Int J Earth Sciences (Geol Rundsch) (2001) 90 : 157±167 DOI 10.1007/s005310000158

O R I G I N AL P AP E R

M.S. Paterson

Relating experimental and geological rheology

Received: 20 July 1999 / Accepted: 9 October 2000 / Published online: 9 March 2001  Springer-Verlag 2001

Abstract This paper is concerned with the question of how to relate laboratory measurements of the rheology of rocks to the rheological assumptions that need to be made in geodynamic modelling. First, there is a brief resumØ of the principal types of rheological behaviour that have been studied in the laboratory, both pressure-dependent, strain rate-independent and pressure-independent, strain-rate-dependent. Then, the generalization of the results from the relatively simple stress states of the experiments to general stress states is discussed, followed by consideration of the extrapolation of the experimental results to geological strains and strain rates. Finally, the problems associated with spatial scale are considered, leading to the question of how to model the rheological behaviour of large-scale rock masses, using the rheological measurements of laboratory specimens and taking into account the heterogeneity of geological-scale rock masses. Keywords Tectonic modelling ´ Extrapolation ´ Stress ´ Rheology ´ Scale

Introduction The geodynamic modelling of deformation in the Earth requires definition of both the boundary conditions for the deformation and the mechanical properties of the rock masses involved. This paper is concerned with the latter aspect, that is, with the material or ªconstitutiveº properties that come into play in the geological deformation. M.S. Paterson ()) Research School of Earth Sciences, Australian National University, Canberra, Australia E-mail: [email protected] Phone: +61-62-62492497 Fax: +61-2-62490738

The choice of rheological laws and parameters for the modelling of deformation in the Earth has been strongly influenced by rheological measurements in the laboratory. However, the timescale of the experiments rarely exceeds a few tens of hours, and the specimens used in the laboratory are, at most, tens of millimetres in size. Therefore, an extrapolation of six to nine orders of magnitude is commonly required to relate the laboratory measurements to the geodynamic situation in both time and space. Also, the amount of strain reached in the laboratory is often less than that in geological deformations, by an order of magnitude or so. Geodynamic modelling encompasses a range of spatial scales, including the following: P1Mantle-scale processes such as mantle convection, plumes, sea-floor spreading P1Continent-scale processes such as subduction, diapiric uplift, rifting P1Formation-scale processes such as folding, faulting, mylonitization In each case, the models commonly treat the rock mass as being initially a homogeneous continuum at that scale. That is, it is assumed that the rock mass can be notionally subdivided as finely as is relevant to the resolution of the deformation pattern being modelled, without affecting the applicable rheological properties. However, if this subdivision is carried further, that is, the rock mass is observed at greater ªmagnificationº, heterogeneity will generally become evident at some scale that is still much larger than that of the laboratory specimen. The problem therefore arises of relating the behaviour of this complex body to the material properties of laboratory-scale specimens, that is, of modelling the behaviour of the continuum from known properties of the component materials of the rock mass. This problem is discussed in the latter part of the paper. The properties of the component materials of the rock mass are obtainable by extrapolation from the

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laboratory to geological conditions. These conditions will involve the nature and magnitude of stress and strain rate, the environmental conditions such as pressure, temperature, pore fluid and chemical factors, and any evolution in behaviour or structure of the rock during deformation. The problems of such extrapolation have been the subject of a previous paper (Paterson 1987), and have been addressed by Carter and Tsenn (1987) and other authors quoted below. It is recognized that there are many mechanisms by which a rock can deform, and that different rheological descriptions apply for different mechanisms. The rheological properties that characterize the various layers or regions in the Earth will therefore depend strongly on depth because of the influence of both pressure and temperature. This dependence has often been depicted on a ªfir treeº or ªstrength envelopeº diagram, following Bird (1978), Brace and Kohlstedt (1980), Kirby (1980), Kohlstedt et al. (1995), Ord and Hobbs (1986), Ranalli (1997) and others. In some respects, such a diagram corresponds to a deformation mechanism map of the Ashby type (Ashby 1972). However, it mixes strain rate-dependent and strain rate-independent behaviour, and it does not take history into account. It is therefore over-simplistic, and needs considerable qualification, especially at shallower depths. In this paper we shall review very briefly the range of rheological or constitutive relationships that might come into consideration in modelling deformation in the Earth, and then discuss the problems of obtaining continuum parameters suitable for geodynamic modelling.

Rheological regimes observed in the laboratory The term ªrheological regimeº is used here to denote the range of conditions under which a particular constitutive relationship can be used to describe the deformation behaviour. A given regime can be expected to involve a particular deformation mechanism, but ultimately its definition is based on the constitutive relationship itself. Primarily, the constitutive relationship defines the dependence of the strain or the strain rate on the stress to which the body of rock is subjected. However, there are a number of other dependencies that have to be taken into account in particular situations: 1.1Pressure dependence. This is likely to be important where brittle or cataclastic processes are involved at some scale in the rock. The pressure is often identified with the gravitational loading at the depth of burial, but more strictly can be taken as the mean stress in the rock (given in frame-independent terms by the first invariant of the stress tensor). There is also a small pressure dependence of the strain rate under given stress in the case of

strain-rate-dependent deformation, but it can generally be ignored at crustal pressures. 2.1Temperature dependence. Apart from minor effects related to the temperature dependence of the elastic constants, the strain rate tends to derive from underlying thermally activated processes, the rate of which is an Arrhenius exponential function of the temperature. In lithospheric problems it can generally be taken as a rule that strongly pressuredependent deformation will be relatively temperature insensitive and vice versa. Only in applications in the deeper mantle will some pressure dependence become significant at the same time as temperature dependence. Consequently, where temperature dependence is important, the primary rheological emphasis is generally on the strain rate, whereas when pressure dependence is important, the primary emphasis is on the strain itself. 3.1Strain and history dependence. Since the strain may modify the microstructure of the rock, it can influence the flow strength, resulting in work hardening or work softening. Similarly prior thermal history can also affect the microstructure, and so influence the flow strength. When the changes brought about by the strain reach saturation or achieve a balance with thermal recovery effects, the rock is said to be deforming in a steady state, in which there is no longer a strain dependence. 4.1Pore fluid and chemical effects. Pore fluid effects may be purely mechanical, as expressed in a dependence on pore fluid pressure, or chemical interaction between pore fluid and solid constituents of the rock may affect the flow strength. In other cases, reactions between the solid constituents may produce phase changes that affect the strength. The following sections list the principal types of constitutive relationship that might have to be taken into account in geodynamic modelling involving the penetrative deformation of bodies of rock, where the rock is treated as a continuum. Fault friction is not included because, at the scale at which a fault is considered as an individual entity, its sliding does not constitute a penetrative deformation but rather it has to be treated as a discrete entity or as determining a boundary condition for a body of rock. Strain-rate-independent deformation Elasticity Elastic deformation occurs whenever a rock mass is under stress. It is of particular interest in connection with phenomena such as earthquakes and post-glacial rebound, but is commonly not of interest in tectonic deformation unless the amount of strain is relatively small. When elastic deformation has to be taken into account, and the density r of the rocks can be inde-

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pendently estimated, the elastic parameters are most reliably obtained from seismologically determined elastic wave speeds Vp and Vs . Where elastic isotropy applies, the LamØ elastic parameters l and G are given by l ˆ r…Vp2 2Vs2 † and G ˆ rVs2 (Jaeger 1962). The long wavelength of seismic waves ensures an averaging over the effects of porosity, fluid content and other structural factors that tend to reduce the elastic moduli to values below those determined in the laboratory on compact samples but more likely to be representative of rocks at the Earth scale. Perfect plasticity Perfect plasticity is an idealized rheological model in which irreversible deformation occurs at constant stress when the stress tensor reaches a limit known as the yield stress. There is no strain or strain-rate dependence of the deformation, the rate being determined by the boundary conditions, and temperature dependence is minimal. The yielding condition is expressed as a scalar function f …sij † of the stress state sij (the yield function or yield surface), usually in the form of either the von Mises criterion p f …sij † ˆ J2 k1 ˆ 0 or the Tresca criterion
f sij ˆ 12 js1 s3 j k2 ˆ 0

for

s 1  s2  s3

where J2 ˆ

1n …s1 6

s2 †2 ‡…s2

s3 †2 ‡…s3

o s1 † 2 ; s 1 ; s 2 ; s 3

are the principal stresses, and k1, k2 are material parameters (Chen and Han 1988). If the von Mises and Tresca criteria are chosen to coincide for uniaxial tension, k1 ˆ ps03 and k2 ˆ s20 , respectively, where s0 is the uniaxial yield stress …s2 ˆ s3 ˆ 0†. In addition to the yield function, it is necessary to postulate a ªflow rule' to determine the plastic strain …p† tensor increment deij that results when the yield condition is reached. This rule is usually formulated in terms of a scalar ªplastic potentialº g, a function of the stress components sij , as …p†

deij ˆ dl

@g @sij

where dl is a scalar constant (this quantity is not a material property, but is determined by boundary conditions). When the function g is taken to be identical with the yield function, the flow rule is said to be an ªassociated flow ruleº. In more general cases, the yield stress will not remain constant during deformation but will tend either to increase (work hardening) or to decrease (work softening). To allow for this effect, the perfect

plasticity theory is modified by introducing a strain dependence in the yield function, involving at least one additional material parameter (ªhardening parameterº, often taken as operating on dl; Chen and Han 1988). Pressure-dependent plasticity The most likely type of penetrative deformation in surficial and shallow crustal regions will involve finescale, penetrative brittle processes, constituting some sort of cataclastic or soil-like flow. Because of the effects of friction and of changes in volume associated with the relative movement of grains or blocks in such flow, a strong dependence on the confining pressure or mean stress will be an important characteristic. The most appropriate model is therefore some sort of pressure-dependent rheology, as invoked in soil mechanics or concrete plasticity, and based on similar theory to perfect plasticity but with additional terms introducing a pressure dependence of the yield surface. The simplest and most frequently used forms of yield function are the Drucker-Prager and Coulomb, which are, respectively, generalizations of the von Mises and Tresca criteria (Chen and Han 1988). The DruckerPrager yield criterion is p J2 k ‡ mi sm ˆ 0 where sm ˆ 13 …s1 ‡ s2 ‡ s3 † is the mean stress, k and i are material parameters, and tensile stresses are reckoned positive. The Coulomb criterion is independent of the intermediate principal stress and can be written as 1‡sin f s1 2c cos f

sin f s3 12c cos f

1 ˆ 0 for

s1  s 2  s 3

where c and f are material parameters, known, respectively, as the ªcohesionº and the ªangle of internal frictionº. When the two criteria are made to coincide for axisymmetric triaxial compression tests, the two sets of material parameters are related by p mi k 2 3 cos f ˆ ˆ 3 sin f tan f c An important aspect of the theory of pressure-dependent plasticity is that, in general, it predicts a change of volume (dilatancy or compaction) during deformation. The amount of the predicted volumetric strain depends on whether an ªassociatedº or ªnon-associatedº flow rule is used. In the latter case, additional material parameters, such as a ªdilatancy factorº b or an ªangle of dilatancyº c, are introduced in the flow rules derived from the Drucker-Prager or Coulomb yield functions, respectively. Measurements of dilatancy can be used to constrain these parameters.

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An important special case of non-associated flow arises in the ªcritical stateº (Schofield and Wroth 1968; Wood 1990) in which deformation proceeds at constant volume, as in prefect plasticity, even though the yield surface is pressure-dependent. Rudnicki and Rice (1975) treat a more general case of non-associated flow in which, while retaining dilatancy or compaction, they also introduce a second hardening parameter to deal with non-isotropic hardening. Their model is applied to the prediction of localization.

Strain rate-dependent deformation Linear viscosity Linear viscous or Newtonian creep is often observed in materials under simple stress states, the strain rate being proportional to the stress difference, and the volume remaining constant. Thus, for an axisymmetric triaxial test, we can write 1 …p† e_ 1 ˆ …s1 Z

s3 †

…1†

where Z is the viscosity. There are various situations in the lithosphere where linear viscous rheology might be expected, especially where granular flow at elevated temperatures is involved. Granular flow is taken here to mean the deformation of a material by the relative movement of its constituent grains or particles, as in the deformation of sand or gravel and in ªsuperplastic-typeº flow at high temperature (Borradaile 1981, uses the term ªparticulate flowº in a similar sense). When the deformation can be assumed to be rate-independent, as in classical soil mechanics, various forms of plasticity theory can be applied, as reviewed above, with suitable values for the parameters usually being determined empirically. However, over geological periods of time, and at elevated temperatures in the Earth, such a type of deformation can become rate-dependent, the rate being basically determined by diffusion or local reaction processes involved in the accommodation processes that relieve the ªlock-upº stresses arising when the particle motions interfere with each other. This category of flow can encompass a wide range of cases, from surficial hill creep and low-temperature solution-precipitation creep to ªsuperplasticº flow in deep-seated rocks. A general characteristic of granular flow models is that they contain (1) geometric factors that relate the local movements on the particle boundaries to the macroscopic strain resulting from the integration of these movements, and (2) rate-determining factors deriving from the accommodation processes. We can expect that after passing through any transient stages associated with establishing a repeatable pattern of relative grain movement, the accommodation rate will

be linearly related to the deviatoric stress, as in (1) above, provided that the accommodation processes are driven by potential differences directly proportional to the principal deviatoric stress components, and that no stress-dependent structural factors come into play. Granular flow accommodated by solution transfer processes is often invoked as an important or even the primary deformation mechanism in wet sediments undergoing diagenesis and consolidation, and in lowgrade metamorphic rocks (Durney 1972; 1976; Elliott 1973; Gratier 1987; McClay 1977). Consequently, there have been many attempts by these and other authors to formulate flow laws for this situation (see also Angevine and Turcotte 1983; Fletcher 1982, 1998; Gratz 1991; Paterson 1995b; Pharr and Ashby 1983; Raj 1982; Rutter 1976, 1983; Stocker and Ashby 1973; Weyl 1959). The form of the flow law depends on the details of the structure and rate-limiting processes in the model assumed. Granular flow may again come into consideration at relatively high temperatures as a grain-size-sensitive flow mechanism in fully compact rocks, especially when they are fine-grained (Ashby and Verrall 1973). In the absence of porosity, the accommodation mechanism could involve grain boundary diffusion (Coble mechanism) or through-the-grain diffusion (NabarroHerring mechanism), and the viscosity would then depend on the grain boundary and bulk diffusion coefficients, respectively, as well as on the grain size (Paterson 1995a). Another linear-viscous relation is that of HarperDorn creep, which is observed in some materials at relatively low stresses (Harper and Dorn 1957; Harper et al. 1958; Wang 1996). Non-linear ªviscosityº The most commonly observed steady state flow law for polycrystalline solids is the so-called power law which, in the case of the axisymmetric triaxial test, can be written as …p† e_ 1 ˆ

1 …s1 Z0

s3 †n

…2†

where n is a numerical constant usually between 1 and 10, and Z9 is a ªpseudo-viscosityº which no longer has the usual dimensions of viscosity. The pseudoviscosity term Z9 can be expanded to include explicit dependence on temperature T, grain size d, and other variables (for example, the fugacity f of a component such as oxygen or water) in the empirical form Q 1 ˆ Adp f q e RT Z0

…3†

where A is an empirical multiplier, Q is an empirical activation energy, and p, q are numerical exponents.

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The relationships (2) and (3) are most commonly associated with crystal plastic deformation mechanisms based on the movement of dislocations. Here, the value of n is usually between 3 and 10, and the flow is not sensitive to grain size (p=0). An fq term may be needed to express dependence on oxygen fugacity when cations of variable valence are involved, or on water fugacity when hydrolytic weakening of silicate bonding can occur. The expressions (2) and (3) have also been applied to the grain-size-sensitive deformation of very finegrained materials in the ªsuperplasticº field, where it is often found that n has a value of between 1.5 and 2.5 and p is between ±2 and ±3. It seems probable that the primary mechanism in these cases is granular flow, but the accommodation mechanism is not well understood; possibly there is accommodation by crystal plastic flow within the grains as an alternative to the diffusion accommodation for which n is expected to be 1 (see, for example, Schmid et al. 1987).

Extrapolation to general stress states We now consider the problems of extrapolation to geological conditions. Our first concern is to relate laboratory measurements of stress-strain-time relationships under relatively simple stress states to the stressstrain-time relationships under the more general stress states that may be involved in geological deformations. For this purpose, it may be convenient to define ªequivalentº or ªeffectiveº stress and strain variables (the term ªequivalentº is preferable to ªeffectiveº if confusion with the Terzaghi effective stress in systems with pore fluid pressure is to be avoided, and it will be used here). That is, we may wish to generalize from an experimental relationship e or e_ ˆ f …s; Ci † to a corresponding constitutive relationship ee or e_ e ˆ f …se ; Ci † where e or e_ is the strain or strain rate variable and s is the stress variable used in the experiment, ee or e_ e and se are ªequivalentº strain or strain rate variable and stress variable, respectively, and Ci represents the material parameters in the stress-strain relationships (for comments on the most general forms of the function f, see Glen 1958). Such a generalization depends basically on knowing or assuming the form of the constitutive relationships that applies in the general case. These relationships can, of course, only be established firmly by further, more difficult experimental studies but, in the absence of such studies, their form must be assumed by analogy with materials for which the constitutive behaviour has been established. If a von Mises-type constitutive relationship applies, the equivalent stress and strain can be defined as follows: r o p 1n …s1 s2 †2 ‡…s2 s3 †2 ‡…s3 s1 †2 ˆ 3J2 se ˆ 2

e_ e…p†

 s  2  2  2  2  …p† …p† …p† …p† …p† …p† e_ 1 ˆ e_ 2 ‡ e_ 2 e_ 3 ‡ e_ 3 e_ 1 9

Note that se is identical with the ªgeneralized deviator stressº used in soil mechanics (Wood 1990). If a Tresca failure criterion applies, the equivalent stress and strain will be of the form s e ˆ s1 s 3 …p† …p† e_ e ˆ e_ 1

for

s1  s2  s3

In the case of materials undergoing strain-rate-dependent plasticity, it is usual to assume that the creep flow stress is governed by an equivalent of a von Mises or Tresca criterion in which the yield stress is replaced by the flow stress (Odqvist 1935, 1966).

Extrapolation in the strain and time domains In the case of strain-rate-independent deformation it is implicit, because of the time independence, that the same stress-strain relationships will apply just as directly in the geological situation as in the laboratory when the appropriate influence of pressure is taken into account. In particular, any strain hardening observed in the laboratory can be expected to occur also in nature. In practice, however, there may be, in addition, a small amount of creep (ªbrittle creepº) even in this case, introducing an element of time dependence, so that the stress may be slightly lower in nature than in the laboratory, and the strain slightly higher. In the case of strain-rate-dependent flow, extrapolation of the laboratory flow laws to the geological timescale is now necessary, and there are several questions that need to be addressed: 1.1Which rheological regime or flow law established in the laboratory is to be identified as being relevant in a particular situation in nature? One may try to decide this question on the basis of observations on the microstructural imprint surviving from the geological deformation, comparing it with the microstructures seen in the laboratory. In making this comparison, however, it must be borne in mind that the microstructures seen in the natural rocks represent the ªclosureº situation at which the microstructure became frozen in. The extent to which the latter represents the deformation regime itself will depend on whether the rock was cooled while under stress or whether the loading was removed before cooling, as well as on the kinetics and scale of microstructural re-arrangement during cooling or annealing. The flow regime predominating during the main part of a geological deformation may therefore be difficult to identify unambiguously. In general, as set out above, the choice will be between a strongly stress-dependent but grain-size insensitive regime based primarily on crystal plasticity, and a more weakly stress-depend-

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ent but grain-size sensitive regime involving primarily relative grain movement (granular flow). Grain-size sensitive flow tends to become more favoured relative to grain-size insensitive flow when the experimentally determined flow laws are extrapolated towards geological conditions, because of both the lower strain-rate sensitivity and the lower temperature sensitivity of the former. This trend may mean that the geological deformation in calcite rocks is often near-Newtonian (Paterson 1987; Rutter 1995; Schmid et al. 1977, 1980; Walker et al. 1990); in particular, in its highly deformed setting, the classical Carrara marble of negligible crystallographic preferred orientation may have been deformed in a granular flow regime, and may possibly have undergone some subsequent grain growth, since it is questionable whether the crystallographic preferred orientation that would be expected to form during predominantly crystal-plastic flow would be so thoroughly removed in annealing. However, Rutter and Brodie (1994) conclude that quartzite rheology under geological conditions will still be grain-size-insensitive, with an n greater than 3, a conclusion that is supported by the observation of crystallographic preferred orientations. 2.1Can strain dependence generally be ignored? For this to be the case, it would have to be assumed that substantial deformations in nature proceed more or less in a steady state, and that a steady state has also been attained in the experiments. The latter assumption is implicit in most attempts to extrapolate creep properties from the laboratory to nature (Brace and Kohlstedt 1980; Goetze and Evans 1979; Kawamoto 1996; Kirby 1977, 1980; Kohlstedt et al. 1995; Ord and Hobbs 1986; Paterson 1987). However, it is known that recrystallization and the development of preferred orientations occur gradually, and so may affect the flow stress over large strains. Recent experiments to large shear strains in torsion (Pieri and Olgaard 1997; Pieri et al. 1998a, 1998b; Stretton and Olgaard 1997) have shown that strains of 1 or more may be required to reach a steady state, and that, at least in marble and anhydrite, there may be some softening beyond this strain after an earlier strain hardening. Therefore, whether a true ªsteady stateº has always been attained in experiments to axial strains of only 0.2 or 0.3 is open to question, as is the corresponding presumption that geological conditions involve steady state deformation. 3.1Will a steady state assumption still be valid when conditions of temperature, stress, etc. are changing with time in a geological situation? That is, will reequilibration effectively keep pace during these changes? Probably no general answer can be given, since it will depend on the relationship between the rates of deformation and the rates at which temperature and stress are changing in the geological situation.

Extrapolation in the spatial domain Even if a valid extrapolation can be made from laboratory conditions to geological conditions for the small piece of rock material that the laboratory specimen represents, it is still necessary to relate the strength of such a piece to the strength of a rock mass on the scale of a geological formation or a major part of the Earth's crust. It will be rare that such a rock mass will be homogeneous on all scales down to that of the laboratory specimen. Thus, for example, there could be a system of shear zones on a mesoscopic scale that is too fine to be taken into account in the geodynamic model, but the scale of the experimental specimen may be such that it would either sample the material within the shear bands or that of the matrix between them. Therefore, we need procedures for estimating the rheological behaviour of larger masses of a given rock, as well as of formations involving more than one kind of rock. It is well known that there are scale effects in the brittle fracture strength of rocks, at least up to the metre scale (for review, see Paterson 1978). These effects can be related to the statistical occurrence of grain-scale flaws of varying severity, the more severe flaws being of sparser occurrence (Weibull 1952). However, while such a factor may be of some importance in the case of pressure-dependent plasticity, it is probably not so in the case of strain-rate-dependent flows. The major factor in most cases of geological application is likely to be the petrological and structural heterogeneity of the rock mass. However, in modelling using continuum mechanics, it is implicit that the rock mass is at least statistically homogeneous on the scale of the model. The smallest relevant sample of this mass can be called the representative elementary volume (REV). The REV contains all the component aspects of the rock mass that are averaged out in the continuum approximation, and the required scale of the REV will correspond to the minimum scale of relevance in the model. In the general case that the rock mass is heterogeneous when viewed at smaller scales than the required REV scale, the material parameters required for the geodynamic model will have to be obtained by a further modelling process which calculates the material parameters for the REV from the experimental parameters obtained from laboratory specimens. Only if the rock is statistically homogeneous at all scales, from that of the geodynamic model down to that of the laboratory specimen, will the experimental parameters be directly applicable to the geodynamic model. In the absence of fine-scale information, this is often assumed to be the case for the convecting upper mantle, but it is clearly not the case in the crust. The heterogeneities in the rock mass of the Earth's crust vary greatly in nature and scale. Accordingly, it

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is difficult to define general rules for the modelling of the REV. However, in the following paragraphs, we attempt to set out some basic considerations that may have relevance in many crustal situations, distinguished somewhat arbitrarily by scale (local, regional and global) Local scale Heterogeneities at this scale include features such as joints, local faults and shear zones within a given type of rock, most of which will act as sites of relative weakness, possibly leading to strain concentration. The weakness of shear zones may be associated with local changes in grain size or in mineralization relative to the main body of rock. At this scale the rock mass can probably often be represented as an assemblage of blocks of a particular rock type, separated by weaker zones, for example, by the shear zones or local faults already mentioned. Then it may be possible to model the REV as a ªgranularº medium, in rough analogy to sand or to fine-grained materials undergoing superplastic flow (Holdsworth et al. 1997; Scott 1996; Turcotte 1986; Twiss and Unruh 1998). In such a ªblock flowº view, the bulk strain would be contributed to largely by the relative movement of the blocks, but the dynamics of the flow could be determined either by the resistance to sliding of the blocks over one another or by the local flow within the blocks involved in accommodation processes required for strain compatibility. In pursuing this question further, we consider separately the cases of strain-rateindependent and strain-rate-dependent materials. Strain rate-independent flow Where the dynamics of the flow is determined by the resistance to sliding of the blocks over one another, this sliding might be expected to occur with a coefficient of friction that is determined by Byerlee's rule, although the normal force on the interfaces will depend on the orientation and distribution of forces between neighbouring blocks. Also, the situation will tend to be complicated by factors such as the accumulation of gouge and the occurrence of pressure solution effects. Where the dynamics of the flow is determined by accommodation processes within the blocks, the principal factors are likely to be dilatancy and cataclasis, as in sand and similar materials, or penetrative small-scale pressure-dependent deformations within the blocks. In either dynamic case, it will be seen that the requirements for modelling the REV are, more or less, the same as those for modelling pressure-dependent or ªmicro-brittleº granular flow on the specimen scale. Therefore, the strain-rate-independent flow properties of the rock mass are likely to be of similar

form to those applicable to laboratory specimens, that is, of Coulomb or Drucker-Prager form in the first approximation. However, although the pressure sensitivity factors tanf or i, respectively, still reflect frictional effects to an important degree, their relationship to the physical coefficient of friction is not a direct one, and the Byerlee rule is not applicable. The important question is how the constitutional parameters c, tanf and tanc or k, i and b, respectively, that are applicable to the local crustal scale can be related to those determined at the laboratory specimen scale. In a very general way, because of the scale effects in brittle fracture mentioned above, and because of minor rate-dependent factors deriving from pressure solution or local crystal plasticity effects in the geological situation, it seems likely that the relevant values of the constitutional parameters will be somewhat less than the laboratory values. Further, as the confining pressure or mean stress is increased, a departure from linear Mohr-Coulomb behaviour in the direction of lower pressure sensitivity is commonly observed (Paterson 1978), giving further emphasis to the choice of a lower pressure sensitivity factor or coefficient of internal friction than that determined in the laboratory. In the absence of proper information, and on purely intuitive grounds, one might select values of perhaps around 80% of laboratory values as a first approximation for the constitutional parameters for geodynamic modelling with Drucker-Prager or Coulomb constitutive relationships where strain rate-independent plasticity is thought to be relevant. A comment is appropriate at this point on the invocation of Byerlee's rule in drawing the upper part of the ªChristmas treeº or ªfir treeº diagrams that purport to describe rheological behaviour as a function of depth in the Earth, following Bird (1978), Brace and Kohlstedt (1980), Goetze and Evans (1979), Kawamoto (1996), Kirby (1980), Kohlstedt et al. (1995), Molnar (1992), Ord and Hobbs (1986), Ranalli (1997), Sibson (1983) and others. Such a procedure is oversimplistic for describing strain-rate-independent behaviour that is penetrative on the scale of the geodynamic model, which should be dealt with as described in the previous paragraphs. Where Byerlee's rule may have direct application is in dealing with displacements on discrete faults, as opposed to distributed brittle behaviour. However, discrete faults are more appropriately dealt with in terms of boundary conditions for the model. Strain rate-dependent flow In the case of strain-rate-dependent flow, the dynamics of the ªblock flowº could again be determined either by the resistance to sliding of the blocks over one another or by the local flow involved in the accommodation processes. The block flow would now be analogous to the temperature-sensitive granular

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flow that occurs in the ªsuperplasticityº of finegrained materials on the laboratory specimen scale, in which the accommodation is either by diffusive material transfer processes or by local crystal-plastic deformation of the grains. Grain scale diffusive accommodation is unlikely to be important for relative block movement at the local crustal scale, although largerscale solution transfer may be important in some cases, as suggested by observations on quartz veins. However, the main accommodation process is likely to be localized strain-rate-dependent plastic flow of the same type as observed in laboratory specimens. The resistance to sliding of the grains over one another is generally not considered to be a rate-controlling factor in granular flow in fine-grained laboratory specimens. Similarly, it would seem unlikely that the resistance to relative sliding of the blocks would be a controlling factor in the block flow on the local crustal scale. For example, if the relative displacement of the blocks occurred on shear zones in which fine grain size resulted in a relatively low flow stress, the interference in motion between blocks would involve flow in coarser-grained parts of the blocks themselves at higher stresses, the latter flow then being likely to be rate-determining for the whole rock mass. Therefore we shall assume here that the resistance to relative sliding of the blocks does not determine the strain-rate dependence in the flow law for the local scale. There are two factors involved in relating the macroscopic flow law to the rate-determining accommodation flow. First, there will tend to be a stress concentration at the accommodation site relative to the applied macroscopic or ªfar-fieldº stress, and second, there will be a strain magnification in the macroscopic strain relative to the accommodation strains (a small accommodation strain may permit a large amount of relative block sliding). The combination of these two factors will result in a considerable increase in the preexponential factor A in (3), or of the corresponding quantity for linear viscosity in the case of solutiontransfer control, relative to that determined on laboratory specimens. However, the temperature and stress sensitivities arise from the accommodation processes, which are envisaged to be the same as in the laboratory specimens, and so we can expect the stress exponent and empirical activation energy to be much the same as for the laboratory specimens. Therefore, for strain rate-dependent flow on the local scale, the form of the flow law can be expected to be the same as that determined on laboratory specimens of the same rock, with similar values of stress exponent and activation energy, but with a larger preexponential factor, leading to lower stresses in the crust than in the laboratory for a given strain rate. The pre-exponential factor may be larger than the laboratory value by an order of magnitude or so (cf. Ashby and Verrall 1973; Paterson 1995a). Thus the rock mass will probably be weaker than the rock material.

Regional scale On this scale, the heterogeneity will primarily involve variation in rock type, each individual rock type acting as a local rock mass on the scale considered in the previous section. Thus the rock mass at the regional scale may be made up of the various elements of a sedimentary sequence, contain igneous intrusions, or consist of metamorphosed versions of any of the former, as in a basin, a craton or a mountain province. The rheological requirement for geodynamic modelling is therefore to arrive at a regional-scale rheology that subsumes the individual rheologies of the local-scale rock masses that make up the regional-scale mass. In many respects this project is analogous to that of determining the rheological behaviour of a polyphase material for which the properties of the individual phases is known. There is a considerable body of studies on the latter topic in the materials science literature, covering various relative configurations of the phases. In some cases, one phase can be treated as forming inclusions within a matrix phase, the inclusions being either of more or less equant aspect ratio, or alternatively, of fibrous or lamellar shape. In other cases, the different phases may be intimately interwoven, with various degrees of continuous connection or percolation. The modelling of the properties of the provincial-scale ªpoly-rockº mass can probably follow in general lines the modelling of such materials in terms of the properties of their constituent phases. As with the polyphase materials, it is difficult to proceed without some more-specific description of what the general structure and boundary conditions are, but there are some general principles that can be recalled from materials science. In the absence of chemical reactions or other interactions between the phases that affect their individual constitutive properties, there are two limits between which the properties of the polyphase material will lie. These bounds correspond to the cases in which the stress and the strain, respectively, are uniform within an REV of the composite body. Various names have been given to the bounds: in elasticity, the Reuss and Voigt bounds; in polycrystal plasticity, the Sachs and Taylor cases; and in discussing lamellar or similar composites, the ªseriesº and ªparallelº cases, respectively. However, in general, the limiting cases themselves are not physically realistic because of incompatibilities in strain and stress, respectively, between the components, and the real situation should lie between the limits. Theoretical models that attempt to reconcile these incompatibilities so that both stress and strain are physically realistic are known as ªself-consistentº models. Thus, when the REV for a crustal region of irregular structure contains more than one type of rock, its modelling may be based on analogy to the modelling of an individual poly-mineralic rock (cf. Bao et al. 1991; Cho and Gurland 1988; Handy 1990, 1994; Ji

165

and Zhao 1993; Jordan 1988; Poech and Fischmeister 1992; Tullis et al. 1991). In other cases, where the structure is of a more regular nature, the modelling may follow that of fabricated composites such as sandwich structures and fibre-reinforced materials (cf. Clyne and Withers 1993). Thus there are many different cases that could be considered. The general problem is to find a flow law for the regional poly-rock mass, the parameters of which can be expressed in terms of the parameters of the flow laws for the component rock masses. The geological region may consist of rock masses which variously undergo either strain-rate-independent or strain-ratedependent flow, depending on the nature of the rocks and on the location of the region in the pressure-temperature field in the Earth, and so the problem can be very complex. For illustration, we consider here only the case that all the component rock masses are undergoing strain-rate-dependent deformation following a power law. In the absence of chemical interaction between the component masses, we might expect bounds to be set by ªseriesº and ªparallelº cases. Then, for an axisymmetric deformation with axial strain rate e_ and stress difference s, we would have: For the parallel case: X X Qi 1 1 Q 1 1 fi si ˆ fi Ai ni e_ ni eni RT s ˆ A n e_ n enRT ˆ

above, will probably be somewhat smaller. However, it is clear that further studies on the modelling of local and regional rheologies in terms of laboratory behaviour are required. Global scale The global scale comprises major crustal provinces such as orogenic belts, cratons, subducting slabs, etc. Viewing the crust as a whole would require treating it as a composite of several regional-scale rock masses, with compositions such as deduced by Christensen and Mooney (1995) and Rudnick and Fountain (1995). However, it is probably better to deal with this scale by defining different provinces in the geodynamic model as part of its boundary conditions, such as recognizing upper, middle and lower crustal provinces, with REVs determined as for the regional scale above. Acknowledgements I am grateful to Professors Georg Dresen and Mark Handy for the invitation to take part in the Workshop on Rheology and Geodynamic Modelling in Neustadt an der Weinstrasse, Germany, which was held on 25 and 26 March 1999, and for encouragement to write up this contribution. Discussions with Professor Dresen and Dr. Erik Rybacki and reviews by Professors Stefan Schmid and Tom Blenkinsop have been very helpful.

i

For the series case: X X Q fi e_ i ˆ fi Ai sni e e_ ˆ Asn e RT ˆ

Qi RT

i

where the quantities without suffix refer to the REV of the regional mass, the quantities with suffix i refer to the ith component rock mass type, and fi is the volume fraction of each component. The problem then becomes to obtain expressions relating the regional A, n and Q values to the local Ai, ni and Qi values for the above cases. Evidently some further assumptions or inputs are needed, and the problem becomes even more complex when more realistic cases are considered. In the absence of proper solutions, one might attempt to proceed by estimating the regional parameters using the law of mixtures but there is no theoretical justification for doing so. In practice the above problem has often been circumvented by assuming that the regional rheology is dominated by that of a single mono-mineralic rock (quartz, feldspar or olivine) for which laboratory-determined rheological parameters are used. This procedure is clearly over-simplistic but, in the absence of better modelling of the local and regional rheologies, it may be acceptable in a first approximation because, intuitively, one may feel that the parameters for the regional rock mass are probably going to be very broadly similar to those from the laboratory except for the pre-exponential factor which, for reasons given

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