## 1. The comment and our reply

As explained in Gagliardini's comment (Gagliardini 2008), from now on simply referred to as ‘the Comment’, the papers by Faria (2006*a*,*b*) and Faria *et al*. (2006) present an application of the thermodynamic theory of mixtures with continuous diversity to the modelling of creep and recrystallization of large polycrystalline masses. In this theory, the polycrystal is regarded as a continuum composed of a huge number of crystallites (grains) in such a manner that each material particle of this continuum1 still contains a large amount of grains within it. Grains with the same lattice orientations constitute what is called a *species*.

The main argument of the Comment is summarized in its last paragraph, here reproduced verbatim (our remarks appear within square brackets).

The FFT results in Lebensohn

*et al*. (2004) [are] clearly contradicting the statement made by Faria and co-authors that stress and strain rate of a species should be independent of its orientation due to the huge number of grains belonging to the same species. The assumption made by Faria and others is not insignificant and is comparable to a uniform strain rate or Taylor-type assumption in the framework of the homogenization model.(Gagliardini 2008)

The following is our reply.

Faria and co-authors have never stated or derived any result suggesting that the species stress should be independent of orientation, so

*Gagliardini's deductions are irrelevant*.Gagliardini's definition of strain rate of a species is different from the definition used by Faria and co-authors. This inconsistency renders his consequent conclusions unfounded. In particular, his direct comparison of the representative volume element (RVE)-phase-averaged strain rate2 introduced by Lebensohn

*et al*. (2004) with the species strain rate used in Faria (2006*a*,*b*) and Faria*et al*. (2006) is invalid because these are distinct quantities. Likewise, his claim that ‘the assumption made by Faria and others […] is comparable to a uniform strain rate or Taylor-type assumption’ has no basis.

More detailed discussions of these points appear below.

## 2. Details

The Comment (Gagliardini 2008) can be epitomized in six statements as follows (in verbatim quotations, our remarks appear within square brackets).

If

represents the average value of a physical quantity inside a grain, then its species average is defined by , where denotes averaging over one species (called ‘per species average’ by Gagliardini). The polycrystal (‘macroscopic’) average follows as , where 〈.〉 denotes averaging over all species (i.e. all crystallographic orientations).*y*The hypothesis of negligible grain shifting assumed by Faria and co-authors is that the strain rate of every species

*D*^{*}is equal to that of the polycrystal, i.e. , where*D*is the strain rate of a grain.*d**Verbatim quotation*. Faria and co-authors argue that ‘this assumption [of negligible grain shifting] is negligible [sic] because in their theory stress and strain inhomogeneities on the grain level are smeared out and the average response of the grains*belonging to a given species*[our emphasis] does not depend any more on the species orientation [sic]’ (see footnote 11 in Faria 2006*a*).*Verbatim quotation*. ‘According to Faria, all crystallites may undergo arbitrary deformations, but the orientation independence of the per-species averaged strain rate arises because each species is composed of a very large number of crystals.’Gagliardini takes for granted that the species fields of strain rate and stress defined in Faria (2006

*a*,*b*) and Faria*et al*. (2006) are identical to the RVE-phase-averaged fields introduced by Lebensohn*et al*. (2004).*Verbatim quotation*. ‘A polycrystal law derived from the theory [presented in Faria (2006*a*,*b*) and Faria*et al*. (2006)] is implicitly built assuming an orientation-independent strain rate, as shown by eqn (3.3)_{2}() in Faria (2006*b*), resulting in an expression for the stress equivalent to that obtained using a Taylor-type assumption in the framework of the homogenization model.’

Now we note the mistakes in these assertions.

*Ad 1*. The definition of average used in the Comment is unclear and misused. It is well known that many types of average exist in mathematical physics and generally one cannot average indiscriminately*any*physical quantity with the same averaging formula (or homogenization rule); there are, for example, extensive and intensive quantities, densities and fluxes, primary and derived quantities, each requiring a particular type of averaging. This is evident, for instance, in the many homogenization rules introduced by Faria (2006*a*), e.g. eqns (3.27)–(3.34).3 Thus, as discussed below, the averaged fields used in Gagliardini (2008) are not consistent with the fields introduced in Faria (2006*a*,*b*) and Faria*et al*. (2006). Consequently, the assertions in the Comment are unfounded.*Ad 2*. The description of ‘negligible grain shifting’ presented in the Comment (Gagliardini 2008) is wrong and has never been invoked by Faria (2006*a*,*b*) or Faria*et al*. (2006). The mistaken description of this concept in the Comment is derived from the aforementioned misuse of averages combined with a disregard of the genuine definition of ‘grain shifting’ proposed by Faria (2006*a*,*b*) and Faria*et al*. (2006).

In simple terms, the concept of grain shifting can be roughly understood as ‘transport of certain grains through the polycrystalline matrix’ or, in other words, ‘relative motion of certain grains with respect to others’ (fig. 4 of Faria 2006*a*). In practice, such relative motions are usually related to extensive grain boundary sliding, diffusion creep or other similar mechanisms. Mathematically, Faria (2006*a*) has defined grain shifting as the relative velocity field (see text below eqn (3.33) in Faria 2006*a*)(2.1)where *v*_{i} is the barycentric velocity field of the polycrystal and is the barycentric velocity field of a given species (i.e. a group of grains possessing the same lattice orientation). Thus, negligible grain shifting simply means . The analogy between grain shifting in large polycrystalline masses and diffusive motion in multiphase or multicomponent mixtures is therefore obvious.

At this point, we may identify the key mistake in Gagliardini's arguments: he disregards the fact that the works by Faria (2006*a*,*b*) and Faria *et al*. (2006) deal with a *continuum theory involving a single size scale*. Microstructural effects are captured in the theory through the introduction of an additional abstract space of species (*viz*. lattice orientations), and not by increasing the spatial resolution of the physical space down to the (sub-)grain scale, as in statistical approaches (e.g. Lebensohn *et al*. 2004). Consequently, all homogenization rules in Faria (2006*a*,*b*) and Faria *et al*. (2006) define averages over the space of orientations *and not* over a ‘microscopic’ physical space.

In order to make this remark more precise, let us be reminded that to each material particle in the continuum theory by Faria and co-authors we can associate only one spatial position, denoted by the position vector *x*_{i}. Consequently, since every material particle is supposed to contain a large number of grains, any attempt to introduce a quantity like ‘the strain rate of a grain ** d**’ is

*logically invalid*, seeing that such a strain cannot be mathematically defined within the context of the theory (figure 1). By the same token, the claim that the species strain rate

*D*^{*}used in Faria (2006

*a*,

*b*) and Faria

*et al*. (2006) obeys the identity is false, no matter which kind of esotheric averaging operator 〈.〉

^{*}Gagliardini may have in mind. Actually, from the structure of the continuum theory presented in Faria (2006

*a*,

*b*) and Faria

*et al*. (2006), and the fact that the strain rate is just the symmetric part of the gradient of velocity, we immediately conclude that the correct definition of the species strain rate

*D*^{*}is (in Cartesian components)(2.2)which is clearly distinct from Gagliardini's implicit average and from the RVE-phase-averaged strain rate used in the statistical theory of Lebensohn

*et al*. (2004).

*Ad 3*. This assertion is unfounded for two reasons. First, as stated in §1, Faria and co-authors have never stated or derived any result suggesting that the species stress should be independent of orientation. Second, it can be verified below that the notions really discussed in footnote 11 of Faria (2006*a*) are conspicuously different from those spuriously attributed to it in the Comment (below, the quantity denotes the Cartesian components of the Cauchy stress tensor of a particular species).

The assumption has

*no relation at all* to artificial constraints on the strain of individual grains (e.g. Voigt–Taylor/Sachs–Reuss upper/lower bounds, […]). 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 and , which describe the average response of the grains belonging to a given species.(Faria 2006

*a*, footnote 11)

Admittedly, the last sentence of the afore-cited footnote is not sufficiently clear and may allow misinterpretation. The original intention of that sentence was to summarize how the theory of continuous diversity deals with the following fact (here, we focus on polar ice because this is the material Gagliardini is interested in, but the same arguments hold also for other large polycrystalline masses): recent microstructural studies of polar ice cores have evidenced the high inhomogeneity of the *intracrystalline* deformation of polar ice (e.g. Kipfstuhl *et al*. 2006). According to these studies, the natural deformation of polar ice is mainly characterized by stress concentrations at grain boundaries, leading to continual formation of dislocation walls and subgrain boundaries, subgrain rotation, etc. Thus, the internal structure of polar ice grains is very intricate and heterogeneous, and may notably contribute to the accommodation of deformation. These observations highlighted the inadequacy of polar ice deformation models based on the assumption of nice homogeneous grains without internal structure. By contrast, the theory developed by Faria (2006*a*,*b*) and Faria *et al*. (2006) does not suffer from such drawbacks because it is a ‘large-scale’ approach, that is, such grain inhomogeneities caused by deformation occur on size scales that are orders of magnitude smaller than the size of a single ice-sheet particle (figure 1). Thus, just as individual molecular interactions cannot be resolved by, and are intrinsically incorporated into, the fields of the continuum theory of viscous and viscoelastic media, the inhomogeneities of individual grains caused by deformation cannot be resolved by, and are intrinsically incorporated into, the fields of species stress , species velocity , species strain rate , etc. of the theory presented by Faria (2006*a*,*b*) and Faria *et al*. (2006; cf. figure 1).

*Ad 4*. False. As already discussed, the orientation independence of the species strain rate is a consequence of the hypothesis of negligible grain shifting and the fact that each*material particle*has a large number of grains (figure 1). Whether a species contains just a few or many grains does not matter at all.*Ad 5*. As already commented in our replies to items 2 and 3, the species fields of strain rate and stress defined in Faria (2006*a*,*b*) and Faria*et al*. (2006) are*not*equivalent to the RVE-phase-averaged fields used by Lebensohn*et al*. (2004). Microstructural effects are captured in our continuum theory through the introduction of an additional abstract space of species (*viz*. lattice orientations), and not by increasing the spatial resolution of the physical space down to the (sub-)grain scale, as in the statistical approach of Lebensohn*et al*. (2004).*Ad 6*. In our notation,4 the below mentioned eqn (3.3)_{2}in Faria (2006*b*) reads (Einstein summation convention is adopted for repeated indices)with , where stands for the components of a species-dependent viscosity tensor and*δ*_{ij}is the Kronecker delta. Again, Gagliardini's statement is false, because eqn (3.3)_{2}in Faria (2006*b*) is*not*a ‘Taylor-type’ constitutive relation, seeing that the quantity above is neither the stress acting on a grain nor an RVE-phase average of it (cf. footnote 2). Rather, as explained in figure 1 (see also fig. 3 of Faria 2006*a*), the quantity denotes the components of the Cauchy stress that drives the large-scale flow of a particular species, according to the balance equation of linear momentum (3.18) of Faria (2006*a*).

## Acknowledgments

The authors thank Leslie W. Morland and Sir Michael Berry for their discussions. S.H.F. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG) Priority Program SPP-1158 grant FA 840/1-1.

## Footnotes

The accompanying comment can be viewed on page 289 or at http://dx.doi.org/doi:10.1098/rspa.2007.0187.

↵To facilitate the comparison of this work with cited references, note that in homogenization and statistical approaches the term ‘material particle’ is often replaced by its counterpart ‘infinitesimal volume element’ or ‘representative volume element’ (RVE).

↵By

*RVE-phase average*, we mean a statistical average in the sense used by Lebensohn*et al*. (2004),*viz*. a volume average over a ‘phase’*within*a suitable RVE of the polycrystal, where a phase is a region within the RVE characterized by a particular lattice orientation.↵As a simple example, let the notation hold for the dislocation density

*ρ*_{D}. Thus, from eqns (3.27) and (3.34) of (Faria 2006*a*), we conclude that the identity is*not true*for the components of heat flux vector*q*_{i}, except in pathological situations.↵The core of the Comment (Gagliardini 2008) is a comparison of Lebensohn

*et al*. (2004) with Faria (2006*a*,*b*) and Faria*et al*. (2006). In all these works, the deviatoric (i.e. symmetric traceless) part of the Cauchy stress is denoted by. For some unimportant reason, the symbol*σ*has replaced*S*in the Comment. In this reply, we do not follow Gagliardini's notation, but rather maintain the original notation*σ*, in order to facilitate comparison with the afore-referred works.*σ*- Received May 9, 2008.
- Accepted May 30, 2008.

- © 2008 The Royal Society

## References

## Note from the Editor

Through an oversight, the authors of this Invited reply were not given the opportunity to respond to the Comment on their papers before it was published. I apologise for this discourtesy. Michael Berry.