J. Non-Newtonian Fluid Mech. 134 (2006) 33–43 Flow-induced anisotropy in polar ice and related ice-sheet flow modelling  Fabien Gillet-Chaulet a , Olivier Gagliardini a, ∗ , Jacques Meyssonnier a , Thomas Zwinger b , Juha Ruokolainen b a Laboratoire de Glaciologie et G´ eophysique de l’Environnement, CNRS, UJF-Grenoble I, BP 96, F-38402 Saint-Martin d’H` eres Cedex, France b Scientific Computing Ltd., P.O. Box 405, 02101 Espoo, Finland Received 1 July 2005; received in revised form 11 November 2005; accepted 11 November 2005 Abstract As fibers or other crystalline materials exhibiting hexagonal symmetry, the crystal of ice can be orientated by using only one single vector, i.e. its c-axis. Such a characteristic allows to apply specific methods to deal with the properties of the polycrystalline aggregate. Among others, the fabric (texture) of the ice polycrystal can be described by an ODF, i.e. a scalar function of two angles that gives the distribution of the orientation of all the constituents (grains). This paper presents a strain-induced anisotropic flow law for polycrystalline ice and the associated equations describing the evolution of its fabric. This constitutive law is formulated at the polycrystal scale and tabulated using a micro–macro model. The fabric is defined by the second- and fourth-order orientation tensors for the c-axes, assuming the so-called “invariant-based optimal fitting closure approximation”. Both the anisotropic constitutive law and the fabric evolution equations have been implemented in a finite element code in order to solve large scale ice flow problem. As an application, the flow of an idealized ice sheet over a bumpy bed is studied. © 2005 Elsevier B.V. All rights reserved. Keywords: Polar ice anisotropy; Ice-sheet flow modelling; Invariant-based optimal fitting 1. Introduction In the terrestrial environment ordinary ice crystallizes in the hexagonal system (ice Ih). The viscoplastic deformation of the ice Ih single crystal results essentially from dislocation glide on the basal plane (see Fig. 1a), perpendicular to the hexagonal symmetry axis, called the c-axis [1]. As a consequence, ice Ih is one of the most anisotropic natural materials. For a given equivalent stress, the strain rate for shear parallel to the basal plane is between three and four orders of magnitude larger than under compression perpendicular to the basal plane [2]. Polycrystalline ice at the ice-sheet surface results from the transformation of deposited snow: since the ice crystals are dis- tributed at random, its mechanical behaviour is isotropic. During the gravity driven flow of polar ice, the polycrystal of ice devel-  Expanded version of a talk presented at the Second Annual European Rhe- ology Conference, Grenoble, February 2005. ∗ Corresponding author. E-mail address: gagliar@lgge.obs.ujf-grenoble.fr (O. Gagliardini). ops a strain-induced fabric (or texture), that is, a preferred orien- tation of the c-axes of its grains. Observations of deep ice cores drilled in Antarctica and Greenland have shown very different fabric patterns, corresponding to different flow conditions. In the Vostok core (Antarctica), the deep ice exhibits girdle type fab- ric patterns [3], whereas in the Dome C (Antarctica) and GRIP (Greenland) cores, the c-axes concentrate along the vertical di- rection to form a fabric with a single maximum [4,5]. Strain- induced fabrics, combined to the strong anisotropy of the single crystal, result in a macroscopic behaviour of polycrystalline ice that is also strongly anisotropic and varies from place to place depending on the flow conditions. As shown experimentally by Pimienta et al. [6], shearing perpendicular to the mean direction of the c-axes of a polycrystal that exhibits a single maximum fabric is about 10 times easier than when shearing an isotropic specimen. To construct ice-sheet flow models aimed at obtaining ac- curate information on the origin and the age of ice from deep ice cores, the strong and evolving viscoplastic anisotropy of po- lar ice must be taken into account. This implies to consider the fabric as an unknown of the ice-sheet flow problem. This paper 0377-0257/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2005.11.005