JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 102, NO. B8, PAGES 17,951-17,965,AUGUST 10, 1997
Study of plate deformation and stressin subductionprocesses
using two-dimensionalnumerical models
Riad Hassaniand DenisJongmans
Laboratoires
de G6ologiede l'Ing6nieuret d'Hydrog6ologie,
Universit6de Libge,Libge,Belgium
JeanCh6ry
Laboratoirede G6ophysique
et de Tectonique,CentreNationalde la Recherche
Scientifique,
Montpellier,France
Abstract. Two-dimensionalfinite elementmodelingis usedto modelsubduction
of an oceanic
lithosphericplatebeneathcontinentallithosphere.The subduction
processis initiatedalonga
preexistinginclined fault and continuesuntil reaching400 km of total convergence.The
lithosphereis assumedto be underlainby an inviscidasthenosphere.
Differentrheologicallaws
havebeenconsideredfor the lithosphere,includingelasticityand elastoplasticity.The modeling
showsthat both the stresssystemin the platesand the surfacetopographyare stronglydependent
on two main parameters:the densitycontrastbetweenlithosphereandasthenosphere
( Ap = Pz - PA) andthecoefficient
of frictionalongthesubduction
plane. Varyingthesetwo
parametersallowsexplanationof the main characteristics
of real subduction
zonesand resultsin
two majorregimesmanifestedby extensionor compression
in the arc-backarc system.Extension
and backarc rifting corresponds
to a positivedensitycontrastand a low coefficientof friction,
while negativeAp valuesand/orhigh frictionleadsto a compressional
regime. The coexistence
of trencharc compression
andbackarc tensionis onlypossiblefor a coefficientof frictionlower
than0.1. The resultsof the numericalexperiments
agreewith thoseof experimentalmodeling
conducted
undersimilarphysicalassumptions.
Introduction
presentedtwo-dimensionalmodelingof the subductionprocess,
including studies of the horizontal motions and of the state
The existenceof two basicallydifferent typesof subduction of stresses in different subduction conditions. Initiation of
was first suggested
by Uyedaand Kanamori [1979] on the basis
subductionand progressive
sinkingof the slab were not usually
of both seismological
and geologicaldata [3/1olnarand Atwater,
modeledin theseworks. At presentand to our knowledge,the
1978]. Chilean-typesubduction
is characterized
by compression
most completesimulatedsubductionresults are those obtained
of theoverriding
plate,whiletheMariana-type
is associated
with
extension
oftheupper
plate
and
back
arcspreading.
One
ofthebyShernenda
[1993,
1994]
viaanalog
models,
which
yielded
principal
problems
related
tosome
subduction
zones
ofthegeneral
conclusions
about
the
generation
of
tension
in the
overridingplate andthe creatiøn of backarcbasins.
secondtype is the coexistenceof tensionaland compressional
In this paper we aim to explain the main features of
stresses[Otsuki, 1989; Whittaker et al., 1992]. In order to
subduction zones through two-dimensional finite element
explain this association,it appearsnecessaryto studythe stress
modeling and to show how stress,strain, and horizontalmotion
regime generatedby the subductionprocess,and some efforts
are influencedby the slab weight and intraplatefriction. First,
have recently been made to model lithospheric subduction
we conducta generalnumericalstudywith a homogeneous
plate,
throughlaboratoryexperiments
andnumericalcomputations.
in order to determinethe range of key parameters,suchas the
Two possible approachesare used in the framework of
densitycontrastbetweenthe lithosphereand the asthenosphere,
numericalmodeling.In the first, lithosphereis viewed as a
and the friction along the subduction surface.Different
viscousfluid at long timescale[Channeland Mareschal, 1989;
rheological behaviors are tested in order to obtain a correct
Moretta and Sabadini, 1995; Zhongand Gumis, 1994], whose
deformedplate geometryas well as realistic stresses.Second,
behavior is expressed by the well-known Navier-Stokes
taking into accountthe results of the parametricstudy, we
equation.General featuressuch as temperaturefield and dip
simulatethe subductionprocessand the associated
phenomena
angleof the slab can be computed,but topography
and elastic
with a more realistic plate rheology.About 100 computations
flexurecannot be directlyderivedfromsuchanalysis.
were carriedout under a variety of conditions,and only a small
The secondapproach,which is usedhere, derivesfrom solid
part of the resultsare presented.The contributionof this paper
mechanics,where the material behavior is governedby
is that,for the first time, platedeformationandstressduringthe
elastoviscoplastic
laws. Severalauthors[e.g.,Bott et al., 1989;
whole subduction
processfrom the onsetto the maturestagesis
Whittakeret al., 1992;Giunchiet al., 1994; Wanget al., 1994]
modeledusingrealisticrheology.
Copyright
1997bytheAmerican
Geophysical
Union.
MechanicalandNumericalAspects
Paper number 97JB01354.
A schemeof the model settingis shownin Figure 1. We
assume
a generalhypoelastic
constitutive
law for thelithosphere:
0148-0227/97/97
JB-01354509.00
17,951
17,952
HASSANIET AL.: STRESSIN SUBDUCTIONPROCESSES
Vo
Io lg
IO
Free
vertical
slip
Imposed
horizontal
CoA0Solid
body
(lithosphere)
velocity
Hydrostatic
pressure
9a
Fluid body(asthenosphere)
Figure 1. Schematicrepresentationof the model used for the computations.Lithosphereis modeled by a
deformablebodyoverlyingan inviscidasthenospheric
fluid. This fluid actson the baseof the lithosphere(and on
anypart of its boundaryin contactwith it) throughan hydrostaticpressure.Vertical displacementsare free on the
two edgeswhile horizontaldisplacements
arelockedat the left edgeandsubmittedat a constantvelocity(5 cm/yr)
at the right edge. Along the inclinedfault, the frictionalCoulomblaw is used.
Do'
Dt
- (cr,
The interplatefrictioncorresponds
to the Coulomblaw
d)
Io1 -
where (5 is the Cauchy stresstensor,d is the eulerian strain rate
tensor,D/Dt is an objectivetime derivative(Jaumannrate), and
9/gis a functional,isotropeandlinear in d [e.g.,Mandel, 1966].
Three models are used in order to mimic lithospheric
rheology: linear elastic, Maxwell viscoelastic, and time
independentelastoplastic.Elasticity is neededto model small
deformationof oceaniclithospherelike flexure. This behavioris
controlledby two elasticparametersand by the elasticthickness
0
C6)
where oN and or are the normal and the shear stresses,
respectively, on the subduction plane. The only control
parameteris the staticfriction coefficientg which, in the frame
of this work, is considered as an effective friction coefficient
which is a blend of the effects of the intrinsic friction coefficient
andpore-fluidpressure.
One of the primary forces that influencesthe subduction
of the lithosphere.All experimentsareconducted
with a Young's systemis the slab pull force due to densitycontrastbetween
Becauseour lithosphereis a
modulusE -- 10• Pa anda Poisson's
ratiov = 0.25 [Turcotteand lithosphereand asthenosphere.
Schubert,1982]. Becausewe do notmodela specificsubduction compressiblesolid due to its elastic properties,an equally
zone,we alsofix for all experimentsthe elasticthicknessto H = compressible asthenosphereis assumed.We also model
as an inviscid fluid. This secondassumption
30 km that approximatelycorresponds
to a 60 m.y. old oceanic asthenosphere
lithosphere[Mc Nutt and Menard, 1982].
meansthat viscousinteractionbetweenthe subducted
plate and
Beyondelasticdomain,threeregimesof rockbehaviorcanbe the surroundingmantle is neglected.This simplificationis
consideredfor the lithosphere:brittle, semibrittle, and ductile similar to that made by Shemerida[1993] and is partially
is
[Kirby, 1980]. For the sakeof simplicitywe assumethatmostof supportedby the fact that the viscosityof the asthenosphere
the lithospheric
strengthis retainedby the brittlepart. This zone several ordersless than that for the lithosphere.Under these
on the
is characterizedby a pressure dependentyield strength two assumptions,the interactionof the asthenosphere
pressurewhich
[Paterson, 1978]. We mimic the brittle behaviorusing an lithospheremaythenbe reducedto an hydrostatic
elastoplasticpressure dependentlaw for which the failure actson eachpart of the lithosphereboundaryin contactwith the
and the pressureat
criterion is the Drucker-Pragerone [Desai and Siriwardane, fluid. The densityof the asthenosphere
depthz are computedby solvingthe stateequations'
1984]:
F(o') = d• (o') + ad 1(o') - aP o
don
(2)
withJ3(o')= [(3/2) s:s]m thesecond
invariant
of thedeviatoric
stresss, J• (r•)= (1/3) r•: I the meanpressureand
ot-
- fldps
dps -
(7)
g dz
where[3is the compressibility
moduluswhichis assumed
to be
6sin ½
3-sin
p,4
' Po= c/ tan(p
½
(3) constant and equal to the lithospheric value. Noting by
•- z-z h the depthbeneaththe top of the hydrostatic
column,
zh, andknowingthatp.q= pøAat •- 0, we obtain
wherec is thecohesion
and 0 thefrictionangle.
The plasticpotentialis givenby
G(O.)= J• (o.) + otpJ1(O.)
(4)
where
sin
••
3-sin
(5)
and W is the dilatancyangle. The plasticflow rule is then non
associative
whenW,• ½.
PJ
1-1fip,•g•,
P.4
(z)- --• ln(1,8p,•g
(8)
Since[• is assumed
constant,
theserelationships
are
approximate
andarevalidin therange
• <<1/ fl,o,•g
(9)
HASSANI ET AL.: STRESSIN SUBDUCTION PROCESSES
For 13-- 10'll Pa4, pøa= 3200kgm'3, g = 10ms'2, this
17,953
Hassani, 1994] and of the Coulomb law [Jean and Touzot,
restrictionbecomes• << 3125 km, which is largely sufficient 1988]. This is the mosttime-consuming
part of the computation.
for our purpose.
For eachnumericalexperimentpresentedin this work, a few
Because the lithosphere-asthenosphere
boundary changes thousands
of degrees
of freedom
and5x105to7x105timesteps
during the subductionprocess,due to the evolution of slab were usedfor a total durationof 8 to 19 m.y. The lengthsof the
contour,the part of the plate boundary3F•t submittedto the time stepsare then 13 and 26 years.
pressurePA is redefined at each time step (see details in
AppendixA).
Subductionof an Elastic Homogeneous
Plate
The mechanicalboundaryvalueproblemto solveis governed
The model used for the simulation is displayedin Figure 1.
by quasi-staticequilibriumequations:
The lithospheric
platehasa thickness
H anda densitypz,andlies
over an asthenospheric
fluid with a densityPA. The subduction
by an
(10) plane initially dips at an angle{} and is characterized
div o + pœg = 0 over•'-•t
v = Vo On3v•lt
o.n = pAn on3F• t
effectivecoefficientof friction}.t. The wholemodelis subjectto
gravity. The left edge of the plate is constrainedto zero
wheref•t is the spacedomainof the plate at time t, 3d2t is the horizontaldisplacement,and the right edge is assignedto a
velocityvo. Bothedgesare freevertically,andtheplate
part of the boundaryassignedto a constantvelocity,Og•t is the constant
part of the boundaryin contactwith the asthenosphere,
and n is surfaceis a free boundary.
This first series of experimentsis conductedfor a purely
its outward unit vector.
elastic
plate. Along the interface,the frictioncoefficient}.t= 0
We use a finite element method for space discretization
associated with a dynamic relaxation method for time
discretization[Underwood,1983;Belytschko,1983; Cundalland
Board, 1988; Poliakov et al., 1993]. This methodleads to a set
of nonlinearequations:
and O= 15ø. The asthenospheric
densityPA= 3200kgm'3
andthe subduction rate Vo- 5 cm/yr. In the first test the
lithospheredensity is identical to that in the asthenosphere
(Ap= p•,- PA= 0 kg m-3).The computations
wereperformed
until a shorteningof 400 km is reached,after a time of 8 m.y.
Mii= Fint(U,6,t)+ Fext(U,t)+ Fc(u,6,t)
(11)
Figure2 presents
the evolutionof the structure
with time. Since
Ap= 0 kgm'-a,theplatesubducts
witha constant
angle,close
where the vectors Fret and Fcxtare the internal and external nodal
to O, resultingfrom a perfect hydrostaticequilibriumwith the
surrounding
fluid. In this elasticcase,the plate is stronglybent
vectors of nodal displacements,nodal velocities, and nodal
whensubducting
underthe upperplate,andit recoversits initial
accelerations,
respectively. M is a fictitiousmassmatrix. The
plane shapeat depth.
quasi-staticsolutionis reachedwhen the inertial regularizing
term Mii is negligiblecomparedto the externalforces.
Influenceof the DensityContrastAp = pL - PA
The time discretizationof (11) is performed using an
The assumption
that pr = pAmaynot be true [Oxburghand
inexpensiveexplicit finite differencescheme(see AppendixB)
which allows to update at each time step velocities and Turcotte, 1976; England and Wortel, 1980] and the oceanic
densitypr stronglydependson its age,varyingfrom a
displacementsat each node of the mesh. Internal and contact lithosphere
forcesare locally computedby using an implicit integrationof little less than p.n to values larger than pA by more than
-3 [Shernenda,
1993].The two followingsets of
the constitutivelaw in contextof largestrain[Pinskyet al., 1983; 100kgm
forces, F½is the vector of contactreaction, u, 6, and ii are the
(km)
400
800
I
I
I
'
I
'
I
I
'
I
I
'
1200
1600
I
I
I
2000
I
I
I
I
I
E
ß
0
'
'
I
400
I
'
'
800
'
i
1200
'
'
'
i
1600
2000
(km)
Figure2. Test1, subduction
of anelastichomogeneous
platewithAp - 0 anda shortening
rateof 5 cm/yr.From
top to bottom,for time = 0, 2.4, 4.8, and8 m.y.
17,954
HASSANI
ETAL.'STRESS
IN SUBDUCTiON
PROCESSES
(a)
0
(km)
400
800
i
i200
i
1600
i
i
i
i
f
2000
I
(b)
i
i
i
i
c)
I
'
'
'
400
I
'
ß
800
'
,I
'
'
1200
'
I
'
'
'
1600
2000
(km)
Figure3. Subduction
ofanelastic
homogeneous
plate:
(a)Test2, Ap- -100kgm-, (top)4.8m.y.and(bottom)
8 m.y. and(b) Test3, Ap = 100kg m-, (top)4.8 m.y.and(bottom)8 m.y.
experimentsare designedto studythe influenceof the density overridingplate, reachingan elevationof about4 km at the end
of the simulation.
ccJntrast
Ap on the subduction
evolution.
In the first one
The arc width is around 200 km.
If there is no
(Test2) the lithosphere is lighter than the asthenospheredensitycontrast,the arc still appearsbut its amplitudeis quite
(/Xp=-100kgm-3) and a significant
change
appears
in the lower (1 kin), as can be seenon Figure 4. However, for a dense
forms(Figure4), attaining
a
deformationpattern(Figure3a). The platebeginsto subductlike plate,a forearcbasinprogressively
in the precedingsimulationbut, as soonas a sufficientlengthis final depthof 4.5 km. The basinbottomis alwayslocatedover
engaged, the slab tends to rise and to be stuck below the the deeper contactpoint betweenthe two plates. In this latter
overridingplate. When the densitycontrastis positive (Test 3, case, the resulting relief is quite similar to the dynamic
•p = 100kgm-3),
the subduction
angleincreases
with the topographyobtainedby Zhongand Gurnis[1994] with a viscous
flow model. However, the basinformingwithin the upper plate
drivingof theslab(Figure3b). If •helengthof thesubducted
plate is sufficient,the slab will reach a vertical position.The is not produced by extension in our model, as it occurs even
whenthe stressregimeis totallycompressive
(high friction). We
thinnerthe elasticplate, the soonerthis phenomenon
occurs.
Thesenumericalsimulationsare in very goodagreementwith addressthis issuefurtherlater in the paper.
the analogexperimentsof Shemenda[1993]. In particular,both
An importantadvantageof numericalmodelingcomparedto
modelingmethodsexhibit the same surfacetopographyfeatures physicalmodelingis that the former is able to providethe stress
for the three cases.When the subductingplate is lighter regimeat any time anywherein the structure.Figure 5 presents
(Figure4), a frontalarc quicklybuildsup at the surfaceof the the magnitudeand the orientationof the principal deviatoric
i
i
bp=o _
,
400
i
i
.......Ap=-100
,
, ;
800
!
1200
1600
(km)
Figure 4. Influenceof the densitycontrastbn the surfacetopography.Comparisonbetweenthe topography
surfaces obtained at the end of the first three tests.
HASSANI ET AL.' STRESSIN SUBDUCTIONPROCESSES
17,955
300 MPa
el•l > 300 MPa
o1{331
> 300 MPa
ß ß
ß
•
ß
300 MPa
+ +
+
+ + + +
ß1•1 > 300 MPa
oI•3 > 300 MPa
+ +
300 MPa
i,i>3OOMPa
'
o
o131
>300
MPa
,
0
.
,
1 O0
.
,
200
.
,
300
.
,
.
400
,
500
.
,
600
,
700
.
800
(kin)
Figure5. Ma•n•tudc
andorientation
o•theprincipal
dcviatofic
stresses
in theovc•idin•plateat theendo• the
numerical
s•mu]afions
•o•th•ccdifferent
values
o•thedensity
contrast
(&p- {top)-•00,(m•dd]c)
O,and(bottom)
•00• m'•).Vertical
cxa•c•afion
•s3.6.Note
thatthestress
cross
•snotdistorted
by
Stresses
withmagnitude
grcatc•than300MPa arcrapresented
byc•c]cs.
stressesin the overridingplate at the end of the simulationfor
distribution.For g > 0.2, horizontal
extension
no longeroccurs
in the overriding
platefar fromthe couplingzoneandthe outer
and minimum compressivestresses,respectively.For clarity, bulge. Horizontal {3• with magnitude around 50 MPa is
the three cases studied. Variables {3• and {33 are the maximum
only stresseswith magnitudelower than 300 MPa are displayed observed.For }.t = 0.5, the horizontal stress {3] reaches a
on the figure (greatervaluesare markedby circles).It is seen magnitude
of 400 MPa.On •theotherhand,for }.t- 0.1,
from Figure5 that the overridingplate beyonda certaindistance horizontal {33 with magnitude lower than 100 MPa are
from the subduction zone is under horizontal tensional stresses,
stillpresentin the plate.In the forearcregion,cc;existent
whose
magnitudes
arebetween
40 (Ap-0 kgm'3)and150MPa compression
with relativelysmallmagnitudeis alsoobserved.
(Ap= 100kgm'3).For the caseAp= -100kgm'3, the stress Within the limits of our models, these results show that the
systemis stronglydisturbedby the flat subduction,resulting formationof a back arc basinrequires,in additionto a positive
fromthe floatingup of the light slab. The extensiondeveloping densi[ycontrast,
theexistence
of relativelyweak}.tvalues(lower
in thebackarcresultsfromthes•nkingof thesubducting
plate than 0.2) at the interfacebetweenthe two plates. If }.t > 0.2,
and the so-called suction effect, which is causedby the compressivestressesdevelopfrom the trench and may totally
hydrostatic
pressing
of the*two
plates[Shemenda,
1993].Dueto suppressthe tensionin the overridingplate.
this suction, the overriding plate remains attachedto the
High friction, however,doesnot prohibit the formationof a
subducting
plate. If the stressvaluesin the upperplateappearto forearc
basin
which
appears
at anyg if Ap> 0. Thisbasin
be quite realistic in magnitude,this is not the case in the formation is therefore not directly related to the stressregime
subducting
platewheretheymayreacha few gigapascals,
at the within the overridingplate. Figure 7 simply suggeststhat this
outerbulgeof the bentelasticplate.
deflectionis causedby the wheelingmotionof the denseslabdue
In all abovetests,thecoefficient
of friction•t is 0. Belowwe to its sinkingwhichdragsthe overridingplate with a downward
present
results
forexperiment•
with•t > 0.
component.
As thedeeperpointof thein.traplate
zoneundergoes
thegreatest
displacement,
themaximaldeflection
is locatedover
Influence of the Coefficient of Friction
this point.
Three numerical simulations have been carried out with the
elastic
parameters
detailed
above
andwithAp= 100kgm'3,for
Bendingof an ElastoplasticPlate
threevaluesof the frictioncoefficient(g = 0.1, 0.2, and 0.5).
The resultsare displayedin Figure 6. In all cases,subduction Althoughmodelingwith an elasticplate is able to explain
occurssimilarly to the preyiousexperiments(Figure 3b). some significant features of subductionzone morphology,it
Thereare, however,importantdifferencesin terms of stress leads, however,to high stresslevels in the oceaniclithosphere
17,956
HASSANI ET AL.: STRESS IN SUBDUCTION PROCESSES
300 MPa
c•
ßlo•l > 300 MPa
o1•31> 300 MPa
c•
200 MPa
ß1o•1> 200 MPa
olo31> 200 MPa
200 MPa
•
,
•
++++ + + + ++
i
i
i
i
i
i
i
i
i
i
i
i
' o o • o1O31>
200MPa
!
i
i
i
I
i
i
i
•
•
õ00 MPa
,
"
.. o1%1> 500 MPa
c:)
0
1 O0
200
300
400
500
600
700
800
(kin)
Figure6. Magnitude
andorientation
of theprincipal
dcviatoric
stresses
in theoverriding
plateat theendof the
simulation
for fourdifferentvaluesof thecoefficient
of friction(It = O,0.1, 0.2, and0.5, fromtopto bottom).
Verticalcxa[[crationis 3.6. Notethatthestress
crossis notdistorted
by theverticalcxa[•cration.Stresses
with
largerma[nitudcarcrepresented
by circles.
whicharenotcompatible
withcommonly
accepted
yieldstrength curvature,Burov and Diarno•t [1992] concludedthat there is no
envelopes
[Me Nutt andMenard,1982].As we wantto deal elastic kernel for radius values lower than 300 km. On the basis
with large deformation,
we have to introduceirreversibleof their results extrapolatedto an oceanicplate, we chose to
theological
behavior
suchasviscoelasticity
andelastoplasticity. modelthe initial geometryof the fault with a curvatureradiusof
The application
of a plasticity
law for theupperpartof the 750 km (rather than a rectilinear fault) which is closeto observed
lithosphere
is well suitedto limit thestress
amplitudes
during values [Mc Nutt and Alenard, 1982; Comte and Suarez, 1995].
the subduction
process.
The Drucker-Prager
criterion(2), used Keeping the same parametersas in the precedingtest, we
with c - l0 MPa, 0 - 30øandzerodilatancy(• - 0ø),allowsus obtained the results (Test 5) of Figure 9, which can be
to keepthe deviatoricstressinsideyield strength
envelopes.comparedto thoseof Figure 8. With this fault shapethe plate
Figure8 presents
theresultsof thesimulation
(Test4) carried doesnot stronglybend when subductingand thusdoesnot coil.
out with the same elastic parametersas above and with It subductswith a mean dip angle of 20ø to 25ø. All the
Ap- 0 kgm'3. Duringthefirststageof thetest,theplateis followingtestshavebeenperformedwith the sameradiusvalue
stronglybent during subduction
but, unlike the previous of 750 km. With the elastoplastic
theology,the deviatoricstress
simulations,the slabdoesnot recoverits initial rectilinearshape in the slab at a depth lower than 30 km are then limited to a
but remainsbent. As a result,the plate coils;that is generally magnitudeof 500 MPa, which are muchmorerealisticthan the
not expectedfrom a geodynamical
viewpoint.On the same valuesof a few gigapascals,
obtainedin the elasticcase.
figurethe yield zoneis represented
as a shadedarea,andit
appearsthat the whole subducted
plate experiences
plastic Back Are Opening,Roll-Back, and Frontal
deformation.One can deducethat this disappearance
of the
Compression
elastickernelmaypreventthecoiledplateto unbend.
The curvatureradius of the coiled slab (Figure 8) is very
Differentmechanisms
have been proposedto explainthe
small, around 100 km. In a study of the equivalentelastic locationof back arc rifting, suchas mantlediapirism[Karig,
convection[ToksSzand Hsui,
thicknessof a layeredcontinentalplate as a functionof the 1971] or inducedasthenospheric
HASSANIET AL.' STRESSIN SUBDUCTIONPROCESSES
17,957
regime could develop in the overriding plate when a dense
lithosphere is used in conjunction with a low coefficient
of friction. This stress situation in the back arc region is
compatible with basin opening, and we discuss now the
rheologicalassumptions
that can be madein orderto generatea
largehorizontalextensionin the overridingplate.
As shownby numericalmodelingof rifting [e.g.,Bassi, 1991],
a material or a geometricalweaknessin the lithospheremay lead
to rifting if extensionalstressor velocitiesare applied. Our goal
is not just to open a basin but rather to study the interaction
between the acting forces due to subductionprocessand a
possibleback arc extension.To achievesuch a coupling,we
introducea weak part made of a viscoelasticmaterial (Maxwell
rheology) in the overriding plate (Figure 10). This zone is
i
200kmlong,witha viscosity
of 1022
Pas. Werealizethatsuch
an inclusion is rather artificial, and an initial geometrical
thinningof the plate would be moreelegant.However,this last
solutionwouldlead to intenseplasticneckingof the meshafter a
few tenths of kilometers, and the numerical solution is no more
Figure 7. Kinematicinterpretationof forearcbasin formation.
tractable.By opposition,our choice allows a more diffuse
The increasinglength of the dense slab betweentwo distinct
deformation
to developthat corresponds
to a larger amountof
instantscausea wheelingmotionof the slab. The overriding
finite extension.
plate is then draggedowing to the hydrostaticsuctionthat
The computations
are madewith a coefficientof friction
maintainsthe two platesin contact.The verticalcomponent
of
this motion explains the surface deflection which has its of0.05anda density
contrast
Ap= 100kgm'3. Various
steps
of
maximumover the deepercontactpoint. Arrowsstandfor total the model evolutionare shownin Figure 11, while Figure 12
displacements
of the contactpoints,dottedand solid lines show presentsthe evolutionof topography
with time. At 5.7 m.y. the
the boundaryof the plates before and after the forearc basin sinkingdenseslab becomeslong enoughto transmittensionto
formation,respectively.
the subducting
plate(slabpull force)andto the overridingplate
(suctionforce). As a consequence,
a backarc basinbeginsto
openin the weakzone. At the endof the simulation
(7.2 m.y.)
1978].Assuggested
byShemenda
[1993],a simplevariation
of the amountof convergence
is almost350 km andthe basin
crustalthickness
in the overriding
platemayinfluence
the extension
is closeto 110km. It is associated
with foredeep
location
of thebackarcrifting.Ournumerical
simulations
with roll-back
ofthesame
orderofmagnitude
andtothedevelopment
elasticplateshave demonstrated
that an extensional
stress of a forearcbasin.The velocityfield in the slab and the
•
]
,
I
I
I
I
I
,
I
I
I
I
I
'
I
0
I
200
I
400
I
600
•
I
800
I
t
I
I
1000
1200
(km)
Figure 8. Test 4, subductionof an elastoplastichomogeneous
plate. The area whereplasticdeformationoccurs
is shaded:(top) 2 m.y., (middle)4.7 m.y., and(bottom)9.5 m.y.
17,958
HASSANI ET AL.' STRESS IN SUBDUCTION PROCESSES
I
I
I
I
I
I
I
I
I
I
I
i
i
I
I
I
I
I
,
i
,
I
[
!
i
,
200
i
400
,
i
,
600
800
I
i
1000
1200
(kin)
Figure9. Test5, subduction
of anelastoplastic
homogeneous
plate.Theinitialgeometry
of thecontactsurface
between
thetwoplatesis specified
to havea curvedshapewitha curvature
radiusof 750 km: (top)2 m.y.,
(middle)4.7 m.y., and(bottom)9.5 m.y.
overriding
plateis displayed
onFigure13. It showsthewheeling obtaininglargeforearccompression
andlargebackarcextension
motionof the denseslabwhichincreases
with the lengthof the with modelinginvolvingeither a locked or an unlockedfault.
subducted
plate(Figures13ato 13c). Thecorresponding
growth Figure 15a displaysthe lengthevolutionof the weaknesszone
of the slabpull forcecausean accelerated
openingof the basin. during the subduction for different values of the friction
At the endof the simulation(7.2 m.y.) thevelocityfield in the coefficient.Except for }.t= 0, all experimentsshow an initial
overridingplate is almosthorizontal(Figure 13d) and exhibits shortening
stageof the weak zone. As the slabpull increases,
valueshigherthan the shortening
rate imposedat the edges shortening
is shiftedto opening(exceptthe caseg = 0.2 whichis
of the model(5 cm/yr).The stressfield at 7.2 m.y. shows onlyassociated
with compression
until thecomputation
fails) and
significanttensionwith a magnitudeof 300 MPa in the weak the lengthvariationis dramaticallyincreased.Figure15bshows
zone (Figure 14a) and a horizontalcompression
in the frontal the variation of the net horizontal deviatoric forces on the vertical
zone (Figure 14b) with magnitudearound 100 MPa in the boundariesof the model, that is, the integral over the plate
shallow part. This simulation illustrates the coexistenceof thicknessof the horizontaldeviatoricstress
trencharc compression
and backarc tension,which is observed
1991].
Thecompressive
stress
amplitude
isstrongly
controlled
in
subduction
zones
[Nakamura
and
Uyeda,
1980;
Doglioni,
by the coefficientof frictiong alongthe subduction
surface.
R//=
dz
(12)
With}.t= 0.07themagnitude
ofthecompressive
stresses
reacheswithconvention
thatnegative
values
holdforcompression.
The
100to 150MPain theforearc,
while,with}.t=0.1, theentire higher
theshear
resistance
along
thesubduction
plane,
thelonger
overriding
plateexperiences
a compression
regimeassociated
thetimeof thecompressive
phase.
Whenthelengthof the
withsignificant
shortening
of theweakzone.Thedramaticdowngoing
slabreaches
a givenvalueLo (depending
onthe
effectof thecoefficient
of friction
(seeFigure
15)explains
the friction
coefficient)
theforce
RHvanishes,
becomes
positive,
and
apparent
contradiction
pointed
outby Whittaker
etal. [1992]in theextensional
phase
begins
(except
for}.t= 0.2). Thisforce
can
400 km
400 km
1000 km
200 km
Figure10. Test6, model
used
forthesimulation
ofa subduction
associated
witha backarcrifting.A weakzone
formed
byaviscoelastic
Maxwell
body
of10TM
Pasviscosity
(shaded)
ispresent
intheoverriding
plate.
HASSANI ET AL.: STRESS IN SUBDUCTION PROCESSES
17,959
o
o
c)
o
i
0
400
i
,
,
800
'
i
1200
,
,
,
i
1600
,
,
,
2000
(kin)
Figure 11. Test 6, deformedgeometryat varioustime steps.The black arrow marksthe trenchlocationwhich
retreatswith increasing
subducted
slablength. Fromtopto bottom,time- 0, 1.9, 3.8, 5.7, and7.2 m.y.
alsobe interpretedas the forceneededto movethe right edgeof
the plate with a constantvelocity of 5 cm/yr (our kinematics
boundarycondition).As the slablengthis lessthanLo, this force
is negative,denotingthat the plate is pushed.When the slab
lengthexceedsLo, the forceRH becomes
positive,denotingthat
theplate,whichsinksby its ownweight,is heldbackat its right
[e.g.,Doglioni, 1993;Dvorkin et al., 1993;Scholzand Campos,
1995]. Accordingto Doglioni'smodel, there is an eastward
mantle flow on a global scale. As a consequence,
the west
dippingsubductionzonesopposethis flow, generatingback arc
extensionwhichdoesnotoccurin eastdippingsubduction
zones.
Based on structuraland geodeticdata, this mechanismhas,
edge.
however, never been modeled. On the other hand, Scholz and
Campos[1995], followingUyedaand Kanamori[1979], propose
Discussion
andConclusions
thattheseaanchor
force
wasresponsible
forback
arcspreading.
This force resultsfrom the hydrodynamicresistanceof the slab
The subductionof an oceanicplate has been numerically motionthrougha viscousfluid. The viscousresistance
was not
modeledusinga finiteelementmethod.The basicassumptions
takenintoaccount
in oursimulations
in whichtheasthenosphere
arerelativelysimple.A greatvarietyof platedips,topographicalwasassumed
to beinviscid.In theworkbyDvorkinet al. [1993]
features,and stressregimeshave been obtainedby varyinga the formationof backarc basinwaslinkedto the stabilityof the
few parameters.Within the limits of our modeling,the two slab which was determinedby its width. If the slabis narrow,
significantparametersappearto be the densitycontrastbetween sidewaysasthenospheric
flow reducesthe suction,resultingin a
the lithosphereand the asthenosphere
(Ap = Pt- P,•), and the rapid sinkingof the slab. This three-dimensional
effectcan not
friction coefficientg betweenthe two plates. For a positive be accounted
by the methodused.
Ap value, a striking feature is that the stressregime in the
Our modeling results are in good agreementwith the
overridingplate may be totally tensional,totally compressive,
or numericalsimulationsof Zhongand Gumis [1994] in terms of
compression
and tensionmay coexist,dependingon the friction trenchtopography.Also both studiesshowthat the shearstress
coefficient.Varying these two parametersprovides a simple alongthe subductionfault must be limited to allow the onsetof
quantitativeexplanationof the differentstraincharacteristics
of back arc extension.On one hand, the similaritiesbetweenour
real subductionzones,from the extensiveMariana arc to the resultsandthoseof ZhongandGurnisarenotsurprising,
because
compressive
Chileanarcregion,aswell asthedip of theslab.
in both studiesthe main actingforceis the slab pull. On the
However,more complexmechanisms
have beenproposedto otherhand,similarresultsmay be unexpected
due to two major
explain these observations,in particular, back arc spreading differencesbetweenthesemodels.
17,960
HASSANI ET AL.' STRESSIN SUBDUCTION PROCESSES
'
'
'
i
...............
'
'
!
i
............
i
.
,
i
!
!
,
i
,
,
,
i
ß
.
,
I
,
.
,
i
,
i
!
,
,
i
*
!
,
i
,
,
!
i
,.
,
i
i
i
i
!
i
........
400
i
i
800
1200
i
1600
i
2000
(km)
Figure 12. Test 6, evolutionof the surfacetopography
with time. Fromtop to bottom,time - 0, 1.9, 3.8, 5.7,
and7.2 m.y.
First,ourmodeling
explicitlyincludes
elasticity
thatallowsus coefficient
of frictionbetween
thetwoplates.It wasshown
that
to describe
elasticbending,
especially
theouterrise. However, theexistence
of backarctension
requires
a verylowcoefficient
thesametopographic
featureis alsopresent
in thepurelyviscousof friction,lessthan0.1. Thisresultis consistent
withlowshear
modelof Zhongand Gumis [1994]. Such a result indicates stressvaluesestimatedby Bird [1978] for MarianaandTonga
thattopographic
arguments
are inappropriate
to decidewhat andwith seismiccoupling
valueswhichappearto be low for
rheological
modelshould
beusedfortheoceanic
lithosphere. mostsubduction
zones[14•mg
et al., 1995].However,
valuesof
Second,
we do not account
for the resisting
viscous
force thefrictioncoefficient
between
twoplatesarerarelyfoundin the
appliedon the sinkingslab, assuming
that asthenospheric
literature.On theotherhand,laboratory
measurements
andthe
coupling
hasa smalleffecton the subduction
process.
During studyof basalfrictionin accretionary
wedges
haveusually
ledto
trenchroll-backandhorizontalmotionof the slabrelativeto the intrinsiccoefficientvaluesmuchhigherthan0.1. In their study
mantle,this assumption
couldnot be true, due to the anchor of the fold and thrustbelt of westernTaiwan,Dahlenet al.
force.We suspect
that accounting
for this forcein our model [1984] adoptedI,t = 0.85, which is basedon laboratory
wouldpreclude
a verylargevelocityfieldof theslab(30 cm/yr), measurements
of manyrocks[Byeflee,1978].Effectivebasal
aspicturedinFigure13.
friction coefficienthas also been deducedfrom structural
Ourresultsarealsoin verygoodagreement
withthephysical considerations
in theAleutian
accretionary
prism[DavisandVon
modelingof Shemenda
[1993, 1994],who thoroughly
studied Huene,1987].This method,whichwas alsoappliedin other
thesubduction
of an elastoplastic
lithosphere
overlying
a low- prisms by Lallemand et al. [1994], led to a mean value
to thefluidpressure.
viscosity
asthenosphere
(usuallywaterin hisexperiments).
Both g = 0.24 + 0.09, with a veryhighsensitivity
numericaland analogmodelingrevealthe importantinfluence The reasons of the differences between the friction values in
of the densitycontraston the subduction
regime, and the prismsandfor the wholeareaof platecoupling
arenoteasyto
importance
of the suctioneffectin generating
tensionin the explain.Frictionis a complexphenomenon
anddepends
on the
and
overriding
plate.If a weakzoneexistsin theoverriding
plate,a presenceof fluid, the natureof materials,the temperature
backarcbasinis likelytodevelop
witha corresponding
roll-back the asperitiesalong the surfaceand is certainlynot constant
of the foredeep.The excellentagreementbetweenthe two alongthe fault. All theseparameters
are poorlyknownat the
differentmodelingtechniques
increasethe reliabilityof the plate interface,and the friction conceptmight be seenas an
obtained results. However, it must be mentioned that neither
oversimplificationat the fault scale. On the other hand, the
technique
hastakenintoaccount
theinfluence
of temperature. representativity
of laboratorytestsmaybe questionable.Despite
In contrast
to the analogmodeling,thenumerical
simulations all these uncertainties and limits, the values determined in this
haveallowedthe stressandstrainregimesto be determined
at studyconstitutea first approximation
of the frictionprocessin a
eachstep,anda quantitativeinvestigation
of the effectsof the subductionzone with active back arc spreading.They are
HASSANIET AL.' STRESSIN SUBDUCTIONPROCESSES
(a)
(c)
(km)
600
500
I
700
•
I
800
,
I
900
,
I
1000 1100
,
I
,
I
500
(km)
600
,
17,961
I
,
700
800
900
1000
1100
I
I
I
I
I
c)
"
ß
i
T=3.8
m.y.
5 cm/yr
i
ß
[
I
,
I
i
I
i
(b)
•
I
•
,
I
•
I
,
I
T-- 7.2 m.y.
15 cm/yr
z(•,l•"'
c:)
c)
T: 6.4m.y.
_
C)
143
0
i
i
i
ß
i
ß
i
CO
'500
600
700
800
900
1000
1100
(km)
(d)
0
I
I
I
I
I
C)
i
i
'7600
7O0
800
900
1000
11O0
(km)
Figure13. Test6, velocityfieldat differentstepsin theslabandin theoverriding
plate.(a) Earlystageof
subduction
with no backarc extension,
(b) beginning
of backarc extension,
and(c andd) final stepof the
simulation
(T = 7.2 m.y.),thevelocity
fieldis almost
horizontal
in theoverriding
plateandexhibits
valuesmuch
higherthantheshortening
rateimposed
at theedgesof themodel(5 cm/yr).
supportedby the work of Tichelaar and Ruff [1993] which Appendix A: Computation of Hydrostatic Forces
studiedthe critical temperaturedefiningthe depthof seismic From the Asthenosphere
couplingalong subductionzones.In order to find this critical
temperature,they determinedthe bestvalue for the coefficientof
Forces related to the applicationof pressureon the
friction matchingthe heat flow data, and they found values lithosphere-asthenosphere
limit (equation(8)) are computedas
rangingbetween0.074 and 0.127 for the different zones. These follows:
approximateresults are in the same range as the values we
Let the segment[P],P2] be an element boundaryof the
inferredfrom computations.Modeling,whichincludestectonics, lithosphere
whichprogressively
goesthroughthe asthenospheric
topography,and stressanalysis,then appearsto be a valuable fluid. Assuming,for example, that z•. > z] where zi are the
tool for constrainingfriction coefficient'values in subduction verticalcoordinates
of the nodePi andnotingby [P,P•.]the part
zones.
of the segment[P•,P2] which entirelylies in the fluid at the
17,962
HASSANI ET AL.: STRESSIN SUBDUCTION PROCESSES
oa
o
oa-
(w•l)
o1•-
o9-
HASSANI ET AL.- STRESSIN SUBDUCTIONPROCESSES
0
100
(a)
•-g
200
300
•=o
400
500
•(s)=z5+(•2- • )'•
pt=o.o5
17,963
(A3)
By a change
of variable(• = s /L) andnoting•o is thelocal
coordinateof P on the segment[P1,P2], the nodalforcescan be
expressed
by
'
1
F,- LnI P,4
(•)N,(•)d• i=1,2
G
C)
(A4)
[o decreases
from1 to 0 duringthesubduction
process.
Accordingto (8) we obtain
0
1
0
-. ._•1
F,=-• n In(1-[3
p,•
g•(•))
Ni(•)d?,i=1,
2 (A5)
which can be easilyintegrated.
With this formula a nonzeroforce is applied at P• even when
!
it doernot lie in the fluid (i.e., when•,o> 0). It allowsus to
computetime-continuousforcesduring the subductionprocess.
These forces are added to the external nodal forces vector Fcxt.
,½,)......
g=O.05
I.L•O.07
•01
Appendix B' Time Discretization of the Finite
Element Formulation
The finite elementdiscretization
of the quasi-static
problem
(equation
(10)) leadsto thefollowingsetof nonlinear
equations:
Fint(U,
li,t)+ Fext(U,t)+
Fc(u,•,t)=O
(B1)
where the vectorsFint, Fext are the internal and external nodal
forces,respectively,Fc is the vector of contactreaction,and
u and • are the vectorsof nodaldisplacements
and nodal
velocities,respectively.
The dynamicrelaxationmethoddampsthe out-of-balance
forces
byintroducing
theregularizing
inertial
j termonthefight
hand side of (B 1):
i
0
100
i
,
200
,
,
!
300
i
,
,
400
Fint(U,(l,l)+Fext(U,t)+Fc(u,•,t)= Mii
,
(B2)
500
where ii is the vector of nodal accelerationsand M a mass
Vot,Amountof convergence(km)
matrixchosen
in a diagonal
formby nodalconcentration
1977]. The quasi-static
solutionof (B1) is reached
Figure 15. Effect of amount of convergenceand friction [Zienkiewicz,
coefficienton the strainandstressregime of back arc zone. (a) when the inertial term is negligiblecomparedto the forces
Lengthvariationof the weakzoneand(b) variationof the net involvedin theproblem.
horizontal'deviatoric forceson the vertical edgesof the model
Usingan explicitfinite differencemethodandwithoutcontact
(seetext for explanation).
phenomenon,
the time discretizationof (B2) leads to 'the
algebraicsystemof equations:
considered
time. Thefluidpressure,
givenby(8), actson[P,P2]
(/•r)f+•- Mi71
[ (•int)/n
+
andtheconsi•'tent
equivalent
nodal
forces
arebydefinition
F•- (F•
•,F•
y)r=n I pn
(•ts))
Ni(s)dsi=1,
2 (Ai)
-adsgn((•Fff
-1/2)(Fint)•
n+(Fext)•;]
I (B3)
(itF
ff+1/2
=it•-•/2+At(iiFff+•
(B4)
(uF)•+l
=u•+At(/tF)•
+¾2
(BS)
(P, v2)
wheres is thecurvilinear
abscissa,
n = (nx,ny)
r is thenormal
vectorof [P•, P2],Ni aretheshapefunction
of nodeP• andP2
N1(
$)=1-•s
N2($)=•s
L is the segmentlength,and • is the distanceto the hydrostatic
levelof pointbelong['P•,P2]:
whereAtis thetimestep,sgn(v) is thesignof v, O•dis a
damping
factor[CundallandBoard,1988],andthesubscript
F
emphasizesthat the corresponding
quantitiesare contact
free (Fc = 0).
The massmatrixis computed
so that the stabilityof the
explicitscheme
is ensured.
In otherwords,
a fictivedensityp
!7,964
HASSANI
ETAL.:STRESS
IN SUBDUCTION
PROCESSES
beltsand
for eachelementis computedfrom the usertime stepAt, the Dahlen,F.A., J. Suppe,andD. Davis,Mechanicsof fold-and-thrust
accretionarywedge:CohesiveCoulombtheory,J. Geophys.Res., 89,
elementsize and the elasticparameters
in sucha way that the
10087-10101, 1984.
travel time of a P wave to crossthe elem6ntis greaterthan the Davis,D.M., andR•vonHuene,Interferences
on sediment
strength
andfault
usertime stepAt:.
frictionfrom structures
at the AleutianTrench,Geology,15, 517-522,
1987.
• = (K+•G)
(B6)
where h is the smallest dimensionof the element, K and G are
Desai, C.S. and H.J. Siriwardane.ConstitutiveLaws for Engineering
Materials, With Emphasison GeologicMaterials, 457 pp., PrenticeHall, EnglewoodCliffs,N.J., 1984.
Doglioni, C., A proposalfor the kinematic modellingof W-dipping
subductions:
Possible
applications
to the Tyn'henian-Apennines
system,
thebulkandtheshearmodulus,
respec•tively,
andr is a factor
Terra Res.,3.423-434, 1991.
Doglioni,C., Geologicalevidencefor a globaltectonicpolarity,J. Geol.
chosen
greaterthan1 to ensurea safetymargin.
Soc.. London, 150, 991-1002, 1993.
'Whencontactbetweenbodiesis invoked,the algorithmmust
Dvorkin,J., A. Nur, G. Mavko, andZ. Ben-Avraharn,
Narrowsubducting
slabsand the origin of backarcbasins,Tectonophysics,
227, 63-79,
be modifiedin the following manner:
1. Predictionphase: The "free" accelerationsiiF and
1993.
England,P., andR. Wortel,Someconsequences
of thesubduction
of young
velocities
•F ateachnodearecompute
d using(B3)and(B4).
slabs,Earth Planet. Sci. Lett.. 47, 403-415, 1980.
2. Contactintegration
phase:The contactfrictionreactionsFc
are computedby solvingthe contactand Coulombfrictionlaw. Giunchi,C., P. Gasperini,R. Sabadini,and G. D'Agostino,The role of
subductionon the horizontalmotionsin the Tyrrhenianbasin: A
This is doneby an implicitalgorithm[Jeanand Touzot,1988]
numerical
model,Geophys.
Res.Lett.,21,529-532, 1994.
which works in the local frame of each contact node and needs
Hassani,R., Mod61isation
num6riquede la d6forrnafion
des systbmes
g6ologiques,
thesis,Univ.deMontpellier,Montpellier,
France,1994.
Jean,M., and G. Touzot,hnplementation
of unilateralcontactand dry
their predictedvelocities/•F.
3. Correctionphase:The correctvelocitiesanddisplacements frictionin computercodesdealingwith largedeformationproblems,J.
Mec. Theor.Appl.. 7, suppl.1, 145-160, 1988.
arecomputed.Thisis donebyconsidering
thatthecontact
forces Karig, D.E., Origin and development
of marginalbasinsin the western
only the knowledgeof the nodepositionat time stationn and
(Fc)i at thedegreeof freedomi producean acceleration
(iic), = Mi71
(F½)i
(n7)
The velocityis thenapproximated
by
ti•+l/2
=(tiF)•+1/2
+At(iic
(BS)
Pacific,J. Geophys.Res.,76, 2542-2561, 1971.
Kirby,S.H., Tectomcstresses
in the lithosphere:
Constraints
providedby the
experimental
deformation
of rocks,J. Geophys.Res., 85, 6353-6363,
1980.
Lallemand,S.E.,P. Schnurle,
andJ. Malavieille,Coulombtheoryappliedto
accretionary
and nonaccretionary
wedges:Possiblecausesfor tectonic
erosionand/orfrontalaccretion,
J. Geophys.Res., 99, 12033-12055,
1994.
Mandel,J.,M•camque desMdieux Conanus,Gauthier-Villars,
Parris,1966.
and the incrementaldisplacementbetweentime stationn and Mc Nutt,M., andH.W. Menard,Constraints
onyieldstrength
in theoceanic
n+l is
•'
lithosphere
derivedfrom observation
of flexure,Geophys.3'.R. astron.
Soc., 71,363-394, 1982.
U•+1=U?+Atti•+•/2
(B9)
Molnar, P., and T. Atwater,Interarcspreadingand cordillerantectonicsas
alternatesrelatedto the age of subductedoceaniclithosphere,
Earth
Planet. Sci. Lett.. 41,330-340, 1978.
Acknowledgments.
We thankFina S.A. who supported
this work
Moretta, A.M., and R. Sabadini,The styleof the Tyrrheniansubduction,
through
a postdoctoral
grant.We thankJ. KringsandD. Demanet
fortheir
Geophys.
Res.Lett.,22, 747-750, 1995.
enthusiastic
technical
help,andwe thankA. Shemenda
andR. Russofor
Nakamura,K., and S. Uyeda,Stressgradientin arc-back-arcregionsand
their helpfulcomments
and for the improvement
of the manuscript.
platesubduction,
3'.Geophys.
Res.,85, 6419-6428,1980.
K. Wang,C. Y. WangandC. Beaumont
provided
a constructive
reviewof
Otsuki,K., Empiricalrelationships
amongthe convergence
rate of plates,
rollbackrateof trenchaxisandisland-arctectonics:
Lawsof convergence
rateof plates,Tectonophysics,
159, 73-94, 1989.
Oxburgh, E.R., and D.L. Turcotte, The physico-chemical
behaviorof
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