On internal friction and cohesion in unconsolidated ice rubble

Cold Regions Science and Technology, 16 ( 1989 ) 237-247 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands

237

ON I N T E R N A L F R I C T I O N A N D C O H E S I O N IN U N C O N S O L I D A T E D ICE RUBBLE Robert Ettema 1 and Gilberto E. Urroz 2 1Institute of Hydraulic Research, College of Engineering, The University of Iowa, Iowa City, IA 52242 (U.S.A.) 2Utah Water Research Laboratory, Utah State University Logan, Utah 84322

(Received December 5, 1988; revised and accepted February 3, 1989)

ABSTRACT Unconsolidated ice rubble is a mass of ice fragments, ofien floating in water, that has not solidified by freezing of pore water or by freeze bonding between individual ice-rubble fragments. It is argued herein that unconsolidated ice rubble undergoing continuous shear deformation is essentially cohesionless; i.e., in terms of the Mohr-Coulomb failure criterion, it should not exhibit a cohesive intercept at zero confining pressure and at large shear deformation. Contacting pieces of submerged ice rubble, however, have a propensity to freeze bond with one another, even at very low confining pressures and in nonfrigid air, such that unconsolidated ice rubble may tend to self-solidify. It is this propensity that significantly contributes to the comparatively large values of internal-friction angle that have been reported for ice rubble, and which may cause relationships between shear strength and confining pressure to be nonunique.

INTRODUCTION Several questions pervade the literature on sheardeformation behavior of unconsolidated ice rubble, which is ice rubble that has not solidified by freezing of pore water between ice-rubble pieces or freeze bonding of contacting ice-rubble pieces. One question is" During continuous shear, does unconsolidated ice rubble shear cohesively, as some, but not all, studies suggest? A related question is: Why are large internal-friction angles, often in excess of 50 o,

0165-232X/89/$03.50

reported for ice rubble when sands and gravels typically have values ranging from 20 to 40 degrees (e.g., as reported in Lambe and Whitman, 1969)? Most studies attempting to elucidate the sheardeformation behavior of ice rubble (e.g., those listed in Table 1 ) have generally treated it as if it conformed to the linear Mohr-Coulomb failure criterion: r = a tan O+c

( 1)

in which r = shear strength; a = compressive stress; = angle of internal friction; and c = cohesive intercept, as indicated in Fig. 1. It is argued herein that unconsolidated ice rubble, undergoing continuous shear, and in the absence of pore-water freezing, behaves as a cohesionless material whose effective internal-friction angle is influenced by confining pressure. Further, during continuous shear of unconsolidated ice rubble, c is not a constant cohesive intercept in the conventional sense, but is rather an expression of shear strength associated with freeze bonding of ice fragments, and is a function of a as well as of other factors including contact period, internal temperature of ice fragments and water salinity. Moreover, it is argued that, when a = 0 , c = 0 during continuous shear. In the literature on ice rubble, however, c is presented as being constant, or zero, for a particular ice rubble. As summarized in Table 1, reported values of c vary from zero to about 4 kPa, and values of O range from as low as 11 ° to as much as 65 °. Wide variations in ~ are, naturally enough, to be expected as ice rubble occurs in diverse forms, roughnesses, sizes and strengths. Additionally, it may exist at ranges

© 1989 Elsevier Science Publishers B.V.

238 TABLE 1 Summary of reported ~, c values O (°)

c (kPa)

v (mm sec -1 )

Keinonen andNyman, 1978 48 0.01 Prodanovic, 1979 47 0.26 0.9--,9 53 0.58 0.1---9 Weiss et al., 1981 13 1.7 4.0 11 1.2 25.0

26 2.3 3.0 25 1.4 24.0 34 4.1 5.0 34 3.4 25.0 Hellmann, 1984 57 0.90 1.6 41 0.90 10.9 65 0.45 1.6 54 0.60 10.9 52 0.40 1.6 52 0.10 10.9 47 0 92.0 Fransson and Sandkvist, 1985 21 0.5 10 34 0.25 10 12 0.5 10 14 0.20 10 Urroz and Ettema, 1987 37 0 2 52 31 51

0 0 0

2 2 2

h (mm)

amax (kPa)

H (mm)

20-23

1.4

300

19 38

2.7 2.7

304 304

Comments

v not reported in reference "[ Reported as Peak Shear Strength

Y

80 80

31 31

1000 1000

]

150 150 200 200

31 31 31 31

1000 1000 1000 1000

10-20" 10-20 10-20 10-20 10-20 10-20 10-20

4 4 4 4 4 4 4

700 700 700 700 700 700 700

Primary Primary Secondary Secondary Tertiary Tertiary Tertiary

39 39 46 46

1.5-2 2 3 4

500 500 500 500

Peak Continuous Peak Continuous

18

0.4

27 70 78

0.4 0.4 0.4

Reported as Peak Shear Strength

76-229

]

76-229 t Reported as Continuous Shear 133-200 152-229

* = ice chips. v= speed of shearing. h = average thickness or characteristic dimension of ice rubble. H = thickness of ice-rubble layer. trmax= maximum reported value of confining pressure used.

ILl I--

C' O' COMPRESSIVE

STRESS,

Fig. 1. Mohr-Coulomb failure criterion.

G

o f ice, air a n d water temperatures, as well as in or out of water which, i n turn, m a y be fresh, brackish or saline. It is interesting that the substantially n o n zero values o f c are reported b y studies generally inv o l v i n g tests at larger values of a . . . . a n d that the range in reported c values increases with a . . . . These o b s e r v a t i o n s are e v i d e n t i n Fig. 2, which also shows that values of ~ scatter widely with values o f a . . . . b u t are distinctly smaller for tests c o n d u c t e d at the highest value o f area, (Weiss et al., 1981 ); though it is curious that F r a n s s o n a n d S a n d k v i s t ( 1 9 8 5 ) re-

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ITIOX

Fig. 2. Reported values of~ and c vs. maximum confining pressure, a .... used in testing.

port values of O as low as 11-13 ° for comparatively moderate values of a . . . . It would seem that reported values ofc are somehow influenced by a. And so they are, but they are also apparent (nonactual) values from misinterpreted data, as indicated in the subsequent discussion of the influences of a on and apparent c. The writers are by no means the first to ponder the perplexing shear-strength behavior of ice rubble. Mellor (1980), for example, gives a thoroughgoing review on this subject. The present paper is an attempt to interpret values of ~ and c reported for unconsolidated ice rubble. Before further appraising the values of ~ and c, reported for ice rubble, it is useful to review briefly their meanings.

ON THE MEANING OF INTERNAL FRICTION Internal-friction angle, ~, is a measure of both contact friction and interlocking among juxtaposed

particles undergoing shear displacement. For a volume of granular material, these two facets of shear resistance are affected by confinement (whether the volume is fully constrained on all sides, or is constrained only on some surfaces), magnitude of confining or compressive pressure acting, strengths of granular pieces, porosity of granular volume, size distribution of particles, their shape, and presence of contaminants (which might affect contact friction). The simplest measures of ¢ are those associated with contact friction only, ~,. Two measures of ¢~ are required in mechanics; its value at static contact, ~,s, and at moving or kinetic contact ~ , u k o Reams have been written on tribology, the science of contact friction (e.g., Bowden and Tabor, 1974), and its dependency on the micromorphology and material properties of contacting surfaces. Values of contact friction, Ous, or Ouk, are usually measured by means of a direct shear device comprising a fixed plane of the material and a moveable plane. In the case of quartz particles sliding on a planar quartz surface, Bowles (1984) shows that values of 0~s

240 range from about 22 ° to 30 °, being larger for smaller particles. Values of 0u are much smaller for ice fragments sliding on ice. Bowden and Hughes (1939), and also Hobbs (1974), indicate values of Ouk for smooth ice on ice ranging from about 1.1 ° to 4.6 ° as ice temperature varies from 0 ° to - 8 0 ° C , respectively. Prodanovic ( 1979 ) reports ~us values as high as 13 ° to 25 ° for pieces of ice sheet sliding on ice sheet. He also reports values of Ouk being about 30% less than 0us. Incidentally, there seem to be comparatively few reported values of ~u for ice on ice. More frequently reported are values of Ou for diverse metals and wood surfaces sliding against ice (e.g., Oksanen, 1983; Tatinclaux et al., 1986). Also, there presently exists some debate concerning the sensitivities of q~uk and 0¢,s to normal pressure. Hoffmann (1985 ), for example, presents data that infer 0~,k decreases to a lower asymptotic value with increasing contact pressure. Recent reappraisal (Tatinclaux, pers. c o m m u n , 1988) of data such as those of Hoffmann, indicate that no such effect actually exists between ice and contact surfaces. For some materials, principally ductile ones, sufficiently large compressive stresses cause plastic deformations of contacting asperities and may lead to a process variously called junction growth, cold welding/fusion or sintering. An increase in the effective internal-friction angle results, or at least is interpreted to result, if increasing compressive stress creates stronger or more fusion bonds. Values of ~ associated with both contact friction and interlocking of particles are more complicated than those solely attributable to contact sliding friction. Two values are customarily assigned for 0 in accordance with the interlocking behavior of particles and density of particle packing. One value is taken at the peak of the stress-strain curve (see Fig. 3), and is termed the peak-friction angle, 0p. Its value is affected by the initial porosity and confining pressure prior to shearing. The other value is taken after considerable straining of the accumulation, when further straining occurs without significant change in either porosity or confining pressure, and is termed continuous-shear, or constantvolume, friction angle, 0c. Values of Op are not attained for "loosely" packed particles; looseness, or conversely denseness, of particle packing is influ-

/~ 112

/

/

EAK STRENGTH

PARTICLES

/1~ CONTINUOUS SHEAR //~LOOSE-PACKED PARTICLES STRENGTH

TIME

Fig. 3. Representative records of shear stress vs. time for densely-packedparticles and loose-packed particles. enced by porosity and particle shape. Minimum values of ~c can be evaluated as the angle of static repose of heaped particles. When estimating the strength of ice rubble, selection of an appropriate internal-friction angle depends on the nature of the ice-rubble process under consideration. Use of 0p is appropriate when assessing the internal strength of relatively compact ice rubble at small strains, such as may pertain for ice rubble comprising a stationary ice jam, or for initial straining of ice rubble confined against the side of some structure. Continuous-shear internal-friction angle, 0c, is appropriately used for assessing the internal strength of ice rubble at large strains, such as in the case of forming, deforming or collapsing icerubble covers; e.g., moving ice jams or ice gorges. A factor to be borne in mind for ice rubble is that, at present, it is difficult to assess beforehand if values of Op are attainable; "loose" ice rubble may undergo large strains and only muster a shear strength commensurate with Oc (see Fig. 3 ). In evaluating ice-rubble friction against a rough surface (with roughness topography exceeding icefragment size ), 0c may be appropriately used, as ice fragments would shear against ice fragments. For a comparatively smooth surface, either Ous or Ouk (for ice against surface material) is appropriately used, as ice rubble rubs directly against the surface.

ON THE NATURE OF COHESION True cohesion is shear resistance that occurs when as in Eq. 1, a = 0 . It gives rise to the shear intercept,

241

c, on Mohr-Coulomb failure envelopes, and is attributable to adhesive bonding, or attractive forces acting among surface atoms, at points of contact between particles. Values of c have generally to be estimated via extrapolation of linear Mohr-Coulomb failure envelopes to the condition a = 0 . They are not measured directly for particulate materials because, due to hydrostatic pressure, the condition a = 0 is unattainable. At least some compressive force acts, no matter how small, on a shear plane through ice rubble as test samples cannot be infinitesimally small. Several studies (Hellmann, 1984; Weiss et al., 1981; Prodanovic, 1979), however, do present data points for a = 0.

SHEAR-BOX TESTING OF ICE RUBBLE Tests on the shear-strength behavior of ice rubble have been mainly conducted using various forms of direct-shear box. Urroz and Ettema (1987), however, used a simple-shear box; Cheng and Tatinclaux (1977) used a cylindrical shear vane (which did not facilitate direct estimation of ~ or c). Some boxes, or testing devices, have fully confined their test volume of ice rubble, whereas others have confined only the sides of the test volume, leaving the top and bottom sides unconfined. Among the former group are the shear boxes used by Keinonen and Nyman ( 1978 ), Prodanovic ( 1979), Weiss et al. (1981), Hellmann (1984), and Fransson and Sandkvist (1985). The latter group includes the shear boxes used by Merino (1974), Cheng and Tatinclaux (1977) and Urroz and Ettema (1987). Most fully-constrained shear boxes were intended to be regulated such that compressive stress was constant during shearing. Weiss et al. apparently were able to achieve this by means of a servo-feedback system. Hellmann's shear box was unable to maintain constant normal pressure because of mechanical difficulties. However, he was able to measure the variation in compressive stress during shearing. Vertically unconfined shear boxes do not impede dilation of ice pieces, and therefore are more limited in the range of a values that they can exert on a given volume of ice rubble. Most studies do not indicate clearly whether reported values of internal resistance are either ~p or

0c. Hellmann (1984), though, meticulously describes the temporal variation of shear strength during shearing of fully confined ice rubble. He identifies three "modes" of shearing, and assigns unique values of shear strength to each. The first mode is associated with denser packing of ice fragments as shear stress (or strain) initially increases. During the second mode, the compressive and shear stresses increase, peaking as the test volume of ice fragments dilates. The third mode represents continuous shearing at large shear strain. The strength value obtained during Hellmann's second mode of shear is essentially equivalent to peak shear strength, yielding 0p, and his third mode corresponds to continuous shearing and yields 0c. At times, as pointed out above for comparatively loose-packed rubble, no peak-shear strength will occur, as was found by Urroz and Ettema using their simple-shear box. Weiss et al. and Prodanovic state only that they determined shear strength based on a peak shear force, measured during each experiment. It is unclear from their studies whether the peak shear strength was actually determined, or whether some local peak during continuous shear was selected. Plots of ~ vs. strain presented by Prodanovic ( 1979 ) suggest that the latter was the case, or that peak and continuous shear strengths were mixed.

THE PECULIAR SHEAR-DEFORMATION BEHAVIOR OF ICE RUBBLE Shear-strength behavior of many materials may not be adequately characterized by means of a unique and linear Mohr-Coulomb failure criterion (see Eq. 1 ). Unconsolidated ice rubble may well be one such material. Its effective internal-friction angle and, at times, the apparent cohesiveness it gains through self-solidifying, may vary with increasing confining stress such that a nonunique relationship exists between a and 3. At comparatively low values of confining stresses, even for barely contacting ice fragments, freeze bonding leads to an effective increase in angle of internal friction. At sufficiently large values of a, significant localized crushing of contact points may occur, thereby diminishing the extents of both interlocking and freeze-bonding, and effectively re-

242 ducing ~. Thus, the peculiar shear-deformation behavior of ice rubble revolves around its propensities to freeze bond and crush. Freeze-bond development in ice rubble is a process incompletely understood. Notably deficient are insights of the time scales associated with freezebond development and of the effect on bond strength of confining pressure, a. As soon as ice rubble comes to rest after its formation, or deformation, adjacent ice fragments may start freezing together and begin to form a rigid matrix known as consolidated ice rubble. In a sense, a surface diffusion of submerged-ice boundaries commences and continues, so that, with time, the bonds strengthen and the shear and compressive strengths of ice rubble increase. For longer periods, heat conduction through the ice leads to further freezing of the pore water until, at the limiting condition and in frigid air, ice rubble may become a monolithic (though not homogeneous) mass. Consolidation of ice rubble may also cause its shear and compressive strengths to increase, because the surface-contact area between fragments may grow as rubble poros-

co (23 u3 rr F03

/ LAW~~°~

DATA: Pi (o~, ri) , i = l , 2,. APPARENT MOHR-COULOMB

&

.................

ity is decreased. Shear strength, T, of ice rubble, and thus ¢ and c values, are functions of time, normal pressure, a, porosity, as well as shape and packing of ice fragments (which may also be partly represented in terms of porosity). Ice/water/air temperatures, and water salinity are additional factors which may affect 3. The limiting case at t i m e = 0 corresponds to the condition when ice rubble is undergoing continuous shear deformation. It is noteworthy that freeze bonds between ice pieces develop so rapidly that ice-rubble shear strength is dependent on the rate of deformation. Strength tests by the authors (Urroz and Ettema, 1987) and others (Cheng and Tatinclaux, 1977; Prodanovic, 1979; and Hellmann, 1984) have demonstrated that ice rubble exhibits a peculiar shear-thinning, in that its shear strength diminishes to a lower asymptote with increasing deformation rate. The writers propose that the influences of freeze bonding and contact crushing on shear-strength behavior of unconsolidated ice rubble, undergoing continuous shear deformation as interpreted from shear-box tests, can be demonstrated as follows.

. . . .

~,U_~-_3 f l ~.,...~.. . I . "0 "~ ~ / . I / I I

I

c (%)c (~z)-J C (o-1/

"7' /

O-

l

o-4 LOW STRESS (FREEZE BONDING) COMPRESSIVE

(7

HIGH STRESS (CRUSHING)

STRESS

Fig. 4. Apparent internal friction and cohesion in shear testing of unconsolidated ice rubble.

243

t

i

i

I

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i

i

600

I

o

If ci varies linearly with ai, i.e., c,= fiG, Eq. 2 could be restated as:

°°

zi = ai (tan~, + fl) = a~ t a n #

500 0. 400 o 300 iz cq 2OO

o

100 0

I 0

I lO0

I

I

I

200

I

300

I

I

I

400

500

CONFINING PRESSURE, o-(Pn)

Fig. 5. Shear strength data from Urroz and Ettema ( 1987).

PRECONSOLIDATION LINE ~ (PRECONSOLIDATION STRESS,(73)j

)

'

/ c(%) ""

~

O"1

~r2 COMPRESSIVE ;TRESS

Fig.6. Preconsolidation linein Mohr-Coulombdiagramof peak shear-strength behavior. Consider a series of data points P,= (a~, z~), with i = 1, ..., n, plotted as shown in Fig. 4. If freeze bonding is affected by ai, term c in Eq. 1 is not constant and should be replaced with ci, or:

c,=f(a,)

(2)

Equation 1 becomes: r,= tr~ tanOi + f ( a i )

(3)

(4 )

in which ~' =effective angle of internal friction which is associated with an apparent Mohr-Coulomb curve, or law, as indicated in Fig. 4 for i = 13. The effects on freeze bonding and r of shear rate, period between shear events, temperatures of ice/ air/water, ice-rubble porosity, and shape of ice fragments should also be included in the functional relationship between c and a,, or in ft. However, in order to simplify the description for the moment, it is expedient to take those factors as being constant and to consider a as the principal independent variable. Also indicated in Fig. 4 are Mohr envelopes associated with lines of c~= constant. These envelopes are of lesser slope, O,, than ~'. Some justification for assuming c, ~- flai, at least for comparatively small values of a, may be gained by inspection of Fig. 5 which shows continuous-shear data conforming fairly tightly to a straight line. The data were obtained by Urroz and Ettema ( 1987 ) using a simpleshear box, ice blocks from a commercial ice machine, and a range of a well below values needed to crush ice. In addition, Ettema and Schaefer ( 1986 ) presented data showing a linear relationship between freeze-bond strength and a, for ice blocks and a < 4 kPa. If c~ values do not vary linearly with or, the relationship between r and a is commensurately nonlinear, as is indicated in Fig. 4 for i = 3-5. Such nonlinear relationships may result if the extent of freeze bonding varies nonlinearly with a, or if ice is crushed at inter-fragment contacts. The solid curve shown in Fig. 4 applies for values of continuous shear strength, with 0c being assessed as ~'. Insofar as values of fl may vary with such factors as shear rate or internal ice temperature, thus a family of envelopes giving a range of ~' would result in accordance with the sensitivities of shear strength and freeze bonding to those factors. A figure similar to Fig. 4 could be prepared using peak values of shear strength, with ~p being assessed as 0'- Because peak shear strength is associated with initial compaction and dilation of ice rubble, and is attained at commencement of continuous shear for

244 [

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38mm

iI

COARSE RUBBLE 0

REPACKED

0

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RUBBLE

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4

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ACTUAL SOMEWHERE

1

ZERO HERE

1

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1

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CONFINING PRESSURE

(kPa)

Fig. 7. Data presented by Prodanovic (1979). densely packed ice rubble (see Fig. 3 ), it is likely to be more sensitive to the influence of self-solidification, or freeze bonding between ice fragments, than is continuous shear strength. Consequently, for Mohr-Coulomb diagrams based on peak shear strength, it is possible that Mohr envelopes associated with ci = constant, could delineate preconsolidation lines. As indicated in Fig. 6, a volume of ice rubble could be initially confined at a pressure 0.3, which is then relaxed, and sheared at lesser confining pressures 0.2 and 0.1, such that its peak shear strengths are greater than if the volume was confined only at 0"2 or al. An explanation for the comparatively large values reported for internal-friction angles (see Table

1 ) is that shear-box testing to date has essentially yielded values of @', both for 0c and 0p. As is suggested in Fig. 4, values of @' are larger than Oi for values of 0.i sufficiently low such that little or no ice crushing occurs. The procedures followed in shearbox testing have not enabled estimation, or exclusion, of the influences of 0. on freeze-bonding or crushing. In addition to performing shear-box tests producing values of 0', it would be of interest, in assessing the relative importance of freeze bonding, to perform tests with preconsolidated ice rubble so as to yield @~,as indicated in Fig. 6. It is difficult to offer a clear-cut explanation why studies involving larger 0.maxshould result in larger values of c (see Fig. 1 ). A plausible, though yet to

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CONFINING PRESSURE (kPa)

Fig. 8. Data presented by Weiss et al. ( 1981 ).

be substantiated, explanation may simply be as follows. Smaller internal-friction angles occur at larger a when crushing of ice at inter-fragment contacts becomes so significant that it diminishes interlocking and, concomitantly, diminishes freeze bonding at ice contacts. Presumably when crushing occurs at ice contacts, freeze bonding will not occur. Thus, if a linear Mohr envelope is fitted through data points predominantly at large a, and forced through a few points at comparatively low a, and it is then extrapolated to a = 0 , a larger c value could conceivably result. Generally, Figs. 4 and 6 indicate that the vagaries of freeze bonding, notably through its dependency on a, time, and crushing may produce nonunique relationships between z and a.

REAPPRAISAL OF DATA FROM WEISS ET AL. A N D PRODANOVIC Weiss et al. ( 1981 ), Prodanovic ( 1979 ), and also Hellmann (1984), presented data that were ostensibly obtained under conditions of a = 0 , i.e., amounting essentially to unconfined test conditions. Data obtained by Weiss et al., Prodanovic, and Hellmann are shown in Figs. 7, 8, and 9, respectively. However, their shear boxes were not configured for unconfined testing of ice rubble. Weiss et al. used a direct-shear box which confined a 1 m deep volume of ice rubble. Prodanovic used a direct shear box that confined a 0.30 m deep volume of ice rubble. Hellmann used a direct shear box, nominally 0.71 m deep (he describes the box as being square-sided with cross-sectional area 0.5 m2). The average vertical, or hydrostatic, pressure existing in each shear box would be approximately: az= 0.5 g ( 1 - q ) (Pw -pi)h

(5)

246 I

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SHEAR VELOCITY = IO.9 mm/s ICE CHIPS O 5

/ /'

PRIMARY SHEAR MODE

-

A /"

SECONDARY SHEAR MODE -----'D---- TERTIARY SHEAR MODE

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CONFINING PRESSURE

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5

(kPo)

ACTUAL ZERO SOMEWHERE

HERE

Fig. 9. Data presented by Hellmann ( 1 9 8 4 ) •

in which g = gravity acceleration 9.81 m s-2, ?/= porosity (i.e. 0.30, using a value given by Weiss et al. ), pi=ice density (i.e. 0 . 9 2 × 1 0 3 kg m -3) and h = depth of ice-rubble volume. Therefore, for the boxes used by Weiss et al., and Prodanovic, and Hellmann, az -~ 0.32, 0.09, and 0.22 kPa respectively. The horizontal pressure during shearing can be estimated tentatively as: o'.
(6)

in which Kp = ( 1 + sin0) / ( 1 - sin0), coefficient of (Rankine) passive stress. Using 0 = 39 ° for Weiss et al., 53 ° for Prodanovic, and 54 ° for Hellmann (0 values assessed by them), estimates o f a ( = ah) for

tests conducted by Weiss et al., and Prodanovic are 1.2, 0.8, and 2.0 kPa, respectively. When these values are used in Figs. 7, 8, and 9, the fitted Mohr envelopes are offset, such that cohesive intercepts likely do not occur at zero for data by Prodanovic or Hellmann, and either no, or reduced, cohesive intercepts occur for the data presented by Weiss et al. Figures 7, 8, and 9 also reveal that data scatter is another significant source of error for determining 0 and c. Though the foregoing calculations contain several assumptions (derived primarily by the writers' uncertainty as to exact test conditions and procedures, and difficulty of precisely postdicting a in

247 terms o f az and ah), they do indicate the uncertain veracities o f reported c values.

CONCLUDING REMARKS The shear-strength b e h a v i o r o f a c c u m u l a t e d ice rubble is i m p o r t a n t in estimating ice loads likely to be exerted against diverse structures and vessels destined to function in ice-covered waters, and in evaluating the equilibrium thickness o f ice jams. However, there is at present scant guidance for selecting appropriate values o f shear strength, and reported values vary widely. As is indicated in this paper, several factors m a y complicate assessment o f shear strength or, in M o h r - C o u l o m b terms, assessm e n t o f ~ and c. The m o s t i m p o r t a n t factors being the complicated shear-deformation b e h a v i o r o f ice rubble, and, relatedly, difficulties in shear testing o f ice rubble. Wide variations in ~ and c are reported for other particulate materials such as soils (e.g., Bowles, 1984; L a m b e and W h i t m a n , 1969 ). Soils engineers, nevertheless, usually have the option to c o n d u c t geomechanic tests enabling values o f ¢ and c to be determined for specific sites; such as the foundation o f a prospective structure. U n c o n f i n e d compression tests, as well as direct shear tests, are customarily used to measure c for cohesive soils. Ice engineers, unfortunately, do not have recourse to use this option. T h e y are prevented f r o m doing so by the impracticality o f either retrieving ice-rubble samples, or performing in-situ tests, and the sheer size o f rubble samples that would have to be tested. O f course, an a p p r o x i m a t e and lower-bound estimate o f 0, or ¢c, can be obtained by measuring the angle o f repose o f an inverted heap o f submerged ice rubble. The explanations o f the shear-strength behavior o f ice rubble outlined herein are admittedly somewhat speculative. Their verification awaits the development, and appropriate use of, well-designed shear boxes for shear-strength testing o f ice rubble.

ACKNOWLEDGEMENTS The writers' research on the shear-strength behavior o f ice rubble was f u n d e d by the U.S. National Science F o u n d a t i o n under G r a n t No. CEE 81-

09252. Helpful c o m m e n t s are m a d e by Drs. W.A. Nixon, and J.-C. Tatinclaux. M. Mellor guided the writers, and they are all gratefully acknowledged.

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