Formation of Lithospheric Shear Zones

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 on two-phase grain damage ⇑ Elvira Mulyukova a, , David Bercovici a a Yale University, Department of Geology and Geophysics, New Haven, CT, USA article info abstract

Article history: Shear localization in the lithosphere is a characteristic feature of plate tectonic boundaries, and is evident Received 17 February 2017 in the presence of small grain mylonites. Localization and mylonitization in the ductile portion of the Received in revised form 15 June 2017 lithosphere can arise when its polymineralic material deforms by a grain-size sensitive rheology in com- Accepted 24 July 2017 bination with Zener pinning, which can impede, or possibly even reverse, grain growth and thus pro- Available online 1 August 2017 motes a self-softening feedback mechanism. However, the efficacy of this mechanism is not ubiquitous and depends on lithospheric conditions such as temperature and stress. Therefore, we explore the con- Keywords: ditions under which self-weakening takes place, and, in particular, the effect of temperature and defor- Shear zones mation state (stress or strain-rate) on these conditions. In our model, the lithosphere-like polymineralic Grain damage mechanics Mylonites material 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 growth and the rate of grain size reduction by damage are in balance. Damage involves processes by which some of the deformational energy gets transferred into surface energy. This can happen by (i) dynamic recrystallization (grain damage) and (ii) stretching, deforming and stirring the material interface (interface damage). The influence of temperature enters through rheological laws (which govern the rate of work and damage), grain growth kinetics, and the damage partitioning fraction, which is the fraction of deformational work that goes into creating new surface energy. We demonstrate that a two-phase damage model, in which the partitioning fraction depends on both temperature and roughness of the interface between the phases, can successfully match the field data, including the reported correlation of grain size and temperature, the increasing dominance of dislocation creep at higher temperatures and a large range of grain sizes observed across the depth of a single shear zone. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction is, for example, geologically evident in the presence of small grain mylonites and ultramylonites (White et al., 1980). Such subdivi- The history of deformation of the Earth’s crust and lithosphere sion of the Earth’s surface into largely undeformed tectonic plates, is recorded in the microstructure of its constitutive rocks, which separated by narrow, strongly deformed plate boundaries, suggests deform in response to the differential stress imposed by forces in that its constitutive materials experience shear localization, as can the lithosphere and underlying convecting mantle. The heat and for example be caused by some self-weakening mechanism (Kaula, the mechanical work supplied by the mantle drives deformation 1980; Bercovici, 1993; Bercovici, 1995; Tackley, 2000; Bercovici, on a multitude of scales: from the movement of the crystalline 2003; Montési, 2013; Bercovici et al., 2015). To understand how defects on the subgrain scale, associated with the solid state diffu- Earth evolved to have such an exotic style of surface dynamics, sion and dislocation creep, to the formation of mylonitic shear we need to understand the mechanisms responsible for strain zones, and further to the planetary scale deformation of the Earth’s localization across the crust and the lithosphere, including their lithosphere, manifested as plate tectonics. brittle and ductile regions. Localization in the brittle portion of The relative motion of the plates is almost entirely accommo- the Earth’s outer shell is an expected feature at low crustal and dated by highly localized deformation at the plate boundaries, as lithospheric pressures and temperatures (Kohlstedt et al., 1995). However, the ductile portion of the lithosphere is where the peak

⇑ Corresponding author. lithospheric strength occurs and is thus a bottle-neck to localiza- E-mail addresses: [email protected] (E. Mulyukova), david.bercovici@ tion; in this region, deformation is also enigmatic since it involves yale.edu (D. Bercovici). http://dx.doi.org/10.1016/j.pepi.2017.07.011 0031-9201/Ó 2017 Elsevier B.V. All rights reserved. 196 E. Mulyukova, D. Bercovici / Physics of the Earth and Planetary Interiors 270 (2017) 195–212 combined brittle-ductile behavior as well as mylonitization (1998) and Linckens et al. (2015). The reported microstructure, (Kohlstedt et al., 1995; Karato, 2008). including grain size and crystallographic orientation, together with A crystalline rock deforming viscously can experience self- the independent temperature measurements, are used to constrain weakening and localization due to a number of processes, such the dominant creep mechanism. In combination with the olivine as shear heating, presence of multiple phases (including melt) flow laws and grain growth kinematics obtained from previous and grain size reduction (via phase transformation or dynamic experimental studies (Hirth and Kohlstedt, 2003), we are able to recrystallization), to name a few (e.g., Hobbs et al., 1990; Hobbs deduce the deformation conditions, such as stress or strain rate, et al., 1990; Montési and Zuber, 2002; Montési and Zuber, 2002; for a given tectonic environment. Some of the physical parameters Karato, 2008; Karato, 2008, pp. 288–301; Bercovici et al., 2015; that enter the model are less well constrained, such as the grain Bercovici et al., 2015, and references therein). The microstructure growth activation energy and the partitioning fraction of the that develops due to the action of a given strain-localizing mecha- mechanical work that goes into creating new grain boundaries. nism can reveal whether and to what degree it is active during Here, we provide further constraints, by comparing the grain size lithospheric deformation. Such interpretations require a thorough dependence on temperature predicted by our model with that quantitative understanding of these mechanisms at the conditions reported in the observational data. representative of the lithospheric crust and mantle. The physical Field observations demonstrate a correlation between grain conditions, characterized by temperature, composition and the size and temperature (with the grain size increasing by over magnitude of the mechanical forcing (e.g., stress), vary spatially two orders of magnitude across a few hundred degrees increase and through time, so that the dependence of shear localizing mech- in temperature), increasing dominance of dislocation creep at anisms and their associated microstructure on these properties higher temperatures, and a coexistence of different deformation must also be understood. regimes across small spatial scales (Jin et al., 1998; Linckens A common geometrical quantity used to characterize the et al., 2015). These observations are used as constraints in the microstructure of a crystalline material is its grain size distribu- model presented below. tion. It can be readily measured in laboratory experiments, as well as in the natural samples collected from the field. From the physical point of view, the size of the grains reflects the amount of inter- 2. A simple shear zone model nal energy that is stored as surface energy on the grain boundaries. In particular, when the material deforms by dislocation creep and A detailed mathematical description of the two phase grain undergoes dynamic recrystallization, by mobilization of a pre- damage model was presented previously (e.g.Bercovici and existing grain boundary or by the formation of subgrain bound- Ricard, 2012; Bercovici et al., 2015; Bercovici and Ricard, 2016). aries, its grain boundary area increases (Karato et al., 1980; Van We provide a brief overview of the equations here, to introduce der Wal et al., 1993). This reduces the grain size and thereby stores the governing physical processes and to highlight the main the deformational work in the form of increased grain boundary assumptions made along the way. area (Bercovici and Ricard, 2005; Austin and Evans, 2007; Ricard We consider the deformation of a polycrystalline material sub- and Bercovici, 2009; Rozel et al., 2011). In polyphase materials, ject to simple shear. The material consists of a non-dilute mixture the relative movement of grains of different phases can also lead of two immiscible fluids, or phases (e.g., a peridotite mixture with to distortion of the inter-phase boundaries, which further increases volume fractions of 40% and 60% pyroxene and olivine, respec- the surface energy (Bercovici and Ricard, 2012). tively). In the general formulation of the model, the phases have For shear localization to occur, there needs to be a mechanism different densities, viscosities and other properties, although in by which deformation can cause weakening, which would increase this study, for simplicity, the phases are assumed to have identical deformation, and which would further enhances weakening. This material properties. is, for example, the case when a material deforms by dislocation- The two-phase material is characterized by a mean grain size accommodated grain boundary sliding and undergoes dynamic within each phase, as well as the geometry (i.e., roughness or, recrystallization (Hirth and Kohlstedt, 1995; Warren and Hirth, specifically, the characteristic radius of curvature) of the interface 2006; Hansen et al., 2011; Hansen et al., 2013; Kohlstedt and that separates the two phases. The grain size and the interface Hansen, 2015; Skemer and Hansen, 2016). Another possible mech- roughness directly or indirectly influence the mechanical proper- anism, which is the focus of this study, is when a polyphase mate- ties of the material, but also evolve in response to the applied rial deforms by grain size sensitive diffusion creep and undergoes mechanical or kinematic forcing. This introduces a non-linear cou- grain size reduction due to interface-damage and pinning, which pling between the applied deformation and the effective rheology are induced by the presence of multiple phases (Bercovici and of the material. Ricard, 2012; Cross and Skemer, 2017). The grain size distribution within each phase is assumed to fol- There is abundant support from field (Jin et al., 1998; Montési low a lognormal distribution, characterized by a mean grain size. and Hirth, 2003; Warren and Hirth, 2006; Skemer et al., 2010; Moreover, the grain size evolution is assumed to be slaved to that Herwegh et al., 2011; Linckens et al., 2011; Linckens et al., 2015; of the interface roughness r, as the grain growth within each phase Hansen and Warren, 2015), experimental (Evans et al., 2001; is blocked by the presence of the other phase. This is known as the Hiraga et al., 2010; Farla et al., 2013; Platt, 2015; Cross and pinned state limit (Bercovici and Ricard, 2012), in which the grain Skemer, 2017) and theoretical studies (Bercovici and Ricard, size is proportional to the interface roughness. For the grain-size 2012; Bercovici and Ricard, 2016) that secondary phases play an distribution and phase volume fraction (e.g., 60% olivine, 40% important role in the mechanical properties of lithospheric materi- pyroxene) assumed in this study (following Bercovici and Ricard als. The goal of this study is to use this observational evidence to (2013, 2014, 2016, 2015)), the mean grain size is approximately test and further constrain the theoretical models of grain damage, p=2 times the interface roughness (the limitations of the pinned- with a particular focus on the effect of temperature, and effectively state assumption are discussed in Appendix A.). depth, on the grain-size dependent shear-localizing mechanisms The model has a two-dimensional plane geometry, with shear (see alsoLanduyt et al., 2008; Behn et al., 2009; Gueydan et al., acting parallel to the x-axis, and assumed to be infinite and uni- 2014). form in the x-direction. The temperature varies with depth, To constrain the models presented here, we use the data from increasing in the negative z-direction, while the applied stress or four different upper mantle shear zones, published in Jin et al. strain rate are constant and uniform in all directions (Fig. 1). E. Mulyukova, D. Bercovici / Physics of the Earth and Planetary Interiors 270 (2017) 195–212 197

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.)

In a natural shear zone, both stress and strain rate are likely to grain size by the interface roughness in the pinned state limit, is vary in space and time. Constant stress and strain-rate conditions included in the prefactor B0 entering the diffusion creep compliance approximate the end-member tectonic states for how driving B. forces or loads are supported by the lithosphere and/or mantle. The evolution of the interface roughness can be expressed as When the lithosphere is strong and provides the primary resis- (Bercovici and Ricard, 2012; Bercovici and Ricard, 2016): tance to plate-driving forces like slab-pull or effective ridge push, it is approximately deforming under constant stress conditions. dr gG f r2 ¼ I I W ð2Þ Once the lithosphere undergoes localization and weakens, the dt qrq1 cg resistance to the driving forces is taken up by the viscous drag from the mantle. This force balance dictates the plate velocity, which where thus imposes a constant mean strain-rate on a deformation zone R m of a given width. W ¼ 2e_s ¼ 2Asnþ1 1 þ F ð3Þ The spatial uniformity of stress or strain-rate is also not guaran- r teed for field or experimental conditions. However, for the basic and case of simple shear, stress is uniform. While it is difficult to 1=m impose a uniform strain rate in simple shear when viscosity is vari- B R ¼ ð4Þ able, the mean strain rate can be controlled. F Asn1 When deformation is driven by an applied stress, the variations is the field boundary roughness (or equivalent grain-size) at which in temperature and grain size with depth may induce vertical the strain-rates from diffusion and dislocation creep are equal. The strain rate variations and thus vertical shear e_ @v =@z. How- xz x compliances A and B are dependent on temperature, and thus the ever, our assumed width of the shear zone in the y-direction field boundary relationship between stress and interface roughness (e.g., 1kmin Linckens et al., 2015) is much smaller than the varies with temperature. width of the lithosphere in the z-direction (of order 100 km), hence The diffusion-driven process of grain growth (first term on the we assumed e_ e_ . xy xz right hand side of (2)) is counteracted by the grain size reduction In the two-dimensional simple shear set up, the only non-zero driven by deformational work (second term on the right hand side component of the symmetric strain rate tensor is @v of (2)). A more detailed mathematical and physical description of e_ ¼ e_ ¼ e_ ¼ 1 x, where v is the rate of displacement in x- xy yx 2 @y x these processes is presented in Bercovici and Ricard (2012). The direction. Similarly, there is only one non-zero component of the grain size reaches a steady state when the grain growth and grain stress-tensor, which we denote as s. The stress - strain rate rela- size reduction processes are in balance. tionship for our model with composite (diffusion and dislocation The general formulation for the fraction of deformational work creep) rheology is then given by: that goes towards distorting the interface in multiphase materials (f ) is: B I e_ ¼ Asn þ s ð1Þ rm b R ¼ m ð Þ f I f 5 where A and B are the dislocation and diffusion creep compliances I b þ b Rm r (assumed equal for both phases), respectively (see Table 1 for the list of all the parameters featured in our model, together with their where f I is given by: ranges of values), and r is the interface roughness (or effective grain f ¼ F expðC ðT=T Þkf Þ¼F f ðTÞð6Þ size). The scaling factor ðp=2Þm, which arises when we represent I 0 f ref 0 0 198 E. Mulyukova, D. Bercovici / Physics of the Earth and Planetary Interiors 270 (2017) 195–212

Table 1 Material and model properties.

Property Symbol Value/deﬁnition Dimension Rate of deformational work W 2se_ Pa s 1 Surface tension c 1Jm2 ¼ : ; ¼ : Phase volume fraction /i /1 0 4 /2 0 6 1 : Phase distribution function g 3/1/2 0 72 Reference temperature Tref 1000 K 2 Damage partitioning fraction Rb 6 f ¼ f m f I 1 I I b þ b Rm r Temperature-dependent prefactor ð ð = Þkf Þ f I F0 exp Cf T Tref ; P Experimentally constrained constants Cf kf Cf 0

Constant prefactor F0 0 < F0 < 1 Mixing transition roughness Rm RmRF m 6 < Mixing transition prefactor Rm 0 Rm 1 Mixing transition exponent b 4

3 _ n Dislocation creep edisl ¼ As

1 Activation energy Edisl 530 kJ mol 5 n 1 Prefactor A0 1:1 10 MPa s Stress-Exponent n 3 n 1 Compliance A Edisl MPa s A0exp RT 3 _ ¼ m Diffusion creep ediff BR s 1 Activation energy Ediff 375 kJ mol 9 l m 1 1 Prefactor B0 1:5 10 m MPa s Grain Size Exponent m 3 l m 1 1 Compliance B Ediff m MPa s B0exp RT

Grain growth4

1 Activation energy EG 200–400 kJ mol 4 l p 1 Prefactor G0 2 10 m s Exponent p 2 l p 1 Grain growth rate GG EG m s G0exp RT

Interface coarsening5 Exponent q 4 q qp G q 1 Interface coarsening rate GI ðl Þ G lm s p m 250

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 1 Grain-growth law from Karato (1989) for olivine, but varying EG instead of using EG ¼ 200 kJ mol . 5 Interface coarsening law from Bercovici and Ricard (2013).

The constant 0 < F0 < 1 is to be varied. The temperature- 2.1. Reference state ð Þ¼ ð ð = Þkf Þ dependence of f I is described by f 0 T exp Cf T Tref . For a ; To nondimensionalize the governing equations, we define a ref- temperature-independent f I Cf is set equal to zero. Otherwise, Cf erence state for the dependent variables e_; s and r. It is mathemat- and kf are experimentally constrained constants that are calculated self-consistently for given values of the activation energies (see ically convenient to use a temperature-dependent reference state Appendix B). (denoted by the subscript s) wherein time and grain-size length scales are derived from rheological properties. First, the strain- Eq. (5) includes a mixing transition roughness Rm that determi- rate, time and stress scales are related through the dislocation nes the dependence of f I on the interface roughness (Bercovici and Ricard, 2016). As proposed by Bercovici and Ricard (2016), parti- creep law: tioning of mechanical work into creating new interface surface _ ¼ 1 ¼ ð Þ n ð Þ es ts A T ss 8 energy is possibly more efficient when the interface has high curvature, compared to when it is smooth. This might occur because For systems with constant applied stress, ss is set to that stress, _ inter-grain mixing is more efficient for smaller grains (Bercovici and es and ts are defined by (8). For systems with a constant _ and Skemer, 2017), as is possibly evident in recent experimental applied strain rate, es is set to that strain rate, and ts and ss are like- studies (Cross and Skemer, 2017, Fig. 10). In the limit of wise set by (8). !1; The reference interface roughness is defined as: Rm f I becomes independent of the interface roughness. ¼ = Following Bercovici and Ricard (2016), we use b 4 and assume B 1 m < ¼ ð Þ Rm RF . To ensure that the latter is the case for the entire range of rs n1 9 Ass stress- and temperature-conditions, we further assume that Rm is parallel to RF , such that: The dimensionless field boundary roughness is given by:

ðn1Þ R ¼ R R ð7Þ ðn 1Þ m m m F ¼ RF ¼ ss m ¼ 1 ð Þ rF 0 10 P rs s s where Rm 0 is a constant to be varied. E. Mulyukova, D. Bercovici / Physics of the Earth and Planetary Interiors 270 (2017) 195–212 199

The dimensionless mixing transition roughness Rm (7) is also 2.3. The solution space for the steady state nondimensionalized using rs, and, using (10), is given by:

n1 The steady state interface roughness r is the solution to a fifth ð Þ 0 1 m ¼ ¼ !1 order polynomial (16), using q b 4. When Rm (i.e., f I is R ¼ R ¼ R rF ð11Þ m m s0 m independent of r in (12)), there is only one real solution (Fig. 2). The interface roughness, and thus also the grain size, decrease with The partitioning fraction (5) can be expressed in terms of the increasing qD=C, as expected. As qD=C increases and the grains nondimensional parameters: shrink, diffusion creep becomes increasingly dominant. Obtaining f a steady state solution in the dislocation creep regime requires f ¼ I ð12Þ I b very small values of qD=C (i.e., qD=C < 1). r0 þ 1 R r m F When Rm 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 Rm: dislocation creep at low qD=C, diffusion creep at high qD=C, and, 2.2. Dimensionless governing equations 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- _ ¼ _ _ 0; ¼ 0; ¼ 0 ¼ 0 Substituting e ese s sss t tst and r rsr into (1) and ble solution branch that is in diffusion creep (Fig. 2). The system (2), and dropping the primes, we get the nondimensional govern- will evolve to one of the two stable states depending on the initial ing equations: conditions, specifically on whether the initial roughness is on one s or the other side of the unstable solution branch. Thus, the steady e_ ¼sn þ ð13Þ rm state is history-dependent, or, as termed by Bercovici and Ricard dr C snþ1 þ s2rm (2016), exhibits a hysteresis effect. Due to the observed hysteresis ¼ 2 ð Þ Dr 14 < q 1 b effect, we will from now on refer to the model with Rm 1 as the dt qr r 1 þ RmrF hysteresis model. All of the stable solutions suggest a decrease in grain size for where increasing values of qD=C, as expected. gGI 2rsssf C ¼ ; D ¼ I ð15Þ 2.3.1. Approximate solutions Asnrq cg s s The occurrence of the hysteresis loop can also be predicted from are dimensionless parameters that determine the efficiencies of the asymptotic analysis of the model equations. To do this, we con- interface healing and damage, respectively. sider the limiting cases for r outside and inside the field and mixing For a system with constant stress we have s ¼ 1, and at the field boundaries, and find the approximate solutions to the system in _ _ boundary this gives rF ¼ 1 and e ¼ 2. For constant strain rate e ¼ 1, steady state. and using (13) with r ¼ rF from (10), this leads to the field- For this analysis, it is useful to rewrite (16) in the following = boundary stress s ¼ð1=2Þ1 n and thus a field boundary grain-size form: ¼ð = Þðn1Þ=nm ¼ : 1 m of rF 1 2 , which gives rF 1 1665 for our chosen values qD 1 þðr =rÞ 0 ¼ rqþ1snþ1 F ð18Þ of n and m. Thus, with our choice of rs; r measures the interface b C 1 þðr=RmÞ roughness as a fraction of the field boundary roughness RF , so that the temperature-dependence of the field boundary (through B=A) Here, we use the general definition for the mixing transition is removed from the dimensionless system. When r < rF, diffusion roughness (Bercovici and Ricard, 2016): creep is the dominant deformation mechanism, while when r > r , F R ¼ R sa=b ð19Þ dislocation creep is the dominant one. m m

At steady state, the interface roughness, and thus also the effec- where a ¼ bðn 1Þ=m ¼ 8=3 for Rm parallel to rF. This yields the fol- tive grain size, are solutions to: lowing approximate solutions (Bercovici and Ricard, 2016): b qD nþ1 qþ1 qD 2 qþ1m r For big grains r !1and (18) leads to: 0 ¼ s r þ s r 1 ð16Þ C C R rF m 1 qD = rqþ1bsnþ1a ¼ R b ð20Þ The ratio qD C governs the steady state and the temperature- C m dependence of the dimensionless interface roughness, and can be expressed in terms of stress or strain-rate scales only (i.e., by elim- For intermediate grains between the mixing transition and the inating rs): ﬁeld boundary roughness, we have Rm < r < rF, which is still m ðqþ1Þ in diffusion creep. In this case r < 1 and ðr =rÞ is large, but qD 2q f B m qþ1þmnðqþ1mÞ F ¼ I m ð Þ b þ s 17a 2 ðq 1 mÞ s ðr=RmÞ is also large, and (18) becomes C cg G m I A 1 or qD qþ1mb 2a ¼ b ð Þ r s Rm 21 ðqþ1Þ C qD 2q f B m qþ1þmnðqþ1mÞ ¼ I e_ nm ð17bÞ 2 ðqþ1þmÞ s C cg G nm I A For very small grains r ! 0, and (18) leads to: for constant stress or strain-rate, respectively, where we used (8) to 1 qþ1m 2 qD substitute for in the second relation. r s ¼ ð22Þ ss C With the choice of the reference state parameters introduced in Section 2.1, the dependence of the interface roughness on temperature is governed by the thermal variability of only two quantities: For the big and small grains, and using q ¼ b ¼ 4; m ¼ 3 and the ﬁeld boundary roughness RF , and the damage to healing ratio a ¼ 8=3; r decreases with qD=C. However, the intermediate qD=C. solution has r increasing with qD=C (since q þ 1 m b ¼2), 200 E. Mulyukova, D. Bercovici / Physics of the Earth and Planetary Interiors 270 (2017) 195–212

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 grey field-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 field boundary roughness rF. The grey dashed line indicates the mixing transition roughness Rm. Results for Rm ¼1and two finite values of Rm are indicated. Red, green and blue lines 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 figure legend, the reader is referred to the web version of this article.) delineating a branch that connects the other branches with an (20) by (21), we get that the two branches meet at ðn1Þ=m opposite slope. r ¼ s ¼ rF. Dividing (22) by (21), we get that the two Where the branches intersect can be found by dividing either ¼ a=b ¼ branches meet at r Rms Rm. the big or small grain branch by the intermediate one. Dividing E. Mulyukova, D. Bercovici / Physics of the Earth and Planetary Interiors 270 (2017) 195–212 201

For constant stress case, where s ¼ 1, the big and intermediate 3. Temperature-dependence of grain size = ¼ b grain branches intersect at qD C Rm , while the small and inter- = ¼ ðqþ1mÞ 3.1. Temperature-dependence of the damage to healing ratio mediate grain branches intersect at qD C Rm (Fig. 2). So the lower branch intersection occurs at a smaller grain size and a smaller qD=C than the upper intersection, giving rise to the hys- In the previous section we explored how the damage to healing = teresis loop. The intermediate branch disappears, together with ratio qD C controls the dimensionless interface roughness r, i.e., the roughness relative to the ﬁeld boundary. The damage to heal- the hysteresis loop, when Rm P RF , which is when the big and the small grain branches intersect (see Bercovici and Ricard, ing ratio varies with temperature, and this determines the 2016). temperature-dependence of the dimensionless roughness. = For constant strain rate case we have e_ ¼ 1. Thus, for large In general, qD C is given by 17a and 17b. Inserting for the rhe- _ ¼ n ological compliances A and B, as well as the interface coarsening grains, (13) implies e 1 s , and so s 1. For the small and intermediate grain branches, which are both in diffusion creep, rate GI (Table 1) and the damage partitioning fraction f I (6), the = (13) implies e_ ¼ 1 s=rm and so s rm. The big and intermediate dependence of qD C on stress, strain rate and temperature can be expressed as: grain branches intersect at qD=C ¼ R b ¼ R 4, while the small m m "# and intermediate grain branches intersect at qD=C ¼ qD DE T kf F s2=3 exp C ; DE ¼ð2=3ÞE þ E ð5=3ÞE bðqþ1þmÞ=ðbþamÞ ¼ 8=3 0 s f disl G diff Rm Rm (Fig. 2). As in constant stress case, the C RT Tref lower branch intersection occurs at a smaller grain size and a smal- or ler qD=C than the upper intersection, giving rise to the hysteresis "# qD DE T kf _ 2=9 ; D ¼ð = Þ þ ð = Þ v loop. F0es exp Cf E 8 9 Edisl EG 5 3 Ediff C RT Tref ð23Þ

Fig. 3. Temperature-dependence of the damage to healing ratio and the dimensionless interface roughness. Results for three different partitioning fractions are shown (see _ 13 1 legend). The colored line is the field boundary, colored according to temperature. Left column: constant stress ss ¼ 100 MPa. Right column: constant strain rate es ¼ 10 s . 1 1 1 All models employ Edisl ¼ 530 kJ mol , Ediff ¼ 375 kJ mol and EG ¼ 300 kJ mol . The resulting values for constants entering the partitioning fraction (6) are Cf ¼ 11:2 and kf ¼ 0:68 (see Appendix B). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 202 E. Mulyukova, D. Bercovici / Physics of the Earth and Planetary Interiors 270 (2017) 195–212 for constant stress and strain rate, respectively. Observational constraints favor the models with positive corre- According to the results in Section 2.3 (Fig. 2), r decreases with lation between temperature and grain size, as well as a transition increasing qD=C, and thus (23) shows that the dimensionless inter- from diffusion to dislocation creep at hotter temperatures. Models face roughness (or the effective grain size) decreases with increas- featuring a temperature-dependent f I comply most readily with ing stress or strain rate, which is in agreement with the these observations. experimental and field observations, as parameterized by the classical piezometers (e.g. Van der Wal et al., 1993) and pale- 3.1.2. Grain growth activation energy EG owattmeters (Austin and Evans, 2007). The grain growth activation energy EG, which enters DE (23),is

one of the parameters with the greatest uncertainty in our model 3.1.1. Temperature dependence of the damage partitioning fraction f I (Evans et al., 2001; Warren and Hirth, 2006). Moreover, the avail- The damage to healing ratio is proportional to f . A temperature I able experimental constraints that do exist for EG apply to the = sensitive f I ampliﬁes the temperature-dependence of qD C, which growth of grains, rather than the interface coarsening that is being yields a stronger variation of r with temperature compared to cases modeled here. However, for the lack of better constraints, we with constant f I (Fig. 3). Thus, models with temperature sensitive assume that EG is the same for grain growth and interface coarsen- f I are more likely to predict a grain size proﬁle that transitions ing. With these limitations in mind, we will vary DE through vary- from diffusion creep at colder temperatures to dislocation creep ing EG, in an attempt to constrain the latter. D at hotter ones. The larger the value of f I , the higher is the temper- According to (23), a positive E contributes to a decrease of ature at which the transition in creep mechanism occurs (Fig. 3). qD=C with increasing temperature. For a positive correlation

Fig. 4. Temperature-dependence of the damage to healing ratio and the dimensionless interface roughness. Results for three values of EG are presented (see legend). The _ 13 1 colored line is the field boundary, colored according to temperature. Left column: constant ss ¼ 100 MPa. Right column: constant es ¼ 10 s . All models employ a ¼ 8 ¼ 7 ¼ 1 ¼ ; temperature-dependent partitioning fraction f I 10 f 0, except for the dashed black line, which has f I 5 10 f 0 and EG 200 kJ mol . For EG 200 300 and 1 400 kJ mol , the constants entering the temperature-dependent partitioning fraction (6) are, respectively: Cf ¼ 1:91; 11:2; 22:0 and kf ¼ 2:94; 0:68; 0:099 (see Appendix B). 1 For the constant stress case with low EG (i.e., EG ¼ 200 kJ mol ), the model fails to produce a profile of interface roughness that increases and transitions from diffusion to dislocation creep with increasing temperature. (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) 195–212 203 between temperature and grain size, as suggested by the observa- to a decrease in grain size with increasing temperature, contradic- tions, qD=C needs to be anti-correlated with temperature (Sec- tory to the field observations. Recent experimental results suggest D = D tion 2.3, Fig. 2). The combined effect of E and f I on qD C are a larger value for EG (Evans et al., 2001), yielding a positive E, and such that the grain growth activation energy needs to be thus also a grain size that increases with temperature. 1 D EG P 250 kJ mol in order for qD=C to decrease with increasing For systems with a constant strain rate, E is positive even for temperature, and thus also for the interface roughness to increase low values of EG. These cases are thus not as sensitive to the value with increasing temperature (Fig. 4). of EG, and therefore less useful for constraining it. The grain growth activation energy used in Rozel et al. (2011), We limit our further analysis to models with a temperature- ¼ 1 D EG 200 kJ mol , renders E to be negative, which would lead dependent f I and a grain growth activation energy

Fig. 5. Temperature-dependence of the damage to healing ratio and the interface roughness (dimensionless in the middle row, dimensional in the bottom row). Left column: ¼ _ ¼ 12 1 ¼ 1 ¼ 1 ¼1 constant ss 50 MPa. Right column: constant es 10 s . Black lines: Rm 3 10 ; grey lines: Rm 1 10 and Rm , with the latter featuring no jump in grain size. ¼ 1 ¼ 6 Thin black lines indicate the unstable solution branch. All models employ EG 400 kJ mol and f I 10 f 0. The colored line is the ﬁeld boundary, colored according to ¼ 1 ¼ 1 temperature. The dashed black and grey lines are the mixing transition roughness Rm 3 10 and Rm 1 10 , respectively. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.) 204 E. Mulyukova, D. Bercovici / Physics of the Earth and Planetary Interiors 270 (2017) 195–212

1 EG P 250 kJ mol , such that DE > 0. Specifically, we use dependence of qD=C only affects variation in the dimensionless 1 EG ¼ 400 kJ mol , which, for our choice of the rheological compli- roughness (i.e., effectively normalized by RF ), which increases with 1 ances, is well within this constraint. increasing T so long as EG P 250 kJ mol . This means that dislocation creep becomes more dominant with increasing temperature 3.2. Hysteresis for sufficiently large EG. However, for our chosen values of Edisl and Ediff (Table 1), the field boundary grain size RF decreases with The expression for the damage to healing ratio is unaffected by increasing T for constant stress, but increases for constant strain inclusion of hysteresis. However, there are important differences in rate. Thus, the actual dimensional value of r can potentially the effect that qD=C has on the interface roughness r in the hystere- decrease with increasing T for constant stress, especially if RF is sis model (see Section 2.3). more temperature sensitive than qD=C. The hysteresis effect (see Section 2.3) introduces three distinct For our given temperature range and values of the activation deformation-related states into the model: there are two states energies, the dimensional field boundary roughness RF in the con- on either side of the hysteresis loop that are dominated by either stant stress case decreases by about one order of magnitude from diffusion or dislocation creep, and there is one state inside the hys- the lowest to the highest T. Meanwhile, the normalized roughness teresis loop where the two creep mechanisms coexist. A steady increases by over three orders of magnitude across the same T- state solution with small r in the diffusion creep regime exists at range due to the sufficiently large decrease in qD=C at ¼ 1 P 1 low temperatures, and one with large r in dislocation creep regime EG 400 kJ mol (which satisfies the required EG 250 kJ mol ). exists at high temperatures (Fig. 5). Both regimes can co-exist at As a net effect, the dimensional roughness increases with tempera- intermediate temperatures that span the hysteresis loop. Material ture, in agreement with the observational constraints (Fig. 5, bot- that is deforming at conditions that fall within the hysteresis loop tom row). is expected to feature pockets of small grains in diffusion creep neighboring pockets of large grains in dislocation creep. 4. Discussion: comparison to lithospheric shear zones The temperature at which the steady state solution crosses rF is controlled by Rm. A larger Rm allows diffusion creep (and thus also Our theoretical models make various testable predictions for mylonites and ultramylonites to form) to exist at higher tem- regarding the steady state grain size of a polymineralic material peratures, and pushing the transition to dislocation creep (and thus deformed by simple shear, and these can be compared with pub- also the occurrence of CPO structure) to occur at higher tempera- lished observational data. Specifically, we are interested in the tures (Fig. 5). effect of temperature, and by inference also depth (since pressure A striking feature of the hysteresis model is the abrupt change effects are less pronounced), on the steady state grain size and in interface roughness associated with intersecting the field the dominant deformation regime. The detailed field data from boundary, which involves crossing the hysteresis loop. We showed three different mantle shear zones reported in Linckens et al. = in Section 2.3 that the range of qD C in which the hysteresis loop (2015) enable us to constrain some of the parameters that enter exists depends on Rm. The larger Rm, the smaller is the jump in r the grain damage models presented here. associated with crossing the field boundary from one creep mech- For all of the three shear zones, the grain size appears to > anism to another. At sufficiently large Rm (i.e., Rm 1), the hystere- increase with increasing temperature, ranging from 7 to 60 lm sis loop vanishes (grey lines in Fig. 5). ultramylonites that form at temperatures 900–1030 K, to 10– 350 lm mylonites that form at 1000–1100 K, and to 35–3900 lm 3.3. Complete temperature-dependence of grain size in a shear zone tectonites that form at 1140 1370 K (Linckens et al., 2015). We showed in Section 3.3 that in order to obtain a positive correlation The variation of interface roughness with temperature T enters between grain size and temperature for stress driven models, the through the T-dependence of the damage to healing ratio qD=C 17a grain growth activation energy EG needs to be relatively large 1 1 and 17b and the field boundary roughness RF (4). The temperature- (EG P 250 kJ mol ). For example, EG ¼ 400 kJ mol satisfies this

Fig. 6. Temperature-dependence of grain size for the Oman ophiolite, according to the data from (Linckens et al., 2015) (black and grey vertical lines, with black indicating data-points with secondary phase volume fraction > 0:3) and the hysteresis model (solid black curves, where the thinner curves indicate the unstable solution branch). Modeled results for interface roughness were converted to grain size, according to the pinned state limit assumption (Section 2). Results for two different sets of values for ; ¼ 1 s F0 and Rm are shown (see titles), and using EG 400 kJ mol . The colored line is the field boundary, colored according to temperature. Rm is shown as dashed black curves. (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) 195–212 205 constraint, and we employ this value from now on. For the same tude across a 300–400 K increase in temperature. To reproduce reason, we employ the temperature-dependent f I 17a and 17b,as such a variation in grain size for the given temperature range with it is more robust in producing a grain size that increases with the paleowattmeter-type models (Austin and Evans, 2007; Rozel temperature. et al., 2011) or the simpler non-hysteretic two-phase grain damage The natural samples studied by Linckens et al. (2015) exhibit a model (Bercovici and Ricard, 2012) would require prescribing a large range of grain sizes, increasing by over two orders of magni- significant variation in stress. The hysteresis grain damage model,

Fig. 7. Temperature-dependence of grain size, according to the data from (Linckens et al., 2015) and the hysteresis model. Explanation for different line-types and colors are given in Fig. 6. For constant strain rate cases, results for two different Rm-values are shown, with the grey dashed curve indicating the smaller Rm-value. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.) 206 E. Mulyukova, D. Bercovici / Physics of the Earth and Planetary Interiors 270 (2017) 195–212 on the other hand, predicts a large range of grain sizes even for the Mylonites from the shear zone samples presented in Jin et al. constant stress models. Therefore, since we consider models where (1998) exhibit a coexistence of both diffusion and dislocation creep stress and strain rate are assumed to be constant and uniform, we regimes, where a grain size dependent rheology, typical of diffu- use the hysteresis model for our further analysis. sion creep, coexists with a CPO-microstructure, typical of disloca- The crystallographic orientation measurements in the natural tion creep. These observations indicate that mylonites are samples suggest that the largest grains (Lanzo-protomylonites deforming mainly by diffusion creep, but can be close to the ﬁeld and Oman-porphyroclasts) deform by dislocation creep, as they boundary such that some dislocation creep occurs within the have a strong crystal preferred orientation (CPO) (Linckens et al., grain-size distribution, as was shown in Rozel et al. (2011), and 2015). In the mylonitic samples, CPO is generally weaker, suggest- as is also assumed in our model. (An alternative model to explain ing that dislocation creep may be occurring, but is less dominant. this observation is deformation by dislocation-accommodated

Fig. 8. Temperature-dependence of the damage to healing ratio, strain rate (for constant stress model), stress (for constant strain rate model) and viscosity for the Oman ophiolite (third row in Fig. 7). For constant strain rate cases, results for two different Rm-values are shown, with the grey curve indicating the result with a smaller Rm. E. Mulyukova, D. Bercovici / Physics of the Earth and Planetary Interiors 270 (2017) 195–212 207 grain boundary sliding (disGBS), e.g., Warren and Hirth (2006, ?).) for a lithospheric shear zone setting, perhaps with the exception In the ultramylonites, no CPO is observed, suggesting that the of the somewhat low viscosity ( 1017 Pa s) obtained for the con- smallest grains are deeply in the diffusion creep regime. Thus, stant stress case at high temperatures. Prediction of such low vis- the field data suggest that as the grain size increases with increas- cosity values in the modeling results is due to the limitation of ing temperature, the dislocation creep becomes increasingly more applying a constant stress across the entire range of depths and dominant. temperatures. For the model to match the data, we need to constrain three At approximately 1000 K or deeper, the model predicts a sharp parameters: (a) the driving stress or strain rate, (b) the scaling fac- decrease in viscosity with decreasing temperature. This jump is tor F0 (see 6) for the damage partitioning fraction, and (c) the mix- associated with the transition from dislocation to diffusion creep, ing transition ratio Rm (see 7). and involves a viscosity-decrease of up to three orders of magni- The applied forcing, i.e., stress or strain rate, determines the tude for the constant stress cases, and up to one order of magni- field boundary grain size at different temperatures. The lower limit tude for the constant strain rate cases. Thus, the variation in on the applied stress or strain rate is constrained such that the lar- grain size and temperature, and in particular the transition in gest grains are on the dislocation creep side of the field boundary deformation regime, helps to weaken the lithosphere and offset (Fig. 6). The upper limit on the applied forcing comes from con- the increase in viscosity induced by the decrease in temperature. straining the ultramylonites to lie in the diffusion creep regime.

The value for F0 is chosen such that the modeled small grain solu- 5. Conclusions tion branch coincides with the ultramylonites. Finally, we choose We presented applications of the grain-damage model Rm such that the large grain solution branch matches the largest grain sizes observed in the field (Fig. 6). (Bercovici and Ricard, 2012) for deformation of a polymineralic material at different thermal conditions that are relevant for study- The value of Rm also determines the temperature at which the transition in deformation mechanism takes place. The deformation ing shear localization in the ductile portion of the lithosphere. Our transition lies within the hysteresis loop, which has the coexis- ultimate goal is to test and constrain the grain damage model with tence of different deformation regimes associated with it. Thus, if geological and experimental observations. We demonstrated that the hysteresis loop lies in the vicinity of the mylonitic samples, it the hysteresis model, in which the partitioning fraction (i.e., the may explain the observed coexistence of deformation regimes fraction of work going to create new grain surfaces) depends on within their grain sizes range. both temperature and interface, can successfully match the field data, including the reported correlation of grain size and tempera- Given these constraints, the values for F0 and Rm predicted by our model fitting are dependent on the prescribed driving stress ture, the increasing dominance of dislocation creep at higher tem- or strain rate conditions. As an example, we show how the field peratures and the large range of grain sizes observed across the data from the Oman ophiolite can be matched by the hysteresis depth of a single shear zone. model for two different stress values, which are within the range In agreement with the classical piezometers, our model predicts of the stress-values inferred by Linckens et al. (2015). The model a decrease of grain size with increasing stress. However, our model with a lower stress value (5 MPa) permits a good match for the does not require large variations in stress across the lithosphere in large grained tectonites, but does not match the ultramylonites order to match the observed variations of grain size. Moreover, our as well (Fig. 6). Moreover, it puts the mylonites relatively deep into results highlight the challenge of inferring the deformational stress diffusion creep regime, making it unlikely to observe a microstruc- or strain rate from the grain sizes observed in a mantle shear zone. ture indicative of dislocation creep within their range of grain To match the observations, the activation energy for grain P 1 sizes. The solution with a higher stress value (100 MPa) agrees well growth needs to be relatively large (EG 250 kJ mol ). Interest- both with the observed ultramylonites and the large grained tec- ingly, the grain growth activation energy predicted by the model tonites. Further, it puts the mylonites in the vicinity of the field is similar to that for diffusion creep. The latter is better constrained boundary, in support of the observed coexistence of deformation from published experimental studies, ranging between 260 and 1 ¼ 1 regimes. However, this model puts the mylonites just outside the 450 kJ mol (Hirth and Kohlstedt, 2003), with Ediff 375 kJ mol predicted steady state solution branches, inferring that they have being a commonly used value (e.g. Rozel et al., 2011). The similar- not reached the steady state yet, but are possibly still shrinking ity in values of EG and Ediff may be unsurprising, given that they (Fig. 6). The misfit between the mylonites and the modeled steady both involve atomic scale diffusion-limited processes. state solution can also be explained by the fact that the natural The previously poorly constrained (or unconstrained) values of shear zones are not necessarily at constant stress; a better match the mixing transition ratio Rm and the partitioning fraction f I have between the model and the data can be obtained by permitting been constrained by fitting our model to the field observations of small variations in stress (Bercovici and Ricard, 2016). lithospheric shear zones at three different tectonic settings. Our 6 5 We applied the same fitting procedure to the field data from the proposed range of values are F0 ¼ 2 10 5 10 and other two locations described by Linckens et al. (2015): the Othris ¼ 1 1 Rm 10 5 10 . However, these values are sensitive to the and Lanzo mantle shear zones. The reported range of grain sizes assumed value for EG and the driving stress or strain rate. The pre- observed in these shear zones is more narrow than in Oman, and dicted values for f I and Rm can be tested in future experimental thus serve as less stringent constraints for the model (Fig. 7). studies, as well as against other geological observations of shear Importantly, no abrupt or physically inconsistent changes in stress zones. or strain rate are needed to be prescribed in order to match the Finally, according to the field constraints used in this study, the reported range of grain sizes observed across each shear zone. hysteresis model predicts that the maximum temperature at which The predicted qD=C and viscosity profiles are expected to be mylonites can form, and thus at which self-weakening and shear similar for the three shear zones, since we used the same set of localization due to the effect of pinning in the ductile portion of parameters for all of them, except for a relatively small differences the lithosphere can occur, is 1350 K. This corresponds to a depth in the driving stress or strain rate. Therefore, we only show the that is well within the lithosphere: a few tens to up to one hundred results for these quantities for the Oman ophiolite (Fig. 8). kilometers, which is sufficient to explain the formation of weak As expected, qD=C and viscosity decrease with increasing tem- plate boundaries, localized deformation of which is responsible perature. The range of our predicted viscosity values is plausible for the occurrence of plate tectonics on Earth. 208 E. Mulyukova, D. Bercovici / Physics of the Earth and Planetary Interiors 270 (2017) 195–212

Acknowledgements Ai Bi ai ¼ ; bi ¼ A B This work was supported by NSF Grants EAR-1344538 and EAR- gGI 2rsssf I CI ¼ ; DI ¼ ð : Þ 1135382. The authors would like to thank the reviewers David n q A 4 Ass rs cg Kohlstedt and Laurent Montesi, who dedicated their considerable ð Þ Gi 2k3 aiRsss/if 0 1 f I expertise to reviewing this paper and made valuable contributions Ci ¼ ; Di ¼ n p 3k c to it. Ass Rs 2 ¼ where Rs rs and kn is the n-th moment of lognormal grain-size Appendix A. Two-phase grain damage model ¼ n2r2 ¼ : distribution, given by kn exp 2 (r 0 8 is the variance). See Section 2.1 and Table 1 for the definitions of the other parameters. We use the simplifying assumption of a pinned state limit for The Zener pinning factor, appearing in (A.3), is defined as: the system studied in the main body of this paper. In this limit, the grain growth rate in the two-phase material is dictated by 2 ¼ Ri ; ¼ 3 k4 ð Þð: Þ the coarsening rate of the interface, or, equivalently, the growth Zi 1 hi where hi 1 /i A 5 r 160 k2 rate of the pinning bodies. Bercovici and Ricard (2012) argue that a two-phase system evolves towards a pinned state limit with In the steady state, the interface roughness is given by: time. This lends support for using the steady state models together 1 1 C qþ1 1 qþ1 with the pinned state limit, as we do in the current study. ¼ I ð : Þ r W A 6 It is, however, important to test if the steady state solution of qDI the full two-phase system is consistent with the one obtained for The steady state grain size is given by: the pinned state limit. Thus, here we find the steady state solution 4 2 4 pD r þ r r for the fully coupled two-phase system. We do this for the ¼ 4 i nþ1 p 1 2 þ ð : Þ 0 Ri 2 si Ri 2 Ri 2 A 7 Ci hi most simple case: a stress driven model without the effect of hi hi hysteresis. For simplicity, we will from now on assume that the two phases have equal dislocation and diffusion creep compliances (A ¼ A ¼ A A.1. Dimensionless two phase grain damage model i and Bi ¼ B ¼ B, respectively), as well as equal grain growth rates ¼ The mathematical model of the grain size evolution in a two- Gi GG. The two-phase grain damage model features three nondi- = phase material was presented in Bercovici and Ricard (2012). The mensional damage to healing ratios: one for the interface (qDI CI) and two for the grains of each of the two phases (pD =C ). For a sys- dimensionless equations for rheology, interface roughness and i i grain size evolution are (respectively): tem driven by a constant applied stress ss, we get: ðqþ1Þ m qþ1þmnðqþ1mÞ qDI 2q f I B m si ¼ s ðA:8Þ e_ ¼a n þ b ðA:1Þ 2 ðqþ1mÞ s i isi i m CI cg GI A m Ri ðpþ1Þ pD 2k p / ð1 f Þf B m pþ1þmnðpþ1mÞ dr CI 2 i ¼ 3 i I 0 m ð : Þ ¼ W ð : Þ þ ss A 9 q1 DIr A 2 ðp 1 mÞ dt qr Ci 3k2 c GG A m dR C i ¼ i Z D R2snþ1Z 1 ðA:3Þ dt p1 i i i i i pRi A.2. Solution space for the steady state where subscript i (i ¼ 1; 2) enumerates the two phases and a bar To simplify the solution space analysis for the two phase grain over a quantity indicates volume average of that quantity. Further, damage model, we assume an equal volume fraction for both ¼ : we introduced the dimensionless parameters: phases (/i 0 5), reducing the two phase-specific steady state

Fig. A.9. Zero-levels for the growth rate of the grain size as a function of the damage to healing ratio in a two-phase grain damage model (black solid lines). White and grey field-colors indicate the sign of the grain size growth rate: grey is negative and white is positive. Black dashed line indicates the field boundary grain size. Green dashed line indicates the grain size above which the Zener pinning factor is negative. (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) 195–212 209 grain size expressions in (A.7) to just one equation. Similarly, the The solution branch in the stable dislocation creep regime pre- = = two damage to healing ratios pDi Ci (A.9) for the grains reduce to dicts an increase in grain size with increasing pDG CG. This is = ¼ = a single parameter pDi Ci pDG CG for both phases. counter-intuitive, since we expect that increasing the damage rate Two-phase grain damage model can have one, two or even to the grains would cause them to shrink. However, the steady three stable solutions for the steady state, depending on the values state grain size depends on the Zener pinning factor Zi (A.7), which = = of qDI CI and pDG CG (Fig. A.9). At high rates of interface damage becomes negative when the grain sizep isffiffiffiffiffiffiffiffiffiffi larger than the interface = (high qDI CI), there is only one stable solution, which is in diffusion roughness by a factor of more than 1=hi (A.5), at which point = creep. At low qDI CI, there are two stable solutions: one in diffu- the roles of the damage and the coarsening terms in (A.3) become sion and one in dislocation creep regimes. There exists a limited reversed. We do, indeed, see that the Zener pinning factor is nega- = range of intermediate qDI CI-values where three stable solutions tive in this part of the solution space (Fig. A.9). Specifically, Zi 0 are available (as long as pD =CG > 50): two in diffusion creep, = = G when the value of qDI CI is smaller than or is comparable to pDi Ci. and one in dislocation creep regimes. = = For Earth-like materials, pDi Ci qDI CI (Bercovici and Ricard, = For all of the stable solutions, an increase in qDI CI (which 2016), rendering the branches with Zi 0 unlikely to occur in implies a decrease in the interface roughness (A.6)) results in smal- nature. ler grain size. Thus, a decrease in the interface roughness drags the grain size down with it, which is intuitive. = = A.3. Temperature-dependence of the grain size in a shear zone For all of the tested values of qDI CI and pDG CG, there exists a stable diffusion creep solution that suggests a decrease in grain = = In the full two-phase grain damage model, the grain size and size for increasing qDI CI- and pDG CG-values. This solution can be readily interpreted: an increase in the damage rate of either the interface roughness parallel each other at cold temperatures = the interface or the grains leads to a smaller grain size. (Fig. A.10). This is the case when qDI CI is significantly larger than = pDi Ci, which happens to be at cold temperatures. At such condi-

= = Fig. A.10. Temperature-dependence of (clockwise from top left) the damage to healing ratio (pDi Ci for the grains, qDI CI for the interface, see legend), the Zener pinning factor, and the dimensionless and dimensional grain size R and interface roughness r (see legend) for the two-phase grain damage model. Dashed lines indicate the solution < ¼ 1 ¼ 8 branches with Zi 0. The model employs EG 400 kJ mol and f I 10 . 210 E. Mulyukova, D. Bercovici / Physics of the Earth and Planetary Interiors 270 (2017) 195–212

tions, the model conforms to the pinned state limit, with Zi close to experimental constraints on f I and F0, we can use the published zero. experimental data to constrain f 0, which is done below. Grain size = As qDI CI approaches and eventually becomes smaller than evolution for a single phase material is given by (Rozel et al., 2011): pD =C at higher temperatures (and the Zener pinning factor i i 2 W becomes negative), the second stable solution emerges for the dR ¼ GG k3 R f ð : Þ p1 B 1 grain size and the interface roughness, as was predicted in Sec- dt pR 3k2 c tion A.2. For this solution branch, there is a divergence in grain size See Table 1 for the deﬁnitions of the other parameters. and interface roughness with increasing temperature, contrary to We assume that the partitioning fraction f of deformational what is expected if the system is approaching a pinned state limit. work going into creating new grain boundary energy, is suppressed The steady state solutions for the interface roughness that cor- when the grains are deforming by diffusion creep (i.e., only the respond to the branches with Z 0 are consistent with the pinned i mechanical work associated with dislocation creep strain can be state solutions found in the main part of this paper. However, the used to form new grains; see Rozel et al. (2011)): additional solution branches that come forth when we solve the f full two-phase model, which has a much larger grain size and f ¼ 0 ðB:2Þ þ B 1 Zi 0, are dramatically different from the prediction of the pinned 1 Asn1 Rm state model. As was already mentioned in Section A.2, the branches where f 0 is the maximum partitioning fraction. The quantity f 0 is with Zi 0 are unlikely to occur in nature. inferred to be temperature-dependent, and can be estimated using piezometers from the published experimental studies, as outlined Appendix B. Temperature-dependence of the partitioning below. fraction The steady state grain size is given by: 2 nþ1 G ¼ 2k3 R f 0As ð : Þ The temperature-dependent part of the partitioning fraction for p1 B 3 pR 3k2 c interface damage (f I ) is assumed to be similar to that of the grain damage (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 their fitting to Eq. B.5 (colored lines). Bottom: Temperature-dependence of f for two : 0 : 1 1 ¼ ð ð = 3Þ2 9Þ different values of EG 200 kJ mol (black solid line) and 400 kJ mol (black dashed line). The relation f 0 exp 2 T 10 presented in (Rozel et al., 2011) (grey dashed 1 line) is shown for comparison, to demonstrate that it is only a good fit when EG ¼ 200 kJ mol . (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) 195–212 211

This equation can be rewritten in the form of the following thermal variation of Clab as well as that of the dislocation creep stress and grain size dependence (i.e., a piezometer): compliance A and the rate of grain growth G (B.8). We can express f in terms of its temperature-dependent 1 0 3k cG pþ1 (C ; A and G) and temperature-independent factors. Inserting R ¼ 2 sðnþ1Þ=ðpþ1Þ ðB:4Þ lab 2k3 pAf0 expressions for A and G, we write: The stress and grain size dependence can be directly measured 3k cG 1 E E f ¼ 2 0 exp disl G in the laboratory experiments, which typically report their results 0 ðpþ1Þ 2k3 A0 pC RT together with a ﬁtted piezometric curve of the form: lab ðB:9Þ ¼ ð ; ; ; ; ; Þ 1 dE f 0 F n p r c A0 G0 ðpþ1Þ exp ¼ klab ð : Þ RT R Clabs B 5 pClab where Clab and klab are determined by ﬁtting to the experimental where we introduced the temperature-independent parameter results. Comparing (B.4) and (B.5), the experimentally approxi- 3k2 cG0 Fðn; p; r; c; A0; G0Þ¼ . According to (B.5), the value of Clab also 2k3 A0 mated constants Clab and klab can be expressed in terms of the depends on n and p, through klab (B.7). This means that F and Clab parameters that enter our theoretical model: are not independent. Additionally, there is a constraint that

1 f 6 1. Therefore, we approximate the entire right hand side of pþ1 0 3k2 cG Clab ¼ ðB:6Þ (B.9) with a single temperature-dependent function of a form that 2k3 pAf0 6 guarantees f 0 1, rather than inverting for F and Clab separately. n þ 1 k ¼ ðB:7Þ Following (Rozel et al., 2011), we approximate the temperature- lab p þ 1 ¼ dependence of f 0 with an exponential function f 0 T kf Using the experimentally determined value of the constant C , exp Cf ð Þ . This function satisfies f 6 1, as long as Cf P 0. lab Tref 0 we can deduce that: The constants Cf and kf are determined by fitting this function to the values of f 0 deduced from the experimentally determined ¼ 1 3k2 cG ð : Þ f 0 pþ1 B 8 quantity Clab and other model-specific parameters in (B.8). Since C 2k3 pA lab Cf is empirically determined, there is still no guarantee it will be a positive number. Bearing this limitation in mind, we find a Note that the relationship between f 0 and Clab depends on the assumed values of the dislocation creep compliance A and the rate parameterization for f 0 for the range of the physical parameters 6 of grain growth G. that satisfy f 0 1. This range is well within what is suggested from Here we assume that the exponents n and p are fixed at n ¼ 3 the previously published experimental studies. nþ1 ¼ ¼ ¼ The parameterization presented in (Rozel et al., 2011)(f 0 and p 2. Thus, for consistency, we fix the exponent klab pþ1, 2:9 and then use the raw data from the published experimental results expð2ðT=103Þ Þ) is only valid for one specific choice of relations ð Þ ð Þ to redo the fitting of the piezometric curves to obtain Clab, similar for A T and G T . to what was done by Rozel et al. (2011). With klab fixed, the varia- To generate the results in the main body of this paper, we per- tion in Clab between different experiments is entirely attributed to formed the fitting of Cf and kf for various possible dE. However, to the variation in temperature (Fig. B.11). Thus, the temperature- facilitate the use of this model in future studies, we present here a dependence of the partitioning fraction f 0 is dependent on the parameterized model of f 0 as a general function of dE and F.

Fig. B.12. Temperature-dependence of the partitioning fraction f 0 for different values of dE (left column) and F (right column). The experimentally deduced f 0 for each set of T kf dE and F (colored circles (B.8)) are approximated with a relation f ¼ exp Cf ð Þ . Constants Cf and kf can be found by performing the fitting for each individual set of dE 0 Tref and F (solid lines). After performing the fitting of f 0 for a range of dE- and F-values, we parameterize Cf and kf as functions of dE and F (dashed lines, (B.10)). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 212 E. Mulyukova, D. Bercovici / Physics of the Earth and Planetary Interiors 270 (2017) 195–212

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