Physics of the Earth and Planetary Interiors...Introduction The history of deformation of the Earth’s crust and lithosphere is recorded in the microstructure of its constitutive

Physics of the Earth and Planetary Interiors 270 (2017) 195–212

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Physics of the Earth and Planetary Interiors

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Formation of lithospheric shear zones: Effect of temperature ontwo-phase grain damage

http://dx.doi.org/10.1016/j.pepi.2017.07.0110031-9201/� 2017 Elsevier B.V. All rights reserved.

⇑ Corresponding author.E-mail addresses: [email protected] (E. Mulyukova), david.bercovici@

yale.edu (D. Bercovici).

Elvira Mulyukova a,⇑, David Bercovici a

aYale University, Department of Geology and Geophysics, New Haven, CT, USA

a r t i c l e i n f o

Article history:Received 17 February 2017Received in revised form 15 June 2017Accepted 24 July 2017Available online 1 August 2017

Keywords:Shear zonesGrain damage mechanicsMylonites

a b s t r a c t

Shear localization in the lithosphere is a characteristic feature of plate tectonic boundaries, and is evidentin the presence of small grain mylonites. Localization and mylonitization in the ductile portion of thelithosphere can arise when its polymineralic material deforms by a grain-size sensitive rheology in com-bination with Zener pinning, which can impede, or possibly even reverse, grain growth and thus pro-motes a self-softening feedback mechanism. However, the efficacy of this mechanism is not ubiquitousand depends on lithospheric conditions such as temperature and stress. Therefore, we explore the con-ditions under which self-weakening takes place, and, in particular, the effect of temperature and defor-mation state (stress or strain-rate) on these conditions. In our model, the lithosphere-like polymineralicmaterial is deformed in a two-dimensional simple shear driven by constant stress or strain rate. The min-eral grains evolve to a stable size, which is obtained when the rate of coarsening by normal grain growthand the rate of grain size reduction by damage are in balance. Damage involves processes by which someof the deformational energy gets transferred into surface energy. This can happen by (i) dynamic recrys-tallization (grain damage) and (ii) stretching, deforming and stirring the material interface (interfacedamage). The influence of temperature enters through rheological laws (which govern the rate of workand damage), grain growth kinetics, and the damage partitioning fraction, which is the fraction of defor-mational work that goes into creating new surface energy. We demonstrate that a two-phase damagemodel, in which the partitioning fraction depends on both temperature and roughness of the interfacebetween the phases, can successfully match the field data, including the reported correlation of grain sizeand temperature, the increasing dominance of dislocation creep at higher temperatures and a large rangeof grain sizes observed across the depth of a single shear zone.

� 2017 Elsevier B.V. All rights reserved.

1. Introduction

The history of deformation of the Earth’s crust and lithosphereis recorded in the microstructure of its constitutive rocks, whichdeform in response to the differential stress imposed by forces inthe lithosphere and underlying convecting mantle. The heat andthe mechanical work supplied by the mantle drives deformationon a multitude of scales: from the movement of the crystallinedefects on the subgrain scale, associated with the solid state diffu-sion and dislocation creep, to the formation of mylonitic shearzones, and further to the planetary scale deformation of the Earth’slithosphere, manifested as plate tectonics.

The relative motion of the plates is almost entirely accommo-dated by highly localized deformation at the plate boundaries, as

is, for example, geologically evident in the presence of small grainmylonites and ultramylonites (White et al., 1980). Such subdivi-sion of the Earth’s surface into largely undeformed tectonic plates,separated by narrow, strongly deformed plate boundaries, suggeststhat its constitutive materials experience shear localization, as canfor example be caused by some self-weakening mechanism (Kaula,1980; Bercovici, 1993; Bercovici, 1995; Tackley, 2000; Bercovici,2003; Montési, 2013; Bercovici et al., 2015). To understand howEarth evolved to have such an exotic style of surface dynamics,we need to understand the mechanisms responsible for strainlocalization across the crust and the lithosphere, including theirbrittle and ductile regions. Localization in the brittle portion ofthe Earth’s outer shell is an expected feature at low crustal andlithospheric pressures and temperatures (Kohlstedt et al., 1995).However, the ductile portion of the lithosphere is where the peaklithospheric strength occurs and is thus a bottle-neck to localiza-tion; in this region, deformation is also enigmatic since it involves

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196 E. Mulyukova, D. Bercovici / Physics of the Earth and Planetary Interiors 270 (2017) 195–212

combined brittle-ductile behavior as well as mylonitization(Kohlstedt et al., 1995; Karato, 2008).

A crystalline rock deforming viscously can experience self-weakening and localization due to a number of processes, suchas shear heating, presence of multiple phases (including melt)and grain size reduction (via phase transformation or dynamicrecrystallization), to name a few (e.g., Hobbs et al., 1990; Hobbset al., 1990; Montési and Zuber, 2002; Montési and Zuber, 2002;Karato, 2008; Karato, 2008, pp. 288–301; Bercovici et al., 2015;Bercovici et al., 2015, and references therein). The microstructurethat develops due to the action of a given strain-localizing mecha-nism can reveal whether and to what degree it is active duringlithospheric deformation. Such interpretations require a thoroughquantitative understanding of these mechanisms at the conditionsrepresentative of the lithospheric crust and mantle. The physicalconditions, characterized by temperature, composition and themagnitude of the mechanical forcing (e.g., stress), vary spatiallyand through time, so that the dependence of shear localizing mech-anisms and their associated microstructure on these propertiesmust also be understood.

A common geometrical quantity used to characterize themicrostructure of a crystalline material is its grain size distribu-tion. It can be readily measured in laboratory experiments, as wellas in the natural samples collected from the field. From the physi-cal point of view, the size of the grains reflects the amount of inter-nal energy that is stored as surface energy on the grain boundaries.In particular, when the material deforms by dislocation creep andundergoes dynamic recrystallization, by mobilization of a pre-existing grain boundary or by the formation of subgrain bound-aries, its grain boundary area increases (Karato et al., 1980; Vander Wal et al., 1993). This reduces the grain size and thereby storesthe deformational work in the form of increased grain boundaryarea (Bercovici and Ricard, 2005; Austin and Evans, 2007; Ricardand Bercovici, 2009; Rozel et al., 2011). In polyphase materials,the relative movement of grains of different phases can also leadto distortion of the inter-phase boundaries, which further increasesthe surface energy (Bercovici and Ricard, 2012).

For shear localization to occur, there needs to be a mechanismby which deformation can cause weakening, which would increasedeformation, and which would further enhances weakening. Thisis, for example, the case when a material deforms by dislocation-accommodated grain boundary sliding and undergoes dynamicrecrystallization (Hirth and Kohlstedt, 1995; Warren and Hirth,2006; Hansen et al., 2011; Hansen et al., 2013; Kohlstedt andHansen, 2015; Skemer and Hansen, 2016). Another possible mech-anism, which is the focus of this study, is when a polyphase mate-rial deforms by grain size sensitive diffusion creep and undergoesgrain size reduction due to interface-damage and pinning, whichare induced by the presence of multiple phases (Bercovici andRicard, 2012; Cross and Skemer, 2017).

There is abundant support from field (Jin et al., 1998; Montésiand Hirth, 2003; Warren and Hirth, 2006; Skemer et al., 2010;Herwegh et al., 2011; Linckens et al., 2011; Linckens et al., 2015;Hansen and Warren, 2015), experimental (Evans et al., 2001;Hiraga et al., 2010; Farla et al., 2013; Platt, 2015; Cross andSkemer, 2017) and theoretical studies (Bercovici and Ricard,2012; Bercovici and Ricard, 2016) that secondary phases play animportant role in the mechanical properties of lithospheric materi-als. The goal of this study is to use this observational evidence totest and further constrain the theoretical models of grain damage,with a particular focus on the effect of temperature, and effectivelydepth, on the grain-size dependent shear-localizing mechanisms(see alsoLanduyt et al., 2008; Behn et al., 2009; Gueydan et al.,2014).

To constrain the models presented here, we use the data fromfour different upper mantle shear zones, published in Jin et al.

(1998) and Linckens et al. (2015). The reported microstructure,including grain size and crystallographic orientation, together withthe independent temperature measurements, are used to constrainthe dominant creep mechanism. In combination with the olivineflow laws and grain growth kinematics obtained from previousexperimental studies (Hirth and Kohlstedt, 2003), we are able todeduce the deformation conditions, such as stress or strain rate,for a given tectonic environment. Some of the physical parametersthat enter the model are less well constrained, such as the graingrowth activation energy and the partitioning fraction of themechanical work that goes into creating new grain boundaries.Here, we provide further constraints, by comparing the grain sizedependence on temperature predicted by our model with thatreported in the observational data.

Field observations demonstrate a correlation between grainsize and temperature (with the grain size increasing by overtwo orders of magnitude across a few hundred degrees increasein temperature), increasing dominance of dislocation creep athigher temperatures, and a coexistence of different deformationregimes across small spatial scales (Jin et al., 1998; Linckenset al., 2015). These observations are used as constraints in themodel presented below.

2. A simple shear zone model

A detailed mathematical description of the two phase graindamage model was presented previously (e.g.Bercovici andRicard, 2012; Bercovici et al., 2015; Bercovici and Ricard, 2016).We provide a brief overview of the equations here, to introducethe governing physical processes and to highlight the mainassumptions made along the way.

We consider the deformation of a polycrystalline material sub-ject to simple shear. The material consists of a non-dilute mixtureof two immiscible fluids, or phases (e.g., a peridotite mixture withvolume fractions of 40% and 60% pyroxene and olivine, respec-tively). In the general formulation of the model, the phases havedifferent densities, viscosities and other properties, although inthis study, for simplicity, the phases are assumed to have identicalmaterial properties.

The two-phase material is characterized by a mean grain sizewithin each phase, as well as the geometry (i.e., roughness or,specifically, the characteristic radius of curvature) of the interfacethat separates the two phases. The grain size and the interfaceroughness directly or indirectly influence the mechanical proper-ties of the material, but also evolve in response to the appliedmechanical or kinematic forcing. This introduces a non-linear cou-pling between the applied deformation and the effective rheologyof the material.

The grain size distribution within each phase is assumed to fol-low a lognormal distribution, characterized by a mean grain size.Moreover, the grain size evolution is assumed to be slaved to thatof the interface roughness r, as the grain growth within each phaseis blocked by the presence of the other phase. This is known as thepinned state limit (Bercovici and Ricard, 2012), in which the grainsize is proportional to the interface roughness. For the grain-sizedistribution and phase volume fraction (e.g., 60% olivine, 40%pyroxene) assumed in this study (following Bercovici and Ricard(2013, 2014, 2016, 2015)), the mean grain size is approximatelyp=2 times the interface roughness (the limitations of the pinned-state assumption are discussed in Appendix A.).

The model has a two-dimensional plane geometry, with shearacting parallel to the x-axis, and assumed to be infinite and uni-form in the x-direction. The temperature varies with depth,increasing in the negative z-direction, while the applied stress orstrain rate are constant and uniform in all directions (Fig. 1).

Fig. 1. A sketch of the shear zone model used in this study. The three inserts are cartoons of some of the possible grain size distributions, with grain-colors indicating grain-size (blue is small and red is large). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

E. Mulyukova, D. Bercovici / Physics of the Earth and Planetary Interiors 270 (2017) 195–212 197

In a natural shear zone, both stress and strain rate are likely tovary in space and time. Constant stress and strain-rate conditionsapproximate the end-member tectonic states for how drivingforces or loads are supported by the lithosphere and/or mantle.When the lithosphere is strong and provides the primary resis-tance to plate-driving forces like slab-pull or effective ridge push,it is approximately deforming under constant stress conditions.Once the lithosphere undergoes localization and weakens, theresistance to the driving forces is taken up by the viscous drag fromthe mantle. This force balance dictates the plate velocity, whichthus imposes a constant mean strain-rate on a deformation zoneof a given width.

The spatial uniformity of stress or strain-rate is also not guaran-teed for field or experimental conditions. However, for the basiccase of simple shear, stress is uniform. While it is difficult toimpose a uniform strain rate in simple shear when viscosity is vari-able, the mean strain rate can be controlled.

When deformation is driven by an applied stress, the variationsin temperature and grain size with depth may induce verticalstrain rate variations and thus vertical shear _exz � @vx=@z. How-ever, our assumed width of the shear zone in the y-direction(e.g., � 1 km in Linckens et al., 2015) is much smaller than thewidth of the lithosphere in the z-direction (of order 100 km), hencewe assumed _exy � _exz.

In the two-dimensional simple shear set up, the only non-zerocomponent of the symmetric strain rate tensor is_e ¼ _exy ¼ _eyx ¼ 1

2@vx@y , where vx is the rate of displacement in x-

direction. Similarly, there is only one non-zero component of thestress-tensor, which we denote as s. The stress - strain rate rela-tionship for our model with composite (diffusion and dislocationcreep) rheology is then given by:

_e ¼ Asn þ Brms ð1Þ

where A and B are the dislocation and diffusion creep compliances(assumed equal for both phases), respectively (see Table 1 for thelist of all the parameters featured in our model, together with theirranges of values), and r is the interface roughness (or effective grainsize). The scaling factor ðp=2Þm, which arises when we represent

grain size by the interface roughness in the pinned state limit, isincluded in the prefactor B0 entering the diffusion creep complianceB.

The evolution of the interface roughness can be expressed as(Bercovici and Ricard, 2012; Bercovici and Ricard, 2016):

drdt

¼ gGI

qrq�1 �f Ir

2

cgW ð2Þ

where

W ¼ 2 _es ¼ 2Asnþ1 1þ RF

r

� �m� �ð3Þ

and

RF ¼ BAsn�1

� �1=m

ð4Þ

is the field boundary roughness (or equivalent grain-size) at whichthe strain-rates from diffusion and dislocation creep are equal. Thecompliances A and B are dependent on temperature, and thus thefield boundary relationship between stress and interface roughnessvaries with temperature.

The diffusion-driven process of grain growth (first term on theright hand side of (2)) is counteracted by the grain size reductiondriven by deformational work (second term on the right hand sideof (2)). A more detailed mathematical and physical description ofthese processes is presented in Bercovici and Ricard (2012). Thegrain size reaches a steady state when the grain growth and grainsize reduction processes are in balance.

The general formulation for the fraction of deformational workthat goes towards distorting the interface in multiphase materials(f I) is:

f I ¼ f �IRbm

Rbm þ rb

ð5Þ

where f �I is given by:

f �I ¼ F0 expð�Cf ðT=TrefÞkf Þ ¼ F0f 0ðTÞ ð6Þ

Table 1Material and model properties.

Property Symbol Value/definition Dimension

Rate of deformational work W 2s _e Pa s�1

Surface tension c 1 Jm�2

Phase volume fraction /i /1 ¼ 0:4;/2 ¼ 0:6Phase distribution function1 g 3/1/2 � 0:72Reference temperature Tref 1000 K

Damage partitioning fraction2f I ¼ f �I

Rbm

Rbmþrb

f I 6 1

Temperature-dependent prefactor f �I F0 expð�Cf ðT=Tref Þkf ÞExperimentally constrained constants Cf ; kf Cf P 0Constant prefactor F0 0 < F0 < 1Mixing transition roughness Rm R�

mRF mMixing transition prefactor R�

m 0 6 R�m < 1

Mixing transition exponent b 4

Dislocation creep3 _edisl ¼ Asn

Activation energy Edisl 530 kJ mol�1

Prefactor A0 1:1 � 105 MPa�n s�1

Stress-Exponent n 3Compliance A A0exp � Edisl

RT

� �MPa�n s�1

Diffusion creep3 _ediff ¼ BR�ms

Activation energy Ediff 375 kJ mol�1

Prefactor B0 1:5 � 109 lmm MPa�1 s�1

Grain Size Exponent m 3Compliance B B0exp � Ediff

RT

� �lmm MPa�1 s�1

Grain growth4

Activation energy EG 200–400 kJ mol�1

Prefactor G0 2 � 104 lmp s�1

Exponent p 2Grain growth rate GG G0exp � EG

RT

� �lmp s�1

Interface coarsening5

Exponent q 4Interface coarsening rate GI q

p ðlmÞq�p GG250

lmq s�1

1 Phase distribution function from Bercovici and Ricard (2012).2 Damage partitioning fraction from Rozel et al. (2011),Bercovici and Ricard (2016), and Appendix B of this study.3 Olivine creep laws from Hirth and Kohlstedt (2003), using n ¼ 3 instead of n ¼ 3:5 (refitting the experimental data for n ¼ 3 does not result in significant changes of A0

and Edisl).4 Grain-growth law from Karato (1989) for olivine, but varying EG instead of using EG ¼ 200 kJ mol�1.5 Interface coarsening law from Bercovici and Ricard (2013).


The constant 0 < F0 < 1 is to be varied. The temperature-

dependence of f �I is described by f 0ðTÞ ¼ expð�Cf ðT=Tref Þkf Þ. For atemperature-independent f �I ; Cf is set equal to zero. Otherwise, Cf

and kf are experimentally constrained constants that are calculatedself-consistently for given values of the activation energies (seeAppendix B).

Eq. (5) includes a mixing transition roughness Rm that determi-nes the dependence of f I on the interface roughness (Bercovici andRicard, 2016). As proposed by Bercovici and Ricard (2016), parti-tioning of mechanical work into creating new interface surfaceenergy is possibly more efficient when the interface has high cur-vature, compared to when it is smooth. This might occur becauseinter-grain mixing is more efficient for smaller grains (Bercoviciand Skemer, 2017), as is possibly evident in recent experimentalstudies (Cross and Skemer, 2017, Fig. 10). In the limit ofRm ! 1; f I becomes independent of the interface roughness.

Following Bercovici and Ricard (2016), we use b ¼ 4 and assumeRm < RF . To ensure that the latter is the case for the entire range ofstress- and temperature-conditions, we further assume that Rm isparallel to RF , such that:

Rm ¼ R�mRF ð7Þ

where R�m P 0 is a constant to be varied.

2.1. Reference state

To nondimensionalize the governing equations, we define a ref-erence state for the dependent variables _e; s and r. It is mathemat-ically convenient to use a temperature-dependent reference state(denoted by the subscript s) wherein time and grain-size lengthscales are derived from rheological properties. First, the strain-rate, time and stress scales are related through the dislocationcreep law:

_es ¼ t�1s ¼ AðTÞsns ð8Þ

For systems with constant applied stress, ss is set to that stress,and _es and ts are defined by (8). For systems with a constantapplied strain rate, _es is set to that strain rate, and ts and ss are like-wise set by (8).

The reference interface roughness is defined as:

rs ¼ BAsn�1

s

� �1=m

ð9Þ

The dimensionless field boundary roughness is given by:

rF ¼ RF

rs¼ ss

s

� �ðn�1m Þ

¼ 1s0

� �ðn�1m Þ

ð10Þ


The dimensionless mixing transition roughness Rm (7) is alsonondimensionalized using rs, and, using (10), is given by:

R0m ¼ R�

m1s0

� �ðn�1m Þ

¼ R�mrF ð11Þ

The partitioning fraction (5) can be expressed in terms of thenondimensional parameters:

f I ¼f �I

1þ r0R�mrF

� �b ð12Þ

2.2. Dimensionless governing equations

Substituting _e ¼ _es _e0; s ¼ sss0; t ¼ tst0 and r ¼ rsr0 into (1) and(2), and dropping the primes, we get the nondimensional govern-ing equations:

_e ¼sn þ srm

ð13Þdrdt

¼ Cqrq�1 � Dr2

snþ1 þ s2r�m

1þ rR�mrF

� �b ð14Þ

where

C ¼ gGI

Asns rqs; D ¼ 2rsssf �I

cgð15Þ

are dimensionless parameters that determine the efficiencies ofinterface healing and damage, respectively.

For a system with constant stress we have s ¼ 1, and at the fieldboundary this gives rF ¼ 1 and _e ¼ 2. For constant strain rate _e ¼ 1,and using (13) with r ¼ rF from (10), this leads to the field-

boundary stress s ¼ ð1=2Þ1=n and thus a field boundary grain-size

of rF ¼ ð1=2Þ�ðn�1Þ=nm, which gives rF ¼ 1:1665 for our chosen valuesof n and m. Thus, with our choice of rs; r measures the interfaceroughness as a fraction of the field boundary roughness RF , so thatthe temperature-dependence of the field boundary (through B=A)is removed from the dimensionless system. When r < rF, diffusioncreep is the dominant deformation mechanism, while when r > rF,dislocation creep is the dominant one.

At steady state, the interface roughness, and thus also the effec-tive grain size, are solutions to:

0 ¼ qDC

� �snþ1rqþ1 þ qD

C

� �s2rqþ1�m � r

R�mrF

� �b

� 1 ð16Þ

The ratio qD=C governs the steady state and the temperature-dependence of the dimensionless interface roughness, and can beexpressed in terms of stress or strain-rate scales only (i.e., by elim-inating rs):

qDC

¼ 2qcg2

f �IGI

Bðqþ1m Þ

Aðqþ1�mm Þ

sqþ1þm�nðqþ1�mÞ

ms ð17aÞ

or

qDC

¼ 2qcg2

f �IGI

Bðqþ1m Þ

Aðqþ1þmnm Þ

_eqþ1þm�nðqþ1�mÞ

nms ð17bÞ

for constant stress or strain-rate, respectively, where we used (8) tosubstitute for ss in the second relation.

With the choice of the reference state parameters introduced inSection 2.1, the dependence of the interface roughness on temper-ature is governed by the thermal variability of only two quantities:the field boundary roughness RF , and the damage to healing ratioqD=C.

2.3. The solution space for the steady state

The steady state interface roughness r is the solution to a fifthorder polynomial (16), using q ¼ b ¼ 4. When R�

m ! 1 (i.e., f I isindependent of r in (12)), there is only one real solution (Fig. 2).The interface roughness, and thus also the grain size, decrease withincreasing qD=C, as expected. As qD=C increases and the grainsshrink, diffusion creep becomes increasingly dominant. Obtaininga steady state solution in the dislocation creep regime requiresvery small values of qD=C (i.e., qD=C < 1).

When R�m is finite (i.e., f I varies with r in (12)), there can be up to

three steady state solutions to (16), depending on qD=C and R�m:

dislocation creep at low qD=C, diffusion creep at high qD=C, and,over a limited range of intermediate qD=C-values, there is a possi-bility for co-existence of the two regimes, together with an unsta-ble solution branch that is in diffusion creep (Fig. 2). The systemwill evolve to one of the two stable states depending on the initialconditions, specifically on whether the initial roughness is on oneor the other side of the unstable solution branch. Thus, the steadystate is history-dependent, or, as termed by Bercovici and Ricard(2016), exhibits a hysteresis effect. Due to the observed hysteresiseffect, we will from now on refer to the model with R�

m < 1 as thehysteresis model.

All of the stable solutions suggest a decrease in grain size forincreasing values of qD=C, as expected.

2.3.1. Approximate solutionsThe occurrence of the hysteresis loop can also be predicted from

the asymptotic analysis of the model equations. To do this, we con-sider the limiting cases for r outside and inside the field and mixingboundaries, and find the approximate solutions to the system insteady state.

For this analysis, it is useful to rewrite (16) in the followingform:

0 ¼ qDC

� ��1

� rqþ1snþ1 1þ ðrF=rÞm1þ ðr=RmÞb

ð18Þ

Here, we use the general definition for the mixing transitionroughness (Bercovici and Ricard, 2016):

Rm ¼ R�ms

�a=b ð19Þwhere a ¼ bðn� 1Þ=m ¼ 8=3 for Rm parallel to rF. This yields the fol-lowing approximate solutions (Bercovici and Ricard, 2016):

� For big grains r ! 1 and (18) leads to:

rqþ1�bsnþ1�a ¼ qDC

� ��1

R��bm ð20Þ

� For intermediate grains between the mixing transition and thefield boundary roughness, we have Rm < r < rF, which is stillin diffusion creep. In this case r < 1 and ðrF=rÞm is large, but

ðr=RmÞb is also large, and (18) becomes

rqþ1�m�bs2�a ¼ qDC

� ��1

R��bm ð21Þ

� For very small grains r ! 0, and (18) leads to:

rqþ1�ms2 ¼ qDC

� ��1

ð22Þ

For the big and small grains, and using q ¼ b ¼ 4;m ¼ 3 anda ¼ 8=3; r decreases with qD=C. However, the intermediatesolution has r increasing with qD=C (since qþ 1�m� b ¼ �2),

Fig. 2. Growth rate of interface roughness (dr/dt, (14)) as a function of r and qD=C for constant stress (left column) and constant strain rate (right column). White and greyfield-colors indicate the sign of dr/dt: grey is negative and white is positive, black solid lines are the zero-levels (i.e., steady state). The black dashed line indicates the fieldboundary roughness rF. The grey dashed line indicates the mixing transition roughness Rm . Results for Rm ¼ 1 and two finite values of Rm are indicated. Red, green and bluelines are the approximate solution branches for r ! 0;Rm < r < rF and r ! 1, respectively, shown in (20)–(22). (For interpretation of the references to colour in this figurelegend, the reader is referred to the web version of this article.)


delineating a branch that connects the other branches with anopposite slope.

Where the branches intersect can be found by dividing eitherthe big or small grain branch by the intermediate one. Dividing

(20) by (21), we get that the two branches meet atr ¼ s�ðn�1Þ=m ¼ rF. Dividing (22) by (21), we get that the twobranches meet at r ¼ R�

ms�a=b ¼ Rm.


For constant stress case, where s ¼ 1, the big and intermediate

grain branches intersect at qD=C ¼ R��bm , while the small and inter-

mediate grain branches intersect at qD=C ¼ R��ðqþ1�mÞm (Fig. 2). So

the lower branch intersection occurs at a smaller grain size and asmaller qD=C than the upper intersection, giving rise to the hys-teresis loop. The intermediate branch disappears, together withthe hysteresis loop, when Rm P RF , which is when the bigand the small grain branches intersect (see Bercovici and Ricard,2016).

For constant strain rate case we have _e ¼ 1. Thus, for largegrains, (13) implies _e ¼ 1 � sn, and so s � 1. For the small andintermediate grain branches, which are both in diffusion creep,(13) implies _e ¼ 1 � s=rm and so s � rm. The big and intermediate

grain branches intersect at qD=C ¼ R��bm ¼ R��4

m , while the smalland intermediate grain branches intersect at qD=C ¼R��bðqþ1þmÞ=ðbþamÞm ¼ R��8=3

m (Fig. 2). As in constant stress case, thelower branch intersection occurs at a smaller grain size and a smal-ler qD=C than the upper intersection, giving rise to the hysteresisloop.

Fig. 3. Temperature-dependence of the damage to healing ratio and the dimensionless ilegend). The colored line is the field boundary, colored according to temperature. Left coluAll models employ Edisl ¼ 530 kJ mol�1, Ediff ¼ 375 kJ mol�1 and EG ¼ 300 kJ mol�1. The rekf ¼ 0:68 (see Appendix B). (For interpretation of the references to colour in this figure

3. Temperature-dependence of grain size

3.1. Temperature-dependence of the damage to healing ratio

In the previous section we explored how the damage to healingratio qD=C controls the dimensionless interface roughness r, i.e.,the roughness relative to the field boundary. The damage to heal-ing ratio varies with temperature, and this determines thetemperature-dependence of the dimensionless roughness.

In general, qD=C is given by 17a and 17b. Inserting for the rhe-ological compliances A and B, as well as the interface coarseningrate GI (Table 1) and the damage partitioning fraction f �I (6), thedependence of qD=C on stress, strain rate and temperature canbe expressed as:

qDC

� F0s2=3s expDERT

� CfTTref

� �kf" #

; DE ¼ ð2=3ÞEdisl þ EG � ð5=3ÞEdiff

or

qDC

� F0 _e2=9s expDERT

� CfTTref

� �kf" #

; DE ¼ ð8=9ÞEdisl þ EG � ð5=3ÞEdiffv

ð23Þ

nterface roughness. Results for three different partitioning fractions are shown (seemn: constant stress ss ¼ 100 MPa. Right column: constant strain rate _es ¼ 10�13 s�1.sulting values for constants entering the partitioning fraction (6) are Cf ¼ 11:2 andlegend, the reader is referred to the web version of this article.)


for constant stress and strain rate, respectively.According to the results in Section 2.3 (Fig. 2), r decreases with

increasing qD=C, and thus (23) shows that the dimensionless inter-face roughness (or the effective grain size) decreases with increas-ing stress or strain rate, which is in agreement with theexperimental and field observations, as parameterized by the clas-sical piezometers (e.g. Van der Wal et al., 1993) and pale-owattmeters (Austin and Evans, 2007).

3.1.1. Temperature dependence of the damage partitioning fraction f �IThe damage to healing ratio is proportional to f �I . A temperature

sensitive f �I amplifies the temperature-dependence of qD=C, whichyields a stronger variation of rwith temperature compared to caseswith constant f �I (Fig. 3). Thus, models with temperature sensitivef �I are more likely to predict a grain size profile that transitionsfrom diffusion creep at colder temperatures to dislocation creepat hotter ones. The larger the value of f �I , the higher is the temper-ature at which the transition in creep mechanism occurs (Fig. 3).

Fig. 4. Temperature-dependence of the damage to healing ratio and the dimensionlesscolored line is the field boundary, colored according to temperature. Left column: cotemperature-dependent partitioning fraction f �I ¼ 10�8f 0, except for the dashed bla400 kJ mol�1, the constants entering the temperature-dependent partitioning fraction (6For the constant stress case with low EG (i.e., EG ¼ 200 kJ mol�1), the model fails to proddislocation creep with increasing temperature. (For interpretation of the references to c

Observational constraints favor the models with positive corre-lation between temperature and grain size, as well as a transitionfrom diffusion to dislocation creep at hotter temperatures. Modelsfeaturing a temperature-dependent f �I comply most readily withthese observations.

3.1.2. Grain growth activation energy EG

The grain growth activation energy EG, which enters DE (23), isone of the parameters with the greatest uncertainty in our model(Evans et al., 2001; Warren and Hirth, 2006). Moreover, the avail-able experimental constraints that do exist for EG apply to thegrowth of grains, rather than the interface coarsening that is beingmodeled here. However, for the lack of better constraints, weassume that EG is the same for grain growth and interface coarsen-ing. With these limitations in mind, we will vary DE through vary-ing EG, in an attempt to constrain the latter.

According to (23), a positive DE contributes to a decrease ofqD=C with increasing temperature. For a positive correlation

interface roughness. Results for three values of EG are presented (see legend). Thenstant ss ¼ 100 MPa. Right column: constant _es ¼ 10�13 s�1. All models employ ack line, which has f �I ¼ 5 � 10�7f 0 and EG ¼ 200 kJ mol�1. For EG ¼ 200;300 and) are, respectively: Cf ¼ 1:91;11:2;22:0 and kf ¼ 2:94;0:68;0:099 (see Appendix B).uce a profile of interface roughness that increases and transitions from diffusion toolour in this figure legend, the reader is referred to the web version of this article.)


between temperature and grain size, as suggested by the observa-tions, qD=C needs to be anti-correlated with temperature (Sec-tion 2.3, Fig. 2). The combined effect of DE and f �I on qD=C aresuch that the grain growth activation energy needs to beEG P 250 kJ mol�1 in order for qD=C to decrease with increasingtemperature, and thus also for the interface roughness to increasewith increasing temperature (Fig. 4).

The grain growth activation energy used in Rozel et al. (2011),EG ¼ 200 kJ mol�1, renders DE to be negative, which would lead

Fig. 5. Temperature-dependence of the damage to healing ratio and the interface roughnconstant ss ¼ 50 MPa. Right column: constant _es ¼ 10�12 s�1. Black lines: R�

m ¼ 3 � 10�1; gThin black lines indicate the unstable solution branch. All models employ EG ¼ 400 kJ mtemperature. The dashed black and grey lines are the mixing transition roughness R�

m ¼ 3in this figure legend, the reader is referred to the web version of this article.)

to a decrease in grain size with increasing temperature, contradic-tory to the field observations. Recent experimental results suggesta larger value for EG (Evans et al., 2001), yielding a positive DE, andthus also a grain size that increases with temperature.

For systems with a constant strain rate, DE is positive even forlow values of EG. These cases are thus not as sensitive to the valueof EG, and therefore less useful for constraining it.

We limit our further analysis to models with a temperature-dependent f �I and a grain growth activation energy

ess (dimensionless in the middle row, dimensional in the bottom row). Left column:rey lines: R�

m ¼ 1 � 10�1 and R�m ¼ 1, with the latter featuring no jump in grain size.

ol�1 and f �I ¼ 10�6f 0. The colored line is the field boundary, colored according to� 10�1 and R�

m ¼ 1 � 10�1, respectively. (For interpretation of the references to colour


EG P 250 kJ mol�1, such that DE > 0. Specifically, we useEG ¼ 400 kJ mol�1, which, for our choice of the rheological compli-ances, is well within this constraint.

3.2. Hysteresis

The expression for the damage to healing ratio is unaffected byinclusion of hysteresis. However, there are important differences inthe effect that qD=C has on the interface roughness r in the hystere-sis model (see Section 2.3).

The hysteresis effect (see Section 2.3) introduces three distinctdeformation-related states into the model: there are two stateson either side of the hysteresis loop that are dominated by eitherdiffusion or dislocation creep, and there is one state inside the hys-teresis loop where the two creep mechanisms coexist. A steadystate solution with small r in the diffusion creep regime exists atlow temperatures, and one with large r in dislocation creep regimeexists at high temperatures (Fig. 5). Both regimes can co-exist atintermediate temperatures that span the hysteresis loop. Materialthat is deforming at conditions that fall within the hysteresis loopis expected to feature pockets of small grains in diffusion creepneighboring pockets of large grains in dislocation creep.

The temperature at which the steady state solution crosses rF iscontrolled by R�

m. A larger R�m allows diffusion creep (and thus also

for mylonites and ultramylonites to form) to exist at higher tem-peratures, and pushing the transition to dislocation creep (and thusalso the occurrence of CPO structure) to occur at higher tempera-tures (Fig. 5).

A striking feature of the hysteresis model is the abrupt changein interface roughness associated with intersecting the fieldboundary, which involves crossing the hysteresis loop. We showedin Section 2.3 that the range of qD=C in which the hysteresis loopexists depends on R�

m. The larger R�m, the smaller is the jump in r

associated with crossing the field boundary from one creep mech-anism to another. At sufficiently large R�

m (i.e., R�m > 1), the hystere-

sis loop vanishes (grey lines in Fig. 5).

3.3. Complete temperature-dependence of grain size in a shear zone

The variation of interface roughness with temperature T entersthrough the T-dependence of the damage to healing ratio qD=C 17aand 17b and the field boundary roughness RF (4). The temperature-

Fig. 6. Temperature-dependence of grain size for the Oman ophiolite, according to the ddata-points with secondary phase volume fraction > 0:3) and the hysteresis model (soModeled results for interface roughness were converted to grain size, according to the ps; F0 and R�

m are shown (see titles), and using EG ¼ 400 kJ mol�1. The colored line is the fie(For interpretation of the references to colour in this figure legend, the reader is referre

dependence of qD=C only affects variation in the dimensionlessroughness (i.e., effectively normalized by RF), which increases withincreasing T so long as EG P 250 kJ mol�1. This means that disloca-tion creep becomes more dominant with increasing temperaturefor sufficiently large EG. However, for our chosen values of Edisl

and Ediff (Table 1), the field boundary grain size RF decreases withincreasing T for constant stress, but increases for constant strainrate. Thus, the actual dimensional value of r can potentiallydecrease with increasing T for constant stress, especially if RF ismore temperature sensitive than qD=C.

For our given temperature range and values of the activationenergies, the dimensional field boundary roughness RF in the con-stant stress case decreases by about one order of magnitude fromthe lowest to the highest T. Meanwhile, the normalized roughnessincreases by over three orders of magnitude across the same T-range due to the sufficiently large decrease in qD=C atEG ¼ 400 kJ mol�1 (which satisfies the required EG P 250 kJ mol�1).As a net effect, the dimensional roughness increases with tempera-ture, in agreement with the observational constraints (Fig. 5, bot-tom row).

4. Discussion: comparison to lithospheric shear zones

Our theoretical models make various testable predictionsregarding the steady state grain size of a polymineralic materialdeformed by simple shear, and these can be compared with pub-lished observational data. Specifically, we are interested in theeffect of temperature, and by inference also depth (since pressureeffects are less pronounced), on the steady state grain size andthe dominant deformation regime. The detailed field data fromthree different mantle shear zones reported in Linckens et al.(2015) enable us to constrain some of the parameters that enterthe grain damage models presented here.

For all of the three shear zones, the grain size appears toincrease with increasing temperature, ranging from 7 to 60 lmultramylonites that form at temperatures 900–1030 K, to 10–350 lm mylonites that form at 1000–1100 K, and to 35–3900 lmtectonites that form at 1140� 1370 K (Linckens et al., 2015). Weshowed in Section 3.3 that in order to obtain a positive correlationbetween grain size and temperature for stress driven models, thegrain growth activation energy EG needs to be relatively large(EG P 250 kJ mol�1). For example, EG ¼ 400 kJ mol�1 satisfies this

ata from (Linckens et al., 2015) (black and grey vertical lines, with black indicatinglid black curves, where the thinner curves indicate the unstable solution branch).inned state limit assumption (Section 2). Results for two different sets of values forld boundary, colored according to temperature. Rm is shown as dashed black curves.d to the web version of this article.)


constraint, and we employ this value from now on. For the samereason, we employ the temperature-dependent f I 17a and 17b, asit is more robust in producing a grain size that increases withtemperature.

The natural samples studied by Linckens et al. (2015) exhibit alarge range of grain sizes, increasing by over two orders of magni-

Fig. 7. Temperature-dependence of grain size, according to the data from (Linckens et algiven in Fig. 6. For constant strain rate cases, results for two different R�

m-values are showthe references to colour in this figure legend, the reader is referred to the web version o

tude across a 300–400 K increase in temperature. To reproducesuch a variation in grain size for the given temperature range withthe paleowattmeter-type models (Austin and Evans, 2007; Rozelet al., 2011) or the simpler non-hysteretic two-phase grain damagemodel (Bercovici and Ricard, 2012) would require prescribing asignificant variation in stress. The hysteresis grain damage model,

., 2015) and the hysteresis model. Explanation for different line-types and colors aren, with the grey dashed curve indicating the smaller R�

m-value. (For interpretation off this article.)


on the other hand, predicts a large range of grain sizes even for theconstant stress models. Therefore, since we consider models wherestress and strain rate are assumed to be constant and uniform, weuse the hysteresis model for our further analysis.

The crystallographic orientation measurements in the naturalsamples suggest that the largest grains (Lanzo-protomylonitesand Oman-porphyroclasts) deform by dislocation creep, as theyhave a strong crystal preferred orientation (CPO) (Linckens et al.,2015). In the mylonitic samples, CPO is generally weaker, suggest-ing that dislocation creep may be occurring, but is less dominant.

Fig. 8. Temperature-dependence of the damage to healing ratio, strain rate (for constanophiolite (third row in Fig. 7). For constant strain rate cases, results for two different R�

m

Mylonites from the shear zone samples presented in Jin et al.(1998) exhibit a coexistence of both diffusion and dislocation creepregimes, where a grain size dependent rheology, typical of diffu-sion creep, coexists with a CPO-microstructure, typical of disloca-tion creep. These observations indicate that mylonites aredeforming mainly by diffusion creep, but can be close to the fieldboundary such that some dislocation creep occurs within thegrain-size distribution, as was shown in Rozel et al. (2011), andas is also assumed in our model. (An alternative model to explainthis observation is deformation by dislocation-accommodated

t stress model), stress (for constant strain rate model) and viscosity for the Oman-values are shown, with the grey curve indicating the result with a smaller R�

m .


grain boundary sliding (disGBS), e.g., Warren and Hirth (2006, ?).)In the ultramylonites, no CPO is observed, suggesting that thesmallest grains are deeply in the diffusion creep regime. Thus,the field data suggest that as the grain size increases with increas-ing temperature, the dislocation creep becomes increasingly moredominant.

For the model to match the data, we need to constrain threeparameters: (a) the driving stress or strain rate, (b) the scaling fac-tor F0 (see 6) for the damage partitioning fraction, and (c) the mix-ing transition ratio R�

m (see 7).The applied forcing, i.e., stress or strain rate, determines the

field boundary grain size at different temperatures. The lower limiton the applied stress or strain rate is constrained such that the lar-gest grains are on the dislocation creep side of the field boundary(Fig. 6). The upper limit on the applied forcing comes from con-straining the ultramylonites to lie in the diffusion creep regime.The value for F0 is chosen such that the modeled small grain solu-tion branch coincides with the ultramylonites. Finally, we chooseR�m such that the large grain solution branch matches the largest

grain sizes observed in the field (Fig. 6).The value of R�

m also determines the temperature at which thetransition in deformation mechanism takes place. The deformationtransition lies within the hysteresis loop, which has the coexis-tence of different deformation regimes associated with it. Thus, ifthe hysteresis loop lies in the vicinity of the mylonitic samples, itmay explain the observed coexistence of deformation regimeswithin their grain sizes range.

Given these constraints, the values for F0 and R�m predicted by

our model fitting are dependent on the prescribed driving stressor strain rate conditions. As an example, we show how the fielddata from the Oman ophiolite can be matched by the hysteresismodel for two different stress values, which are within the rangeof the stress-values inferred by Linckens et al. (2015). The modelwith a lower stress value (5 MPa) permits a good match for thelarge grained tectonites, but does not match the ultramylonitesas well (Fig. 6). Moreover, it puts the mylonites relatively deep intodiffusion creep regime, making it unlikely to observe a microstruc-ture indicative of dislocation creep within their range of grainsizes. The solution with a higher stress value (100 MPa) agrees wellboth with the observed ultramylonites and the large grained tec-tonites. Further, it puts the mylonites in the vicinity of the fieldboundary, in support of the observed coexistence of deformationregimes. However, this model puts the mylonites just outside thepredicted steady state solution branches, inferring that they havenot reached the steady state yet, but are possibly still shrinking(Fig. 6). The misfit between the mylonites and the modeled steadystate solution can also be explained by the fact that the naturalshear zones are not necessarily at constant stress; a better matchbetween the model and the data can be obtained by permittingsmall variations in stress (Bercovici and Ricard, 2016).

We applied the same fitting procedure to the field data from theother two locations described by Linckens et al. (2015): the Othrisand Lanzo mantle shear zones. The reported range of grain sizesobserved in these shear zones is more narrow than in Oman, andthus serve as less stringent constraints for the model (Fig. 7).Importantly, no abrupt or physically inconsistent changes in stressor strain rate are needed to be prescribed in order to match thereported range of grain sizes observed across each shear zone.

The predicted qD=C and viscosity profiles are expected to besimilar for the three shear zones, since we used the same set ofparameters for all of them, except for a relatively small differencesin the driving stress or strain rate. Therefore, we only show theresults for these quantities for the Oman ophiolite (Fig. 8).

As expected, qD=C and viscosity decrease with increasing tem-perature. The range of our predicted viscosity values is plausible

for a lithospheric shear zone setting, perhaps with the exceptionof the somewhat low viscosity (� 1017 Pa s) obtained for the con-stant stress case at high temperatures. Prediction of such low vis-cosity values in the modeling results is due to the limitation ofapplying a constant stress across the entire range of depths andtemperatures.

At approximately 1000 K or deeper, the model predicts a sharpdecrease in viscosity with decreasing temperature. This jump isassociated with the transition from dislocation to diffusion creep,and involves a viscosity-decrease of up to three orders of magni-tude for the constant stress cases, and up to one order of magni-tude for the constant strain rate cases. Thus, the variation ingrain size and temperature, and in particular the transition indeformation regime, helps to weaken the lithosphere and offsetthe increase in viscosity induced by the decrease in temperature.

5. Conclusions

We presented applications of the grain-damage model(Bercovici and Ricard, 2012) for deformation of a polymineralicmaterial at different thermal conditions that are relevant for study-ing shear localization in the ductile portion of the lithosphere. Ourultimate goal is to test and constrain the grain damage model withgeological and experimental observations. We demonstrated thatthe hysteresis model, in which the partitioning fraction (i.e., thefraction of work going to create new grain surfaces) depends onboth temperature and interface, can successfully match the fielddata, including the reported correlation of grain size and tempera-ture, the increasing dominance of dislocation creep at higher tem-peratures and the large range of grain sizes observed across thedepth of a single shear zone.

In agreement with the classical piezometers, our model predictsa decrease of grain size with increasing stress. However, our modeldoes not require large variations in stress across the lithosphere inorder to match the observed variations of grain size. Moreover, ourresults highlight the challenge of inferring the deformational stressor strain rate from the grain sizes observed in a mantle shear zone.

To match the observations, the activation energy for graingrowth needs to be relatively large (EG P 250 kJ mol�1). Interest-ingly, the grain growth activation energy predicted by the modelis similar to that for diffusion creep. The latter is better constrainedfrom published experimental studies, ranging between 260 and450 kJ mol�1 (Hirth and Kohlstedt, 2003), with Ediff ¼ 375 kJ mol�1

being a commonly used value (e.g. Rozel et al., 2011). The similar-ity in values of EG and Ediff may be unsurprising, given that theyboth involve atomic scale diffusion-limited processes.

The previously poorly constrained (or unconstrained) values ofthe mixing transition ratio Rm and the partitioning fraction f I havebeen constrained by fitting our model to the field observations oflithospheric shear zones at three different tectonic settings. Ourproposed range of values are F0 ¼ 2 � 10�6 � 5 � 10�5 andR�m ¼ 10�1 � 5 � 10�1. However, these values are sensitive to the

assumed value for EG and the driving stress or strain rate. The pre-dicted values for f I and Rm can be tested in future experimentalstudies, as well as against other geological observations of shearzones.

Finally, according to the field constraints used in this study, thehysteresis model predicts that the maximum temperature at whichmylonites can form, and thus at which self-weakening and shearlocalization due to the effect of pinning in the ductile portion ofthe lithosphere can occur, is � 1350 K. This corresponds to a depththat is well within the lithosphere: a few tens to up to one hundredkilometers, which is sufficient to explain the formation of weakplate boundaries, localized deformation of which is responsiblefor the occurrence of plate tectonics on Earth.


Acknowledgements

This work was supported by NSF Grants EAR-1344538 and EAR-1135382. The authors would like to thank the reviewers DavidKohlstedt and Laurent Montesi, who dedicated their considerableexpertise to reviewing this paper and made valuable contributionsto it.

Appendix A. Two-phase grain damage model

We use the simplifying assumption of a pinned state limit forthe system studied in the main body of this paper. In this limit,the grain growth rate in the two-phase material is dictated bythe coarsening rate of the interface, or, equivalently, the growthrate of the pinning bodies. Bercovici and Ricard (2012) argue thata two-phase system evolves towards a pinned state limit withtime. This lends support for using the steady state models togetherwith the pinned state limit, as we do in the current study.

It is, however, important to test if the steady state solution ofthe full two-phase system is consistent with the one obtained forthe pinned state limit. Thus, here we find the steady state solutionfor the fully coupled two-phase system. We do this for themost simple case: a stress driven model without the effect ofhysteresis.

A.1. Dimensionless two phase grain damage model

The mathematical model of the grain size evolution in a two-phase material was presented in Bercovici and Ricard (2012). Thedimensionless equations for rheology, interface roughness andgrain size evolution are (respectively):

_ei ¼aisni þ bisiRmi

ðA:1Þ

drdt

¼ CI

qrq�1 � DIr2 �W ðA:2ÞdRi

dt¼ Ci

pRp�1i

Zi � DiR2i s

nþ1i Z�1

i ðA:3Þ

where subscript i (i ¼ 1;2) enumerates the two phases and a barover a quantity indicates volume average of that quantity. Further,we introduced the dimensionless parameters:

Fig. A.9. Zero-levels for the growth rate of the grain size as a function of the damage to hfield-colors indicate the sign of the grain size growth rate: grey is negative and white is pindicates the grain size above which the Zener pinning factor is negative. (For interpretatversion of this article.)

ai ¼ Ai

A; bi ¼ Bi

B

CI ¼ gGI

Asns rqs

; DI ¼ 2rsssf �Icg

Ci ¼ Gi

Asns Rps

; Di ¼ 2k33k2

aiRsss/if 0ð1� f �I Þc

ðA:4Þ

where Rs ¼ rs and kn is the n-th moment of lognormal grain-size

distribution, given by kn ¼exp n2r2

2

� �(r ¼ 0:8 is the variance). See

Section 2.1 and Table 1 for the definitions of the other parameters.The Zener pinning factor, appearing in (A.3), is defined as:

Zi ¼ 1� hiRi

r

� �2

; where hi ¼ 3160

k4k2

ð1� /iÞ ðA:5Þ

In the steady state, the interface roughness is given by:

r ¼ CI

qDI

� � 1qþ1 1

�W

� � 1qþ1

ðA:6Þ

The steady state grain size is given by:

0 ¼ R4i �

pDi

Ci

� �r4

h2i

snþ1i Rpþ1

i � 2r2

hiR2i þ

r4

h2i

ðA:7Þ

For simplicity, we will from now on assume that the two phaseshave equal dislocation and diffusion creep compliances (Ai ¼ A ¼ Aand Bi ¼ B ¼ B, respectively), as well as equal grain growth ratesGi ¼ GG. The two-phase grain damage model features three nondi-mensional damage to healing ratios: one for the interface (qDI=CI)and two for the grains of each of the two phases (pDi=Ci). For a sys-tem driven by a constant applied stress ss, we get:

qDI

CI¼ 2qcg2

f �IGI

Bðqþ1m Þ

Aðqþ1�mm Þ

sqþ1þm�nðqþ1�mÞ

ms ðA:8Þ

pDi

Ci¼2k33k2

pc/ið1� f �I Þf 0

GG

Bðpþ1m Þ

Aðpþ1�mm Þ

spþ1þm�nðpþ1�mÞ

ms ðA:9Þ

A.2. Solution space for the steady state

To simplify the solution space analysis for the two phase graindamage model, we assume an equal volume fraction for bothphases (/i ¼ 0:5), reducing the two phase-specific steady state

ealing ratio in a two-phase grain damage model (black solid lines). White and greyositive. Black dashed line indicates the field boundary grain size. Green dashed lineion of the references to colour in this figure legend, the reader is referred to the web


grain size expressions in (A.7) to just one equation. Similarly, thetwo damage to healing ratios pDi=Ci (A.9) for the grains reduce toa single parameter pDi=Ci ¼ pDG=CG for both phases.

Two-phase grain damage model can have one, two or eventhree stable solutions for the steady state, depending on the valuesof qDI=CI and pDG=CG (Fig. A.9). At high rates of interface damage(high qDI=CI), there is only one stable solution, which is in diffusioncreep. At low qDI=CI , there are two stable solutions: one in diffu-sion and one in dislocation creep regimes. There exists a limitedrange of intermediate qDI=CI-values where three stable solutionsare available (as long as pDG=CG > 50): two in diffusion creep,and one in dislocation creep regimes.

For all of the stable solutions, an increase in qDI=CI (whichimplies a decrease in the interface roughness (A.6)) results in smal-ler grain size. Thus, a decrease in the interface roughness drags thegrain size down with it, which is intuitive.

For all of the tested values of qDI=CI and pDG=CG, there exists astable diffusion creep solution that suggests a decrease in grainsize for increasing qDI=CI- and pDG=CG-values. This solution canbe readily interpreted: an increase in the damage rate of eitherthe interface or the grains leads to a smaller grain size.

Fig. A.10. Temperature-dependence of (clockwise from top left) the damage to healingfactor, and the dimensionless and dimensional grain size R and interface roughness r (sebranches with Zi < 0. The model employs EG ¼ 400 kJ mol�1 and f �I ¼ 10�8.

The solution branch in the stable dislocation creep regime pre-dicts an increase in grain size with increasing pDG=CG. This iscounter-intuitive, since we expect that increasing the damage rateto the grains would cause them to shrink. However, the steadystate grain size depends on the Zener pinning factor Zi (A.7), whichbecomes negative when the grain size is larger than the interfaceroughness by a factor of more than

ffiffiffiffiffiffiffiffiffiffi1=hi

p(A.5), at which point

the roles of the damage and the coarsening terms in (A.3) becomereversed. We do, indeed, see that the Zener pinning factor is nega-tive in this part of the solution space (Fig. A.9). Specifically, Zi 0when the value of qDI=CI is smaller than or is comparable to pDi=Ci.For Earth-like materials, pDi=Ci qDI=CI (Bercovici and Ricard,2016), rendering the branches with Zi 0 unlikely to occur innature.

A.3. Temperature-dependence of the grain size in a shear zone

In the full two-phase grain damage model, the grain size andthe interface roughness parallel each other at cold temperatures(Fig. A.10). This is the case when qDI=CI is significantly larger thanpDi=Ci, which happens to be at cold temperatures. At such condi-

ratio (pDi=Ci for the grains, qDI=CI for the interface, see legend), the Zener pinninge legend) for the two-phase grain damage model. Dashed lines indicate the solution


tions, the model conforms to the pinned state limit, with Zi close tozero.

As qDI=CI approaches and eventually becomes smaller thanpDi=Ci at higher temperatures (and the Zener pinning factorbecomes negative), the second stable solution emerges for thegrain size and the interface roughness, as was predicted in Sec-tion A.2. For this solution branch, there is a divergence in grain sizeand interface roughness with increasing temperature, contrary towhat is expected if the system is approaching a pinned state limit.

The steady state solutions for the interface roughness that cor-respond to the branches with Zi � 0 are consistent with the pinnedstate solutions found in the main part of this paper. However, theadditional solution branches that come forth when we solve thefull two-phase model, which has a much larger grain size andZi 0, are dramatically different from the prediction of the pinnedstate model. As was already mentioned in Section A.2, the brancheswith Zi 0 are unlikely to occur in nature.

Appendix B. Temperature-dependence of the partitioningfraction

The temperature-dependent part of the partitioning fraction forinterface damage (f �I ) is assumed to be similar to that of the graindamage (f 0), but scaled by a constant F0 (6). While there aren’t any

Fig. B.11. Top: Published experimental piezometers (colored circles, see legend) and thdifferent values of EG : 200 kJ mol�1 (black solid line) and 400 kJ mol�1 (black dashed lineline) is shown for comparison, to demonstrate that it is only a good fit when EG ¼ 200 kJ mis referred to the web version of this article.)

experimental constraints on f �I and F0, we can use the publishedexperimental data to constrain f 0, which is done below. Grain sizeevolution for a single phase material is given by (Rozel et al., 2011):

dRdt

¼ GG

pRp�1 �k33k2

R2fWc

ðB:1Þ

See Table 1 for the definitions of the other parameters.We assume that the partitioning fraction f of deformational

work going into creating new grain boundary energy, is suppressedwhen the grains are deforming by diffusion creep (i.e., only themechanical work associated with dislocation creep strain can beused to form new grains; see Rozel et al. (2011)):

f ¼ f 01þ B

Asn�11Rm

ðB:2Þ

where f 0 is the maximum partitioning fraction. The quantity f 0 isinferred to be temperature-dependent, and can be estimated usingpiezometers from the published experimental studies, as outlinedbelow.

The steady state grain size is given by:

G

pRp�1 ¼ 2k33k2

R2f 0Asnþ1

cðB:3Þ

eir fitting to Eq. B.5 (colored lines). Bottom: Temperature-dependence of f 0 for two). The relation f 0 ¼ expð�2ðT=103Þ2:9Þ presented in (Rozel et al., 2011) (grey dashedol�1. (For interpretation of the references to colour in this figure legend, the reader


This equation can be rewritten in the form of the followingstress and grain size dependence (i.e., a piezometer):

R ¼ 3k22k3

cGpAf0

� � 1pþ1

s�ðnþ1Þ=ðpþ1Þ ðB:4Þ

The stress and grain size dependence can be directly measuredin the laboratory experiments, which typically report their resultstogether with a fitted piezometric curve of the form:

R ¼ Clabs�klab ðB:5Þ

where Clab and klab are determined by fitting to the experimentalresults. Comparing (B.4) and (B.5), the experimentally approxi-mated constants Clab and klab can be expressed in terms of theparameters that enter our theoretical model:

Clab ¼ 3k22k3

cGpAf0

� � 1pþ1

ðB:6Þ

klab ¼nþ 1pþ 1

ðB:7Þ

Using the experimentally determined value of the constant Clab,we can deduce that:

f 0 ¼ 1Cpþ1lab

3k22k3

cGpA

� �ðB:8Þ

Note that the relationship between f 0 and Clab depends on theassumed values of the dislocation creep compliance A and the rateof grain growth G.

Here we assume that the exponents n and p are fixed at n ¼ 3and p ¼ 2. Thus, for consistency, we fix the exponent klab ¼ nþ1

pþ1,

and then use the raw data from the published experimental resultsto redo the fitting of the piezometric curves to obtain Clab, similarto what was done by Rozel et al. (2011). With klab fixed, the varia-tion in Clab between different experiments is entirely attributed tothe variation in temperature (Fig. B.11). Thus, the temperature-dependence of the partitioning fraction f 0 is dependent on the

Fig. B.12. Temperature-dependence of the partitioning fraction f 0 for different values ofdE and F (colored circles (B.8)) are approximated with a relation f 0 ¼ exp �Cf ð T

TrefÞkf

� �. Co

and F (solid lines). After performing the fitting of f 0 for a range of dE- and F-values, we parof the references to colour in this figure legend, the reader is referred to the web versio

thermal variation of Clab as well as that of the dislocation creepcompliance A and the rate of grain growth G (B.8).

We can express f 0 in terms of its temperature-dependent(Clab;A and G) and temperature-independent factors. Insertingexpressions for A and G, we write:

f 0 ¼ 3k22k3

cG0

A0

� �1

pCðpþ1Þlab

expEdisl � EG

RT

� �

f 0 ¼ Fðn;p;r; c;A0;G0Þ 1

pCðpþ1Þlab

expdERT

� � ðB:9Þ

where we introduced the temperature-independent parameterFðn; p;r; c;A0;G0Þ ¼ 3k2

2k3cG0A0. According to (B.5), the value of Clab also

depends on n and p, through klab (B.7). This means that F and Clab

are not independent. Additionally, there is a constraint thatf 0 6 1. Therefore, we approximate the entire right hand side of(B.9) with a single temperature-dependent function of a form thatguarantees f 0 6 1, rather than inverting for F and Clab separately.

Following (Rozel et al., 2011), we approximate the temperature-dependence of f 0 with an exponential function f 0 ¼exp �Cf ð T

TrefÞkf

� �. This function satisfies f 0 6 1, as long as Cf P 0.

The constants Cf and kf are determined by fitting this function tothe values of f 0 deduced from the experimentally determinedquantity Clab and other model-specific parameters in (B.8). SinceCf is empirically determined, there is still no guarantee it will bea positive number. Bearing this limitation in mind, we find aparameterization for f 0 for the range of the physical parametersthat satisfy f 0 6 1. This range is well within what is suggested fromthe previously published experimental studies.

The parameterization presented in (Rozel et al., 2011) (f 0 ¼expð�2ðT=103Þ2:9Þ) is only valid for one specific choice of relationsfor AðTÞ and GðTÞ.

To generate the results in the main body of this paper, we per-formed the fitting of Cf and kf for various possible dE. However, tofacilitate the use of this model in future studies, we present here aparameterized model of f 0 as a general function of dE and F.

dE (left column) and F (right column). The experimentally deduced f 0 for each set ofnstants Cf and kf can be found by performing the fitting for each individual set of dEameterize Cf and kf as functions of dE and F (dashed lines, (B.10)). (For interpretationn of this article.)


We vary dE by varying EG between 200� 400 kJ mol�1, whilekeeping Edisl constant at ¼ 530 kJ mol�1. F is varied within therange F � �104 � 2 � 105. The results are presented in Fig. B.12.

To summarize, the general model for the temperature-dependent part of the partitioning fraction is given by:

dE ¼ Edisl � EG

F ¼ 3k22k3

cG0

A0

Cf ¼ 35� 9 � 10�5dE� 0:5log10ðFÞkf ¼ 1:7 � 10�19dE3:4F0:081

f 0 ¼ exp �CfTTref

� �kf !

ðB:10Þ

which guarantees f 0 6 1 for the range of parameters listed above.

References

Austin, N.J., Evans, B., 2007. Paleowattmeters: a scaling relation for dynamicallyrecrystallized grain size. Geology 35 (4), 343–346.

Behn, M.D., Hirth II, G., 2009. Implications of grain size evolution on the seismicstructure of the oceanic upper mantle. Earth Planet. Sci. Lett. 282 (1–4), 178–189. J.R.E. http://www.sciencedirect.com/science/article/pii/S0012821X09001575.

Bercovici, D., 1993. A simple model of plate generation from mantle flow. Geophys.J. Int. 114 (3), 635–650.

Bercovici, D., 1995. On the purpose of toroidal motion in a convecting mantle.Geophys. Res. Lett. 22, 3107–3110.

Bercovici, D., 2003. The generation of plate tectonics from mantle convection. EarthPlanet. Sci. Lett. 205 (3), 107–121.

Bercovici, D., Ricard, Y., 2005. Tectonic plate generation and two-phase damage:void growth versus grainsize reduction. J. Geophys. Res. 110, B03401. http://dx.doi.org/10.1029/2004JB003181.

Bercovici, D., Ricard, Y., 2012. Mechanisms for the generation of plate tectonics bytwo-phase grain-damage and pinning. Phys. Earth Planet. Inter. 202, 27–55.

Bercovici, D., Ricard, Y., 2013. Generation of plate tectonics with two-phase grain-damage and pinning: Source–sink model and toroidal flow. Earth Planet. Sci.Lett. 365, 275–288.

Bercovici, D., Ricard, Y., 2014. Plate tectonics, damage and inheritance. Nature 508(7497), 513–516.

Bercovici, D., Ricard, Y., 2016. Grain-damage hysteresis and plate tectonic states.Phys. Earth Planet. Inter. 253, 31–47.

Bercovici, D., Schubert, G., Ricard, Y., 2015. Abrupt tectonics and rapid slabdetachment with grain damage. Proc. Nat. Acad. Sci. 112 (5), 1287–1291.

Bercovici, D., Skemer, P., 2017. Grain damage, phase mixing and plate-boundaryformation. J. Geodynamics 108, 40–55.

Bercovici, D., Tackley, P.J., Ricard, Y., 2015. The generation of plate tectonics frommantle dynamics. In: Bercovici (Gerald Schubert editor-in-chief), David (Ed.),Treatise on Geophysics. . second ed., Mantle Dynamics second ed., Vol. 7Elsevier, pp. 271–318. Ch. 7.

Cross, A., Skemer, P., 2017. Ultramylonite generation via phase mixing in high strainexperiments. J. Geophys. Res-Solid Earth 122, 1744–1759.

Evans, B., Renner, J., Hirth, G., 2001. A few remarks on the kinetics of static graingrowth in rocks. Int. J. Earth Sci. 90 (1), 88–103.

Farla, R.J., Karato, S.-I., Cai, Z., 2013. Role of orthopyroxene in rheological weakeningof the lithosphere via dynamic recrystallization. Proc. Nat. Acad. Sci. 110 (41),16355–16360.

Gueydan, F., Précigout, J., Montesi, L.G., 2014. Strain weakening enables continentalplate tectonics. Tectonophysics 631, 189–196.

Hansen, L.N., Cheadle, M.J., John, B.E., Swapp, S.M., Dick, H.J., Tucholke, B.E., Tivey, M.A., 2013. Mylonitic deformation at the kane oceanic core complex: Implicationsfor the rheological behavior of oceanic detachment faults. Geochem. Geophys.Geosyst. 14 (8), 3085–3108.

Hansen, L.N., Warren, J.M., 2015. Quantifying the effect of pyroxene on deformationof peridotite in a natural shear zone. J. Geophys. Res.: Solid Earth 120 (4), 2717–2738.

Hansen, L.N., Zimmerman, M.E., Kohlstedt, D.L., 2011. Grain boundary sliding in sancarlos olivine: Flow law parameters and crystallographic-preferred orientation.J. Geophys. Res.: Solid Earth 116 (B8). http://dx.doi.org/10.1029/2011JB008220.

Herwegh, M., Linckens, J., Ebert, A., Berger, A., Brodhag, S., 2011. The role of secondphases for controlling microstructural evolution in polymineralic rocks: areview. J. Struct. Geol. 33 (12), 1728–1750.

Hiraga, T., Tachibana, C., Ohashi, N., Sano, S., 2010. Grain growth systematics forforsterite ± enstatite aggregates: effect of lithology on grain size in the uppermantle. Earth Planet. Sci. Lett. 291 (1), 10–20.

Hirth, G., Kohlstedt, D., 2003. Rheology of the upper mantle and the mantle wedge:a view from the experimentalists. In: Eiler, J. (Ed.), Subduction Factor Mongraph.Vol. 138. Am. Geophys. Union, Washington, DC, pp. 83–105.

Hirth, G., Kohlstedt, D.L., 1995. Experimental constraints on the dynamics of thepartially molten upper mantle: 2. deformation in the dislocation creep regime. J.Geophys. Res.: Solid Earth 100 (B8), 15441–15449.

Hobbs, B., Mühlhaus, H.-B., Ord, A., 1990. Instability, softening and localization ofdeformation. Geol. Soc., London, Special Publications 54 (1), 143–165.

Jin, D., Karato, S.-I., Obata, M., 1998. Mechanisms of shear localization in thecontinental lithosphere: inference from the deformation microstructures ofperidotites from the ivrea zone, northwestern italy. J. Struct. Geol. 20 (2), 195–209.

Karato, S., 1989. Grain growth kinetics in olivine aggregates. Tectonophysics 168(4), 255–273.

Karato, S., 2008. Deformation of Earth Materials: An Introduction to the Rheology ofSolid Earth. Cambridge Univ Pr.

Karato, S.-I., Toriumi, M., Fujii, T., 1980. Dynamic recrystallization of olivine singlecrystals during high-temperature creep. Geophys. Res. Lett. 7 (9), 649–652.

Kaula, W.M., 1980. Material properties for mantle convection consistent withobserved surface fields. J. Geophys. Res.: Solid Earth 85 (B12), 7031–7044.

Kohlstedt, D., Evans, B., Mackwell, S., 1995. Strength of the lithosphere: constraintsimposed by laboratory experiments. J. Geophys. Res.: Solid Earth 100 (B9),17587–17602.

Kohlstedt, D., Hansen, L., 2015. 2.18 – constitutive equations, rheological behavior,and viscosity of rocks. In: Schubert, G. (Ed.), Treatise on Geophysics. second ed.Elsevier, Oxford, pp. 441–472.

Landuyt, W., Bercovici, D., Ricard, Y., 2008. Plate generation and two-phase damagetheory in a model of mantle convection. Geophys. J. Int. 174 (3), 1065–1080.

Linckens, J., Herwegh, M., Müntener, O., 2015. Small quantity but large effect? Howminor phases control strain localization in upper mantle shear zones.Tectonophysics 643, 26–43.

Linckens, J., Herwegh, M., Müntener, O., Mercolli, I., 2011. Evolution of apolymineralic mantle shear zone and the role of second phases in thelocalization of deformation. J. Geophys. Res. 116. B06210, 21 pp.

Montési, L., Zuber, M., 2002. A unified description of localization for application tolargescale tectonics. J. Geophys. Res. 107. http://dx.doi.org/10.1029/2001JB000465.

Montési, L.G., 2013. Fabric development as the key for forming ductile shear zonesand enabling plate tectonics. J. Struct. Geol. 50, 254–266.

Montési, L.G., Hirth, G., 2003. Grain size evolution and the rheology of ductile shearzones: from laboratory experiments to postseismic creep. Earth Planet. Sci. Lett.211 (1), 97–110.

Platt, J.P., 2015. Rheology of two-phase systems: a microphysical and observationalapproach. J. Struct. Geol. 77, 213–227.

Ricard, Y., Bercovici, D., 2009. A continuum theory of grain size evolution anddamage. J. Geophys. Res. 114, B01204. http://dx.doi.org/10.1029/2007JB005491.

Rozel, A., Ricard, Y., Bercovici, D., 2011. A thermodynamically self-consistentdamage equation for grain size evolution during dynamic recrystallization.Geophys. J. Int. 184 (2), 719–728.

Skemer, P., Hansen, L.N., 2016. Inferring upper-mantle flow from seismicanisotropy: an experimental perspective. Tectonophysics 668, 1–14.

Skemer, P., Warren, J.M., Kelemen, P.B., et al., 2010. Microstructural and rheologicalevolution of a mantle shear zone. J. Petrol. 51 (1–2), 43–53.

Tackley, P., 2000. Self-consistent generation of tectonic plates in time-dependent,three-dimensional mantle convection simulations 1. Pseudoplastic yielding.Geochem. Geophys. Geosyst. 1 (8), 1021.

Van der Wal, D., Chopra, P., Drury, M., Gerald, J.F., 1993. Relationships betweendynamically recrystallized grain size and deformation conditions inexperimentally deformed olivine rocks. Geophys. Res. Lett. 20 (14), 1479–1482.

Warren, J.M., Hirth, G., 2006. Grain size sensitive deformation mechanisms innaturally deformed peridotites. Earth Planet. Sci. Lett. 248 (1), 438–450.

White, S., Burrows, S., Carreras, J., Shaw, N., Humphreys, F., 1980. On mylonites inductile shear zones. J. Struct. Geol. 2 (1–2), 175–187.

http://refhub.elsevier.com/S0031-9201(17)30082-1/h0005












http://dx.doi.org/10.1029/2004JB003181

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