A Critical Review of the Mechanics of Polycrystalline Polar Ice

GAMM-Mitt. 29, No. 1, 80 – 117 (2006)

A Critical Review of the Mechanics of PolycrystallinePolar Ice

Luca Placidi∗ 1, Kolumban Hutter2, and Sergio H. Faria3

1 Laboratorio di Materiali e Strutture Intelligenti, Palazzo Caetani (Ala Nord), Cisterna diLatina, Italy

2 Bergstraße 5, 8044 Zurich, Switzerland, formerlyDepartment of Mechanics, Darmstadt University of Technology, Hochschulstr. 1, D-64289Darmstadt, Germany

3 Max Planck Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103 Leipzig,Germany

Received 15 July 2004, revised 25 May 2005

Key words Ice sheet, Ice, Fabric, Texture, Recrystallization, Polygonization.MSC (2000) 86-02, 86A40, 74A10, 74E15, 74E25

We review the developments of the constitutive theories of polar ice both from a viewpointof mechanics as well as materials science, as it was studied during the last approximately 50years. First proposals were based on the postulation of ice as a very viscous, non-Newtonianand heat conducting fluid. Today’s understanding of polar ice sees it as a creeping polycrystalwith induced anisotropy and a nontrivial grain-size distribution, both evolving in time. Phe-nomena related to recrystallization, namely grain rotation, grain-size redistribution by normaland abnormal grain growth, nucleation, polygonization and grain boundary migration havealso to be accounted for to obtain a stress constitutive law that may reliably predict ice flowin large ice masses. The main topics on which we focus our efforts are the role of anisotropyand its evolution. We review the models that take into account the microscopic as well as themacroscopic point of view. Numerical models are also considered. A novel graphical methodis proposed for the comprehension of the rules that drive the rotation of the crystallites ina polycrystal, not only for the simple cases of compression and extension but also for thesimple shear mode of deformation. Particular emphasis is given to the constitutive theoriesrelated to recrystallization phenomena, for which the paper can be considered an introductoryoverview. The research methods on which all proposed laws are based borrow concepts fromthe mechanics of solids and materials science but belong also to the class of anisotropic fluids.

c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

The reconstruction of the climate on Earth from time series of the isotope ratios of O2, D,CH4, CO2, and other trace gases (see e.g. Fig. 1), which are determined by analyzing theice and the air trapped within it from ice cores of Antarctica, Greenland and several glaciers,requires knowledge of the age of an ice sample that was located at a certain position in anice sheet, Fig. 2. The age of the sample can be roughly defined as the difference between

∗Corresponding author; e-mail: [email protected], Phone: +39 06 44 585 297, or +39 3294936262 Fax:+39 06 48 848 52


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the times when the ice is analyzed in the laboratory and when it fell as snow on the surfaceof the ice sheet or glaciers and started its motion along its own trajectory. Reconstructing theclimate in the past approximate 500 000 years thus boils down to the determination in an icecore of the depth-age relationship, h = h (t), that allows the transformation of an isotope-depth relation f = f (h) to an isotope-age relation f = f (t). Since the Deuterium D and thecarbon-dioxide CO2 isotope data in ice are pretty good proxy-data of the mean temperatureon Earth (approx. 15◦C, today), ice cores are ideal objects to reconstruct the climate throughthe last ice ages.

Down to depths of 1500m or somewhat more (in Antarctica and Greenland) the f (t)-relation can fairly reliably be determined by methods of stratigraphy (e.g., counting layers,performing electrical conductivity measurements on the ice core, correlating dark impuritylayers with volcanic eruptions, etc.). At larger depths the annual layers are so thin (< 100μm)that computational methods, using, e.g., continuum mechanical methods, are needed to inferfrom them the age of the ice. This can be done by either solving the age equation A = 1(A for age, see [56]), more explicitly,

∂A

∂t+ (∇A) · v = 1, (1)

Fig. 1 Time series of (a) atmospheric CO2, (b) the surface temperature (deviation from today) recon-structed from D, (c) atmospheric CH4, (d) δ18Oatm of atmospheric oxygen and (e) the insolation at65N as obtained from the Antarctic Vostok Ice core [102] for a time span of 420, 000 years into thepast. Courtesy of Nature, reproduced with permission.


82 L. Placidi, K. Hutter, and S.H. Faria: A Critical Review of Polycrystalline Ice

or by determining the ice particle trajectories from

dxdt

= v (x, t) , x (t0) = x0. (2)

Both equations require knowledge of the velocity field v through time t, and this is obtainedby determining the motion of the ice through space for all times. There are many subtletiesthat make this problem a very challenging one. Among these, a significant contribution to theuncertainty of the h (t)-relationship is an inadequate knowledge of the material behaviour ofice under thermomechanical creeping processes on geological time scales.

Classically, ice of the Earth’s large ice masses1 has been treated as a heat conducting andnon-Newtonian fluid with a power law viscosity, strongly dependent on temperature (see e.g.equation (5) and the parameterization shown in (6)). The absence of privileged directionsin the distribution of lattice orientations is generally assumed and the isotropic mechanicalresponse is inferred. On the other hand, fabric profiles in many ice cores show a transition withdepth of the ice from isotropy to strong anisotropy and often, at very large depths, a near returnto isotropy. Fig. 3 shows data through the GRIP2 ice core in Greenland with measured profilesfor the mean grain size and the Schmidt diagrams determined from analyses of thin sectionsof the samples. An anisotropic constitutive model that was constructed for this situation [87]showed that at a certain depth (in GRIP below approx. 2800m) differences in age of theice, computed with an isotropic and anisotropic stress-stretching relationship may differ byseveral tens of thousands of years (40000 to 100000 years, depending on basal topography).This makes the development of the theories regarding the material response of polycrystallinepolar ice an urgent and extremely challenging research topic.

(a) (b)

Fig. 2 (a) Ice specimen at a certain depth of an ice core. (b) Ice sheet and ice shelf bounded by theatmosphere, the ocean and the solid earth. The figure also shows the trajectories of the ice which falls assnow onto the free surface and is subsequently transported within the ice until it surfaces again and thenmelts or breaks off. (Courtesy: R. Greve and R. Calov [68], reproduced with permission).

1A useful reference about the dynamics of planetary ices can be found in this volume, see [57]2Greenland Ice Core Project, drilling the core in the years 1990-1992. The core was taken at summit, the highest

point of the Greenland ice sheet. It is 3029 m long.


GAMM-Mitt. 29, No. 1 (2006) 83

This paper reviews the efforts that have been taken since the early 80s of the last century byothers and ourselves to arrive at an adequate constitutive relation of a very complex anisotropicfluid. Our review comprises works up to and including 2003. It demonstrates that there is veryactive research going on in the geophysical sciences that use the most advanced methods ofmechanics and materials sciences.

What is presented in Section 2 is a review of the various flow laws proposed for isotropicice, with emphasis on the parameterization of the Glen–Steinemann flow law in the geophys-ically relevant regimes of temperatures (210-273.15 K) and stresses (1-5 bars). The influenceof the grain size (Section 3) on the flow law is reported and its parameterizations are dis-cussed. The law accounts only for a mean grain diameter. Studies on the GRIP ice core byTh. Thorsteinsson [123] have, with good accuracy, shown that grain sizes are log-normallydistributed which ought to be accounted for. The influence of lattice orientations is also anal-ysed in Section 3. In early studies of the flow law for polar ice, the orientation of the crystal-lites, the dust content, grain growth and grain size are only accounted for in the isotropic flow

Fig. 3 Data from the Greenland Ice Project (GRIP) drill core, 1990–1992. Diagrams depict (a) meangrain size, (b) orientation of the c-axis of the crystals, (c) degree of anisotropy (a/a0 = 1 correspondsto isotropy, a uniform distribution of fabrics over the entire Schmidt circle, a/a0 = 0 correspondsto a single maximum fabric), and (d) temperature. Each point inside the circles shown in (b) is thevertex of a unit vector designating the c-axis direction of a crystal grain (a so called Schmidt diagram).from [123] with modifications. [Courtesy of Alfred Wegener Institut fur Polar- und Meeresforschung,Bremenhaven, reproduced with permission.]



law by an enhancement factor, thus not accounting for the nonuniformity of the orientationdistribution in a Representative Volume Element (RVE).

In order to model anisotropy induced by the applied load, both the Taylor and the Sachsassumptions in microscopic models have been used in the literature; thus, either the strainincompatibility or the stress incompatibility are avoided, but generally not both (Sections 4).

Macroscopic models, which assume a statistical distribution of c axes within the RVE aredeveloped in glaciology since approximately 1995. They are based on the assumption of theexistence of an Orientation Distribution Function (ODF), which represents the probabilitydensity of finding a crystallite with its c-axis orientation within a given solid angle. The-oretical models then consist of an evolution (Fokker–Planck) equation for the ODF, a set ofconstitutive equations for the macroscopic quantities, including a certain number of anisotropytensors of second and fourth order, and a thermodynamic exploitation for the proposed con-stitutive class (Section 5).

These models account for crystal rotation but not anisotropy produced by recrystallization,i.e. by normal and abnormal grain growth, by nucleation, polygonization and Grain BoundaryMigration (GBM). For this part of anisotropy the linear defects in the crystal lattice (namelydislocations) are responsible. This requires the description of the evolution of the dislocationdensity, whereby its growth rate must be attributed to the stretching, whilst its wastage is due tograin boundary migration and polygonization and both also depend on grain size (Section 6).

This is the state of the art that we shall further detail in the text that follows.For the sake of simplicity, when Cartesian tensor notation is used, the usual summation

rule for repeated indices will be implemented with which the reader is supposed to be fa-miliar. Parentheses, square brackets and angular parentheses define rules of symmetrization,antisymmetrization and traceless symmetrization, respectively, defined for a general tensorquantity Ahij , i.e.,

Ah(ij) ≡ 12 (Ahij + Ahji) , Ah[ij] ≡ 1

2 (Ahij − Ahji) , Ah〈ij〉 ≡ Ah(ij)− 13Ahkkδij . (3)

2 The Legacy of Isotropic Ice

In order to describe the flow of an ice sheet, polycrystalline ice is often considered as anisotropic very viscous fluid. This led to the celebrated Glen (or Glen–Steinemann) flow law[41, 119] and its representation as a three-dimensional viscous power-law fluid [99],

Dij = A (T ) σn−1 t〈ij〉, (4)

valid for effective stresses in the range of 100 to 300 kPa where Dij is the stretching (strainrate), t〈ij〉 the deviatoric part of the Cauchy stress tensor tij , 2σ2 = tijtij is related to thesecond invariant of tij , and A (T ), a temperature dependent parameter, is also called the ratefactor. Hence, as already observed by Steinemann [119] almost 50 years ago, the exponent ndepends on σ (see also [2, 27, 33, 119, 134]). Besides this awkward dependence of the powerlaw exponent on stress, Equation (4) suffers from its infinite apparent viscosity at zero stressdeviator, η = dt〈ij〉/dDij → ∞ (no sum over i,j) as tij → 0. In laboratory experiments,when transient creep has not been fully eliminated, the apparent viscosity is a finite quantity;furthermore, this singularity would cause subtle difficulties whenever shallow ice asymptoticsare considered [56]. Thus, it transpires that Glen’s flow law need be generalized. Among


GAMM-Mitt. 29, No. 1 (2006) 85

others, Lliboutry [83], Colbeck and Evans [17], Hutter et al. [66, 67, 69] have amended thesimple Glen’s flow law by writing

Dij = A (T )f (σ) t〈ij〉, (5)

with f(σ) > 0 for all σ – called the creep response function or fluidity – obeying the finiteviscosity requirement 0 < f(0) < ∞ and f(σ) otherwise chosen to match experiments in theentire stress range of 0 to 1 MPa. Parameterizations for the fluidity f and for the rate factor Aare

f (σ) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

a σ4 + b σ2 + η−1,

σn−1 + η−1,

1|σ| sinhn (ασ) + η−1,

A (T ) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

A0 exp(− Q

kT

),

2∑j=1

Aj exp [αj (T − TM)] ,

B0 exp{

T0

T− C

(Tr − T )s

},

(6)

in which bounded η �= 0 accounts for Newtonian behaviour at zero effective stress; k is theBoltzmann constant3, T the absolute temperature and TM its melting point temperature, while

Fig. 4 Steady state creep for the secondary flow of polycrystalline ice in uniaxial compression againstapplied stress. � −2◦C, � −8◦C, •−14◦C, −22◦C, � −34◦C, ◦−48◦C. The curves are calculated via sinh

functions based on all experimental points between −2◦C and −48◦C. (from Hutter [67] according to Barnes etal. [9]).

3This requires Q to be measured in electron volts; if Q is measured in joules, then k is replaced by R, theuniversal gas constant.



a, b, n, α, Aj , αj , B0, Tr and s are temperature-independent parameters to be identified byexperiments. Specifically, the polynomial law, the first line of (6)1, was recommended byLliboutry [83] in 1969, while the second line of (6)1 is a slight modification of (4) to accountfor finite apparent viscosity. The power-sinh law in the third line of (6)1, also known as theGarofalo law [40], matches uniaxial compression tests by Barnes et al. [9] very well in theentire stress range 0 < σ < 1 MPa (see Fig. 4). The Arrhenius relationship in the first line of(6)2 represents measurements of Mellor and Testa [93] only for T < 263 K, whilst the secondline of (6)2, due to Morland and Smith (see e.g., [67]) is valid also for higher temperatures.In 1981, Hooke proposed a different parameterization for A (T ), represented by the third lineof (6)2.

As shown by Hutter [67] it is possible to give a formal derivation of (5) from first principlesof continuum mechanics by assuming that the polycrystalline ice is an incompressible non-linear viscous fluid. To this end, two other assumptions are necessary: the third invariant of thedeviatoric stress has no effect on the material response and the stress deviator must be collinearto the stretching tensor. The validity of these two assumptions is doubtful [8, 42, 67, 100, 131];however the precision of the measurements is not clear and a conclusion that (5) would beinappropriate seems to be premature.

Notwithstanding, in attempts to achieve a better parameterization of the flow law (5), mea-surements often disagree with previous investigations, and so such contradictions still mustbe explained. The reason seems to be the high number of variables that consistently affectequation (5), and these variables must be found in the microstructure. Impurity content, rockparticles, lattice orientations and grain size of all crystallites can in fact play a significantrole. An important requirement in deformation studies is to understand the influence of suchvariables to acquire knowledge of the conditions in which they occur.

The usual method is to develop constitutive relations for the various microscopic defor-mation processes and to construct the so called deformation-mechanism map to delineate theprecise set of conditions where each individual mechanism is dominant [77]. Generally, it isconvenient to plot the macroscopic variables (e.g. temperature, deviatoric stress, strain rate) asthe axes for such maps, holding the microscopic variables (e.g. grain size, dislocation densityetc.) constant [51]. A first classification of deformation mechanisms differentiates betweenboundary and lattice mechanisms. The boundary mechanisms are related to the behaviour ofthe grain boundary and therefore to grain size. The lattice mechanisms refer to the behaviourof the crystalline structure, and since this presents a strong anisotropy for ice, a randomlessdistribution of the lattice orientations destroys the isotropic structure of the flow law (5).

3 Grain Size and Lattice Orientation

3.1 The Influence of Grain Size

The concept of crystal-size-dependent rheology is not new in glaciology. In fact, in 1971Barnes et al. [9] suggested a grain-size-dependent deformation mechanism for temperatureshigher than 263 K and attributed it to Grain Boundary Sliding (GBS). The experimental dataof [9] indicate a marked softening of polycrystalline ice above that temperature, and a similarbehaviour does not occur in creep of a single ice crystal. This and the formation, in that rangeof temperature, of liquid phase pockets at grain boundary sites suggest that the softeningmechanism might be associated with GBS.


GAMM-Mitt. 29, No. 1 (2006) 87

Moreover, in 1997 Goldsby and Kohlstedt [48] published results based on innovative labo-ratory techniques for fabricating ice samples with very fine grain size in such a way to be ableto parameterize measurable4 strain rates in terms of grain size. They proposed a new flow lawfor ice, namely a macroscopic superplastic law5 of the form

Dij = A (T )σn−1

Dpt〈ij〉, (7)

in which D is the grain size and the power exponent p is positive. This flow law is still ofGlen-type, i.e., it only applies to isotropic polycrystalline ice. Further investigations on grain-size control of the strain rate were conducted by the same authors in 2001 [50] and in 2000by Cuffey et al. [18, 19]. These last two papers are interesting because they investigate theconnection of the flow law with the impurity content and rock particles; this is also the subjectin Thorsteinsson et al. [125] (see e.g. eq. (20)).

The development of a grain-size-dependent flow law, such as equation (7), has alwayscome parallel with investigations of its microscopic interpretation, about which there is noagreement in the glaciological community (see also [29, 49]). The main criticism concernsthe compatibility between such a mechanism and the formation of fabric, i.e., the change ofthe distribution of lattice orientations. Moreover, in 1995 de la Chapelle et al. [23] interpretedthe softening due to the formation of the liquid phase at grain boundaries as an attenuationof the internal stress field that should promote glide on basal planes, see also [25]. This isa deformation mechanism that is different from the sliding of grain boundaries, but it is themost important mechanism describing the creep deformation of ice. At present, it seems thatdeformation is indeed dominated by lattice mechanisms (basal glide) but an influence of im-purities and some amount of GBS (yet not enough for superplastic flow) cannot be discarded.As the microscopic mechanism of crystal glide on basal planes is considered responsible forthe nonuniform distribution of lattice orientations, we shall investigate the latter in the nextSection.

Furthermore, we remark that, even if the main microscopic deformation mechanism can beclassified as a lattice mechanism, the role of the dislocation density ρd must not be forgotten,and, indeed, explicit evolution equations for ρd have always had grain size as one of the fun-damental governing parameters (see e.g. subsection 6.3). Moreover, from a microscopic pointof view, grain size varies abruptly from a crystallite to its neighbours; thus, the phenomeno-logical description will obviously become very complicated if all grain sizes are accountedfor. Generally, this difficulty is overcome by accounting in a Representative Volume Element(RVE) only for the mean value of the grain size of the crystallites present in the RVE. It is,however, the correct belief that this approximation is nevertheless too rough and one is forcedto deal with the whole distribution of grain sizes. Measuring the grain-size distribution, alsocalled texture, is now a common procedure in investigations of ice cores [123]. Generally, thedistribution n�(D) of grain sizes follows a log-normal distribution. Analyses of the propertiesof this kind of distribution can be found, e.g., in the 1988 review by Marsh [90]. A theoret-ical investigation on grain-size distributions applied to the case of ice and of polycrystallinematerials can also be found in [103].

4We can observe from (7) that the higher the strain rate, the smaller will be the grain size.5The terminology plastic and viscous is not as strict in materials science as it is in mechanics, where (7) would

be called viscous.



3.2 The Influence of Lattice Orientations

It is widely accepted that the reason of the mechanical anisotropy of polycrystalline ice liesin the microstructure and comes from the strong anisotropy of the monocrystals (see e.g.[11, 27, 33, 60, 88, 91, 100, 127, 131, 134]). Thus, the most important characteristic, generallybelieved to affect the flow law of polycrystalline ice, is the distribution of lattice orientations.If it is uniformly random, then the behaviour is isotropic and the framework of (5) or (7)is useful. On the other hand, deformations induce a change in such a distribution and ananisotropic flow law must be formulated. So, whereas formulas (5) or (7) indeed may matchan individual stretching component with an individual stress deviator component, their three-dimensional extrapolation is simply false. Nonetheless, early attempts to account for thedifferences of the fluidity in Holocene and Pleistocene ice maintained (5), amending it toincorporate an enhancement factor6 E, viz.,

Dij = EA (T ) f (σ) t〈ij〉, (8)

and attributed to E all those properties one did not properly understand but is forced to accountbecause of obvious observational evidence [2, 3, 7, 11, 15, 33, 37, 54, 55, 65, 80, 81, 86, 100,104, 123, 124]. An early internal variable theory was also proposed to thermodynamicallyjustify the introduction of E [70]. The failure of this kind of fudging procedure in the analysesof the influence of lattice orientations was quickly realized as with (8) no directional variationof the effective fluidity can be modelled. So, (8) is also a false three-dimensional amendmentof the classical Glen flow law.

Single crystals deform in fact by slip along the crystallographic planes (Fig. 5), and forusual temperatures (210K < T < 273.15K) the slip resistance is up to 60 times smallerparallel to the basal planes than to the other slip planes of the system. Thus, as commented byMcConnel [91] in 1891:

“[...] the crystal behaved as if it consisted of an infinite number of indefinitely thinsheets of paper, normal to the optic axis, attached to each other by some viscous

ccc

Fig. 5 Sketch of a single ice crystal in the form of a right hexagonal prism. Three crystallographicplanes are indicated. The optic axis is indicated here by c.

6The strain rate enhancement or enhancement factor is the ratio between the measured strain rate and the strainrate as considering by isotropic behaviour, i.e., the Glen flow law.


GAMM-Mitt. 29, No. 1 (2006) 89

substance which allowed one to slide over the next with great difficulty. This com-parison proved to be the key to the whole question of the plasticity of a crystal ofice.” (p. 325)

Following this simple idea, which has since then been repeatedly confirmed by experiments[27, 43, 60, 61, 89, 110, 119, 136] it is possible to comprehend the failure of the isotropyassumption through the gedankenexperiments illustrated in Figs. 6 and 7.

The “thin sheets of paper” mentioned by McConnel can be envisaged as cards in a pile,with their normal vectors, representing the c-axis of the crystal, initially inclined by someangle (φ0 in case (a), θ0 in case (b) of Fig. 6) with respect to the vertical. Independently ofthe value of this initial angle (φ0 or θ0), the c-axis of the crystal will tend to rotate towards theprincipal axis of compression, whenever a load is applied on the crystal. On the other hand, itis easy to show by a similar gedankenexperiment, see Fig. 7, that the c-axes of the crystallitestend to rotate away from the principal axis of tension. This means that there is a connectionbetween the sense of the rotational velocity of the crystallites and their strain rate.

As far as we have explained, only the deformation mechanism of glide along basal planesis coherent with the development of fabrics. On the other hand, the superplastic flow lawrepresented in (7) is not compatible with such a mechanism. To overcome this problemGoldsby, Kohlstedt et al. [48, 101] suggested a compromise between the different mecha-nisms. However, a crucial experiment on the basis of which one could identify a mechanism

φ0

W

Present configurationInitial configuration

c

φ

c c

W

θ

c

(a)

(b)

0θW

W

Fig. 6 An ice crystal as a “deck of cards”: when subjected to vertical compression due to the load Wthe crystal deforms by basal slip, forcing a rotation of its c-axis towards the principal compressive axis(vertical), regardless what the starting orientation of its c-axis in the initial configuration may be. In spiteof some differences, the grains of a polycrystalline piece of ice are expected to deform in a somewhatsimilar manner, building so a concentration of c-axes about the vertical (called single-maximum/single-pole fabric).



that justifies equation (7) has not yet been found. We claim that a justification of (7) in termsof a microstructure process is not clear, but its validity in a certain range of the deformationmechanism map is supported by experimental data.

Mechanical tests on single crystals disclose behaviour similar to that of (5) for the com-ponents relative to simple shear (i.e., D13 or D23 with the third component of the frame ofreference parallel to the c-axis of the crystal). Models and experiments of ice single crys-tals [27, 33, 60, 134, 136] lead in fact to creep rate equations having power law form andan Arrhenius type temperature dependence: since viscous sliding of the “cards” is driven bythe resolved shear stress on the basal plane τ , the strain rate is proportional to τn, and theexponent is preferably n = 2 and less convincingly n = 3 [27, 43]. This supports, for apolycrystalline ice sample, the supposition that the controlling mechanism of deformation forT < 263K should also be basal sliding via dislocation glide. However, this raised for a longtime a number of objections [28, 43, 60, 134]. Indeed, as we will discuss in the next Section,the mere mismatch of the crystal orientations in a polycrystal seems to suggest that besidesthe planes of easy glide also the prismatic and pyramidal planes of hard glide should becomeactive because of geometric compatibility (see e.g. the two-dimensional example in Fig. 8).This could qualitatively explain why polycrystalline ice is harder than a single crystal of iceunder easy glide, whereas it is softer under hard glide conditions. It must be remarked herethat the geometrical incompatibility between grains that are deformed only by glide on basalplanes can be adjusted not only by the activation of pyramidal and prismatic planes, but also by

(b) (c)(a)

W

W W

Fig. 7 The analogue of Fig. 6 for the case of tension. It is evident that the crystallite tends to rotate awayfrom the principal axis of tension. On the left-hand side of the figure (a), the crystallite is schematicallyrepresented as a rectangle. The lines in the rectangle represent the basal planes and the weight Wgenerates a tension of the crystal. Since gliding along the basal planes is the only possible deformationmechanism, the rectangle can be deformed only as shown in part (b). In this position, the crystal is not inequilibrium, since a couple is acting on it. (c) Rigid body rotation re-establishes equilibrium and showsthat under tension, the crystallites rotate away from the principal axis of tension.


GAMM-Mitt. 29, No. 1 (2006) 91

the mobility of grain boundaries through the phenomenon of grain boundary migration (seee.g. Section 6). Besides, the above arguments also make it obvious that the flow resistancewill be orientation dependent according to the actual distribution of the crystal orientationswithin a sample.

We discussed the importance of grain size and orientation distributions and their role inthe flow law. As we already pointed out, these concepts are often referred to in other words,namely as texture and fabric, respectively. Semantically, these two words have the samemeaning. However, it is common in geophysics to refer by fabric to a characterization of adistribution of lattice orientations and by texture to a distribution of grain sizes.

4 Modelling Induced Anisotropy: Microscopic Models

In this Section we review the various different model attempts to account for the anisotropicbehaviour of the polycrystal. An important issue is hereby the accommodation of the geomet-ric misfit that develops due to the different alignments of the individual crystals in a RVE. Toaccommodate for these misfits, several hypotheses have been proposed: Taylor’s assumptionconjectures that all crystals in an RVE suffer the same strain; according to Sachs, the stress suf-fered by each of the crystallites is supposed to be the same; and the self-consistent viscoplasticmodel (VPSC) assumes a compromise between the two. The attempts of constructing a ho-mogenized anisotropic flow law aimed to upscale deformation and/or stress relationships thatare assumed inhomogeneous on the level of the single crystal but need be representative forthe polycrystal at the RVE-scale. It will be seen that many proposed relationships are unableto describe the evolution of the anisotropic flow law.

4.1 The Taylor Hypothesis

If an incompressible body suffers uniform strain, it is obvious that stretches in five suitablydistinct directions suffice to determine the whole deformation of the specimen simply becausethe stretching tensor of a density preserving body has five independent components. Thisled von Mises [133] to conclude in 1928 that five independent slip systems7 are required toimpose an arbitrary strain in any crystal which deforms only by glide along these slip planes.Just ten years later, Taylor [121] used von Mises’ inference to develop his celebrated theory ofplastic strain in polycrystalline metals. Taylor’s hypothesis can be reported as follows: eachcrystal of the aggregate should suffer exactly the same strain as the bulk (avoiding so the oc-currence of microcracks and voids, see Fig. 8b, in the material due to incompatibilities amongneighbouring grains). On these grounds he calculated deformation and fabric development ofa polycrystalline medium in terms of its single crystalline rigid-plastic behaviour. In Fig. 8a,we show an undeformed polycrystal in which the activation of the basal planes only causesa V-shaped void in Fig. 8b. On the other hand, the activation of both (basal and prismatic)planes avoids the occurrence of the geometrical incompatibility, see Fig. 8c.

Because of its simplicity, the hypothesis proposed by Taylor still continues to be widelyused (sometimes with slight modifications) to predict the evolving anisotropy of polycrystals[3, 59, 132]. In spite of this, it is now recognized that the assumption underlying Taylor’s

7A slip system is defined by the combination of a crystallographic plane and a particular slip direction on thisplane.



model builds just a simplified picture of a commonly much more complex reality. Patchydeformations, grain boundary sliding and migration, shear and kink bands, diffusional flow,polygonization and dynamic recrystallization are just some of the presently known processeswhich can occur, allowing inhomogeneous strains [3, 24, 26, 48, 54, 58, 65, 89] that frustratethe requirement of five independent slip systems for each crystal [12, 85, 109, 130].

4.2 The Sachs Hypothesis

When applied to ice, Taylor’s model achieves a very deceptive performance [12, 14, 16]:it demands a contribution to the total strain of about 60% arising from non-basal slip and,consequently, it predicts strain rates which are too low in comparison with experimental data,as well as rather loose fabrics (i.e., low anisotropy). This outcome, added by the lack ofexperimental evidence on non basal slip and the observation that hard glide is at least 60 timesmore difficult to activate than basal easy glide [27, 86], led glaciologists to prefer modelswhich do not strictly require strain compatibility among neighbouring grains. The hope wasthat additional mechanisms (usually not considered in the modelling), like grain boundarymigration or polygonization, can fulfil the necessity of coherence in the aggregate.

The simplest alternative is to reckon the deformation as being produced entirely by basalglide and to assume that stress, instead of strain, is homogeneous in the polycrystal. This lasthypothesis, which can be traced back to Sachs [112] and Reuss [111], is commonly calledSachs’ hypothesis. The Sachs’ hypothesis has been considerably more successful for ice thanTaylor’s hypothesis. Many of the already proposed models for anisotropic ice are based onSachs’ hypothesis even though some with modifications [13, 38, 39, 45, 46, 47, 85, 125, 131].

(a) (b) (c)

POLYCRYSTAL INCOMPATIBILITYGEOMETRICAL

COMPATIBILITYGEOMETRICALUNDEFORMED

Fig. 8 (a) Schematic two-dimensional picture of a three-dimensional polycrystal made of two crys-tallites. Thick lines represent grain boundaries; thin lines and dashed lines are the representatives ofbasal and prismatic planes, respectively. Such planes are always orthogonal and the distance betweenthe planes of the same kind is thought to be infinitely thin. The basal planes of the crystallite on theleft-hand side are not drawn, because the c axis of that grain is orthogonal to the plane of the figure.(b) If the crystallite on the right-hand side is deformed via glide on basal planes, then the geometricalcompatibility cannot be fulfilled because the deformation of the other crystallite can not adjust the de-formation of the first one, i.e., a V-shaped vacuum is created. We remark that a rotation of the crystalliteon the left-hand side could adjust the deformation of the other grain, but it is forbidden by the presenceof further surrounding crystals. (c) If the prismatic planes are active (i.e. the deformation is allowed alsovia glide on these prismatic planes), then the compatibility of deformation between the two crystallitescan be achieved.


GAMM-Mitt. 29, No. 1 (2006) 93

4.3 Models Including Fabric Evolution

A first effort to describe the fabric evolution of ice under prescribed uniaxial compressivestrain was given by Azuma and Higashi [6]. Through direct observation of the crystallographicorientations in thin ice specimens compressed at constant rate, they found that the compressivestrain ε(g)(< 0) in the grains was not the same, but correlated with the orientation of their c-axis, given by the angle φ0 relative to the direction of compression of the sample (cf. Fig.6(a)). Hence, crystals more favourably aligned for deformation by basal glide should exhibitlarger strains. The fitting of their data led Azuma & Higashi to suggest an empirical relationbetween ε(g) and the bulk axial compressive strain ε of the form

ε(g) =S(g)

Sε, (9)

where S(g) = cosφ0 sin φ0 is the Schmidt factor (of the grain g, of which the c-axis is orientedat φ0) and S is its average, viz.,

S =1N

N∑g=1

S(g), (10)

where N is the number of grains. To compute the rotation of the c-axis of a crystallite, theyused the simple geometric relation valid for compression,

sin φ =(1 + ε(g)

)sin φ0, (11)

where φ is the angle between the sample’s axis of compression and the c-axis after rotation(see e.g. [113, 123, 131]). This model was extended to uniaxial tension ε(g)(> 0) by Fujitaet al. [37] and Lipenkov et al. [82]: they used another simple geometric relation (see e.g.[113, 123, 131]) valid for tension, i.e.,

cosφ =(1 + ε(g)

)−1

cosφ0, (12)

whereas Alley [1] generalized the model to pure shear by superimposing uniaxial tensile andcompressive strains. Further, Alley also combined pure shear with a subsequent rigid bodyrotation to enable the simulation of simple shear. A weak point of these models is that theempirical law (9) is based on laboratory observations where the strain rate is almost fourorders of magnitude larger than in ice sheets. On the other hand, laboratory tests can not beperformed at such small strain rates: the time that one must wait to have reasonable strainswould be simply too long8.

The evolution of fabric is only part of the problem of the anisotropic behaviour. The otherproblem concerns directly the form of the flow law. In the next subsection, we analyse modelsbuilt to this aim.

8This is a dilemma of many attempts to identify the characteristic parameters of mechanical processes of geo-physical time scales. They last much longer than human life spans.



4.4 Homogenized Anisotropic Flow Laws

In 1994 Azuma [4] attempted to improve his early approach and presented in 1996, togetherwith Goto-Azuma [5], an anisotropic generalization of Glen’s flow law (4) in the form

Dij = AS(ij)

(S(kl)tkl

)n, (13)

with

A = A′ exp(− Q

kT

), S(ij) =

12

(Sij + Sji

), Dkk = Sjj = 0. (14)

in which Q, k, T are the activation energy, the Boltzmann constant, the absolute temperatureand A′ is a constant. Dij and tij are, respectively, the stretching and the Cauchy stress of thepolycrystal, while Sij denote the averaged Cartesian components of the Schmidt tensor, viz.,

Sij =1N

N∑g=1

S(g)ij , S

(g)ij = m

(g)i c

(g)j , (15)

where c(g)i and m

(g)i are, respectively, unit vectors parallel to the c-axis and to the resolved

shear stress in the basal plane of the g-th grain, viz.,

m(g)i =

T(g)i(

T(g)k T

(g)k

) 12, T

(g)i = tijc

(g)j −

(c(g)k tklc

(g)l

)c(g)i . (16)

Thorsteinsson et al. [124] suggested that the implicit idea of the above model should be toreplace the crystallites of the aggregate by a single other one, of which the c-axis orientationis given by the averaged Schmidt tensor S. However, this claim is false because S cannot berelated to the c-axis of a single crystallite. To prove this, it suffices to give a simple example:imagine a polycrystal is made of two orthogonal crystallites with slip systems

(m(1), c(1)

),(

m(2), c(2))

and respective Schmidt tensors S(1), S(2). For the sake of simplicity, considermatrix notation:

m(1) = (1, 0) , c(1) = (0, 1) , S(1) = m(1) ⊗ c(1) =(

0 10 0

);

m(2) = (0, 1) , c(2) = (1, 0) , S(2) = m(2) ⊗ c(2) =(

0 01 0

);

the averaged Schmidt tensor, S, is easily evaluated,

S =12

2∑g=1

S(g) =(

0 12

12 0

); (17)

its determinant is non-zero and so it is not a dyad. This means, S can not be represented bya tensor product between two vectors as every Schmidt tensor can, e.g., S(1) and S(2) (seealso the definition (15)2). This implies that it is impossible to extract from S the c-axis of ahypothetical single crystal.


GAMM-Mitt. 29, No. 1 (2006) 95

The same authors also proposed that the model of Azuma–Goto-Azuma [5], mentionedabove, could be derived from the Sachs hypothesis expressed in terms of the flow law

Dij = AGij , where G(g)ij = S

(g)(ij)

(S

(g)(kl)tkl

)n

, tij = t(g)ij , n > 0, (18)

where Gij denotes the average of G(g)ij over all grains. The derivation is accomplished through

replacing in (18) the average of the (n+1)-power tensorial product of the symmetric Schmidttensor S

(g)(ij) by the (n + 1)-power tensorial product of its averages; that is, for the case n = 3

(say):

S(g)(ij)S

(g)(kl)S

(g)(pq)S

(g)(rs) ⇒ S(ij)S(kl)S(pq)S(rs). (19)

Nevertheless, the validity of (19) is certainly questionable. Indeed, Thorsteinsson et al. [125]applied both models (Sachs’ and Azuma’s) to account for effects of anisotropy in the iceextracted from the Dye 3 borehole in Greenland and, as expected, sensible differences betweenthe two approaches were observed, even though the authors could not identify which modelhad the best performance. They concluded also that no model was able to describe the strainrate enhancement at Dye 3 well. To justify the discrepancies of ≈ 25%, they replaced the ratefactor A in (13) and in (18) by AT , i.e.,

AT = A (1 + f) , f =C

D, (20)

where f is a phenomenological impurity factor depending in the most trivial manner on theimpurity content C and grain size D. Besides, the analogy with the superplastic flow law (7)with respect to the grain-size dependency is noted.

The weakest point in Azuma–Goto-Azuma’s model [5] (as well as in its adaptation toapplications) is its inability to describe fabric evolution. More useful models are developed infact with the target to solve both problems, i.e. the fabric evolution and the anisotropic flowlaw.

4.5 Complete Microscopic Model

One of the first approaches to reach the goal to describe, at the same time, the fabric evolutionand the anisotropic flow law, was proposed by van der Veen and Whillans in 1994 [131]; theyadopted Sachs’ hypothesis and assumed that each crystal should deform only by basal slipdriven by a viscous power law. The idea is the following: A certain state of stress is appliedto the whole polycrystalline material. The Sachs hypothesis assures that this is transferredto all crystallites. The viscous power law that they assume for the single crystallite is thethree-dimensional generalization of that derived by Weertman in 1973 [135], i.e.,

eiν = A(τ2ην + τ2

μν

)n−12 τiν , i = η, μ no summation over ν, (21)

where eiν and τiν are the strain rate9 and the stress tensors of the crystallite in the frame ofreference Oημν, the third axis of which (with index ν) is parallel to the c-axis and coherent

9We write eiν for stretching and call it strain rate despite the fact that we used Diν before and prefer to call itstretching. We do it, to make evident a common error that was also made when van der Veen and Whillans introducedtheir method. The error is that the integral over time, eiν , is interpreted as a strain, which it is not. This is a quitecommon misinterpretation of the time integral of the stretching written as eiν .



with the crystallite itself; A is a temperature-dependent rate factor and n is a parameter setequal to 3. Let us remark that the strain rates related to the other components (eηη, eμμ, eνν ,eημ = eμη) are zero. This reflects the assumption that each crystal is allowed to deform onlyby basal slip. If Δt is a small time interval, the authors interpreted the accumulated stretching

Δeij = eijΔt, (22)

as the strain contribution of the crystallite in the frame of reference Oημν in the time intervalΔt. The relation between Δekl, the strain contribution of the single crystallite in the frameof reference Oημν, and the strain Δεk

ij , the strain contribution of the single crystallite withrespect to the external frame of reference Oxyz is

Δεkij = ωkiωljΔekl, (23)

where ωij is the rotation from the external frame of reference Oxyz to the reference frameOημν that matches the crystal orientation at the start of each time step. The strain of the bulkpolycrystal Δεb

ij is the mean over all the crystallites of Δεkij , i.e.,

Δεbij =

1N

N∑k=1

Δεkij , (24)

where N is the number of crystallites. It is clear that the explicit definition of ωij is of greatimportance for the fabric evolution. At the initial step, ωij is defined by the latitude and thelongitude of the single crystallite to which it is referred. At the second step these angles aredefined again in order to be coherent not only with the bulk rigid rotation of the polycrystalbut also with the rotation of the single crystallite due to the mechanism described in Figs. 6and 7 (this additional rotation mechanism is called tilt). The routine is then done for manysteps, each representing the evolution in the time interval Δt. The result is of course thedevelopment of fabric and of a numerical anisotropic flow law. We advise the reader that wehave to substitute the symbols eij , εk

ij and εbij , used in the original paper of van der Veen and

Whillans, by Δeij , Δεkij and Δεb

ij in order to emphasize once more that in a non-linear theoryintegrating stretching or strain rate in time does not yield a strain measure. Accumulatingstrains by (24), therefore does not yield a strain tensor. So the entire process yielding (24) isdubious. On the other hand, the lower the value of Δt, the better will be the approximationexpressed by equation (22).

In spite of some reasonable qualitative agreement of the numerical simulations with lab-oratory and field observations, some strong anomalies – as for example the minimum of thestrain rate at about 10% of the strain, instead of 1% as observed in practice [11, 71, 80, 81] –evidenced the distinction between simulation and real mechanisms taking place during thedeformation. Although the similarity of this approach with that of Azuma and Higashi [6]was recognized for simple regimes of deformation, the greatest merit of the van der Veen andWhillans model lies undoubtedly in their pioneering attempt to simulate dynamic recrystal-lization in polycrystalline ice, even though only numerically and through artificial criteria. Wewill discuss later on their approach to this problem.

A more sophisticated mechanical model for ice, capable also of describing fabric develop-ment, was proposed by Duval, Castelnau and co-workers [12, 14, 15, 16] through the adap-tation of the so called visco-plastic self-consistent approach (VPSC) developed for metals by


GAMM-Mitt. 29, No. 1 (2006) 97

Lebensohn and Tome [79]. In this method, both stress equilibrium and strain compatibilityamong neighbouring grains tend to be satisfied in a recursive manner. Each grain is assumedto be an inclusion embedded in an homogeneous matrix, which ought to represent an av-erage state of the polycrystalline environment. Hence, neither the stress nor the strain areconstrained to be the same for all crystallites.

Despite the natural superiority of the VPSC approach in comparison with simpler methodsof homogenization of the microstructural behaviour, some shortcomings still persist. Forinstance, the method is strictly limited to linear constitutive laws and requires the postulationof a constitutive equation for the response of the individual grains. Consequently, appropriatevalues for all the microscopic parameters involved in this equation must be determined, so asto provide physically acceptable results at the microscopic as well as at the macroscopic levels.In particular, a critical point is the stipulation of the yield shear stress for each slip system. Forice, the best fits of experimental data suggest a resistance to glide in non basal planes 70 timesgreater than in the basal plane [12, 15]. Although this sounds tolerable for the case of prismaticslip (as commented before, the observed resistance of ice single crystals to hard glide is morethan 60 times higher than for easy glide [27, 86]) the contribution of pyramidal slip to thepolycrystalline deformation would be inadmissibly high: about 9%. This is rather unphysical,since the occurrence of pyramidal slip in polycrystalline ice is supported neither by theorynor by experiments [22, 61]. The probable reason for this unwanted result is the limitednumber of deformation mechanisms presently available in the VPSC: it considers only glidingon crystallographic planes, being therefore incapable of adequately treating inhomogeneousintracrystalline deformation (kink and deformation bands, polygonization, etc.) as well asrecrystallization and intergranular interactions, which all are likely to occur in ice [48, 89,137].

In order to account for this fault, even more sophisticated (and complex) descriptions of mi-crostructural processes occurring in ice have been recently addressed. For instance, Meysson-nier and Philip [95] have utilized finite element simulations. In this case, each crystal istreated as a continuum, and complex inhomogeneous grain deformations can be simulatedwithout violation of either strain compatibility or stress equilibrium. Due to its complexity,however, only a two-dimensional array with a little more than thousand grains was considered.

5 Modelling Induced Anisotropy: Macroscopic Models

Common to all approaches discussed above is the fact that they constitute discrete modelswhich reckon the polycrystal as an aggregate of a finite number of grains. The advantageof such discrete models is, of course, the potential accuracy of the description: each grainof the polycrystal is individually considered. The consequential disadvantage is that discretemodels can only handle a small number of ice crystals, since the required storage capacityand computational time tend to increase considerably with the dimension of the aggregate.A further disadvantage is that the number, size and arrangement of grains need be knownif the influence of the crystallites on the homogenized behaviour at the RVE-scale is to befollowed through time. For most situations, this is unrealistic. In fact, it is rather so that themacroscopic response of the polycrystal is made up as the response of a very large number ofcrystals arranged in orientation, size and other properties according to statistical randomness.In other words, the individual crystal properties are not so significant, it is their statistical



distribution. This viewpoint frees the scientist from having to prescribe the number, size and(orientational) distribution of the grains within an RVE and to follow these through time;instead it suffices to describe the initial statistical description of the orientation of the grainsat the microscale. In such cases the best alternative is to resort to a macroscopic continuumdescription, able to account somehow for the microstructural anisotropy of the medium.

5.1 Lliboutry’s Model

One model of this sort has been proposed by Lliboutry in 1993 [84, 85]. He considered thatthe stress acting on every crystallite was the same (Sachs’ hypothesis), assumed that eachgrain deforms only by basal glide and introduced for each of it the dissipation potential Ψ∗,i.e.,

Ψ∗ = 12k1τ

2n + 1

4k3τ4n, (25)

where τn is the respective resolved shear stress on the basal plane, while k1 and k3 are con-stants. For a given prescribed fabric, the model is suited to a homogenization procedure, i.e.,the dissipation potential Ψ of the polycrystal is assumed to be the mean of Ψ∗, weighted byits relative volume. Thus,

Ψ =∫ π/2

0

Ψ∗ V ∗(θ) dθ, (26)

where V ∗(θ) is the relative volume10 of the crystals with the c-axis having an angle with thevertical that is larger than θ.

The relevance of working with the dissipation potential is the form taken by the stretching,

Dij =∂Ψ∂Sij

, (27)

where Sij = t〈ij〉 is the Cauchy stress deviator. Thus, Lliboutry obtains a three-dimensionalanisotropic generalization of the classical flow law of polynomial type, which contains a lineardependence on stress (and therefore viscous Newtonian behaviour) in addition to the usualcubic dependence of Glen’s law. The greatest weakness of this theory is that it deals only witha static and prescribed fabric, being unable to determine its evolution.

5.2 The ODF Approach

To overcome the problem of the evolution of fabrics, two other approaches to describe theinduced anisotropy of ice were proposed ten years ago by Meyssonnier and Philip [94] andSvendsen and Hutter [120] through consideration of a continuous statistical distribution ofcrystallographic orientations within the polycrystal. The microstructural variable adopted inboth works was a typical measure of anisotropy in complex media, namely, an OrientationDistribution Function f∗(n) (ODF, see e.g. [75]) which gives the fraction of crystallites of

10The relative volume of the crystals V ∗(θ) gives the same kind of information we have from the OrientationDistribution Function (ODF), see the next Section. It is possible to compare, in fact, the two different homogenizationrules given by equations (26) and (34). However, let us remark that the interpretations of V ∗(θ) and ODF aredifferent.


GAMM-Mitt. 29, No. 1 (2006) 99

which the c-axes are directed towards every particular orientation in space. More specifically,f∗(n) d2n represents the fraction of the number of crystallites that have orientations directedtowards n within the solid angle d2n, respectively. Thus, integration of the ODF over the unitsphere, S2, gives unity, i.e.,

∫S2

f∗ (n) d2n = 1. (28)

It should be remarked that using a macroscopic continuum theory based on the concept of anODF demands to regard each representative point of the body in the model as an assemblageof an “infinite” number of crystallites in order to ensure continuity to ODF. This requirementis the so called continuum hypothesis, and it is trivially satisfied for the large ice masses inGreenland and Antarctica, since their dimensions (> 106 m)11 are by far much larger than thetypical grain sizes (∼ 10−3 m).

The first model, by Meyssonnier and Philip [94], employed a simpler version of the VPSCmodel already discussed in the previous Section, by regarding each crystallite as a continuoustransversely isotropic medium. Instead of treating the polycrystalline body as an aggregate ofindividual grains, they used the ODF to determine the fabric evolution. However, in order tosimplify the calculations, only Newtonian viscous behaviour for the grains was considered.An even more simplified version of the Meyssonnier–Philip model was presented by Gagliar-dini and Meyssonnier [38, 39], with the aim at a numerical implementation to simulate icesheet flow.

The second model, by Svendsen and Hutter [120], treated ice as a non-linear inelasticmaterial, and utilized the conventional formalism of continuum mechanics [128, 129] to derivethe fundamental equations of their model. They used a dissipation potential for the entirepolycrystal (which was regarded as a transversely isotropic material) of the form

Ψ = Ψ(t〈ij〉, Aij , T

)(29)

where T is the absolute temperature, t〈ij〉 is the deviatoric stress tensor and Aij is an additionalsymmetric tensor measuring the degree of transversal isotropy: the second moment of theODF, named by different authors alignment, conformation, anisotropy or also structure tensor[21, 78],

Aij (xi, t) =∫S2

A∗ij (ni) f∗ (xi, ni, t) d2n, A∗

ij = ninj , (30)

where f∗ is the ODF, ni is a unit vector parallel to the c-axis of each crystallite and the integra-tion is performed over the whole range of orientations S2. Implementing the general principleof material objectivity, equation (29) can be better represented by a simpler functional depen-dence, in which only scalar quantities are involved. It assumes that (29) is a scalar isotropicfunction of two symmetric tensors and of a scalar, and so

Ψ = Ψ(IItD , IIItD , IIA, IIIA, ItDA, I(tD)2A, ItDA2 , I(tD)2A2 , T

), (31)

11Compare with the usual dimensions of common continuous bodies (> 10−2 m) and their constituent molecules(> 10−10 m)



where IB , IIB , IIIB denote the first, second and third scalar invariants of B = BT andBD is the deviatoric part of a general symmetric tensor B. Furthermore, neglecting the thirdscalar invariant of the Cauchy stress deviator is an approximation that is already employedin Glen’s flow law; thus, Svendsen and Hutter implement this approximation and assume, asLliboutry did, the validity of equation (26). As a consequence, they are able to write a generalform for the anisotropic flow law,

D = γ1tD +γ2A

D +γ3

[tDA + AtD

]D

+γ4

(A2

)D+γ5

[tDA2 +A2 tD

]D, (32)

where 12

γ1 =∂Ψ

∂IItD

, γ2 =∂Ψ

∂ItDA

, γ3 =∂Ψ

∂II(tD)2A

, γ4 =∂Ψ

∂ItDA2, γ5 =

∂Ψ∂I(tD)2A2

.

(33)

A common shortcoming of these two theories is that both reckon the polycrystalline ice asa transversely isotropic medium, hence restricting the fabric to remain axially symmetric; inother words, if one represents the unit vector ni in spherical coordinates (θ, ϕ), the depen-dence of the orientation distribution function f ∗ is constrained to vary only with the latitudeθ and not with the longitude ϕ.

The transversal isotropy of ice - even though approximate - is in fact typical only of singlecrystals, and it cannot a priori be taken for granted for polycrystals. Hence, to overcome thisproblem, Svendsen and Hutter [120] suggested to introduce into the functional dependenceof Ψ in (29) other structural tensors thereby being able, following Boehler [10], to take intoaccount other kinds of anisotropies.

A different and even simpler approach to account for diverse anisotropies was suggestedby Godert and Hutter [46] by employing a homogenization technique similar to that used byLliboutry [85]. They assigned a dissipation potential to each grain in the form (25) and, usingthe ODF as a statistical distribution (see e.g. [63]), they proposed the dissipation potential forthe entire polycrystal to be

Ψ (xi, t) =∫S2

Ψ∗ (xi, ni, t) f∗ (xi, ni, t) d2n. (34)

This kind of connection between the microscopic and the macroscopic level is used system-atically by the authors, e.g., for the structural tensor as in (30); besides, on the polycrystallinelevel, no compromise is made in this model with the degree of material anisotropy, which isgiven by the eigenvalues of Aij , viz. λ1, λ2 and λ3, as follows:

λ1 �= λ2 �= λ3 �= λ1 =⇒ orthotropic behaviour,λ1 �= λ2 = λ3 =⇒ transversally isotropic behaviour,λ1 = λ2 = λ3 =⇒ isotropic behaviour.

12We remark that in the paper a simplified version of the general anisotropic flow law comes from an incorrectevaluation of the tensor A2, the square of the structure tensor.


GAMM-Mitt. 29, No. 1 (2006) 101

Motivated by the ideas of the VPSC approach [16], the same authors [45, 47] introducedinteractions among neighbouring grains and implemented their model numerically in a cou-pled finite element-finite volume scheme. Comparison of the results of their model with theGRIP ice-core data and the predictions of the VPSC were performed. Curiously, in spite ofthe fact that Godert and Hutter [45, 47] employed simply a linear (i.e. Newtonian) viscousrelationship between stress and strain rate for basal slip, they showed that their model couldbetter fit the GRIP data than the non-linear VPSC method. However, the authors found thatthe maximum increase of the fluidities due to crystal alignments does not achieve the valueobserved in laboratory tests [11]. They claimed that this was due to an overestimation of theshear viscosity in the application of their model to the case of single maximum fabric; they infact suggested that the observed flow law should change according to the loading and align-ment. We remark that the error of this overestimation must be solved in order to be able tosimulate correctly the anisotropic behaviour of ice in large ice masses.

On the basis of the theory of mixtures with continuous diversity [32], Placidi and Hutter[106] have formulated a new enhancement-type flow law for anisotropic ice. This law dependson the ODF and yields the Glen flow law for a random distribution of lattice orientations.Moreover, it is independent upon the frame of reference and reproduces all experimental re-sults for girdle fabrics (all the crystallites are inclined with respect to the vertical with a givenangle) and single maximum fabrics.

In the ODF-approach, the evolution of fabric is simply prescribed by an evolution equa-tion of the ODF. A useful scheme to find an appropriate evolution equation of ODF can befound in [20]. Besides, Placidi [103] presented a set of constitutive equations (independent ofthe observer and coherent with the Second Law of Thermodynamics [108]) that explicitly ac-counts for grain rotation and grain boundary migration in the evolution of the orientation massdensity, a function that is associated to the ODF. Such an evolution equation is reasonable butstill needs to be implemented in a more realistic numerical simulation.

material plane

normal vector

deformation

inverse deformation

Fig. 9 Cubic sample subjected to an arbitrary cyclic deformation. The panels show material planesarranged diagonally, with their respective normal vectors. At the initial configuration (left), the orien-tational distribution of planes is assumed isotropic. During deformation, an anisotropic distribution ofnormal vectors is induced, but at the end of the cycle this distribution becomes isotropic again, as thefinal configuration returns to the identical (initial) one.



5.3 The Morland–Staroszczyk Model

A purely phenomenological approach to anisotropic ice dynamics, which ignores any mi-cromechanical processes and microscopic interactions at the grain level, was proposed in aseries of papers by Morland and Staroszczyk [98, 115, 116, 117, 118]. The fundamental ideabehind this theory lies in the expectation to obtain a sketchy picture of the anisotropy of thepolycrystal from its instantaneous state of deformation (Fig. 9) without any explicit referenceto fabric or grain size. More precisely, the model is based on the following flow law,

tD = HI (D)+HA

(D, B, M (r)

), HA

(D, I, M (r)

)= 0, (r = 1, 2, 3) (35)

where tD is the Cauchy stress deviator, H I and HA are, respectively, the isotropic and theanisotropic part of the material response, D is the stretching tensor, I is the identity, B is theleft Cauchy–Green deformation tensor and M (r) are the three structural tensors that take intoaccount the anisotropy, and are related to the principal directions of B,

M (r) = e(r) ⊗ e(r), (36)

where e(r) are the eigenvectors of B. Hence, whenever B = I (i.e., no instantaneous defor-mation), the material response is isotropic. The assumption that the evolving anisotropy doesnot depend on the deformation history ensures the straightforwardness of the model but it im-plies also unpleasant consequences, like the reversibility of the induced anisotropy (see Fig.9). In fact, according to the Morland–Staroszczyk model, described by (35) and Fig. 9, an ini-tially isotropic sample of ice, deformed by any cyclic process (of shear, compression/tension,etc.), such that B = I at the final configuration, would manifest again an isotropic mechani-cal behaviour. This does agree neither with experience nor with the mechanisms of basal slipdepicted in Figs. 6 and 7.

Of course, the supposition of a direct correlation between polycrystalline deformation andinduced anisotropy is not new, it can be traced back at least to the early model of Azumaand Higashi [6] and of van der Veen and Whillans [131]. The difference is that, in this phe-nomenological approach, no compromise with microscopic mechanisms is assumed.

The combination of simplicity and flexibility found in this model is in fact its greatest at-tractiveness, but also its principal weakness. While it presents itself as a suitable approach tomodel ice sheet flow with comparatively low computational costs, it is in fact physically fee-ble: the microscopic processes occurring within the ice – including changes in grain size andshape, as well as in the fabric itself – are all cryptically subsumed in rheological coefficients.

6 Anisotropy Induced by Recrystallization

In this section we collect the most important facts regarding micromechanical processes whichinfluence the macroscale deformation and go beyond the mere rotation of the lattice. Amongthese is e.g. the dislocation density. It is the total length of linear lattice defects per unit vol-ume and makes the polycrystal generally softer than the perfect crystal would be. The famousOrowan relation describes the total deformation due to dislocations as the product of disloca-tion density, Burgers vector and grain size. However, there is also a counter-effect expressedas dislocation interactions or pile-ups that makes the material actually stiffer. Recrystallization


GAMM-Mitt. 29, No. 1 (2006) 103

is a term under which a whole class of processes is subsumed that are all somehow connectedwith temporal changes of the grains and their sizes. Among these are grain boundary mi-gration with their associated normal and abnormal grain growths, nucleation, polygonizationand recovery. A characteristic feature of polar ice is that the time rate of change of the graindiameters is governed both by a growth rate and an annihilation rate which produce togetheran equilibrium grain size13. Similarly, the rate of change of the dislocation density is governedby a growth and decay mechanism that, apart from a Burgers vector dependence, are linkedto grain size. We close by showing that available models which include recrystallization arenot satisfactorily describing the deformation of polycrystalline ice in polar ice masses.

6.1 The Role of Dislocations

In spite of the symmetry of the crystalline lattice of ice illustrated in Fig. 5, the mechanicalbehaviour of a real crystal of ice is very different from that expected for a hypothetical, perfectcrystal. In 1926 Frenkel (see [65]) put forward a simple method of estimating the theoreticalshear strength of perfect crystals and found that it is much larger than the observed values. Ascommented by Kittel [74], this cannot be explained without the presence of defects.

The defects may be grouped into different types according to the dimension of the regionwhere the crystalline structure is destroyed [44]: if this region is a point, its dimension is zeroand we refer to it as a point defect; if it is a line, the dimension is one and we refer to it asa line defect or dislocation. Two-dimensional regions of defects are called plane defects, andlarger defects, where the structure is seriously disrupted in a three-dimensional region, arecalled gross defects.

We have pointed out that dislocation is a line defect in the lattice structure. Owing to this,we can define the dislocation density (ρd) as the total length of line defects per unit volume,which means that it has the dimension of the inverse of a surface area. The curve on which theline defect lies is the domain of a vector field, characterized by the so called Burgers vectorb, that defines the displacement which the lattice must undergo in order to produce the defect.Depending upon whether b is parallel or orthogonal to that line, we call the dislocation screwor edge, respectively. Hybrid dislocations (combinations of screw and edge types) are alsofrequent.

Since the early works on dislocations, many attempts were concerned with the experimen-tal verification of the idea that dislocations should not only occur, but must also play a centralrole in the deformation of polycrystalline materials. The glide of dislocations during defor-mation suggests in fact a relation between the total deformation (the strain ε) and the storeddislocation density in the material. This is for instance given by the Orowan relation, i.e.,

ε = ρd b D, (37)

where b is the length of the Burgers vector of a certain dislocation that moves the distance D,approximately identified by the grain size14.

Moreover, the mobility of dislocations has often some relation with the softness of thematerial. We remarked that a perfect crystal is very hard and the introduction of dislocations

13We remark that such an equilibrium grain size occurs only as long as the nucleation and growth of new grainsdoes not occur.

14The grain size is in fact the upper bound for the distance that a dislocation can move.



makes the material softer. Thus, in principle, the larger the dislocation density is, the greaterwill be the ductility of the material. However, this simple correlation is proved to fail inmany cases, including ice. The interaction of dislocations with one another plays in fact avery important role. Dislocations can for instance pile up, which decreases their mobility andmakes the material harder. Therefore, it is customary to use a relation between the stress, σ,and the dislocation density as follows:

σ = c G b√

ρd, (38)

where c is a constant, generally equal to 12 and G is the shear modulus. Relation (38) has

been shown to be valid for a wide range of materials by McElroy and Szkopiaz [92]; however,Montagnat and Duval pointed out that it only represents an upper bound for ice because ofthe occurrence of Grain Boundary Migration (GBM) [96]. GBM makes, in fact, ρd smallerbut has practically no effect on σ. It is a phenomenon belonging to the general class ofrecrystallization mechanisms, that we will discuss in the next subsection.

6.2 Phenomena Related to Recrystallization

Recrystallization is a word associated with a wide number of processes, and there is no agree-ment on its precise definition. The reason lies perhaps in the high complexity of the physicalphenomena that scientists want to refer to with this word and the impossibility to build a math-ematical theory that associates to it a measurable quantity. In this paper, we try to associatewith this word all the phenomena that can be interpreted by what the word itself suggests. Inparticular, we will address grain boundary migration (leading to normal and abnormal graingrowth), nucleation, polygonization and recovery.

A first example is given when a portion of mass belonging to a given crystallite in a poly-crystal is transferred to a neighbouring crystallite. In this case we say that the boundary be-tween the two crystallites has moved (see e.g. Fig. 10). We call this process grain boundary

12

12

Fig. 10 Schematic two-dimensional illustration of grain boundary migration. The boundary betweengrains “1” and “2” migrates so that grain “1” grows while grain “2” shrinks. The parallel thin linesrepresent the basal planes of the two grains and the arrows indicate directions normal to them. Besides,where the boundary passes, the dislocation density decreases to almost zero value. Basal planes repre-sented by thin dashed lines indicate, in this figure, the fact that the dislocation density in that region isvery low.


GAMM-Mitt. 29, No. 1 (2006) 105

migration and regard it as a recrystallization phenomenon. Notice that grain boundary migra-tion can easily adjust the geometrical incompatibility shown in Fig. 8b without the necessityto activate the pyramidal and the prismatic planes.

The migration of grain boundaries is driven essentially by the reduction of the Gibbs freeenergy stored either in the grain boundaries or within the grains by dislocations. The migrationof grain boundaries driven by the grain boundary energy is generally quite uniform, implyingthat certain crystallites grow while others shrink. This process is normal grain growth. If onlya few grains grow in a very accelerated pace, then we refer to this as abnormal grain growth;in this case the microstructure becomes unstable and a few grains may grow excessively.The driving force for abnormal grain growth is a reduction in grain boundary energy (whichis, incidentally, the same force that drives normal grain growth). However, an importantquestion to consider is whether or not normal or abnormal grain growth can occur. In a three-dimensional polycrystalline material the main factors which lead to abnormal grain growthare impurities and fabrics.

Impurities are in fact generally considered to prevent the growth; they provide a pinningpressure against the usual pressure that drives the growth. However, this pinning pressure,also called the Zener pinning pressure, is inversely proportional to the grain size; this meansthat the larger the grain size, the lower the Zener pinning pressure. Thus, the crystallites withlarge grain sizes are favoured to grow and those with small grain sizes are favoured to bepinned by the impurities, see [65].

If the grain boundary divides crystallites with very small c-axis misorientation angle, thenit is called a low angle grain boundary. If the c-axis misorientation angle across the boundaryis higher, then we call the boundary a high angle grain boundary. An important characteristicof the grain boundary energy is that it is higher for high angle grain boundaries and lowerfor low angle grain boundaries. Therefore also strong fabrics, such as a single maximumfabric, can drive abnormal grain growth; this is possible because in a sample with a singlemaximum fabric we have mostly low angle grain boundaries which possess low energy andlow mobility. On the other hand, if some grains with relatively different lattice orientationsare present, then these introduce high angle grain boundaries (possessing high energy andhigh mobility) and these grains may grow preferentially, providing room for the abnormalgrain growth. Abnormal grain growth can in a very rough manner be viewed as a kind ofaccelerated normal grain growth, see e.g. [65], of only certain privileged grains and thisgrain-growth process is driven by the grain boundary energy.

The phenomenon of nucleation is different from abnormal grain growth. Generally werefer to it when a new crystallite nucleates and rapidly grows within a polycrystalline matrixfrom which it is separated by a high angle grain boundary. Migration (or dynamic) recrys-tallization follows the nucleation of new grains. This is the recrystallization process whichoccurs on the bottom of an ice sheet. It is driven by the difference between the high disloca-tion density ρd stored in old grains and the low value of ρd in newly nucleated grains.

Polygonization is a rather different phenomenon but nevertheless quite common in icesheets. When a polycrystal is deformed, dislocations are produced and start to move. Thetendency of dislocations to be arranged into a state of lower energy configuration makes pos-sible the formation of regular arrays of dislocations. Such arrays generate low angle grainboundaries, also called subgrain boundaries. Thus, the crystallite is bent or twisted, subgrainboundaries are formed and subsequently the grain is broken (it is said that it is polygonized);the newly formed low angle grain boundaries divide the new crystallites. The result is that the



mean grain size is reduced. Moreover, the new boundaries decrease the amount of disloca-tions; hence, the formation of subgrain boundaries is also a recovery process.

Recovery is a general characterization of recrystallization phenomena. In fact, in all pro-cesses where the grain boundary moves (i.e., grain boundary migration, normal and abnormalgrain growth and nucleation), the recrystallized material swept by the boundary has, tempo-rarly, a much lower dislocation density. Thus, we can claim that, where the boundary passes,the dislocation density will momentarily decrease to a very low value (see Fig. 10 ).

6.3 Dislocation Density Dynamics

In this subsection we discuss some models to predict the dynamics of dislocation density. Forthe sake of simplicity, we will review only the models that are proposed for ice15 (see, e.g.[22, 26, 96, 97]).

We have seen that dislocation density increases with deformation. From the Orowan re-lation (37) one can easily derive an equation for the time rate of change of the dislocationdensity by straining (see e.g. [22, 26, 96, 97])

dρ+d

dt=

|ε|b D

, (39)

where ε is the strain rate and the superscript + indicates that the production of dislocation dueto straining is always positive. To generalize this to the multidimensional case, Morland [97]suggested to replace ε by the positive square root of the second invariant of the stretching,√

DhkDhk.The negative contribution to the production rate of the dislocation density is given by the

recovery processes of recrystallization. The mechanisms that are generally taken into accountare those explained in the previous section, i.e., grain boundary migration (including alsonucleation, normal and abnormal grain growth) and polygonization.

The reduction in the dislocation density due to grain boundary migration is generally mod-elled by the following formula, proposed by many authors [22, 26, 96, 97], i.e.,

dρ−d

dt= −αρdK

D2, (40)

where α is a coefficient exceeding 1, which makes it possible to take into account a higherdislocation density near the grain boundaries, and K is the grain boundary migration rate. Thesuperscript ”−” indicates that the production of dislocations due to GBM is always negative.

The grain boundary migration rate is a typical parameter related, for example, to normalgrain growth. It was introduced in metallurgy in 1948 by Smith [114] and its adequacy forpolar ice was supported by Gow [52] in 1969, who fitted a parabolic growth relationshipbetween grain size and time t for several sites in Antarctica and Greenland, i.e.,

D2 = D20 + Kt =⇒ dD

dt=

K

2D. (41)

D0 is the value of D at t = 0. It should be remarked here that expression (40) for the reductionof the dislocation density due to grain boundary migration can be easily derived from the

15A complete review of the models that were proposed for different materials is given in the book of Humphreysand Hatherly [65].


GAMM-Mitt. 29, No. 1 (2006) 107

Orowan relation (37), if (41) is taken into account. In 1994 Jacka and Li [72] proposeda differential equation having the ingredients of an evolution equation of grain size. Thisequation was able to reproduce the steady state grain-size that realistically occurs in the middlepart of an ice sheet, i.e.,

dD

dt=

K

D− PD. (42)

This equation balances the time rate of change of the grain diameters by a growth term due to,essentially, grain growth rate proportional to D−1 and an annihilation term proportional to Dwith strength P . In [105], we proposed to assign to this term the meaning of polygonization.For constant K and P , (42) can easily be integrated as follows

D2 = D20 exp (−2Pt) +

K

2P[1 − exp (−2Pt)] . (43)

In Fig. 11 the graph of D (t), together with an explicit comparison of the data from the Byrdice core is given. A macroscopic continuum model was recently presented by Placidi [103]to account for the evolution equation of the size mass density, that is a function somehowrelated to the Crystal Size Distribution (CSD). Such a law reproduces the general result givenin (42), whenever the evaluation of the mean of grain size is deduced from the form of theirdistribution.

Montagnat and Duval [96] noted in 2000 that a combination of equations (39) and (40),by assuming the usual evolution equation for normal grain growth (41), would provide acontinuous increase of dislocation density and grain size. An equilibrium state for them canbe reached only when polygonization is taken into account. It provides in fact new boundariesformed by dislocations that are subtracted from the interior of the crystallites. A relationbetween the dislocation density ρsgb

d stored by the subgrain boundary (sgb), the misorientationangle θ formed by it, the magnitude of the Burgers vector b of the dislocations and the averagedgrain size D has been proposed in 1972 by Dillamore et al. [24] for metals, and reads

ρsgbd

= βθ

bD, (44)

where β is a dimensionless parameter that Dillamore et al. [24] set equal to 1. Montagnatand Duval [96] use β = 2, apply this relation to ice and deduce a reduction of dislocationdensity due to polygonization by simply differentiating (44). Thus, loss of dislocations bypolygonization is assumed to be expressed by the following formula,

dρ−(pol)d

dt= −dρsgb

d

dt=

2θ

bD2

[dD

dt

]. (45)

The weak point of the formulation by Montagnat and Duval is that they use expression(41) for the rate dD/dt of (45), which is valid for normal grain growth but not for polygo-nization. They in fact did not use (45) as contributing to the dislocation loss, but rather onlyto establishing a condition for the steady state of the dislocation density and grain size, viz.,

dρ−(pol)d

dt+

dρ−d

dt=

dρ+d

dt. (46)



Byrdg=2.0g=3.0

g=0.5g=1.0

Fig. 11 The curves based on (42) fit well the experimental data given for grain size of the ice core atByrd Station, Antarctica. The values K = gK and P = gP with k = 9mm2ka−1 and P = 0.23ka−1

guarantee that they all reach the same steady state Deq = 6.3mm for each value of the nondimensionalfactor g. Let us remark that the figure gives us the physical interpretation of this nondimensional factor:it is the velocity in which the grain size D reaches the steady state Deq .

On the other hand, the advantage of the model of Montagnat and Duval [96] is that it includesboth the contributions on dislocation density of polygonization, as well as of straining and ofgrain boundary migration.

6.4 A Review of Models Including Recrystallization

In this subsection, we review some models for polar ice which take recrystallization intoaccount. The problem one needs to solve is the determination of the fabric and the mechanicalresponse of ice after the occurrence of nucleation and dynamic (migration) recrystallization.

The main idea is due to Kamb [73] and states that the polycrystal tends to assume a mi-crostructural configuration with a minimum amount of free energy. A general correlationbetween the onset of nucleation and dynamic recrystallization and the acceleration in creeprate (i.e., an acceleration of the strain rate) is generally accepted in the literature. This canbe attributed to crystal reorientations produced by nucleation and dynamic recrystallization[73]: the newly formed crystals and those that grow after the occurrence of dynamic re-crystallization have orientations that are favourable for intracrystalline plastic flow, and theorientations that are preferentially eliminated are those for which intracrystalline plastic flow


GAMM-Mitt. 29, No. 1 (2006) 109

is more difficult. Basal glide in the newly-nucleated, well-oriented crystallites is easier andthe microstructural configuration is more stable. This means that it is very important to under-stand which directions are well oriented for deformation in any state of the stress deviator orstretching. In the states of compression or extension such orientations are those at 45◦ fromthe principal axes of deformation (see Fig. 12a). In Fig. 12, we show a simple graphicalmethod to extract this information even for simple shear. Moreover, it has been recognizedthat, to describe the recrystallization processes, we need a parameter that is able to switchthem on, e.g. dislocation density or its associated lattice distortion energy.

As we have already pointed out, a first attempt to achieve this goal was done by van derVeen and Whillans in 1994 [131]. The parameter that switched recrystallization on was forthem the accumulated strain. This means that those crystallites reaching a certain thresholdvalue for the strain were constrained to be recrystallized, and the recrystallization replacedstrained crystals by new ones at the optimum orientation for basal glide. In uniaxial compres-sion, for example, the new crystallites had a c-axis oriented at 45

◦with respect to the axis of

compression.The authors recognized the following problem: the crystals unfavourably oriented for basal

glide do not accumulate any strain and they do not recrystallize. On the other hand, it isbelieved that in nature awkwardly aligned crystals are stressed more than others and they arethe first ones to recrystallize. However the Sachs model, that they used, does not allow todistinguish between different states of stress. To overcome this problem the authors proposedthat it is not the strain that must reach a certain threshold value to allow recrystallization, butthe difference of this and the mean strain.

A second problem arises when the recrystallization fabric was analyzed for simple shear.Experiments [64, 73] on ice samples subjected to simple shear show a two-maxima fabric.The first maximum is normal to the shear plane and is called M1. The second maximum isinclined at about 70◦ from M1 and is called M2. The main characteristic is that M1 is strongerthan M2. The failure of the model of van der Veen and Whillans is that M1 is weaker than M2.The reason is the following: in principle the two maxima should place M1 normal to the shearplane and M2 orthogonal to M1 (see Fig. 12). These are in fact two easy glide orientationsfor simple shear. However the tilt-rotation of the crystallites and the whole rotation of thepolycrystal16 determine a net clockwise rotation for M2 and a negligible rotation for M1,because the experimental apparatus is such that the rotation of the polycrystal is clockwise.This means that the crystallites oriented in M1 accumulate more strain than those in M2, andthey are more easily recrystallized. Thus, the residence time of crystals in M2 is larger, andthis maximum appears to be stronger. To overcome this problem, van der Veen and Whillansassumed that new crystals are formed preferentially in the direction containing most crystals.Thus, recrystallized grains are formed at the easy glide orientations not with equal probabilitybut in proportion to the number of crystals that are already near that orientation. The easy glideorientation relative to the M2 maximum is at 90◦ from M1 and does not correspond to thatmaximum because of the effect of the tilt and the bulk rotations; it contains fewer grains thanthe other easy glide orientation, that is in fact the place of the maximum M1. Therefore, M1

is favoured by this third mechanism and ultimately all crystals accumulate on this maximum.

16The sense of the tilt-rotation is shown in Fig. 12 by curved arrows. We remark that the whole rotation of thepolycrystal is not present in the figure because we do not show the form of the sample after the deformation



The criteria shown in this paper seem to be artificial. However, they represent a first ap-proach of the problem and were of great importance in the development of this field.

Another step towards an understanding of recrystallization processes in the dynamics ofpolycrystalline ice has been done by Ktitarev et al. [76] and by Faria et al. [36]. In this casethe approach is essentially numerical and based on a cellular automaton algorithm to simulatesimultaneously grain growth and fabric evolution, including recrystallization and polygoniza-tion processes. The algorithm proved to be quite robust and the simulation results show sat-isfactory agreement with the fabric and texture data from GRIP. However, this approach is sofar restricted to a prescribed stress state and a limited number of crystallites.

Furthermore, recrystallization processes have been included in 2001 in the model of Mor-land and Staroszczyk [118], but through an artificial mechanism: a smooth scaling function,dependent on temperature and on the second invariant of strain rate (not strain!), such that it isunity for low strain rates and falls off to zero when the strain rate surpasses a prescribed criticalvalue. When this scaling function vanishes, the fabric of the medium becomes isotropic again.

(a)

(b)

Fig. 12 Schematic, two-dimensional representation of a volume element. Straight arrows inside thevolume element stand for the c-axes of crystallites well oriented for deformation. Curved arrows indicatethe rotational motion of such crystallites, due to the tilt mechanism explained in Figs. 6 and 7. Dottedlines mark the boundary inside the volume element where the validity of the information given by thecurved arrows is ensured. The arrows outside the volume element indicate the state of the deviatoricstress or stretching acting on it. In case (a), the state of deviatoric stress (or stretching) is pure shear.The c-axes are well oriented at 45◦ with respect to the axes of tension and compression and the tiltmechanism rotates the crystallites towards the axis of compression and away from the axis of tension. In(b) a rigid anticlockwise rotation of the volume element is performed. The result is a sample subjectedto a deformation of simple shear. The situation exposed in panel (b) is shown by two equivalent volumeelements.


GAMM-Mitt. 29, No. 1 (2006) 111

In 2002 Morland [97] generalized the model and replaced the smooth scaling function by thelattice distortion energy. An evolution equation for the lattice distortion energy was proposed;it has the same form as the equation for the dislocation density (39) and (40) to take into ac-count the effects of straining and grain boundary migration. The effect of polygonization was,however, not accounted for. We finally remark again that a continuum macroscopic model,based on the theory of mixtures with continuous diversity [32] was proposed by Placidi [103]to account for grain boundary migration in an evolution equation for the orientation massdensity, a function that is associated to the ODF.

7 Summary and Discussion

In this article we presented a review of the thermomechanical description of the stress–deformation relationship of polycrystalline ice in polar ice masses. The purpose of this anal-ysis was and still is the determination of the proper flow of the ice in the large ice masses onEarth and, in particular, of the age of the ice at any position within the ice sheet. Knowing thedepth–age relationship in ice cores allows reconstruction of the climate in the past millennia(ca. 800000 years for Antarctica).

Until about 20-25 years ago, the ice in glaciers and ice sheets was treated as a non-Newtonian heat conducting incompressible power law fluid. We reviewed the parameteri-zation of this simple model that is known as Glen’s flow law in Glaciology, Norton–Hoff’slaw in Mechanics and Materials Science or Metallurgy and Ostwald-de Waele law or Reinerlaw in Rheology (eq. (5)). First observational recognitions that this simple law would beinsufficient in explaining the differences in creep behaviour of Holocene and Pleistocene icewere attributed to crystal size and impurity dependences of the fluidity as chiefly advocatedfor in the 60s to 80s of the last century and culminating in the concept of a scalar enhance-ment factor. Orientation dependence, i.e., anisotropy of the stress–deformation relationshipwas soon recognized and became evident when thin cuts of ice specimens from deep icecores in Greenland (Dye3, Camp Century, GRIP, GISP2, North GRIP) and Antarctica (Vos-tok, Dome Concordia, Dome Fuji, Dronning Maud Land) became available. The treatmentof the ice as a polycrystalline material with induced anisotropy became inescapable. Precur-sory work on this had started already in the 60s and 70s of the last century. Early attemptsconcentrated on the account of the lattice orientations of the crystallites partly by restrictingthe crystal slip to the direction of easy glide (along the basal planes). The deformationalmisfit of the individual crystals was first eliminated by imposing the Taylor assumption withpoor reproduction of measurements. Imposition of the Sachs–Reuss assumption or of theself-consistent visco-plastic model brought much improved coincidence with data, but notsufficient to restrict possible anisotropy mechanisms to lattice rotation alone. Additional mi-cromechanical processes, subsumed as recrystallization, must be accounted for, if satisfactoryagreement of upscaled anisotropy with data is to be reached. This requires to account for linedefects via dislocation balances, various processes influencing the grain size, among themgrain boundary migration, polygonization, nucleation and recovery. Embedded in a conceptin which the anisotropy tensors of the macroscopic constitutive model are obtained from adistribution function of orientation (and grain size), such formulations have acquired a stateof completeness and maturity that may allow realistic applications in ice flow problems oflarge ice masses.



To this aim, on the basis of the theory of mixtures with continuous diversity [32], a numberof papers has been published in recent years by the present authors and their co-workers[30, 31, 34, 35, 103, 107, 108]. These works provide balance equations and a coherent systemof restrictions for the assignment of constitutive equations. Further contributions addressedthe postulation and reduction of constitutive equations [103, 105, 106]. The target has been togive an appropriate set of equations that can usefully be implemented in a realistic numericalsimulation for the creep and recrystallization of large ice masses. However, there still remainsimportant work to be done. Some of the topics addressed above have not yet been derivedin the context of thermodynamics. Moreover, the deduced constitutive equations must beimplemented in numerical simulations and the results of such simulations must be comparedwith observations. Such comparison will provide new impetus for the refinement of the theoryand the suggestion of new constitutive relations. Finally, a proper, complete derivation ofthese new model equations requires these equations to be in conformity with the Second Lawof Thermodynamics.

Acknowledgements The research reported here has been generously supported throughout the yearsby the Deutsche Forschungsgemeinschaft. We thank Dr. N. Kirchner for her constructive review of anearlier version of this paper.

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A Critical Review of the Mechanics of Polycrystalline Polar Ice

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