the model. In

ourmodels,themasterfaultis thistypeof interface and is assignednormal and shearstiffnessesand a coefficientof slidingfriction.

We usethe followingmaterialproperties for the hanging wall:Elasticshearmodulus G = 2.1 GPa, elasticbulkmodulus K = 2.1 GPa,density p= 2600

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ERICKSONET AL.: MECHANICS OF EXTENSION AND INVERSION to 10 MPa at an equivalent plastic strain of 0.2, above which it remains at 10 MPa.

These material

propertiesare based on experimentalvalues for sandstone[Edmondand Paterson, 1972]. However, the cohesion in our models is ---50% weaker than ex-

Figure 2. Model geometry and boundaryconditions. A velocity boundaryconditionis applied to the left side of the hangingwall. The top surfaceis stressfree. The footwallhaszero horizontalvelocity on the sidesand zero verticalvelocityon the bottom. Varying the radiusof curvaturer createsthe different fault geometryin eachmodel. Elementson the left side of the hangingwall are progressivelydeletedasthey reachthe left edgeof the footwall.

kg/m 3, tensile strength T- 10MPa(formostmodels), friction angle 4=23 ø, and dilation angle q/= 11ø. The friction angle is constantas a function

of meanstress,andthe linearshearandtensilepans of the envelopintersectat a tensilestressof 10 MPa. Cohesionsoftenslinearly from an initial value of 15

perimentalvaluesin orderto accountfor scale,pore pressure,and incomplete lithification. Reduced strengthpropertiesare generally necessaryto produce realistic large-scale structuresin numerical mechanicalmodels and is to be expected when scalingpropertiesthat are derivedfrom centimeterscale experiments to kilometer-scale geological structures[Schultz,1996]. The modelsalsoprovide a limit to the range of values that can be used for material properties;gaps open along faults if the materialis too strong.In the modelspresentedhere, somegapsopenalongfaultsfor modelswith sharp fault bendsbecausethe materialis too strongto conform to the fault shape.A weaker material could havebeenusedfor thesemodels,but the properties for all simulationswere kept the same for consistency. Althoughit would be possibleto model the

Figure3. Elasticmodels, displaying distributions of differential stress (grayscalecontours, contour inter-

val= 25 MPa)andmeanstress (solid-line contours, contour interval - 25 MPa,tension ispositive). (a) Sharp faultbend. (b)Faultbend withr - 2 kmconnecting linearflatandrampsegments. Thelinear partof theramp hasa60ødip.(c)Ramp withr - 2 kmanda90ødipatthesurface. (d)Ramp withr - 4 kmanda 60ødipatthesurface. (e) Rampwithr - 6.8 km anda 45ødipatthesurface.

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hardeningandsofteningbehaviorsof cohesion,friction, anddilationon the basisof experimentalresults in an attemptto be realistic[e.g., Ord, 1991], we have chosento use a simple softeningrule. The only softeningbehaviorwe presentis a linear softening relationship for cohesion with increasing plasticstrain, althoughtestswith more complicated material behaviorsgive similar results.We do not considerthe effects of compactionor isostasyin these models.

The models are plane strain and consist of a hangingwall with elastic-plasticmaterialproperties and a footwall with elastic properties(Figure 2). The hangingwall is movedover the footwall by a constanthorizontalvelocity of 2 cm per time step, appliedat the left sideof the hangingwall, for a total displacement of 2 km. At the left sideof the model, hangingwall elementsare deletedas they reachthe left edge of the footwall. The hangingwall is initially 2 km thick and 10 km long. Hanging wall elementsare 25 m on a sidein the part of the hanging wall that undergoessignificantdeformationand 100 m on a sidein otherpartsof the model. There are -15,000 elementsin each model. A fine mesh of

elements is used in order to study in detail the hangingwall deformationby allowingthe development

of small-scale

shear bands.

The

use of a

cohesion-softening material enhancesthe development of these shear bands. A gravitational body force is appliedto eachelement,and the top surface is stress free.

In the models, we assume that the

fault is a frictionlessinterface (/a = 0ø) in order to studythe effectsof fault shapeon hangingwall deformation. However, friction on the master fault is

likely to be importantin determiningthe characterof hanging wall structures.For example, a frictional masterfault would causefaults in the hangingwall to propagatedown to the master fault and would producemore internalextensionwithin the hanging wall. Friction along the masterfault alters the orientation of hanging wall shear bands near thrustfault ramps [Ericksonet al., 200!]. In the current

Figure 4. Principalstresses in elasticmodels.See Figure3 for modelgeometries. Maximumstressis 350 MPa. Regionsof tensilemeanstressareshaded gray andtensileprincipalstresses are shownwith thicker lines.

models, the master normal fault either consists of

planar flat and ramp segmentsseparatedby a sharp fault bendor of a combinationof planar and arcuate segments.Arcuatefault segmentswith differing radii of curvature

are used to create different

fault

geometries.In this paper,the horizontalpart of the fault is referredto asthe flat, andthe dippingpart of the fault is referredto asthe ramp. 3. Results

vature2 km (Figure 3b). (3) An arcuatesegment with radiusof curvature2 km dips90ø at thetop surface(Figure3c). (4) An arcuatesegmentwith radius of curvature4 km dips60ø at the top surface(Figure 3d). (5) An arcuatesegmentwith radius of curvature6.8 km dips45ø at the top surface(Figure3e). 3.1.1. Elastic models. As a first step,we investigate elasticmodelsat small displacements (20 m), which allows observation of the stress field and in-

3.1. Influence of Fault Shape

In orderto investigatethe effectsof fault shape on hangingwall deformationwe use five different fault geometriesin the models (Figure 3). In all modelsthe fault containsa horizontalflat segment. The ramp segmentconsistsof one of the following geometries:(1) A planar segmentdips 45ø (Figure 3a). (2) A planar segmentdips 60ø and is connected to the flat by an arcuatesegmentwith radiusof cur-

sight into fault initiationwithoutthe complications of plasticityand shearbanddevelopment.However, stresses from the elasticregimecannotbe extrapolatedinto the plasticregimebecausethe introduction of faultschangesthe stateof stress[Gerbaultet al., 1998]. For all fault shapes,compressive meanstress is highestalongthe baseof the hangingwall, above the flat andju.stto the left of the baseof the ramp (Figure3). Tensile(positive)meanstressdominates

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Mohr-Coulomb

Failure Criterion

Figure 5. Mohr-Coulombfailure criterionappliedto elasticmodels,showingthe regionsof likely fault initiation.See Figure3 for modelgeometries.Contoursof the largestcohesionfor which shearfracture initiatesat that stateof stressasgivenby (3), which indicateregionsof mostlikely shearfracture(contour intervalof 25 MPa). A frictionangleof 23ø is usedto coincidewith thatusedin the elastic-plastic models.

the upper--500 m of mostof the hangingwall. The maximum tensile mean stressis located at the top surfaceand abovethe flat, just to the left of the base of the ramp. Differential

stress in each elastic model has two

maxima,one at the baseof the hangingwall andjust to the right of the maximum in compressivemean stress,andthe otherat the top surfaceand coincident with the maximum

in tensile mean stress.

With

a

sharpfault bendat the baseof the ramp(Figure3a), the upper maximumin differentialstressis small, and the distributionof differentialstressis asymmetric, forming an antitheticband of high differential stress. In elastic-plasticmodels this band of high differential

stress results in a series of antithetic

shearbands. With increasingradiusof curvature, the maxima in differential stressbecomeseparated and are no longerconnectedby a bandof high differentialstress.With increasingradiusof curvature, the distribution

of differential

stress also becomes

more symmetric. In elastic-plasticmodels this symmetricdistributionof differentialstressproduces symmetricgraben. With high radiusof curvatureof the ramp, the maximumin differentialstressat the base of the hanging wall becomestwo distinct

maxima,andwith a radiusof rampcurvatureof 6.8 km (Figure3e), the maximaare symmetrically disposedabout the maximumin compressive mean stress.

Principalstresses in theelasticmodelsarehighest at thebaseof therampandlowalongtheramp(Figure 4). Orientations of maximumcompressive stress radiatefromthe baseof the rampandareparallelto thefaultalongmostof itslength.In theupperpartof the hangingwall, maximumtensilestresses are subhorizontal.

A Mohr-Coulomb fracture criterion can

be appliedto theseelasticmodelsin order to deter-

minethe regionsin whichfractures are mostlikely to initiate. Regionsof likely shearfractureare contouredon the basisof the largestcohesionat which shearfracturewill initiatefor a given frictionangle and stateof stress,where

C=sec • crxx -2 Cry.),, +CrxY 2 2

/

determinesthat cohesionfor a linear failure enve-

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Strain

Figure 6. Distributionof shearstrainin elastic-plastic modelswith a varietyof masterfault shapes.Shear strainin Figures6, 8, 10, 11, 13, and 14 is measuredin a coordinateframe that moveswith the mesh.Dis-

placementin eachmodelis 2 km, exceptfor Figure6a, whichhasa displacement of 1.2 km. Figures6a-6e havethe samefault shapesasthosein Figure3. Numberscorrespond to timing of faults;1 is the oldestand 3 is the youngest.

lope [Ericksonand Wiltschko,1991]. This fracture criterion(Figure 5) is a functionof both the mean and differential stress (Figure 3). For all fault shapesthe site of most likely fractureis at the top surfaceand just to the left of the base of the ramp. There is also a band of likely fractureextendingup from the baseof the ramp. This bandis bestdefined for a sharpfault bendat the baseof the ramp and re-

stressand low mean stress(Figures 6 and 7). Becausethe material has a tensile strengthand because the upperpart of the hangingwall is in tension(Figure 4), there is a region near the surface that undergoestensile failure (Figure 6). This region of tensilefailure is 650-850 m deepif T = 10 MPa and increasesin thicknesswith increasingradius of cur-

sults in antithetic

become less important. In the model with a sharp

shear bands

if the

material

is

elastic-plasticinsteadof elastic. The regionof likely fractureat the top surfaceis greatestfor higherfault curvaturesbecausedifferential stressis higher and mean stressis more tensile(Figure 3). 3.1.2. Elastic-plastic models. With a frictional, dilatant, cohesion-softening, elastic-plasticmaterial, shear bands develop in regions of high differential

vature of the fault, for which localized shear bands fault bend the shallow-level

band of tensile failure

is

less well developed. The thicknessof the zone is also a functionof the tensilestrengthof the material (Figure 8). If the tensile strengthis high enough, there is no tensile failure, and all deformation is ac-

complishedby shearbands(Figure 8a). With a high tensile strength,the shearbandsreach the top sur-

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?

l•/q [ Illl •ll Illl

III

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II

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faceof the modelto producefault scarps.Although thesefault scarpsmay be realistic,the high shear strainswhereshearbandsmeetthe top surfacecause elementalcollapsein theseregions.It is therefore necessary to includea lower tensilestrengthto produce a region of tensile failure at shallowlevels (Figure8b) eventhoughthis tensileregionmay be unrealisticallythick. In the modelwith a sharpfault bend(Figures6a and7a), antitheticnormalshearbandsdevelopin the hangingwall. Theseshearbandspropagate up from the concentration in differential stress at the fault

bend and are initiated and then abandonedas they moveover andaway from the fault bend. A similar

process occursin materialas it movesovera thrust-

c

fault bend [Ericksonet al., 2001]. These antithetic shear bands form one boundaryof a graben;the other boundaryis formed by the master fault. The spacingof shearbandsin thesemodelsis determined by the friction and dilation anglesbut also by the mesh size relative to the model dimensions[Ord, 1990]. Althoughthe meshmay influencethe orientations of shear bands [Tvergaard et al., 1981; Artnero and Garikipati, 1996], shearbandsdo not developstrictlydiagonallyto the meshand are controlledby the frictionanddilationangles. The angle betweenthe conjugateshearbandsthat bound the grabenis-77 ø (Figure 8a), which is closeto the 75ø predictedfor the friction and dilation anglesused here,with the relationshipdescribedby Vardoulakis [1980]. The hangingwall fold that developsin this model is monoclinal,with an approximatelyplanar dippinglimb formedby the antitheticfaults, which separatestwo subhorizontallimbs. In modelswith arcuateramps(Figures6b-6e and 7b-7e), shear bands develop below the zone of shallow-level

tensile failure.

Because of the smaller

concentration

in differential

stress at the base of the

ramp in thesemodels,shearbandsdo not propagate up from the baseof the rampbut insteaddevelopas symmetricgrabenabovethe baseof the ramp and in the stratigraphicmiddle of the hanging wall. The graben boundedby antithetic and syntheticshear bandsare displacedto the left with increasingdisplacementon the master fault, and as they move away from the fault bend,they are abandonedin favor of new shear bands, which form similar

symmetricalgraben. The newly formedshearbands crosscutpreviouslyformedbands. With increasing radius of curvatureof the ramp, the stressconcentration at the base of the ramp becomeslower in magnitude,and there is less localizationinto shear bands.

Figure 7. Deformedgrids of elastic-plasticmodels with a varietyof masterfault shapes.Models in Figures 7a-7e are the same as those in Figures 6a-6e. Shadedlayersarepassivemarkerlayers.Contoursof shearstrainare alsoshown(contouris 0.4).

With

a fault radius of curvature

of 6.8 km

(Figure 6e), there is no localizationof strain into shearbands.With a ramp that is partly planar and partly arcuate(Figure 6b), a small grabendevelops at the toe of the hangingwall, with the masterfault as oneboundingfault andan antitheticshearbandas the other. Faultswith planarramps(Figures6a and 7a) producemonoclinalhangingwall folds,whereas faultswith arcuateramps(Figures6c-6e and 7c-7e) producerounded,convexupward folds. Thesefault

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Shear Strain

Figure 8. Shearstrain(contourintervalof 0.5) in modelswith differenttensilestrengthT. (a) Plot showing thatwith T = 15MPa, shearbandsreachthetop surface.(b) Plotshowingthatwith T - 10 MPa, thepartof the hangingwall nearthe top surfaceundergoestensilefailure and shearbandsare restrictedto lower levels.

patternsandfold geometries are similarto thoseobservedin seismicprofilesandanaloguemodels.

uniform materialproperties,a more roundedrollover anticlinedevelopsas a resultof the bedding-parallel shear,althoughthe limb dips are approximatelythe

3.2. Stratigraphic Layering

same for both models.

In the model with stratigraphiclayering (Figure 9), therearethreeweaklayers,eachoneelement(25 m) thick, initially at depthsof 500, 1000, and 1500 m. Cohesionin theselayersis initially 6 MPa, softeningto 3 MPa, andtensilestrengthis 5 MPa. The remainderof the hangingwall hasthe sameproperties as other models,with cohesioninitially 15 MPa

softeningto 10 MPa andtensilestrengthof 10 MPa. The fault geometryin this model is the same as that in Figure 6b. In this modelwith weak layers,syn-

thetic faults do not develop. Instead, the weak layersare sitesof bedding-parallel shear. Although two antitheticshearbandsdevelop,they have lower shearstrainand lessdisplacementthan shearbands in the modelwith uniformmaterialproperties.Sense of shearon bedding-parallel shearbandsis top to the right in mostlocations,althoughthe top weak layer hastop to the left shearin the steeplydippingbeds near the fault. This bedding-parallelshearaccommodates flexural slip folding of the rollover anticline. Slip on the bedding-parallelshearbands increases with increasingstratigraphic heightin the hangingwall, and,in general,with increasingdip of the hangingwall layers. Comparedto modelswith

3.3. Sequential Development

By followingthe sequentialdevelopmentof the model in Figure 6b the timing of initiation and growth of faults can be determined. Initially, a symmetricgraben and smaller nested graben develop abovethe fault bend (Figures10a and 11a). The shear bands boundingthe graben meet at a common point-100 m above the master fault, and

each boundingshearband extendsupward to the base of the zone of tensile failure.

Antithetic

and

syntheticshearbandsare activesynchronously, and all shearbandsof a nestedgrabenare active simultaneously. With continued displacementon the master fault, new graben develop, and antithetic shearbandscrosscut syntheticfaults(Figure11d). A smallgrabenboundedby the masterfault and an antithetic shearband also developsat the toe of the hangingwall. As the boundingshearbandsof the largergrabenare carriedto the left by displacement on the master fault, the antithetic shear band that boundsthis grabenis abandonedin favor of a new antitheticshearband,while the syntheticshearcon-

tinuesto be active (Figures10b and l lb).

With

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Shear Strain

Deformed

Grid

......: ........::.

IIIIIII1111111111111111111'111 b Figure 9. Model with weak layersto representweak beddingunits.The fault geometryis the sameas that in Figure6b. Weak layersaccommodate layer-parallelshear,whichreplacesthe syntheticfaultsobserved in Figure6b. Displacement of themodelis 2 km. (a) Contoursof shearstrain(contourintervalof 0.25). (b) Deformedgrid. Weak layersareshaded.Contoursof shearstrainarealsoshown(contouris 0.4).

continueddisplacement,the synthetic shear band shallows, becomesconcave upward, and becomes inactivebecauseof folding of the hangingwall during movementover the listric masterfault (Figures 10c and 1l c). New syntheticshearbandsdevelop and curve into the overlying,preexistingsynthetic fault, and antitheticfaults crosscutthe preexisting syntheticfault (Figures10d and 1ld). Someold antithetic faults are reactivatedat late stagesafter a period of inactivity (Figure 11d). Thereforesome early formed shearbandsare usedfor long periods, whereasothersare abandonedandcrosscutby newly formedshearbands. There is a generalprogression of oldestto youngestfaults from left to right in the hangingwall as faultsdevelopabovethe baseof the ramp and then are carriedaway from the baseof the rampby displacementon the masterfault. 3.4. Inversion

Reversingthe drivingvelocity on the left side of the hangingwall, so that the velocity is directedto the right,producesinversion.This boundarycondition causeshanging wall deformationto change from subhorizontal

extension to subhorizontal

short-

ening. In an elasticmodelof shorteningwith a fault shapethe sameas that in Figure3b, high differential stressis concentratedmidway up the ramp adjacent to the fault and at the top surfaceabovethe base of the ramp (Figure 12a). Compressivemeanstressis

concentratedin thesesameregions,whereastensile mean stressis highestalong the fault at the base of the ramp. Becauseof this stressdistribution,regions of most likely fault initiationare at the top surface abovethe baseof the ramp,alongthe fault abovethe baseof the ramp,and alongthe fault midway up the ramp(Figure 12b). During inversionof the model in Figure 10d, preexisting faults that developedduring extensionare either

reactivated

as reverse

shear bands or are

crosscutby new reverseshearbands(Figures13 and 14). Becausethe shear bandshave been weakened by cohesionsoftening from an initial value of 15 MPa to a minimumvalue of 10 MPa during extension, they are likely sites of reactivationduring inversion. Becauseor'the new stateof stressduring inversion(Figure 12), the toe and baseof the hanging wall becomesitesof deformation.Theseregions were left relatively undeformedduring the extensionalphaseof the deformation. The upper part of the hanging wall, which underwenttensile failure during extension,developsthrust faults during inversion.

Antithetic

normal faults are reactivated and

propagateto the surfaceas reversefaults. During inversion, shorteningis accompaniedby higher mean stressrelative to the extensionalphaseof the deformation. The higher mean stressresultsin no zone of tensile failure during this phase,and shear bandsreach the top surface. The reversesenseof offset on the shearbandscan be seenby the offset of

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strongerin shortening,moreof the displacement is accommodated by displacementon the masterfault

duringshortening thanduringextension. Dilatancy of---15%duringextension and shortening alsodecreases the extensional heave and increases the

shortening heave.Thetopof thehangingwall hasa heaveof 0.95 km during2 km of extensionon the

left side.Duringinversion, 2 km of displacement on theleft sideof thehangingwall produces 1.35km of heaveof the top of the hangingwall, so that it fin-

Shear Strain

Incremental

..-...:.:-7:.T.:.-:-;.•..:-.:•:•:•:..-::-:•..-..-•..--...... ........................... :.:...•

d

Figure 10. Distributionof cumulativeshearstrain (contourinterval of 0.25) during sequentialdevelopment of the model in Figure 6b. Symmetrical grabenform abovethe baseof the rampand are progressivelyabandoned as displacement on the master fault movesthem away from the base of the ramp. Four stagesof developmentare shown,at displacementsof (a) 0.5 km, (b) 1 km, (c) 1.5 km, and (d) 2 km. Numberscorrespond to timing of faults; 1 is the oldestand3 is the youngest.

the top surfaceand of layerswithin the deformed grid (Figures13d and 15). The pressuredependence of the plasticyield surface results in the material being stronger in shorteningthan in extensionbecausethe higher mean stressrequiresa higher differential stressto produceplasticstrain. Thereforea displacementof 2 km is accommodated in differentways during extension and shortening. Becausethe material is

Figure 11. Distribution of incrementalshear strain that accumulatesduring each 0.5 km of displacement (contour interval of 0.1) during sequential developmentof the modelin Figure6b. Active shear bands can be recognizedduring each increment. Early grabenare movedto the left by displacement on the masterfault and are replacedby new graben that form abovethe baseof the ramp.Four stagesof developmentare shown,at displacements of (a) 0.5 km, (b) 1 km, (c) 1.5 km, and (d) 2 km.

26,666

ERICKSON

ET AL.: MECHANICS

OF EXTENSION

AND INVERSION

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