The following article appears in the journal JOM,
51 (2) (1999), pp. 21-27.

Featured Overview

The Structure and Mechanical Behavior of Ice

Erland M. Schulson
Author's Note: The Ice Research Laboratory at Dartmouth College was founded by the author in 1983 through a development grant from Mobil Corporation. It was expanded in 1984 through an Army Research Office-URIP, expanded again in 1986 through an Office of Navy Research-URI, and expanded again in 1994 through a second Army Research Office-URIP. The IRL is a materials research facility housed within cold rooms. The laboratory currently consists of ten separate cold rooms, some capable of reaching below -40°C. Situated within are facilities for growing and characterizing ice of different kinds, preparing test specimens, and measuring mechanical and electrical properties.

Since icebergs were first proposed as potential aircraft carriers in World War II, research has led to a better understanding of the mechanical behavior of ice. While work remains, especially in relating fracture on the small scale to that on the larger scale and to the appropriate structural features, the groundwork in materials science has been laid. This paper presents an overview of the structure and mechanical behavior of polycrystalline terrestrial ice.


During the Second World War, G. Pyke proposed that an iceberg be used as an aircraft carrier in the Atlantic.1 When it became apparent that natural icebergs were too small, proponents then launched a plan to build a giant "bergship." They found that the mechanical properties of ice were not up to the task, howeverthe tensile strength was too low, and the ability to withstand ballistic impacts and explosions was unacceptable. In February 1943 it was discovered that wood pulp could solve the problem; for instance, the addition of four percent Canadian spruce more than doubled the strength and, on a weight-to-weight basis, increased the shock resistance to that of concrete. Events changed, however, and the ship was never built. Still, pykrete was invented, and the study of ice as a material had begun.

Since then ice research has flourished. Nucleation and growth from both vapor and liquid states has been studied and placed within the context of classical thermodynamics and kinetics. The structure of ice and natural ice formations has been examined and then related to the thermal-mechanical history of the material. Electrical properties have been measured and explained in terms of the number density and mobility of protonic charge carriers. Optical and thermal behaviors have been explored. Mechanical behavior has been thoroughly studied, from the flow and fracture of single crystals to the creep of glaciers and the fracturing of Arctic sea-ice covers. Indeed, over the past ten years alone more than 10,000 papers on ice have appeared in the scientific and engineering literature.

Why the interest? Ice, it turns out, is a factor in activities as diverse as skiing and skating, rainmaking, polar marine transportation, and cold ocean oil exploration. It is an element in the degradation of cold concrete and other porous materials. It forms as "icing" on airplanes and electrical transmission lines. Ice is also a factor in global climate, evident perhaps from the facts that the Antarctic and Greenland ice sheets cover about 10% of the earth's land area and sea ice covers about 10% of the ocean surface either seasonally or perennially. In addition, the ice sheets and the air bubbles entrapped therein are the storehouses of the pa-leoclimate record. Ice is also a major constituent of the moons of Jupiter and of other extraterrestrial bodies.

This article reviews the structure and mechanical behavior of polycrystalline terrestrial ice. Fuller accounts are given in the literature cited and in the following references: ice physics by Hobbs2 and by Petrenko and Whitworth;3 ice-structure interactions by Sanderson;4 sea ice by Weeks;5 ice mechanics by Michel;6 and plastic flow and fracture in the Johannes Weertman Symposium.7 Curtin8 offers a historical perspective through the eyes of the U.S. Navy, and Levi9 describes the role of ice in the global heat budget. Durham et al.10 discuss the creep of planetary ice.


Figure 1: A schematic of the crystal structure of hexagonal ice Ih. Each H2O molecule has its four nearest neighbors arranged near the vertices of a regular tetrahedron (shaded) centered about the molecule of interest. The stacking sequence is . . . ABBAABBA . . . and may be seen from the numbers on the oxygen atoms: numbers 1-7 = A, 8-10 = B, 11-13 = B, 14-20 = A, 21-27 = A, 28-30 = B, 31-33 = B. Near the melting point the O-O distance is 0.276 nm, and the lattice parameters are a = 0.4523 nm and c = 0.7367 nm.
Ice possesses 12 different crystal structures, plus two amorphous states. At ordinary (low) pressures the stable phase is termed ice I. There are two closely related variants: hexagonal ice Ih, whose crystal symmetry is reflected in the shape of snowflakes, and cubic ice Ic. Ice Ih is obtained by freezing water; ice Ic is formed by depositing vapor at low temperatures ( -130°C). Amorphous ice can be obtained by depositing vapor at still lower temperatures and by compressing ice Ih at liquid nitrogen temperature. In addition to the elemental phases are clathrate hydrates. These are crystalline compounds composed of a large H2O cage in which Xe, Ar, or CH4, for instance, is entrapped. Clathrates are of economic interest because they offer an abundant source of natural gas.

Crystal Structure

Ice Ih, termed ordinary ice, is the common terrestrial form. Figure 1 shows its crystal structure. Each H2O molecule has four nearest neighbors arranged near the vertices of a regular tetrahedron centered about the molecule of interest. The oxygen atom of each molecule is strongly covalently bonded to two hydrogen atoms, while the molecules are weakly hydrogen bonded to each other. When projected onto the plane perpendicular to the c-axis, the molecular stacking sequence is . . . ABBAABBA . . . . It can alternatively be viewed as . . . A'B'A'B'A'B' . . ., where the prime denotes pairs of molecules joined in a dumbbell manner along the c-axis, reminiscent of the close packing of hexagonal metals. The lattice parameters near the melting point are a = 0.4523 nm and c = 0.7367 nm. The c/a ratio (1.628) is very close to the ideal ratio (1.633) and is independent of temperature. The ice Ih unit cell is relatively open (packing factor less than 0.34), and this accounts for ordinary ice being less dense than water.

The relationship to Ic lies in a common tetrahedral arrangement of H2O molecules. Ic, however, has the diamond cubic crystal structure, in which the stacking sequence relative to the {111} plane is . . . AABBCCAABBCC . . . ; its lattice parameter (at -130°C) is 0.635 nm.

The hydrogen atoms are arranged randomly11 according to the Bernal-Fowler rules.12 First, two protons must be located near each oxygen. Second, only one proton must lie on each O-O bond. The random arrangement persists at low temperatures, owing to the extraordinarily slow reorientation of the H2O molecule (greater than 100 years at liquid nitrogen temperature), and this leads to a large amount (3.41 J/mol.) of zero-point entropy. Ice does not violate the third law of thermodynamics.

Defect Structure

Point Defects

Figure 2: A composite x-ray topograph showing slip bands in two adjacent grains of an ice Ih polycrystal slowly strained under uniaxial compression at -6°C. The bands formed during in-situ straining through slip on basal planes. The images of each grain were obtained from separate Laue spots (the diffraction vectors are indicated), and, hence, an image of the grain boundary is present in both. The dislocations in the slip bands were nucleated at facets on the grain boundaries, such as those arrowed, and then traversed the grains to relieve the stresses associated with grain-boundary sliding. During straining, the dislocations in the slip band starting at S were observed to traverse the grain and pile up at the opposite boundary at P. (From Liu, Baker, and Dudley.16)
Hydrogen and oxygen appear to have the same diffusion coefficient in ice Ih, implying that the H2O molecule moves as a unit. The process is characterized by an activation energy of 0.65 ±0.03 eV. This is similar to the sum of the vacancy formation and migration energies, 0.53 eV and 0.09 eV, respectively,13 suggesting that vacancies are the dominant point defect. However, recent studies14 of the annealing of dislocations indicate that interstitials are the more important defect, at least at temperatures above -50°C.

Violations of the Bernal-Fowler rules create ionic and Bjerrum defects.15 If the proton moves along the O-O bond, then the first rule is violated: one proton near an oxygen atom creates an OH- ion; three create an H3O+ ion. If the proton moves around the oxygen atom, then the second rule is violatedno hydrogen atom on an O-O bond creates L-type Bjerrum defects (L stands for leer, which means empty in German); two protons create a D-type Bjerrum defect (D means doublet). Both kinds of defects contribute to electrical conductivity (the migration of ions allows protons to move from one end of a bond to the other), and the movement of Bjerrum defects allows protons to move from one side of an oxygen atom to another. Without the migration of both defects, long-range protonic conduction could not occur.

Despite the fact that water is a universal solvent, the solubility of substances in ice Ih is very low. The solubility of HCl, for instance, is 3 X 10-6 at -10°C. Exceptions are HF and NH3. These molecules are assumed to dissolve substitutionally, creating L-defects in HF and D-defects in NH3, as well as additional ionic defects. The impurities increase electrical conductivity.


Figure 2 shows a composite x-ray topograph illustrating slip bands in two adjacent grains in a polycrystal. In both crystals slip occurred by dislocation glide on basal or {0001} planes.16 The Burgers' vectors were parallel to the direction <> and of a magnitude defined by the shortest distance between oxygen atoms in the same basal plane (i.e., by the distance not between nearest neighbors, but between next-nearest neighbors, as in Figure 1). Basal slip,17 in principle, can take place on both a more widely spaced set of planes, termed shuffle planes (e.g., the plane between atoms 8 and 11 in Figure 1), and a less widely spaced set, termed glide planes (e.g., the plane between atoms 1 and 8 in Figure 1). It is not clear which set is dominant. The distinction is significant, however, because adjacent planes of oxygen atoms of the glide set relate to each other in a manner similar to that in face-centered cubic and hexagonal close-packed metals, leading to the possibility of dislocation dissociation into partials. To date, however, partial dislocations have not been observed.

A unique feature is worth noting. Because the protons in ice Ih are arranged randomly, the translation of part of the crystal relative to the rest by the Burgers' vector will not exactly reproduce the atomic arrangement.18 Instead, the translation introduces Bjerrum defects. The stress needed to create them (of formation energy 0.68 eV) is orders of magnitude greater than can be accounted for by the actual flow stress.18 This implies that some kind of protonic rearrangement must occur. However, the precise way in which dislocations overcome the obstacle presented by proton disorder is not yet known.

Planar Defects

Stacking faults have been observed in as-grown crystals using x-ray topography.19 They can be eliminated by annealing and so are considered to be unstable defects. Twins have not been observed, in material either well annealed or plastically deformed. Barring free surfaces, grain boundaries are the most prominent planar defect. They exhibit ledges, some as large as 1 mm (Figure 2), and close to the melting point they contain liquid water in submillimeter-sized veins that lie along lines of intersection.20 Within warm sea ice they also contain millimeter-sized brine pockets. Grain boundaries are sites of sliding and crack nucleation and are thus important microstructural features.


The microstructure of a natural ice formation depends on its thermal-mechanical history. Grain size is typically around 1 mm to 20 mm, and the grain shape varies from equiaxed to elongated. Glacial ice, for instance, forms through the sintering of snow under pressure and is often characterized by equiaxed, randomly oriented grains near the upper part. Deeper within, especially in sheets that flow down mountains under gravitational forces, creep deformation may be accompanied by dynamic recrystallization and the development of texture,21 in which case the microstructure becomes more complex.

Arctic sea ice5 forms directly upon the unidirectional solidification of salt water. Floating covers form and consist primarily of columnar-shaped grains elongated in the growth direction, reminiscent of metallic ingots. Once thickened to a few centimeters, the covers develop a strong growth texture in which the crystallographic c-axes are confined more or less to the horizontal plane, but are either randomly oriented within this plane or aligned22 with the ocean current. Sea ice is characterized also by an intragranular porous substructure that consists of submillimeter diameter air bubbles and brine pockets, totaling 4-5 vol.%, arrayed in a plate-like manner parallel to basal planes. Also, cold sea ice may contain precipitates of sea salts (mainly NaCl). Both deformation and growth textures lead to macroscopically anisotropic inelastic behavior.


Figure 3: Schematic stress-strain curves. I, II, and III denote low-, intermediate-, and high-strain rates. The arrows indicate either ductile (horizontal) or brittle (vertical) behavior.

The elastic behavior of ice is characterized by moderate anisotropy. At temperatures near the melting point, Young's modulus23 of single crystals varies by less than 30%, from 12 GPa along the least compliant direction (parallel to the c-axis) to 8.6 GPa along the most compliant direction (inclined to both the c- and a-axes). Along directions within the basal plane Young's modulus is 10 GPa. For randomly oriented polycrystals, typical values of Young's modulus and Poisson's ratio are 9.0 MPa and 0.33 at -5°C. Hobbs2 gives a more complete account and lists both the elastic stiffness and the elastic compliance tensors versus temperature.

Inelastic behavior is markedly anisotropic. The critical resolved shear stress for non-basal slip is 60 times or more greater than that for basal slip,24 and this presents a problem for polycrystals. Basal slip allows only two independent deformation modes. When coupled with the facts that twinning does not occur and four independent deformation modes are required25 (from self-consistent calculations) for extensive, crack-free flow, the plastic anisotropy leads to the build-up of internal stresses on the scale of the grain size. The stresses arise because grains favorably oriented for slip shed load to those less well oriented. The implication is that unless time is allowed for the internal stresses to relax, plastic flow will initiate cracks.26,27 If the cracks are tolerated, the ice will exhibit macroscopically ductile behavior. If not, then the material will exhibit macroscopically brittle behavior.

That ice can be brittle at temperatures right up to its melting point is perhaps surprising. The reason is related to the fact that its melting point diffusivity is around 10-15-10-14 m2/s, compared to higher values of 10-11-10-12 m2/s for elemental metals. Diffusion-assisted stress relaxation thus occurs relatively slowly.

Stress-Strain Curves

Figure 3 shows schematic stress-strain curves. At low rates of deformation, cracks do not form, and the material is ductile (curves I). At high rates, cracks do initiate, and the material is brittle (curves III) independent of stress state. At intermediate strain rates, cracks also develop, and the material is brittle under tension (curve TII) but ductile under compression (curve CII). The ductile-brittle transition occurs at lower strain rates under tension because the applied stress opens the cracks directly. Under compression, the required tensile stress is generated locally through crack sliding. Note that the compressive stress-strain curve at intermediate strain rates displays a peak owing, we believe, to crack-induced localized flow.

Figure 4: Tensile and compressive strengths of equiaxed and randomly oriented fresh-water ice of about 1 mm grain size vs. strain rate. At the top of the figure, I, II, and III correspond to the stress-strain curves of Figure 3.30,48,78-80
Figure 4 shows measurements of tensile and compressive strength obtained from fresh-water ice about 1 mm in grain size loaded uniaxially at temperatures around -10°C. Strength is defined as the highest stress recorded during the experiment. For this material deformed under these conditions, the low strain rate is less than 10-7 s-1 and the high strain rate is greater than10-3 s-1, as noted.

Ductile Behavior

Ductile behavior (regimes TI, CI,and CII in Figures 3 and 4) is characterized by strain-rate hardening and thermal softening. The strain-rate hardening exponent28 m0.3(  m) and the apparent activation energy27 Q ranges between 45 and 90 kJ/mol. (  memQ/RT), but is generally around 60 kJ/mol. At temperatures above -10°C, Q increases to 120-200 kJ/mol. in polycrystals. As expected for a dislocation-based process, confinement raises the compressive flow stress in the manner of a von Mises material,30-36 meaning that at failure the deviatoric component of the applied stress tensor is independent of the hydrostatic component. Brine inclusions 3 vol.%, typical of first-year sea ice, lower the flow stress by a factor of two34,35 and increase the quasisteady-state creep rate by about an order of magnitude. Grain size is a factor only within very finely grained material37 where, for instance, refinement from 89 µm to 8 µm increases the steady-state creep rate at -37°C by a factor of 30 under a compressive stress of 1.4 MPa.

Plastic flow and quasisteady-state creep of coarsely grained ice has been explained24,29 and then modeled quantitatively in terms of dislocation or power-law creep (i.e., by glide and climb of basal dislocations). Supporting this view is the fact that the activation energies for self-diffusion (0.65 eV = 62 kJ/mol.) and creep are essentially the same. Also, the creep rate is independent of grain size, and the dependence of the creep rate on stress (the inverse of the strain-rate sensitivity of the flow stress) is of the correct magnitude. The flow of very finely grained ice of micrometer dimensions can be rationalized in terms of grain-boundary sliding accommodated by dislocation creep. The effect of brine inclusions has been explained by a reduction in internal back stress.

Brittle Behavior

Brittle behavior sets in at higher strain rates. Under tension (regimes TII & TIII, Figures 3 and 4) ice breaks after lengthening 0.01-0.1% through transgranular cleavage.38 The tensile strength is rate independent and is only slightly thermally dependent,39 rising by less than 25% upon decreasing temperature from -5°C to -20°C. The tensile strength decreases with increasing grain size, in the Hall-Petch manner of metals and alloys,39 and decreases with brine inclusions.40 The tensile behavior of virgin ice has been explained and modeled in terms of the nucleation and growth of cracks. Their size is controlled by grain size, and their resistance to propagation is set by the fracture toughness of the material. Accordingly, cracks within finely grained material (around 1-2 mm) are shorter upon nucleation than the critical size, and so the strength is limited by crack propagation. Within more coarsely grained ice, the cracks propagate immediately upon nucleation. The significance of the different mechanisms is that the more finely grained ice exhibits a moderate amount of ductility (0.1%). The tensile strength of the cracked ice41 is controlled by crack propagation, unless the cracks are either short or blunted.

Brittle failure under compression (regime CIII, Figures 3 and 4) is marked by sudden material collapse after shortening less than about 0.5%. The failure mode is generally shear faulting on planes inclined by about 30° to the direction of maximum principal stress, although axial splitting can also occur under unconfined loading. The material now exhibits strain-rate softening, but is still thermally softened. The brittle compressive strength rises sharply under a small amount of confinement in a Coulombic manner.42-47 This implies that the deviatoric stress at failure increases with increasing hydrostatic stress and means that frictional crack sliding is an important element in the failure process. Again, the strength decreases with increasing grain size in a Hall-Petch manner.48 Brine inclusions, however, have no effect at all.47

Failure Envelopes and Failure Surfaces

In practice, ice is generally loaded multiaxially, particularly under compression. The bottom of glaciers, for instance, experiences multiaxial stresses when in contact with rock protrusions, and ice-structure interactions generate multiaxial stress states within the contact zone. The confinement raises the failure stress, as noted above, with the effect being greater within the regime of brittle behavior. Envelopes and surfaces describing both ductile and brittle failure under both biaxial32,34,35,42 and tri-axial30,31,36,43-47,49,50 loading have now been obtained and can be understood within the context of the mechanisms that are described herein. The challenge is to incorporate them in models of ice loads.


It is worth considering brittle compressive failure at some length in recognition of the practical importance of the phenomenon. From the perspective ice-structure interactions, for instance, the average effective strain rate within the contact zone is estimated from the relationship


where v is the velocity of the ice relative to the structure, and L is the width of the structure. Typical values are v = 0.1- 1 m/s and L = 10-100 m, giving strain rates that lead to brittle behavior.

a b c
Figure 5: (a) A compressive shear fault in columnar-grained ice. The columns are perpendicular to the page and the fault runs from the upper right to the lower left. The ice was loaded biaxially such that the major stress (11) was vertical and the minor stress (22) was horizontal. Note the wing cracks (arrowed) in the background field, the zigzag edges of the fault, and the milky zones stemming from one side of parent-inclined cracks (e.g., A and B). (b) A thin-section of the fault. Note the secondary or splay cracks stemming from parent cracks A and B. (c) A stress-strain curve for the above test specimen. (From Schulson, Iliescu, and Renshaw.51)

High-speed photography42,48 has revealed that brittle compressive failure is a multistep process. It begins with the nucleation of cracks at grain boundaries at applied stresses around one-quarter to one-third of the terminal failure stress; continues with a progressive and generally uniform increase in the crack density throughout the body as a whole as the load rises; and terminates, as noted, through the sudden formation near the peak of the stress-strain curve of one or more macroscopic shear faults. The important issue is the initiation of the fault, which limits the strength.

Consider the most recent observations.51 Figure 5a shows a typical terminal shear fault; Figure 5b shows a thin section of the same fault, and Figure 5c shows the corresponding stress-strain curve. The fault was created by loading coarsely grained (10 mm) columnar fresh-water ice biaxially across the columns under a moderate degree of confinement (minor stress/major stress = 22/11 = 0.1) at -10°C at 5 X 10-3 s-1. In Figure 5, the long axis of the grains is perpendicular to the page.

Figure 6: A schematic sketch of the comb-crack mechanism.
Several points are noteworthy. First, the fault is a zone of intense damage about 2-3 grain diameters wide, inclined by = 26° to the maximum principal stress (vertical). Second, wing-cracks (arrowed) are distributed across the overall field of damage. They stem from the tips of intergranular parent cracks, which are inclined by about 45° to the direction of maximum principal stress and tend to be aligned with this direction. Third, the fault has zigzag edges, implying that wing cracks are an element in its structure. Fourth, milky features (e.g., A and B in Figure 5a) stem from one side of some parent-inclined cracks. These features are actually sets of closely spaced secondary cracks of about one-half the grain diameter, evident from the thin section (Figure 5b). These are termed splay cracks in recognition of the term used to describe similar features in faulted rock.52-55

Experiments and analyses have shown that the parent cracks nucleate through grain-boundary sliding.56-59 The wing cracks initiate as a result of frictional sliding of the parent cracks.50 The splay cracks, it is thought, initiate from Hertzian contact stresses across the parent-crack faces and then propagate within a tensile field created most likely by nonuniform displacements across the sliding crack.

Schulson et al.51 propose that splay cracks are critical features in initiating the fault. Upon forming, they create sets of closely spaced microcolumns fixed on one end and free on the other. The free end contacts the sliding crack, which induces a moment that causes the columns to bend and break, rather like the breaking of teeth in a comb under a sliding thumb (Figure 6). It is the failure of these microcolumns under frictional shear loading, they suggest, that initiates the fault. Near-surface microcolumns probably break first, owing to less constraint there. It is imagined that growth then follows along a band of reduced shear strength that is composed of splay cracks formed prior to fault initiation plus fresh splay cracks created within a kind of process zone just ahead of the advancing fault front (Figure 7). The front moves rapidly across the section, creating "gouge" in its wake.

An estimate of the stress to initiate the fault may be obtained as follows. Assume that the fault is initiated when a microcolumn breaks. Assume also the scenario sketched in Figure 6, where M and P, respectively, are the induced moment and axial load per unit depth of the microcolumn; and n are the shear stress and normal stresses, respectively, acting on the microcolumn; and is the inclination of the parent crack. Then, by invoking the analysis of Thouless et al.60 for the propagation of an edge crack in a brittle plate, one can show that for = 45° the initiation stress, f, under uniaxial loading is approximated by the relationship51

Figure 7: A schematic of the proposed development of a compressive shear fault, from a point very close to terminal failure. The hexagonal network denotes the microstructure. Wing cracks sprout from the tips of some inclined, intergranular cracks, and splay cracks stem from one side of some of the parent cracks. The fault grows through a kind of process zone and creates gouge in its wake as splay-crack-induced microcolumns break. (From Schulson, Iliescu, and Renshaw.51)
where KIc and µ are the fracture toughness and friction coefficient, respectively; is the slenderness ratio of the microcolumns; and h is their length. Taking61 KIc = 0.1 MPa·m0.5 and µ = 0.5862 and using (from these observations) = 5.3 ±2 and h = D/2, where D is the grain size, the estimated initiation stress (lower limit) is 1.0 ±0.5 MPa. The actual uniaxial strength for coarsely grained ice is about two to three times this estimate. The difference may reside in the fact that the model ignores crack/crack interactions and column/column frictional interactions. Nevertheless, it gives the correct order of magnitude of the strength, and it captures the (grain size)-0.5 dependence of the compressive failure stress by relating the microcolumn length to the grain size. Also, through the effects of temperature and sliding speed on the friction coefficient,62 the model correctly captures the magnitudes of both thermal and strain-rate softening.

It is not a new idea that failure of deformation-induced microcolumns is the micromechanical event accounting for the initiation of a shear fault. Others have advanced a similar view.63-65 Previously, however, failure was imagined to occur by elastic buckling of columns fixed on both ends, created, for instance, by echelon arrays of wing cracks. Given the dimensions of the splay-induced microcolumns created in ice, the Euler buckling stress is estimated to be 630 MPa to 3,000 MPa, and this is two to three orders of magnitude greater than the strength of the material. Hence, it is our opinion that elastic buckling is not the event that triggers the fault.

Ductile-Brittle Transition under Compression

Consider again the ductile-brittle transition under compression (CII-CIII, Figure 4). It marks the point where the compressive strength reaches a maximum, and so is a factor in limiting the forces induced by a floating ice cover when pushing against the sides of an engineered structure.

The transition can be understood in terms of the competition between stress relaxation and stress build-up at crack tips. At intermediate rates of deformation crack-tip stresses relax through creep deformation, and so the mode-I stress intensity factor KI, at either the tips of wing cracks or splay cracks, never reaches the critical level. At high rates, on the other hand, stress build-up dominates, and KI quickly reaches the critical level Kic. The transition occurs when the competition between stress relaxation and stress build-up is in balance.

Schulson34,48 modeled the process by invoking Ashby-Hallam63 frictional sliding-crack mechanics and Riedel-Rice66 crack-tip creep. By assuming that cracks propagate when the crack-tip creep zone size falls below a small fraction f of the crack length, he obtained the transition strain rate in terms of the independently measurable parameters of fracture resistance (KIc,), creep constant B, (  B 1/m) friction (µ), and crack length (D); f must be calculated from the Riedel-Rice model. The transition strain rate may then be expressed by the relationship

Figure 8: (a) A satellite image of large cracks, termed "leads," in the Arctic sea-ice cover. Note the general alignment of the pattern and the wing-like cracks (two are arrowed). (From Schulson and Hibler.70) (b) Wing cracks formed within columnar ice loaded to failure in the laboratory. The major stress was vertical and the minor stress (0.05 of major stress) was horizontal. (From Iliescu and Schulson [unpublished].)

where R is the ratio of the confining stress to the most compressive stress. A comparison with experiment34,35,67 shows that the model correctly captures the effects of crack size (set by grain size68 in virgin material), confinement, and brine pores and that it predicts for the conditions of Figure 4 a transition strain rate of 10-3 s-1, which is close to that observed. The model also holds that through the effects of temperature on friction and creep, the transition strain rate is only slightly dependent upon temperature, at least over the range -40°C to -3°C, again in accord with experiment. Moreover, by including the crack size, the model accounts for the fact that sheets of first-year sea ice, which are laced with meter-sized (and larger) cracks and wind loaded under compression, exhibit macroscopic brittle behavior69 even though they are deformed at rates as low as 10-7 s-1.

Small-Scale vs. Large-Scale Failure

The brittle compressive failure mechanism, we believe, may be both scale- and material-independent. For instance, Figure 8 compares wing-like cracks that are tens of kilometers long seen by satellite70 in the Arctic sea-ice cover with similar millimeter-sized features seen in the laboratory. The parent sliding cracks on the larger scale may have been thermal cracks or refrozen leads. Also, shear-like faults characterized by kilometer-wide bands of damage have also been seen in the sea ice cover.71

Moreover, there is new evidence72 that fracturing and fragmentation of ice exhibit fractal organization in the lab and in the field. Within faulted rock, both splay cracks (as noted above) and zigs and zags denoting wing cracks73,74 have been seen on small and large scales. While the physics may not change with size, the terminal compressive failure stresses will probably be lower in larger features, possibly scaling as (crack size)-0.5. Consistent with this notion, at least for ice, is Sanderson's4 observation that large fractures fail at lower stresses than small ones. Also consistent are recent measurements of stresses within floating covers,71,75 which are usually within the kPa range as compared with the MPa range of lab measurements. The ductile flow of glaciers, on the other hand, reflects the power-law creep relationship of small test specimens, implying that dislocation-based processes are scale-independent.

Failure under tension is size-dependent, owing in part to the larger flaws within the larger features. Dempsey76,77 has discussed this aspect of the subject, from the perspective of applied mech-anics.


This work was supported by the Army Research Office and Office of Naval Research and by Exxon and Mobil.

1. M.F. Perutz, J. Glaciol., 1 (1948), p. 95.
2. P.V. Hobbs, Ice Physics (London: Oxford University Press, 1974).
3. V.F. Petrenko and R.W. Whitworth, Physics of Ice (London: Oxford University Press, in press).
4. T.J.O. Sanderson, Ice Mechanics Risks to Offshore Structures (London: Graham & Trotman, 1988).
5. W.F. Weeks, Physics of Ice-Covered Seas, ed. M. Leppäranta (Helsinki: Helsinki University Printing House, 1998), pp. 1-24 and 25-104.
6. B. Michel, Ice Mechanics (Quebec, Canada: Les Presses de l'Université Laval, 1978).
7. R.J. Arsenault et al., eds., Johannes Weertman Symposium (Warrendale, PA: TMS, 1996).
8. T.B. Curtin, Naval Research Reviews L, Arctic Studies (1998), p. 6.
9. B.G. Levi, Physics Today (November 1998), p. 17.
10. W.B. Durham, S. Kirby, and L.A. Stern, J. Geophys. Res., 102 (1997), p. 16293.
11. L. Pauling, J. American Chemical Society, 57 (1935), p. 2680.
12. J.D. Bernal and R.H. Fowler, J. Chem. Physics, 1 (1933), p. 515.
13. J.W. Glen, Cold Regions Science and Technology, Monograph II-C2a (Hanover, NH: U.S. Army Corps. of Engineers, 1974).
14. T. Hondoh et al., J. Chem. Physics, 87 (1983), p. 4044.
15. N. Bjerrum, Det Kongelige Danske Videnskabernes Selskab Matematisk-fysiske Meddeleiser, 27 (1951), p. 56.
16. F. Liu, I. Baker, and M. Dudley, Phil. Mag. A, 71 (1995), p. 15.
17. R.W. Whitworth, Phil. Mag. A, 41 (1980), p. 521.
18. J.W. Glen, Phys. Kondens. Mater., 7 (1968), p. 43.
19. M. Oguro and A. Higashi, Physics and Chemistry of Ice, ed. E. Whalley, S.J. Jones, and L.W. Gold (Ottawa, Canada: Royal Society of Canada, 1973), p. 33.
20. J.F. Nye, Physics and Chemistry of Ice, ed. N. Maeno and T. Hondoh (Sapporo, Japan: Hokkaido University Press, 1992), pp. 200-205.
21. S. de la Chapelle et al., J. Geophys. Res., 103 (1998), p. 5091.
22. W.F. Weeks and A.J. Gow, J. Geophys. Res., 84 (1978), p. 5105.
23. N.H. Fletcher, The Chemical Physics of Ice (New York: Cambridge University Press, 1970), p. 271.
24. P. Duval, M.F. Ashby, and I. Anderman, J. Phys. Chem., 87 (1983), p. 4066.
25. J.W. Hutchinson, Metall. Trans. A, 8 (1977), p. 1465.
26. L.W. Gold, Canadian J. of Physics, 44 (1966), p. 2757.
27. L.W. Gold, Phil. Mag., 26 (1972), p. 311.
28. N.K. Sinha, J. Materials Sci., 23 (1988), p. 4415.
29. J. Weertman, Annu. Rev. Earth Planet Sci., 11 (1983), p. 215.
30. S.J. Jones, J. Glaciol., 28 (1982), p. 171.
31. J.-P. Nadreau and B. Michel, Cold Regions Science and Technology, 13 (1986), p. 75.
32. R. Frederking, J. Glaciol., 18 (1977), p. 505.
33. J.A. Richter-Menge, J. Offshore Mechanics and Arctic Engineering, 113 (1991), p. 344.
34. E.M. Schulson and S.E. Buck, Acta Metall., 43 (1995), p. 3661.
35. E.M. Schulson and O.Y. Nickolayev, J. Geophys. Res., 100 (1995), p. 22383.
36. J.S. Melton and E.M. Schulson, J. Geophys. Res., 103 (1998), p. 21759.
37. D.L. Goldsby and D.L. Kohlstedt, Scripta Materialia, 37 (1997), p. 1399.
38. E.M. Schulson et al., J. Materials Sci. Ltrs. 8 (1989), p. 1193.
39. E.M. Schulson, P.N. Lim, and R.W. Lee, Phil. Mag. A, 49 (1984), p. 353.
40. J.A. Richter-Menge and K.F. Jones, J. Glaciol., 39 (1993), p. 609.
41. E.M. Schulson, S.G. Hoxie, and W.A. Nixon, Phil. Mag. A, 59 (1989), p. 303.
42. T.R. Smith and E.M. Schulson, Acta Metall., 41 (1993), p. 153.
43. R.E. Gagnon and P.H. Gammon, J. Glaciol., 41 (1995), p. 528.
44. M.A. Rist and S.A.F. Murrell, J. Glaciol., 40 (1994), p. 305.
45. J. Weiss and E.M. Schulson, Acta Metall., 43 (1995), p. 2303.
46. E.T. Gratz and E.M. Schulson, J. Geophys. Res., 102 (1997), p. 5091.
47. E.M. Schulson and E.T. Gratz, Acta Metall. (in press).
48. E.M. Schulson, Acta Metall., 38 (1990), p. 1963.
49. S.K. Singh and I.J. Jordaan, Cold Regions Science and Technology, 24 (1996), p. 153.
50. B. Zou, J. Xiao, and I.J. Jordann, Cold Regions Science and Technology, 24 (1996), p. 213.
51. E.M. Schulson, D. Iliescu, and C.E. Renshaw, J. Geophys. Res. (in press).
52. T.-F. Wong, Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 19 (1982), p. 49.
53. R.R. Gottschalk et al., J. Geophys. Res., 95 (1990), p. 21613.
54. S.J. Martel and D.D. Pollard, J. Geophys. Res., 94 (1989), p. 9417.
55. K.M. Cruikshank et al., J. Struct. Geol., 13 (1991), p. 865.
56. H.J. Frost, Proc. in Joint Applied Mechanics and Materials Summer Conference, ed. J.P. Dempsey and Y.D.S. Rajapakse (Los Angeles, CA: University of California, 1995), pp. 1-8.
57. R.C. Picu, V. Gupta, and H.J. Frost, J. Geophys. Res., 99 (1994), p. 11775.
58. R.C. Picu and V.J. Gupta, Acta Metall., 43 (1995), p. 3791.
59. J. Weiss, E.M. Schulson, and H.J. Frost, Phil. Mag. A, 73 (1996), p. 1385.
60. M.D. Thouless et al., Acta Metall., 35 (1987), p. 1333.
61. J.P. Dempsey, Ice Structure Interactions, ed. S.J. Jones (New York: Springer-Verlag, 1991), p. 109.
62. D.E. Jones, F.E. Kennedy, and E.M. Schulson, Ann. Glaciol., 15 (1991), p. 242.
63. M.F. Ashby and S.D. Hallam, Acta Metall., 34 (1986), p. 497.
64. C.G. Sammis and M.F. Ashby, Acta Metall., 34 (1986), p. 511.
65. Z.P. Bazant and Y. Xiang, J. Eng. Mech., 2 (1997), p. 162.
66. H. Riedel and J.R. Rice, ASTM-STP-7700, (1980), p. 112.
67. R.A. Batto and E.M. Schulson, Acta Metall., 41 (1993), p. 2219.
68. D.M. Cole, Proc. Fourth Int. Symp. on Offshore Mech. Arctic Engng. (New York: ASME, 1985), p. 220.
69. J.R. Marko and R.E. Thomson, J. Geophys. Res., 82 (1977), p. 979.
70. E.M. Schulson and W.D. Hibler, III, J. Glaciol., 37 (1991), p. 319.
71. J.A. Richter-Menge et al., Proc. of the ASYS Conference on the Dynamics of the Arctic Climate System, ed. P. Lemke (Gotteborg, Sweden: World Meterological Org., 1996), pp. 327-331.
72. J. Weiss and M. Gay, J. Geophys. Res., 103 (1998), p. 24005.
73. D.E. Moore and D.A. Lockner, J. Struct. Geol., 17 (1995), p. 95.
74. R. Bilham and P. Williams, Geophys. Res. Lett., 12 (1985), p. 557.
75. W.B. Tucker, III and D.K. Perovich, Cold Regions Science and Technology, 20 (1992), p. 119.
76. J.P. Dempsey, contribution to Research Trends in Solid Mechanics, a report from U.S. National Committee on Theoretical and Applied Mechanics (in press).
77. J.P. Dempsey, Johannes Weertman Symposium, ed. R.J. Arsenault et al. (Warrendale, PA: TMS, 1996), p. 351.
78. R.W. Lee (M.S. thesis, Thayer School of Engineering, Dartmouth College, 1985).
79. E.M. Schulson and N.P. Cannon, Proc. IAHR Ice Symp. (Hamburg, Germany: Hamburgische, Schiffbau-Versuchanstahlt GmbH, 1984), p. 24.
80. I. Hawkes and M. Mellor, J. Glaciol., 11 (1972), p. 103.


E.M. Schulson is currently a professor of engineering at Thayer School of Engineering at Dartmouth College.

For more information, contact E.M. Schulson, Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire 03755; (603) 646-2888; fax (603) 646-3856; e-mail erland.schulson@dartmouth.edu.

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