JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 95, NO. B4, PAGES 4909-4923, APRIL 10, 1990
A Finite AmplitudeNeckingModel of Rifling in Brittle Lithosphere
JIAN LIN
Departmentof Geologyand Geophysics,
WoodsHole Oceanographic
Institution,WoodsHole, Massachusetts
E. M. PARMENTIER
Departmentof GeologicalSciences,Brown University,Providence,Rhode Island
We formulatea mechanicalmodel describingthe formationof rifts as finite amplitudenecking of an
elastic-plasticlayer overlying a fluid substrate. A perfectly plastic rheology is a continuum
descriptionof faulting in rift zones. Two importantaspectsof rift evolutionare illustratedby this
model:the evolutionof the rift width as extensionproceedsand the finite strainthat occurs. A region
at yield initially developswith a width determinedby the thicknessof the brittle layer, and the
internal deformationwithin this yield zone is proportionalto the topographicslope. As extension
proceeds,the surfacewithin the rift subsides,and the width of the subsidingyield zone decreases.At
any stage of rifting, matedhalin regionsjust outside the yield zone is deformed but no longer
deforming. The width of thesedeformedregionsincreaseswith increasingextension. Vertical forces
due to the massdeficit of the rift depression
will flex the elasticlayer outsidethe yield zone, creating
flanking uplifts. The externalforce requiredto maintainactive rifting increaseswith the amountof
lithosphericstretching,
indicatingthat rifting is a quasi-static,
stableprocess. Becausethe yield zone
will revert to elasticbehaviorif the externalforce causingextensionis removed,the model predicts
that the rift depressionand flanking uplifts will be preservedafter extensionstops. Our simple
mechanicalmodel demonstratesthe inherent relationshipamong graben formation, lithospheric
thinning,and rift shoulderuplift in rift zones.
INTRODUCTION
when
Rifting occurs where lithospheric extension is localized
into a narrow zone.
Rift zones have been identified both in
an initial
normal
fault
causes elastic
flexure
in a
brittle lithosphere, a second normal fault will form where
the
flexural
stresses
are
maximum.
The
initial
and
continentsand on ocean floor [Burke, 1978; Ramberg and
Neumann,1978; Easton, 1983;Rambergand Morgan, 1984;
Rosendahl, 1987]. Since continentalrifts are a precursorof
new oceanbasinsand mid-oceanridge rifts occurwherenew
oceanic crust is generated,understandingthe evolution of
rifts is of fundamentalsignificanceto the study of plate
tectonics.
Models which attempt to express our
understandingof rifting mechanismscan be put into two
major categories. The first categoryattributesrift structural
and morphologicalcharacteristicsto thermal processesin
the lower lithosphereand asthenosphere.For example,rift
shoulderuplift has been explainedas the resultof (1) lateral
conduction
of heat from extendedto unextended
lithosphere
[e.g., Cochran, 1983]; (2) small-scale mantle convection
driven by a lateral thermal gradient[e.g., Buck, 1986]; and
(3) reequilibrium of a thermally stratified lithospherein
which the amount of extensionvaries with depth [e.g.,
Royden and Keen, 1980; Keen, 1985, 1987]. These models
are concerned primarily with the thermal and mechanical
behaviorof the lower lithosphereand asthenospherewhich
deformin ductile (viscous)flow at high temperatures
rather
than brittle deformationof shallowlithosphere.
The secondcategoryof modelsinvestigateexplicitly the
secondarynormal faults thus comprise the boundary of a
sinking rift graben. Bott [1976] and Bott and Mithen
[1983] modified Vening Meinesz'smodel to includea ductile
layer in the lower crust. In the modified model, while the
brittle upper crustis stretchedby normal faulting and graben
subsidence, the ductile lower crust flows horizontally
beneath the subsiding graben.
These models, while
attractive for their simplicity, cannot describe the extensive
faulting and crustal thinning commonly observed in rift
0148-0227190/89JB-03059505.00
analysis assumes a uniform state of horizontal extension
zones.
Deformation of brittle lithosphere in rift zones has also
been treated as necking of a stronglayer overlying a weaker
substrate. Arternjev and Artyushkov [1971] qualitatively
considered rifting as necking of a ductile crust which is
mechanically strong near the surface and weaker at depth.
They proposedthat a perturbationin crustal thicknesscould
induce nonuniform extension, which localizes stresses and
generatesa narrow zone of normal faults in the strongupper
crust. This hypothesiswas further developedand testedby
Zuber and Parrnentier [1986] and Parrnentier [1987], who
modeledlithosphereas a perfectly plastic layer, a continuum
description of deformation on many faults. Zuber and
Parrnentier [1986] found that an initial perturbation in
deformationof brittle lithospherein rift zones,especially lithosphericthicknesswill amplify under extension. When
thoseat the initiation stage. Vening Meinesz [1950] first this occurs, extension in the lithosphere will concentrate
suggested that a rift forms in an extending elastic into a zone with a width comparable to the dominant
lithospherewhich fails by normalfaulting. He arguedthat wavelength, which dependsonly on lithospheric thickness
and theology. These necking instability models, however,
are
basedon the assumptionof small deformationamplitude
Copyright1990 by the AmericanGeophysicalUnion.
and are not applicableto later stagesof rifting when large
Paper number 89JB03059.
finite strains have developed. In particular, since the
4909
4910
LIN AND PARMENTIER: RIFTING OF BRrlqLE LITHOSPHERE
onto which small-amplitudeperturbationsare superimposed, postseismicrelaxation of crustal earthquakes[Rundle, 1982;
extensional
deformation
is not confined to a narrow zone
Savage and Gu, 1985] but shouldhave a secondorder effect
on long-term (--1 m.y.) rift basin-scale deformation [e.g.,
within the rift [Parmentier et al., 1987].
In this paper, we formulate a finite amplitude necking Phipps Morgan et al., 1987; Lin and Parmentier, 1989].
model of rifting in brittle lithosphere overlying ductile Ductile (viscous) flow of the crust and mantle along with the
(viscous) substrate.
Brittle failure and lithospheric resulting temperature variations could be treated by
extension are localized in a plastic yield zone beneath the combining various ductile lithospheric stretching models
rift basin.
We demonstrate that the formation of such a
[e.g., Keen, 1985, 1987] with our present analysis for
For
region of necking makes it possibleto explain lithospheric necking in a strong brittle lithospheric layer.
thinning, graben subsidenceand rift shoulder uplifts in a simplicity, we treat the ductile lower lithosphere beneath
continentalrifts and the asthenospherebeneathoceanic rifts
self-consistentfashion. In the following sections we will
examine the major characteristicsof the plastic zone, its as inviscid fluids.
interaction with adjacent undeforming brittle lithosphere,
and its control on rift evolution.
LithosphericFaulting and Perfect Plastic Flow
In an extensively faulted media, if the characteristic
Momm MI•cmu•rICS
spacing between faults in the media is much smaller than
Studies of the rheology of the continental lithosphere the size of the deforming region, we can approximate the
[e.g., Goetze and Evans, 1979; Brace and Kohlstedt, 1980; motion on faults by perfect plastic flow. Closely spaced
Kirby, 1983] suggest a strong variation of strength with fractures and faults occur in many continental and oceanic
depth. Brittle deformationoccursin the upper crust, where rifts. In the Rhine graben, for example, the rift graben is
lithospheric strength is controlled by frictional sliding on framed by two inward facing masterfault zones40 km apart,
faults and the mode of faulting [Byerlee, 1968]. At greater while the rift valley itself is broken into internal tilted
depths, dislocation creep, which is chiefly a function of blocks by many sets of secondary normal faults. The
strain rate and temperature, is the dominant deformation characteristicdistancebetween thesesecondarynormal faults
mechanism. We idealize lithosphereas a brittle layer with is only 1 km [lilies, 1972], far smaller than the 40 km
thickness
H anddensity
Pl overlying
a fluidwithdensity
Pa width of the rift valley. Since we are concernedprimarily
(Figure 1). This brittle layer representsthe upper 12-15 km with the overall characteristics of rifting instead of the
of a prerift continentalcrust or the upper few kilometers of specific motion on individual faults, perfect plastic flow is
oceanic lithosphere at a mid-ocean ridge. In general, an acceptableapproximationto motion on faults. Based on
stresses associated with ductile deformation
in the lower
this same rationale, perfect plasticity has also been applied
continental crust and oceanic upper mantle depend on the to describe deformation in mountain belts and accretionary
width of the necking region and the rate of lithospheric wedges [Elliott, 1976; Chapple, 1978; Davis et al., 1983].
extension.
Viscous
stresses associated
with
ductile
In our model, the mechanical behavior of a rifted brittle
mantle
lithosphereis characterizedby a plasticyield zone embedded
flow can influence greatly the short-term (100-200 years)
ß
0.SWs
'
•'Fx
....................................
•.__::__:
-
DUCTILE SUBSTRATE :-:=-:•i
Fig. 1. A model of rift zone and definitionof geometricparameters. Rheologicalstratumof rifts is modeledas a
brittle layer overlyinga ductile substrate. The rift zone is characterizedas a plastic yield zone embeddedin an
otherwiseelasticlithsophere. In general,it is possiblefor an upperplasticyield zone and a lower plasticzone to
coexist,with N as the connecting
point. The relativesizesof the two plasticzonesis describedby the locationof
connecting
point
N. .H.[
istheriftbasin
depth,
andHb istherelifonthelower
plastic
zone.WA isthewidth
ofthe
activerift valley,andw S is thedistance
between
rift shoulders.
LIN AND PARMENTIER: RIFTING OF BR1TrLE LITHOSPI•RE
in an otherwise elastic lithosphere(Figure 1). This model
does not describethe initiation stagesof rifting but assumes
that many faults defining the yield zone are already present.
In general, plastic solutions allow the coexistence of an
upper plastic zone (area A'NA in Figure 1) and a lower
plasticzone (area I'NI in Figure 1). The locationof point N
is not determined by mechanical equilibrium conditions
alone but by energy considerations,as will be discussed
later. The importantgeometricparametersinclude the depth
of rift basin, Ht; the relief on the bottom of the plastic
layer, Hb; the width of rift valley, WA; and the distance
between the flanks of rift shoulders,WS. Since the
mechanical properties of the lithospheric material are
different inside and outside the plastic zone, different
mathematical formulations are needed, accordingly.
Solutions will be developed first for the plastic zone and
then for the elastic plates.
Perfect plasticity is characterizedby the existence of a
mechanical yield point beyond which permanent strain
appears. While most metals at low temperaturesobey the
pressure-independentvon Mises failure criterion, soils and
sands commonly exhibit pressure-dependent yielding
behavior in accord with the Coulomb (frictional-type)
criterion. Since the strength of the britfie lithosphere is
controlledby frictional propertiesof rocks [e.g., Brace and
Kohlstedt, 1980], the Coulomb criterion is more appropriate
for crustalmaterial. Although the two failure criteria predict
deformation patterns which are different in detail, the
fundamental
features
are
the
same.
To
maintain
both
generality and simplicity, the mathematical formulations
that we give are applicableto both von Mises and Coulomb
materials,while the calculatedexamplesemploy the simpler
von Mises
material.
4911
Y
A'
0
A
X
lines -'"
•
•
o• lines
N
Fig. 2. The x-y coordinatesystemand plasticcharacteristic
lines
in a rift yield zone. The inclined lines are lines of stressor
velocitycharacteristics.For avon Misesmaterial,the stressand
velocity lines coincide. For a Coulomb-typematerial,however,
stress characteristic lines have different inclination angle from
velocity characteristiclines. The parametersct and [5 represent
conjugate characteristiclines.
plates can be regarded as rigid and the changes in the
plastic-elastic boundaries due to elastic deformation can be
neglected [Hill, 1950]. Therefore the deformation of the
elasticplates is consideredonly when we later calculatethe
uplift of rift shouldersdue to the formation of a rift basin.
A complete descriptionof flow within the plastic zones
requires the determination of stress and velocity at each
point.
For plane strain deformation there are five
The yield stress of avon Mises material is equal to a
pressure-independentparameter k, whose value depends
mainly on the material type. In this type of material, unknowns:
threecomponents
of stresses
Ox,Oy,'•xy and
brittle failure occurs along planes on which the maximum two components
of velocityUx, Uy. Five independent
resolved shear stress '• = k. The von Mises criterion can be
equationsare therefore necessaryto solve for both stresses
written
as
and velocities.
(•3-ol)2/4= (Ox-Oy)
2/4+•xy2=k2
(1)
where Ol and o3 are the minimum and maximum tensile
stresses,
Ox,Oy,and'•xyarethestress
components
in a x-y
The yield criteria, equation (1) or (2), provide one of the
basic equationsfor determiningstressfield. The other two
basic equations required for the determination of stresses
come
from
the
consideration
that
stresses
must
be
at
equilibrium at each point:
coordinatesystem(as definedin Figure 2).
The yield stressof a Coulomb(frictionaltype) materialis
3Ox/3X+ 3•:xy/3y= 0
(3a)
pressure-dependent. The frictional sliding on a typical
failureplane is governedby h:l= Co + (-On) tan•, whereCO
•y/By + 3•xy/BX= - plg
(3b)
and • are the cohesionand angle of friction of the material
where g is the accelerationof gravity.
and On is the normal stressacrossthe failure plane (Figure
Although equations(1), (3a) and (3b) are the fundamental
3a). In an x-y coordinatesystem,this criterionis expressed equationsin the plastic problem,the mathematicalnatureof
as
these equations is difficult to understandwithout further
derivations. We therefore expressthe stressesas functions
(Ox-(•y)
2/4+'•xy
2 = [(Ox+C•y)
sin2(])
+Col
2 (2) of two new parameterso and 0 [Sokolovskii, 1965], where
• =- p + Co/tan•, p is an averagestress,and 0 represents
The von Mises criterion can be regardedas a special caseof
the angle between the maximum principal stress direction
the Coulomb criterion, when Co = k and • = 0.
and the x- axis (Figure 3b),
Stress Field
in Plastic
Yield Zone
The magnitude of the plastic flow strains within the
centralplastic zone is far greaterthan the elastic strainsin
the adjacent lithospheric plates. When calculating the
deformationof the plastic zone, these elastic lithospheric
Ox = o (1 + sin• cos20) - Co/tan•
(4a)
Oy= o (1 - sin•cos20)- Co/tan•
(4b)
Xxy= o sin•sin20
(4c)
4912
LIN AND PARMENTIER: RIFTING OF BR1TIZE LITHOSPHERE
•T
(1 + sin lb cos20)ao/ax + sinlbsin20 ao/ay - 20 sinlb
(sin20aO/ax - cos20ao/ay)= 0
(5a)
sin lb sin20ao/ax + (1 - sin• cos20) ao/ay + 20 sinlb
(cos20
aO/ax+ sin20aO/ay)= - pig
(5b)
This system of equationsis hyperbolic and can thereforebe
solved by the method of characteristics[Sokolovskii, 1965,
p.29].
The solutionsto equations(5a) and (5b) are describedin
terms of stress characteristic lines [Hill, 1950]. In von
Mises material, the stress characteristic lines (also called
slip lines) coincide with directions of maximum resolved
shear stress. Slip lines with a right-handedor left-handed
(a)
senseof shear are called ot lines or [• lines, respectively
(Figure 2). In an x,y coordinatesystem, they form a righthanded
rxy
o-n
set
of
curvilinear
coordinate
axes
where
the
algebraically largest (least compressive)principal stresslies
in the first and third quadrants(Figure 3c).
Except for a few geometricallysimple casesfor which full
analytical solutions can be obtained, the hyperbolic
equations (5) must be solved numerically. Sokolovskii
[1965] showsthat theseequationsare reducibleto a simpler
set of equationsof characteristicsß
dy = dx tan(0 • •;)
(6a)
(b)
do= 2(Itanq•
dO+ plg/cosq•
(ß sinq•
dx+ cosq•
dy) (6b)
where e = •:/4- q•/2. The first equation of equation (6a)
(with minus sign in front of e) describesthe geometryof o•
lines, and the first equationof equation(6b) (with plus sign
i
I
in frontof plg/COSq•)
describes
therelationship
between
(•
and 0 along the same o• lines. Similarly, the second
equationsof equations(6a) and (6b) apply to the [1 lines.
From these basic equations, a slip line field can be
constructed numerically
using finite
difference
approximations.
The stress boundary conditions are known only at the
upper and lower surfacesof the plastic yield zone. At the
upper surface,stressfree conditionsapply, namely,
(c)
Fig. 3. (a) Failure criteria of frictional or Coulomb-type (pressure
dependent) material. At any point in space the Mohr failure
envelopeconsistsof two straightlines with slopesof :L-tan•(where
(•n = 0
(7a)
(•s = 0
(7b)
At the lower surface (the interface between the plastic zone
• is theangleof internalfriction)andintercepts
of :kC
o. (b) Stress and ductile substrate), the shear stress is zero due to the
convention. Compressivestress is negative, with 03>01
assumed inviscid ductile substrate, and the hydrostatic
algebraically
(03 is theleastcompressive
stress).(c) Relationof normal stressis given by
stresscharacteristiclines to principal stresses. Stresscharacteristic
lines are designatedeither •t lines or 13 lines dependingupon
whether they have a right-handed or left-handed sense of shear
across them, respectively. The angle formed by the stress
characteristiclines is •/2-•; 0 is the counterclockwiseangle from
thex direction
to theo3 direction.
In this functional transformation, we have assumed that the
material is at yield so that the failure criteria (equation(2))
canbe employed
to reduce
thethreeunknowns
(•x,(•y,and
Xxyto onlytwounknowns
(5and0. Substitution
of (4) into
(3) results in a quasi-linear system of first order partial
differential equationin (•(x,y) and 0(x,y):
(•n = -Plg H + PagHb
(8a)
Os = 0
(8b)
where Hb is the vertical displacementof the lower surface
from its initial height.
Since the stress boundary
conditions in equations (7) and (8) do not involve
velocities, the stressfield in the plastic zone is determined
independentof velocity field [Hill, 1950]. The procedureto
determinenumerically the slip line field in the plastic zone
is given in Appendix A.
LIN AND PARMENTffiR: RIFTING OF BRITrLE IATHOSPItERE
Velocity FieM in Plastic Yield Zone
Once the stressfield has been determined,a velocity field
can be calculated given sufficient velocity boundary
conditions. To determinevelocity boundaryconditions,we
consider that the two elastic lithosphericplates move away
from the center with a speedUo.
For an incompressible material, the divergence of the
velocity field must vanish:
3Ux/3X+ 3Uy/3y= 0
S' A'
BRITTLE
4913
A S
ZONES
•'
Fx
iLAYER
EL STIC
I
(9)
A material property of an isotropic plastic material is that
the principal axes of stress and strain rate coincide [Hill,
19501:
2'Cxy/(Ox-Oy)
= (SUy/SX
+ 8Ux/Sy)/(SUx/SXBUy/By)
(10)
Substitutionof (6) into (10) gives
(3Ux/•Jy
+ 3Uy/•Jx)tan20+ (3Ux/•Jx-3Uy/•Jy)
=0
(11)
Equations (9) and (11) form a system of hyperbolic
equations for velocity field, whose characteristic lines
coincide with the stress characteristiclines (slip lines) for
von Mises material only. Similar to the stress equations,
these velocity equations are reducible to the equations of
characteristics [Salencon, 1974]'
duct-ucttan•)d(0-•)/2)- u[3/cos
•) d(0-•)/2)= 0
along an ct hne
(12a)
du[3+ uct/cos•)
d(0+ •)/2)+ u[3tan½d(0+ •)/2)= 0
alonga 13line
(12b)
From (12) it follows that velocity characteristic lines are
lines
of zero incremental
extension
and that instantaneous
Fig.
4. (a) Calculated plastic zone geometry and stress
characteristiclines for a rift zone with inclined topography. The
solid lines are 0t lines and dashedlines, are [5lines. ( b) Calculated
velocity field corresponding
to stresssolutionsof Figure 4a. Note
that major velocity discontinuitiesoccur at the boundarybetween
the plastic zone and elastic plates, while secondarydiscontinuities
occur along severalplanes inside the plastic zone.
total surfaceextensionAL. At AL = 0, the upper and lower
boundariesof the yield zone have not deformed, and both
the ct and I• lines are inclined to the horizontal surfacesat
angle •;/4. The velocity field at A L = 0 indicates four
blocks of distinctive instantaneousmotion (Figure 5b).
Simple Shear DiscontinuitiesFraming
Plastic
Yield Zones
discontinuitiesin the velocity field can occur only in the
As the two elastic lithosphericplates move apart at a
tangential planes across these lines. Finite-difference
equivalences of equation (12) are used to determine velocity uo, the material in the upper plastic zone moves
numerically the velocity field in the plastic zone (see downwardat speeduo, while the material in the lower part
moves upward at the same speed. The change of the rift
Appendix A).
zone geometry can be readily observedby comparisonof
the rift configuration at A L = 0 and that at A L -- 0.2 H
RESULTS
(Figure 5). At AL = 0.2 H, both the upper and lower plastic
The calculated geometry of plastic yield zones with a zones become smaller from the initial configulations, but
prescribedrift valley topographicslope is shown in Figure they are still connectedat the same point. Notice that the
4a for avon Mises material. The correspondingvelocity material within the narrow, shaded bands at AL = 0.2 H was
field in the plastic zones in Figure 4b shows several within the plastic zone at AL = 0 but now has become a part
velocity discontinuity planes in the plastic zones. The of the elastic plates. A finite amplitude simple shear
most prominent velocity discontinuity is along the develops discontinuouslyacrossthe slip lines defining the
boundary between the plastic zones and the elastic edge of the yield zone. In the example illustratedin Figure
lithospheric plates (line ABCN and line A'B'C'N). 5, point N is assumedfixed, so are the location of the slip
Secondary velocity discontinuities occur along several lines with simple shear. Clay experiments illustrated in
planes inside the plastic zones (e.g., line OB and line OC). Figure 6 [Cloos, 1968] show similar velocity fields to those
The magnitudeof the secondaryvelocity discontinuitiesis in Figure 5b with narrow lines across which velocity
proportional to the topographicslope.
vectors change direction dramatically. Similar lines of
If the top and bottom of the plastic zones are initially velocity discotinuities have also been observed in sandbox
flat, deformation within the these zones is considerably experiments [Horsfield, 1980].
simpler. The evolution of such plastic zones is shown in
According to mechanical characteristicsand deformation
Figure 5 for a rift with symmetricupper and lower plastic history, a rifted brittle lithospherecan be divided into three
zones. Both the stress patterns (left) and instantaneous subregions(Figure 5a). Region 1 is within the plastic yield
velocity fields (right) are illustratedfor different amountsof zone. The upper plastic zone is the rift graben. The size of
4914
LIN AND PARMENTIER:RIFTING OF BRrITLE IiTHOSPItERE
VELOCITY
DEFORMATION
PATTERN
A'
FIELD
A
AL=O
AL = 0.2H
AL = 0.4H
AL = 0.6H
-H
(A)
REGION
1- DEFORMING
REGION
2: DEFORMED
(B)
PREVIOUSLY
REGION
3: UNDEFORMED
Fig. 5. Evolutionof rift geometry
in absence
of regionalisostatic
compensation.
(a) Plasticzonegeometryat various
stagesof rifting. Thereare threeregionswith differentdeformation
characteristics
andhistory. Region1 is within
the plasticzone. The inclinedlinesin theplasticzonesare thelinesof velocitycharacteristics.
Internaldeformation
canoccurin thisregion. In region2 thematerialhaspreviously
deformed
butis presently
a partof theelasticplate.
Region3 is theuncleformed
partof thebrittlelithosphere.
(b) Velocityfieldsat variousstages
of rifting. In general,
the velocityfieldsin theplasticzonesare notuniform.The exampleshownis for a specialcasewhenthe top and
bottomboundaries
of the plasticzonesareflat. Notethatvelocities
arediscontinuous
at the boundaries
betweenthe
plastic zones and the elastic plates.
Region
3 is undeformed
material.In
this region decreasesas rifting progresses. If the progresses.
rifts,majoractivenormal
faultswerefoundto
topography
of rift valley is not flat, internaldeformationcontinental
zones
framing
rift grabens
[lilies,1972;
will occur within this region and velocity will not be be withinnarrow
uniformas in this simpleexample. The inclinedlines are
Ebinger
etal.,1984;
Shudofsky,
1985;
Rosendahl,
1987].
velocitycharacteristic
lines. Region2 is presently
partof It wasalsofoundthatall of theminoractivefaultsoccur
theriftgraben,
andthesurface
scarps
beyond
the rigid lithospheric
plate,although
it deformed
in thepast onlywithin
faultsystem
correspond
to inactive
by plasticflow. The sizeof thisregionincreases
asrifting themajorboundary
LINANDPARMENTIER:
RIFTING
OFBRITTLE
LrrHOSPHERE
4915
J
%•.
_
•
r,,
J
:,,
/,,r/l,;-
4--•--.•
2x Fg'
,,
,T
::
•,
•//•
I
I
..
--_•
I
•
I
• .... •
I
I
• • •cm
b
•ts
duringderogation
Fig. 6. (a) Grabenformationin clay experiment,
in whichclay Scale• •. (b) Migrationof reference
photographs.
Note the
grabenslid downbetweenblocksafter symmetrical
pull [Cloos, [Cloo•, 196S]as tracedfrom successive
1968]. In this particularexperiment,displacementalong the close sim•afity betweenthis velocity field and the predicted
grabenboundary
is 20 mm on fightsideand22 mm on left side. vcl•ity fieldin theupperplastic•nc in Figure
faults [Gibbs, 1984; Ebingeret al., 1987]. The deformation be expressedin terms of the layer thicknessH, the
patternof our model is generallyin agreement
with these thicknessof the upper plastic zone HN, and the total
amount of surface extension AL:
observations.
The predictedsurfaceslopeT (Figure1) of the deformed
Ht = (1/2) AL (if theupperplasticzoneis present) (14a)
region (in absenceof erosion)is given by
tanT = 0.5
(13a)
or
T = tan-1(0.5)= 27ø
Hb = (1/2)AL (if thelowerplasticzoneis present)(14b)
(13b)
This slopeis inherentlydetermined
by the initial surface
slope (0ø) and the inclinationof the shearslip lines
framing plastic zones (45ø) and is independent
of the
separation
velocityof the two elasticplates.
Althoughthe stressandvelocitylines coincidefor the
yon Mises material as in the example of Figure 5, they
might not in other cases. For the same boundary
conditions,
if a Coulomb-type
material
with30ø of angleof
internal friction was considered instead, the stress lines
AH = Ht + Hb
(14c)
WA = 2HN - AL
(14d)
WS = 2HN + AL
(14e)
where A H is the amount of lithospheric thinning due to
rifting. We note that while the width of rift valley WA
narrows, the distance between the flanks of rift shoulder,
W S, increases.The difference
betweenthesetwo widths,
therefore, reflects the amount of rift extension. More
examplesare shownin Figure 10.
velocitylinesremainat 45ø inclination.Sincethe finite
Even for rift zones at their early stagesof rifting (suchas
strain in plastic zonesis determinedby the incremental
the
Lake Tanganyikaof the EastAfricanRift), stratigraphic
accumulation of displacements, it is the velocity
records
on sedimentarybasinsindicatenarrowingof active
characteristiclines, not the stresscharacteristic
lines, which
deformationzones [Ebingeret al., 1984, 1987;Rosendahl,
controlthe evolvinggeometryof a rift zone. In addition,
1987]. Other examplesof narrowingof extensionzone can
which of the velocity characteristiclines is active is
be found in the Rhine graben[Villemin et al., 1986], the
determined
by specificvelocityboundaryconditions.It is
U.S. Atlantic passivemargin [Stecklerand Watts, 1982;
thereforeincorrectto interpretstresscharacteristic
lines as
would incline 60ø to the flat surface(equation(6a)) while
faultsor evenas planesof potentialfaults,aswassuggestedKligfield et al., 1984], and the Baikal rift [Zorin and
in the earliermodelof Hafner [1951].
Rogozhina, 1978].
Althoughthe exampleillustratedin Figure5 is for a
DISCUSSION
specialcasewhen the upperand lower plasticzonesare Stress Distribution in Plastic Zones
symmetric,the conclusions
hold for other rifts with
It is easy to verify that the stressfield in the upper
asymmetric
upperandlowerplasticzones.SeeFigure8 for
plasticzone (Ht < y < HN) is givenby
anotherspecialexamplein which there is only an upper
plastic zone.
Thinningof Lithosphereand Shrinking
of Plastic Zone
As rifting progresses,
the plasticzonesbecomenarrower.
This reduceslithosphericthicknessbeneaththe rift valley
Ox= -PlgY+ PlgHt+ 2k
(15a)
Oy= - plgy+ plgHt
(15b)
Xxy = 0
(15c)
The Similarly,in the lower plasticzone (HN < y < H-Hb) the
geometric
variablesof the rift valley andplasticzonescan stressfield is given by
and narrows the width of rift valley (Figure 5).
4916
LIN AND PARMEN'I•R:
RIFTING OF BR1TrLE LITHOSPHERE
Ox= -plgY + (Pa-Pl)gHb
+ 2k
(15d)
Oy= - plgy+ (pa-Pl)gHb
(15e)
Xxy- 0
-2
(15f)
0
(A)
\L-]A
/
!
st
!
i
s
i
Forconvenience,
we definedeviatoric
stresses
•x, •y,
•xyasthestresses
inexcess
ofthelithostatic
overburden,
O
namely,in theupperplasticzone,
•x = Ox+ plgy= piglit+ 2k
(16a) •
•y = Oy+ plgy= plgHt
(16b)
•xy = Xxy= 0
(16c)
j'-'N•
-$
I
I
i
i
-8
/
/
t
10
Similarly, in the lower plastic zone,
•x = øx + PlgY= (Pa-Pl)gHb
+ 2k
(16d)
•y = Oy+ plgy= (pa-Pl)gHb
(16e)
•xy = Xxy= 0
continental crust [Brace and Kohlstedt, 1980].
Work
,
I
0
o00
300
(MPa)
(16f)
Therefore, when the top and bottom boundaries of the
plastic zones are flat and when the boundary stressesare
uniform, the deviatoric stresses in the plastic zones are
uniform. An example of the deviatoric stress solutions
(equations (16)) are plotted in Figure 7 for a 10-km brittle
lithospherewith a yon Mises yield limit of 2k = 80 MPa, a
value that correspondsto the average brittle strength of
Minimum
,
I
!
-2
0
I•
-4
i•
/'i\
o
!i •:A
/
,'
II/
-$
Solution
i
For the deformation of perfect plastic material the
-8
/
equilibrium equations alone do not always lead to unique
I
!
stress and velocity fields. An example of this nonunique
nature can be seen in Figure 1, where plastic solution
-10
I
!
I
permits point N at any level. The correspondingsolutions
satisfy all boundary conditions as well as the failure
-300
0
300
criterion. Therefore it is necessaryto include supplementary
mechanical considerations.
A preferred solution is
determinedby energy considerations.
The rate of increaseof gravitationalpotential U of plastic Fig. 7. Vertical distributionof deviatoricstressesalong line SANIJ
(Figure 1). (a) The long dashedline showsthe horizontaldeviatoric
zones is given by
stress•_ as a functionof depth. Lithostatic
stressis plottedby a
(MPa)
dU/dt
=Jv- Plguy
dxdy
........................
,•,,•.•. to/The long dashednne showsthe
(17a)vertical
deviatoric
stress
•y asa function
of depth.Again
lithostaticstressis plotted by a short dashedline.
whereUyis thevertical
velocity,
andtheintegration
should
be performedover the entire plastic zones(areasA'NA and
I'NI in Figure 1).
The rate of energy dissipationE by the plastic zones in
resisting the shear strength of the lithosphericmaterial can
dW/dt
=Js=•)v
uitids
(17c)
whereti (i = 1,2) are the normaland tangentialcomponents
of tractionforceson plasticzoneboundaryand ui (i = 1,2)
are the normaland tangentialcomponents
of the velocity
vector.
The
integration
should
be
performed
over the entire
dE/dr
=Iv(1/2)(;}ui/;}x
j +;}uj/;}xi
) oijdxdy(17b)
be evaluatedby
where ui (i = 1, 2) are the x, y componentsof velocity field
andoij (i,j = 1, 2) arecomponents
of stress
tensor.
The rate of work W performedon the plasticzonesby the
two adjacent elastic plates and the inviscid substrate is
given by
boundaryof the plastic zones (lines A'A, AN, and NA and
linesNI, II', andI'N in Figure 1).
We know that the work done by the forces transmitted
throughthe elasticplatesand by the substratemustequal
energydissipated
in the plasticdeformation
plusthe change
in potential energy:
LIN AND PARMENTIER: RIFTING OF BR1TrLE IZrHOSPItERE
dW/dt = dE/dt + dU/dt
4917
stress distribution in the plastic zones should be
reevaluated. In Appendix B, we derive analytic solutionsfor
It is straightforward to evaluate the rate of increase of the stressand velocity fields of a viscous half-spacewith a
gravitational energy of plastic zones during rifting using flat surface. We show that although the stressfield in the
viscous half-space is a function of the width of the lower
equations(15) and (17a):
plastic zone, the total rate of energy dissipationin resisting
dU/dt
=- uoPlg
[(HN-Ht)
2- (H-HN-Hb)
2] (19a)viscous flow does not depend on the geometry of the lower
plastic zone. Therefore, the conclusionswe have reached
Similarly, the rate of work performed by the two elastic for inviscid substrate hold valid for viscous substrate as
plates and the inviscid substrate can be elavulated using well.
equations(15) and (17c):
The forces and bending moment acting on the elastic
(18)
plate are calculatedby directly integratingthe stressesgiven
dW/dt = 2uok (H-Ht-Hb)
in equations(15) along the edge of the elastic plate (line
- uoPlg
[(HN-Ht)
2 - (H-HN-Hb)
2]
(19b)
SANIJ in Figure 1):
To evaluate the rate of energy dissipation by plastic
zones (equation (17b)), we note that velocity gradient is
presentonly along the plastic zone boundaries(lines AN,
A'N, NI, and NI' in Figure 1). The resultant tangential
velocitycomponent
alongtheseboundaries
is •
FxTM
ISANIJ•x dy
=2k(H-Ht-Hb)
+(1/2)[plgHt
2+(pa-Pl)gHb
2]
uo, and
the resultant shear stressalong these boundariesis k. The
integration in equation (17b) is therefore replaced by the
summationof stresswork along lines AN, A'N, NI, NI' in
Figure 1, which gives
dE/dt = 2uok (H-Ht-Hb)
+ plgHt(HN-Ht)
+ (pa-Pl)gHb(H-HN-Hb) (20a)
Fy=ISANiJ
O'-y
dy
(19c)
From the expressionsof equations(19a), (19b) and (19c), it
is obvious that the fundamental relationship of equation
(18) is satisfied.
Examining the deformationhistory in Figure 5, we notice
that the gravitational potential of the upper plastic zone is
reducedduringextensionwhile that of the lower plasticzone
is increased. The reduction rate of the gravitational
potential is maximum when there is only an upper plastic
zone (H=HN in equation(19a)). We argue that this is the
preferredmode of deformationin rifts sincethe rate of work
required by the elastic plates and ductile substrate is
minimumized (equation(19b)). In this preferredsolutionfor
rifting (Figure 8), the upper plastic zone collapsesunder its
own weight; the gravitationalpotential releasedis converted
into energy dissipationin plastic flow.
= (1/2)[piglit
2+(pa-Pl)gHb
2]
+ plgHt(HN-Ht)
+ (pa-Pl)gHb(H-HN-Hb)(20b)
For a specificexamplewith HN = H andpa = Pl' the
calculated
forcesFx andFy areplottedin Figure9a asa
function
of
the
surface
extension
AL.
We
note
that the
required driving force Fx increaseswith the amount of
extension. The increasein the requireddriving force is due
to the deepeningof the surfacedepressionat the rift and the
resulting horizontal forces createdby the topography. This
implies that a greater driving force is required to achieve a
greater amount of extension. In this case quasi-static,
stable rifting of brittle lithosphereoccurs. If HN • H, the
required extensional force depends also on the density
difference between the lithosphereand asthenosphere(Figure
When a viscous substrate is considered, additional terms 9b). For a specified rift plastic zone geometry (a given
due to viscous stresses must be added to the boundary H N), smaller driving force is required to causerifting in a
(Pa < Pl).
conditions of lower plastic zone (equation (8)), and the lithosphereoverlying a less denseasthenosphere
Uplift of Rift Shoulders
The formation
of a rift basin causes a mass deficit
at rift
zone and unloads the lithosphere. Since the lithospheric
plates outside the rift zone have elastic strength, the
lithosphere responses to the deepening rift basin by
regional isostatic compensation:flanks of the rift shoulder
deflect upward by elastic flexure, and the wavelengthof the
deflection is wider than the rift graben itself. The amount
of
elastic
deflection
can be calculated
as a function
of
extension.
DUCTILE
SUBSTRATE
Thecalculated
verticalforceFyincreases
astherift basin
deepens(Figure 9a). This vertical force causesuplift of rift
Fiõ. $. A rift model in which the work requiredfor lithospheric shouldersso that the unloading due to rift basin formation
extensionis minimized. In this rift there is only one plastic zone
can be isostaticallycompensated. We have also calculated
which forms the rift valley. The motion in the viscoussubstrateis
inducedby the horizontalseparationof the lithosphericplates. The the bending moment Mo , which can cause uplift of rift
direction of viscousflow is shownby arrows on the streamlines.
shoulders as well. When the thin elastic plate outside the
4918
LIN AND PARMENTIER: RIFTING OF BR1TrLE LITHOSPHERE
whereD = EH3/[12(1-•2)]is theflexuralrigidityof the
2.0
HN/H = I
elasticplate, E is Young'smodulus,and• is Poisson's
ratio.
(A)
Theflexure
length
oftheelastic
plate
isl =[4D/(p!g)]
1/4.
For plausiblevaluesof E and• (Table 1), the flexurallength
I of a 10-km-thickelasticplate (H = 10 km) is about35
1.5
km.
1.0
Vening Meinesz[1950] predicteda rift valley width which
is controlledby the flexural length of an elasticplate. For
a 10-km-thick elastic plate with flexural length of 35 km,
the predictedVening Meinesz width of rift valley is about
28 km. In the presentmodel, however,the initial width of
the rift valley is proportionalto the original thicknessof
0.5
theupperplasticzoneHN. Fortheminimum
worksolution
in whichHN = H (Figure8), we predicta 20-kminitialrift
0.0
0.0
0.2
0.4
0.6
0.8
I
1.0
/tL/H
valley width when the brittle plate thicknessH is 10 km.
For other casesin which HN < H, even narrowerinitial
width of rift valley is expected.
The calculateddeflection for different stagesof rifting is
superimposedon lithosphericstretchingin Figure 10. It is
found that every kilometer of plate thinning will create
roughly 600 m of flank uplift. We also found that about
95%of rift shoulder
upliftis dueto theverticalforceFy and
the effect of bending moment M o can be neglectedin
1.10
AL/H
= 0.1
1.05
.............
,•.•o......,.
...... "'
1.00
?xh\9
%/" , .,'"
0.95
calculating first-order elastic deflection of the brittle layer.
This approximation has been made in simple models of
Vening Meinesz [1950], Heiskanen and Vening-Meinesz
[1958], and Jackson and McKenzie [1983].
Several authors have consideredother two processesthat
modify the uplifted rift shoulders: (1) erosion of the rift
shoulders and sedimentary filling of the rift basin [e.g.,
King et al., 1988; Stein et al., 1988], and (2) additional
long-term subsidenceor uvlift due to reequilibrium of the
thermally stratified lithosphere [e.g., Salveson, 1976,
0.90
0.85
0.0
0.2
0.4
0.6
0.8
1.0
HN/H
Fig. 9. (a) Inset illustratesthe forces and bendingmomentacting
on the elastic lithosphere, which are calculated by integrating
deviatoricstresses
alongline SAN. Solid line showsthe horizontal
deviatoricforce Fx as a functionof total surfaceextensionAL.
Dashed
lincshows
thevertical
loadFy asa fun_•io_n_
o.f•L. This
example
is shown
for HN = H, Pa= Pl = .•.UxlO"Mg/m-',k = 40
MPa. (b) Dependenceof the requiredextensionalforce on rift
geometryand the density differencebetweenthe lithosphereand
asthenosphere
. Thisexampleis shownfor AL = 0.1 H. WhenHN
1978; McKenzie, 1978; Keen, 1985, 1987; Hellinger and
Sclater, 1983; Steckler, 1985; Buck, 1986]. In this paper
we attemptto focus only on the fundamentalcharacteristics
of rifting in brittle lithosphere. The erosion and thermal
processeshave not been integratedinto the presentmodel.
A more obviousconsequence
of erosionis the smoothingof
rift topographydue to the sedimentaryfilling of rift valley.
Results of Figure 9a suggest that a rift with smoother
topography(correspondingto smaller AL) requires smaller
driving force Fx. It is plausiblethat continuous
erosion
might facilitate furtherdevelopmentof a rift by reducingthe
required extensionalforce.
Purely mechanicalmodelssuch as thoseexaminedin this
paper describerapid rifting in which the rate of lithospheric
extension is much faster than that of thermal conduction, so
that isothermsin the lithospheremove with the deforming
material.
In this case, the thickness of the undeformed
= H, thereis onlyan upperplasticzone. WhenHN = 0, thereis
onlya lowerplasticzone. Notethatfor a givenHN, the required lithospheredependsonly on the initial thermal state of the
lithosphere. On the other hand, if the rate of extensionis
extensional
forceis smaller
for smallervaluesof pa - PI'
slow, thermal conduction and advection could significantly
alter the thickness of the brittle lithosphere during rifting
[e.g., England, 1983;Sawyer, 1985]. Mid-oceanridge rifts
rift plasticzoneis subjected
to a lincloadFy anda bending are examplesof thermallycontrolledrifts, in which a steady
moment Mo at x-0, positionlocationof the rift shoulder, state lithosphere thickness is maintained [e.g.,Phipps
the equilibriumsmalldeflectionõ(x) can be evaluatedbased Morgan et al., 1987;Lin and Parmentier, 1989]. The width
on thin plate bending equations[Turcotte and Schubert, of mid-ocean ridge rifts often varies significantly along the
axis of slow spreadingridgeswith wider rift valley observed
1982, p. 128]:
in the vicinity of cold transforms [e.g., Fox and Gallo,
1984].
15(x)
=/2/(2D)
exp(-x/t)
[-Mosin(x/t)
+ (Fyt +Mo)cos(x//)]
Since the width of a rift is proportional to the
thickness of the brittle lithosphere, the wider rift valley
LIN AND PARMENTIER: RIFTING OF BR1TrLE LITHOSPHERE
Table
1.
Notation
Value Used
Symbol
Quantity
Ol
maximum compressivestres
minimum compressivestres
normal stress in the x-direction (horizontal)
normal stressin the y-direction (vertical)
o3
Ox
Oy
_xy_
Cx, Cy,
shear stress
k
von Mises
Co
cohesive strength
normal stressat top or bottom boundary
shear stressat top or bottom boundary
averagestress,equal to - (Ol + 03)/2
angle of internal friction
counterclockwiseangle between x-direction and 03
velocity componentin the x-direction
velocity componentin the y-direction
velocity componentalong an ct line
on
os
p
0
Ux
Uy
uot
Fy
Mo
8
4919
Pa
Pa
Pa
Pa
Pa
stresses in excess of lithostatie
failure
Units
overburden
coefficient
Pa
40
MPa
Pa
0
Pa
Pa
Pa
o
o
ms
ms
ms
velocity componentalong a [• line
horizontal force per unit length
vertical force per unit length
bending moment per unit length
vertical deflectionof elastic plate
surfaceslopeof deformedregions
ms
N
m
o
x
horizontal
vertical
WA
WS
width of active rift (rift graben)
km
distance between
rift shoulders
km
H
initial thicknessof brittle layer
thicknessof upper plastic zone
depth of rift valley
relief of plastic zone at bottom surface
amount of lithosphericextensionat surface
flexural length of elastic plate
density of brittle lithosphere
density of ductile substrate
accelerationof gravity
gravitational potential of plastic zones
rate of energy dissipationin resistingmaterial
shear strength
rate of work by externalsurfaceforceson
plastic zones
Young'smodulus
km
AL
1
Pl
Pa
g
U
E
W
coordinate
km
coordinate
Poisson's
-1
-1
Nm-1
Nm-1
y
HN
Ht
Hb
-1
-1
km
km
km
km
km
km
Mgm-3
Mgm-3
9.8
js-1 m-1
Js-1 m-1
3.0x1010
To consider the role of faulting and brittle lithospheric
deformation, we have formulated a model describing the
formationof rifts as finite amplitudenecking of an elasticplastic layer overlying a fluid substrate. A perfectly plastic
theology is a continuumdescriptionof faulting. This model
illustrates two important aspects of rift evolution that
previousmodels do not treat explicitly: the evolutionof the
Pa
Nm
flexural rgidity of elastic plate
viscosity of ductile substrate
CONCLUSIONS
-2
j s-lm-1
0.25
ratio
near transformsmight simply indicate colder thus thicker
lithosphere there.
ms
Pa s
rift width as extension proceeds and the finite strain that
occurs. A region at yield initially develops with a width
determinedby the thicknessof the layer and the geometry
of the bounding slip lines. As extension proceeds, the
surface within
the rift
subsides, and the width
of the
subsidingyield zone decreases. At any stage of rifting,
material in regions just outside the yield zone is deformed
but no longer deforming. The width of these deformed
regions increaseswith increasingextension. Vertical forces
due to the mass deficit of the rift depressionwill flex the
elastic layer outside the yield zone, creating flanking
uplifts.
4920
LIN AND PARMENTIER: RIFTING OF BRITTLE LITHOSPHERE
RIFT
TOPOGRAPHY
10000
RIFT
SHOULDER
RIFT
SHOULDER
i
5000
Ws
I
-5000
!
I
RIFT
VALLEY
-10000
-20
-10
0
10
20
DISTANCE (KM)
Fig. 10. Predicted successiverift topographyin a 10-km-thick lithospheric plate. The model includes the
subsidenceof rift graben and uplift of rift shouldersby n;gional isostaticcompensation. Note that as the rift basin
deepens,
theactiverift valleyWA narrows,
whilethedistance
between
rift shoulders
WS widens.
o = p' sinA/sin(A-/5')
This model makes several important predictions for the
morphology and structural evolution of rift zones. First,
faulting should extend to the rift shouldersbut not beyond,
and the rift
shoulders
will
widen
as extension
accumulates.
However, the rift, defined as the region in which the surface
is subsiding,narrows with increasingextension. Models in
which neckingis controlledby ductile flow predict that rifts
should widen with increasingextension[e.g., Keen, 1985].
Field examplesin rift zones show that the region of active
faulting and subsidencenarrows'with increasingextension.
This indicates the important role of deformation within a
brittle lithospheric layer in controlling rift structure and
evolution. Second, since the required extensional force
increases with the amount of extension, lithospheric
stretching in rift zones is a quasi-static, stable process.
Finally, because the yield zone will revert to elastic
behavior if the external force causingextensionis removed,
the model predicts that the rift depressionand flanking
uplifts will be preservedindefinitely after extensionstops.
APPENDIX A: CALCULATION OF STRESS AND
VELtXrrI•
FIELDS IN PLASTIC ZONE
Stress Field
The stress field in plastic zones is constructed in a
piecewise fashion downward from the upper surface and
upward from the lower surfaceof the plastic zones. As an
example, we show how to construct stress and velocity
fields for the upper plastic zone (area ABCNC'B'A'O in
Figures 4a and 4b). The procedurefor constructingthe
lower plastic zone is the same except that different stress
boundaryconditions(equation(8)), shouldbe used instead.
Since the upper surface is a free surface, On and Os are
zero along AOA'. To convertthesevaluesinto the variables
o and 0, we use the following expressions[Sokolovskii,
1965, p. 14]:
0 = )• + •
(A1)
(A2)
where
p'= [(on + Cn/tanO)
2 + Os]1/2
(A3)
15'=tan-l[os/(On
+ Cn/tanO)]
(A4)
A = sin-1(sin/5'/sin0)
(AS)
•, = •r/2 + 1/2(-A-/5')
(A6)
and 0t is the slopeof the upper surface.
Starting from line AO, we can successivelyobtain stress
fields for blocks ABO, BCO and CON according to
following procedure.
Step I (block ABO). Supposewe are to calculateboth the
x-y position of point P and the stressesat P from the given
positions and stressesof points 1 and 2 along line AO
(Figure 4a). To get the x-y coordinatesof point P, we
apply the the first recurrence formula (with + sign) of
equation (6a) along line 1P (0t line) and the secondformula
(with negativesign) alongline 2P ([• line). The two setsof
recurrenceformula in equation (6b) are then applied along
line 1P and 2P to determine the values of o and 0 at point
P. From the known stressesalong boundary AO, such
procedure can be continued to obtain stresssolutionsat any
point in block AOB.
Step 2 (block BOC). Due to a discontinuityin surface
slope at point O, there is a stresssingularityat the point O,
for which a special formula for stress is needed
[Sokolovskii, 1965, p. 26]:
o = (1 + sin0) exp [(n-20) tan0]
(A7)
LIN AND PARMENTIER: RIFTING OF BR1TrLE I.J'DtOSPHERE
4921
Approachingpoint O from boundaryBO and boundaryCO, where I.t is the mantle viscosity, z = x + iy and • = x- iy
the 0 value change from n/2-lx to n/2. Using the known are complex variables. The velocity field is given by
stressconditions along boundaryBO and formula (A1), we
Uo
....
l+i
-1
can constructthe slip line field in block BOC.
2n
z+s
z-s
Step 3 (block CON). Due to the symmetric requirement
(B4a)
about the center line ON, 0 is •;/2 along boundaryON. This
condition, together with the known stressvalues at OC, is where
sufficient for calculating stresses at all the points inside
block CON. Readersare referred to Sokolovskii [1965, p.
117] for more details in numerical procedures.
ux+iUy
=(•)[-iA+iA-B-B+iC-iC+(z-z)(+ -i)]
A=ln(
t-z)Itt_7_}-_soo (B4b)
Velocity Field
Once the slip line field is determined, the velocity field
can be constructed in a similar piecewise fashion (Figure
4b). This time, however, we need to calculate velocity in
the following order: first at block COC'N, secondat blocks
OBC and OC'B', and finally at blocks AOB and A'OB'.
Horizontal velocity is equal to +uo along boundaryABCN
and -Uo along A'B'C'N'. Along both boundaries,the vertical
velocity is zero. The finite-difference forms of recurrence
equations(12a) and (12b) are used in velocity calculations.
APPENDIX B: SOLUTIONS OF STRESS AND VELOCITY
.
-
(B4c)
..t--> + oo
(B4d)
C= in(t-z)lt=+ s
The rate of energy dissipation in resisting the viscous
motion
of the substrate is
dEviscous/dt=
OyyUy
dx
XxyU
xdx
(B5a)
where the first term on the right-hand side, the energy
dissipation due to the work of the vertical normal stress,is
FIEI2)S IN VISCOUS SUBSTRATE
In this section we derive the rate of work required to given by
deform the viscous substrate underlying the brittle
lithosphere. The mantle substrateis idealized as a viscous
half-space with flat top surface. If no slip occurs at the
interface between the brittle lithsosphere and the viscous
I_oo
OyyUy
dx
=(z•tu•)[
_ln(t)lt._
>0+ln(2s)l
(SSb)
subtrate, the velocities of the mantle substrate match those
of the deforming brittle lithosphere at the interface. and the secondterm, the energy dissipationdue to the shear
Considering a rift in which the base of the lower plastic stress, is given by
zone is 2s wide, the requiredvelocitiesat y = 0 are
ux = -uo,Uy=0
for-oo< x < -s
(Bl a)
ux = 0, Uy= uo
for -s < x < +s
(Blb)
Ux= Uo,Uy= 0
for+s< x< +oo (Blc)
XxyU
xdx= (
Batchelor, 1967].
We have obtained the stress and velocity solutions for
this problem using the complex-variable techniques of
Muskhelishvili [1953, p.584]. The stressfield is given by
2--(OyyOxx)
+iXxy
=
(B5c)
SubstitutingEquations(B5b) and (B5c) into (B5a), we arrive
at
whereux andUy arethehorizontal
andverticalvelocities
and Uo is the horizontal velocity of the undeformingplates.
For these boundary conditions,stressand velocity fields in
the half-space can be determined uniquely by solving the
equationson stressequilibrium and mass conservation[e.g.,
)[21n(t)lt-•
oo-ln(t)lt.->
0- ln(2s)]
dEviscous
tdt=
(8gug)[h(01t-->
ooln(t)lt-->
0l (B5d)
We note that the right-handside of equation(B5d) is not a
functionof the plastic zone width s. This implies that the
total rate of energy dissipation of the viscous flow is the
same for different sizes of the lower plastic zone.
Acknowledgments. We thank David Sparks, Jason Phipps
Morgan, Robert Detrick, Tim Byrne, and Paul Hess for comments
on the manuscriptand RossStein and GeoffreyKing for furnishing
preprints of their papers. W. Jason Morgan, Roger Buck, and
YongshunJohn Chen providedthoroughreviewsof the manuscript.
Ms. Mona Delgado typeset the manuscript. This researchwas
supported
by NASA grantNSG 7605 and supplemented
by the WHOI
Culpeper Young Scientist Award (J.L.) from the Culpeper
Foundation. Wood Hole OceanographicInstitution contribution
(gUo)[.(z+•)+i(-z+•)+2s
+(z+•)+i(z-•)-2s.]
(B2)
n
(z+s)
2
(z-s)
2
7151.
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