The Effect of Anisotropy on the Creep of Polycrystalline Ice

Abstract

Quantitative effects of crystallographic orientation fabrics are incorporated into the flow law for isotropic polycrystalline ice by the introduction of an enhancement factor applied to the isotropic fluidity. An aggregate is viewed to a first approximation as a collection of grains deforming independently by basal glide. The influence of preferred orientations on the mean intragranular rate of strain is treated in terms of a redistribution of the magnitude and orientation of resolved basal shear stress. A quantitative measure of this effect on the fluidity of the aggregate is provided through the development of a geometric tensor and a stress configuration parameter. Intergranular interference is then considered as a dissipative process modifying the fluidity of the aggregate.

Empirical justification for the model at low octahedral shear stresses is provided by several laboratory creep tests on naturally anisotropic bore-hole specimens under both in situ and anomalous stress situations. Predicted enhancement factors ranged from approximately 0.2 to 2.8 and agree well with measured values. The tests were carried out in uniaxial compression and simple shear.

Résumé

Des effets quantitatifs liés à l'orientation cristallographique résultant de la texture sont pris en considération dans la loi d'écoulement généralement utilisée pour la glace polycristalline isotrope en associant un facteur d'accroissement à la fluidité isotrope. Un agrégat est considéré en première approximation comme un ensemble de grains indépendants se déformant d'abord par glissement basal. L'influence des orientations préférentielles sur la vitesse moyenne de déformation intergranulaire est traitée en terme d'une redistribution de la grandeur et de l'orientation de la contrainte de cisaillement réduite dans le plan basal. Une mesure quantitative de cet effet sur la fluidité de l'agrégat est obtenue à l'aide du développement d'un tenseur géométrique et d'un paramètre lié à la configuration de la contrainte. L'accomodation intergranulaire est ensuite considérée comme un processus dissipatif modifiant la fluidité de l'agrégat.

La justification expérimentale du modèle dans le cas de faibles contraintes de cisaillement octaédriques est faite grâce à plusieurs essais de fluage en laborátoire sur des carottes de glace naturelle anisotropes à la fois sous des contraintes, correspondant à celles existant dans les glaciers, et, des contraintes exceptionnelles. Les facteurs d'accroissement prévus ont des valeurs comprises entre 0,2 et 2,8 et correspondent bien aux valeurs mesurées. Les essais sont faits par compression uniaxiale et par cisaillement simple.

Zusammenfassung

Quantitative Einflüsse der kristallographischen Orientierungsstruktur werden in das Fliessgesetz für isotropes polykristallines Eis einbezogen, indem ein Steigerungsfaktor der isotropen Fliessfähigkeit eingeführt wird. Ein Aggregat wird in erster Näherung als eine Ansammlung von Körnern betrachtet, die unabhängig voneinander durch basales Gleiten verformt werden. Der Einfluss bevorzugter Orientierungen auf die mittlere Dehnungs-geschwindigkeit in den Körnern wird durch eine Neuverteilung der basalen Schubspannung nach Grösse und Orientierung berücksichtigt. Ein quantitatives Mass dieser Auswirkung auf die Fliessfähigkeit des Aggregats wird durch die Entwicklung eines geometrischen Tensors und eines Spannungskonfigurations-parameters gegeben. Gegenseitige Beeinträchtigung der Körner wird dann als dissipativer Vorgang betrachtet, der die Fliessfähigkeit des Aggregats verändert.

Das Modell wird bei niedrigen oktaedrischen Schubspannungen durch mehrere Kriechversuche im Labor an natürlich anisotropen Bohrlochproben empirisch bestätigt, sowohl unter in situ- als auch unter anomalen Spannungsbedingungen. Die vorausgesagten Steigerungsfaktoren erstrecken sich von ungefähr 0,2 bis 2.8 und stimmen gut mit gemessenen Werten überein. Die Versuche wurden unter einachsigem Druck und einfachem Schub ausgeführt.

I. Introduction

Empirical flow laws for polycrystalline ice presently in use are based for the most part on the suite of existing data relating to tesselate, fine-grained, isotropic polycrystals. Quantitative effects of preferred crystallographic orientation fabrics are here incorporated into the flow law for an isotropic polycrystal by the introduction of an enhancement factor applied to the isotropic fluidity.

The flow law for polycrystalline ice relates the rate of strain tensor ?_ij to the stress tensor σ_ij. Assuming that hydrostatic pressure does not effect the flow law, Glen (1958) showed that ?_ij is related to the deviatoric stress tensor σ_ij’ through functions of the second and third deviatoric stress invariants

(1)

(2)

In constructing a rheological model to investigate the effects of anisotropy it is useful to recognize the physical characteristics described by these invariants. Σ₂' is frequently specified in terms of the octahedral shear stress

(3)

where the τ_i are the principal shear stresses related by the expression (Jaeger, [^c1969])

(4)

A zero subscript hereafter denotes the octahedral value of the associated tensor defined as in Equation (2). As Σ₂' is (5/3) times the mean square shear stress evaluated over all surface orientations, τ₀ is proportional to the root-mean-square (rms) shear stress.

The third deviatoric stress invariant contains information related to stress configuration. When dependence on Σ₂' is eliminated from Σ₃' by scaling the components of σ_ij' to an octahedral value of 2^1/6, the resulting normalized value of Σ₃’ ranges from +1 for uniaxial tension through zero for pure shear in two dimensions to —1 for uniaxial compression. A more convenient stress configuration parameter is realized by defining the third deviatoric stress invariant in terms of the principal shear stresses. The invariant

(5)

is non-dimensional and independent of Σ₂' while retaining the symmetry and unity limits exhibited by the normalized form of Σ₃’.

Specification of the independent physical invariants τ₀ and Λ together with the orientation of the principal axes of stress is sufficient to define an arbitrary deviatoric stress situation. A precise definition of an arbitrary orientation fabric fwill also be required. Let f be defined as the volume fraction per steradian of an aggregate of total volume V possessing optic axes oriented within the elemental solid angle dΩ:

(6)

It follows that the orientation density of an isotropic aggregate, which we shall call f, is $$

Consider an aggregate in which the octahedral strain rate $$ is a function of the rms resolved basal shear stress, as may be expected if the rate-controlling process is basal glide. Jaeger [^c1969]) shows that the shear stress resolved on a basal plane with direction cosines l_i relative to the principal axes of stress is

(7)

Eliminating the τ_i in Equation (6) using Equations (2), (3), and (4) yields

(8)

where

(9)

is the geometric stress factor (Weertman, 1973) for the grain.

Defining the rms resolved basal shear stress as

(10)

and substituting Equations (5) and (7) gives

(11)

Equation (10) may be evaluated for an isotropic aggregate using

(12)

The result is

(13)

notably independent of Λ. If, as proposed above, $$ is a function of τ_rms, the implication of Equation (12) is that for an isotropic aggregate $$ is a function of τ₀ and not of stress configuration. This suggests a flow law of the form (Nye, 1953)

(14)

The large suite of empirical creep data applicable to isotropic polycrystals presently available indicates (cf. Weertman, 1973) that $$ τ₀ ⁿ, implying (Nye, 1953; Langdon, 1973) that λ $$ where G is the shear modulus for ice.

Weertman (1963) viewed an ice polycrystal as a collection of grains deforming independently by basal glide governed by a monocrystalline flow law of the form

(15)

where $$_B is the basal glide rate and λ_? $$ is the basal fluidity. Assuming an isotropic aggregate in uniaxial compression under an axial stress $$ and using an axial geometric strain-rate factor (b_a) to resolve $$ into its axial component, he found the average resolved axial strain-rate to be

(16)

An additional factor (β) was introduced to account for effects of intergranular interference; the factor $$ is the axial fluidity of the aggregate. As the axial geometric strain-rate factor b_a is only one component of a general transformation tensor b_ij , Equation (15) may be written for each component of $$

(17)

Equation (16) may be regarded as a flow law for anisotropic ice deforming in an arbitrary stress situation, the tensor fluidity λ_ij = λ _Bβb_ijα accounting for the effects of fand Λ through the composite geometric tensor

(18)

If the symmetric part of ay is written $$_ij then the octahedral deformation rate of the aggregate is $$₀ = λ₀τ₀, where

(19)

Finally, consider two aggregates differing only with regard to their orientation fabrics and deforming under identical conditions of temperature, stress situation, etc. If one is isotropic, the ratio of their octahedral deformation rates is

(20)

Equation (19) provides a flow law for an arbitrary orientation fabric and stress situation through application of an enhancement factor E to the flow law for an isotropic aggregate.

As a beginning, the present paper examines the results to be expected at small octahedral shear stresses where both laboratory and field data indicate an approach to a linear stress-strain-rate dependence (n = 1). The geometric tensor, interference factor, and enhancement factor are considered further in Sections 2, 3, and 4, respectively. In Section 5 the results of several specially designed creep tests are provided to establish an empirical justification for the model.

2. The geometric tensor

If it is assumed that the basal glide rate is directed parallel to (Kamb, 1961) and is linearly proportional to the resolved basal shear stress, then $$ while λ_? is independent of stress. Let $$_ij ^B be the strain-rate of the grain in a coordinate system x₁ ^B is parallel tol_i and x₃ ^B is oriented parallel to the resolved shear stress. The strain-rate of the grain in the principal stress coordinates is then given by the transformation

(21)

where c_pq is the cosine of the angle between the positive x_{p B} axis and the positive x_q axis. As the only non-vanishing component of $$, Equation (20) simplifies, becoming

(22)

where c_3i = l_i and c_ij = c_ij(Λ, _l) is a unit vector parallel to the resolved shear stress. Let the geometrically related factors on the right-hand side of Equation (21) be absorbed into the composite geometric tensor α_ij = c_3ic_1jα = b_ijα.

The bulk strain-rate of the non-interacting aggregate is obtained by forming the volumetric mean granular rate of strain:

(23)

Substitution of Equations (21), (17), and (5) into (22) gives

(24)

It follows that the octahedral deformation rate of the non-interacting aggregate is

(25)

where

(26)

3. The interference factor

A sufficient condition for accommodation at grain boundaries is that each grain conform individually to the bulk How situation (Taylor, 1956). Then each grain must carry out a rotational and deformational adjustment $$, where $$_ij is the granular rate of interference. The specific rate of dissipation associated with the adjustment is q =⅓$$_ijσ_ij ’. Since the process is dissipative, the rate of dissipation for the aggregate is given by the root-mean-square granular rate

(27)

The simplifying approximation

(28)

retains first-order interference effects. Hence, substituting Equation (27) into Equation (26),

(29)

where

(30)

The effect of this dissipation is an increase in the aggregate viscosity. If λ_0N ^–1 is the ambient viscosity prior to the inclusion of interference effects and λ₀ ^–1 is the inclusive viscosity, we may write after Batchelor (1967, equation 4.11.16)

(31)

where $$ is the ambient non-interacting octahedral deformation rate given by Equation (24). Eliminating $$ between Equations (28) and (30) and introducing Equations (24) and (25) yields

(32)

Finally, eliminating the ratio λ₀/λ_B between Equations (18) and (31) yields a first-order estimate of the interference factor

(33)

4. The enhancement factor

The analytical integration of Equation (17) for an isotropic orientation density using Equations (11) and the subsequent evaluation of the aggregate octahedral geometric factor yields ^*

(34)

Combining Equations (29), (10), and (12) leads to the result

(35)

The isotropic interference factor may then be found by substituting the above values into Equation (32), obtaining

(36)

It follows, using Equation (19), that the octahedral enhancement factor for an anisotropic aggregate is

(37)

For a given orientation fabric and stress configuration Equations (17), (29), and (32) must then be integrated numerically to obtain E(f, Λ).

The typical creep set-up yields only that strain-rate component associated with the direction of the applied load. Thus, if the model is to be tested conveniently or applied generally to past anisotropic creep results, a definition of component fluidity enhancement is needed. A practical definition is complicated by the need to identify a coordinated system in which the applied load may be expressed as a single tensor component. For uniaxial and simple shear (Jaeger, [^c1969]) tests between parallel opposed platens, the required system Xi^P is clearly associated directly with platen orientation. Let x₁ ^P be directed parallel to the applied shear stress (if any) and x₃ ^P be directed parallel to the platen normal and away from the specimen. It follows that the measured uniaxial and simple shear components are $$ and $$ , respectively. If $$_ij ^P is the geometric tensor expressed in the x_i ^P system, the appropriate component enhancements are ^*

(38)

for uniaxial compression, and

(39)

for simple shear.

5. Results

In view of the paucity of suitably documented creep tests on anisotropic polycrystals, several experiments were designed specifically to provide empirical as well as model-derived enhancements in uniaxial compression (Λ = — 1) and simple shear (Λ = o). The results of these tests are reported in this section.

Anisotropic ice cores were obtained from two quite different stress situations at Law Dome, Antarctica. These cores have been subjected in the laboratory to stress configurations simulating both in situ and anomalous conditions. Three specimens were prepared from the 318 m core at site SGD, the Dome Summit, a region of uniaxial compression (Λ = — 1). The girdle fabric of the parent core is illustrated in Figure 1c. Two of the specimens (318D1 and 318D2) were loaded in uniaxial compression as in situ. The third specimen (318D3) was tilted 90° to the in situ case with the axis of compression normal to the axis of symmetry of the girdle fabric. The octahedral shear stress and temperature in each of the tests were 0.005 MN m^–2 and —10.2°C, respectively.

Fig. 1. Fabrics of (a) laboratory-prepared isotropic ice, (b) Cape Folger core from 200 m depth sectioned at 45° to the horizontal and (c) a horizontal section of Dome Summit core from 318 m depth.

A second set of three specimens was prepared from the 200 m core at site SGF near Cape Folger. The parent core exhibited a strong single-pole fabric (Fig. 1(b)), indicative of its history of simple shear (Jaeger, [^c1969]) flow in that region. In order to reinstate the in situ stress configuration (Λ = o), these specimens were mounted between parallel platens and a constant shear load applied. The plattens were constrained to remain equidistant to simulate the in situ situation of simple shear How. Specimen 200F1 was oriented as m situ (single-pole axis normal to the platens) while the single pole axes of specimens 200F2 and 200F3 were tilted at 22.5^° and 45°, respectively, to the platen normal into the direction of the applied shear stress. These tests were carried out at 0.04 MN m^–2 octahedral and -6.0°C.

Fig. 2. Creep curves of axial strain versus time for specimens in uniaxial compression.

Fig. 3. Creep curves of shear strain versus time for specimens in simple shear.

With each set, an isotropic aggregate of similar grain size was tested under identical conditions. A typical orientation fabric for these laboratory-prepared samples is shown in Figure 1(a). Figures 2 and 3 show the creep curves obtained for the uniaxial and shear tests, respectively.

Steady-state secondary creep rates were required to evaluate the observed component enhancements. Reference to Figure 3 suggests that the samples deforming in simple shear had settled to steady-state creep rates during the latter half of the experiment. Slopes of the creep curves for the period from 140 to 240 h were thus used to calculate the observed shear enhancements.

At the lower temperature and smaller octahedral shear stress used in the uniaxial experiment the duration of the primary stage of creep was much longer. The residual curvature at the conclusion of the experiment (Fig. 2) appears to be less for the pre-strained bore-hole material than for the isotropic sample. Therefore it may be expected that the observed axial enhancements, based on the slopes of the creep curves between 1000 and 1200 h, slightly underestimate the actual steady-state axial enhancements. The observed component enhancement for each of the six anisotropic specimens with respect to its associated isotropic control sample is entered in Table 1.

Table 1. High-Frequency Conductivity of Ice

For each test, the measured orientation fabric, stress configuration, and tilt orientation of the specimen were presented as inputs to a computer program which performed the numerical integrations and calculations necessary to evaluate the component enhancements. The resulting estimates of the component enhancements calculated from the model are listed in Table I above the observed enhancements.

6. Conclusions

The close agreement between observed and model-derived enhancements given in Table 1 points to the validity of the present linear model at the small octahedral stresses involved. The restriction of the present model to linear stress-strain-rate dependence suggests a further examination, theoretical and empirical, of non-linear effects. A preliminary model incorporating a power-law rheology predicts that an equivalent power exponent be applied to the linear enhancements. This result seems to be supported by additional uniaxial and simple-shear tests on isotropic and natural anisotropic aggregates now completed in the octahedral stress range between 0.05 and 1.6 MN m^–20.

It may be concluded that the use of a flow law derived from laboratory studies of isotropic ice to model flow in regions of natural ice masses exhibiting strong crystallography anisotropy will lead to underestimates of actual strain-rates by a factor likely to be in excess of 10. The enhancement factor described here offers an effective quantitative means of accounting for the major effects of crystallography. If, as field results suggest, the crystallographic fabric of natural ice masses is distributed in a systematic way throughout, then the enhancement factor, which depends on the local crystallographic fabric and stress situation, could be used in a flow law as a function of location.