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Meteoritics & Planetary Science 41, Nr 10, 1497–1508 (2006) Abstract available online at http://meteoritics.org

The fracture of Ih: A short overview

Erland M. SCHULSON

Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire 03755, USA E-mail: [email protected] (Received 25 October 2005; revision accepted 09 April 2006)

Abstract–This paper presents a short overview of the fracture of water ice Ih. Topics include the ductile-to-brittle transition, tensile and compressive strength, compressive failure under multiaxial loading, compressive failure modes, and brittle failure on the geophysical scale (Arctic cover, Europa’s icy crust). Emphasis is placed on the underlying physical mechanisms. Where appropriate, comment is made on the formation of high-latitude impact craters on Mars.

INTRODUCTION (Weeks and Gow 1980). (The terms S2 and S3 are taken from the classification of texture by Michel and Ramseier 1971). Water ice can adopt one of 13 crystalline forms or one of Subsurface Martian ice may be a textured polycrystal as well, least two amorphous states (Petrenko and Whitworth 1999). should it have formed from the directional solidification of Under low pressure and at temperatures above about −120 °C, water. water ice (or simply ice) adopts a hexagonal structure, Whether in the form of a single crystal or a polycrystal, ice denoted Ih, reflected in the shape of snowflakes. Each exhibits two kinds of inelastic behavior. When slowly loaded it molecule within the Ih unit cell is connected to four others flows plastically (i.e., creeps). This is manifested, for instance, (owing to the 104.5° H-O-H bond angle), and each unit cell by strains in excess of unity within alpine where the contains four molecules. As a result of the molecular bend, deformation rate is equal to or less than about <10−9s−1 and the structure is relatively open as opposed to the closely stresses are of the order of 0.1 MPa and lower. Creep of clean packed structure of hexagonal metals: upon melting it ice, i.e., free from dust and rock, has been studied extensively collapses and this accounts for the density of ice Ih and is well understood (for reviews see Duval et al. 1983; (917 kg m−3 at 0 °C) being lower than the density of water. Weertman 1983; Durham and Stern 2001). Dirty ice is Given the relatively high temperatures near the surface of currently under investigation (Durham et al. 2006). On the Mars (>−120 °C, Armstrong et al. 2005), ice Ih is probably other hand, when rapidly loaded/deformed, but well below the crystalline form of interest in relation to the role of dynamic rates, ice fractures, as evident from the rubble field volatiles in the evolution of impact craters. that forms when a sheet of sea ice is pushed up against an With the exception of snowflakes, ice Ih generally forms engineered structure or from the fracture patterns that develop as a polycrystal, at least on Earth. Glaciers and , for within the winter sea ice cover on the Arctic Ocean (Marko and instance, form through the pressure-sintering of , and Thomson 1977; Kwok 2001; Schulson 2004; Marsan et al. may be characterized as bubbly aggregates of equiaxed grains 2004). Brittle behavior has been less extensively studied. 1–20 mm in diameter. The as-formed aggregates are However, significant progress has been made during the past randomly oriented (i.e., the crystallographic c-axes exhibit no decade or so, and that progress forms the focus of this preferred orientation), but can develop a strong texture discussion. through creep deformation and dynamic recrystallization In relation to Martian features, creep/ductile behavior (Castelnau et al. 1996). In comparison, the ice cover on the appears to govern the evolution of lobate debris aprons and Arctic Ocean forms through unidirectional solidification, and softened craters (Mangold et al. 2002). On the other hand, is comprised of columnar-shaped grains (~2–20 mm in brittle fracture probably governs the initial formation of high- diameter) whose crystallographic c-axes tend to lie within the latitude impact craters and subsequent rim collapse, and plane of the sheet. The c-axes are randomly oriented within almost certainly controls the initiation of polygonal features that plane in most locations within the Arctic Ocean (giving that form through tensile overloads that result from thermal S2 ice), but aligned within the plane (giving S3 ice) in others contraction (Mellon 1997; Mangold 2005).

1497 © The Meteoritical Society, 2006. Printed in USA. 1498 E. M. Schulson

To some extent, the present discussion continues where volume fractions, the nature of that transition is probably earlier and more thorough reviews of brittle failure ended different from the transition in ice per se. (Schulson 2001, 2002a). For details beyond the scope of this Ice, incidentally, is unusual in exhibiting brittle behavior overview, we encourage the reader to consult the earlier right up to the melting point, at deformation rates well below reviews. dynamic rates. At root are sluggish dislocation kinetics (e.g., see Ahmad and Whitworth 1988; Shearwood and Whitworth DUCTILE-BRITTLE TRANSITION 1991). Owing to its crystal structure, ice Ih slips preferentially on the basal planes, and this is impeded by the unique To begin, a few comments on the ductile-brittle transition requirement of protonic rearrangement (Glen 1968). may be helpful. This transition is often defined, especially Molecular diffusivity is low as well—about three orders of under monotonically increasing loads, in terms of a critical magnitude lower than atomic diffusivity through metals at an strain rate, and is marked by a pseudo-linear stress-strain equivalent homologous temperature. (The melting −15 curve that ends abruptly with a sudden drop in load at terminal point diffusion coefficient of H2O in ice Ih is around 10 to failure. Under tension, the transition strain rate is of the 10−14 m2/s compared to 10–12 to 10–11 m2/s for elemental order of 10−7 s−1 for warm (−10 °C) polycrystalline metals.) This factor, however, appears to play more the role of aggregates of 1–2 mm grain size; under compression it is suppressing diffusion creep than of impeding dislocation higher and typically falls within the range 10−4 s−1 to 10−3 s−1 mobility. Interestingly, the “degree of brittleness” is rather for granular ice of the same grain size (Schulson 2001). For high, as evident from the relatively low ratio of toughness to S2 columnar ice compressed across the columns, the surface energy. For structural ceramics and rock, this ratio of transition strain rate is essentially the same as it is for granular toughness (~1 J/m2) to surface energy (~0.1 J/m2) is generally ice, but when loaded along the columns it is an order of greater than 30. The implication is that the creation of new magnitude lower (Kuehn and Schulson 1994). For larger surface is the major process underlying the toughness of ice bodies that contain larger stress concentrators or for more (Nixon and Schulson 1987). coarsely grained ice which also contains larger stress The ductile-brittle transition can be understood in terms concentrators, the transition strain rate is lower, scaling of the competition between crack-tip creep and crack roughly as (length of concentrator)−1.5 (Schulson 2001). The propagation (Schulson 1990, 2001). When creep occurs (compressive) transition strain rate decreases with decreasing sufficiently rapidly to relax stresses that concentrate at crack temperature, to the extent that it is about an order of tips, ductile behavior follows. When creep occurs too slowly, magnitude lower at −40 °C than at −10 °C, for both fresh- stress builds up to the point of local rupture and brittle water ice and salt-water ice of typical salinity 4–5‰ (Qi and behavior sets in. The transition cannot be accounted for Schulson 1998). At −80 °C, the transition (compressive) simply in terms of the onset of cracking, for given the occurs at a strain rate of ~10−5 s−1, for finely grained (d = plastically anisotropic nature of the Ih crystal (Duval et al. 1 mm) polycrystals of fresh-water ice (Aragawa and Maeno 1983), cracks form at quite low strain rates within the ductile 1997). In addition to temperature, grain size, and texture, regime, but are stable there. Nor can the transition be other factors are salinity (more below), confinement, and explained in terms of a critical density of deformation preexisting damage: an increase in each raises the critical damage, for ice that is rather slowly compressed (but not so strain rate (Schulson 2001). slowly as to avoid crack nucleation altogether) is riddled with Little is known about the role of hard particles. This gap short cracks that change the optical appearance of the material is unfortunate, given that Martian ice probably contains dust from transparent to snowy white; yet the damaged material and rock. Indeed, the role of second phases in general has not can be shortened by more than 10% without collapsing. Only been well studied. About all that is known with certainty is when cracks begin to propagate does the macroscopic that inclusions of brine and porosity in salt-water ice (~5% by behavior change. For a physically based model that volume) raise the transition strain rate by about an order of incorporates the resistance to creep, to crack growth and to magnitude with respect to fresh-water ice, owing to a brine- frictional sliding across crack faces, as well as grain size and induced increase in the creep rate or, correspondingly, to a confinement, see the earlier discussions (Schulson 1990, reduction in viscosity under a given stress at a given 2001). The model given there appears to account for all temperature (compare Schulson and Nickolayev 1995 with experimental observations to date. It also accounts for the Schulson and Buck 1995). Concerning the effect of rock, ductile-to-brittle transition within rocks and minerals as well Mangold et al. (2002) claim that a ductile-brittle transition (Renshaw and Schulson 2001), but for the reason mentioned occurs when the ice content of ice-sand mixtures is lower than above is not directly applicable to rock-rich, ice-rock ~28% (or the sand content exceeds 72%). This is based upon mixtures of the Mangold (2002) kind. However, it should be triaxial compression creep experiments on sand-rich mixtures applicable to leaner dust-rock-ice mixtures where ice is the (52–75% sand by volume) confined under 12 MPa at −10 °C. major phase, provided that the effects of the particles on However, given the likelihood of particle contact at such high creep, friction, fracture toughness, and grain size are known. The fracture of water ice Ih 1499

FAILURE CRITERIA strength thus reaches a maximum at the ductile-to-brittle transition. It is important to recognize the dual mechanical character of ice when considering failure criteria under multiaxial states TENSILE FAILURE of stress, particularly compressive stress states (discussed below). Within the ductile regime, Jones (1982) found from When deformed at quasi-static strain rates, bubble-free compression experiments on equiaxed and randomly oriented virgin ice (i.e., crack-free) of grains about 1 mm in diameter polycrystals (termed granular ice) of finely grained (d ~1 mm) fracture via transgranular cleavage under stresses of ice loaded triaxially at −11 °C that the differential stress at ~1.5 MPa. The tensile strength scales with (grain size)−0.5 in a yield (i.e., the difference between the greatest and the least Hall-Petch manner, and is generally governed by the unstable, principal stress) is independent of confining pressure, opening-mode (termed mode-I) propagation of a single crack implying conformity with the classical von Mises criterion for whose size is set by grain size. In addition to crack plastic yielding, as discussed by Nadreau and Michel (1986). propagation, crack nucleation is important within finely Similarly, Schulson and Nickolayev (1995) and Melton and grained material, slowly loaded. The fracture toughness of Schulson (1998) found from compression experiments on ice, which measures the resistance to crack propagation (more both fresh-water and salt-water S2-textured columnar-grained below), is not very sensitive to temperature and strain rate, material of ~6 mm column diameter loaded at −10 °C under and so quasi-static tensile failure is relatively insensitive to both biaxial (across-column) and triaxial compressive stresses these parameters. The dynamic tensile strength, in that textured ice obeys Hill’s criterion (1950); comparison, is not governed by the propagation of a single correspondingly, the associated flow rule applies, which crack, but by the interaction of many cracks. As a result, the means that the “strain vector” is essentially normal to the tensile strength increases with strain rate and reaches a failure envelope. Hill’s criterion extends von Mises’ criterion level as high as ~17 MPa for finely grained ice (d <1 mm?) at by incorporating the fact that, owing to crystallographic 104 s−1 at −30 ± 10 °C (Lange and Ahrens 1983). texture and to the preference for glide on basal [0001] planes, For a more in-depth discussion, please see the earlier the yield strength of S2 ice is an anisotropic property. On the reviews (Schulson 2001, 2002a). other hand, when multiaxially loaded under moderate degrees of confinement within the brittle regime, ice obeys Coulomb’s UNCONFINED COMPRESSIVE FAILURE failure criterion. This is evident from measurements on both granular ice (Jones 1982) and columnar ice (Schulson and In comparison to tensile fracture, brittle compressive Nickolayev 1995; Gratz and Schulson 1997; Schulson et al. failure is a more complicated process. Consider first the 2005, 2006). Coulomb’s criterion incorporates the fact that important factors and then the mechanism. owing to internal frictional sliding—a process that does not Temperature has a large effect: the colder the ice, the play a role in volume-conserving plastic deformation and stronger it is, reminiscent of ductile behavior. For instance, hence is not part of either von Mises’ or Hills’ criterion—the the unconfined brittle compressive strength (UCS) of differential stress at terminal failure increases with increasing equiaxed polycrystals of ~1 mm grain size increases from hydrostatic component of the stress state. Correspondingly, ~3 MPa at 0 °C (Carter 1971) to ~72 MPa at −173 °C failure occurs through the development of Coulombic shear (Arakawa and Maeno 1997). Strain rate, in comparison, has a faults inclined by about 30° to the direction of shortening small effect, and one that is opposite to the strain-rate (more below). In this case, the “strain vector” is not normal to hardening that operates within the ductile regime. For the failure envelope (Schulson, unpublished). Under still example, at −10 °C the UCS of the same material decreases higher confinement within the brittle regime, but not so high from about 10 MPa at 1 × 10−3 s−1 to 7 MPa at 10−1 s−1 (Carter as to cause a phase change (through either melting, should the 1971; Schulson 1990). Microstructure and crystallographic temperature of the ice exceed −22 °C, or transformation to a texture are also important: the finer the grains, the stronger is more dense ), polycrystalline ice Ih fails either by the ice; and columnar-grained material of S2 texture loaded spalling out of the loading plane under biaxial loading or by along the columns is about three to four times stronger than plastic faulting across planes inclined by about 45° to the material loaded across the columns (Kuehn and Schulson direction of shortening under triaxial loading (Schulson 2002). 1994). For instance, at −10 °C at 10−3 s−1 the UCS of granular It is erroneous to use one failure criterion to account for ice increases from about 4 MPa to 10 MPa upon reducing the both ductile and brittle behavior, as some authors do, because grain size from 8 mm to 1 mm, again in accord with Hall- the physics are different. Ductile behavior is based upon Petch behavior (Schulson 1990); and under the same dislocation mechanics, while brittle behavior depends upon conditions, the UCS of S2 saline ice of 4–5 ppt salinity and of crack mechanics. As a result, ice exhibits marked strain-rate 4–6 mm column diameter is 10–14 MPa along the columns hardening within one regime (ductile) and moderate strain- and 3–5 MPa across the columns (Kuehn and Schulson 1994). rate softening within the other (brittle). The compressive In comparison, grain size has either little or no effect on the 1500 E. M. Schulson

Frictional sliding is key here. The kinetic coefficient of friction for ice sliding upon itself increases with decreasing temperature and with decreasing sliding speed (Kennedy et al. 2000, Fortt and Schulson 2004). As discussed earlier (Schulson 2001), this can account quantitatively for the effects of both temperature and strain rate on the UCS. The effect of grain size is accounted for in terms of stress intensification at crack tips, which scales as (crack size)½, and by the fact, already noted, that crack size is controlled by grain size. Incidentally, the difference between compressive and tensile strength under uniaxial loading may now be understood. Under compression crack growth is a quasi- stable process. This means that the applied stress must continue to increase to effect crack growth. Under tension, on the other hand, cracks grow in an unstable manner once KI = KIc, as already mentioned. Thus, this difference in crack behavior accounts for the order of magnitude difference in the uniaxial strength (for material of the same grain size). So far, our discussion has focused on behavior under quasi-static deformation rates. How does ice behave when loaded within the dynamic regime, as would be the case Fig. 1. A photograph showing a specimen of fresh-water, columnar during Martian impact? There have been few measurements (S2) ice loaded across the columns to terminal failure under uniaxial − − compression (vertical), at –10 °C at 5 × 10–3 s–1. The long axes of the of compressive strength at strain rates above 10 1 s 1, and columnar-shaped grains are perpendicular to the plane of the page. where data exist they are scattered. For instance, Jones Note the splits oriented along the loading direction, as well as the (1997) compressed fresh-water, S2 columnar ice of ~6 mm wing cracks. Arrows point to wing cracks that originated from column diameter by loading uniaxially along the columns sliding along parent cracks inclined to the loading direction. The (Jones, personal communication 2005) at −11 °C at strain splits were created by the linking up of the wing cracks (from Iliescu − − − and Schulson 2004). rates from 10 1 s 1 to 10 s 1. He claimed that the strength increased slightly with increasing strain rate, scaling as ductile compressive strength (Cole 1987, 2001), although (strain rate)0.15. Similarly, through uniaxial compression texture is still important (Kuehn and Schulson 1994). Other experiments on the same kind of material loaded along factors that may be important, but which have not been the columns at −10 °C at strain rates between 10−2 s−1 and studied in a systematic manner, include preexisting damage 1.6 s−1, Schulson et al. (2005a) found that the strength and porosity. increased slightly with increasing strain rate, scaling as Concerning the failure mode, when care is taken to (strain rate)0.16. Finally, from a series Hopkinson-bar tests at eliminate lateral constraint at the ice-platen interface, −10 °C and −30 °C on disc-shaped specimens of a variety of terminal failure occurs through the development of a series of microstructures, Shazly et al. (2006) again reported macroscopic splits aligned with the loading direction. (Under moderate strain-rate hardening at strain rates from 50 to lateral constraint, either shear faults develop or the ice 1000 s−1. An interesting point from Shazly’s study is that, “explodes.”) The axial splits form through the linking up of unlike the static strength, the dynamic strength of the ice wing cracks. Wing cracks are deformation features that appears to be relatively insensitive to microstructure. The develop within both columnar ice (Cannon et al. 1990) and problem with all three sets of data is the high degree of granular ice (Schulson 1990), and are defined as out-of-plane scatter: in one case (Jones) the regression coefficient was extensions to primary cracks. Figure 1 shows examples in S2 only r2 = 0.46; in another (Schulson), r2 = 0.61. Thus, while columnar ice. Wing cracks initiate to relax the tensile stress the results appear to be suggestive of moderate strain-rate that builds up at the tips of primary cracks as a result of hardening, more work is needed before firm conclusions can frictional sliding across the crack faces (opposing faces of be drawn. Perhaps the strongest point that can be made is which are in contact). Through additional sliding, the that these higher-rate data imply that the moderate strain-rate extensions lengthen when crack-mouth opens and the mode-I softening that characterizes compressive behavior at lower stress intensity factor at the crack tip reaches the critical value strain rates within the regime of brittle behavior does not for crack propagation (i.e., KI = KIc). Mode-I loading refers to continue to characterize the material as it enters the regime the opening of a crack under external load, as opposed to of dynamic loading. New gas-gun experiments at NASA shearing (mode-II) or to tearing (mode-III) of the material (Carney et al. 2006) may shed further insight into dynamic adjacent to the crack tip. compressive failure. The fracture of water ice Ih 1501

Fig. 2. The brittle compressive failure envelope for the ice of Fig. 1 proportionally loaded biaxially across the columns to terminal failure, at –10 °C at 5 × 10–3 s–1. (The strain rate is along the direction of higher stress.). Along the lower segment of the rising branch, failure occurs by shear faulting (see Fig. 4.) Along the descending branch, failure occurs by spalling out of the loading plane (from Iliescu and Schulson 2004).

BRITTLE COMPRESSIVE FAILURE UNDER MULTIAXIAL LOADING

Under multiaxial compressive stress states, confinement plays two roles: it increases the normal stress acting across inclined cracks, and thus impedes frictional sliding, and it tends to close up wing cracks, thus lessening the mode-I stress intensity factor and impeding crack growth. As a result, a small amount of confinement markedly raises the terminal failure stress. To illustrate this point, Fig. 2 shows a failure envelope for S2 columnar-grained, fresh-water ice of 2–4 mm column diameter that was biaxially compressed across the columns to terminal failure, at −10 °C at 5 × 10−3 s−1 (Iliescu and Schulson 2004). Figure 3 shows the microstructure of the ice. The envelope has both a rising or a confinement- strengthening branch and a descending or confinement- weakening branch. Our interest at the moment is the rising branch. Along this branch a confining stress of only 1 MPa increases the compressive strength by a factor of two, from ~5 MPa to ~10 MPa. (The uniaxial strength is lower than noted above because the grain size is larger.) Correspondingly, the mode of failure changes from the axial Fig. 3. Photograph showing the microstructure of columnar-grained splitting shown in Fig. 1 to Coulombic shear faulting, shown S2 fresh-water ice. The ice was formed through unidirectional in Fig. 4. solidification. 1502 E. M. Schulson

σ Fig. 4. Photographs showing a shear fault in the S2 ice of Fig. 3. In these images the major stress 11 was applied vertically and the minor stress σ22 horizontally. The columnar-shaped grains are again perpendicular to the plane of the page (from Schulson et al. 1999). a) Fault as σ seen in thick section. Note its inclination with regards to 11. Note the wing cracks (arrowed) distributed within the background field of damage. The fault is bordered by wing cracks, as evident from its zig-zag edges. b) The same fault as seen in a thin section. The fault is comprised of a band of damage that includes other wing cracks plus comb cracks. c) A section of (b) at higher magnification showing comb cracks. Note that the comb is comprised of a set of closely spaced secondary cracks that stem from one side of a parent crack that was inclined σ to 11 owing to nonuniform sliding across the parent crack. σ At this juncture, the reader familiar with the theory of principal stress, 11, is taken as the most compressive brittle compressive failure under multiaxial loading may applied stress). That tenet, however, applies to materials that wonder whether S2 ice violates a basic tenet of that theory. exhibit inelastically isotropic behavior. S2 ice exhibits The tenet holds that shear faults should be parallel to the anisotropic behavior, and under uniaxial loading is stronger second principal stress (Rudnicki and Rice 1975), whereas when loaded along the columns than when loaded across the faults within the S2 ice loaded biaxially are parallel to them, as already noted. Hence, S2 ice does not violate this σ the third principal stress (i.e., parallel to 33 = 0; the first basic tenet. The fracture of water ice Ih 1503

Shear faults appear to develop through interactions amongst another kind of secondary crack, termed a comb crack (Schulson et al. 1999). A sketch of this feature is shown in Fig. 5, and an actual example taken from within a Coulombic fault is shown in Fig. 4c. Comb cracks are comprised of sets of fixed-free microcolumns that form through the creation of sets of secondary cracks that stem from one side of a parent crack owing to nonuniform sliding across the parent (for discussion see Cooke 1997; Schulson et al. 1999; Renshaw and Schulson 2001). The microcolumns are loaded mainly by frictional drag across their free ends, like the teeth in a comb under a sliding thumb. Once the appropriate combination of mode-I and mode-II stress intensity factors for the attendant secondary cracks reaches a critical level, the most favorably situated columns break (actually, the secondary cracks grow). Load is then transferred to adjacent columns, which fail, and so on, leading to a kind of failure cascade. Terminal failure is then marked by a fully developed shear fault inclined at ~30° to the direction of shortening. Modeling of the process in terms of the comb mechanics developed by Renshaw and Schulson (2001) leads to failure stresses that are in good agreement with measurement (Schulson et al. 2006). The model, incidentally, accounts as well for the low-confinement strength of rocks and minerals within which comb cracks (there termed “splay cracks,” “horse-tail cracks,” and “feather cracks”) also develop. An important implication of frictional sliding is that the process is completely suppressed once the hydrostatic component of the stress tensor reaches a critical level. From the comb crack mechanism—indeed, from any frictional sliding mechanism—one can show that for the across-column biaxial loading of S2 ice described above, sliding is suppressed when the following criterion is satisfied: Fig. 5. Sketch of comb crack mechanics. The comb consists of a set of slender microcolumns, fixed on one end and free on the other. The ⁄ microcolumns are loaded axially, P, and by frictional drag across ()μ2 12 μ i + 1 – i their free ends. The drag is the greater force and this creates a Rt = ------⁄ (1) ()μ2 12 μ moment, M, which loads tips of the secondary cracks under mixed i + 1 + i mode-I and mode-II. The secondaries propagate when the mixed- σ σ mode stress intensity reaches the critical level. Columns/teeth near where Rt is the ratio of the minor to major stress (Rt = 22/ 11) the free surface “break” first, owing to lower constraint there, leading μ and i is the internal friction coefficient. Under the conditions to “blowout.” of the experiments underlying the above failure envelope μi = 0.98 ± 0.03 (Schulson et al. 2006). This leads to Rt = 0.18 ± In the example shown in Fig. 2, the failure envelope is σ σ 0.01, in excellent agreement with observation (Iliescu and symmetrical about the loading path 22 = 11. This is a Schulson 2004). Under biaxial confinements greater than this manifestation of the fact that S2 ice is mechanically isotropic level, a new failure mechanism sets in, characterized by the within the X1-X2 plane (defined in Fig. 3). In the X3 or confinement-weakening shown in Fig. 2 and by spalling out along-column direction the ice is stronger, as already of the loading plane. Iliescu and Schulson (2004) describe this mentioned. In the absence of S2 texture (i.e., for randomly new process and present a physical model that is based upon oriented granular ice), a single confining stress has either little the competition between wing crack growth and Euler or no effect at all (Weiss and Schulson 1995). In that case, the buckling of deformation-produced thin plates. Under triaxial biaxial and uniaxial compressive strengths are essentially confinement greater than that required to suppress frictional equal. To strengthen such polycrystals through confinement, sliding, plastic faults develop (Schulson 2002b), a failure- an additional confining stress, σ33, must be applied. In other mode transition that is again seen in rocks and minerals words, triaxial loading is required to impart to randomly (Renshaw and Schulson 2004). oriented aggregates the kind of confinement-induced 1504 E. M. Schulson

Table 1. Brittle compressive failure modes for polycrystals of ice Ih. S2 columnar S2 columnar Granular Granular Ice/stress state low-R high-R low-R high-R Biaxiala Coulombic faults whose Spalling out of the plane Splitting across planes Splitting across planes trace lies in the plane of of loading parallel to direction of parallel to direction of loading shortening and shortening and perpendicular perpendicular to no-load to no-load direction direction Triaxiala Coulombic faults whose Plastic faults Coulombic shear faults Plastic faults trace lies in the plane of the two most compressive stresses aBiaxial stresses applied across the columns of S2 columnar ice; triaxial stresses applied both across and along the columns where the along-column stress is the least compressive stress strengthening that is imparted to columnar-grained, S2 Forthcoming). The difference, we suggest, is related to stress textured ice through biaxial loading across the columns. In concentrators that are a million times longer in the ice sheet that case, the critical degree of confinement that marks the than in laboratory specimens, and thus a thousand times more transition from shear faulting to a new failure mode is still effective in reducing the magnitude of the applied stress to given by the above equation, but the stress ratio is now activate faulting. defined as R = σ33/σ11. Another large-scale feature that, like the Arctic sea ice To summarize, Table 1 lists the brittle compressive cover, is riddled with cracks is the icy crust of Europa. While modes for both S2 columnar ice and granular ice loaded under many different mechanisms appear to have been active, and both low and high degrees of confinement under either biaxial may still be, a recent analysis (Schulson 2002c) suggests that or triaxial loading. Although listed as a brittle mode, plastic the frictional sliding wing-crack mechanism may be amongst faulting (on planes of maximum applied shear stress) is better them and that this mechanism can account for the formation viewed as a brittle-like mode: the attendant stress-strain curve of wedge-shaped cracks. The analysis also leads to crustal exhibits a sudden drop upon fault formation, but the damage stresses that are in keeping with earlier estimates based upon within the fault is comprised mainly of plastic deformation elasticity theory (e.g., Greenberg et al. 1998). and of dynamically recrystallized grains as opposed to The question of size and its role on compressive failure microcracks that characterize Coulombic faults (Schulson is currently one of the outstanding issues in ice mechanics. 2002b). The transition from low-to-high confinement is Some investigators (Overland et al. 1995; McNutt and defined by Equation 1. Overland 2003) argue that new physics enters the picture as spatial scale increases, and they support that argument with BRITTLE COMPRESSIVE FAILURE the observation (Sanderson 1998) that the contact pressure at ON THE LARGER SCALE terminal failure decreases as size increases. The idea is that there are emergent scales, as in hierarchy theory of Turning to the larger scale, and specifically, to the ice complexity as applied to biological systems (O’Neill et al. cover on the Arctic Ocean—a floating plate that is comprised 1986), above which new mechanisms set in. The problem is largely of S2 columnar ice—failure there also appears to that the emergent scales seem to be defined arbitrarily. We occur through Coulombic shear faulting, at least in some take a different view, as discussed elsewhere (Schulson 2004, scenarios (Schulson 2004). Figure 6 shows an example. The Forthcoming). Based upon the observation that certain figure shows a time series of satellite images that captured the fracture features—wing cracks, comb cracks, and Coulombic development of a set of subparallel, left-lateral faults that shear faults—look the same on both the smaller and larger formed during the 1997-1998 SHEBA experiment (see scales (Schulson and Hibler 1991; Schulson 2004), and based Perovich et al. 1999 for discussion of SHEBA). Note the pair upon the reasonable quantitative agreement between the of wing cracks in Fault BB′. At the time the images were wing-crack based prediction of failure stresses and measured obtained, wind-driven loading had induced a biaxial state of stresses within the Arctic ice sheet, noted above, we suggest compression within the plane of the cover: the major in situ that the physics of fracture may be scale independent. stress was measured to be 30 kPa (Richter-Menge et al. 2002). Consistent with this view is the result from fractal analysis of An independent analysis (Schulson 2004) of the faults based fragmentation patterns in ice (Weiss 2001, 2003) that shows upon wing crack mechanics generated a stress level of that in over nine order of magnitude, from 10−4 m to 105 m, ~19 kPa, in fair agreement with the measurement. It is worth there is no characteristic length. Moreover, similar analysis noting that these stresses are three orders of magnitude lower (Marsan et al. 2004) suggests that localized deformation on than the failure stress of liter-sized specimens of first-year sea all scales accounts for the deformation of the Arctic ice ice harvested from the parent ice cover (Schulson et al., cover. The fracture of water ice Ih 1505

Fig. 6. RADARSAT images of the sea ice cover on the Arctic Ocean taken during the SHEBA experiment, courtesy of R. Kwok of Jet Propulsion Laboratory and H. Stern of University of Washington. RADARSAT images (from Schulson 2004). a) Day 302 of 1997. Grid is 50 km × 50 km. Each division is 10 km. b) Day 314 of 1997. Note minor distortion of grid, indicative of shear deformation. c) Day 317 of 1997. Note major distortion of grid caused by development of a set of subparallel, left-lateral (from plan view) shear faults. Two major faults labeled AA′ and BB′. Note also pair of wing cracks (boxed) in fault BB′.

FRACTURE TOUGHNESS Particles of kaolinite up to 3% by volume have little effect (Smith et al. 1990). Implicit in all of the above discussion is fracture Size may also be a factor. For instance, the apparent toughness (i.e., the resistance to crack propagation), or more fracture toughness of test specimens harvested from floating specifically the plane strain fracture toughness (equivalently, ice sheets doubles in value as size increases up to ~3 m, and the critical mode-I stress intensity factor), denoted KIc. The then remains more or less constant, at least up to 80 m value usually taken for ice under conditions on Earth is KIc = (Dempsey 1996). However, it is not clear whether the higher 0.1 MPa m0.5. The fracture toughness increases slightly with values were obtained under crack-tip loading rates high both decreasing temperature (Liu and Miller 1979; Goodman enough to suppress significant contributions from creep. For 1979; Nixon and Schulson 1987) and decreasing grain size loading rates too low, the apparent fracture toughness can be (Nixon and Schulson 1988), but decreases with increasing up to three times as high as the plane strain value (Bentley porosity (Timco and Frederking 1982; Rist et al. 1999, 2002). et al. 1989). 1506 E. M. Schulson

As already noted, the energy consumed in creating new polycrystalline ice under uniaxial compression. Cold Regions surface can account for a major fraction of the plane strain Science and Technology 26:215–229. fracture toughness of ice (Nixon and Schulson 1987). The Armstrong J. C., Titus T. N., and Kieffer H. H. 2005. Evidence for subsurface water ice in Korolev crater, Mars. Icarus 174:360–372. remainder is probably related to either crack tortuosity or Bentley D. L., Dempsey J. P., Sodhi D. S., and Wei Y. 1989. Fracture inelastic processes yet to be specified. toughness of columnar fresh-water ice from large scale DCB tests. Cold Regions Science and Technology 17:7–20. IMPACT CRATERS ON MARS Cannon N. P., Schulson E. M., Smith T. R., and H. J. 1990. Wing cracks and brittle compressive fracture. Acta Metallurgica et Materialia 38:1955–1962. Finally, what insight can be gained about the formation Carney K. S., Benson D. J., DuBois P., and Lee R. 2006. A of impact craters within the polar ice caps and subsurface phenomenological high strain rate model with failure for ice. permafrost on Mars from our knowledge and understanding International Journal of and Structures, doi:10.1016/ of ice on Earth? That’s a difficult question to answer, given j.ijsolstr.2006.04.005. that we know so little about the character of Martian ice. Carter D. 1971. Lois et mechanisms de l’apparente fracture fagile de la glace de riviére et du Rac. Ph.D. thesis, L’universite Laval, However, the impact-on-ice experiments of Kawakami et al. France. (1983) offer some guidance. Through a systematic study in Castelnau O., Duval P., Lebensohn R. A., and Canova G. 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