## Abstract

This is the first of a series of works on the continuum mechanics and thermodynamics of creep and recrystallization of large polycrystalline masses. The general continuum theory presented here is suited to mono- and multi-mineral rocks. It encompasses several symmetry groups (e.g. orthotropic and transversely isotropic) and diverse crystal classes of triclinic, monoclinic and rhombic systems, among others. The cornerstone of the current approach is the theory of mixtures with continuous diversity, which allows one to regard the polycrystal as a ‘mixture of lattice orientations’. Following this picture, balance equations of mass, linear momentum, lattice spin, energy, dislocations, and entropy are set forth to describe the response of the polycrystal (i.e. the ‘mixture’), as well as of a group of crystallites sharing the same lattice orientation (*viz*. a ‘species’). The connection between the balance equations for a species and those for the mixture is established by homogenization rules, formulated for every field of the theory.

## 1. Introduction

Geomorphological processes often involve the deformation of large masses of rock, flowing in a slow and continuous viscoplastic regime named *creep*. In nature, creeping rocks are seldom monomineral—like ice in glaciers and ice sheets, or pure halite (rock–salt) in salt domes and beds (Handin *et al*. 1986; Paterson 1994)—but sometimes one mineral may be predominant, constituting what is called the *primary* or *connected phase*: e.g. olivine in the upper mantle or anhydrite-rich halite in large salt deposits in the crust (Hobbs *et al*. 1976; Chopra 1986). The corroboration of such a prevailing, connected phase is essential for regarding the medium (in a good approximation) as a monomineral rock. Nevertheless, in most common situations the material is indeed multi-mineral, and requires therefore a multiphase description. In this work, both cases (single- and multiphase modelling) will be addressed.

Independently of the mono- or multi-mineral character of the rock, its structure is typically *crystalline*. This means that minerals and rocks are, generally, composed of *crystallites* (also called *grains*) possessing a highly ordered atomic structure: the *lattice*. The peculiar symmetry of the lattice, geometrically represented by *crystallographic axes* (Kocks *et al*. 1998), causes crystallites to be remarkably anisotropic. As a consequence, the orientational distribution of crystallographic axes—called *texture* or *fabric*1—is of prime importance for the mechanics of the rock. On the other hand, during deformation, the lattice of some crystallites may bend, twist, break, and rotate, changing the original fabric. The latter may also be modified by the growth and shrink of grains, as well as by *dynamic recrystallization*, which involves the *nucleation* of new grains and the irregular *migration* of their boundaries (Poirier 1985; Humphreys & Hatherly 2004).

Besides all issues mentioned so far, the deformation of large rock masses is also complicated by its experimental unattainability: usual geomorphological processes last for millennia, reaching remarkably large strains at such a slow pace that it is impossible to reproduce analogous conditions in laboratory. Further, field observations are often hindered by the environment, either because of inhospitality (e.g. ice sheets) or due to natural barriers (e.g. Earth's mantle). Hence, to proceed on the subject, we are forced to rely on judicious theories.

This series is an attempt in that direction. Its objective is to present a general continuum theory for the mechanics and thermodynamics of large polycrystalline masses,2 including fabric (i.e. texture) evolution, anisotropic response and recrystallization. The theory is intended for mono- and multi-mineral rocks and encompasses several symmetry groups, including transversely isotropic and orthotropic, as well as any crystal class whose symmetry is susceptible to being described in terms of three orthogonal axes. In this Part I, general balance equations and homogenization rules are presented. The ensuing Part II (Faria *et al*. 2005) deals with the construction of a thermodynamically consistent constitutive theory for a whole class of polycrystalline media. Finally, Part III (Faria submitted) illustrates the usefulness of the theory for the particular case of anisotropic ice sheets.

The structure of this article is as follows: §2 introduces the general formalism of mixtures with continuous diversity, which represents the cornerstone of the present theory, and shows how to incorporate polycrystals into this scheme. In §3, balance equations and homogenization rules are derived for monomineral rocks. Finally, §4 ends the article with remarks on the extension of the theory to multi-mineral rocks. Important information about *notation* is provided in appendix A.

## 2. Polycrystals as mixtures with continuous diversity

At first sight, we could naively conjecture that polycrystalline minerals are simply dull gatherings of single crystals. However, the truth is much more complex than that: polycrystals are made of grains that interact continually through exchanges of mass, energy, momenta and entropy. Thus, from a thermodynamic point of view, we may say that crystallites are *mutually interacting open systems*.

### (a) Mixtures of crystallites

Examples of interacting open systems are abundant in nature: granules with different sizes in a polydisperse granular medium, distinct phases of a material undergoing phase changes, incompatible populations disputing a common territory, chemical substances reacting in a mixture, just to mention some. All these examples have in common the fact that their dynamics are described by the same set of fundamental laws, which form the basis of the *standard theory of mixtures*. Of course, the word ‘mixture’ is used here in its broadest sense, as a ‘mixture of grain sizes’, a ‘mixture of phases’, a ‘mixture of populations’, or a ‘mixture of chemical substances’. It should be noticed that the identification of the constituents (or *species*) in these examples also differs from mixture to mixture. For instance, in mixtures of populations we can use either taxonomic or physiological attributes to identify the members of a species, whereas in a chemical mixture the species are distinguished by chemical composition. For polycrystals, there are many crystallite properties which can be used to identify a species, like grain size and shape, orientation of crystallographic axes, etc. depending on the characteristics of the material and the problem. Experience shows, however, that crystallographic orientation is frequently the most significant distinctive property. Hence, by adhering to such a characterization we may portray the polycrystal as a *mixture of orientations*, in the sense that grains with similar lattice orientations within the same aggregate should behave alike.3

At this moment, we face a technical problem: an appropriate mixture theory for polycrystals should be able to cope with all possible species, *viz*. an infinite number of them, for the lattice orientation may vary continuously in space. Standard mixture theory is not suitable for this case, since it can deal solely with a limited number of constituents. Hence, we are forced to resort to a different kind of theory, apt to model less orthodox types of mixtures possessing a *continuous diversity* of species.

### (b) Résumé of the theory of mixtures with continuous diversity

Succinctly, a mixture with continuous diversity can be regarded as a multi-component medium made up of an infinite number of mutually interacting species whose distinctive properties vary smoothly from one to another. As a matter of fact, the intuitive notion of continuous diversity is remarkably old (see Asimov 1979), and also its mathematical modelling is long-established, being formally rooted in Euler's (1767) pioneering work on the demography of structured populations. Since then indeed, the same concept has been improved and/or independently rediscovered in diverse contexts, ranging from chemical mixtures (de Donder 1931; Aris & Gavalas 1966) and gas dynamics (Curtiss 1956; Dahler 1959) to anisotropic fluids (Condiff & Brenner 1969) and sea ice (Coon *et al*. 1974). Eventually, the term ‘mixture with continuous diversity’ was coined (Faria 2001) in an effort to incorporate all those formerly unrelated approaches into a unified thermodynamic theory.

Mathematically, the idea of continuous diversity can be readily grasped by considering the example of an ordinary chemical mixture of *N* components. In this simple case, the mass density field of the *α*th species at position *x*_{i} and time instant *t* is denoted by *ϱ*^{α}(*x*_{i}, *t*), with *α*=1, 2, …, *N*. Notice that the species label *α* is not just a counter: the mixture can only have a physical meaning if there exists a one-to-one relation between *α* and the distinctive properties of the constituents, e.g. *α*=1↦liquid, *α*=2↦solid, etc. Now, to derive the respective mass density field in a mixture with continuous diversity, we must simply allow the species label *α* to be a real variable, defined in a compact interval called *species assemblage*. The end points *α*_{min} and *α*_{max} are, generally, chosen so that has *complete diversity*, i.e. it accounts for all possible species in the medium.

A result of the procedure outlined above is that *α*∈ has acquired the status of a new variable, in addition to *x*_{i} and *t*, in such a manner that the mass density field of the *α*th species4 is given by *ϱ*^{*}(*x*_{i}, *t*, *α*) and should be interpreted as a density on . The superscript asterisk indicates that the respective field is a function not only of *x*_{i} and *t*, but also of *α*. Of course, the same procedure can be extended to all other physical quantities of interest, enabling the definition of the species fields of stress *t*_{ij}^{*}(*x*_{k}, *t*, *α*), internal energy *e*^{*}(*x*_{i}, *t*, *α*), velocity *v*_{i}^{*}(*x*_{j}, *t*, *α*), etc.

### (c) Definition of a crystalline species

On many occasions, it may happen that the specification of a species requires more than one distinctive property, in such a manner that multiple labels (*γ*=1, …, *ν*) must be introduced (Faria 2001). This is for instance the case of polycrystals modelled as ‘mixtures of crystallographic orientations’, for which we generally need *ν*=3, as explained below (see also figure 1).

We all know that lattice orientations can be represented in a number of ways, e.g. through *Euler angles*, *quaternions*, *Cayley–Klein parameters*, *Rodrigues vectors*, etc. (Synge 1960; Sutton & Balluffi 1995; Goldstein *et al*. 2002). Suppose we decide to use Euler angles, *viz*. φ, θ and ψ, which are the standard choice for the analytical treatment of fabrics. Thus, we could in principle set *α*_{1}=φ, *α*_{2}=θ and *α*_{3}=ψ but this choice of labels makes calculations rather cumbersome. Instead, we follow the standpoint of Liu (1982, 2002) that constitutive relations for anisotropic media—including crystalline matter—are best expressed in terms of *anisotropic invariants*. In diverse situations, these invariants account for anisotropy by means of an *orthogonal triad of unit vectors* , which determines the axes of symmetry of the lattice. This is the case of transversely isotropic and orthotropic symmetries, as well as diverse crystal classes of triclinic, monoclinic and rhombic systems, among others (for a comprehensive list see Liu (1982)).

Equivalence between the nine components of the triad and the three Euler angles φ, θ and ψ is established through the orthonormality conditions,5(2.1)(cf. appendix A), which imply that only three components of the triad are in fact independent: *n*_{1}^{1}, *n*_{2}^{1} and *n*_{1}^{2}, say. As illustrated in figure 1*b*, we can easily identify these three key components with the labels *α*_{1}, *α*_{2} and *α*_{3} which specify a species:(2.2)An instructive interpretation of (2.1) and (2.2) is provided by the *drifting boat metaphor* (figure 1*c*,*d*; see also Kocks *et al*. 1998). The first two Euler angles φ and θ specify, respectively, the longitude and colatitude of a ‘fictitious boat’ drifting on the surface of the unit sphere . Clearly, these two angles establish the orientation of the unit radius vector *n*_{i}^{1} of . On the other hand, rotations about *n*_{i}^{1} are described by the third Euler angle ψ, which defines the instantaneous direction of the ‘bow of the boat’ with respect to the *local southward direction*. In other words, the angle ψ determines the orientation of the unit vector *n*_{i}^{2}, which specifies a point in the unit circle , where is the tangent space of at *n*_{i}^{1} (appendix A; Abraham *et al*. 1988). Of course, any orthonormal triad is completely determined through the knowledge of *n*_{i}^{1} and *n*_{i}^{2}, seeing that *n*_{i}^{3} is given by (2.1)_{3}.

To sum up, any species in a polycrystal modelled as a ‘mixture of orientations’ is uniquely determined by three species labels, which can be related to two mutually orthogonal unit vectors and . The appropriate species assemblage for the kind of polycrystals considered here is, therefore, , also called *orientation space*, since it comprises all possible orientations of the lattice. Accordingly, the vectors *n*_{i}^{1}, *n*_{j}^{2} and *n*_{k}^{3} are also named *orientation vectors*. Hence, by using just two of such vectors *n*_{i}^{A} (*A*=*1*, *2*) we can introduce, in conformity with §2*b*, thermodynamic fields that are *orientation dependent*: the species mass density *ϱ*^{*}(*x*_{i}, *t*, *n*_{j}^{A}), the species Cauchy stress *t*_{ij}^{*}(*x*_{k}, *t*, *n*_{l}^{A}), etc.

The fact that just *n*_{i}^{1} and *n*_{i}^{2} suffice to define a lattice orientation does not mean that *n*_{i}^{3} is dispensable: all three vectors are needed to describe material symmetries in an intelligible manner. Further, *n*_{i}^{3} is also requisite to distinguish between crystallites with right- and left-handed symmetries, as occurring, e.g. in *deformation twins* (Humphreys & Hatherly 2004). It must be noticed, however, that from the viewpoint of the present theory no continuous process can transform a right-handed lattice into a left-handed one, and vice versa: twinning is a discontinuous transformation. Consequently, crystallites with left- and right-handed symmetries must be treated as distinct materials, requiring a *multiphase theory*, just like the case of multi-mineral rocks. This topic is examined in §4.

In the simple case of a polycrystal made of *transversely isotropic grains*, just one crystallographic axis turns out to be relevant (figure 1*a*). The angle ψ becomes superfluous, and the triad {*n*_{i}^{1}, *n*_{j}^{2}, *n*_{k}^{3}} reduces to a single orientation vector: *n*_{i}^{1}=*n*_{i}. Accordingly, the orientation space is restricted to and the species fields assume the forms *ϱ*^{*}(*x*_{i}, *t*, *n*_{j}), *t*_{ij}^{*}(*x*_{k}, *t*, *n*_{l}), etc. Materials of this sort are examined in Parts II and III (Faria submitted; Faria *et al*. 2005).

## 3. Balance equations for several crystal classes

Continuous diversity inevitably implies some kind of ‘species hierarchy’, which is intuitively expressed by the notion of *familiarity*: two species are said to be *familiar* if their distinctive properties, and consequently their behaviours—are alike, though not identical. The concept of familiarity stems from the existence of a metric in , and it is clearly the counterpart in of the usual notion of closeness in . As discussed below, familiarity plays a key role in the construction of balance equations, owing to its relevance for interactions and mass exchanges between species.

### (a) Transition rate, lattice spin velocity and orientational gradient

One of the greatest virtues of familiarity is that it allows us to treat the species labels *α*_{γ} and the position vector *x*_{i} at the same footing. For instance, we can conceive a situation in which, besides usual mass transfers by transport phenomena, the mass of constituent *α*_{γ} varies in time through *inter-species transitions*. From the obvious similarity of these transitions in with ordinary motions in , we immediately conclude that the rate at which such continuous mutations occur can be described by a kind of ‘velocity’, called *transition rate* and denoted by *u*_{β}^{*}(*x*_{i}, *t*, *α*_{γ}), with *γ*, *β*=1, …, *ν*. In fact, by considering a unit volume in a medium at rest (*viz*. *v*_{i}^{*}(*x*_{j}, *t*, *α*_{γ})≡0), we readily infer that *u*_{β}^{*} determines the rate at which the amount of mass *ϱ*^{*} performs a continuous transition from the constituent *α*_{γ} to some other *familiar species*, by altering its distinctive properties.

Now, let us apply the concepts of familiarity and transition rate to polycrystals. The starting point is a suitable interpretation of *familiarity in polycrystalline media*: within a given material particle, two crystalline regions made of the same substance and possessing the same symmetry are said to belong to familiar species if their relevant crystallographic axes are closely oriented. Clearly, what is meant by ‘closely oriented’ depends on the medium and the problem under consideration (familiarity in , like neighbourhood in , is a relative notion). Be that as it may, the formation of subgrain boundaries can be used to establish a natural upper bound for familiarity in polycrystals. Hence, in this work we assume the simple convention that *subgrains of a given crystallite belong to familiar species*, whereas highly misoriented grains (greater than 10°, say) pertain to disparate species.

Turning attention now to transition rates, we conclude from §2*b* and the discussion above that continuous transitions in must correspond to smooth changes of orientation. We may express such changes by two transition rate vectors (cf. Goldstein *et al*. 2002)—*viz*. *u*_{i}^{A*}(*x*_{j}, *t*, *n*_{k}^{B}), with *A*, *B*=*1*, *2*—related to the time rates of the Euler angles φ, θ (for *A*=*1*) and ψ (for *A*=*2*). It is evident that only three of the six components of *u*_{i}^{A*} are independent: *u*_{1}^{1*}, *u*_{2}^{1*} and *u*_{1}^{2*}, say, while the other three are determined by the conditions (cf. (2.1)),(3.1)Hence, both transition rate vectors are tangent to the sphere , while in addition, *u*_{i}^{2*} must also be tangent to the circle (cf. figures 1*b* and 2*d*). On the other hand, from the notion of familiarity we immediately infer that the continuous transitions expressed by the rates *u*_{i}^{A*} must correspond to *rotations* of the crystalline lattice. This conclusion is illuminating, because it implies that *u*_{i}^{A*} can be expressed in terms of a more fundamental quantity: the *lattice spin velocity s*_{i}^{*} (cf. figure 2),(3.2)(3.3)Of course, in the case of transverse isotropy we have the vector *n*_{i}^{2} becomes superfluous, and hence we can set and , as it should be.

From a different perspective, we may interpret the introduction of transition rates as a direct generalization of the notion of velocity, by replacing *v*_{i} with {*v*_{i}^{*}, *u*_{j}^{1*}, *u*_{k}^{2*}}. Evidently, such a generalization stems from a related extension of the concept of position, from *x*_{i} to {*x*_{i}, *n*_{j}^{1}, *n*_{k}^{2}}. Now, it is obvious that any change in the description of position entails upon a corresponding extension of the spatial gradient operator *∂*/*∂**x*_{i}, which becomes {*∂*/*∂**x*_{i}, *∂*_{j}^{1}, *∂*_{k}^{2}}, where *∂*_{i}^{A} (with *A*=*1*, *2*) denote the *orientational differential operators* in and ,(3.4)respectively. The first term on the right-hand side of (3.4)_{1} represents the usual directional derivative along *n*_{i}^{1}, while the second term arises from the normalization condition (2.1)_{1}, which implies that *∂*_{i}^{1} cannot have a component in the *n*_{i}^{1}-direction. Likewise, it follows from (2.1)_{1,2} that *∂*_{i}^{2} cannot have components in the directions given either by *n*_{i}^{1} or by *n*_{i}^{2}—there remains just the *n*_{i}^{3}-direction available—as expressed by (3.4)_{2}. Of course, in the simple instance of transversely isotropic crystallites, we have *n*_{i}^{1}=*n*_{i} and consequently *∂*_{i}^{1}=*∂*_{i}, since *∂*_{i}^{2} is not defined in this case.

### (b) Balance equations for polycrystals

Within the framework of continuum theories, crystals and polycrystals have sometimes been modelled as *polar media*6 (e.g. Forest *et al*. 2000), a supposition which dates back to Voigt (1887) and the Cosserat brothers (1909). Presently, the equations of polar theory are well-known (Dahler & Scriven 1963; Truesdell & Noll 1965; Capriz 1989; Svendsen 2001) and consist of the balance equations of mass, linear momentum, angular momentum (spin) and internal energy, respectively:(3.5)(3.6)(3.7)(3.8)where *ϱ*, *ϱv*_{i}, *ϱIs*_{i} and *ϱe* denote the densities of mass, linear momentum, spin momentum and internal energy, respectively. Other fields occurring in (3.5)–(3.8) are defined in appendix A. In some situations, it may be advantageous to replace (3.8) by the more fundamental balance equation of total energy(3.9)(3.10)from which (3.8) can be derived with the help of (3.6) and (3.7).

In practice, however, the classification of polycrystals as polar media is often unnecessary. Experience shows that in many situations *m*_{ij}=*I*=*c*_{i}=0 and *s*_{i}=*w*_{i} (where *w*_{i}≔(1/2)*ϵ*_{ijk}*∂**v*_{k}/*∂**x*_{j} denotes the local angular velocity of the continuum, i.e. one half of vorticity) may be good assumptions, in such a manner that (3.5)–(3.8) reduce to the balance equations of ordinary (non-polar) continua:(3.11)(3.12)(3.13)(3.14)Consequently, (3.9) and (3.10) simplify to(3.15)

(3.16)

Equations (3.11)–(3.13) and (3.15) are also known as the continuity equation, *Euler's first and second laws of motion*, and the *first law of thermodynamics*, respectively.

The set of balance equations (3.11)–(3.16) has long since been used in many theories for polycrystals as a rule of thumb. Notwithstanding, in the current approach we do not need to *postulate* the validity of (3.11)–(3.16); rather, we may adopt, *a priori*, the more general equations (3.5)–(3.10) and then, through an appropriate constitutive theory, it is possible to *prove* that (3.11)–(3.16) are valid for some particular problem (see Part III, Faria submitted).

That grains in a polycrystal should, generally, be modelled as polar media is justified by the micromechanics of crystals (see, e.g. Asaro 1983; Forest *et al*. 2000): torsion/bending of grains and the rotation of crystallographic axes relative to the matrix are clear indications of couples and asymmetric stresses acting on the grains. In this sense, polycrystals can also be seen as ‘mixtures of polar media’. What remains questionable is if such mixtures of polar media do behave themselves as polar media, or if the couples and asymmetric stresses acting on distinct species cancel each other on average, resulting in no net outcome.

In ordinary mixture theory (e.g. Faria & Hutter 2002), species balance equations are obtained from (3.5) to (3.10) in two simple steps. First, every field in (3.5)–(3.10) is replaced by its respective species field, characterized by the label *α*=1, …, *N*, *viz*.: *ϱ*(*x*_{i}, *t*) becomes *ϱ*^{α}(*x*_{i}, *t*), and *s*_{i}(*x*_{j}, *t*) becomes *s*_{i}^{α}(*x*_{j}, *t*), etc. Second, a production/exchange term describing *inter-species interactions* is added to every balance equation, since mixed species are in fact interacting open systems (cf. §2*a*).

In contrast, polycrystals modelled as mixtures with continuous diversity need slightly more complex species balance equations, in view of the generalizations discussed in §3*a*. Hence, we must convert the two steps mentioned above into four.

Every field in (3.5)–(3.10) is replaced by its respective species field,

*viz*.:*ϱ*(*x*_{i},*t*) becomes*ϱ*^{*}(*x*_{i},*t*,*n*_{j}^{A}), and*s*_{i}(*x*_{j},*t*) becomes*s*_{i}^{*}(*x*_{i},*t*,*n*_{k}^{A}), etc.A production/exchange term describing interspecies interactions is added to every balance equation, since mixed species are in fact interacting open systems.

As explained in §3

*a*, the velocity*v*_{i}is replaced by {*v*_{i}^{*},*u*_{j}^{1*},*u*_{k}^{2*}}, with*u*_{i}^{A*}given by (3.2), while the gradient operator*∂*/*∂**x*_{i}is replaced by {*∂*/*∂**x*_{i},*∂*_{j}^{1},*∂*_{k}^{2}}, with*∂*_{i}^{A}defined in (3.4).Following the same reasoning as the last item, also fluxes and stresses must be extended to have their counterparts in the species assemblage . These extensions are the

*interspecies stresses and fluxes τ*_{ij}^{A*},*ϖ*_{ij}^{A*}and*ξ*_{i}^{A*}, with*A*=*1*,*2*(see appendix A for the definitions of these fields).

Through these four steps, we derive from (3.5) to (3.10) the species balance equations for polycrystals modelled as mixtures with continuous diversity (summation convention applied to lowercase *and* capital repeated indices, cf. appendix A):

mass(3.17)

linear momentum(3.18)

lattice spin momentum(3.19)

internal energy(3.20)

It must be noticed that (3.20) is not derived directly from (3.8), but rather from (3.9) and (3.10) through the species balance equation of total energy(3.21)(3.22)Again, all fields in (3.17)–(3.22) are defined in appendix A. Details of the derivation of (3.20) from (3.9), (3.10), (3.21) and (3.22) are given in Faria & Hutter (2002).7

Two points are worthy of notice concerning (3.17)–(3.21): first, no balance equation is proposed for *u*_{i}^{A*}, since these fields can be derived from *s*_{i}^{*} via (3.2). Second, the corresponding balance equations for transversely isotropic crystallites are much simpler, seeing that in this case we can drop the superscripts ‘*A*’ out of all equations (e.g. *ξ*_{i}^{A*} becomes *ξ*_{i}^{*}, etc. see also remark 2.2 and Faria *et al*. (2005)).

At first sight, (3.17)–(3.22) may look somewhat formidable in comparison to (3.5)–(3.10). However, a careful analysis shows that the differences are actually not so striking. There is even an interpretation of (3.17)–(3.22), without direct reference to that can be valuable in certain situations. Consider for instance the species balance equation of linear momentum (3.18). We can readily rearrange it as(3.23)The left-hand side of (3.23) has exactly the form of the usual balance equation of linear momentum (3.6) or (3.12). Now, the right-hand side of (3.23) can, as a whole, be interpreted as an *effective production*/*exchange rate* of linear momentum within a polycrystalline particle (cf. figure 3): the first term describes interactions between highly misoriented crystallites, i.e. *interactions across high-angle grain boundaries*; on the other hand, the term can be interpreted as an specialized production/exchange rate in that describes *interactions across low-angle grain boundaries* (i.e. *subgrain boundaries*). Finally, the last term on the right-hand side of (3.23) does not represent a production/exchange by interactions, but rather by mass transfer: when the lattice of a grain rotates, its mass is in effect transferred from one orientation to another, and consequently its inherent properties (e.g. stored energy, etc.) are carried with it through a kind of ‘rotational convection’.

### (c) On the irreversibility of recrystallization: the balance equation of dislocations and the second law of thermodynamics

Mass, momenta and energy are clearly not enough to model the thermodynamics of creep and recrystallization. Indeed, the basic fields *ϱ*^{*}, *v*_{i}^{*}, *s*_{i}^{*} and *e*^{*} that are solutions of the system (3.17)–(3.20), do suffice to describe fabric, motion, lattice spin and temperature, respectively, but they fail to define uniquely a recrystallization process. The cause of this failure lies in the fact that the driving force for recrystallization is closely related to a particular part of the internal energy, which is stored during deformation in linear lattice defects called *dislocations*. Consequently, what we need is a balance equation of dislocations.

Kröner (2001) has shown that the notion of dislocation density as a scalar internal variable can be illuminatingly introduced in continuum mechanics by means of statistical arguments. Using a similar approach, Faria *et al*. (2003) proposed a species balance equation of dislocations for ice sheets. Here, based on the results derived so far, such an equation can be generalized for diverse classes of polycrystals, by proposing the following *species balance equation of dislocations:*(3.24)Succinctly, the species dislocation density *ρ*_{D}^{*} is an internal variable representing the total length of dislocations in crystallites with crystallographic axes directed towards *n*_{i}^{A} and enclosed in a unit volume of the polycrystal. Hence, its dimension is length^{−2}. The dislocation production rate *Π*_{D}^{*} represents the production/consumption of dislocations by Frank–Read sources, dipole annihilation, etc. (Asaro 1983; Poirier 1985). Finally, the interspecies dislocation fluxes *j*_{Di}^{A*} portray the dislocation exchange between subgrains. Clearly, such interspecies fluxes are relevant only in specialized models of dislocation–subgrain-boundary interactions, so that *j*_{Di}^{A*} may be neglected in most common applications.

Comparison of (3.24) with (3.17)–(3.21) reveals that there are two terms absent in (3.24), namely the (spatial) divergence of a conductive flux and an external supply. The absence of the latter is obvious: dislocations cannot be supplied from external sources to the bulk of the polycrystal. In contrast, the absence of a conductive flux of dislocations is less obvious and was discussed in detail by Faria *et al*. (2003). Succinctly, it is a particularity of the theory of mixtures with continuous diversity applied to *large polycrystalline masses*:1 in this case the mean free path of mobile dislocations turns out to be many orders of magnitude smaller than the size of a single material particle, in such a manner that—on a large-scale perspective—all dislocations seem to be ‘tied’ to the material.

The last but not least fundamental quantity to be introduced in this theory is the *entropy*, which expresses the irreversibility of natural processes. Its species balance equation arises as part of the so-called *entropy principle*, which sets up the *second law of thermodynamics* in a suitable mathematical form for continuum theories:

There exists for every species in a mixture with continuous diversity an additive scalar quantity called entropy, such that:

it evolves according to the species balance equation of entropy,(3.25)

for every species of the mixture, the specific entropy

*η*^{*}, its fluxes*ϕ*_{i}^{*}and φ_{i}^{A*}, as well as its specific production rate*ς*^{*}are all given by constitutive relations; andthe net entropy production rate density of the mixture is non-negative for all thermodynamic processes.

As regards the last item, it should be emphasized that the entropy production rates of some species can be *negative*—provided that any such losses are compensated for with simultaneous positive productions by other species—so that *ς*^{*} may have a non-vanishing lower bound. The crucial question is thus whether such a lower bound can be mathematically expressed in a tractable form, *viz*. by a *conventional* constitutive equation.8 The answer is fortunately affirmative.9

*There exists for every species in a mixture with continuous diversity a scalar quantity δ*^{*}*, called specific entropy deviation rate, such that:*

*it is given by a conventional constitutive equation; and**the inequality δ*^{*}≤*ς*^{*}*holds for all thermodynamic processes.*

Proposition 3.3 offers an interpretation of the second law of thermodynamics that is as general as—and is much simpler to be exploited on the species level than—the one presented in (iii) of axiom 3.2 (see Part II, Faria *et al*. 2005). A formal proof of it can be found in Faria (2001). The fact that the lower bound for *ς*^{*} can indeed be expressed by a conventional constitutive equation is only possible because the species entropy production rate can *always* be written as , where denotes the specific entropy production rate of the *pure species*, i.e. in the limiting case when all other species are absent. For polycrystals modelled as mixtures with continuous diversity, such a pure species is evidently a *single crystal* (with crystallographic axes parallel to *n*_{i}^{A}).

### (d) Homogenization rules

The basic strategy of the theory of mixtures with continuous diversity applied to polycrystals is to solve the coupled problem of creep, evolving fabric and recrystallization first on the species level, where a solution is easier to be found. Then, once all species fields are determined, the behaviour of the polycrystal (i.e. of the mixture) can be derived by accounting for the response of all species. The connection between species and mixture responses is set out by certain averaging relations, called *homogenization rules*. Such rules can be derived in a similar manner as done for ordinary chemical mixtures, namely by exploring the additivity of density fields, combined with the expected forms of the mixture balance equations. For polycrystals, such equations are (3.5)–(3.9), together with the mixture balance equations of dislocations and entropy (cf. Groma 1997; Acharya & Beaudoin 2000; Liu 2002),(3.26)where all quantities are defined, as usual, in appendix A.

Homogenization rules appropriate for polycrystals made of transversely isotropic grains have been discussed by Faria & Hutter (2002). Here, we extend those rules to more general crystal symmetries. First, we notice that all species density fields have been defined with respect to a common volume, *viz*. a unit volume of the *mixture*. Consequently, all such fields are additive, in the sense that mixture densities result from the combination of the densities of all species. From this reasoning, we obtain the first three series of homogenization rules, valid for the fundamental density fields (D_{μ}) and their respective production/exchange rate densities (P_{λ} and C_{ζ}),(3.27)The last integral in (3.27) vanishes because of the *conservation* of mass, momenta and total energy of the mixture. The explicit forms of the integrals (3.27) depend on the domains of their respective integrands: for polycrystals in general, we have10(3.28)whereas for the particular case of polycrystals made of transversely isotropic grains:(3.29)Notice that in both instances the integrals are normalized, so that they yield unity when the integrand is just a unit constant.

Now, to derive further homogenization rules we will need the following result.

*Let* *be a compact manifold and* *a vector field of class* *, with* *, k*≥1*, and γ*=1, …, *ν. Then*(3.30)

This proposition is a direct specialization of the divergence theorem in *ν* dimensions (Abraham *et al*. 1988). In order to apply it to polycrystals, we observe first that the diversity completeness of the orientation space implies the *tangentiality* of interspecies fluxes, i.e. —otherwise the interspecies fluxes could reach extraneous species outside —so that(3.31)where D_{μ} is defined in (3.27) and the explicit form of the integral above is given by (3.28) or (3.29). Hence, integration of the balance equations (3.17)–(3.21), (3.24) and (3.25) over the whole orientation space , combined with (3.27) and (3.31), with subsequent subtraction of the resulting expressions from (3.5)–(3.9) and (3.26), leads directly to the last homogenization rules of interest. Such a procedure is described in detail in Faria & Hutter (2002) and references therein. In short, its outcome is the set of homogenization rules for external supplies(3.32)and for stresses and fluxes(3.33)where the relative velocities and are, respectively, called *grain shifting velocity* and *lattice deflecting rate*. Finally, from (3.10), (3.22), (3.27), (3.32) and (3.33) we obtain the homogenization rules for the heat flux and for the density, production/exchange and supply of internal energy, respectively(3.34)

Thus, with the help of (3.27), (3.31)–(3.34) we can recover the balance equations (3.5)–(3.9) and (3.26), for the polycrystal through integration of the species balance equations (3.17)–(3.21), (3.24) and (3.25) over all possible lattice orientations.

The homogenization rules presented in this section are very general, occasionally *too* general even, in the sense that *C*_{i}^{*}=*I*=0 are often reasonable assumptions. Indeed, the velocity *C*_{i}^{*} is only relevant when a pronounced, selective shifting of grains with some particular lattice orientation takes place (figure 4). This is most likely to occur during *superplastic flow* (Poirier 1985). Nevertheless, superplastic flow with *selective grain shifting* constitutes a very particular situation: in most common cases we may set instead, which represents a prodigious simplification to the theory.11 In contrast, it is obvious that we cannot expect to hold in general, because this would imply a ‘frozen fabric’, i.e. no texture development. Nevertheless, it is evident that the *rotational inertia* of the lattice must be extremely small—the lattice does not continue rotating after cessation of the applied torques—so that *I*=0 is valid as a rule.

## 4. Closing remarks

This work presented a general continuum theory for the thermomechanics of large polycrystalline masses, including fabric (i.e. texture) evolution, anisotropic response and recrystallization. It encompasses several symmetry groups, including transversely isotropic and orthotropic, as well as all crystal classes whose symmetries can be described in terms of three orthogonal axes (cf. Liu 1982). The theory is based on the concept of a mixture with continuous diversity (Faria 2001), by regarding the polycrystal as a ‘mixture of lattice orientations’. Its strategy consists in solving the coupled problem of creep, evolving fabric and recrystallization on the species level (i.e. for every ‘component of the mixture’) as described by the system of equations (3.17)–(3.20) and (3.24). Once all species fields are determined, the original initial/boundary-value problem for the polycrystal can thus be solved by application of the homogenization rules (3.27) and (3.31)–(3.34).

It should be observed, however, that the approach studied so far is suitable only to monomineral rocks, in contrast to the assertion in §1 (cf. also remark 2.1). Nevertheless, the extension of the theory for *multi-mineral rocks* is formally direct: it represents what was named by Faria (2001) and Faria & Hutter (2002) a *hybrid mixture with continuous diversity* (*viz*. involving discrete and continuous species labels). Effectively, the essential difference between mono- and multi-mineral rocks is that in the latter case we must deal with *N* phases (every rock-forming mineral being regarded as a particular phase), i.e. with *N* distinct ‘mixtures of orientations’. Thus, in a multiphase polycrystal we need to add a discrete species label *α*=1, …, *N* to every field of the theory presented so far. For instance, the species fields of mass density and Cauchy stress in a rock made of *N* minerals read *ϱ*^{*α}(*x*_{i}, *t*, *n*_{j}^{A}) and *t*_{ij}^{*α}(*x*_{k}, *t*, *n*_{l}^{A}), respectively, with *α*=1, 2, …, *N* and *A*=*1*, *2*. The fundamental equations (3.17)–(3.22), (3.24) and (3.25) remain exactly the same as before (except, of course, for the appearance of the superscript *α* in every field), whereas the homogenization rules (3.27) and (3.31)–(3.34) now incorporate a sum over *α*. For instance, the homogenization rule (3.33) for the Cauchy stress becomes(4.1)Further details about balance equations and homogenization rules for hybrid mixtures with continuous diversity are available in Faria & Hutter (2002).

Finally, the construction of constitutive equations is studied in the subsequent parts of this series (Faria submitted; Faria *et al*. 2005).

## Acknowledgments

This work was conceived during the EPICA-DML 2003/04 deep-drilling expedition in Dronning Maud Land, Antarctica. It was resumed in Brazil, and finished in Leipzig. I am grateful to D. Freche, K. Hutter, G. M. Kremer, P. Shipman and E. Zeidler for suggestions and assistance, as well as to I. Hamann, S. Kipfstuhl, H. Miller and the science group of the mentioned expedition for enjoyable discussions *in situ*. Financial support is acknowledged from the Alfred Wegener Institute for Polar and Marine Research (Bremerhaven) and the Darmstadt University of Technology. This work is a contribution to the ‘European Project for Ice Coring in Antarctica’ (EPICA), a joint ESF (European Science Foundation)/EC scientific programme, funded by the European Commission and by national contributions from Belgium, Denmark, France, Germany, Italy, the Netherlands, Norway, Sweden, Switzerland and the United Kingdom. This is EPICA publication no. 140.

## Footnotes

↵In order to avoid the vocabulary conflict between geology and materials science, the terms ‘texture’ and ‘fabric’ are used here as synonyms to the preferred orientations of the lattice. No particular word is employed in reference to grain sizes and shapes.

↵Here, ‘large’ means ‘big enough to allow the description of fabric through a continuous function’. In practice, we can bluntly estimate it as ‘10

^{10n}crystallites, with*n*≥1’ (cf. Part III).↵The notion of a ‘mixture of orientations’ is well-established in the literature. In rheology, for instance, it has been invoked by Prager (1955), Curtiss (1956), Dahler (1959) and many others (see references in Faria 2001 and Faria & Hutter 2002). Within the context of crystal mechanics, models based on the concept of ‘orientational distribution function’ bear a noticeable resemblance to the present approach (cf. Zhang & Jenkins 1993; Kumar & Dawson 1996; Raabe & Roters 2004), although the analogy between polycrystals and ‘mixtures of orientations’ seems to have been first explicitly exploited by Faria

*et al*. (2003).↵As a matter of fact, the continuity of

*α*renders the precise determination of a single, definite species impossible: only references to an ‘infinitesimal range of species’ d*α*(which includes*α*itself) have strict meaning. Notwithstanding, we adhere here for simplicity, to the common shorthand ‘the species*α*’ when referring to such a ‘species range’ (cf. Aris & Gavalas 1966; Faria*et al*. 2005).↵The ± sign in (2.1)

_{3}stands for +1 for a right-handed triad and −1 for a left-handed one. Of course, both triads are related by an inversion transformation. In this sense, the symbol ‘±’ can be interpreted as an axial unit scalar, which ensures that*n*_{i}^{3}is an absolute vector.↵Roughly, polar media are microstructured continua characterized by couple stresses, body couples, and additional degrees of freedom subsumed in an intrinsic angular momentum called

*spin*(Dahler & Scriven 1963; Truesdell & Noll 1965; Capriz 1989; Svendsen 2001).↵The cited authors considered only the particular case of transverse isotropy, but the procedure is exactly the same for more complex symmetries considered here.

↵Here, ‘conventional constitutive equation’ means a nonlinear, constitutive function (or functional) of the same general type supposed to hold for

*ς*^{*},*η*^{*},*e*^{*}, etc. (cf. Faria*et al*. 2005).↵This result is trivial for ordinary mixtures, but not for mixtures with continuous diversity, since in the latter case the entropy production rate of the mixture is given by a non-conventional, integral relation (see (3.27)) that can hardly be exploited on the species level.

↵Application of (3.28)–(3.27) implies , etc.

↵The assumption

*v*_{i}^{*}=*v*_{i}has no relation at all to artificial constraints on the strain of individual grains (e.g. Voigt–Taylor/Sachs–Reuss upper/lower bounds, cf. Asaro (1983), Humphreys & Hatherly (2004)). In the present theory, all crystallites may undergo arbitrary deformations, since each material particle is large enough to contain a huge number of grains. Thus, stress and strain inhomogeneities on the grain level are already smeared out in the definitions of*v*_{i}^{*}and*t*_{ij}^{*}, which describe the average response of the grains belonging to a given species (cf. Part III).- Received October 25, 2004.
- Accepted November 8, 2005.

- © 2006 The Royal Society