Modeling viscoplastic behavior and heterogeneous intracrystalline deformation of columnar ice polycrystals

doi:10.1016/j.actamat.2008.10.057

Acta Materialia

Volume 57, Issue 5, March 2009, Pages 1405-1415

https://doi.org/10.1016/j.actamat.2008.10.057 Get rights and content

Abstract

A full-field formulation based on fast Fourier transforms (FFT) has been adapted and used to predict the micromechanical fields that develop in two-dimensional columnar Ih ice polycrystals deforming in compression by dislocation creep. The predicted intragranular mechanical fields are in qualitative good agreement with experimental observations, in particular those involving the formation of shear and kink bands. These localized bands are associated with the large internal stresses that develop during creep in such anisotropic material, and their location, intensity, morphology and extension are found to depend strongly on the crystallographic orientation of the grains and on their interaction with neighboring crystals. The predictions of the model are also discussed in relation to the deformation of columnar sea and lake ice, as well as with the mechanical behavior of granular ice of glaciers and polar ice sheets.

Introduction

Ih ice single crystals deform plastically in the dislocation glide regime essentially by $(0 0 0 1) 〈 1 \bar{2} 1 0 〉$ basal slip. The yield point observed during the early stage of plastic flow, associated with the formation of slip lines, is related to the multiplication of basal dislocations by slip, cross-slip and/or dislocation climb [1]. The stress required to produce a given effective strain-rate along a crystallographic direction not lying on the basal plane is between one and two orders of magnitude greater than the stress necessary to produce the same strain-rate along a direction belonging to the basal plane [2].

The single crystals that form glacier ice and polar ice sheets exhibit a wide range of sizes and morphologies, but, in general, the structure of this polycrystalline ice can be characterized as being “granular” or “three-dimensional” (3-D). Another natural form of ice is the so-called “columnar” or “two-dimensional” (2-D) polycrystalline ice (also referred to as “S2” ice in glaciological literature [3]), consisting of an aggregate of columnar grains with the 〈c〉-axis of each single-crystal randomly oriented in the plane perpendicular to the direction of the columns. This kind of aggregate is obtained when ice grows from the surface of calm water in an unidirectional temperature gradient. This type of ice forms the natural covers of the Arctic Ocean and northern large rivers. Two-dimensional ice samples can be also prepared in the laboratory, for controlled testing [4], [5], [6], [7].

The aforementioned very large viscoplastic anisotropy of ice single crystals has consequences on the mechanical response of ice polycrystals. On the one hand, the development of lattice preferred orientations (crystallographic textures) as ice deforms (e.g., when ice is transported into the depths of a polar ice sheet) determines the striking differences in the viscous response of textured ice polycrystals to stresses applied along different directions (e.g., [8]). On the other hand, the fulfillment of both compatibility and stress equilibrium across grain boundaries results in heterogeneous intragranular deformation patterns [4], [5], [6], [7], [9], [10], [11]. High orientation gradients were observed in ice crystals extracted from the Antarctic ice sheet [12]. Dynamic continuous and discontinuous recrystallization, which is very active in ice sheets [13], contributes to the reduction of the long-range internal stresses field induced by such intragranular deformation heterogeneities.

Texture development in polar ice sheets and the resulting anisotropic response of polycrystalline ice have been intensively studied using mean-field models (e.g., [14], [15], [16]). This kind of approach is based on the statistical characterization of the intragranular mechanical fields (in terms of average grain stresses and strain-rates, and, in the most advanced formulations, also through the determination of the intracrystalline average field fluctuations [16]), but the actual micromechanical fields remain inaccessible to these homogenization approaches.

The modeling of the intracrystalline heterogeneity that develops in ice polycrystals (which requires the use of full-field approaches) has been, on the other hand, much less investigated. To fill this gap, this work is devoted to the study of the correlation existing between the heterogeneous deformation patterns that appear inside the constituent single-crystal grains of an ice aggregate and their corresponding crystallographic orientations, along with the influence of other factors, such as orientation and size of neighboring grains. To this end, a full-field formulation based on the fast Fourier transform (FFT) [17], [18], [19] has been adapted to obtain the micromechanical fields that develop in polycrystalline ice deforming by dislocation creep.

We have chosen to pursue this study on columnar ice polycrystals, for various reasons. On the one hand, dealing with a 2-D problem allowed us to use a higher resolution (i.e., more discretization points) to characterize the intracrystalline fields, and to fully visualize the results in a 2-D representation. Another advantage is that the mathematical representation of this kind of polycrystals is easier since each crystallographic orientation is almost fully characterized by only one angular parameter (rather than by three Euler angles, as in the case of 3-D polycrystals). Also, most importantly, we have available a comprehensive set of experimental results on crystal orientation and neighborhood type dependence of the intracrystalline localization patterns observed in laboratory grown and tested columnar ice specimens with different microstructures [4], [5], [6], [7], which can be used for validation of our model predictions.

The plan of this paper is as follows. In Section 2 we review the available experimental evidence on the effective and local viscoplastic behavior of polycrystalline ice and recall some experimental results obtained by Mansuy [5] on the orientation- and microstructure-dependent deformation localization patterns in columnar ice polycrystals. In Section 3 we provide details of the model utilized and the unit cell used in this study. In Section 4 we present the results of our simulations and compare them with the experimental evidence. In Section 5 we conclude discussing possible improvements of the modeling of natural polycrystalline ice, based on the capabilities of the present micromechanical formulation.

Section snippets

Effective and local viscoplastic behavior of polycrystalline ice

The secondary creep of polycrystalline ice is reached at strains of about 1%. The corresponding stress exponent is close to 3 for deviatoric stresses higher than 0.2 MPa [2]. Otherwise, for conditions prevailing in polar ice sheets (deviatoric stresses lower than 0.2 MPa and strain-rates lower than 10⁻¹⁰ s⁻¹) the stress exponent for steady-state creep is lower than 2, as suggested by borehole deformation measurements [20], bubbly ice densification [21] and laboratory tests [22]. Under these very

The FFT-based formulation

The intracrystalline states that develop during creep of polycrystalline ice can be obtained using an extension of an iterative method based on FFT, originally proposed by Moulinec and Suquet [17] and Michel et al. [18] for linear and nonlinear composites. This formulation was later adapted to polycrystals and applied to the prediction of texture development of fcc materials [19], and in turn used for the computation of field statistics and effective properties of power-law 2-D polycrystals [37]

Results and discussion

A FFT-based calculation was run to obtain the overall and local mechanical response of the above-described unit cell representing a columnar ice polycrystal, to the following imposed strain-rate tensor (see also Fig. 3): ${\dot{E}}_{ij} = [\begin{matrix} 1 \times 10^{- 8} & 0 & 0 \\ 0 & - 1 \times 10^{- 8} & 0 \\ 0 & 0 & 0 \end{matrix}] s^{- 1} .$

The computed effective equivalent stress reached a value of 0.01875 in units of τ^bas, resulting in a normalized reference equivalent stress σ_o (see Eq. (2)) [14] of 9.11 × τ^bas. This roughly represents an effective response twice as soft for this kind

Concluding remarks

A full-field formulation was adapted and used to predict the micromechanical fields that develop in columnar ice polycrystals deformed under plane-strain compression. This formulation, conceived as a very efficient alternative to FE methods (which calculation times usually scale with N², where N is the number of discretization points), is based on the repetitive use of the FFT algorithm, whose computing time scales with N × log N. This high numerical efficiency combined with the resolution of the

Acknowledgments

This work was partially supported by the Office of Basic Energy Sciences, Project FWP 06SCPE401 (USA), and by CNRS (ST2I Department) and University Joseph Fourier, Grenoble (France).

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Published by Elsevier Ltd.

Modeling viscoplastic behavior and heterogeneous intracrystalline deformation of columnar ice polycrystals

Abstract

Introduction

Section snippets

Effective and local viscoplastic behavior of polycrystalline ice

The FFT-based formulation

Results and discussion

Concluding remarks

Acknowledgments

Comput Mater Sci

Tectonophysics

Earth Planet Sci Lett

Acta Mater

Comput Methods Appl Mech Eng

CR Phys

Earth Planet Sci Lett

Scr Mater

Acta Metall

Acta Mater

Scr Mater

Int J Solids Struct

Acta Mater

Acta Mater

Comput Geosci

Mec Ind

Eng Fracture Mech

Scr Mater

Philos Mag

J Phys Chem

Can Geotech J

Ann Glaciol

Contribution à l’étude du comportement viscoplastique d’un multicristal de glace: hétérogenéité de la déformation et localisation, expériences et modèles

J Phys IV

Constitutive properties of ice at Dye 3, Greenland

Deformation mechanisms in ice