Low steady-state stresses in the cold lithospheric mantle inferred from dislocation dynamics models of dislocation creep in olivine

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Low steady-state stresses in the cold lithospheric mantle

inferred from dislocation dynamics models of dislocation

creep in olivine

Francesca Boioli, Andrea Tommasi, Patrick Cordier, Sylvie Demouchy,

Alexandre Mussi

To cite this version:

Francesca Boioli, Andrea Tommasi, Patrick Cordier, Sylvie Demouchy, Alexandre Mussi.

Low

steady-state stresses in the cold lithospheric mantle inferred from dislocation dynamics models of

dislocation creep in olivine. Earth and Planetary Science Letters, Elsevier, 2015, 432, pp.232-242.

�10.1016/j.epsl.2015.10.012�. �hal-01277178�

Contents lists available atScienceDirect

Earth

and

Planetary

Science

Letters

www.elsevier.com/locate/epsl

Low

steady-state

stresses

in

the

cold

lithospheric

mantle

inferred

from

dislocation

dynamics

models

of

dislocation

creep

in

olivine

Francesca Boioli

a

,

Andrea Tommasi

b

,

Patrick Cordier

a

,

∗

, Sylvie Demouchy

b

,

Alexandre Mussi

a

a_{UnitéMatériauxetTransformations,UMR8207CNRS- UniversitéLille1,F-59655Villeneuved’Ascq,France} b_{GéosciencesMontpellier- UniversitédeMontpellier&CNRS,F-34095Montpellier,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received4July2015

Receivedinrevisedform28September 2015

Accepted6October2015 Availableonline3November2015 Editor: J.Brodholt Keywords: olivinerheology creep deformation lithosphericmantle numericalmodelling

Transmission electron microscopy observations on olivine crystals deformed at moderate (≤1273 K) temperatureevidenceddislocationsinteractionsexplainingthehardening observedintheexperiments, butalsorecoverymechanismsbytheabsorption oremissionofpointdefects.Thusweinvestigatethe possibility that, atgeological strain-rates, theserecovery processesallow steady-state deformationby dislocation creepatlowtomoderate temperaturesinthelithospheric mantle.We testthishypothesis usinga2.5-D dislocationdynamics(DD)model,whichcombinesdislocationglideandrecoverybyclimb. This model shows that diffusion-controlled recoveryprocesses allow for steady-state deformation by dislocationcreepinthelithosphericmantleatstresses<500 MPa.Forstressesof50–200 MPa, steady-statestrain-ratesof10−15s−1maybeattainedattemperaturesaslowas900 K.FittingoftheDDmodel produces aflow law,whichrepresents alowerbound forthe lithospheric mantlestrength, since the modelsdescribethedeformationofanolivinesinglecrystalinaneasysliporientation.Comparisonof strain-ratesandMohotemperaturesinferredfordifferentgeodynamicenvironmentsandthepredictions ofthismodel-basedflowlawimplies,nevertheless,that,exceptinincipientrifts,mostoftheobserved deformationmaybeproducedbystresslevels_≤200 MPa,consistentwiththoseinferredtobeproduced byconvection. Thisconvergencesuggeststhatthe presentmodels,whichexplicitlycalculatethe time-dependentdislocationdynamics,mayprovideacorrectfirstorderestimateofthemechanicalbehaviour ofthelithosphericmantle,whichcannotbederiveddirectlyfromanyexistingdata.

©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

The strength and the active deformation mechanisms in the lithospheric mantle are one of the major open questions in platetectonics.Extrapolation ofempiricalflow lawsderived from high-temperature (

>

1473 K)deformation experiments on olivine single-crystals and polycrystals to geological strain-rates (10−12 to 10−15 _s−1_{) predicts stresses largely exceeding 1 GPa in the}

cold uppermost levelsof the lithosphericmantle.Indeed, for de-forming the lithospheric mantle at temperatures of 873 K and strain-rates of 10−14 s−1, anhydrous flow laws for olivine poly-crystalspredictstressesrangingfrom1.6 GPato5.07 GPa(Fig. 1). Lower stresses (375–800 MPa) are predicted by extrapolation of hydroushigh-temperatureflowlaws(Fig. 1),butthesevalues

prob-*

Correspondingauthorat:UniversitéLille1,UnitéMatériauxetTransformations, UMRCNRS8207- BatC6, 59655Villeneuved’AscqCedex,France.Tel.:+33320 434341.

E-mailaddress:[email protected](P. Cordier).

ably overestimateboth thewatercontentsinolivine inthe litho-sphericmantle(BellandRossman,1992; IngrinandSkogby,2000; Demouchy et al., 2006; Peslier, 2010) and the effect of wa-ter on the olivine rheology, which is probably limited to a re-duction in strength by a factor 2–3 (Demouchy et al., 2012; Feietal.,2014).

The predicted strength of a 100 km thick plate is therefore significantly higher than the stresses which may be produced in a viscoelastic lithosphere by mantle convection or those in-ferred by modellingthedeformation inresponseto crustalloads, such as volcanic chains (100–200 MPa; e.g., Beuchert and Pod-ladchikov, 2010; Zhong and Watts, 2013). Yet, continental plates deform. This paradox may be partially solved by proposing that the cold uppermost lithospheric mantle deforms by brittle fail-ure or low temperatureplasticity (Fig. 1). When implemented in numerical models, flow laws simulatingthese processes and the associated thermaldissipation allowindeed successfulsimulation of manygeodynamic processes,in particularcontinental breakup (e.g., Brune, 2014). However, the rarity of mantle earthquakes in

http://dx.doi.org/10.1016/j.epsl.2015.10.012

Fig. 1. Strengthenvelopemodelfora100km-thickcontinentallithosphere deform-ingataverticallyuniformstrain-rateof10−14_s−1_{.Thegeothermisplottedingrey.}

Yieldstressesinthecrustwerecalculatedusing:the frictionalslidinglawfrom

Byerlee (1977)and Goetze’scriterionwithadensityof2700 kg m−3_;theflowlaw

foranhydrousquartzitefromGleasonandTullis (1993);andtheflowlawfor an-hydrousdiabasefromMackwelletal. (1998).Inthemantle(belowtheMoho),are displayed:thelowtemperatureflowlawsfromDemouchyetal. (2013),Raterronet al. (2004),andEvansandGoezte(1979),aswellastheanhydrousandhydroushigh temperatureflowlawsfromHirthandKohlstedt (2003),theanhydroushigh tem-peratureflowlawfromFauletal. (2011),andtheanhydroushightemperatureflow lawfromChopraandPaterson (1984).

continental domains(Maggiet al., 2000; Jackson, 2002) tendsto falsifytheassumptionofacoldmantlerheologycontrolledby fric-tionalprocesses. Indeed, the rare earthquakes nucleating at sub-Moho depths beneath the Himalayas (Monsalve et al., 2006) or activeriftzones(LindenfeldandRumpker,2011)wereascribedto theunderplating ofthecold Indianlithosphere ortomagma mi-grationratherthantectonicprocesses.

Ontheother hand,thechangefroma powerlawto an expo-nentialdependenceofstrain-rateonstress(i

.

e.,powerlaw break-down)is consistent withthe few dataon deformation of olivine single andpolycrystals attemperatures

<

1300 K (Raleigh,1968; Phakeyetal.,1972; DurhamandGoetze,1977; EvansandGoetze, 1979; Raterronet al., 2004; Demouchy etal., 2009, 2013, 2014). Thesedata also indicate that the shallow cold lithospheric man-tlemaydeformatgeologicalstrain-ratesundersignificantlylower stressesthanpredictedfromtheextrapolationofhigh-temperature data(Fig. 1).Indeed,ata temperatureof873 K andastrain-rate of10−14 s−1,theflow lawfromDemouchyetal. (2013) predicts stressesof

∼

270 MPa.

However, conducting deformation experiments on olivine at

T

<

0

.

5Tm (Tm isthe meltingtemperature, i.e. forolivine Fo90

≈

1973 K) suffers from major shortcomings. As temperature de-creases,higherpressures areneededtoavoidbrittledeformation. Gas-medium apparatus, where confining pressures are limitedto

<

500MPa,provideaccuratemechanicaldata,butdonotallowto deform plasticallyolivine below1100 K (Demouchy etal., 2013). Steady-state is rarely achieved in low temperature experiments; moststress strain-ratecurves are characterised bymarked strain hardening. Flow laws have been therefore established by fitting the data at a given strain, which is in all cases lower than the strain valuesexpectedinthemantle.Demouchyetal. (2013),for instance, used themaximum stress attained in each experiment, whichcorrespondstototalshorteningsrangingfrom4.7to23.5%.

Below 1100 K, ductile deformation can only be achieved by applyinghighconfiningpressures,forinstanceusingtheD-DIA ap-paratus,whichhashighuncertaintyinthestressdata(Raterronet al.,2004; Longetal.,2011),orbyusingindentationmeasurements (EvansandGoetze,1979),whichgiveonlyindirectrheological in-formation.In spiteof theselimitations,exponential flowlaws, in particulartheonefromEvansandGoetze (1979),havebeenwidely usedtomodelthemechanicalbehaviouroftheuppermantle.Yet, therearemanyreasonstoquestionwhetherthisstrain-dependent data may be used to predict the mechanical behaviour of the uppermostlithospheric mantleinsituations inwhichlarge defor-mationsareexpected, such asconvergent plateboundaries, litho-sphericscaleshearzonesandtransformfaults,orcontinentalrifts. Inthisarticle,weproposeanewstrategytoinfertherheology ofolivineatlow temperatureandnaturalstrain-ratesby combin-ingexperimental dataandnumericalmodellingofintracrystalline plasticity. First, we rely on a detailed analysis by Transmission ElectronMicroscopy(TEM)ofthemostrecentlow-temperature de-formationexperimentsofolivinebyDemouchyetal.(2013, 2014), whichallow unravellingthe elementarydeformationmechanisms at work. These observations permit to identify the mechanisms, whichleadto hardeningandbrittleness, butalsotohighlightthe recoveryprocesses,whicharehinderedathighstrain-rates.These recovery processes are diffusion-driven and, thus, both time and temperature-dependent.Atgeologicalstrain-rates,recoveryshould bemuchmoreeffectivethanintheexperiments,possiblyallowing steady-state to be achieved. To test for this hypothesis, disloca-tion dynamicsmodels are used toanalyse the interplaybetween dislocation glide anddiffusion-drivenrecovery processes,such as climb,inolivine.Withinthismodelwecanpredictthesteady-state strain-rates, which might be achieved at the stress-temperature conditions expected to prevail in the lithospheric mantle. These predictions, which represent a lower bound since we model the deformationofawell-orientedsinglecrystal,notapolycrystal,are validated by comparison to strain-rates and Moho temperatures observedinavarietyofgeodynamicenvironments.

2. Experimentalconstraints

2.1. Deformationexperiments

The rheology of mantle rocks at lithospheric temperatures has been essentially constrained by deformation experiments on olivinecrystalsandaggregatesundervariableconfiningpressures, fromatmospheric(indentationexperiments)toupto3–8 GPa (D-DIAexperiments),andtemperatures(300–1623 K).Sincetheslow strain-rates relevant for geodynamics (10−12 s−1 to 10−18 s−1) cannot be achieved in the laboratory, in mechanical testing of geomaterials time is often traded for temperature. However, the validityofthisassumptionisstilltobetested.

Recently, tri-axial compression experiments on oriented sin-gle crystals of San Carlos olivine were performed at

tempera-tures relevant of the uppermost mantle (1073 to 1363 K; De-mouchy et al., 2009, 2013). These experiments were carried out at a confining pressure of 300 MPa in a high-resolution gas-medium mechanicaltestingapparatusunder constantstrain-rates from7

×

10−6 s−1 to 1

×

10−4 s−1.Olivinecylinders with differ-entcrystallographicorientations(closeto

[

101

]c

,

[

110

]c

and

[

011

]c

orientations)were compressedin ordertoactivateone ortwo of thedominant slipsystemsin olivinein each experiment, namely

[

100

](

001

)

,

[

001

](

100

)

and

[

001

](

010

)

,or

[

100

](

010

)

.Finalfinite strains ranged from4 to 23%. The experiments yielded differen-tial stress often higher than the confining pressure: from 88 to 1076 MPa. Although high, these stresses are significantly lower than those predicted by the extrapolation of the flow laws de-rivedfromhighertemperatureexperiments(1473–1573 K)tothe presenttemperatures(Fig. 1).Thisresultimpliesthat timecannot be simplytraded fortemperatureinstudyofthe mechanical be-haviourofthelithosphericmantle.

However,thesefaststrain-rateexperimentsatlowtemperature (1073 K

=

0

.

47Tm)underlowconfiningpressurefacedseveral lim-itations:stress hardeningwas observed inmostexperiments,the samplesoftenbrokebeforeachievingasteady-stateregimeoreven before 5% of strain, and stick-and-slip-like behaviour was occa-sionally observed (Demouchy et al., 2013). Postmortem study of the samples by EBSD revealed micro-fracturing superimposed to plasticdeformation(Demouchyetal.,2013, 2014).These observa-tionsindicatetransitional brittle–ductiledeformationmechanisms at1073 Kandlaboratorystrain-rates.Olderstudies(Raleigh,1968; Phakeyetal.,1972; EvansandGoetze,1979) despiteusing differ-entapproachesmusthavebeenconfrontedtothesamelimitations. Insummary,below1073 K,thecurrentexperimentalmethodsfor characterising therheology ofolivine reachtheir limits. New ap-proaches are needed to properly determine the rheology of cold uppermostlithosphericmantle.

2.2. Transmissionelectronmicroscopy(TEM)analysisofthe

deformationmechanisms

Thedeformedcrystalsdescribedabovehavebeencharacterised by TEM. Doubly polished thin sections (30–25 μm thick) were glued on a Mo grid and ion milled at 5 kV under a low beam angleof15◦ untilelectrontransparencyisreached.TEM observa-tionswerecarriedoutusingaPhilipsCM30microscopeoperating at300 kVanda FEITecnaiG20 microscopeoperating at200 kV. Dislocationsmicrostructureshavebeenanalysedusingweak-beam dark-field(WBDF)andelectrontomographymethods(Mussietal., 2014, 2015a).

Here we summarise the observations made on all samples. Rather than emphasising the differences between samples de-formed along differentdirections, we highlightthe generaltrend andthedeformationmechanismswhichhavebeenfoundto oper-ateineverycase. Asexpected, atca. 0

.

5Tm,evidenceforglide of

[

001

]

dislocationson

{

110

}

and

(

100

)

planespredominates.Ifthe crystalsarenotfavourablyorientedforactivationoftheseslip sys-tems,

[

001

]

dislocationsglidein other

{

hk0

}

planes,butevidence forcrossslipeventsintheeasiestplanesiscommon.In

{

110

}

and

(

100

)

,non-screw segments slipmuch faster.Their curvedshapes indicate that they bear little lattice friction (Fig. 2a). They leave behindlongstraightscrewsegments;theirslowermotioncontrols theglidestrain-rates.Elastic interactionsbetweennon-screw seg-mentsresultindislocation dipoles,which areextremelycommon in the deformed samples (Fig. 2a and b). These dipoles are sta-bleandform obstacles forother gliding dislocations, resultingin progressive building of entanglements (Fig. 2b andc). We inter-pretthismechanismastheprimarysourceofhardeninginolivine deformedatlowtemperature.

Despite the low temperature, we also observe pervasive ev-idence for onset of recovery mechanisms involving ionic diffu-sion. Every sample displays evidence for breakdown of disloca-tion dipoles into strings of prismatic loops (Fig. 2c). This well-known recovery mechanism, which has been described in ce-ramics deformed at high temperature (Junqua and Grilhé, 1984; Lagerlöf et al., 1989), works as follows. Under the influence of pipe diffusion, lineinstabilities develop along the dipole by self-climb.Eventually,theseinstabilitiesresultinpinchingofthedipole (Fig. 2d)andinformationofalignedprismaticloops.Theseloops progressively shrink andultimatelydisappear by absorbing point defects that diffuse toward them. In fact, Goetze and Kohlstedt (1973) hadalreadynoticed thecollapse ofdislocation loops dur-inghigh-temperatureannealingexperimentsinolivine.

Insamplesdeformedatlaboratorystrain-rates,thatis,at10−4 to 10−6 _s−1_{, these recovery mechanisms are not fast enough.}

The densityofloopsincreases.The interactionsoftheloopswith gliding dislocations create numerous non-glissiledislocation seg-mentsandcontribute veryefficientlytohardeningasdescribedin Mussietal. (2015b).Asaresult,carryingdeformationexperiments atlaboratorystrain-rates becomesdifficultattemperaturesbelow 1273 K.Steady-stateis notreached, instabilitiesdevelop, andthe samplesbecomebrittle.Anobvioushypothesisisthatatgeological strain-rates, that is,at10−12 _{to 10}−16 _s−1_{,recoverymechanisms}

may have enough time to operate and allow steady-state creep, butuntilwhichtemperatureandwithwhichefficiency?Inthe fol-lowing, wetake advantageofrecentdevelopmentsindislocations dynamicsmodellingtoinvestigatethispossibility.

3. Dislocationdynamicsmodelling

3.1. Descriptionofthemodel

Weuse2.5-DimensionalDislocationDynamics(2.5-DDD) sim-ulations tomodeldislocation creepin olivineintemperatureand stress ranges relevant for the lithospheric mantle. The present work follows the methodology recently established to study the hightemperature(T

>

1400 K) creeppropertiesofolivine (Boioli et al., 2015). This previous study showed that the interplay be-tween glide and climb dislocation motion leads to steady-state creep behaviour, following a power law that is in good agree-ment with high-temperature experimental data. The aim of the present models is to understand to which extent the interaction betweenglideandclimbmayallowsteady-statecreep atthelow temperatures (800 K

<

T

<

1400 K)which prevailinlithospheric mantle.

Discrete DD is a well-established simulation technique to de-scribe thecollectivebehaviour ofdislocationsandtomodel plas-tic flow in crystalline solids. In this method, the forces acting on dislocations are calculated using the linear elasticity theory. This provides a description of the long-range elastic strain field induced by dislocations, their reciprocal interaction, and the in-teraction oftheselinedefects withan external stress field(Hirth and Lothe, 1992). These long- and short-range interactions are key ingredients to describe the collective behaviour of disloca-tions. Dislocations are then moved in the directionof the forces according to mobilitylaws,which take into account the relevant atomisticprocessescontrollingdislocationmotion(Devincreetal., 2011).Inthe 2.5-Ddislocationdynamicsmodel(Gomez-Garciaet al., 2006), dislocations are materialised by parallel straight seg-ments perpendicular to a reference plane and their dynamics is followedonlyinthereferenceplane.Additionalrulesareincluded toreproduceimportant3-Ddislocationmechanisms,asdislocation multiplication, dipole annihilation or junction formation. First, a multiplication rule is used to reproduce the general observation that dislocationloopsexpandunderan externalloadingandthat,

Fig. 2. TEMimagesof:(a)OlivinecrystalPOEM19deformedat900◦C,ε˙=1.01×10−5 _s−1 _{displayingstraight[001]screwsegments(horizontal)andcurvednon-screw}

segmentsthatinteractelastically(arrow).(b)POEM19–900◦C,ε˙=1.01×10−5_s−1_{.Theseinteractionsleadtotheformationofstabledipoles(arrow),whicheventually}

leadtotheformationofentanglements.(c)POEM11–850◦C,ε˙=7.06×10−6_s−1_{.Withinthoseentanglements,onecanfindevidenceofrecovery.Followingthemechanism}

depictedin(d),dipolespinch(P)andbreakdowninstrings(S)ofsessileloopswhichshrinkanddisappearbydiffusion.(d)Undertheactionofpipediffusion,instabilities developonthedislocationlines,whichpinchandbreakupintosessileloops.Theloopsaresinksforpointdefectsandprogressivelyshrinkanddisappearasaresultofbulk diffusion(afterJunquaandGrilhé,1984andLagerlöfetal.,1989).

asaconsequence,thedislocationdensity

ρ

increaseslinearlywith the plasticstrain

ε

produced. Dislocation dipoles are inserted at random positions in the simulation box, provided that the local effectivestress has the same signof the external stress, accord-ingtothe multiplicationrate:d

ρ

/

d

ε

=

m, wherem isa constant (Gomez-Garciaetal.,2006).Thisreflectstheobservationthatmost ofthedislocationsemerginginareferenceplane,orslice,ofa3-D volume originate from sources in the surrounding volume. Here, thevaluem

=

2

×

1015_m−2 _{hasbeentakentoreproducethe}

evo-lutionofthedislocationdensitymodelledby 3-D-DD simulations (Durinck et al., 2007). Second we allow dislocations involved in adipole tomutuallyannihilate when thedistancebetweenthem issmallerthan a criticaldistancera

=

10b (b istheBurgers vec-tor modulus). These annihilation events result in a reduction of the dislocation density during the simulations. More details on the2.5-DDD approachandthe simulationsettingsformodelling olivine dislocation creep are presented in Boioli et al. (2015). In thefollowing,wedescribetheglideandclimbmobilitylawsused inthepresentmodel.

Akeypointtomodeldislocationcreepisthedescriptionofthe interplay between the glide velocity of dislocations in the most favourable crystallographic planes and the displacement rate of dislocationsoutside the glide planesby climb, i.e. the motionof dislocation induced by the absorption/emission of point defects, whichcontrolsrecovery.In highlatticeresistance materials, such

asolivine,itiscommonlyassumedthatplasticstrainiscontrolled by themotionofslowscrewsegments left bytherapid propaga-tionofnon-screwsegmentswithproducelittlestrain.Screw dislo-cationsmoveaccordingtoathermally-activatedkink-pairs mecha-nismandtheoveralldislocationglidevelocity,whichislimitedby themotionofthescrewsegments,canbeexpressedbyan Arrhe-niusrateequation(Kocksetal.,1975):

vg

=

b L lc

ν

D b lc exp



−



H

(

τ

∗

)

kBT



(1) where L isthelength ofthedislocation, b is themodulus ofthe Burgersvector,lcisthecriticallengthforkinknucleation,

ν

D isthe Debyefrequency,kB istheBoltzmannconstant,andT isthe tem-perature (in K).



H

(

τ

∗

)

= 

H0

(

1

− (

τ

∗

/

τ

P

)

p

)

q is the activation enthalpy of kink formation, which depends on the effective re-solvedshearstress

τ

∗(ateachdislocationposition

τ

∗

=

τ

app

+

τ

int iscalculatedasthesumoftheappliedstressandtheelastic inter-actionstress inducedby all theotherdislocations, bothprojected along the slipdirection). In the expressionof



H

(

τ

∗

)

,

τ

P is the Peierls stress,i

.

e.the stressrequiredto movedislocationsat0K,



H0 isthe activation enthalpy at0stress, and p and q are

em-piricalparameters.Theglidemobilityfor

[

001

]

dislocationsinthe present simulations is definedusing the parameters obtainedby Durincketal. (2007),whicharereportedinTable 1.

Table 1

Parametersusedtodefinetheglidemobilitylawfor[001]dislocationsasdefined byDurincketal. (2007). τP (MPa) p q ν0 (m/s) lc (nm) H0 (eV - kJ/mole V) 1730±41 0.5988 1.1506 2.75×103 _{1.95 5.4 - 521}

The second important mobilitylaw is the one describing the climb velocity, which controls the elimination of the prismatic loops. Climboccursthrough theabsorption/emissionofpoint de-fects (dominantly vacancies) by the dislocations. Here, we as-sumethat the dislocation lines are saturatedwith jogs, i.e. steps

along the dislocation lines in the climb direction, where vacan-cies can be absorbed or emitted instantaneously (with no acti-vation barrier). Withinthisassumption the climbprocess is lim-ited by diffusion of vacancies from the bulk to the dislocation (or vice-versa). Assuming steady-state conditions, thenet flux of vacancies from andto the dislocation core is calculatedby solv-ing the diffusion equation; this allows an analytical expression of the climb velocity vc to be derived (Hirth and Lothe, 1992; MartinandCaillard,2003):

vc

=

η

Dsd b



exp



τ

_c∗

Ω

kBT



−

c∞ c0



(2) where Dsd is the vacancy self-diffusion coefficient,

Ω

is the va-cancy formation volume,

η

is a geometrical factor that depends on the geometry of the flux field, and c_∞ and c0 are the

va-cancy concentration far fromthe dislocation andthe equilibrium vacancyconcentration inthebulk volume,respectively.Following Boioli et al. (2015), we assume that far away from the disloca-tionsthevacancyconcentrationisconstantandequaltothe equi-librium concentration in the bulk volume (c_∞

=

c0). The climb

process in olivine, which is an ionic crystal composed by four different elements, clearly involves the diffusion of several va-cancy species. Here, sincesilicon isthe slowest diffusingspecies, we consider that thediffusion rateis limitedby the silicon self-diffusivity. Hence, in Eq. (2),

Ω

=

72

.

4 Å3, that is, the vacancy formation volume for a Schottky defect estimated from the unit cellvolume ofolivine, and Dsd

₌

_Dsd

Si

=

D0exp

(

Hsd Si

kBT

)

isthe

self-diffusion coefficient of silicon taken from diffusion experiments athigh pressure and hightemperature. We haveconsidered dif-ferent sets of diffusion equations for Si self-diffusivity, obtained underboth anhydrous andhydrousconditions,to span the max-imum range of Si diffusion coefficient and environmental condi-tions to properly assess its influence on the calculated stresses. First, in analogy with the simulations reported in Boioli et al. (2015),weusedtherecentSidiffusivitiesforanhydrous,iron-free, forsteritefromFei etal. (2012),which yields foranhydrous con-ditions 3

.

74

×

10−24 _m2

_/

_{s at 1273 K (D}

0

=

2

.

51

×

10−7 m2

/

s;



Hsd_Si

=

4

.

25 eV (410 kJ/mol),



V

=

1

.

7 cm3

/

mol, ambient pres-sure).Second, weused theSidiffusivityforhydrousiron-bearing olivine from Costa and Chakraborty (2008), which yields for hy-drous conditions 3

.

44

×

10−22 _m2

_/

_{s at 1273 K (D}

0

=

1

.

68

×

10−7 m2

/

s;



Hsd_Si

=

3

.

71 eV (358 kJ/mol) fwater

=

0

.

98 GPa),and foranhydrous olivine (

∼

Fo93) from Dohmen etal. (2002),which

yields 1

.

78

×

10−26 _m2

_/

_{s at 1273 K (D}

0

=

6

.

31

×

10−5 m2

/

s;



Hsd_Si

=

5

.

48 eV (529 kJ/mol)).The valuefortheSidiffusion acti-vationenthalpyinanhydrousforsteritereportedinFeietal. (2012) is comparable with the one measured in iron-bearing olivine,



Hsd_Si

=

4

.

03

±

0

.

31 eV (389 kJ/mol) (Karato and Ogawa, 1982), but smaller than the value reported by Dohmen et al. (2002),



Hsd_Si

=

5

.

48

±

0

.

42 eV (529 kJ/mol). Areduced efficiencyofthe recoveryprocessesisexpectedifthelatterdiffusivitydataare con-sidered.Anupperandalower boundestimateofstrain-rateswas

obtainedbyusingSidiffusivitydatameasuredforhydrousand an-hydrousolivine,respectively.

3.2. Creepsimulations

Inorder toaddressthe plasticbehaviouratlower strain-rates, we performedcreepsimulationsby couplingglideandclimb dis-location motion. In all simulations, we imposed single slip and uniaxial loading conditions on a single crystal. The geometry of thesimulation boxissketchedinFig. 3a.Theloadingaxisis par-alleltothe y directionanditformsanangleof45◦ withtheglide directionand withthenormal tothe glide plane. Theclimb dis-placementdirectionisperpendiculartotheglideone.Differentbox dimensions: L

=

Lx

=

Ly, between2and8 μm,anddifferent ini-tialdislocationdensityvalues

ρ

0,between1012and2

×

1013m−2,

havebeenconsidered.Inparticular,

ρ

0hasbeensettobe,

approxi-mately,from3to10timessmallerthantheequilibriumdislocation densitypredictedfora givencreep stress inthe considered tem-peratureregimeasinBoiolietal.(2015).

In Fig. 3, we compare the dislocation microstructure and the stress field induced by both the dislocation microstructure and the externalloadingpredictedby DDmodels inwhichonly glide was activated or where glide was coupled to recovery by climb. Both models were run for a temperature of 1100 K and an ap-plied stress of 50 MPa. In the initial configuration (Fig. 3b), dis-locations are randomly distributed. An initial dislocation density

ρ

0

=

2

.

5

×

1012m−2isassumedinbothruns.Thefinaldislocation

densityis

ρ

=

6

×

1012m−2intheglideonlymodels(Fig. 3c).Itis lower,

ρ

=

5

.

5

×

1012_m−2_{,whenclimbisactivated(Fig. 3d).}

Dur-ing the deformation process, dislocations with similar sign tend to align perpendicularly to the glide direction creatingsub-grain boundaries, which can be seen in thefinal configuration ofboth models(arrowsinFig. 3candd).However,themostcommon fea-turesinthemicrostructurearepairsofdislocationswithopposite signs,formingdipoles.InFig. 3candd,wehighlightafewdipoles withcircles, butmostdislocationsareinvolved indipole interac-tions.

The analysisof the strain vs. time curves (Fig. 4) shows that when dislocations are restricted to move by glide only, after a shorttransient,they becometrappedindipoles,blockingany fur-ther plasticstrainproductionifstress isnot increased.The effect of climb is to eliminate these “jammed”configurations, allowing for steady-state deformation. Indeed, glide produces dislocation multiplication,whereasclimbpromotesdipoleannihilation.Atthe steady-state,the dislocation densityfluctuatesaround an equilib-rium value, denoting balance between these competing mecha-nisms. This is in agreement with the creep behaviour ofolivine predictedinthehightemperatureregime (Boioli etal., 2015)and demonstrates the importance of the climb mechanism even at moderatetemperature.

First,weperformedafewsimulationsintheconditionswhere experiments were done to check for the validity of the model before addressing thedeformation behaviour atgeological strain-rates(10−12–10−15s−1),relevantforthelithosphericmantle.The first tests were carried out ata constant strain-rate of 10−5 _s−1

andfor800 K

≤

T

≤

1400 K. Thepredictedflowstressesare con-sistent withthe experimental mechanicaldata, aswe can see in Fig. 5awherethecalculatedstressescorrespondingtoashortening

ε

=

2% (empty boxes) are compared with the experimental val-ues (circles). Under those conditions, diffusion processes are too slow to sustain such high strain-rates andglide is the dominant deformation mechanism. At larger deformations (

ε

>

2%), strain hardening due to the interaction of dislocation in different glide planes occurs. Here we donot focus on the plastic behaviour in this advanced deformation stage because the 2.5-DDsimulations are not welladapted todescribe thehardening observedinmost

Fig. 3. (a)Sketchofthe2Dsimulationbox.Singleslipconditionsanduniaxialloadingalongthe y direction(loadingdirection)areapplied.Theanglebetweentheloading axisandglidedirectionis45◦.Dislocations(plusormultiplicationsymbols)aresuperimposedonacolourbackground,whichrepresentsthecomponentσy y ofthestress

fieldinducedbythedislocationmicrostructureandtheimposedexternalforces.(b)Initialconfiguration.(c,d)Finaldislocationmicrostructureinamodelwithglideonly(c) andinwhichglideiscoupledwithclimb(d).Arrowsindicatesub-grainboundariesformedbyarraysofdislocations.Thedislocationdensityρisequalto2.5×1012_m−2_in

(b;initialdislocationdensity)andto6×1012_m−2_and5

.5×1012_m−2_{inpanels(c)and(d),respectively(endofsimulationdislocationdensities,whicharequasistationary}

in(d),butstillevolvingin(c)).

Fig. 4. Plasticdeformationεvs.time,obtainedincreepconditions(σ=50 MPa,

T=1100 K),assumingglideonly(solidgreyline)andglidecoupledwith climb (solidblackline)dislocationmotion.Theaverageslopeoftheblacklinerepresents thesteady-statestrain-rate(dashedline).

low-temperatureexperiments.Thecomplexityofdislocation inter-actionsunderthesehigh-stressconditionsarenotfullycapturedin thepresentmodel. 2.5-DDsimulations produce, however,precise estimatesofthe criticalstresses implied indislocation glide. The agreement between the simulated and experimentally measured mechanicalbehaviour atlow finite strainsinthe highstrain-rate regimesubstantiatesthepredictive powerofthe2.5-DDDmodel andjustifiestheuseofthismethodtopredicttheflowstressesin thelowstrain-rateregime.

Alargersetofsimulationshasthenbeenperformedunder con-stant loadinorder tocalculatethe creepstrain-rate inolivine in the temperature range between 800 and 1400 K.For each tem-perature,we performedseveralsimulations byincreasing the ap-plied stressfrom 43to 500MPa.Each simulationwas runup to steady-state conditions.Creep rates were determined by a linear least-squarefitofplasticstrainvs.timecurves,astheoneshown inFig. 4.TheresultsareplottedinFig. 5(flowstressescalculated at 10−14 s−1 as a function oftemperature), Fig. 6(strain-rate as afunctionoftemperature),andFig. 7(strain-rateasafunctionof stress).

4. DiscussionofDDsimulationresults

The main question addressedinthe presentstudyis whether steady-statedislocationcreepmaybeattainedatlowtomoderate temperaturesinthe uppermantle.Athightemperature, diffusion processes are fast andclimb is expectedto occur. In olivine, DD simulationsperformedbetween1400Kand1700 Kconfirmedthat climbisakey mechanismto reachsteady-stateconditions(Boioli etal., 2015). Climb governstherateatwhichdislocationsbypass obstaclesoratwhichdislocationdipolesannihilate,controllingthe steady-state. However, recovery mechanisms promoted by diffu-sion mayoperate even atintermediate/low temperatureand fast strain-rates, as indicated by the TEM data in Fig. 2, which dis-plays evidence for breakdown of dislocation dipoles into strings ofprismaticloopsinducedbytheabsorption/emissionofpoint de-fects. Here,we showthat steady-statecreepis observeddownto 800 K even under low stresses (50 MPa). In this extreme case, steady-state strain-rates are extremelylow (

∼

10−18 _s−1_{) and do}

Fig. 5. Criticalflowstressσ asafunctionofthetemperatureT .(a)Flowstresses obtainedby2.5-D DDsimulationsforstrain-rates of10−5 _{(emptyboxes) andof}

10−14_s−1 _{(full boxes)comparedwithmechanicaldataforolivinesinglecrystals}

from Demouchyet al. (2013)(circles, D&al2013single crystals)and olivine ag-gregatefromDemouchyet al. (2014)(diamonds,D&al2014agreg).Theflow law derivedfromthe2.5-D DDresults at10−14_s−1_{isshownassolidline.(b)Flow}

stresscalculatedby2.5-D DDmodelandthe resultingflowlawcomparedwith modelsobtainedbyextrapolatingtheexperimentalresultsfromlaboratory strain-ratesto10−14_s−1_{strain-ratefromH&K:HirthandKohldstedt(2003)inanhydrous}

andhydrousconditions,E&G1979:EvansandGoetze (1979),F&al2011:Fauletal. (2011),D&al2013:model3fromDemouchyetal.,2013.

not produce deformations measurableby geodetic tools. This re-sultdemonstrates,however,that,whentimeisavailable,interplay between glide andclimb may result in steady-state deformation evenunderverylowtemperatures(800 K

=

0

.

35Tm).

Analysis of the full dataset allows further examining the in-terplay between climb and glide and their respective influence on the creep strain-rates. In the temperature range considered in oursimulations (800–1400 K),climb controls thesteady-state strain-rate up to stresses of 200 MPa. The dislocation creep be-haviour in this ‘climb-controlled’ regime is illustrated in Fig. 3 and4.Afractionofmobiledislocationsmovesbyglideuntilthey reach a quasi-equilibrium configuration. Thanks to climb events, some dislocations escape fromsuch ‘jammed’ configurations, be-comemobile,andproducefurtherplasticstrain.Thissuccessionof eventsisperiodicallyrepeatedintimeleadingtosteady-state con-ditions,i.e. a linearincrease oftheplasticdeformation withtime (constantstrain-rate). In thisregime the dislocation density fluc-tuates around an equilibrium value, which depends only on the

Fig. 6. Calculated 2.5-D DD strain-rate valuesε˙ as a functionof the reciprocal temperatureT .Coloursindicatecalculationsfordifferentstresses.Thecreep strain-rates obtainedbyusing the Si self-diffusion coefficientmeasured in anhydrous forsterite (Feiet al., 2012) areshown byfull symbolsand dashed lines (D0=

2.51×10−7_m2

/s andHsd

Si=4.25 eV (410 kJ/mol)).Theupperandthelower

lim-itsofthecolouredbandsrepresentthe2.5-D DDresultsobtainedbyconsideringthe Siself-diffusioncoefficientvaluesof:D0=1.68×10−7m2/s andHsdSi=3.71 eV

(358kJ/mol)(CostaandChakraborty,2008) andD0=6.31×10−5m2/s;HsdSi=

5.48 eV (529kJ/mol)(Dohmenet al.,2002),for anhydrousandhydrous olivine, respectively. Squares delimitatestrain-rate–temperature conditionsinferred for lithosphericplatesindifferentgeodynamicsituations(seetextfordetails).

Fig. 7. Calculated2.5-D DDstrain-ratevaluesε˙asafunctionoftheappliedcreep stressσ.Fulllinesshow thefitobtainedusingEq.(5),with A=0.002 andthe glideparametersasdefinedinTable 1.

applied creep stress. The influence ofclimb becomes less impor-tant withincreasing appliedstresses. Inthe simulations where a stress of500 MPa is imposed, we observe that climb events are rare.However,whiletheclimbvelocitydependslinearlyonstress, the glide velocity increases exponentially with it (see Eqs. (1) and (2)), eventually allowing to reach large deformations before dislocation–dislocation interactions become important and ‘jam’ the microstructure. Thus, although climb events are rare in the simulations where a stress of 500 MPa is imposed, steady-state strain-rates are achieved, because thestress level ishigh enough toallowdislocationstobypasslocalattractiveorrepulsive config-urationswithouttheabsorptionoremissionofpointdefects.These high-stresssimulationswerecarriedoutuptostrainsof10% with-out developmentofa‘quasi-equilibrium’ dislocationconfiguration that would require climb to overcome obstacles andsustain the plasticflow.

Athightemperatures,creepstrain-ratesarecommonlyanalysed byusingapowerlawequation:

˙

ε

=

A

σ

nexp



−

Q kBT



(3) whereQ isthecreepactivationenthalpy,

σ

theappliedstressand

T thetemperature(inK). Q ,n and A areassumedtobeconstant and depend on the physical mechanism controlling the macro-scopic creep for a givenmaterial. To characterise the creep acti-vationenthalpy, weplotthe creep strain-ratesobtainedfromthe DDsimulationsasafunctionofthereciprocaltemperature(Fig. 6). Thecurvesofthelogarithmofthestrain-rateasafunctionofthe reciprocaltemperaturepresentthesamenegativeslope,indicating aconstantactivationenthalpy Q forcreepintheexaminedstress range. By averaging the values for different stresses, we obtain

Q

=

5

.

2

±

0

.

3 eV (502

±

29 kJ

/

mol), whichisinagreement with theactivationenergydeterminedinhightemperaturecreep( Q

∼

5

.

4 eV

=

520 kJ

/

mol, Kohlstedt and Goetze,1974; Q

∼

4

.

7 eV

=

453 kJ

/

mol, GueguenandDarot,1982; Q

∼

4

.

9 eV

=

470 kJ

/

mol, Mei and Kohlstedt, 2000) and in numerical simulations ( Q

=

5

.

07 eV

=

489 kJ

/

mol, Boioli etal.,2015). The Q value extracted fromthesimulationsliesbetweentheactivationbarriersforclimb, hereequivalentto theSiself-diffusion enthalpy(410 kJ/mol) and forglidemotionof

[

001

]

dislocations(521 kJ/mol).

Inordertoillustratetheinfluenceofthediffusioncoefficienton ourresults,we compare thecreep strain-rates obtainedby using the Si self-diffusion coefficient measured in anhydrous forsterite (Fei et al., 2012), dashed lines in Fig. 6, and the one measured in olivine for hydrous (Costa and Chakraborty, 2008) and anhy-drous(Dohmenetal.,2002) conditions.Weobserveanincreaseof thestrain-ratesandadecreaseoftheactivationenthalpy Q when

Sidiffusivitydata forhydrousconditionsare considered. The op-posite behaviour (decrease ofthe strain-rates and an increase of the activation enthalpy Q ) is predicted when Si diffusivity data for olivine in anhydrous condition are used. This shift is shown inFig. 6byusingcolourbands.Thelowerandtheupperlimitof thebandsrepresentthestrain-ratevaluesforolivine under anhy-drousandhydrousconditions,respectively.Wenoticethat,evenif thediffusioncoefficientmeasuredbyCostaandChakraborty,2008 (Dohmen et al., 2002) is

∼

2 orders of magnitude larger (lower) thantheonemeasuredbyFeietal. (2012)atT

=

1273 K,we ob-servean increase (decrease)ofthestrain-rates valuesby afactor of2–4at50 MPaandbylessthan1orderofmagnitudeat800 K and50MPa,conditionsatwhich weobserve themaximumshift inthe strain-rate values.This highlights that, to evaluate the ef-fect of the diffusion coefficient on the strain-rates, the interplay between glide and climb must be considered and that estimat-ing creep strain rates by simply assuming a linear relationship between strain-rates values and diffusion coefficient is not pos-sible. At low stress (

≤

200 MPa),when climb plays a key role in theplastic deformation process, thefaster Sidiffusion coefficient reportedfor hydrousconditions enhancesthe recoveryprocesses and produces larger strain-rates. On the contrary, at high stress (500 MPa),we observenoinfluenceoftheSidiffusioncoefficient onthestrain-rates.Thisconfirmsthat,atthesehighstresses,climb isnot effectiveanddislocationmotionby glidecontrolsthecreep propertiesofolivine.

InFig. 7,theevolutionofthestrain-rateasafunctionofstress isplottedfordifferenttemperatures.Apowerlawbehaviourwould resultinstraight lines.As expected, thepredicteddependenceof thestrain-rateonstress inthe temperaturerangeof800–1200 K deviatesfrom thepower law behaviour observed at higher tem-perature. Above 1400 K a constant stress exponent n

∼

3–3

.

5 is obtained for experiments in a wide range of stress conditions (Jin et al., 1994; Bai et al., 1991; Hirth and Kohlstedt, 2003) andit isassumedto characterise dislocation creep behaviour inolivine.

A similar stress exponent, n

∼

3

±

0

.

2, is obtained by 2.5-D DD simulations of olivine creep behaviour at temperatures

>

1400 K andappliedstress

<

100 MPa(Boiolietal.,2015).However,atlow temperature,thecreepbehaviourofolivinecannotbemodelledby apowerlawsincetheapparentstress exponentn wouldincrease from 3.5 to 10 when the applied stress is increased from50 to 500 MPa(Fig. 7) at1000 Kandtoevenlarger valuesat800and 900 K.

Mechanicalpropertiesofolivineatlowtemperaturearebetter described byintroducing thedependenceofstrain-ratesonstress withintheexponentialterm:

˙

ε

=

A exp



−

Q kBT



1

−



σ

σ

P



p



q



(4) wherekB istheBoltzmann constant,

σ

P is thePeierls stress and p andq are empiricalparameters (e.g., EvansandGoetze, 1979). This equation is assumedto be moreadapted when plastic flow iscontrolled by glideinvolvinglattice friction.ByusingOrowan’s equation we can expressthe strain-rateasa function ofthe dis-location density

ρ

andtheaverage dislocationvelocity:

ε

˙

=

ρ

bv.

By further assuming that the average dislocation velocity is pro-portionaltotheglidevelocitywecanrewriteEq.(4)asfollows:

˙

ε

=

ρ

bv

=

ρ

b Avg

=

A

ρ

bv0exp



−

Q kBT



1

−



0

.

5

σ

σ

P



p



q



(5) where p, q,

σ

P, and Q are the parameters usedto describe the glide velocity vg (Eq. (1)) reported in Table 1, v0 is the

pre-exponentialfactorintheexpressionof vg

,

b istheBurgersvector modulus,

ρ

is thedislocation density, and A is a fitting param-eter. In Eq. (5), we set the resolved shear stress

τ

=

0

.

5

σ

, that is, the applied stress

σ

reduced by the Schmid factor. Following Boioli etal.(2015),we assumethat therelationship betweenthe equilibriumdislocation density

ρ

andthe stress isgiven by

ρ

=

(

0._μ5σ

)

sC

/

b2, with the stress exponent s

=

1

.

31 and C

=

0

.

104. Then,theremainingadjustableparameter Aisfittedtoreproduce the modelled dependence of the strain-rates on stress, as illus-trated in Fig. 7.By fitting thesedata (full symbolsin Fig. 7) we findA

=

0

.

002

±

0

.

0003.ThecurvesinFig. 7areobtainedby plot-tingEq.(5)fortemperaturesvaryingfrom800 Kto1200 K using thevaluesdefinedabove.Thecorrectfitoftheresultsofthemodel fordifferentstressesandtemperatureswithasingleparameterset indicatesthatitispossibletodescribetheaveragedislocation ve-locity asthe glide velocity reduced by a scaling factor( A). This reflectstheobservationthatdislocationscanbetrappedindipole interactionsandneedtobereleasedbyclimbinordertoflowfrom one“jammed”configurationtoanotherone.Wenoticethat,inthe examined (T

,

σ

) range,the dependenceof thestrain-rate on the appliedstress ismainlygoverned bythedependenceoftheglide velocityonthestress.Infact,byassumingaconstantaverage dis-location density

ρ

=

3

×

1013 m−2, we can still reproduce quite welltheDDsimulationresults.

Inordertoproposeanewflowlawforthelithosphericmantle, we focusonthenumericalresultsatstrain-ratesof10−14,which are relevant for the lithospheric mantle. The curve obtained by substituting A

=

0

.

002 inEq.(5)(otherparameters asinTable 1) andbyassuminganaveragedislocationdensity

ρ

=

3

×

1013_m−2

is shown with a solid line in Fig. 5. This flow law obtained by fittingthe resultsofthe modelrepresentsa lower boundfor the strengthofan olivinerockatmoderatetolow temperature,since wemodelthedeformationofasinglecrystalinaneasy glide ori-entation,not ofanolivine-richrock,composedby crystalswitha varietyoforientations.Comparisonbetweenexperimentaldatafor olivine crystals andpolycrystals atmoderate temperature(circles

and diamonds in Fig. 5a, respectively) shows that the polycrys-tals are stronger by a factor two or less. A similar difference is obtained by analysing the high-temperature data (e.g., Mackwell etal., 1985).Thus the model-basedflow lawprobably underesti-matesthe strengthof amantlerock,butby a factortwo orless. It probably captures the correct order of magnitudes of stresses andstrain-ratesinthelithosphericmantle.Weemphasizethatthe noveltyofourmodelistotakeintoaccountdiffusion-driven recov-eryprocesses,which,basedonoursimulationresults,areexpected tobeimportantforstresses

≤

200 MPaevenatmoderate tempera-ture(T

∼

0

.

35Tm).Flowlawsbasedonextrapolationofmechanical datafromlaboratoryto geologicalstrain-rates cannotcapturethe influenceofsuchprocessesthatarenegligibleatlaboratory strain-rates.Thisexplainsthediscrepancybetweenthe2.5-DDD model resultsandexistingflowlawsbasedontheextrapolationof exper-imental data fromlaboratory to geological strain-rates, shownin Fig. 5b.

5. Geodynamicimplications

The2.5DDmodelssubstantiate theintuitivepredictionthat if recoveryprocessesarealreadyactive during lowtemperature de-formationexperimentsatstrain-ratesof10−5 _s−1_{,atmuchslower}

geologicalstrain-rates,diffusionalprocessesareeffectiveenoughto allowforsteady-statedislocationcreep evenintheshallowlevels ofthelithosphericmantle.Atlaboratorystrain-rates,thestrengths predictedby thesemodelsare lowerbya factor2than those ob-tainedusingtheexponential flowlawofDemouchyetal. (2013), corroboratingthatthisflowlawrepresentsanupperboundforthe strengthofthelithosphericmantle(Demouchyetal.,2014).

To evaluate if the predictions of these models are consistent with the strain-rates observed in nature, we plotted in Fig. 6 strain-rates calculated based on GPS data (Kreemer et al., 2014) andMohotemperaturesforavarietyofgeodynamicenvironments. Thissimpleapproach,whichimplicitlyassumesvertically homoge-neousstrain-rates within a plate, allows by comparisonwiththe modelpredictionsatthesameconditions,toverifyiftheobserved strain-ratesmaybeproducedbystresslevelsconsistentwiththose estimatedfromplateboundaryforcesanalysesorconvection mod-els(

≤

200 MPa).Inmostcases,thisconditionissatisfied.Cratonic environmentsare characterised byextremelylow temperaturesat Mohodepths(

<

800 Kconstrainedbyxenolithsthermobarometry, heatflowdata,andPn-velocitiesinavarietyofcratons;e.g.Boyd, 1973; Brazieretal.,2000; MareschalandJaupart,2004; McKenzie et al., 2005; Hyndman et al., 2009; Schutt et al., 2011; Baptiste et al., 2012). Consistently, they do not deform (non-measurable strain-rates, i.e.

<

10−18 s−1). High strain-rates (

>

10−15 s−1) are observed in active collisional boundaries, such asthe Himalayas, the Andes, the Alps, the Cordillera, or Anatolia (Kreemer et al., 2014). All these domains are characterised by high Moho tem-peratures,ranging between900and1100 K (Calvertetal., 2000; Al-Lazkietal.,2004; LiangandSong,2006; Hyndmanetal.,2009; Díazetal.,2013).Thepresentmodelsindicatethatatsuch temper-atures,strain-ratesof10−14s−1 to10−15s−1 maybeattainedfor stressesintherange50–200 MPa,that is,forstresslevels consis-tentwiththoseproducedbymantleconvection.LowPnvelocities indicatinghighMohotemperatures(

>

900K;BuehlerandShearer, 2012) and high strain-rates (

>

10−14 s−1; Kreemer et al., 2014) also characterise the San Andreas fault and the mature domains of the East Africanrift system, such as the Afars (Mechie et al., 1994).

In contrast, incipient continental rifts, in which extension af-fects thick and hence cold plates, such as the southern part of the East African rift, where high P-wave velocities and xenolith data indicates Moho temperatures 800–900 K (Henjes-Kunst and Altherr,1992; Mechieetal., 1994; Brazieretal.,2000; Baptisteet

al.,2015),arecharacterised bystrain-rates

>

10−16_s−1_,which,

ac-cordingly to the present models cannot be produced by stresses

≤

200 MPa.Thus,ifthepresentmodels predictcorrectlythe rhe-ologyoftheshallowlithosphericmantle,additionalsoftening pro-cesses, such aslocalised advectiveheating by magmapercolation (Baptiste et al., 2015), orgeometrical softening due to preferred orientation ofolivine inthe lithospheric mantle (Tommasi et al., 2009) areneededforallowingtheinitiationofcontinentalrifting.

As an additionaltestof themodels,we comparetheir predic-tions with temperature and paleopiezometric data derived from the analysis ofperidotitic mylonites from the Beni Bousera mas-sif in the Rif belt, Morocco (Frets et al., 2014). These mylonites constitutetheupperpartofakm-scaleshearzone,which accom-modated ca. 30 km of thinning of the lithospheric mantle in a transtensionalsetting.Paleothermometrydataontheserocks indi-catethattheyequilibratedattemperaturesrangingbetween1173 and1473 K.Recrystallised grainsizesforthelow-temperature my-lonitesareintherange100–150 μm,whichcorrespond,basedon the paleopiezometer of Van der Walet al. (1993), to stresses of 30–40 MPa. 2.5 DD models predict, for such temperatures and stresses, strain-rates of ca. 10−14 s−1 (Fig. 6). These values are lowerbyafactor10thanthoseinferredforthisshearzonebased on thermal considerations (preservation of a temperature gradi-ent of

>

200 K/km; Frets et al., 2014). Yet, much higher strain-rates,consistentwiththepreservedthermalgradient,arepredicted forthecoarse-grainedperidotites,whichaccommodatedthelatter stagesofthefunctioningoftheshearzone(Fretsetal.,2014). Sim-ilarly, if we use the present models to estimate the strain-rates in California based on paleothermometric and paleopiezometric data formantlexenoliths intheCima Volcanicfield in California (1150–1220 K andstresses

<

20 MPa;BehrandHirth, 2014), we obtainstrain-rates of10−14–10−15s−1,whichareconsistentwith thosederivedfromGPSdataforthisregion(Kremeeretal.,2014). In summary, although the model-based flow law is a lower bound forthe strength ofthe lithospheric mantle,the predicted behaviouroftheshallowmantleisremarkablyconsistentwith ge-ological observations. It is also more coherentthan the previous flow laws with the low strength of the shallow subcontinental lithosphere inferred from the analysis of post-seismic deforma-tion(e.g.,Freedetal.,2012).Modelsoftheflexureassociatedwith Hawaiipredict evenlowerstrengths intheshallowandcold sub-oceanicmantle (Zhong and Watts,2013). Thevery low strengths inferredfromthesedataareneverthelessdifficulttoreconcilewith ourpresentknowledgeofdeformationmechanismsinolivine-rich rocks.Deformationofmostofthesubcontinentallithospheric man-tleatstresses

≤

200 MPaaspredictedbythepresentmodelsmay havemultipleimplicationsforthethermo-mechanicalevolutionof theplates.Softeningprocessessuchasshearheatingandgrainsize reduction wouldbe lesseffective. Brittle processesshould notbe activated. On the other hand, extension of the dislocation creep field to lower temperatures wouldimply development of olivine crystal preferred orientationsand, hence,anisotropy in the shal-lowsubcontinentallithosphericmantle.

6. Conclusions

Recent deformation experimentsonolivine single crystalsand polycrystals showed that olivine-rich rocks have much lower strengths than those predicted from the extrapolation of high-temperature,low-stressexperimentaldata(Demouchyetal.,2013, 2014). This observation falsifies the common assumption that a temperature enhancement may compensate for the high strain-rates in the experiments, calling for new data on the rheology ofthe lithosphericmantle.However, deformationexperiments on olivineattemperatures

<

1273 Khavemanyshortcomings.In par-ticular, steady-state is almost never reached, most data showing

a continuous hardening up to activation of brittle deformation processes. Nevertheless, TEM analysis of deformed olivine single crystalsdeformed underlow/intermediate temperature evidences dislocationinteractionsproducinghardening,butalsotheirpartial recoveryby theabsorptionoremission ofpoint defects.As diffu-sion is a time-dependent process, we propose that at geological strain-rates theserecoveryprocesses allow steady-state deforma-tionby dislocationcreep atlowto moderatetemperaturesinthe lithosphericmantle.

Wetestedthishypothesisusingadislocation dynamicsmodel, which associates dislocation glide and recovery by climb. This modelshowsthatdiffusion-controlledrecoveryprocessesallowfor steady-state deformation by dislocation creep in the lithospheric mantle. Ifreasonable stress levels (50–200 MPa) are considered, steady-state strain-rates of 10−15 s−1 may be attained at tem-peratures as low as 900 K. Fitting of the DD model produces a flow law, which represents a lower bound for the lithospheric mantle strength, since the models simulate the deformation of an olivine crystal in an easy slip orientation. Experimental data at both moderate and high temperature indicate that an olivine polycrystal may be up to two times stronger than single crys-tals. Comparison of strain-rates andMoho temperatures inferred fordifferentgeodynamicenvironmentsandthepredictionsofthis model-basedflow law imply that, except in incipient rifts, most of the observed deformation may be produced by stress levels

≤

200 MPa,consistentwiththoseinferredtobeproducedby con-vection.Modelpredictionsarealsoconsistentwithdirectestimates ofstresses and temperatures in naturally deformedmantle peri-dotitesandtheinferredstrain-ratesforeachcontext.Insummary, thereisa clearneed, ifwe wanttocorrectlymodelthe mechan-ical behaviour of the lithospheric mantle, to go beyond a sim-pleextrapolationoflaboratorydatatogeological strain-rates.The presentstudyillustrateshowmodellingofthedislocation interac-tionsmayhelpconstrainingthisscaletransfer(10ordersvariation instrain-rate).

Acknowledgements

A Marie Curie fellowship awarded to S.D. (PoEM: Plasticity ofEarthMantle, FP7-PEOPLE-20074-3-IRG, N◦230748-PoEM) sup-portedthe experimental work used inthis study.This work was supported by funding from the European Research Council un-dertheSeventhFrameworkProgram(FP7),ERCgrantN◦290424– RheoMantoP.C.TheTEMnationalfacilityinLilleissupported by theCNRS(INSU)andthe ConseilRégionalduNord Pasde Calais, France.Themanuscriptbenefitedfromcarefulreviewsanduseful commentsfromR.DohmenandI.Jackson.

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