Review articleMultiscale modeling of ice deformation behavior
Graphical abstract
Introduction
Ice is a common mineral on the Earth's surface, where it occurs as ice Ih. As ice is relatively close to its melting temperature, glaciers and polar ice sheets deform by ductile dislocation creep at strain rates in the order of 10−12 to 10−6 s−1. Research on the flow of ice is of direct importance to society as it is needed to understand and predict the effects that global warming could have on sea level rise, glacier retreat, etc. There is also an increasing awareness that ice is a valuable analogue for other minerals and crystalline materials, as it is the only common mineral where this creep can be readily observed in nature and in the laboratory. Numerical modeling has become a key method to link the mechanics of ice from the dislocation scale to that of flowing ice masses.
Most of the efforts made to simulate the ductile mechanical behavior of polycrystalline ice are related to the modeling of ice flow and fabric evolution in the conditions of polar ice sheets or glaciers. Ice is increasingly considered as a model material to validate micro-macro mechanical approaches for materials with a high viscoplastic anisotropy. Most of the modeling techniques presented in this paper are currently used or further developed for other materials. For geological applications, one main limitation could be related to the “one phase” approach for most of these techniques, well adapted to ice. The reader will find, at the end of the paper, a table summarizing the main aspects of each techniques, with application ranges and limitations.
Ice Ih has a hexagonal crystal structure with a c/a ratio of 1.628. This c/a ratio is very close to the 1.633 value for a closely packed structure, but ice is not closely packed (see Schulson and Duval (2009) for a recent review). The elastic anisotropy of ice single crystals is small. The Young modulus E only varies by about 30%, depending on the direction of the loading axis with respect to the c-axis. The highest value is along the c-axis with E = 11.8 GPa at −16 °C (Gammon et al., 1983).
Single crystals deform plastically essentially by glide of dislocations on the basal plane. There are three equivalent directions for the Burgers vector, but slip on the basal plane is almost isotropic. In conditions where basal slip is favored, the stress-strain rate relationship after a strain of about 5% can be expressed by a power law with a stress exponent n = 2 ± 0.3 (Higashi et al., 1965; Jones and Glen, 1969; Mellor and Testa, 1969). At similar strain rates, the equivalent stress requested for non-basal slip is about 60 times larger than for basal slip (Duval et al., 1983).
For ice polycrystals deformed under the laboratory conditions (strain rate between about 10−8 s−1 and 10−6 s−1 and temperature generally higher than −30 °C), strain is essentially due to intracrystalline dislocation glide. The transient creep regime is characterized by a strong directional hardening until the strain-rate minimum is reached for an overall strain of 1% (Duval et al., 1983). This strain-rate decrease can reach three orders of magnitude. It is associated to the development of a strong internal stress field due to plastic incompatibility between grains (Ashby and Duval, 1985; Duval et al., 1983; Castelnau et al., 2008b). A significant part of the transient creep is recoverable, i.e., on unloading a creep specimen, a reverse creep is observed, with reverse strain which can be more than ten times the initial elastic strain (Duval, 1976; Duval et al., 1983). In the secondary creep regime, isotropic polycrystals deform (at similar stress levels) a 100 times slower than a single crystal optimally oriented for basal slip. In this regime, the minimum strain rate and the stress are linked by a power law, referred to as Glen's law in glaciology (Glen, 1955), expressed through a relationship of the form (1) for temperatures lower that −10 °C.with the applied stress, Ep = 0.72 eV and the stress exponent n = 3 (Barnes et al., 1971;Budd and Jacka, 1989). A is a constant, kB the Boltzmann constant and T the temperature. Above −10 °C, rises more rapidly with increasing temperature and cannot be described by this equation (Morgan, 1991). No grain-size effect is expected for power-law secondary creep at laboratory conditions (see (Duval and Le Gac, 1980; Jacka, 1994) for instance). But a grain size effect was, however, measured during transient creep (Duval and Le Gac, 1980).
At strains larger than 1–2% (tertiary creep regime), dynamic recrystallization is predominant, and new grain microstructures and crystal orientations are generated (Jacka and Maccagnan, 1984; Duval et al., 2000).
At stresses lower than 0.1 MPa, relevant to deformation conditions in glaciers, ice sheets or planetary bodies, there is a clear indication of a creep regime with a stress exponent lower than two. This indication results from both the analysis of field data and laboratory tests, although the difficulty of obtaining reliable data at strain rates lower than 10−10 s−1 is at the origin of contradictory results (Mellor and Testa, 1969; Barnes et al., 1971; Dahl-Jensen and Gundestrup, 1987; Pimienta et al., 1987; Lipenkov et al., 1997; Goldsby and Kohlstedt, 1997). In particular, Goldsby and Kohlstedt (1997) suggest a grain-size dependence of the ice viscosity associated with this low stress regime, based on laboratory experiments performed on very small grain-size samples. This grain-size effect would be associated with a grain boundary-sliding dominated creep. Its extrapolation to polar ice-core deformation conditions remains controversial (Duval and Montagnat, 2002). Diffusional creep, commonly associated with such conditions in many materials yields a viscosity much higher than that deduced from field data (Lliboutry and Duval, 1985). For a review on ice behavior, see (Duval et al., 2010).
Ice as a model material exhibits a challenging viscoplastic anisotropy owing to the presence of only two independent easy slip systems for the dislocations (basal plane). While five independent systems are required to accommodate an arbitrary deformation in a single crystal (Taylor, 1938), Hutchinson (1977) showed that four systems are required for allowing a hexagonal polycrystal such as ice to deform. Being able to represent and to take into account this anisotropy in micro-macro models which aim at linking the single crystal scale to the polycrystal scale, is of primary interest to the material science community. This anisotropy needs to be accounted for at the dislocation scale in order to build physically-based model for the activation of (poorly known) secondary slip systems. The impact of dislocation induced internal stress fields, but also the characterization and development of highly heterogeneous strain and stress fields within polycrystals, and their impact on fabric development turn out to be of strong importance (Castelnau et al., 1996a; de la Chapelle et al., 1998).
During gravity-driven flow of glaciers and ice sheets, the macroscopic behavior of ice becomes progressively anisotropic with the development of fabrics (or textures, c-axis preferred orientations). This anisotropy and its development depends on the flow conditions, but strongly influences the response of ice layers to imposed stress (see Gundestrup and Hansen (1984); Van der Veen and Whillans (1990); Mangeney et al. (1997) for pioneer field work and modeling on the subject). Indeed, a polycrystal of ice with most of its c-axes oriented in the same direction deforms at least ten times faster than an isotropic polycrystal, when sheared parallel to the basal planes.
Fabrics basically develop as the result of lattice rotation by intracrystalline slip (Azuma and Higashi, 1985; Alley, 1988, 1992). Dynamic recrystallization can have a major impact on fabric development, especially at temperatures above −10 °C close to bedrocks or within temperate glaciers (Alley, 1992; Duval and Castelnau, 1995; de la Chapelle et al., 1998; Montagnat et al., 2009), see Section 5. Questions, however, remain to what extent different recrystallization processes operate as a function of depth in polar ice sheets (Kipfstuhl et al., 2006, 2009; Weikusat et al., 2009).
Accurate modeling of ice flow under natural conditions is relevant for many scientific objectives, such as the response of ice sheet to climate changes (Seddik et al., 2012), the interpretation of climate signals extracted from ice cores (Faria et al., 2010), the energy balance in extraterrestrial satellites (Sotin et al., 2009), and since a few years, the accurate prediction of sea-level rise that is linked to the behavior of fast-moving coastal glaciers (Gillet and Durand, 2010). In this context, challenges are mainly (i) to establish an ice flow law adapted to low stress conditions, changes in temperatures and impurity content, (ii) to consider the macroscopic anisotropy due to fabric development at the given conditions, (iii) to be able to integrate processes such as dynamic recrystallization that can strongly influence fabric development and the flow law.
The aim of this paper is to present a general overview of the main modeling techniques adapted to ice, and the main modeling results obtained from the single crystal scale to the large scale that is relevant to ice sheet flow modeling. Techniques are highly diverse, from dislocation dynamics (micron scale) to Finite Element methods that are adapted to the whole ice sheet (km scale), via mean-field and full-field micro-macro approaches and coupling with a microstructure evolution models (cm to m scale, limited to a 2D configuration, see 5.2). We will mostly focus on recent advances and topics that are still under development.
Section snippets
Modeling ice single crystal behavior
Owing to its high viscoplastic anisotropy, with dislocations gliding mostly on the basal plane, studying and modeling ice single crystal behavior is a challenge for regular approaches.
Recent efforts focused on three main objectives; (i) understanding, representing and taking into account the dislocation dynamics, (ii) improving our knowledge about secondary slip systems in ice, (iii) providing an accurate crystal plasticity constitutive law that can be implemented in mean-field and full-field
Microstructure characterization
From the mechanical point of view, polycrystalline materials have to be considered as a specific class of composites. They are composed of many grains, with grain size in the range of mm to cm for natural ice. Grains are assembled in a random way, i.e. their size, shape, and lattice orientation do generally not depend on the size, shape, and orientation of the surrounding grains (Fig. 5). Therefore, the microstructure of ice polycrystals can hardly be described exactly in 3-D, unless one makes
Full field approaches for the polycrystal
Mean-field approaches have been extensively used to predict the mechanical behavior of ice polycrystals, and the fabric development as measured along ice cores. Due to its high viscoplastic anisotropy, deformation in ice is expected to be strongly heterogeneous, with a strong impact of grain interactions and kinematic hardening (Duval et al., 1983; Hamman et al., 2007; Montagnat et al., 2011; Grennerat et al., 2012). The mean-field approaches described above are based on the statistical
Modeling of dynamic recrystallization mechanisms
Under laboratory conditions (described in Section 1), dynamic recrystallization (DRX) dominates the changes of microstructures and fabrics in the tertiary creep regime, that is after about 1% macroscopic strain (Duval, 1981; Jacka and Maccagnan, 1984; Jacka and Li, 1994). During DRX, grain nucleation and grain boundary migration are two processes that contribute to the reduction of the dislocation density, therefore of the stored deformation energy (Humphreys and Hatherly, 2004). In the
Toward large scale ice flow modeling
A number of models have been developed in glaciology to simulate the flow of anisotropic ice and the strain-induced development of fabric within polar ice-sheets. Accounting for ice anisotropy in an ice-flow model implies to (i) build a macroscopic anisotropic flow law whose response will depend on the local fabric and (ii) have a proper description of the ice fabric at each node of the mesh domain and be able to model the fabric evolution as a function of the flow conditions. We hereafter
Synthesis and perspectives
Applications of ice mechanical behavior modeling extend from below the single-crystal scale to the ice sheet scale. This scale range far exceeds that of engineering material sciences but is similar to the geological one. Within this scale range, many physical processes come into play, some of which are not yet very well described. Furthermore, there exist strong interactions between these processes that create bridges between the different levels of complexity. Modeling of ice has strongly
Acknowledgment
Financial support by the French “Agence Nationale de la Recherche” is acknowledged (project ELVIS, #ANR-08-BLAN-0138). Together with support from institutes INSIS and INSU of CNRS, and UJF – Grenoble 1, France. PDB and JR gratefully acknowledge funding by the German Research Foundation (DFG, project BO-1776/7). The authors gratefully aknowledge the ESF Research Networking Programme Micro-Dynamics of Ice (MicroDIce).
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