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J. Phys. Chem. B 1997, 101, 6254-6258
The Brittle Failure of Ice under Compression Erland M. Schulson Thayer School of Engineering, Dartmouth College, HanoVer, New Hampshire, 03755 ReceiVed: October 11, 1996; In Final Form: May 1, 1997
This paper reviews progress during the past decade in understanding the brittle compressive failure of ice. Evidence is presented for the frictional crack sliding-wing mechanism and for the role of localized fragmentation. The ductile-to-brittle transition is explained in terms of the suppression of crack growth and is modeled in terms of Ashby-Hallam wing crack mechanics and Riedel-Rice crack-tip creep. The brittle compressive strength is related to ice-ice friction. Modeling and experiment suggest that the transition strain rate of cracked ice scales as (crack size)-1.5 and that its brittle strength scales as (crack size)-0.5, at least for small specimens. Possible size effects are noted.
Introduction Terrestrial ice (Ih) is loaded under compression in a number of situations, including the interaction of ice floes, the pushing of floating sheets against stationary objects, and the wind-driven movement of arctic ice covers against land. In these cases and others like them, the compressive loads are limited not only by the ice thickness but also by its compressive strength. Brittle failure, it turns out, limits the ice forces. This paper considers the nature of the brittle compressive failure of ice at temperatures of practical interest (>0.8 Tm). Brittle failure is a process. It begins with the nucleation of cracks, continues with their growth and interaction, and terminates in the development of macroscopic splits, faults, and spalls. Most of our understanding of these events has been gained from experiments in the laboratory on submeter-sized specimens. Yet compressive failure is generally important on larger scales, ranging from meters to tens of meters in the case of ice-structure interactions and to kilometers and above for geophysical events (e.g., ridging and leading). The question is whether the mechanisms that govern on the smaller scale also govern on the larger. The issue is not unique to ice and is considered again later. Ductile Vs Brittle Behavior Ice exhibits two kinds of behavior, Figure 1. It is ductile when slowly compressed: it shortens by 10% or more without collapsing, displays marked strain-rate hardening (i.e., where m ≈ 0.3), and is little affected by grain size. When rapidly loaded it is brittle: it collapses (when unconfined) after shortening around 0.1%, is strengthened upon grain refinement, and, albeit somewhat contentious, appears to exhibit moderate strain-rate softening1-5 at least up to the onset of the pseudodynamic range. Ductile behavior is controlled by the glide and climb of basal dislocations and by dynamic recrystallization;6 brittle behavior, by the growth and interaction of cracks. That ice is brittle at temperatures so close to its melting point is perhaps a little surprising but is a reflection, in part, of low molecular diffusivity. A decade ago it was suggested7 that the brittle behavior of ice may be linked to internal tensile fractures and specifically to the frictional sliding-wing cracking mechanism,8-10 Figure 2. Accordingly, owing to the sliding across opposing faces of parent cracks inclined to the direction of highest loading, outX
Abstract published in AdVance ACS Abstracts, July, 1, 1997.
S1089-5647(96)03219-1 CCC: $14.00
Figure 1. Schematic sketch illustrating the ductile-to-brittle transition. The curves show hypothetical compressive stress-strain curves at progressively increasing strain rates reprinted from ref 1. Copyright 1990 Elsevier Science Ltd.)
of-plane tensile extensions or wings sprout from their tips, aligned more or less along the loading direction. Upon additional sliding, the mouths of the wings open, thereby concentrating tensile stress. The wings grow when their mode-I stress intensity factor reaches a critical level (i.e., KI ) KIC). Upon interacting with other wings (and with secondary tensile cracks, more below), macroscopic splits and faults form, marking the point of terminal failure. The mechanism was questioned because, although able to account for certain observations, wing cracks had not been seen in ice. That now has changed. Wing Cracks Cannon et al.11 obtained the first direct evidence. They compressed freshwater columnar (S2) ice by rapidly loading the material uniaxially across its columns at -20 °C. (S2 ice is an orthotropic material whose crystallographic c-axes are randomly oriented12 in the plane perpendicular to the long axis of the columnar grains.) Through pulse loadings, they observed the nucleation of parent inclined cracks at grain boundaries, the initiation of wings from the tips of the inclined cracks, and the lengthening of the wings along the loading direction, Figure 2c-e. Subsequently, Schulson et al.13 filmed wing crack growth in the same material loaded monotonically (at 2 × 10-2 s-1, -10 °C) and found that it agreed well with the dictates of the Ashby-Hallam10 model. Within the short-crack regime at least, © 1997 American Chemical Society
The Brittle Failure of Ice under Compression
J. Phys. Chem. B, Vol. 101, No. 32, 1997 6255
Figure 2. Schematic sketches indicating that sliding cracks nucleate at grain boundaries (a) and upon sliding initiate wings (b) whose tips contain a (circled) creep zone. Photographs showing wing cracks (c-e) in S2 freshwater ice (from ref 11) loaded uniaxially (vertical in figure) under (c) 5.2 MPa, (d) 6.6 MPa, and (e) 6.9 MPa. Winglike leads (f) in first-year ice in the Beaufort Sea (reprinted from ref 28. Copyright 1990 Elsevier Science Ltd.).
the crack length scaled roughly as (applied stress)2. More recently, wing cracks have also been seen5,14 to initiate from grain boundary parent cracks in S2 saline ice of 4-5 ppt salinity loaded under similar conditions. Also, short wings have been found in saline ice at the tips of pores15 but appear not to lengthen much, in agreement with theory.16,17 Wing cracks also develop in granular ice; i.e., material composed of randomly oriented, equiaxed grains. Examples have been found in ice of laboratory1,18 and of glacial origin.19 In this case the parent inclined segments are more difficult to find: they are entrapped more or less within single grains and do not penetrate the specimen in the way they do in columnar ice, making less probable their capture in thin sections. Consequently, mode-II crack sliding is accompanied by modeIII loading,11 and this impedes wing growth. In other words, when the deformation changes from predominantly 2D (as in S2 columnar ice) to 3D (as in granular ice), wing cracks appear not to lengthen by more than one or two grain diameters. Restricted wing growth has also been noted and discussed by Germanovich et al.17 following experiments on the 3D deformation of silica glass and cold PMMA. Having established that the mechanism operates, the question now is the role it plays in compressive failure.
Ductile-to Brittle Transition Consider first the ductile-to-brittle transition. This is important because it defines the conditions under which the compressive strength reaches a maximum.1 The transition occurs when cracks begin to propagate. This view is based upon experiments20 at -10 °C in which alongcolumn “Teflon cracks” were inserted into S2 freshwater ice. The specimens were pulse loaded across the columns to induce wing cracks and then loaded monotonically to failure. At a lower strain rate (3 × 10-5 s-1), the wings did not propagate; instead, new cracks formed as the load increased, and the ice exhibited ductile behavior. At a higher rate (2 × 10-2 s-1), the wing cracks did propagate and the ice was brittle. That wing cracks form but do not propagate under conditions of ductile behavior is also supported from observations by Ple and Weiss21 who deformed S2 freshwater ice by creep (compressed uniaxially across the columns at -10 °C at 1.5 MPa). Nonpropagating wing cracks can also be seen in Sinha’s22 micrographs of freshwater ice that had been deformed under creep conditions. The transition can be understood in terms of a competition between stress buildup and stress relaxation near the wing crack tips. Stress buildup dominates at higher loading rates: the mode-I stress intensity factor eventually reaches the critical level
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Figure 3. (a) Comparison of calculated (dotted line) and observed (solid line) transition strain rates for S2 freshwater ice (from ref 24) and for S2 saline ice (from ref 15) versus across-column confinement. The calculations were limited to R < 0.3 where in-loading-plane frictional sliding operates. (b) Showing the effect of grain size (which sets the sliding crack size) on the transition strain rate for freshwater S2 ice loaded uniaxially across the columns (reprinted from ref 20. Copyright 1993 Elsevier Science Ltd.). The bars show the data. The lines show three different functionalities to compare with the one (d-1.5) predicted.
at which point the wings grow and interact and brittle failure ensues. Relaxation dominates at lower loading rates, leading to crack blunting and to ductile behavior. The transition thus occurs at an intermediate strain rate, typically around 10-410-3 s-1 for small test specimens. Schulson1 modeled the transition by assuming that crack growth begins when the creep zone radius, rc (Figure 2b), is less than a certain fraction, f, of the parent crack length and by incorporating both Ashby-Hallam10 frictional sliding/wing crack mechanics and Riedel-Rice23 crack-tip creep. Accordingly, when modified to include confinement,18 his analysis showed that the transition strain rate is given by
˘ t )
BK3Ic fd1.5{(1 + µ2)0.5 - µ - R[µ + (1 + µ2)0.5]}
where B is the parameter in the power law creep expression ˘ ) Bσn, d is the length of the parent sliding crack (equal to the grain size in virgin ice), and µ is the friction coefficient; the confinement, R, is the stress ratio and is defined by the conditions of the experiment. The model thus holds that the transition strain rate is about an order of magnitude higher for saline ice than for freshwater material, owing to its (saline) lower creep resistance, and that it increases with increasing confinement and with decreasing crack size. Comparison with experiment is found to be good,14,15,20,24-26 Figure 3, as seen upon inserting into the above equation the independently measured parametric values (given in the original papers). (The comparison is only made at low confinement where shear faulting
Schulson occurs; more below.) The model also predicts that over the temperature range of practical interest (-40 to 0 °C) the transition strain rate increases only by about a factor of 3 or 4, owing to the opposing effects of temperature on creep and on toughness and friction, again in agreement with experiment.27 The model incorporates several assumptions: that crack-tip creep is dominated by secondary creep; that the initial wing cracks are proportional in length to the parent sliding crack; that the fracture toughness is independent of crack size. All can be questioned. Nevertheless, the analysis appears to capture the essence of the transition. Does it work in the field? Consider the slow deformation (about 1%/day or 10-7 s-1) of a sheet of columnar-grained firstyear sea ice. Floating ice sheets are typically laced with vertically oriented cracks, some longer than several kilometers. The cracks, termed leads because of their size, are often aligned, not unlike the kind of alignment of splits and faults one associates with brittle compressive failure in the laboratory. Even winglike leads have been reported28 (Figure 2f). It would appear, therefore, that in a global sense the sheet behaves in a macroscopically brittle manner. If we assume that the deformation results from compressive loading, then we must ask: why does the sheet exhibit brittle behavior when the deformation rate is so low? An answer may be found in the model of the D-B transition. Barring concerns about the effect of size on the fracture toughness29 or about the appropriate friction coefficient, the model predicts a transition strain rate of about 10-7 s-1 for ice containing through-thickness cracks about 5 m long should they slide and develop wings. A smaller transition rate is expected for longer sliding cracks. It is not difficult to imagine features this size developing, for instance, from the linking up of thermal cracks and then becoming active under wind-induced stresses. The model thus allows a rationalization. Whether it truly accounts for the geophysical behavior through the (crack size)-3/2 scaling requires further examination. Brittle Compressive Failure Modes Consider next the brittle failure process. As determined from high-speed photography,13,25 cracks first nucleate within virgin material at stresses around one-third to one-half of the failure stress. They continue to nucleate as the stress increases. At about the same time, wings sprout from some of the cracks, presumably from those most favorably oriented for sliding. They then lengthen. Growth occurs in a stable, albeit jerky manner, as the stress increases. Eventually the material fails macroscopically, either by splitting along the loading direction (unconfined ice), by shear faulting along a plane inclined by about 30° to the direction of highest stress (moderately confined ice), or by spalling (more highly confined ice). The strengthlimiting step presumably corresponds to the completion of the macroscopic split/fault/spall. Splitting is the simplest mode. It appears to develop from the axial growth and interaction of wing cracks. Relatively extensive growth occurs in columnar (S2) ice (Figure 2) when loaded uniaxially across the columns, and so only a few wing cracks are needed to create a split in laboratory-sized (say 100200 mm) specimens. Less growth occurs in granular ice, as already noted. As a result, this material is more highly damaged at failure because a split requires the linking up of many short wings. A question not yet answered is whether axial splits also form in very large plates under uniform loading. Faulting is a more complicated mode and is less well understood. It operates when wing crack growth is constrained by an external confining stress. In this situation secondary tensile cracks initiate from the surfaces of the sliding-parent
The Brittle Failure of Ice under Compression cracks,18,25 thereby creating a localized zone of short, thin plates. It is suggested that a macroscopic shearlike fault is triggered when the thin plates become so numerous that while rubbing together they break like the breaking of teeth in a comb. The fault, in other words, is the manifestation of a deformationinduced structural instability. Such a mechanism would account for the intense fragmentation localized to the fault zones and for the rapid fault propagation.5,14,25 Spalling, the third mode, develops through the across-column propagation of cracks. It operates when the ratio of the minor to major stress exceeds a value (R0 defined below) great enough to stop frictional sliding in the loading plane. Spalling can also be explained in terms of wing crack growth and interaction.5 The faulting and spalling modes operate in both columnar and granular ice proportionally loaded. Another faulting mode is activated in granular ice when rapidly loaded triaxially in a pressure cell, e.g., at -40 °C.30,31 In this case, the fault is inclined by about 45° to the direction of highest stress. Rist et al. suggest31 that this kind of shear faulting is triggered not by a critical density of localized tensile failures but by unstable in-plane crack propagation, frictional in character at low hydrostatic stresses (
