Hightemperature deformation and recrystallization - Wiley Online Library

JOURNAL OF GEOPHYSICAL RESEARCH: SOLID EARTH, VOL. 118, 943–967, doi:10.1002/jgrb.50125, 2013

High-temperature deformation and recrystallization: A variational analysis and its application to olivine aggregates Klaus Hackl,1 and Jörg Renner2 Received 1 March 2012; revised 5 February 2013; accepted 10 February 2013; published 29 March 2013.

[1] We develop a framework for a variational analysis of microstructural evolution during inelastic high-temperature deformation accommodated by dislocation mechanisms and diffusive mass transport. A polycrystalline aggregate is represented by a distribution function characterizing the state of individual grains by three variables, dislocation density, grain size, and elastic strain. The aggregate’s free energy comprises elastic energy and energies of lattice distortions due to dislocations and grain boundaries. The work performed by the external loading is consumed by changes in the number of defects and their migration leading to inelastic deformation. The variational approach minimizes the rate of change of free energy with the evolution of the state variables under constraints on the aggregate volume, on a relation between changes in plastic strain and dislocation density, and on the form of the dissipation functionals for defect processes. The constrained minimization results in four basic evolution equations, one each for the evolution in grain size and dislocation density and flow laws for dislocation and diffusion creep. Analytical steady state scaling relations between stress and dislocation density and grain size (piezometers) are derived for quasi-homogeneous materials characterized by a unique relation between grain size and dislocation density. Our model matches all currently available experimental observations regarding high-temperature deformation of olivine aggregates with plausible values for the involved micromechanical model parameters. The relation between strain rate and stress for olivine aggregates maintaining a steady state microstructure is distinctly nonlinear in stark contrast to the majority of geodynamical modeling relying on linear relations, i.e., Newtonian behavior. Citation: Hackl, K., and J. Renner (2013), High-temperature deformation and recrystallization: a variational analysis and its application to olivine aggregates, J. Geophys. Res. Solid Earth, 118, 943–967, doi:10.1002/jgrb.50125.

1. Introduction [2] Processes that alter the characteristics of the arrangement of grain boundaries in polycrystalline aggregates are generally termed recrystallization. Specifically, dynamic (or syntectonic) recrystallization accompanies inelastic deformation at temperatures exceeding roughly half the melting temperature. The work performed on the material by the external loading is partly consumed by (micro-)structural changes. The engineer’s interest in recrystallization stems from the relation between physical properties, particularly mechanical properties, and structure. For the geoscientist, understanding recrystallization bears the potential to lead to estimates for paleo-stress levels in the Earth’s lower crust 1 Mechanik - Materialtheorie, Ruhr-Universität Bochum, Bochum, Germany. 2 Experimentelle Geophysik, Ruhr-Universität Bochum, Bochum, Germany.

Corresponding author: J. Renner, Institute for Geology, Mineralogy, and Geophysics, Ruhr-Universität Bochum, D-44780 Bochum, Germany. ([email protected]) ©2013. American Geophysical Union. All Rights Reserved. 2169-9313/13/10.1029/jgrb.50125

and mantle, of foremost importance for constraining fundamental geodynamic parameters, such as the magnitude of driving forces for plate tectonics. [3] Our objective is to establish a consistent set of microstructural evolution equations for a plastically deforming material. For this end, we employ a variational principle [e.g., Ziegler, 1963; Ziegler and Wehrli, 1987; Hackl and Fischer, 2008] and gain relations between flow laws and microstructural scaling rules. Besides the treatment of the onset of dynamic recrystallization by irreversible thermodynamics [Poliak and Jonas, 1996], we are aware of only a limited number of recent studies devoted to the derivation of evolution equations for dynamically recrystallizing materials from basic thermodynamic principles [see Le and Kochmann, 2009; Ricard and Bercovici, 2009; Rozel et al., 2011]. In our approach, we investigate the rate of change of the free energy of a deforming aggregate accounting for dissipation not only associated with microstructural changes but also related to the movement of the various defects present. The statistical distribution of defects in the aggregate, the microstructure, is a result of the analysis rather than an input. [4] In the following, we first review some of the recrystallization phenomena that shall be addressed by the model, 943

HACKL AND RENNER: HIGH-TEMPERATURE RECRYSTALLIZATION OLIVINE

then outline the basics of the employed variational analysis, and explain our concept of a distribution function for states of grains. The major part of the present work subsequently deals with the specific application of the variational approach to dynamic recrystallization leading to general evolution equations for inelastic deformation accommodated by diffusion and dislocation processes in aggregates characterized by two microstructure elements, grain size, and dislocation density. Finally, we explicitly employ the general model for idealized, quasi-homogeneous materials and quantitatively apply the analytical results to aggregates dominantly composed of olivine, the major rock-forming mineral in earth’s upper mantle.

2. Background and General Approach 2.1. Plastic Flow and Recrystallization Phenomena [5] The kinetics of inelastic deformation are strongly sensitive to grain size. At low to intermediate temperatures, i.e., temperatures below about half the melting temperature, dislocations play an important role in this interrelation [e.g., Hirth, 1972; Lasalmonie and Strudel, 1986; Nabarro, 2000]. Stationary grain boundaries affect the spatial organization and thus the mobility of dislocations by acting as sinks, sources, or obstacles. In turn, grain boundary migration may be induced by gradients in dislocation density. Finally, new grains can nucleate at sites of very high dislocation density leading to recovery of work-hardened materials [e.g., Mitra and McLean, 1967; Glover and Sellars, 1973; Cahn, 1996; Covey-Crump, 1997; Nes et al., 2002]. At temperatures approaching the melting temperature, grain size constitutes the effective length scale of diffusion paths along which transport of the constituents is realized for deformation by mass transport. [6] The link between microstructural state and ratecontrolling deformation processes expresses itself in fundamental and phenomenological scaling relations between external mechanical state variables, such as macroscopic stress or strain rate, and internal state variables, such as average measures of dislocation density or grain size. First and foremost, the Orowan equation kPp k = p bvdis ,

(1)

links the plastic strain rate P p (please see the list at the end regarding our notation) with the density of dislocations, , their unit contribution to strain, expressed by the Burgers vector, b, and their mean velocity, vdis , controlled by external variables, such as stress, temperature, etc., but also internal variables, such as impurity content. The parameter p denotes a proportionality constant related to the fraction of mobile dislocations. [7] At temperatures exceeding about half the melting temperature, dislocation density adjusts to the acting external stress on time scales corresponding to only a few percent of strain in metals but also oxides including minerals [e.g., Takeuchi and Argon, 1976; Kohlstedt and Weather, 1980; de Bresser, 1996]. Physically, a balance is attained between externally applied load and interaction forces between dislocations. Initially proposed for gliding dislocations located on a single glide plane [Taylor, 1934], a power law relation between “steady state” flow stress, ss (some scalar

measure of the deviator of the stress tensor), and (average) dislocation density rp

/ ss

(2)

has been empirically confirmed from a multitude of deformation experiments on single crystals and polycrystalline aggregates [e.g., Bailey and Hirsch, 1960; Takeuchi and Argon, 1976]. Based on considerations of the long range interaction between the stress field of dislocations, Taylor [1934] derived rp = 2. Experiments tend to yield exponents smaller than 2 [e.g., Takeuchi and Argon, 1976; Poirier, 1985]. When used for polycrystalline aggregates, a power law such as (2) has been called a Bailey-Hirsch relation [e.g., Liu and Li, 1989; Takaki et al., 2007]. [8] At temperatures up to about half the melting temperature, the instantaneous yield strength y of materials often correlates inversely with grain size D as expressed by the so-called Hall-Petch relation: y = 0 + kD D–s ,

(3)

where 0 and kD denote material dependent intrinsic strength and proportionality factor. The exponent s is found to be close to 0.5 in many cases in accord with models for dislocation pileups at grain boundaries [e.g., Hirth, 1972]. When large strains are achieved, grain size is not a fixed parameter but evolves towards a steady state value, Drec , inversely related to “steady state” flow stress: –mp

Drec / ss

(4)

[e.g., Poirier and Guillopé, 1979; Kohlstedt and Weather, 1980; Etheridge and Wilkie, 1981; Tullis and Yund, 1985; Drury and Urai, 1990; Derby, 1991; Post and Tullis, 1999; de Bresser et al., 1998; Stipp and Tullis, 2003; Stipp et al., 2010] with an exponent mp close to 1 [e.g., Twiss, 1977]. [9] The so-called piezometric relations or piezometers, equations (2) and (4), have been assumed to exhibit universal scaling character when using the Burgers vector of the dominating dislocation glide system and the shear modulus for normalization [e.g., Derby, 1991]. Yet, recrystallized grain size significantly varies with subtle differences in composition that cause only minute variations in Burgers vector and shear modulus. For example, laboratory experiments yield contrasting piezometers for two nominally pure calcite varieties, Carrara marble [Pieri et al., 2001; Rutter, 1995] and Solnhofen limestone [Rybacki et al., 2003], that also differ significantly in their flow behavior at laboratory conditions. Similarly, the recrystallized grain size is affected by details of composition for Cu [Garcia et al., 2001]. Furthermore, the proportionality factor in equation (4) possibly varies with temperature beyond the dependence of Burgers vector and shear modulus on temperature [e.g., de Bresser et al., 1998; Orlova and Podstranska, 1998; Drury, 2005] also suggesting a relation between the specifics of the involved flow laws and the resulting piezometer. [10] A substantial number of previous approaches relying on a variety of experimental and analytical techniques have collected comprehensive observations on the kinetics of dynamic recrystallization [for a review see for example Humphrey and Hatherly, 1995]. These observations have led to the formulation of classical dislocation density based models [e.g., Kocks and Mecking, 1981; Gottstein and Argon, 1987; Goerdeler et al., 2004] and further attempts to

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derive plausible kinetic equations of state [e.g., Epstein and Maugin, 2002], both providing the basis for computer modeling with a range of calculation techniques [e.g., Song and Rettenmayr, 2007]. In several studies, the flow behavior of materials was modeled with a priori given grain structure [e.g., Kurzydlowski and Bucki, 1993; ter Heege et al., 2004; Berbenni et al., 2007] rather than modeling the evolution towards a recrystallized state. Furthermore, various lines of reasoning have been followed to justify the empirical relation (4) from micromechanical models considering for instance a balance between nucleation and growth of grains [e.g., Derby and Ashby, 1987; Shimizu, 1999], between diffusion and dislocation creep mechanisms, “the boundary hypothesis” [de Bresser et al., 1998], or between the applied and dissipated power, the “paleowattmeter” [Austin and Evans, 2007]; see also Rozel et al. [2011]. Only the second and third directly relate the microstructure evolution to the bulk mechanical behavior. 2.2. Basics of Variational Analysis [11] Open systems are the subject of irreversible thermodynamics. We adopt a variational framework [Hackl, 1997; Hackl and Fischer, 2008; Fischer et al., 2010] that considers a system on which mechanical work is performed by its environment. The mechanical power exerted on the system is consumed by (a) changing the energy state of the system using “storage containers” from which energy can—at least to some extent—be recovered and (b) driving dissipative processes that ultimately convert the incoming power into outgoing heat. Our approach rests on the basic assertion that the system “aims” at minimizing its change in energy at every time step with respect to the rates of change of the chosen state variables (the fluxes) requiring that the dissipation (outgoing heat) is maximal. Notably, the approach constitutes a global extremum principle. When several means of storing energy are accessible and several dissipative processes can be activated, the evolution generally does not minimize energy flux into every single “storage container” or maximize energy dissipation for every single process but the combined action of energy storage and dissipation is extremal. The global maximization is, however, identical to maximizing every single process, for example, when the operating dissipative processes are totally unrelated. Coupled processes lead to an intricate combination of storing and dissipating energy which potentially is far from obvious. [12] Let us specifically consider a system described by (sets of) external, i.e. controllable, state variables X and internal state variables x that might be scalars, vectors, tensors, or a combination thereof. The thermodynamic fluxes P and v = xP are given by the material time derivaV = X tives of the state variables. While the fluxes are physically well defined, the state variables are generally not because their initial values might be arbitrary, as is the case even for X for the application we have in mind (section 3.1) for which X = , the total strain. Let us therefore introduce a set of observable variables z which are physically well defined, an example being elastic strain. These observable variables will be a subset or a combination of the state variables, i.e., z = z(X, x). Note that all physical quantities occurring in the following are functions of z, V, and v, only. [13] Let us now introduce the specific Helmholtz free energy (z). Here “specific” refers to “energy per volume.”

The rate of change in specific free energy reads P = @ V – ı –Q V – ı, @X

(5)

where @ /@X = (@ /@z) (@z/@X). Elementary thermodynamics gives the specific dissipation ı as @ v q v 0, ı(z, v) = – @x

(6)

(“ ” denotes contraction over all indices present) with @ /@x = (@ /@z) (@z/@x), and q is the (set of ) thermodynamic forces. In general, dissipation can be considered to be due to various different processes associated with internal state variables xi , i = 1, : : : , k, i.e., x = {x1 , : : : , xk }. The specific dissipation is then given by k k X X @ ı= – vi qi vi , @xi i=1

(7)

i=1

where vi = xP i . We assume that for each process, some knowledge on the functional form of the dissipation as dependent on the fluxes vi is available by constitutive arguments: ıi = qi vi = ıOi (z, vi ).

(8)

Note that equation (8) implies that the different processes are to a certain extent uncoupled by assuming that the functional form of the dissipation for the ith variable may depend on all internal state variables but only on the ith flux. [14] The variational principle employed in this paper can now be formulated as follows: From all admissible fluxes v, the fluxes assumed by the system are those that minimize P (z, v) subjected to the relations given in (8).

Here minimization is taken with respect to fixed z, and admissibility refers to possible constraints by conservation laws and boundary conditions. Possible constraints can be taken into account by either directly selecting the admissible v from a suitably restricted set, or by using Lagrange multipliers. Because of equation (5), minimizing the rate of free energy is equivalent to maximizing dissipation; see Hackl and Fischer [2008] for details. [15] Introducing Lagrange multipliers i associated with the relations (8), an appropriate Lagrangian L is given by L= P +

k X i=1

k @ X i ıi – ıOi = V+ (i – 1) ıi – i ıOi (9) @X i=1

when using (5). Stationarity of L yields the evolution equations for vi by the following steps. First, the derivative @L/@vi = 0 gives qi =

i @ıOi . i – 1 @vi

(10)

Second, multiplication with vi and using (8) yields

945

ıOi =

i @ıOi vi . i – 1 @vi

(11)


Constraining the Lagrange multipliers i and upon their insertion into equation (10), we arrive at implicit evolution equations for the fluxes vi in terms of the observable variables z: qi =

ıOi

@ıOi vi @vi

@ıOi . @vi

(12)

2.3. Continuum Description of Polycrystalline Aggregates 2.3.1. Distribution Function [16] In extending the variational formulation introduced above to polycrystalline aggregates, we assume that the state variables have constant values within every grain of the aggregate; i.e., to every grain, a state z can be assigned. Let us further consider a control or representative volume V of the aggregate and introduce a distribution function f (z, t) of grain states; i.e., f (z, t) d gives the probability of finding a grain in a state lying in the subset d = dz of the entire space, . The control volume is chosen large enough to contain a sufficient number of grains for the statistical description inherent in f to be valid but small enough to ensure that inhomogeneity of microstructure is still captured by variations in the shape of distribution functions for different control volumes occupying physically different locations in the material. [17] The expectation value of a property g(z) is then given as Z hgi =

g(z) f (z) d.

(13)

Assuming spherical grains of diameter D, we specifically obtain the representative volume as V=

E Z D3 = D3 f (z) d 6 6

D

(14)

and the number N of grains within volume V as

is gained when it is appreciated that one is free in the choice of control volume. This continuity equation constitutes an evolution equation for the distribution function f (z) provided the fluxes w = zP and sources r are known. Thus, this equation is the desired relation between microstructural variables and parameters characterizing the kinematics and dynamics of the deformation state. The evolution equations constraining the fluxes w can be derived by our variational analysis as we will demonstrate in the following subsection. 2.3.2. Variational Analysis for Polycrystalline Aggregates [18] On the level of the control volume, free energy and dissipation are obtained from the specific quantities as D 3 E D , 6

(19)

D E D 3 E O i = D3 ıOi , D ıi as well as 6 6

(20)

‰=

and i =

respectively. We apply now the variational procedure, minP with respect to v under constraints i = Oi imizing ‰ for contributions to dissipation. As an additional constraint, we require inelastic deformation to proceed at constant volume. Here we make two assumptions regarding the relation between diffusive mass transport and the evolution of a control volume. First, it is assumed that diffusive transport remains local; i.e., mass is exchanged between neighboring grains, but diffusive fluxes on the length scale of the control volume do not occur. Second, the control volume is considered large enough for boundary effects to become negligible; i.e., the local diffusive mass transfer between grains across the boundary of a control volume has a zero net balance. These assumptions result in the control volume that is constant in time, i.e., VP = 0. [19] Introducing Lagrange multipliers i and V , the corresponding Lagrangian writes

Z N = h1i =

f (z) d.

(15)

P + L=‰

Z

fP(z) d =

Z

Z r(z) d –

f (z) zP dA ,

(16)

@

where r(z) denotes a source function and the surface integral accounts for grains entering or leaving the state space across its boundary, @. A boundary element of the state variable space is denoted dA with an outward directed normal. Equation (16) reflects that the number of grains may not be conserved during deformation but changes due to spontaneous appearance/disappearance of grains inside and at the boundary of the state space. Applying Gauß’ theorem, the boundary integral can be transformed into a volume integral Z

Z f (z) zP dA =

@

r ( f (z) zP ) d.

(17)

(21)

To explicitly perform the variational analysis, we first have P andV. P In section A, we show how to to further specify ‰ calculate the rate of an expectation value in general and explicitly find VP =

Z @D3 @D3 Vf d + vf d @X @x Z Z @z + D3 r d – D3 f V dA @X @ Z @z – D3 f v dA = 0. @x @

6

Z

(22)

Using the general equations (A4) and (A5) and employing w = (@z/@X)V+(@z/@x)v, the rate of free energy follows as

Here r denotes the Nabla operator in the state variable space given by r (@/@zi ) and “r” is chosen as a short-hand notation for “divergence” where as above, the “” product denotes contraction over all indices present. The local form Pf (z) + r ( f (z) zP ) = r(z)

O i + V V. P i i –

i=1

Balancing the number of grains for the control volume gives NP =

k X

(18) 946

Z @ D3 @ D3 Vf d + vf d @X @x Z Z @z D3 r d – D3 f + V dA @X @ Z @z 3 – D f v dA . @x @

P = ‰ 6

Z

(23)


In (23), we can obviously identify the dissipation as (e.g., see (5)) =– 6

! Z @ D3 @z 3 vf d – D f v dA @x @x @ (24)

Z

2.3.3. Evolution of the Distribution Function [21] With the evolution equations for the state variables as derived in the preceding subsection at hand, we can describe the evolution of the entire aggregate via the distribution function. Let us assume that the evolution equations (28) can be solved for zP to give

composed of the contributions by the respective internal state variables Z @ D3 i = – vi f d, 6 @xi

Introduction of (31) into the continuity equation (18) gives Pf (z) = –r (f(z)b(z)) + r(z).

Z @z @ = D3 f v dA. 6 @ @x

(26)

Using (23), (25), (26), and (22), we obtain for the Lagrangian 6

Z @ D3 (V + ) Vf d + D3 (V + )r d @X Z @z – D3 (V + )f V dA @X Z@ @z 3 – D (V + )f v dA @x @ " # 3 Z Z k X @ D (1 – i ) + vi f d – i D3 ıOi f d @xi i=1 ! Z @D3 +V vf d . (27) @x

Z

Stationarity of L now yields the desired evolution equations for the fluxes vi . Considering the volumetric integrals in (27), variation with respect to vi gives (1 – i )

@ D3 @ D3 @ıOi – i D3 + V =0 @xi @vi @xi

for

z 2 . (28)

Equation (28) constitutes an implicit evolution equation for vi . The boundary integrals in (27) on the other hand give D3 (V +

)f=0

for

(31)

(25)

and a boundary term

L=

zP = b(z).

z 2 @,

(29)

a condition for our minimization problem to be well posed. Usually, equation (29) will be satisfied by imposing f = 0 on that part of @ where D ¤ 0. [20] It remains to determine the Lagrange multipliers. Multiplication of equation (28) with /6 vi f and integration yields + + * * Oi @ D3 @ ı 3 O i – i (1 – i ) D vi + V vi = 0. (30) 6 @vi 6 @xi

Condition (30) together with equation (22) constitute k + 1 in general coupled equations for 1 , : : : , k , V . These equations are usually nonlocal due to the involved expectation values constituting integral terms over the entire control volume: the evolution of an individual grain is coupled to the evolution of all other grains composing the aggregate.

(32)

Equation (32) is an evolution equation of parabolic type for f (z) that can be solved for given source term r(z) by a, preferably implicit, time-integration scheme. Investigations using this kind of procedure will be given in future works.

3. Modeling Recrystallization [22] So far, we kept the description of our approach on a rather general level. In this section, we now specifically introduce (a) the state variables that we consider for the recrystallization problem, (b) the relevant contributions to free energy and dissipation, and (c) the constraints on dissipation forms to be accounted for. Then the specific variational analysis is performed resulting in four evolution equations whose characteristics are investigated in the light of general experimental observations and theoretical considerations. 3.1. State Variables and Associated Distribution Function [23] Individual grains of the polycrystalline aggregate are here characterized by three parameters: elastic strain e , dislocation density , and grain size D. Hence, we have z = {D, , e }. Variations in elastic strain and dislocation density within a grain are neglected, but a grain is characterized by the averages of these parameters. We assume the elastic strains of individual grains to be elements of a given distribution not subject to the variational formulation to be developed. Indeed, the total strain field will accommodate itself to yield a minimum of elastic energy for a specific microstructure of the aggregate and a given global strain. This accommodation occurs with the speed of sound and thus on a completely different timescale than the processes involved in inelastic deformation and structural changes. Vast literature is available on how to calculate the elastic energy of an aggregate [Hashin and Shtrikman, 1963; Suquet, 1997; Milton, 2001]. [24] The current work intends to demonstrate the potential of the thermodynamic approach rather than to attempt modeling the entirety of dynamic recrystallization phenomena. We disregard, for example, shape variations and orientation characteristics of grains. The latter would be essential for addressing dynamic texture evolution or deciphering the state of recrystallization from measurements of physical properties, e.g., elastic wave velocities [Karato, 2003]. Subgrain size is clearly also relevant in recrystallization processes. Yet, we consider subgrain boundaries a special case of grain boundaries that do not deserve separate treatment on the level of our approach because experimentally

947


constrained and theoretically motivated subgrain piezometers [e.g., Edward et al., 1982] are formally similar to the grain size piezometer (4). The formal similarity is likely an expression of the fact that the evolution of either boundary type is fundamentally related to the kinetics of dislocation generation and annihilation. Nothing of the following treatment precludes an extension of state space per se. However, we only consider grain size and dislocation density as internal state variables representing microstructure to avoid obscuring the principle of our approach and further specify the exact choice of variables in the following. [25] The total strain rate of a grain, P , comprises in a unique way elastic (subscript e) deformation and inelastic (subscript i) deformation, P = P e + P i . Furthermore, we distinguish two contributions to the inelastic strain rate of individual grains, P p due to crystal plasticity (subscript p) and P d due to transport of matter (diffusion; subscript d), such that P i = P p + P d . We assume these straining mechanisms to be volume conserving; i.e., the strain rates are trace-less (tr(Pp ) = tr(Pd ) = 0). While the rates can formally be integrated to yield strain measures as Z t k = k0 +

t0

P k dNt,

k = e, p, d, or no subscript

(33)

only e is physically well defined because it is uniquely related to stress via the elastic constitutive law. For the remaining strains, the initial value k0 is arbitrary. Nevertheless, for the sake of formal agreement with the general variational scheme, we will use equation (33) as formal definitions for the various strains. Note that this procedure does not restrict the generality of the following approach since only strain rates will appear in the variational formulation. [26] Similar to the approach for strain, we explicitly introduce the notion that the total change in dislocation density P = PP + PT results from two contributions, PP driven by deformation and PT thermally activated and unaffected by deformation. Furthermore, we assume a direct kinematic relation between plastic strain rate and changes in dislocation density, introducing the deformation-related specific dislocation production, PP = QP (D, , P p ) ,

the appearance of grains with states within the volume of the state space; i.e., they represent nucleation of grains with finite size and dislocation density. Since newly nucleated grains occupy part of the volume of an old grain, the reduction in size of the old grain represents a spontaneous disappearance of a state in the volume of the parameter space, i.e., r < 0, and may lead to a negative Pf for the particular class to which the old grain contributed before the nucleation event. Even for r = 0, disappearance of states may result from the shrinkage of grains to the extent that they ultimately reach the side of the boundary of the state variable space characterized by D = 0. Appearance of grains is in principle possible from the other part of the boundary of the state space corresponding to = 0. We assume however that f = 0 for = 0, i.e., that all grains have a non-vanishing dislocation density (that may however be arbitrarily small for individual grains), and D3 f ! 0 for D ! 0, i.e., that the distribution function does not grow too fast for small grain sizes. Then condition (29) is satisfied, and the boundary integrals encountered in section 2.3.2 vanish in our subsequent analysis. 3.2. Free Energy and Dissipation [29] We consider the specific free energy to separate as (D, , e ) = e (e ) + dis (D, ) + gb (D) ,

accounting for plausible dominating dependences on the chosen state variables of (a) stored elastic energy, e (e ), most strongly affected by elastic strain, (b) potentially grainsize dependent energy associated with the dislocations of given density, dis (D, ) (note that the energy associated with a dislocation depends on the size of the material volume by which it is enclosed [e.g., Hirth and Lothe, 1992], typically either related to the average distance between dislocations or to the grain size), and (c) energy due to grain boundaries, gb (D), often assumed to be determined by a (constant) interfacial energy, , i.e., gb (D) = 6 /D. [30] Inserting (36) into (23), the rate of free energy becomes

Z t t0

PT dNt

(34)

(35)

with the thermally activated processes affecting dislocation density. [27] In summary, we recognize the external state variables as X , and the internal state variables as x {D, T , p , d }, while the observable variables are given as z = {D, , e }. The dimension of the parameter space is thus 7 (two scalars and five components of a symmetric, volume conserving strain tensor). The parameter space has a volume element d = dD d de and a boundary given by @ = {D = 0 or = 0}. [28] We introduce the distribution function f (D, , e , t). In our approach, source function values r > 0 quantify

Z

Z @ P e f d + D3 r d @e ! Z Z @ D3 @ 3 P d + + Df D f P d @D @ Z Z = D3 P f d + D3 r d 6 Z – D3 dev P p + P d f d Z @ D3 ( dis + gb ) P d Df + @D Z @ + D3 dis PT + Qp f d , @

P = ‰ 6

that represents activation of dislocation sources by deviatoric stresses but also annihilation events when dislocations with opposite signs encounter each other on the same glide plane, for example. We formally associate a state variable T = T0 +

(36)

D3

(37)

where we made use of = @ e /@ = @ e /@e [e.g., Coleman and Noll, 1963] and introduced the deviatoric part of the stress tensor, dev = – tr( )I/3, since under the assumption of volume-conserving straining mechanisms inelastic strain rates are trace-less (tr(Pp ) = tr(Pd ) = 0) and driven by deviatoric stresses alone. [31] The dissipation as defined by (24) decomposes in a natural way into parts associated with the rates of the state

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P PT , P p , P d }, i.e., = D + + p + d , variables, xP = {D, given as D p d

Z P d , =– 3D2 D Df 6 Z @ =– D3 dis PT f d , 6 @ Z @ dis = D3 dev P p – Qp f d , 6 @ Z = D3 dev P d f d , 6

Od =

D @( dis + gb ) e + dis + gb + 3 @D

(38b) (38c) (38d)

(39)

is defined as a short notation for the energy terms contributing to the total driving force for grain boundary migration. In the most general case, Qp may be nonlinear in P p . In this respect, equation (38c) actually represents a slight generalization of definition (24). Note that our account for dissipative processes resembles that presented by Ricard and Bercovici [2009, equation 70] who however argued that the elastic component and the component due to dislocation energy are negligible in particular for processes on geological time scales. 3.3. Constitutive Assumptions on Dissipative Processes [32] In general thermodynamical terms, dissipation represents the production of irreversible entropy. Within the specific context of recrystallization, we consider the movement of lattice defects as the major source for dissipation. The generation of defects in contrast stores some of the deformational work as free energy. Three constitutive assumptions are now introduced regarding the dissipation associated with the migration of point and line defects. At this stage, we remain quite general regarding the kinetics of dislocation processes and refrain from choosing specific models. However, the agreement reached on the basic characteristics of the role of point defects in diffusive inelastic deformation leads us to use the well-established specific relations—elaborated on in Appendix B—from the outset. [33] (1) For an individual grain, diffusion-related dissipation is associated with two types of fluxes. A flux of constituents across (?) the grain boundary of width w alters the number of constituting atoms and thus changes the size of a grain. In contrast, fluxes along (k) a grain’s boundary or through its bulk volume—either by vacancy mechanisms through the lattice or along dislocation cores— conserves the number of constituents of the grain that however changes its shape. Dissipation associated with grain coarsening (subscript D) and change in shape controlled by various diffusion processes is explicitly calculated for spherical grains in Appendix B. Guided by that calculation, we formulate Z O D = w P 2 f d D2 D 4M?

Z

1 D5 kPd k2 f d . Meff

(41)

(38a)

where D

and

(40)

The dissipation for grain coarsening (40) corresponds to the one used by Fischer et al. [2003] when the mobility for transport of constituents across grain boundaries assumes the classical intrinsic grain boundary mobility for normal grain growth, i.e., M? /w ! mgb = vm D? /wRT (vm , D? , R, and T denote molar volume, diffusion coefficient for transport across the grain boundary, universal gas constant, and absolute temperature, respectively). Introduction of an P effective mobility Meff = c M in (41) is justified in i i i B2.3 and expresses the general relation between mobility and diffusion coefficient, M / D. For the action of the same driving force, the individual mobilities are additive as are the corresponding diffusion coefficients. The formulation of (40) and (41) is naturally valid for small shape changes of spherical grains under uniaxial tension or compression. Severe flattening by diffusion may require modifications. [34] (2) During plastic deformation, dissipation is first at all associated with the migration of dislocations. Thus, we intuitively formulate Z Op = Rp (D, , P p )D3 f d 6

(42)

introducing the specific dissipation due to dislocation motion, Rp . [35] (3) The thermally activated annihilation of dislocations in the absence of deformation contributes to so-called recovery or annealing driven by the reduction in free energy and causes a dissipation of the general form Z v2dis O = D3 f d , 6 Mdis (D, )

(43)

where vdis and Mdis (D, ) denote the corresponding dislocation velocity and (effective) mobility, respectively. The explicit dependence of the mobility on grain size and dislocation density reflects the average distance to infinite dislocation sinks, the grain boundaries, and the mean free path length for dislocations, respectively. Specific propagation mechanisms previously considered in micromechanical treatments are climb and cross slip among others [e.g., Nes, 1995; Liu and Evans, 1997] and may concurrently occur combining to an effective dislocation mobility (similar to the effective grain boundary mobility). Simple geometric reasoning requires that the characteristic time associated with a change in dislocation density is controlled by the average velocity and thus PT / vdis . Combining the proportionality factor with the mobility to a new function Bdis (D, ), we arrive at Z PT2 O = D3 f d 6 Bdis (D, )

(44)

for the constraint on the form of the dissipation due to thermally activated changes in dislocation density.

949


3.4. Variational Analysis [36] The appropriate Lagrangian for the recrystallization model reads P + D D – O D + d d – Od L =‰ O + p p – O p + V VP + T –

(45)

dislocation production rate, Qp , and the dissipation associated with the migration of dislocations, Rp . However, our theory does not depend on the specific choice in principle but of course, details of the theory’s results depend on it. 3.5.1. Grain Size Evolution [39] Upon insertion of the result for the Lagrangian multipliers, the grain size evolution equation (46a) reads

with multipliers D , d , , p , and V corresponding to the constraints on the forms of D (40), d (41), (44), p (42), and to the constant volume constraint (22), respecP tively. Variation of the Lagrangian (45) with respect to D, P d , PT , and P p immediately gives + (1 – D ) D P = M? V D , D w (d – 1) Meff P d = dev , 12d D2 – 1 @ dis , and P = Qp – Bdis 2 @ p @Rp 1 @ dis @Qp dev = + . p – 1 @Pp 1 – p @ @Pp

(46a) (46b) (46c) (46d)

Substitution of these field equations (46) back into the constraints associated with the Lagrangian multipliers (22), (40), (41), (42), and (44) yields D E D E D E 2 = 0, 2V D2 + 2V D2 D + 1 – 2D D2 D D E D E V D2 + (1 – D ) D2 D = 0 , d = –1 , = –1 , D E @Qp @ D3 dis P p – D3 Rp @ @Pp D p = . E @Rp D3 P p – D3 Rp @Pp

(47a) (47b) (47c) (47d)

(47e)

Solving the first and second equations of (47) for the Lagrangian multipliers D and V leads to 2D = 1

(48)

and ˝ ˛ 8 < ˝ 0 ˛ for D = 1 (D – 1) D2 D D2 D ˝ ˛ . V = = for D = –1 : –2 ˝ 2 ˛ D2 D

P = D

8 ˆ
p 2 = 3 ), i.e., flow stress – = 3/2 kdev k 1 3 p and axial strain rate P = 2/3 kPk. (The scalar description is for example used when reporting empirical piezometers (see Figure C1) and thus the coefficients of the empirical relations are not numerically identical to the coefficients of our model. The same is true for the coefficients of flow laws.) 4.2.3. Flow Laws [68] The flow behavior of the quasi-homogeneous material can be gained from the evolution equations. For example, we consider diffusion creep to be dominated by grain boundary diffusion (Meff ! 720wMk /D), i.e., d = hD6 ikPd k2 /720 wMk . Then the diffusion creep flow law (66a) and the dislocation creep flow law (70) become

955

P d = AQ *d D–3 M dev , and P p = AQ *p kdev knp –1 dev ,

(91a) (91b)

HACKL AND RENNER: HIGH-TEMPERATURE RECRYSTALLIZATION OLIVINE Table 1. Explicit Analytical Expressions for the Pre-factors of the Piezometers (87) and (88) and Flow Laws (91) Derived for a Quasi-Homogeneous Medium Deforming at Steady State 2 6 13mp –np cQp = 42 4

5 3

np –mp

2 dQ p =

4213mp –np 3np +mp

Q* = A d

2909907 wMk 65536

Q *p = A

4

w

3(np –mp ) 4

np +5mp

nuc 4

15+3m+3˛–2m˛ m(˛+1)

3

5+˛–m(4+˛) m(˛+1)

˛+5 m(˛+1)

"

5–m+˛

5 m(˛+1)

respectively. The stress exponent of the dislocation creep flow law relates to the model parameters by np =

˛+5 . m(1 + ˛)

(92)

Note that for the purpose of subsequent parameter identification, we already presented equations (87), (88), and (91) in the forms traditionally used by experimentalists and the involved relations for the pre-factors of the flow laws are listed in Table 1. The asterisk superscript illustrates the thermal activation inherent to these parameters, i.e., AQ *i / exp(–Hi /RT) with an associated activation enthalpy Hi and i = d, p. Since only two micromechanical exponents—˛ introduced in the nucleation law (83) and m introduced in the relation between dislocation velocity and Peach-Koehler force (58)—determine the three macroscopic exponents (rp , np , and mp ), the latter are related by rp =

2(np – mp ) 3

.

0

1np –mp 3 2 3np Q *p C B ap A 7 @ 1 3 A 5 4 g04 M?

3 1 np +3mp 4np np +5mp M? 5 np –mp nuc 1 5 p np –mp b4np wnp +3mp 2mp (1+2np ) a4m Q *p 4mp A p g0

(120 – 154m + 71m2 – 14m3 + m4 ) 2

2np (np –mp )–2np –mp 2

(93)

Thus, our approach provides analytical relations between experimentally constrained bulk parameters and our model parameters that are suitable for a micromechanical interpretation and allow for a comparison with independent experimental evidence on kinetics of defect processes. 4.2.4. Comparison to Previous Analytical Piezometric Relations [69] Analytical piezometric relations have been derived before relying on various approaches [e.g., de Bresser et al., 1998; Shimizu, 1999; Austin and Evans, 2007]. We refer the reader to de Bresser et al. [2001] for a quite comprehensive test of previously reported relations against experimental constraints. The recently presented paleo-wattmeter [Austin and Evans, 2007] specifically rests on the suggestion that grain-size evolution during deformation is determined by the rate of mechanical work. While similarities exist between this and our approach, Austin and Evans [2007] consider grain growth and grain size reducing mechanisms to be uncoupled and grain coarsening during diffusion creep to follow the same kinetics as during static annealing, i.e., driven by reduction in grain boundary energy. Our evolution law (50), in contrast, indicates that the change in size of a specific grain is controlled by the relation of its energy state to an average energy state in the control volume.

# 1 ˛ w3 ˛+1 kp ˛+5 a4p nuc b2 g0 M3? ˛+3

These energy states are affected by the state of crystal plastic deformation, too. Some experimental evidence indeed points towards “dynamic grain growth” even during diffusion creep [e.g., Wilkinson and Caceres, 1984; Kellermann Slotemaker et al., 2004; Rofman et al., 2009]. [70] The scaling relation found by Austin and Evans [2007] is of the general form of equation (4), however, it includes a temperature dependence of the proportionality factor determined by parameters of the normal grain growth law and the dislocation creep law plus the fraction of dislocation creep work invested in change of internal energy. The exponent of this scaling relation is related to the stress exponent. In fact, the stress exponent of the dislocation creep law is the common feature of almost all previous approaches as well as ours (see (93)). Derby and Ashby [1987] deduce mp = np /2; i.e., the stress exponent is actually the sole parameter. In other approaches, exponents related to grain size appear in addition, either from the grain size dependence of the diffusion creep law, P d / D–p [e.g., mp = (np – 1)/p de Bresser et al., 1998], or from the normal grain growth law, Dl / t (e.g., mp = np + 1/l + 1 Austin and Evans [2007]; Ricard and Bercovici [2009]). Neither of the two appears in our relation mp = np – 3rp /2 that instead constitutes a link between the two scaling relations for recrystallized grain size and dislocation density. Only the early work of Twiss [1977] also used a micromechanical parameter directly related to dislocations in addition to the stress exponent. [71] The grain growth exponent l cannot appear in our relation since we explicitly neglected grain boundary energy as driving force for growth in the simplified version (86) of (50). Above we refrained from explicitly tracking the grain size exponent p as a free parameter but only presented the example elaborated here of diffusion creep kinetically controlled by transport along grain boundaries (p = –3). Yet we tested for the role of p in the presented relations by also considering diffusion creep to be lattice diffusion-dominated (Meff ! 240Mvd /). We then find P d / D–2 M , i.e., NabarroHering creep with p = –2 [e.g., Evans and Kohlstedt, 1995; Balluffi et al., 2005], but the grain size exponent p does not enter any of the piezometric relations. [72] Austin and Evans [2007] and subsequently Ricard and Bercovici [2009] provide a physical interpretation for their scaling relations between recrystallized grain size and stress at steady state, leading to the term “paleo-wattmeter”

956


since average grain size scales with the rate of mechanical work done by dislocation creep. By using (70) in (88), it becomes obvious that the dissipation associated with dislocation movement dominates the recrystallized grain size also in our approach.

5. Application to High-Temperature Deformation of Olivine Aggregates [73] We proceed by evaluating our analytical results for a quasi-homogeneous material using previously reported experimental data for aggregates dominantly composed of olivine (see Appendix C), the major constituent of peridotite rocks in the upper mantle [e.g., Anderson, 2007]. After decades of research, flow laws appear fairly constrained and their extrapolation to nature matches independent geophysical observations [Hirth and Kohlstedt, 2003]. Grain growth in olivine aggregates was the subject of a number of studies [see review by Evans et al., 2001] but an unbiased determination of M? appears still lacking. Pores or melt pockets [Faul and Scott, 2006] seemingly affected boundary mobility in all previous experiments. In the absence of unequivocal constraints on normal grain growth in olivine aggregates, we therefore proceed assuming M? = Mk . With respect to piezometric relations, we represent the bulk of the available observations by empirical fits (see Appendix C). [74] The seven experimental parameters are fully constrained by only six model parameters (Table 2) because of relation (93) between the stress exponent and the exponents of the two piezometers. Therefore, we cannot independently fit rp to the theoretical value of 2 and respect the nominal experimental constraints on np and mp . In principle, the relation between the various exponents can be manipulated by altering relation (69) between the dislocation production function and plastic strain rate to Qp / kPp kqp resulting in rp = 2(np qp – mp )/3, i.e., a relation with an additional free parameter that provides more freedom in the choice of exponents. We refrain from adding this additional parameter. Though experimental observations are somewhat ambiguous regarding the exact exponent of the dislocation piezometer (Appendix C) and a tendency for values lower than 2 is observed with increasing stress, the theoretical value of 2 is however compatible with experimental observations when accounting for quoted experimental uncertainties (Figure 2). We investigate two scenarios. On the one hand, a set of exponents, np = 3.5, mp = 1.25, and rp = 1.5 (scenario 1), puts emphasis on respecting the exact value of the experimentally determined stress exponent np . On the other hand, a set of exponents, np = 4, mp = 1, and rp = 2 (sce-

Figure 2. Relation among the various exponents (stress exponents np , mp , and rp in the dislocation flow law (equation (91)), the recrystallized grain size piezometer (equations (4 and 88)), and the dislocation density piezometer (equations (2 and 87)), respectively) typically describing the bulk theological behavior on the basis of experiments. The rectangle indicates the range compatible with experimental uncertainties. Notably, our theory predicts the theoretical Bailey-Hirsch/Taylor value of rp = 2 for mp = 1 and np = 4, a stress exponent compatible with classical, micromechanically motivated dislocation creep laws [e.g., Weertmann, 1957; Poirier, 1985]. nario 2), enables us to match the theoretical exponent of the Bailey-Hirsch relation and is also “appealing” since all scaling relations then exhibit simple integer powers. 5.1. Derived Micromechanical Parameters [75] Values of the introduced micromechanical parameters (Table 3) that bring the model and the general characteristics of the experimental observations to a perfect match were determined using the symbolic formula manipulation software Mathematica. The micromechanical parameters finally permit a process-oriented discussion of the deformation characteristics as expressed by phenomenological bulk parameters. An interesting result is for example that the average dislocation velocity (58) depends nonlinearly on the Peach-Koehler force (vdis / F2PK ) for both scenarios of exponents. The further discussion of the derived parameters is structured such that first the characteristics of a life cycle

Table 2. Compilation of Material Constants (Assumed to be Typically Known to an Order of Magnitude at Least), Experimental Constraints, and Model Parameters. We Assumed M? = Mk M. The Total Number of Constraints and Model Parameters Differs Since the Theory Predicts a Relation Between the Various Exponents rp = 2(np + mp )/3 Material Constants Grain boundary energy, width Shear modulus Burgers vector

Experimental Constraints on Deformation Characteristics , w b

Diffusion creep flow law Dislocation creep flow law Dislocation piezometer Grain size piezometer Total: 7

957

A*d A*p , np cp , rp dp , mp

Parameters of Derived Micromechanical Model Diffusion kinetics Dislocation velocity Dislocation density evolution Nucleation Total: 6

M kp , m ap g0 , ˛

HACKL AND RENNER: HIGH-TEMPERATURE RECRYSTALLIZATION OLIVINE Table 3. Experimental Constraints and Calculated Micromechanical Model Parameters for Steady State Deformation of Olivine Aggregates in the “Quasi-Homogeneous Medium” Approximation. For the Derivation of Model Parameters, We Assumed M? = Mk M and H? = Hk HM Scenarioa 1 and 2

1 2 1 and 2

Experimental Constraints

Model Parameters

Diffusion creepb 1.0 10–15 Ad [m3 /Pa s] Hd [kJ/mol] 335

Diffusion kinetics M [m2 /Pa s] 2.2 10–8 HM [kJ/mol] 335

Dislocation creepb Ap [1/Pa3.5 s] 3.6 10–16 Ap [1/Pa4 s] 6.6 10–20 Hp [kJ/mol] 480

Dislocation velocity kp [m/N2 s] 5.0 10–8 kp [m/N2 s] 4.6 10–8 Hkp [kJ/mol] 240 m [-] 0.5

1

Dislocation piezometerc 2.5 10–3 cp [-]

2

cp [-]

1

Grain size piezometerc dp [-] 25

2

dp [-]

5.0 10–2

100

Z max tmax =

Dislocation density evolution ap [-] 9.6 10–7 Hap [kJ/mol] 145 ap [-] 1.3 10–6 Hap [kJ/mol] 145 Nucleation g0 [m4/3 /s] Hg0 [kJ/mol] ˛ [-] g0 [1/s] Hg0 [kJ/mol] ˛ [-]

scenario of exponents based on actual empirical values is to be favored over the alternative scenario of integer exponents in this context since it predicts the transition to nonphysical conditions with Dmin < DM way outside of any relevant stress states while the transition occurs just at the boundary of relevant laboratory conditions, i.e., stresses on the order of 1 MPa, for the integer-value scenario. Furthermore, the minimum grain size still assumes physically plausible, super-atomistic dimensions even for stresses approaching theoretical strength, i.e., the shear modulus. [78] The predicted lifetime of grains defined as

3.4 10–5 335 13/3 3.3 104 335 3

a

1: np = 3.5, mp = 1.25, rp = 1.5; 2: np = 4, mp = 1, rp = 2. Experimentally derived parameters of the flow characteristics of wet (i.e., for a hydroxyl concentration COH = 1000, see Billen and Hirth [2007]) olivine aggregates [Hirth and Kohlstedt, 2003]; Ap was adjusted to account for the change in stress exponent np between the scenarios. c See Figure C1. b

0

d/Qp

(95)

assumes reasonable values and new grains nucleate with rates that are in accord with typical recrystallization structures (Figure 4). For example, the predicted lifetime of an olivine grain reaches the age of the Earth at stresses of about 1 kPa. At laboratory stresses, the lifetime is comparable to the typical duration of experiments. [79] We are not aware of quantitative estimates for nucleation rates in deforming olivine aggregates but the actual figures determined here appear reasonable in the light of quenched microstructures. A typical laboratory sample has a volume of 10–6 m3 and it takes 104 s to deform it to a strain of 10% at a typical laboratory strain rate of 10–5 s–1 . A nucleation rate of 108 (1010 ) m–3 s–1 corresponds to 106 (108 ) nuclei or 1 (100) nuclei per every “old” grain with a typical grain volume of 10–12 m3 corresponding to a grain diameter of 100 μm. While nucleation rate increases

of individual grains are investigated. Then we analyze the partition of deformation mechanisms in terms of stored energy, dissipation, and strain rate. After investigating the origin of bulk thermal activation, we close by outlining future perspectives of our approach. 5.1.1. Critical Size, Nucleation Rate, and Lifetime [76] We successively analyze quantitative predictions of the model regarding the minimum grain size permitting growth, the nucleation rate, and the average life expectancy of grains to check for internal consistency of our model and plausibility of determined kinetics parameters. Most of the quantitative predictions are not critically sensitive to the choice of the set of exponents. A detailed uncertainty analysis of our model results would be quite involved and we take this insensitivity as evidence for the robustness of our approach. [77] The analytical expression of the ratio between minimum and average grain size reads nuc 1 Dmin =4 DM b cQ p dQ p

kdev k (5mp –2np )/3

(94)

highlighting the crucial role of interfacial energy, nuc , and demonstrating the ratio’s relation to the proportionality constants of the grain size as well as the dislocation piezometer. Our results respect the relation Dmin < DM at least for stresses as applied in the laboratory for both considered scenarios regarding the various exponents (Figure 3). The

Figure 3. Stress dependence of minimum grain size Dmin (solid lines) with a positive growth rate (see equations (82 and 84)) and mean recrystallized grain size DM (dashed lines; see equation (88)) derived from the explicit modeling for olivine aggregates in the “quasi-homogeneous medium” approximation documented in Table 3. The gray and black lines represent scenarios 1 and 2 considered for the various involved exponents, respectively (Table 3). The box indicates the range of stresses covered by laboratory experiments.

958


Figure 4. Stress (and temperature) dependence of (a) the lifetime of a grain, tmax , and (b) the nucleation rate g resulting from the explicit modeling for olivine aggregates in the “quasi-homogeneous medium” approximation (Table 3). The gray and black lines represent scenarios 1 and 2 considered for the various involved exponents, respectively (Table 3). The boxes indicate the range of stresses covered by laboratory experiments. with increasing stress, lifetime decreases. Also, the effect of temperature is opposite for nucleation rate and lifetime. Nucleation rate exhibits standard thermal activation, lifetime decreases with increasing temperature. 5.1.2. Contribution of Dislocation and Diffusion Creep to Bulk Creep Rate [80] Before actually presenting the energy and dissipation ratios modeled for olivine aggregates in the quasihomogeneous approximation, it seems mandated to recall that in the context of the current variational analysis of inelastic high-temperature deformation, it is the creation of the agents of dissipation (grain boundaries and dislocations) that raises the energy state. Thus, a balance is needed that creates agents at reasonable cost whose movements in turn gain the largest possible dissipation. In our model, the coupling between processes related to dislocations and grain boundaries is introduced in the continuity equation of the distribution function (18) and the constraints formulated for dissipation contributions (38) and manifests itself in the coupled evolution equations (46). The general grain growth law (50) documents, for example, the intricate interrelation between the fate of an individual grain and the average state of the aggregate. Nucleation annihilates dislocations at the cost of boundary energy in a deforming aggregate. Here, the nucleation rate directly enters the continuity equation of the distribution function (18) via the source term affecting the evolution of the entire aggregate and is particularly linked to the deformation characteristics via the minimum grain size for growth (82). [81] In our quasi-homogeneous approximation, the ratio of the mean energies stored in the system associated with the two microstructural elements, dislocations and grain boundaries, dis,M gb,M

=

b2 M b Q = cQ p dp 6 /DM 6

kdev k rp –mp ,

(96)

is essentially a product of the two piezometric relations (88) and (87). For all but the highest stresses, the ratio falls below unity, i.e., much more energy is stored in boundaries

than in dislocations (Figure 5a). However, this result cannot be examined in isolation. Firstly, realized deformation characteristics are not controlled by the absolute level of energy reached at steady state but the increase rate during evolution. Secondly, the ratio of dissipation contributions associated with movement of grain boundaries and dislocations has to be concurrently looked at because the movement of defects has to consume all of the incoming power at steady state (when grain size and dislocation density are constant). [82] According to our modeling, the ratio of the two mean dissipation contributions due to migration of dislocations and grain boundaries results to d,M kPd kM AQ *d 1–np kdev k 1–np +3mp = = * , p,M kPp kM AQ p b3 dQ 3p

(97)

Upon insertion of the numerical results, we find this ratio to critically depend on the chosen set of exponents (Figure 5b). For the set of integer exponents, the ratio is independent of stress and varies roughly two orders of magnitude around unity at temperatures representative of laboratory experiments. For the alternative scenario of exponents, dislocation creep constitutes the dominant dissipative mechanism. Since at steady state the ratio of strain rates associated with the two mechanisms is simply identical to the ratio of dissipation contributions (97), this result shows that inelastic deformation is dominantly accommodated by dislocation creep with the actual ratio modified by temperature, stress, and also the set of exponents to various extents. At very high stresses and/or low temperature, diffusion creep may become dominant. The partition of strain rates due to diffusion and dislocation mechanisms at steady state has been a matter of debate. Montesi and Hirth [2003] also note on the basis of the available empirical relations that in general the ratio of the two creep rates is stress dependent when an aggregate assumes steady state grain size. de Bresser et al. [1998] relied on the intuition-appealing assertion that the two concurrent mechanisms have to contribute about equally to bulk strain rate for steady state to be maintained,

959


Figure 5. Stress (and temperature) dependence of the ratio between (a) the energy stored in dislocations and grain boundaries and (b) the dissipation associated with the movement of dislocations and grain boundaries according to our explicit modeling for olivine aggregates in the “quasi-homogeneous medium” approximation (Table 3). The gray and black lines represent scenarios 1 and 2 considered for the various involved exponents, respectively (Table 3). The boxes indicate the range of stresses covered by laboratory experiments. the “boundary hypothesis.” Our approach suggests that this equality of contribution is not a universal feature of hightemperature deformation. 5.1.3. Thermal Activation [83] Thermal activation of bulk processes is often considered the “fingerprint” of a specific atomistic process. To what extent this fingerprint is blurred by the collective action of processes in aggregates can only be predicted on the basis of models. Thus, activation enthalpies provide an important tool for a deeper understanding of aggregate behavior when comparing predictions of micromechanical models of aggregates and independent experimental evidence on the kinetics of isolated processes. For diffusion creep, the signature of the atomistic process realizing deformation is directly reflected by the bulk flow behavior. The thermal activation of the creep rate is identical to the thermal activation of the rate-controlling diffusion process represented by the corresponding mobility, i.e., A*d = Ad exp(–Hd /RT) / Mk / exp(–Hk /RT) and thus Hd = Hk . This model prediction compares favorably with the similarity of experimental constraints on activation energies for grain boundary diffusion and for diffusion creep determined for olivine aggregates [see Hirth and Kohlstedt, 2003]. [84] For dislocation creep, relations between micromechanical and bulk parameters are more complex. To match the temperature dependence A*p = Ap exp(–Hp /RT) of the pre-factor in (91b) the involved micromechanical parameters also have to depend on temperature via Boltzmann terms. The following relation among their activation enthalpies (here denoted as H indexed by the corresponding parameter, i.e., index kp for kp , index ap for ap , etc. and H? denoting the activation enthalpy for the diffusive transport controlling M? ) has to be obeyed Hp =

2np + mp 2(np – mp ) np – mp np – mp Hkp + Hap – Hg0 – H? . 3mp 3np 3mp 2np (98)

Obviously, the bulk thermal activation enthalpy of dislocation creep does not reflect a single atomistic process.

The thermal activation terms quoted in Table 3 ensure the two proportionality factors of the piezometric relations (equations (88) and (87)) to be temperature independent, i.e., 2(np – mp ) 2(np – mp ) np – mp np – mp Hp + Hap – Hg0 – H? 3np 3np 6np 2np 0 , and mp mp np – mp np + mp Hdp = – Hp – Hap – Hg0 + H? 0 . np np 4np 4np (99) Hcp =

When however, these restrictions are not used or independent experimental evidence provides specific constraints on the thermal activation of any of the micromechanical parameters [e.g., Toriumi and Karato, 1978], then the two piezometric relations may become temperature dependent [see for example de Bresser et al., 1998; Austin and Evans, 2007]. If the total spread in experimental data resulted from temperature variations alone, apparent activation enthalpies of |Hcp | 100 kJ/mol and |Hdp | 136 kJ/mol would be possible. [85] The availability of only three easy slip systems for dislocation glide in olivine crystals requires an additional accommodation mechanism, such as dislocation climb or grain boundary sliding [Castelnau et al., 2008]. An advanced analysis of the thermal activation of dislocation creep can thus, for example, start by identifying the thermal activation of parameter kp of the dislocation velocity with the one of self-diffusion of Si in olivine [Dohmen et al., 2002; Costa and Chakraborty, 2008]. This diffusion process constitutes the likely rate-controlling atomic process for dislocation climb that in turn probably controls the time relevant for estimating the average dislocation velocity [e.g., Poirier, 1985]. Note, the activation enthalpy determined for kp under the constraint of temperature-independent piezometers (Table 3) is only about half of the experimental

960


value for the enthalpy of Si-self diffusion [Dohmen et al., 2002]. Alternatively, one could take the recently found coincidence of thermal activation of Si-lattice diffusion in olivine and thermal activation of creep [Dohmen et al., 2002] as the starting point for the search of analytical relations for the dislocation production rate Qp and the dissipation function Rp . Finally, it should be recalled that our quasi-homogeneous approximation neglected annihilation of dislocations, a further potentially thermally activated process. 5.2. Perspectives [86] Our approach provides a full tensorial description of the deformation characteristics of aggregates and is thus applicable to any deformation setup. Large strain deformation is of crucial importance for a number of questions relevant for processing materials [e.g., Segal, 2002] but also for understanding tectonics. Recent efforts in constraining the rheology of olivine aggregates to large strains [e.g., Bystricky et al., 2000; Zhang et al., 2000] led to improved understanding of texture development. Yet, in particular the quantitative evaluation of flow law parameters from torsion tests has proven a challenge. Derivation of the total torque measured for a solid cylinder by integration of the contributions of cylindrical shells potentially deforming by different mechanisms can be performed analytically or numerically employing the evolution equations derived here. A substantial understanding of the similarities and differences between results from different loading geometries will also help to clarify whether the appropriate flow laws should preferentially be formulated on the basis of the von Mises or the Tresca hypothesis. [87] An important area of research where our model can be employed concerns the role of localization in hightemperature deformation [e.g., Skemer and Karato, 2008]. The development of shear zones is crucial for the effective bulk strength of the lithosphere [e.g., Montesi and Zuber, 2002]. A number of studies have aimed to constrain the relation between localization and the evolution of microstructure as characterized by grain size at hightemperature/pressure conditions where brittle processes are ineffective [e.g., Drury, 2005; de Bresser et al., 2001; Kameyama et al., 1999; Montesi and Hirth, 2003; Braun et al., 1999; Rozel et al., 2011]. Our approach is well suited for addressing such transient phenomena. [88] Alternatively, one may use the results of the quasihomogeneous approximation to investigate situations in which deformation proceeds close to steady state, i.e., deformation conditions change so slowly that the material “instantaneously” assumes associated dynamic equilibrium states with Pf = 0. Then the internal variables can be eliminated from the mechanical problem by substituting the scaling relations (87) and (88) into the flow laws (91). The rheology of such a material is then uniquely determined by the external variables, strain rate, stress, and temperature. For the scenario 2 that ascribes exclusively integer values to the involved exponents, the total strain rate of an aggregate becomes proportional to the fourth power of deviatoric stress. For the other scenario, one finds a sum of terms proportional to a power of 3.5 (from dislocation creep) and 4.75 (from diffusion creep) in deviatoric stress. Considering that at steady state dislocation creep is the

dominant mechanism in the quasi-homogeneous approximation, it seems quite reasonable to simply use the dislocation creep law (that is notably independent of internal variables) in geodynamic modeling of “slow” processes in order to implicitly account for the effect of microstructure on rheology. While similar ideas have been presented previously [see for example Montesi and Zuber, 2002; Rozel et al., 2011] large scale geodynamic modeling almost exclusively employs linear viscous behavior. Clearly, the achievement of a steady state microstructure as envisioned here for the quasi-homogeneous model may be hindered in the mantle for example due to pinning of grain boundaries by secondary minerals. Yet, the investigation of flow in a mantle characterized by nonlinear viscosity according to our model appears a worthwhile exercise at least from a conceptual point of view since it provides a well defined reference scenario. [89] In application to mantle dynamics and observations on samples derived from Earth’s mantle, we have to keep in mind the polyphase character of peridotites that may lead to a substantial effect of Zener pinning on grain boundary mobility [e.g., Bercovici and Ricard, 2012] requiring modification of our approach. Indeed, in their study of mantle xenoliths, Chu and Korenaga [2012] found a two order of magnitude discrepancy between modeled stresses and stresses predicted by application of grain size piezometer to the olivine grains possibly related to a substantial suppression of grain growth by an unspecified mechanism for which pinning due to the presence of orthopyroxene is a candidate. Skemer and Karato [2008] deduced a significant difference in grain boundary mobility and activated deformation mechanisms between olivine and orthopyroxene from analysis of mantle xenoliths. It is rather straightforward to extend our model to polyphase aggregates, and future work will address this extension.

6. Summary and Conclusions [90] We presented a variational analysis of the rate of change in energy of a deforming polycrystalline material aiming at a basic but thermodynamically consistent description of its deformation and concurrent recrystallization at high temperature. Our formulation rests on a distribution function that characterizes the constituting grains by their size, their average dislocation density, and strains accommodated by dislocation mechanisms and diffusive mass transport. As a result, we derived a closed and internally consistent set of evolution equations that compares favorably with classic empirically motivated or theoretically formulated relations. The evolution equations rest on clearly defined micromechanical parameters that are suitable for a process-oriented interpretation of bulk behavior and that can be readily compared to independent results on defect kinetics. [91] To demonstrate the relevance of our approach, we investigated quasi-homogeneous materials deforming at microstructural steady state for which grains are characterized by a unique relation between their size and their dislocation density. The derived fully analytical model equations were related to bulk parameters usually determined in experiments and specifically used to model the high-temperature deformation of olivine aggregates. Notably, we presented

961


relations between dislocation density (recrystallized grain size) and flow stress at steady state, so-called piezometers, that are consistent with the rheological behavior as expressed by flow laws. In the case of high-temperature deformation by grain boundary and dislocation migration, the variational principle asserting that the system evolves by minimizing the rate of change in stored energy and maximizing the dissipation translates into a basic relation between the agents of deformation: Creation of boundaries, recrystallization, is costly but necessary to renew the potential for the operation of the effective dissipation agents, the dislocations. [92] The presented framework can be used with dissipation functions alternative to the ones considered here and tailored to specific applications. For example, future work could introduce more explicit coupling between dislocation processes and interfaces by accounting for (sub)grain boundaries in the dislocation production rate (55). Also, the content of second phases could be introduced as an internal variable to enclose pinning effects for grain boundary motion. Numerical treatment will likely be necessary in many cases, in particular for transient phenomena.

7. Notation [93] Vectors and tensors are represented by bold symbols. dev. deviator of second rank tensor, dev . = . – tr( . )I/3 k.k norm of second rank tensor h.i expectation value in state space generic dot product, summation over all indices present O indicates form used as constraint on dissipation for variational analysis Q indicates parameters of flow laws and scaling relations in full tensorial notation rather than in the special case of conventional triaxial compression

8. Nomenclature A, AQ pre-exponential factors in flow laws [Pa–n s–1 ] ap numerical constant for generation of dislocations (see (55)), [-] b Burgers vector, [m] bp numerical constant for annihilation of dislocations (see (55)), [-] Bdis mobility function for dislocations in (44) ci numerical constants in effective mobility cp , cQ p proportionality factor in dislocation piezometer (87), [-] dp , dQ p proportionality factor in grain size piezometer (88), [-] D grain size, [m] D diffusion coefficient, [m2 s–1 ] f (z) distribution function of states, [m] FPK Peach-Koehler force, [N m–1 ] g, g0 nucleation rate, pre-factor in (83), [m–3 s–1 ], [m˛–3 s–1 ] h amplitude of distribution function H enthalpy, [J mol–1 K–1 ] I unit matrix j volume flux, [m s–1 ]

kp material constant in relation between average dislocation velocity and Peach-Koehler force (58), [m2+2/m N–1/m s–1 ] L Lagrangian, [W] m exponent in relation between dislocation velocity and Peach-Koehler force (58), [-] mgb grain boundary mobility, [m s kg–1 ] mp exponent in recrystallized grain size piezometer (4), [-] M mobility, [(m/s)/(N/m3 ) = m3 s kg–1 ] N number of grains in a control volume, [-] n normal vector np stress exponent in dislocation creep law, [-] q thermodynamic forces associated with internal variables Q thermodynamic forces associated with external variables Qp dislocation production rate, [m–2 s–1 ] r(z) source function, [s–1 ] rp exponent in dislocation piezometer (2), [-] R universal gas constant, [J mol–1 K–1 ] Rp specific dissipation associated with the migration of dislocations, [W m–3 ] T absolute temperature, [K] v thermodynamic flux of internal state variables, v = xP v grain boundary velocity, [m s–1 ] vdis average dislocation velocity, [m s–1 ] vm molar volume V thermodynamic flux of external state variables, P V=X V control volume, [m3 ] w grain boundary width, [m] x vector of internal state variables, x {D, T , p , d } X vector of external state variables, X {} z vector of observable variables, z {D, , e } ˛ exponent in nucleation relation (83), [-] ı specific dissipation, [W m–3 ] D ı Dirac-(delta)function dissipation, [W] , P strain, strain rate, [s–1 ] interfacial energy, [J m–2 ] Lagrange multiplier, [m3 ]

shear modulus, [N m–2 ] r Nabla operator in state space specific Helmholtz free energy, [J m–3 ] ‰ Helmholtz free energy, [J] dislocation density, [m–2 ] , stress tensor, scalar measure of stress, [Pa] , @ phase space, boundary of phase space, [m–1 ] p fraction of mobile dislocations (Orowan equation (1))

9. Subscripts bd d dis D e eff

962

bulk boundary diffusion associated with diffusion processes associated with dislocations associated with grain boundary processes elastic effective


ld lattice diffusion n normal p (crystal) plastic, i.e., accommodated by dislocation processes pd pipe diffusion T thermally activated V associated with constant volume constraint y yield P driven by deformation associated with dislocation processes 0 initial or reference ? perpendicular to grain boundary k parallel to grain boundary

Appendix A: Rates of Expectation Values [94] Rates of expectation values are calculated according to

Z Z d d hgi = gf d = g fPd, dt dt

(A1)

spontaneous disappearance of a large grain and the appearance of a grain with an intermediate size. These linked spontaneous Z processes are required to preserve volume, i.e., I(r) D3 r d 0. 6

Appendix B: Dissipation Due to Diffusion [95] Dissipation due to defect activity can be generally quantified as the product of a driving force and a defect flux, i.e., ı = Fj. The flux of a defect, in turn, is determined by the driving force (for example, gradient in chemical potential) and the defect’s general ability to move, expressed by mobility M, often assumed to be independent of the specific driving force, such that j = MF [e.g., Balluffi et al., 2005]. Thus, dissipation due to defect movement reads ı = j2 /M. In case of the irreversible movement of atomistic constituents of the material under consideration, i.e., diffusion, the total dissipation is related to a volume flux j of constituents according to

because g(z) is a constant with respect to time in state space. Using the evolution equation for the distribution function (18), we obtain Z Z d hgi = gr d – g r (f w) d. dt

(A2)

For the second term, Gauß’ theorem yields Z

Z

Z

g r (f w) d =

rg wf d,

g f w dA – @

(A3)

and thus Z Z Z d hgi = gr d + rg wf d – g f w dA. dt @

(A4)

Employing this result (A4), the rate of change of free energy specifically reads P = ‰ 6

Z

r(D3

Z Z 3 ) wf d + D r d –

D3

@

(A5)

=

6

Z

r(D3 ) wf d +

Z

@D3

Z

Z

D3 r d –

Z

D3 f w dA

@

@D3

Vf d + vf d 6 @X @x Z Z Z @z @z + D3 r d – D3 f V dA – D3 f v dA = 0. @X @x @ @ (A6)

For the specific choice of variables presented in cection 3.1, the change in volume explicitly reduces to VP =

Z Z P d . D3 r d + D2 Df 6 2

(A7)

The first integral represents the balance of spontaneous appearances and disappearances of grains having a specific set of observable variables in contrast to slow evolution of grains having one set of observable variables to a neighboring set due to diffusive processes. For example, a nucleus spontaneously occurring in a large grain also requires the

(B1)

where M denotes the appropriate mobility of the involved atoms and Vflux the volume in which the flux takes place [e.g., Svoboda and Turek, 1991]. The following explicit calculations are based on spherical grains of diameter D. The treatment of more realistic grain shapes will obviously lead to modifications in the geometrical factors but will not affect the basics of the relations. B1. Grain Coarsening [96] The flux of atoms towards a grain across its boundP and here ary is related to its change in diameter by j = D/2 Vflux = wD2 , the grain’s “boundary layer” in which diffusion takes place, yielding for the dissipation of an individual grain O grain = w D2 D P2. D 4M?

f w dA .

The rate of change in volume generally calculates as VP =

O d = 1 j 2 Vflux , M

(B2)

Note that the superscript “grain” discriminates dissipation on grain scale from dissipation on scale of the control volume (no superscript). The subscript of the mobility M? emphasizes that transport occurs across a grain boundary. For an assemblage of grains, we arrive at O D = w 4M?

Z

P 2 f d . D2 D

(B3)

B2. Diffusion Creep [97] Changes in shape of individual grains accommodated by diffusive transport of matter are collectively termed diffusion creep. We consider a spherical grain under p uniaxial tension or compression with axial strain rate Pd = 2/3 kPd k (Figure B1). The velocity of grain boundary displacement reads

963

0

1 0 sin Pd x B Pd C B cos cos B C B v=B– 2 yC=B– @ P A @ cos 2sin – dz – 2 2

1

C D C C Pd . A 2

(B4)


determine the dissipation associated with deformation of an individual grain is now performed over the volume of the grain O grain = 2 vd Mvd

ZD/2 Z/2 2 r4 sin2 + 1 cos2 cos d dr Pvd 4 0

0

D5 kPgv k2 . = 240Mvd

(B10)

Finally, we arrive at the dissipation O vd =

Figure B1. Spherical grain under uniaxial tension or compression. Three different rate-controlling transport mechanisms are treated in the following: diffusion along the grain boundary (gb) and through the grain volume (vd) by either lattice diffusion (ld) or pipe diffusion (pd) along dislocations. B2.1. Diffusion Along the Grain Boundary [98] Using of the velocity vn = v the normal component n = Pbd D sin2 – 12 cos2 /2, where the normal vector to the surface is given by 0

1 sin n = @ cos cos A , cos sin

(B5)

the mass balance for a spherical segment with w D reads D2 cos d + wjbd ( + d )D cos( + d ) 2 – wjbd ( )D cos = 0 D d ,vn cos = –w (jbd ( ) cos ) 2 d

bd cos d =

0

Mk

for

i = bd, vd

(B12)

i

where cbd = 1440w/D and cvd = 240/ were derived O d has to be above (B8, B11). Now, the total dissipation maximized with respect to the individual contributing strain rates P under the constraint that the latter have to obey P d = i P i for i = bd, vd, i.e., all individual transport mechanisms add up to the total diffusive strain rate P d . From @ @i

X kPi k2 X + P i ci Mi i

i

(B7) Od =

D6 kPbd k2 . (B8) 720wMk

!

=2

P i + = 0, ci Mi

(B13)

Meff = (B9)

B2.2. Diffusion Through the Grain Volume [99] When transport is accommodated by diffusion through the grain volume, the flux field, and the velocity field coincide, i.e., j = v. The integration necessary to

Z

kP k2 5 P d D f d . i ci Mi

(B14)

240 w 3 Mk + Mvd . D

(B15)

P [101] The effective mobility Meff i ci Mi is dominated by the largest mobility Mi in perfect analogy to the commonly defined effective diffusion coefficients [e.g., Frost and Ashby, 1982]. From the explicit relations for spherical grains derived above, we infer

Z

D6 kPbd k2 f d . 720wMk

X Z kPi k2 D5 f d ci Mi

Od =

For an aggregate, we arrive at O bd =

(B11)

P we find P i = –ci Mi /2 and thus P d = – i P ci Mi /2 yielding / for the Lagrangian multiplier = –2P d i ci Mi . Finally, P inserting kPi k2 = c2i M2i kPd k2 /( Mj )2 into equation (B12), the extremal dissipation results to

upon integration over . The dissipation for an individual grain results from integration over the thin boundary layer to Z/2 j2

D5 kPvd k2 f d

B2.3. Total Dissipation due to Conservative Diffusion Processes [100] The total dissipation results as the sum of the dissipations associated with the various involved transport mechanisms that are here considered to be independent of each other:

(B6)

yielding the local flux

O grain = 2D2 w bd

Z

for an aggregate. Note, the mobility is conceived to be proportional to an effective volume diffusion coefficient, i.e., Mvd / D = Dld + 2 Dpd , that accounts for diffusion through the lattice (ld) as well as diffusion along dislocation cores of width , so-called pipe diffusion (pd) [e.g., Frost and Ashby, 1982].

vn

P D2 jbd = bd sin cos 8w

240Mvd

Appendix C: Experimental Constraints on Flow and Recrystallization of Olivine Aggregates [102] The experimental constraints on the parameters involved in (88) to (91) and employed for our quantitative modeling are listed in Table 3. Note that we aim

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Figure C1. Experimental data for (a) recrystallized grain size and (b) dislocation density as a function of flow stress for various olivine aggregates. For normalization, we used b 5 10–10 m [Deer et al., 1992; Durinck et al., 2007] and 50 GPa [Jackson et al., 2002]. The dashed lines indicate the general trend of the bulk data sets employed for the explicit modeling represented in Table 3. polycrystalline aggregates: ZG79 = Zeuch and Green [1979], vdWea93 = Van der Wal et al. [1993] (Ah: Aheim dunite, A b: Anita bay dunite), Zea00 = Zhang et al. [2000], dK01 = de Kloe [2001]; single crystals: PGC72 Phakey et al. [1972], KG74 = Kohlstedt and Goetze [1974], DGB77 = Durham et al. [1977], Kea80 = Karato et al. [1980], BK92 = Bai and Kohlstedt [1992], JK01 = Jung and Karato [2001]. at demonstrating the general plausibility of our approach and rather avoid here indulging in a lengthy discussion of specific problems associated with deriving piezometric relations (regarding stereological issues, representation of grain size distribution functions by individual parameters, quantification of flow stress from experiments with various loading geometries, etc.) and of the peculiarities of the rheological characteristics of olivine (associated with hydrogen content, oxygen fugacity, and hydrostatic pressure, for example; readers are referred to the seminal work of Korenaga and Karato [2008] regarding the derivation of a robust set of best fit parameters for flow laws from currently available data sets). Nevertheless, two general comments on the various experimental constraints seem warranted to prevent over-interpretation of our results. (1) The piezometer relations and flow laws are derived in independent studies. Thus, details in sample composition and experimental procedure may differ and actually lead to combining slightly disparate results. For example, hardly any information is available on the hydrogen content of samples used to derive the dislocation piezometer. (2) The flow law parameters result from small strain experiments (< 10%) but are here used in model equations for steady state deformation. Such strains are likely sufficient to adjust the dislocation density but certainly not the grain size distribution. [103] We represent the bulk of the available observations with respect to piezometric relations by empirical fits (Figure C1), Dr /b = 100(/ )–1 and b2 = 0.0025(/ )1.5 for scenario 1 of exponents and Dr /b = 25(/ )–1.25 and b2 = 0.05(/ )2 for scenario 2, the integer exponents scenario. Note that we used 50 GPa [Jackson et al., 2002] and b 0.5 nm [Deer et al., 1992; Durinck et al., 2007]. Our trendlines represent the data for recrystallized grain size Dr and average dislocation density as well as previous

fits to subsets of data and pre-factors and exponents are comparable to previously reported values [e.g., Twiss, 1977; Kohlstedt and Weather, 1980; Jung and Karato, 2001]. As further quantitative estimates for material parameters we employed nuc = /1000 with 1 J/m2 [Cooper and Kohlstedt, 1982; Duyster and Stöckhert, 2001] and w 1 nm [White and White, 1981]. [104] Acknowledgments. Generous funding by the German Science Foundation (SFB 526) is appreciated. We are extremely thankful to Laurent Montesi and Yanick Ricard for their constructive and meticulous reviews.

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