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Localization of deformation in polycrystalline ice: experiments and numerical simulations with a simple grain model
Computational Materials Science 25 (2002) 142–150 www.elsevier.com/locate/commatsci
Localization of deformation in polycrystalline ice: experiments and numerical simulations with a simple grain model P. Mansuy, J. Meyssonnier *, A. Philip Laboratoire de Glaciologie et G eophysique de l’Environnement, CNRS et Universit e Joseph Fourier (UJF-Grenoble I), B.P. 96, F-38402 Saint-Martin d’H eres Cedex, France
Abstract Creep tests were carried out on structure-controlled laboratory-made ice specimens to assess a simple constitutive model which accounts for the outstanding viscoplastic anisotropy of ice. A homogeneous deformation was observed when testing a circular monocrystalline inclusion embedded in a fine-grained isotropic ice matrix, whereas severe localization of the deformation, essentially in the form of kink bands, was observed in a multicrystalline inclusion made of a large central grain surrounded by a crown of medium-size grains. Finite-element simulations were performed by assuming that the grain behaves as a transversely isotropic medium. A very good agreement was obtained for the monocrystalline inclusion when using the grain model parameters derived from data on isolated ice single crystals. However, the simulation fails to reproduce accurately the heterogeneous deformation of the multicrystal, although it provides a good prediction of the locations where localization features are susceptible to appear. 2002 Elsevier Science B.V. All rights reserved. PACS: 83.50.Nj; 81.40.Lm; 83.20.)d; 92.40.Sn Keywords: Strain localization; Viscoplasticity, yield stress; Deformation, plasticity, and creep; Constitutive relations; Ice; FE modelling
1. Introduction The ice single crystal possesses a hexagonal P63 / mmc type structure, close to hcp (c=a ¼ 1:629). It deforms essentially by dislocation glide on the basal plane, perpendicular to the rotational symmetry c-axis (see Fig. 1(a)). At high temperature
* Corresponding author. Tel.: +33-47682-4270; fax: +3347682-4201. E-mail address: [email protected] (J. Meyssonnier).
the isotropic polycrystal viscoplastic behaviour is described by a power law with stress exponent n ¼ 3, whereas for the single-crystal stationary creep the exponent is between 1.5 and 2 [1–3]. This discrepancy has been explained by invoking basal dislocation climb [3]. However, neither this mechanism, nor cross-slip, or twinning, or significant dislocation glide on an other system than basal, have been observed yet [4]. The predominance of basal glide results in a marked anisotropy of the ice crystal viscoplastic behaviour: the stress required to produce the same effective strain rate in a non-basal plane is at least 60 times greater than in
0927-0256/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 2 ) 0 0 2 5 8 - 6
P. Mansuy et al. / Computational Materials Science 25 (2002) 142–150
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fx1 ; x2 ; x3 g, and Fig. 1. (a) Hexagonal structure of ice Ih; (b) schematic structure of the specimens; and (c) global reference frame R local reference frame Rfx1 ; x2 ; x3 g attached to the grain (the grain c-axis corresponds to the x3 -axis).
the basal plane [3]. In polycrystalline ice, the lack of glide systems other than basal leads to large incompatibilities between grains, then to the buildup of high internal stresses. Under the conditions prevailing in a polar ice-sheet (very low deviatoric stresses, low temperatures, strain rates in the order of 1012 s1 ), strain induced lattice preferred orientations develop (see e.g. [5]), and, owing to the extremely high anisotropy of the grain viscoplastic behaviour, significant macroscopic anisotropies result [6,7]. Therefore, modelling texture evolution and the associated macroscopic anisotropy of polar ice is a major issue for improving ice-sheet flow simulations performed to study the ice-sheet climate interactions. The micro–macro models currently developed to this aim [8,9], are based on constitutive models for the ice grain behaviour which need to be improved, mainly because the precise way in which accommodation of ice deformation occurs is not yet known. This requires a better understanding of the mechanisms leading to strain heterogeneity and localization inside the grains. The purpose of the present study is to assess the validity of a simple model of grain by comparing experimental data with numerical simulations of the experiments.
2. Experiments In order to achieve a better knowledge of the processes involved in the deformation of polycrystalline ice, and in particular to apprehend the
mechanisms responsible for intragranular strain heterogeneity, creep tests were carried out on specially designed specimens of laboratory-grown ice multicrystals. 2.1. Specimens microstructure and testing conditions The ice specimens were given the shape of thick plates 210 mm high, 140 mm wide and 8 mm thick. Each specimen was a thick section of a mono or multicrystalline inclusion embedded in a finegrained ice matrix (see Fig. 1(b)). The grains of the inclusion were between 7 and 40 mm in diameter, and cut so that their c-axes were in the plane of the specimen. The grains of the matrix were oriented at random, and their size was small compared to the smallest dimension of the specimen, between 0.7 and 1.6 mm in diameter, so that the matrix could be considered as macroscopically isotropic at the multicrystal scale. A detailed description of the specimen preparation procedures is given by [10–12]. The ice specimens were tested under compression creep exerted in their plane. Each specimen was inserted between two glass plates placed on each largest side to avoid buckling and to impose a condition of plane strain (see Fig. 1(b)). The axial strain was obtained from the axial shortening of the specimen measured with a LVDT within 5 lm. To prevent the formation of cracks the compressive stress was limited to 1 MPa. Embedding the studied multicrystal in a macroscopically homogeneous matrix allowed to avoid possible development of localization structures
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generated by stress concentrations at the interface between the specimen and the platens of the testing device. On the other hand, since the boundary conditions are known only at the matrix-platens interface, the interpretation of experimental data required to perform numerical simulations of the tests. The special shape and structure of the specimens allowed to make the deformation of the multicrystal visible during the tests. By placing the deforming specimen between crossed polarizers, with light transmitted from behind, each grain of the multicrystal appeared with a different color, owing to the birefringence of ice (the c-axis is the optical axis). Photographs taken at regular time intervals allowed to follow the evolution of the shape of the inclusion, of the rotation of its basal planes, and of the strain localization features. The tests were conducted in a cold room at a controlled temperature of 10 0:1 C. Their duration was up to 45 days, depending on the level of loading. Different types of inclusions were studied, however the present paper focuses on the two types which enlighten the influence of the neighbourhood of a
grain on its propensity to exhibit a heterogeneous deformation, namely: • cylindrical monocrystalline inclusion, 30 mm in diameter; • rosette-shaped inclusion made of a cylindrical monocrystal, 40 mm in diameter, surrounded by a crown of halved cylindrical monocrystals about 7 mm in diameter. In the following, these specimens are referred to by type-A and type-B, respectively (see Fig. 2(a) and (b)). 2.2. Experimental results 2.2.1. Type-A cylindrical inclusion Several creep tests were carried out with different orientations w of the inclusion c-axis with respect to the loading axis, namely 0, 25, and 45 (see Fig. 1(c)) and different compressive stresses (0.5, 0.75, and 1 MPa). For configurations with non-zero resolved shear stress in the inclusion
Fig. 2. (a) Photograph of type-A specimen with an initially circular monocrystalline inclusion, at 6.3% axial strain: the parallel lines are the traces of the basal planes, and the white areas at the inclusion-matrix boundary are evidence of dynamic recrystallization; (b) rosette-shaped specimen B1 at 0% axial strain (w ¼ 72:5); (c) specimen B1 at 2% axial strain: note the slight flexion of the basal planes and the appearance of faint bands perpendicular to the basal planes; (d) specimen B1 at 6.5% axial strain: the white bands crossing through the central grain are kink bands; (e) basal plane RSS map computed at 6.5% axial strain (the white color corresponds to zero RSS, the grey scale step is 0.005 MPa); and (f) map of c-axis misorientation with respect to average for the central grain (the white color corresponds to the average of 62, the grey scale step is 2).
P. Mansuy et al. / Computational Materials Science 25 (2002) 142–150
basal planes, color undulations, under the form of blurred stripes perpendicular to the c-axis, were observed from the beginning of the tests. These markers of basal glide activation evolved in thin dark lines, corresponding to steps at the surface of the specimen, with increasing strain (see Fig. 2(a)). These dark lines allowed to record the rotation of the inclusion c-axis towards the loading axis during the tests. Whatever its initial crystallographic orientation, the shape of the inclusion, initially circular, remained always ellipsoidal, with a more or less pronounced aspect ratio depending on its orientation and on the specimen axial strain achieved. Meanwhile, the inclusion long axis rotated in the same direction as the basal plane but not at the same rotation rate. Moreover, the dimension of the inclusion in the direction of the caxis did not show any measurable variation during the tests, even in the most deformed configurations (i.e. up to 50% strain in the inclusion). At the beginning of the loading, a pattern of faint blurred bands appeared orthogonal to the basal planes. However, after 1% specimen axial strain these bands tended to disappear, except for two faint bands situated at each far end of the elliptically deformed inclusion. No other strain localization feature were observed during the tests. For the configuration with basal planes orthogonal to the loading axis (w ¼ 0), the inclusion did not show any observable deformation, while the matrix seemed to flow around it. 2.2.2. Rosette-shaped type-B specimens Two tests were carried out with the initially inner cylindrical grain oriented at w ¼ 72:5 (specimen B1) and w ¼ 0 (specimen B2), respectively. Type-B differ from type-A specimens only by the number of grains in contact with the inner large monocrystal: 18 medium-size grains for typeB, against more than 400 small-size matrix grains for type-A. This change in the monocrystal environment resulted in observations departing markedly from that of type-A specimens during its deformation. For specimen B1, the central grain of the multicrystal exhibited severe kinking and bending. Each kink band was initiated at a triple junction of the central grain with two medium-size grains of the surrounding crown. It formed per-
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pendicular to the basal planes of the central grain, and crossed right through it (see Fig. 2(c)). During the tests the kink bands rotated in the direction opposite to that of the bulk central grain basal planes, i.e. they did not remain perpendicular to these planes as the deformation of the specimen increased. At 6.5% specimen axial strain, the shape of the central grain departed from ellipsoidal, and a network of kink bands, 0.1–1.5 mm wide, approximately parallel to each other, delimited long areas of the central crystal (see Fig. 2(d)). One of these areas exhibited polygonization with a misorientation of 8 each side of the sub-boundary (see top of Fig. 2(d)). However, a majority exhibited a progressive change of the lattice orientation from one boundary of the central grain to the other, with a misorientation between 3 and 5. The c-axis orientation with respect to the loading axis was about 90 inside the kink bands, whereas it varied from about 55 to 62 outside, depending on the area considered. The kink bands c-axis misorientation with respect to the adjacent areas was about 27. Dynamic recrystallization occurred mainly in the fine-grained matrix whose grain size at 12.5% axial strain was increased up to about 2 to 5 mm, and in the surrounding medium-size grains, some of them being totally destroyed and replaced with much smaller grains. Specimen B2, whose c-axis was initially parallel to the direction of compression, exhibited no localization features, except for two faintly bended areas on the left and right sides. Its deformation remained small and non-uniform, concentrated in a few activated basal planes.
3. Numerical model The experiments were interpreted through finite-element simulations. 3.1. Multicrystal and matrix behaviour The ice single-crystal deforms essentially by slip on the basal planes, with no preferential glide direction [13]. The present model takes this feature into account by considering each grain of the ice multicrystal as a transversely isotropic continuous medium, whose plane of isotropy corresponds to
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the basal plane of the grain [9], and with a very weak resistance to shear parallel to this basal plane. In the same line, the fine-grained ice matrix is considered as a homogeneous isotropic medium. The interest of this homogeneous continuous formulation is that it is easier to handle in finiteelement computations than a model based on dislocation glide on distinct crystallographic planes which is otherwise not ascertained, while retaining the first order effects of the outstanding anisotropy of ice. The simplest anisotropic constitutive law for the non-linear viscoplastic behaviour of the grain is adopted. It is similar to that proposed by [14], and derives from a dissipation potential (see [15]). In the local reference frame R attached to the crystal with rotational symmetry axis x3 along the c-axis, it is written as s11 s22 ¼ 2gH 12 ðd11 d22 Þ; s33 ¼ 2gH 12
4a1 3
d33 ;
s23 ¼ 2gH 12 bd23 ; s31 ¼ s12 ¼
ð1Þ
2gH 12 bd31 ; 2gH 12 d12 ;
where s and d are the deviatoric stress and the strain-rate tensors, respectively, gH 12 is the apparent viscosity for shear in the (basal) plane of isotropy ðx1 ; x2 Þ, a is the ratio of the axial viscosity 1 along the x3 -axis to that in the plane of isotropy, and b is the ratio of the viscosity for shear parallel to the plane of isotropy to gH 12 . The apparent viscosity is defined as ð1nÞ=n
1=n c_ } gH 12 ¼ A
¼ A1 s1n } ;
ð2Þ
where A is a temperature dependent fluidity parameter, and c_ } and s} are two invariants by rotation about the x3 -axis (but not under any change of reference frame). c_ } and s} are defined by 4a 1 2 c_ ax þ c_ 2? þ bc_ 2k ; 3 3 1 s2 þ s2? þ s2k s2} ¼ 4a 1 ax b c_ 2} ¼
ð3Þ
1 The axial viscosity is defined as the halved ratio of the axial deviatoric stress to the axial strain rate in a uniaxial creep test.
with 2 ; c_ 2ax ¼ 3d33 2 2 ; c_ 2? ¼ ðd11 d22 Þ þ 4d12 2 2 þ d31 Þ; c_ 2k ¼ 4ðd23
s2ax ¼ 3s233 =4;
ð4Þ
2
s2? ¼ ðs11 s22 Þ =4 þ s212 ; s2k ¼ s223 þ s231 ; and they are linked by s} ¼ gH 12 c_ } , thus c_ } ¼ Asn} :
ð5Þ
When a ¼ b ¼ 1 the medium is isotropic, c_ 2} ¼ c_ 2 ¼ 2dij dij , s2} ¼ s2 ¼ sij sij =2, and (1) reduces to Norton–Hoff’s law (known as Glen’s law in Glaciology). This isotropic version was used to describe the fine-grained matrix behaviour with temperature dependent fluidity parameter B, namely: sij ¼ 2B1=n c_ ð1nÞ=n dij :
ð6Þ
Estimates of the parameters involved in relations (1) and (6) were deduced from experimental data. The behaviour of isotropic ice is welldescribed and documented: the exponent n of the power law is 3 for s > 0:1 MPa [3,16], and, according to [16–18], the mean value B ¼ 18:6 MPa3 a1 at 10 C was adopted. The grain parameters in (1) were obtained either by considering the grain as an isolated single crystal [3] or by using data from mechanical tests on polar ice with single maximum fabric (i.e. involving the influence of grain boundaries) [6,7]. Assuming n ¼ 3 in (1), the best set (A,a,b) was obtained by comparing type-A experiments with numerical simulations (see Section 4). 3.2. Numerical scheme The computations were done in a fixed global whose x1 axis is perpendicular reference frame R to the plane of the specimens. Since all the multicrystal c-axes (x3 -axes) lie in this plane, each local reference frame R attached to a grain of the multicrystal was chosen so that its x1 -axis coincides with x1 . The orientation of the x3 -axis (c-axis) is denoted by of R with respect to the x3 -axis of R
P. Mansuy et al. / Computational Materials Science 25 (2002) 142–150
by R. w, and the matrix for passing from R to R The loading conditions (loading along the x3 -axis, plane strain) allowed to consider a two-dimensional (2D) problem. The constitutive models (1) and (6) were implemented in a 2D finite-element code using a velocity–pressure formulation [19] with a quadratic interpolation of the velocity components and a linear interpolation of the pressure. Type-A specimens were discretized with 568 six-nodes triangular elements (T6) for the matrix and 324 for the inclusion, type-B specimens with 634 T6-elements for the matrix, 426 for the central grain of the multicrystal, and 216 for the 18 surrounding medium-size grains. The boundary conditions were prescribed according to the experimental conditions, that is, free lateral boundaries, zero velocity on the lower boundary, and constant vertical velocity (parallel to the loading axis) on the upper boundary adjusted at each time step to ensure a constant compressive stress condition. The gravity forces were neglected. The rotation rate of the multicrystal grain caxes was derived from the transformation formula for the spin tensor under a change of reference frame: R ¼ R1 R_ þ x; R1 x
ð7Þ
of a grain dewhich expresses that the spin x composes into the sum of the viscoplastic spin x measured in its rotating reference frame R, plus [9]. By the rate of rotation of R with respect to R expressing that the basal planes of the ice crystal remain parallel to each other during the deformation (i.e. the component u3 of the velocity along the x3 -axis, measured in the rotating frame R, is independent of x2 ), the only non-zero component of x is x23 ¼ d23 , and since in the 2D flow the sole 23 , is invariant by , x non-zero component of x rotation around the x1 -axis, Eq. (7) leads to w_ ¼ ou3 =ox2 :
ð8Þ
The evolution of the deformation of the specimen was simulated by updating the position of each node from the finite-element computed velocity field, and the multicrystal c-axes orientations from (8), at each time step. This time step was
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chosen so that the increment of axial strain was less than 2 103 .
4. Numerical simulations Numerical simulations carried out for type-A specimens homogeneous deformation allowed to constrain the set of parameters for the grain model (1), by comparing the computed and experimental evolutions of the average c-axis orientation hwi of the elliptical monocrystalline inclusion and of its aspect ratio, versus the specimen axial strain (see Fig. 3). These simulations were performed by assuming a constant value of w inside each triangular element of the inclusion. The maximum misorientation in the inclusion, occurring in the elements at the inclusion-matrix interface, was 3:3 after 6% axial strain (and only 0:6 elsewhere). The parameter a was found to have practically no influence in the range 0.5–10, and was subsequently fixed to a ¼ 1 [10]. As expected the inclusion behaviour is mainly influenced by parameter b which characterizes the crystal anisotropy. From the comparisons shown in Fig. 3, the parameters deduced from isolated single-crystal data (set 1, Table 1) allow to fit the experimental curves quite well (within the accuracy of the experimental measurements), contrary to set 2, derived from polar ice data, which underestimates the inclusion aspect ratio and the c-axis rotation rate (Fig. 3). The simulation of type-B experiment B1 shown in Fig. 2(b)–(d) was performed with the set of parameters suitable for type-A simulations. Further refinement was obtained by using a quadratic interpolation of the c-axis orientation in each element of the multicrystal. Fig. 3 shows the evolution versus the specimen axial strain of the central grain aspect ratio and average c-axis orientation. This latter was measured in the crystal areas delimited by the kink bands, excluding the highly misoriented zones within the kink bands. The simulated evolution of the shape of the central grain is slower than that observed, as well as the simulated average rotation of the basal planes (see Fig. 3). However the rigid-body rotation of this grain (i.e. the rotation of its long axis) is in good agreement with observations.
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Fig. 3. Observed (symbols) and computed (lines) evolutions of the aspect ratio and of the crystallographic orientation as a function of the total strain for: (left) type-A specimens monocrystalline inclusion; (right) specimen B1 central grain. The different sets of grain model parameters are given in Table 1. The thin and dotted lines for B1 specimens correspond to simulations performed with an equivalent homogeneous monocrystalline inclusion.
Table 1 Values of inclusion parameters A=B, a and b used in numerical simulations (n ¼ 3, B ¼ 18:6 MPa3 a1 ) Set
A3 =B3
a
b
1 2 3
0.1 0.17 0.1
1 1 1
0.01 0.13 0.0001
The simulation of localization features was based on the following simple considerations. Each kink band delimits two zones of the crystal with the same basal plane orientations, while its interior basal planes remain parallel to each other, but at high angle from the adjacent zones. The two kink planes act as dislocation walls fed by dislocations which glide on the basal planes [20]. Owing to the fact that only the basal system is activated and the sole efficient obstacles to dislocation motion are the grain boundaries, a dislocation wall can be seen, in a 2D situation, as the line which intersects the basal planes at points where the gliding dislocations come to arrest, i.e. where the resolved shear stress (RSS) vanishes. According to this simple model, basal planes bending, susceptible to evolve into a polygonization wall and eventually into a sub-boundary, should correspond to a change of sign of the RSS, whereas
a kink band should correspond to two successive changes of RSS sign on a short distance (see Fig. 4). The basal RSS computed for the central grain of specimen B1 at 7% axial strain, vanishes along two lines situated at the ends of the grain and along one line which crosses right through the grain (see Fig. 2(e)). These lines are parallel to each other and roughly parallel to the observed kink bands. They are linked to one or two triple junctions at the grain boundary. Elsewhere the RSS
Fig. 4. Schematic of the RSS based model: on the single glide plane, geometrically necessary dislocations are arrested at zero RSS locations.
P. Mansuy et al. / Computational Materials Science 25 (2002) 142–150
map shows a pattern with zones of highest RSS values elongated in the direction of the basal planes and of the simulated kink band. The RSS gradients are not high enough to show any sign of polygonization. The map of the computed misorientation of the elements in the central grain with respect to the average hwi ¼ 62 shows four bands parallel to each other and to the direction of the kink bands (see Fig. 2(f)). The positive misorientation inside these bands is in agreement with that of the observed kink bands, however the maximum misorientation is only 15.
5. Conclusion Compressive creep tests carried out on two types of structure-controlled ice specimens with a mono and multicrystalline inclusion in a homogeneous fine-grained ice matrix allow to enlighten the phenomena leading to strain localization in a polycrystal of ice. The deformation of type-A initially circular monocrystalline inclusions did not exhibit any other strain localization feature than that associated to basal glide. The deformation at the inclusion scale was then very homogeneous. Type-B inclusions exhibited strain heterogeneity, closely related to stress concentrations at the triple junctions between the large central grain and its surrounding medium-size grains, with the development of kink bands associated to lattice bending. Finite-element simulations based on a simple model for the ice grain behaviour which account for the extreme viscoplastic anisotropy of ice, allowed to reproduce the homogeneous deformation of type-A specimens quite well. Since the severe bending occurring at the kink planes could not be described correctly due to the relative mesh coarseness, even with a quadratic interpolation of the c-axes orientations in each element, the simulation of specimen B1 heterogeneous deformation is not satisfactory. However, considering that only the basal system is activated allows to obtain some information on the location of localization features in relatively good agreement with the observations. Further development of the numerical simulations should use this information for adaptative mesh refinement.
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Modelling the macroscopic behaviour of anisotropic polar ice and the evolution of its anisotropy is a crucial matter for ice-sheet flow modelling in relation with climate changes. Up to now, grain to grain interactions are quite ignored in current polycrystal models which aim at modelling polar ice properties, and are based on onesite self-consistent approaches [9,21]. In view of accounting for the heterogeneous deformation of the ice grain in such homogenization schemes, the possibility of simulating type-B specimen behaviour with an equivalent homogeneous inclusion in an isotropic matrix was investigated. Since the observed average lattice rotation is faster for typeB than for type-A specimens, the equivalent inclusion should be modelled with a very low value of the grain-model anisotropy parameter b to simulate the crystallographic evolution of the grain. However, doing so the simulated shape evolution is too fast (see Fig. 3), so that there is no straightforward solution to this problem. However, since crystallographic texture evolution is predominant as regards the macroscopic polycrystal behaviour, this experimental-numerical approach deserves further investigation.
Acknowledgements The financial supports of the Centre National de la Recherche Scientifique (Programme INSUGeomateriaux) and of Universite Joseph Fourier Grenoble-I are greatly acknowledged.
References [1] A. Higashi, S. Koinuma, S. Mae, Plastic yielding in ice single crystals, Jpn. J. Appl. Phys. 3 (10) (1964) 610– 616. [2] A. Higashi, S. Koinuma, S. Mae, Bending creep of ice single crystals, Jpn. J. Appl. Phys. 4 (8) (1965) 575–582. [3] P. Duval, M.F. Ashby, I. Andermann, Rate-controlling processes in the creep of polycrystalline ice, J. Phys. Chem. 87 (21) (1983) 4066–4074. [4] V.F. Petrenko, R.W. Whitworth, Physics of Ice, first ed., Oxford University Press, Oxford, 1999. [5] T. Thorsteinsson, J. Kipfstuhl, H. Miller, Textures and fabrics in the GRIP ice core, J. Geophys. Res. 102 (C12) (1997) 26583–26600.
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[6] P. Duval, H. Le Gac, Mechanical behaviour of Antarctic ice, Ann. Glaciol. 3 (1982) 92–95. [7] P. Pimienta, P. Duval, V.Y. Lipenkov, Mechanical behaviour of anisotropic polar ice, in: Symposium on The Physical Basis of Ice Sheet Modelling, Vancouver, International Association of Hydrological Sciences, Publication 170, 1987, pp. 57–66. [8] O. Castelnau, Modelisation du comportement mecanique de la glace polycristalline par une approche auto-coherente; application au developpement de textures dans les glaces des calottes polaires, These de Doctorat, Universite Joseph Fourier, Grenoble-I, 1996. [9] J. Meyssonnier, A. Philip, A model for the tangent viscous behaviour of anisotropic polar ice, Ann. Glaciol. 23 (1996) 253–261. [10] P. Mansuy, J. Meyssonnier, A. Philip, Modelling the ice single-crystal viscoplastic behaviour, in: K. Hutter, Y. Wang, H. Beer (Eds.), Advances in Cold-Region Thermal Engineering and Sciences, Technological, Environmental and Climatological Impact, Lecture Notes in Physics, vol. 533, Springer-Verlag, Berlin, 1999, pp. 215–224. [11] P. Mansuy, A. Philip, J. Meyssonnier, Identification of strain heterogeneities arising during deformation of ice, Ann. Glaciol. 30 (2000) 121–126. [12] P. Mansuy, Contribution a l’etude du comportement viscoplastique d’un multicristal de glace: heterogeneite de la deformation et localisation, experiences et modeles, These de Doctorat, Universite Joseph Fourier, Grenoble-I, 2001. [13] W.B. Kamb, The glide direction in ice, J. Glaciol. 3 (30) (1961) 1097–1106.
[14] A.F. Johnson, Creep characterization of eutectic composites, in: Mechanical Behaviour of Anisotropic Solids, Proceedings of the Euromech Colloquium 115, J.P. Boehler ed., Martinus Nijhoff Publishers and Editions du CNRS, 1982, pp. 775–788. [15] J. Meyssonnier, A. Philip, Remarks on self-consistent modelling of polycrystalline ice, in: K. Hutter, Y. Wang, H. Beer (Eds.), Advances in Cold-Region Thermal Engineering and Sciences, Technological, Environmental and Climatological Impact, Lecture Notes in Physics, vol. 533, Springer-Verlag, Berlin, 1999, pp. 225–236. [16] L. Lliboutry, P. Duval, Various isotropic and anisotropic ices found in glacier and polar ice caps and their corresponding rheologies, Ann. Geophys. 3 (2) (1985) 207– 224. [17] T.H. Jacka, The time and strain required for development of minimum strain rates in ice, Cold Reg. Sci. Technol. 8 (3) (1984) 261–268. [18] J. Meyssonnier, A. Goubert, Transient creep of polycrystalline ice under uniaxial compression: an assessment of internal state variable models, Ann. Glaciol. 19 (1994) 55– 62. [19] J. Meyssonnier, Ice flow over a bump: experiment and numerical simulations, J. Glaciol. 35 (119) (1989) 85–97. [20] R.W.K. Honeycombe, The Plastic Deformation of Metals, second ed., Edward Arnold Editors, 1984. [21] O. Castelnau, P. Duval, R.A. Lebensohn, G. Canova, Viscoplastic modeling of texture development in polycrystalline ice with a self-consistent approach: Comparison with bound estimates, J. Geophys. Res. 101 (6) (1996) 13851–13868.
Localization of deformation in polycrystalline ice: experiments and numerical simulations with a simple grain model P. Mansuy, J. Meyssonnier *, A. Philip Laboratoire de Glaciologie et G eophysique de l’Environnement, CNRS et Universit e Joseph Fourier (UJF-Grenoble I), B.P. 96, F-38402 Saint-Martin d’H eres Cedex, France
Abstract Creep tests were carried out on structure-controlled laboratory-made ice specimens to assess a simple constitutive model which accounts for the outstanding viscoplastic anisotropy of ice. A homogeneous deformation was observed when testing a circular monocrystalline inclusion embedded in a fine-grained isotropic ice matrix, whereas severe localization of the deformation, essentially in the form of kink bands, was observed in a multicrystalline inclusion made of a large central grain surrounded by a crown of medium-size grains. Finite-element simulations were performed by assuming that the grain behaves as a transversely isotropic medium. A very good agreement was obtained for the monocrystalline inclusion when using the grain model parameters derived from data on isolated ice single crystals. However, the simulation fails to reproduce accurately the heterogeneous deformation of the multicrystal, although it provides a good prediction of the locations where localization features are susceptible to appear. 2002 Elsevier Science B.V. All rights reserved. PACS: 83.50.Nj; 81.40.Lm; 83.20.)d; 92.40.Sn Keywords: Strain localization; Viscoplasticity, yield stress; Deformation, plasticity, and creep; Constitutive relations; Ice; FE modelling
1. Introduction The ice single crystal possesses a hexagonal P63 / mmc type structure, close to hcp (c=a ¼ 1:629). It deforms essentially by dislocation glide on the basal plane, perpendicular to the rotational symmetry c-axis (see Fig. 1(a)). At high temperature
* Corresponding author. Tel.: +33-47682-4270; fax: +3347682-4201. E-mail address: [email protected] (J. Meyssonnier).
the isotropic polycrystal viscoplastic behaviour is described by a power law with stress exponent n ¼ 3, whereas for the single-crystal stationary creep the exponent is between 1.5 and 2 [1–3]. This discrepancy has been explained by invoking basal dislocation climb [3]. However, neither this mechanism, nor cross-slip, or twinning, or significant dislocation glide on an other system than basal, have been observed yet [4]. The predominance of basal glide results in a marked anisotropy of the ice crystal viscoplastic behaviour: the stress required to produce the same effective strain rate in a non-basal plane is at least 60 times greater than in
0927-0256/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 2 ) 0 0 2 5 8 - 6
P. Mansuy et al. / Computational Materials Science 25 (2002) 142–150
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fx1 ; x2 ; x3 g, and Fig. 1. (a) Hexagonal structure of ice Ih; (b) schematic structure of the specimens; and (c) global reference frame R local reference frame Rfx1 ; x2 ; x3 g attached to the grain (the grain c-axis corresponds to the x3 -axis).
the basal plane [3]. In polycrystalline ice, the lack of glide systems other than basal leads to large incompatibilities between grains, then to the buildup of high internal stresses. Under the conditions prevailing in a polar ice-sheet (very low deviatoric stresses, low temperatures, strain rates in the order of 1012 s1 ), strain induced lattice preferred orientations develop (see e.g. [5]), and, owing to the extremely high anisotropy of the grain viscoplastic behaviour, significant macroscopic anisotropies result [6,7]. Therefore, modelling texture evolution and the associated macroscopic anisotropy of polar ice is a major issue for improving ice-sheet flow simulations performed to study the ice-sheet climate interactions. The micro–macro models currently developed to this aim [8,9], are based on constitutive models for the ice grain behaviour which need to be improved, mainly because the precise way in which accommodation of ice deformation occurs is not yet known. This requires a better understanding of the mechanisms leading to strain heterogeneity and localization inside the grains. The purpose of the present study is to assess the validity of a simple model of grain by comparing experimental data with numerical simulations of the experiments.
2. Experiments In order to achieve a better knowledge of the processes involved in the deformation of polycrystalline ice, and in particular to apprehend the
mechanisms responsible for intragranular strain heterogeneity, creep tests were carried out on specially designed specimens of laboratory-grown ice multicrystals. 2.1. Specimens microstructure and testing conditions The ice specimens were given the shape of thick plates 210 mm high, 140 mm wide and 8 mm thick. Each specimen was a thick section of a mono or multicrystalline inclusion embedded in a finegrained ice matrix (see Fig. 1(b)). The grains of the inclusion were between 7 and 40 mm in diameter, and cut so that their c-axes were in the plane of the specimen. The grains of the matrix were oriented at random, and their size was small compared to the smallest dimension of the specimen, between 0.7 and 1.6 mm in diameter, so that the matrix could be considered as macroscopically isotropic at the multicrystal scale. A detailed description of the specimen preparation procedures is given by [10–12]. The ice specimens were tested under compression creep exerted in their plane. Each specimen was inserted between two glass plates placed on each largest side to avoid buckling and to impose a condition of plane strain (see Fig. 1(b)). The axial strain was obtained from the axial shortening of the specimen measured with a LVDT within 5 lm. To prevent the formation of cracks the compressive stress was limited to 1 MPa. Embedding the studied multicrystal in a macroscopically homogeneous matrix allowed to avoid possible development of localization structures
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generated by stress concentrations at the interface between the specimen and the platens of the testing device. On the other hand, since the boundary conditions are known only at the matrix-platens interface, the interpretation of experimental data required to perform numerical simulations of the tests. The special shape and structure of the specimens allowed to make the deformation of the multicrystal visible during the tests. By placing the deforming specimen between crossed polarizers, with light transmitted from behind, each grain of the multicrystal appeared with a different color, owing to the birefringence of ice (the c-axis is the optical axis). Photographs taken at regular time intervals allowed to follow the evolution of the shape of the inclusion, of the rotation of its basal planes, and of the strain localization features. The tests were conducted in a cold room at a controlled temperature of 10 0:1 C. Their duration was up to 45 days, depending on the level of loading. Different types of inclusions were studied, however the present paper focuses on the two types which enlighten the influence of the neighbourhood of a
grain on its propensity to exhibit a heterogeneous deformation, namely: • cylindrical monocrystalline inclusion, 30 mm in diameter; • rosette-shaped inclusion made of a cylindrical monocrystal, 40 mm in diameter, surrounded by a crown of halved cylindrical monocrystals about 7 mm in diameter. In the following, these specimens are referred to by type-A and type-B, respectively (see Fig. 2(a) and (b)). 2.2. Experimental results 2.2.1. Type-A cylindrical inclusion Several creep tests were carried out with different orientations w of the inclusion c-axis with respect to the loading axis, namely 0, 25, and 45 (see Fig. 1(c)) and different compressive stresses (0.5, 0.75, and 1 MPa). For configurations with non-zero resolved shear stress in the inclusion
Fig. 2. (a) Photograph of type-A specimen with an initially circular monocrystalline inclusion, at 6.3% axial strain: the parallel lines are the traces of the basal planes, and the white areas at the inclusion-matrix boundary are evidence of dynamic recrystallization; (b) rosette-shaped specimen B1 at 0% axial strain (w ¼ 72:5); (c) specimen B1 at 2% axial strain: note the slight flexion of the basal planes and the appearance of faint bands perpendicular to the basal planes; (d) specimen B1 at 6.5% axial strain: the white bands crossing through the central grain are kink bands; (e) basal plane RSS map computed at 6.5% axial strain (the white color corresponds to zero RSS, the grey scale step is 0.005 MPa); and (f) map of c-axis misorientation with respect to average for the central grain (the white color corresponds to the average of 62, the grey scale step is 2).
P. Mansuy et al. / Computational Materials Science 25 (2002) 142–150
basal planes, color undulations, under the form of blurred stripes perpendicular to the c-axis, were observed from the beginning of the tests. These markers of basal glide activation evolved in thin dark lines, corresponding to steps at the surface of the specimen, with increasing strain (see Fig. 2(a)). These dark lines allowed to record the rotation of the inclusion c-axis towards the loading axis during the tests. Whatever its initial crystallographic orientation, the shape of the inclusion, initially circular, remained always ellipsoidal, with a more or less pronounced aspect ratio depending on its orientation and on the specimen axial strain achieved. Meanwhile, the inclusion long axis rotated in the same direction as the basal plane but not at the same rotation rate. Moreover, the dimension of the inclusion in the direction of the caxis did not show any measurable variation during the tests, even in the most deformed configurations (i.e. up to 50% strain in the inclusion). At the beginning of the loading, a pattern of faint blurred bands appeared orthogonal to the basal planes. However, after 1% specimen axial strain these bands tended to disappear, except for two faint bands situated at each far end of the elliptically deformed inclusion. No other strain localization feature were observed during the tests. For the configuration with basal planes orthogonal to the loading axis (w ¼ 0), the inclusion did not show any observable deformation, while the matrix seemed to flow around it. 2.2.2. Rosette-shaped type-B specimens Two tests were carried out with the initially inner cylindrical grain oriented at w ¼ 72:5 (specimen B1) and w ¼ 0 (specimen B2), respectively. Type-B differ from type-A specimens only by the number of grains in contact with the inner large monocrystal: 18 medium-size grains for typeB, against more than 400 small-size matrix grains for type-A. This change in the monocrystal environment resulted in observations departing markedly from that of type-A specimens during its deformation. For specimen B1, the central grain of the multicrystal exhibited severe kinking and bending. Each kink band was initiated at a triple junction of the central grain with two medium-size grains of the surrounding crown. It formed per-
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pendicular to the basal planes of the central grain, and crossed right through it (see Fig. 2(c)). During the tests the kink bands rotated in the direction opposite to that of the bulk central grain basal planes, i.e. they did not remain perpendicular to these planes as the deformation of the specimen increased. At 6.5% specimen axial strain, the shape of the central grain departed from ellipsoidal, and a network of kink bands, 0.1–1.5 mm wide, approximately parallel to each other, delimited long areas of the central crystal (see Fig. 2(d)). One of these areas exhibited polygonization with a misorientation of 8 each side of the sub-boundary (see top of Fig. 2(d)). However, a majority exhibited a progressive change of the lattice orientation from one boundary of the central grain to the other, with a misorientation between 3 and 5. The c-axis orientation with respect to the loading axis was about 90 inside the kink bands, whereas it varied from about 55 to 62 outside, depending on the area considered. The kink bands c-axis misorientation with respect to the adjacent areas was about 27. Dynamic recrystallization occurred mainly in the fine-grained matrix whose grain size at 12.5% axial strain was increased up to about 2 to 5 mm, and in the surrounding medium-size grains, some of them being totally destroyed and replaced with much smaller grains. Specimen B2, whose c-axis was initially parallel to the direction of compression, exhibited no localization features, except for two faintly bended areas on the left and right sides. Its deformation remained small and non-uniform, concentrated in a few activated basal planes.
3. Numerical model The experiments were interpreted through finite-element simulations. 3.1. Multicrystal and matrix behaviour The ice single-crystal deforms essentially by slip on the basal planes, with no preferential glide direction [13]. The present model takes this feature into account by considering each grain of the ice multicrystal as a transversely isotropic continuous medium, whose plane of isotropy corresponds to
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the basal plane of the grain [9], and with a very weak resistance to shear parallel to this basal plane. In the same line, the fine-grained ice matrix is considered as a homogeneous isotropic medium. The interest of this homogeneous continuous formulation is that it is easier to handle in finiteelement computations than a model based on dislocation glide on distinct crystallographic planes which is otherwise not ascertained, while retaining the first order effects of the outstanding anisotropy of ice. The simplest anisotropic constitutive law for the non-linear viscoplastic behaviour of the grain is adopted. It is similar to that proposed by [14], and derives from a dissipation potential (see [15]). In the local reference frame R attached to the crystal with rotational symmetry axis x3 along the c-axis, it is written as s11 s22 ¼ 2gH 12 ðd11 d22 Þ; s33 ¼ 2gH 12
4a1 3
d33 ;
s23 ¼ 2gH 12 bd23 ; s31 ¼ s12 ¼
ð1Þ
2gH 12 bd31 ; 2gH 12 d12 ;
where s and d are the deviatoric stress and the strain-rate tensors, respectively, gH 12 is the apparent viscosity for shear in the (basal) plane of isotropy ðx1 ; x2 Þ, a is the ratio of the axial viscosity 1 along the x3 -axis to that in the plane of isotropy, and b is the ratio of the viscosity for shear parallel to the plane of isotropy to gH 12 . The apparent viscosity is defined as ð1nÞ=n
1=n c_ } gH 12 ¼ A
¼ A1 s1n } ;
ð2Þ
where A is a temperature dependent fluidity parameter, and c_ } and s} are two invariants by rotation about the x3 -axis (but not under any change of reference frame). c_ } and s} are defined by 4a 1 2 c_ ax þ c_ 2? þ bc_ 2k ; 3 3 1 s2 þ s2? þ s2k s2} ¼ 4a 1 ax b c_ 2} ¼
ð3Þ
1 The axial viscosity is defined as the halved ratio of the axial deviatoric stress to the axial strain rate in a uniaxial creep test.
with 2 ; c_ 2ax ¼ 3d33 2 2 ; c_ 2? ¼ ðd11 d22 Þ þ 4d12 2 2 þ d31 Þ; c_ 2k ¼ 4ðd23
s2ax ¼ 3s233 =4;
ð4Þ
2
s2? ¼ ðs11 s22 Þ =4 þ s212 ; s2k ¼ s223 þ s231 ; and they are linked by s} ¼ gH 12 c_ } , thus c_ } ¼ Asn} :
ð5Þ
When a ¼ b ¼ 1 the medium is isotropic, c_ 2} ¼ c_ 2 ¼ 2dij dij , s2} ¼ s2 ¼ sij sij =2, and (1) reduces to Norton–Hoff’s law (known as Glen’s law in Glaciology). This isotropic version was used to describe the fine-grained matrix behaviour with temperature dependent fluidity parameter B, namely: sij ¼ 2B1=n c_ ð1nÞ=n dij :
ð6Þ
Estimates of the parameters involved in relations (1) and (6) were deduced from experimental data. The behaviour of isotropic ice is welldescribed and documented: the exponent n of the power law is 3 for s > 0:1 MPa [3,16], and, according to [16–18], the mean value B ¼ 18:6 MPa3 a1 at 10 C was adopted. The grain parameters in (1) were obtained either by considering the grain as an isolated single crystal [3] or by using data from mechanical tests on polar ice with single maximum fabric (i.e. involving the influence of grain boundaries) [6,7]. Assuming n ¼ 3 in (1), the best set (A,a,b) was obtained by comparing type-A experiments with numerical simulations (see Section 4). 3.2. Numerical scheme The computations were done in a fixed global whose x1 axis is perpendicular reference frame R to the plane of the specimens. Since all the multicrystal c-axes (x3 -axes) lie in this plane, each local reference frame R attached to a grain of the multicrystal was chosen so that its x1 -axis coincides with x1 . The orientation of the x3 -axis (c-axis) is denoted by of R with respect to the x3 -axis of R
P. Mansuy et al. / Computational Materials Science 25 (2002) 142–150
by R. w, and the matrix for passing from R to R The loading conditions (loading along the x3 -axis, plane strain) allowed to consider a two-dimensional (2D) problem. The constitutive models (1) and (6) were implemented in a 2D finite-element code using a velocity–pressure formulation [19] with a quadratic interpolation of the velocity components and a linear interpolation of the pressure. Type-A specimens were discretized with 568 six-nodes triangular elements (T6) for the matrix and 324 for the inclusion, type-B specimens with 634 T6-elements for the matrix, 426 for the central grain of the multicrystal, and 216 for the 18 surrounding medium-size grains. The boundary conditions were prescribed according to the experimental conditions, that is, free lateral boundaries, zero velocity on the lower boundary, and constant vertical velocity (parallel to the loading axis) on the upper boundary adjusted at each time step to ensure a constant compressive stress condition. The gravity forces were neglected. The rotation rate of the multicrystal grain caxes was derived from the transformation formula for the spin tensor under a change of reference frame: R ¼ R1 R_ þ x; R1 x
ð7Þ
of a grain dewhich expresses that the spin x composes into the sum of the viscoplastic spin x measured in its rotating reference frame R, plus [9]. By the rate of rotation of R with respect to R expressing that the basal planes of the ice crystal remain parallel to each other during the deformation (i.e. the component u3 of the velocity along the x3 -axis, measured in the rotating frame R, is independent of x2 ), the only non-zero component of x is x23 ¼ d23 , and since in the 2D flow the sole 23 , is invariant by , x non-zero component of x rotation around the x1 -axis, Eq. (7) leads to w_ ¼ ou3 =ox2 :
ð8Þ
The evolution of the deformation of the specimen was simulated by updating the position of each node from the finite-element computed velocity field, and the multicrystal c-axes orientations from (8), at each time step. This time step was
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chosen so that the increment of axial strain was less than 2 103 .
4. Numerical simulations Numerical simulations carried out for type-A specimens homogeneous deformation allowed to constrain the set of parameters for the grain model (1), by comparing the computed and experimental evolutions of the average c-axis orientation hwi of the elliptical monocrystalline inclusion and of its aspect ratio, versus the specimen axial strain (see Fig. 3). These simulations were performed by assuming a constant value of w inside each triangular element of the inclusion. The maximum misorientation in the inclusion, occurring in the elements at the inclusion-matrix interface, was 3:3 after 6% axial strain (and only 0:6 elsewhere). The parameter a was found to have practically no influence in the range 0.5–10, and was subsequently fixed to a ¼ 1 [10]. As expected the inclusion behaviour is mainly influenced by parameter b which characterizes the crystal anisotropy. From the comparisons shown in Fig. 3, the parameters deduced from isolated single-crystal data (set 1, Table 1) allow to fit the experimental curves quite well (within the accuracy of the experimental measurements), contrary to set 2, derived from polar ice data, which underestimates the inclusion aspect ratio and the c-axis rotation rate (Fig. 3). The simulation of type-B experiment B1 shown in Fig. 2(b)–(d) was performed with the set of parameters suitable for type-A simulations. Further refinement was obtained by using a quadratic interpolation of the c-axis orientation in each element of the multicrystal. Fig. 3 shows the evolution versus the specimen axial strain of the central grain aspect ratio and average c-axis orientation. This latter was measured in the crystal areas delimited by the kink bands, excluding the highly misoriented zones within the kink bands. The simulated evolution of the shape of the central grain is slower than that observed, as well as the simulated average rotation of the basal planes (see Fig. 3). However the rigid-body rotation of this grain (i.e. the rotation of its long axis) is in good agreement with observations.
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Fig. 3. Observed (symbols) and computed (lines) evolutions of the aspect ratio and of the crystallographic orientation as a function of the total strain for: (left) type-A specimens monocrystalline inclusion; (right) specimen B1 central grain. The different sets of grain model parameters are given in Table 1. The thin and dotted lines for B1 specimens correspond to simulations performed with an equivalent homogeneous monocrystalline inclusion.
Table 1 Values of inclusion parameters A=B, a and b used in numerical simulations (n ¼ 3, B ¼ 18:6 MPa3 a1 ) Set
A3 =B3
a
b
1 2 3
0.1 0.17 0.1
1 1 1
0.01 0.13 0.0001
The simulation of localization features was based on the following simple considerations. Each kink band delimits two zones of the crystal with the same basal plane orientations, while its interior basal planes remain parallel to each other, but at high angle from the adjacent zones. The two kink planes act as dislocation walls fed by dislocations which glide on the basal planes [20]. Owing to the fact that only the basal system is activated and the sole efficient obstacles to dislocation motion are the grain boundaries, a dislocation wall can be seen, in a 2D situation, as the line which intersects the basal planes at points where the gliding dislocations come to arrest, i.e. where the resolved shear stress (RSS) vanishes. According to this simple model, basal planes bending, susceptible to evolve into a polygonization wall and eventually into a sub-boundary, should correspond to a change of sign of the RSS, whereas
a kink band should correspond to two successive changes of RSS sign on a short distance (see Fig. 4). The basal RSS computed for the central grain of specimen B1 at 7% axial strain, vanishes along two lines situated at the ends of the grain and along one line which crosses right through the grain (see Fig. 2(e)). These lines are parallel to each other and roughly parallel to the observed kink bands. They are linked to one or two triple junctions at the grain boundary. Elsewhere the RSS
Fig. 4. Schematic of the RSS based model: on the single glide plane, geometrically necessary dislocations are arrested at zero RSS locations.
P. Mansuy et al. / Computational Materials Science 25 (2002) 142–150
map shows a pattern with zones of highest RSS values elongated in the direction of the basal planes and of the simulated kink band. The RSS gradients are not high enough to show any sign of polygonization. The map of the computed misorientation of the elements in the central grain with respect to the average hwi ¼ 62 shows four bands parallel to each other and to the direction of the kink bands (see Fig. 2(f)). The positive misorientation inside these bands is in agreement with that of the observed kink bands, however the maximum misorientation is only 15.
5. Conclusion Compressive creep tests carried out on two types of structure-controlled ice specimens with a mono and multicrystalline inclusion in a homogeneous fine-grained ice matrix allow to enlighten the phenomena leading to strain localization in a polycrystal of ice. The deformation of type-A initially circular monocrystalline inclusions did not exhibit any other strain localization feature than that associated to basal glide. The deformation at the inclusion scale was then very homogeneous. Type-B inclusions exhibited strain heterogeneity, closely related to stress concentrations at the triple junctions between the large central grain and its surrounding medium-size grains, with the development of kink bands associated to lattice bending. Finite-element simulations based on a simple model for the ice grain behaviour which account for the extreme viscoplastic anisotropy of ice, allowed to reproduce the homogeneous deformation of type-A specimens quite well. Since the severe bending occurring at the kink planes could not be described correctly due to the relative mesh coarseness, even with a quadratic interpolation of the c-axes orientations in each element, the simulation of specimen B1 heterogeneous deformation is not satisfactory. However, considering that only the basal system is activated allows to obtain some information on the location of localization features in relatively good agreement with the observations. Further development of the numerical simulations should use this information for adaptative mesh refinement.
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Modelling the macroscopic behaviour of anisotropic polar ice and the evolution of its anisotropy is a crucial matter for ice-sheet flow modelling in relation with climate changes. Up to now, grain to grain interactions are quite ignored in current polycrystal models which aim at modelling polar ice properties, and are based on onesite self-consistent approaches [9,21]. In view of accounting for the heterogeneous deformation of the ice grain in such homogenization schemes, the possibility of simulating type-B specimen behaviour with an equivalent homogeneous inclusion in an isotropic matrix was investigated. Since the observed average lattice rotation is faster for typeB than for type-A specimens, the equivalent inclusion should be modelled with a very low value of the grain-model anisotropy parameter b to simulate the crystallographic evolution of the grain. However, doing so the simulated shape evolution is too fast (see Fig. 3), so that there is no straightforward solution to this problem. However, since crystallographic texture evolution is predominant as regards the macroscopic polycrystal behaviour, this experimental-numerical approach deserves further investigation.
Acknowledgements The financial supports of the Centre National de la Recherche Scientifique (Programme INSUGeomateriaux) and of Universite Joseph Fourier Grenoble-I are greatly acknowledged.
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