Introduction

Our objective in this chapter is to consider the ductile-to-brittle transition within polycrystalline ice loaded under compression. We describe a physical model that accounts quantitatively for the effects (described below) of strain rate, temperature, grain size, confinement, salinity and damage and which accounts as well for the transition within a number of rocks and minerals. We express the transition in terms of a critical strain rate, although it could just as well be expressed in terms of other factors. However, strain rate seems to be the most useful, because it spans the widest range of environmental and microstructural conditions encountered in practice.

The critical strain rate marks the point where compressive strength (Chapters 11 and 12) and indentation pressure (Chapter 14) reach a maximum. It is thus a point of considerable importance in relation to ice loads on offshore structures. It is of interest as well in relation to the rheological behavior of the sea ice cover on the Arctic Ocean (Chapter 15) and to the deformation of icy material of the outer Solar System.

Competition between creep and fracture

Three explanations have been offered. An early view, particularly in relation to the transition during indentation failure in the field (Sanderson, 1988), was that the transitional strain rate marks the conditions under which the first cracks nucleate, somewhat analogous to the transition under tension.

Prologue In the late 1980s, following a presentation by one of us during a meeting of the Advisory Board of Dartmouth's Ice Research Laboratory, R. Weaver of the University of Colorado pointed out that he had seen in Defense Meteorological Satellite images of the winter sea ice cover on the Beaufort Sea features that resembled the wing cracks that one of us had just seen in laboratory specimens and had reported to the Board. The features,Figure 15.1, were dispersed amongst a variety of leads and open cracks and had the distinct markings of wing cracks: extensions, wide at the mouth and sharp at the tip, that had formed out-of-plane from the tips of inclined primary cracks. We estimated from wing-crack mechanics (Chapter 11) that a far-field compressive stress of around 3 kPa would have been needed to create them. W. D. Hibler then calculated the probable stress within the ice cover at the time of the sighting, from the historical record of wind fields, and obtained a value of between 7 and 14 kPa. We published the results shortly thereafter (Schulson and Hibler, 1991). Thus began an enquiry that forms the basis of this chapter and continues as we write: Is the physics of fracture independent of spatial scale?

Introduction

The sea ice cover on the Arctic Ocean during winter, as revealed through satellite images, generally contains oriented features.

Introduction

Ice fractures under tension in a number of engineering and geophysical situations. Examples include ice breaking by ships (Michel, 1978), the bending of floating ice sheets against offshore structures (Riska and Tuhkuri, 1995), the formation of thermal cracks (Evans and Untersteiner, 1971) and the building of pressure ridges (Hopkins et al., 1999) within the sea ice cover on the Arctic Ocean. Other examples include the fracture of pancake ice within the southern Atlantic Ocean (Dai et al., 2004), the calving of icebergs (Nye, 1957; Nath and Vaughan, 2003) and the crevassing of ice shelves (Rist et al., 1999, 2002; Weiss, 2004). Extra-terrestrial tensile failures include the initiation of polygonal features within the ground ice on Mars (Mellon, 1997; Mangold, 2005) and the formation of long lineaments within the icy crust of Europa (e.g., Greenberg et al., 1998; Greeley et al., 2000). In many cases, fast crack propagation is at play, which is to say that crack growth occurs so rapidly that the dissipation of mechanical energy through creep deformation is not a major consideration. Linear elastic fracture mechanics is then a valid method of analysis. In other cases, such as the formation of crevasses and the slow propagation of cracks within ice shelves, analysis based upon non-linear processes and/or sub-critical crack growth may be more useful (Weiss, 2004). In still other cases, more in the laboratory than in the field, tensile strength is limited by crack nucleation, as will become apparent.

Introduction

During the gravity-driven flow of glaciers and ice sheets, isotropic ice at the surface progressively becomes anisotropic with the development of textures. Strain-induced textures, combined with the strong anisotropy of the ice crystal, make the polycrystal anisotropic. 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 it is sheared parallel to the basal planes. Depending on the flow conditions, this anisotropy varies from place to place. To construct ice-sheet flow models for the dating of deep ice cores, this evolving viscoplastic anisotropy must be taken into account. Computation with isotropic and anisotropic flow models predicts at depth an age of ice that can differ by several thousand years. From Mangeney et al. (1997), anisotropic ice could be more than 10% younger above the bumps of the bedrock and could be older by more than 100% within hollows. An adequate constitutive relationship must be also incorporated within large-scale flow models to simulate the variation of polar ice sheets with climate.

Various models have been proposed to simulate the evolution of the anisotropy and the behavior of such ices. The increasing numerical capability of computers and the advances in theories that link materials' microstructures and properties have enabled the development of new concepts and algorithms that constitute the so-called multiscale approach for the modeling of material behavior. In this chapter, we restrict our analysis to physically based micro-macro models using a self-consistent approach.

Introduction

Single crystals undergo plastic deformation as soon as there is a component of shear stress on the basal plane. Basal slip is observed for shear stresses in the basal plane lower than 0.02 MPa (Chevy,2005). Evidence of easy basal slip was first shown by McConnel (1891) and confirmed by many authors (Glen and Perutz, 1954; Griggs and Coles, 1954; Steinemann, 1954; Readey and Kingery, 1964; Higashi, 1967; Montagnat and Duval, 2004; see Weertman, 1973, for a review). A clear illustration of basal slip was obtained by Nakaya (1958) by performing bending experiments. Traces of the basal slip lines were made clearly visible by shadow photography (Fig. 5.1). Though some prismatic glide is observed for orientations close to those that inhibit basal slip, no clear observation of any deformation is reported in crystals loaded along the [0001] direction, which inhibits both basal and prismatic slip.

Basal slip takes place from the motion of basal dislocations with the Burgers' vector (Hobbs, 1974). The macroscopic slip direction corresponds to the maximum shear direction in the basal plane (Glen and Perutz, 1954). The slip direction is therefore always close to the direction of the maximum shear stress. From Kamb (1961), the failure to detect a slip direction in ice is explained by the fact that slip can occur in the three possible glide directions on the basal plane with a value of the stress exponent, which relates strain rate to stress, between 1 and 3.

Introduction

In this chapter we review the elastic behavior of ice, friction of ice on ice and mass diffusion. In terms of creep, elastic properties allow the applied stress to be normalized and thus the behavior to be analyzed within the context of physical mechanisms (Chapters 5–8). The mass diffusion coefficient plays a similar role in creep under low stresses. It is important, as well, to the transformation from snow to ice (Chapter 3). In terms of fracture, elastic constants affect fracture toughness (Chapter 9) and, through that property, both the tensile (Chapter 10) and the compressive strength (Chapters 11, 12). Elasticity is also relevant to the ductile-to-brittle transition (Chapters 13) and to ice loads on structures (Chapter 14). Friction is a factor in the DB transition under compression and is a major consideration in brittle compressive failure, on scales small (Chapters 11, 12) and large (Chapter 15). Friction is also fundamental to tidally driven, strike-slip-like tectonic activity on a number of icy satellites within the outer Solar System, including Jupiter's moon Europa (Greenberg et al., 1998; Hoppa et al., 1999; Schulson, 2002; Kattenhorn, 2004), Neptune's Triton (Prockter et al., 2005) and Saturn's Enceladus (Nimmo et al., 2007; Smith-Konter and Pappalardo, 2008). Thermal properties play a less direct role, but we list them for completeness, Table 4.1.

Elastic properties of ice Ih single crystals

Elastic properties have been relatively well studied.

Creep and fracture of ice are significant phenomena with applications in climatology, glaciology, planetology, engineering and materials science. For instance, the flow of glaciers and polar ice sheets is relevant to the global climate system and to the prediction of sea-level change. The ice stored within the Greenland and the Antarctic ice sheets, should it flow into the sea, would raise the level by ∼7 m and ∼60 m, respectively. The creep of ice sheets, which can impart strains that exceed unity in several thousands of years, is also relevant to paleoclimatology. The determination of the age of ice and the age of greenhouse gases (Chapters 2, 3) entrapped within deep ice cores depends in part upon constitutive laws (Chapters 5, 6) that describe the deformation of the bodies – laws that must incorporate plastic anisotropy (Chapters 5 and 7) and the presence of water near the bottom. Fracture, too, plays a role in ice sheet mechanics. When the bed steepens, the ice flows more rapidly and creep can no longer accommodate the deformation (Chapters 9, 10). Fracture ensues, leading to the formation of glacier icefalls and crevasses. Fracture is of paramount importance in the catastrophic failure of icefalls, and it is a key step in the calving of icebergs and thus in the equilibrium between accumulation and loss of mass from polar ice sheets.

Introduction

There are 12 crystalline forms of ice. At ordinary pressures the stable phase is termed ice I, terminology that followed Tammann's (1900) discovery of high-pressure phases. There are two closely related low-pressure variants: hexagonal ice, denoted Ih, and cubic ice, Ic. Ice Ih is termed ordinary ice whose hexagonal crystal symmetry is reflected in the shape of snow-flakes. Ice Ic is made by depositing water vapor at temperatures lower than about −130 ℃. High-pressure ices are of little interest in relation to geophysical processes on Earth, but constitute the primary materials from which many extra-terrestrial bodies are made. We describe their structure and creep properties in Chapter 8.

In addition to the 12 crystalline forms, there are two amorphous forms. One is termed low-density ice (940 kg m−3 at −196 ℃ at 1 atmosphere) and the other, high-density ice (1170 kg m−3, same conditions). The density of ice Ih is 933 kg m−3 at the same temperature and pressure (Hobbs, 1974). Amorphous ices can be made at low temperatures in five ways (see review by Mishima and Stanley, 1998): by condensing vapor below −160 ℃; by quenching liquid; by compressing ice Ih at −196 ℃; by electron irradiation; and through transformation upon warming from one amorphous state to another. Although once thought to be a nanocrystalline material, amorphous ice is now considered to be truly glassy water. Its mechanical behavior remains to be explored.

In this chapter we address the structure of ice Ih.

Introduction

When subjected to high pressures and varying temperatures, ice can form in 12 known ordered phases. On Earth, only ice Ih is present because polar ice sheets are too thin to reach the critical conditions for the formation of ice II and ice III. The situation is very different for the large icy satellites of the outer planets. Temperature and pressure are such that current models of the internal structure of the principal moons of the outer planets suggest that a thick icy shell containing several high-pressure phases of ice surrounds the silicate core. The occurrence and properties of such ices are the subject of numerous studies to understand the tectonics and dynamics of these ices. A review of the main physical properties of high-pressure ices can be found in Klinger et al. (1985), Schmitt et al. (1998) and Petrenko and Whitworth (1999).

The regions of stability of ice crystalline phases on the pressure–temperature diagram are shown in Figure 8.1. Several phases are not mentioned in this figure because they are not stable or not present in icy satellites. In this chapter, the analysis of the mechanical properties of high-pressure ices is restricted to ices II, III, V and VI, the only ices studied in the laboratory. Crystalline structure, density and shear modulus of several phases of water ice are given in Table 8.1.

Introduction

Understanding how polar ice sheets interact with the climatic system is of the highest importance to predict sea-level changes. Ice sheets contain information on the climate and the atmospheric composition over the last 800 000 years (EPICA Community Members, 2004). Interpretation of ice core data is directly dependent on the accuracy of ice sheet flow models used for ice core dating. Knowledge of the rheological properties of ice in the low stress conditions of glaciers and polar ice sheets is therefore needed to improve the constitutive laws that are incorporated in flow models. Due to very high viscoplastic anisotropy of the crystal (Chapter 5), ice is considered as a model material to validate micro-macro polycrystal models used to simulate the behavior of anisotropic viscoplastic materials (Gilormini et al., 2001; Lebensohn et al., 2007).

Ice displays a wide range of mechanical properties, including elasticity, visco-elasticity, viscoplasticity, creep rupture and brittle failure (Schulson, 2001). In glaciers and ice sheets, ice is generally treated as a heat-conducting non-linear viscous fluid.

Ice is assumed here to be incompressible. It will be shown that the main effect of hydrostatic pressure on the ductile behavior of ice is to modify the melting temperature of pure ice Tf with dTf /dP ≈ 0.074 ℃/MPa (Lliboutry, 1971).

In this chapter, we focus the analysis on the mechanical behavior of granular glacier ice. We assume that the behavior is ductile without the formation of cracks.

Introduction

Rectangular Cartesian coordinate systems are well suited for presenting the basic concepts of continuum mechanics since many mathematical complications that arise with other coordinate systems, for example, convected systems, are avoided. However, for the solution of specific problems, the use of rectangular Cartesian coordinate systems may result in considerable difficulties, and the use of other systems may be desirable in order to take advantage of symmetry aspects of the problem. For example, in the analysis of the expansion of a thick-walled cylindrical tube, the use of cylindrical polar coordinates has an obvious advantage. We introduce a simple theory of curvilinear coordinates in this appendix and specialize it for orthogonal curvilinear systems, in particular cylindrical and spherical. We avoid the use of the general theory of tensor components referred to curvilinear coordinates by considering what are known as the physical components of tensors that are derived for orthogonal coordinate systems. Superscripts are used to denote curvilinear coordinates.

Description of Motion

In this chapter we are concerned with the motion of continuous bodies without reference to the forces producing the motion. A continuous body is a hypothetical concept and is a mathematical model for which molecular structure is disregarded and the distribution of matter is assumed to be continuous. Also it may be regarded as an infinite set of particles occupying a region of Euclidean point space E3 at a particular time t. The term particle is used to describe an infinitesimal part of the body, rather than a mass point as in Newtonian mechanics. A particle can be given a label, for example, X, and there is a one-one correspondence between the particles and triples of real numbers that are the coordinates at time with respect to a rectangular Cartesian coordinate system.

There are four common descriptions of the motion of a continuous body:

1. Material description. The independent variables are the particle X and the time t.

2. Referential description. The independent variables are the position vector X of a particle, in some reference configuration, and the time t. The reference configuration could be a configuration that the body never occupies but it is convenient to take it as the actual unstressed undeformed configuration at time t = 0. The term natural reference configuration is used to describe the unstressed undeformed configuration at a uniform reference temperature.

3. […]

Introduction

In this chapter problems of finite deformation elastostatics for isotropic hyperelastic solids are considered. Exact solutions for some problems of finite deformation of incompressible elastic solids have been obtained by inverse methods. In these methods a deformation field is assumed, and it is verified that the equilibrium equations and stress and displacement boundary conditions are satisfied. There are some elastostatic problems that have deformation fields that are possible in every homogeneous isotropic incompressible elastic body in the absence of body forces. These deformation fields are said to be controllable. In an important paper by Ericksen an attempt is made to obtain all static deformation fields that are possible in all homogeneous isotropic incompressible bodies acted on by surface tractions only. Ericksen further showed that the only deformation possible in all homogeneous compressible elastic bodies, acted on by surface tractions only, is a homogeneous deformation. In this chapter solutions are given for several problems of finite deformation of incompressible isotropic hyperelastic solids and two simple problems for an isotropic compressible solid. These solutions with one exception involve controllable deformation fields and are based on the physical components of stress, deformation gradient, and the left Cauchy-Green strain tensor. This is in contrast to the approach of Green and Zerna, who used convected coordinates and tensor components.

Rivlin, in the late 1940's, was the first to obtain most of the existing solutions, for deformation of an isotropic incompressible hyperelastic solid.

This book is based on notes prepared for a senior undergraduate or beginning graduate course that we taught at the universities of Alberta and Victoria. It is primarily intended for use by students of mechanical and civil engineering, but it may be of interest to others. The mathematical background required for the topics covered in the book is modest and should be familiar to senior undergraduate engineering students. In particular it is assumed that a reader has a good knowledge of classical vector mechanics and linear algebra. Also, a background of the classical thermodynamics usually taught in undergraduate engineering courses is desirable. One motivation for the book is to present an introduction to continuum mechanics that requires no background in certain areas of advanced mathematics such as functional analysis and general tensor analysis. The treatment of continuum mechanics is based on Cartesian tensor analysis, but orthogonal curvilinear coordinates and corresponding physical coordinates are considered in appendices.

A list of books that consider tensor analysis and applications is given at the end of chapter 1. Several of the books are out of print but may be useful to students if they can be obtained from libraries. Mathematica is used for symbolic manipulation, numerical computation, and graphs where appropriate, and its use is encouraged.

Chapter 1 is a detailed introduction to Cartesian tensor analysis. It differs from some other treatments of the topic, for example, the early texts by Jeffreys and Temple, by emphasizing both symbolic and suffix notation for first-(vectors) and second-order tensors.