sea ice estimating strength of first year ice crrel 96-11

7/29/2019 Sea Ice Estimating Strength of First Year Ice CRREL 96-11

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Sea IcePart II. Estimating the Full-Scale Tensile, Flexural,and Compressive Strength of First-Year Ice

Austin Kovacs September 199

CRRELREPO

RT

96

-11

r = 0.9412

S D = 0.25

B = 1970 0.950.32

C

90 80 70 60 50 40 10 8

101

010

1

10 610

4

10

,BulkPorosity()B

,Horizontal

CompressiveStrength(MPa)

C

,StrainRate

(s )1

a


2/23

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Cover:Exxon ice floe c values vs. bulk porosity and strain rate.

Abstract: Sea-ice salinity, density, and temperature data

were used to develop new methods for determining the

bulk brine volume and porosity of sea-ice floes. Meth-

ods for estimating full-thickness ice sheet strength, based

on large-scale field tests, are presented. The relation-

ships among bulk sea-ice properties, strain rate, and

strength are illustrated. A new constitutive equation was

developed for predicting the full-thickness horizontal

compressive strength c of first-year sea ice as a func-

tion of the applied strain rate and bulk porosity in the

form c 2l/

B= B n m , where parameters B2, n, and m

are about 2.7 103, 3, and 1, respectively, and and

Bare the ice strain rate and ice floe bulk porosity

of sea ice, respectively. An estimate of the horizontal

force that may develop between first-year sea ice and a

90-m-wide structure is given. Estimating sea-ice strength

based on remote ice conductivity measurements is also

discussed conceptually.


3/23

CRREL Report 96-11

Sea IcePart II. Estimating the Full-Scale Tensile, Flexural,and Compressive Strength of First-Year Ice

Austin Kovacs September 1996

Approved for public release; distribution is unlimited.

US Army Corpsof EngineersCold Regions Research &

Engineering Laboratory


4/23

ii

PREFACE

This report was written by Austin Kovacs, Research Civil Engineer, Applied

Research Division, Research and Engineering Directorate, U.S. Army Cold Regions

Research and Engineering Laboratory (CRREL), Hanover, New Hampshire.

The author acknowledges with thanks the review comments of Dr. Walter Spring

of the Mobil Research and Development Corporation, Dr. Brian Wright of B. Wright

and Associates Ltd., and Drs. Anatoly M. Fish and Devinder S. Sodhi of CRREL. Dr.

Fish provided valuable comments related to ice creepfailure relationships.

The contents of this report are not to be used for advertising or promotional

purposes. Citation of brand names does not constitute an official endorsement or

approval of the use of such commercial products.


5/23

CONTENTSPage

Preface ..................................................................................................................... ii

Introduction ............................................................................................................ 1

Tensile and flexural strength ............................................................................... 4

Compressive strength ........................................................................................... 6

Icestructure interaction force ............................................................................. 14

Summary ................................................................................................................. 15

Literature cited ....................................................................................................... 16

Abstract ................................................................................................................... 19

ILLUSTRATIONS

Figure

1. Comparison between the Cox and Weeks and the Frankenstein

and Garner methods for calculating sea-ice brine volume ............. 1

2. Sea-ice bulk density and conductivity vs. floe thickness ..................... 2

3. Sea-ice bulk brine volume vs. bulk salinity with average ice floe

temperature as a parameter................................................................. 2

4. Sea-ice bulk brine volume vs. average absolute ice temperature

and bulk salinity ................................................................................... 4

5. Combined Arctic and Antarctic ice floe bulk salinity vs. thickness ... 4

6. 3-D presentation of sea ice brine volume vs. DC conductivity andtemperature ........................................................................................... 5

7. EM-31 instrument conductivity reading at ~10 kHz vs. ice floe

snow plus ice thickness ....................................................................... 5

8. Measured sea-ice floe bulk brine volume vs. the calculated bulk

DC conductivity of the ice ................................................................... 6

9. Ice sheet horizontal tensile strength vs. bulk DC conductivity

and brine volume .................................................................................. 6

10. Ice sheet flexural strength vs. bulk DC conductivity and

brine volume.......................................................................................... 6

11. Sea ice bulk salinity vs. floe thickness .................................................... 8

12. Timco and Frederking estimated full ice sheet horizontal uncon-

fined compressive strength vs. Exxons field-measured values .... 813. Ice floe bulk porosity vs. average temperature and bulk salinity ...... 9

14. Modified TimcoFrederking ice floe cstrengths vs. their

original values ....................................................................................... 9

15. Modified TimcoFrederking ice floe cstrengths vs. Exxons

field-measured cvalues ...................................................................... 10

iii


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INTRODUCTION

It is well known that the strength of sea ice is toa large extent dependent on brine volume, andthat brine volume is a function of ice salinity and

temperature. Brine volume may be estimated us-ing the empirical expressions of Frankenstein andGarner (1967) or the theoretical densityvolumefraction equations of Cox and Weeks (1983). Theformer is by far the simplest method, because theonly inputs required in the brine volume equa-tions are ice salinity and temperature. Since thelatter approach is based on phase relationships, itallows for the calculation of the gas volume andthus the total ice porosity and the brine-free icedensity. With this method, the bulk density of thesea ice is also needed and therefore must be deter-

mined. In principle this is a simple task,but it can be difficult to do in the field.Today, ice core weight is measured by elec-tronic scales rather than triple beam bal-ances, which are subject to uneven supportsurface and wind effect errors when usedat unsheltered sites. Electronic scale weightmeasurements can be highly representativeof the in-situ ice if brine drainage has notoccurred during coring or after the ice corewas removed from the bore hole. Deter-mining ice core volume is a bit more trouble-some. Ice core length should be measured

along the center axis using a special caliperor gauging device (Kovacs 1993). A meterrule should not be used for this purpose (itoften is), because the ends of the ice samplemay not be parallel to each other. The moredifficult measurement is ice core diameter.Again, a ruler should not be used. A pietape (Kovacs 1993) or a caliper with long,

wide tongs is generally satisfactory, provided thecore is ridge-free and of uniform diameter. Wherethis is not the case, the diameter measured willnot be accurate and the volume determinationwill be off. This is illustrated in Figure 1, which

gives a comparison between brine volume deter-minations made on ~10-cm-long sections fromseveral first-year Arctic sea-ice cores using theequations of Cox and Weeks and Frankensteinand Garner. The regression curve through the datain Figure 1 did not include the outlier (+). Theoutlier indicates where a measurement error oc-curred, perhaps because of a volume measure-ment error.

It is recommended that when the Cox andWeeks (1983) equations are used, and they must

be for determining gas volume, total porosity, and

Sea IcePart II. Estimating the Full-Scale Tensile, Flexural, and

Compressive Strength of First-Year Ice

AUSTIN KOVACS

Figure 1. Comparison between the Cox and Weeks and theFrankenstein and Garner methods for calculating sea-ice brinevolume. The data point represented by + is considered anoutlier.

250

200

150

100

50

0 50 100 150 200

+

, Ice Brine Volume (Frankenstein-Garner) ()B (F-G)

,IceBrineVolume(Cox-Weeks)()

B(C-W)

r = 1.0002

S D = 1.15

= 1.270 + 1.026B (F-G)B (C-W)


7/23

brine-free ice density, the Frankenstein and Gar-ner (1967) equation also be used to estimate the

ice brine volume. When the two brine-volumeestimates are off by more than 2, a measure-ment or data logging error has occurred. If theproblem cannot be resolved, the ice property val-ues calculated with the Cox and Weeks equationsshould not be used. It should be noted that theFrankenstein and Garner equations for determin-ing brine volume are not valid below 23C. Onlythe Cox and Weeks equations should be used

below this temperature.The physical and electromagnetic properties of

21 winter Beaufort Sea ice cores, obtained by theauthor on various field programs, were deter-mined for each 10-cm increment of ice core. Physi-cal property calculations were made using themathematical expressions of Cox and Weeks(1983). The electromagnetic properties of the icewere determined using the procedures of Kovacset al. (1987) as modified by Kovacs and Morey(1987). Examples of the calculations are given inTables 1 and 2. Also assembled were the authorsdata from an additional 23 Beaufort Sea ice cores,where only the salinity and temperature of thewinter ice were determined for each 10-cm incre-ment of core. For these data, only the brine vol-

ume of the ice could be calculated using the equa-tions of Frankenstein and Garner (1967).From the above data, a number of winter ice

floe property trends can be seen. Ice floe bulkdensity vs. thickness is shown in Figure 2. The

bulk density is seen to decrease with ice floe thick-ness. For the first-year sea ice, the decrease isassociated with brine drainage and growth rate

0.93

0.92

0.91

0.90

0.89

0 200 400 600 800

T , Floe Thickness (cm)F

,BulkDensity(Mg/m

)3

B

0.10

0.08

0.06

0.04

0.02

0

,BulkD

CConductivity(S/m)

B

First-Year IceMultiyear Ice

+

x

B B

0.88

+

+

+

+

+

+

+

++

+

+++

+

x

+

x

x

x

r = 0.9192S D = 0.002

= 0.9363 0.0018TF0.5

B

T /154F = 0.147eBr = 0.9152

S D = 0.008

Figure 2. Sea-ice bulk density and conductivity vs. floe thickness.

processes, which reduce the vol-ume fraction of the heavier brineentrained within the ice (Kovacs1996). The lower multiyear icedensities are the result of this icecontaining proportionally less

brine and more gas, especially inthe freeboard portion, which is for

all intents and purposes low-den-sity fresh ice. While density is theproperty that a materials strengthis often related to, as with thestrength of snow (Kovacs et al.1969), it is not used for this pur-pose in sea-ice mechanics. Tradi-tionally, sea-ice failure strength,elastic modulus, and such have

been referenced to the brine vol-ume of the ice. This brine volume

referencing is still the norm for sea-ice tensile orflexural strength test presentations. The reason-

ing is that the strength of a material is a functionof its solidity. Therefore, the more brine volumethere is, the less solid the ice, and the weaker itwill be. This method, of course, does not take intoaccount the added porosity associated with thesmall gas inclusions. Since brine volume, salinity,and temperature are interrelated, brine volumeneeds to be correlated with these two parameters,as shown in Figure 3. This figure clearly illus-trates that, at any given bulk salinity SB, the bulk

brine volume B of an ice sheet is dependent onthe average ice sheet temperature TA. While thispresentation is enlightening, it does not serve theuser very well. More useful is the presentation inFigure 4, where the relationship provided allowsfor the determination of the bulk brine volume

0 2 4 6

s , Bulk Salinity ()B

8

100

80

60

40

20B

,BulkBrineVolume()

3C

11C

Figure 3. Sea-ice bulk brine volume vs. bulk salinitywith average ice floe temperature as a parameter.

2


8/23


9/23

100

80

60

40

20

0

B

= 41.64 S0.88

T0.67

B A

r2

= 0.995

S D = 1.73

B,

BulkBrineVolume()

TA,Avg.FloeTemp.(|C|) SB

,BulkS

alinity(

)

8

6

4

2

046

810

Figure 4. Sea-ice bulk brine volume vs. average absolute icetemperature and bulk salinity.

vs. ice-sheet average temperature and bulk salin-ity. It should be realized that Figure 3 is an endview of the 3-D data presentation in Figure 4. InFigure 3, the points with no vertical tails are withinone standard deviation (SD) of the brine volumesurface shown in Figure 4. Points with tails lie

between one and two standard deviations of thebrine volume surface, which is located at the tipof the vertical line.

TENSILE ANDFLEXURAL STRENGTH

The equation for predicting the bulk brine vol-umeB in as a function ofSB and the absolutevalue ofTA (Fig. 4),

B B0.88 A0.67= 41 64. S T , (1)along with the expressions of Kovacs (1996)for estimating the bulk salinity () of first-year sea ice vs. ice floe thickness TF (cm)(Fig. 5),

S TB F= +4 606 91 603. . / , (2)offer the opportunity to make other ice prop-erty assessments, which in turn can be used toestimate ice sheet strength.

For example, the horizontal uniaxial tensilestrength t of an ice sheet could be estimatedusing the empirical equation of Dykins (1970):

t B= 0 816 0 069 0 5. . .. (3)

Similarly, ice floe flexural strength fmaybe estimated using the empirical equation(Timco and OBrien 1994)

f 5.88e B= 1 76. . (4)These expressions could be used as fol-

lows. Ice thickness is determined by drill-

hole measurement. Then from eq 2 the bulksalinity of the ice is estimated. Ice tempera-ture is determined next at 2025 cm belowthe ice surface. A linear temperature gradi-ent to the ice/water interface (~1.7C) isassumed, and the average ice sheet tem-perature is calculated. The bulk salinity andaverage temperature values are then usedin eq 1 to determine the bulk brine volume.This value can now be used in eq 3 or 4 todetermine t or f, respectively. It should

be noted that Gavrilo et al. (1995) com-pared various published expressions for deter-

mining the flexural strength of sea ice and foundthat eq 4 gave f values 2 to 4 times higher thanother research results. Further evaluation of thevarious f equations may be in order.

The above procedure still requires a drill-holethickness and a near-surface ice temperature mea-surement. These measurements may not be nec-essary. Using the formulations for determiningthe electromagnetic properties of sea ice (Kovacsand Morey 1987), the low-frequency (DC to ~100MHz) bulk conductivity* B of a model sea-ice

Figure 5. Combined Arctic and Antarctic ice floe bulksalinity vs. thickness (from Kovacs 1996).

20

10

0 50 100 150 200


S,BulkSalinity()

B

30

r = 0.7302S = 4.606 + 91.603/TFB

S D = 1.4

* Sea-ice conductivity is approximately constant between DCand 100 MHz (see Tables 1 and 2).

4


10/23

sheet was determined along with its brinevolume B vs. ice temperature and salinity.An ice sheet bulk density of 0.911 Mg/m3

was used in the determinations. The calcu-lated results are graphically shown in Figure6. There is a decrease in brine volume withdecreasing ice temperature. In addition toice conductivity, Figure 6 indicates that ei-ther ice salinity or temperature is required toestimate.

The complex interrelationship between the

model sea-ice temperature, salinity, brine vol-ume, and conductivity is well revealed inFigure 6. The results suggest that determin-ing B of natural sea ice remotely will not, ofitself, allowB to be estimated. Not revealedis the effect of density, which was fixed at0.911 Mg/m3 for this example. The decreasein the bulk density and conductivity with

increasing ice floe thickness as re-vealed in Figure 2 affects all other iceproperties. It turns out that this ef-fect is favorable to the remote sound-ing of sea ice. For example, Kovacsand Morey (1991) and Kovacs et al.(1996) have shown that a portableelectromagnetic induction sounding

system can be used to measure re-motely an apparent bulk conductivityfrom which an estimate of winterBeaufort Sea ice thickness can bemade (Fig. 7). When the B vs. B atDC values listed in Tables 1 and 2and from 20 other ice floes were com-pared, it was found that the calcu-lated bulk conductivity of BeaufortSea ice was directly related toB, asshown in Figure 8. Therefore, in prin-ciple, it should be possible to esti-mate both undeformed winter Arctic

ice floe thickness and bulk brine vol-ume remotely using a conductivitymeasurement system. Before the lat-ter is possible, the relation betweenthe field- measured apparent bulkconductivity, which is highly affect-ed by the underlying seawater, andthe bulk brine volume of sea ice floesof various thicknesses will need to

be determined. Conductivity mea-surement at a frequency well above50 kHz is likely to be necessary toseparate the sea-ice conductivity con-

Salinity = 7

4

70

60

50

40

30

20

10

0.10

0.08

0.06

0.04

0.02

0

20 16 12 8 4T, Temperature (C)

,BrineVolume()

B

,

BulkDCC

onductivity(S/m)

B

70

60

50

40

30

20

0

0.08 0.06 0.04 0.0220

1612 8

4

T,Temperature(C)

4

Salinity=7

S D = 0.98r = 0.9962

= 9.02 +217.09/T + 528.482B B

,Brin

eVolume()

B

,BulkDCConductivity(S/m)

B

Figure 6. 3-D presentation of sea-ice brine volume vs. DC conductiv-ity and temperature with salinity as a parameter for an ice density of0.911 Mg/m3 (a) and a boresight view from the above temperature

window (b).

Figure 7. EM-31 instrument conductivity reading at ~10 kHzvs. ice floe snow plus ice thickness (from Kovacs et al. 1996).

0.8

0.6

0.4

0.2

0 2 4 6 8

Drill-hole-Measured Snow and Ice Thickness (m)

Instru

mentConductivityReading(S/m)

5


11/23

tribution from the seawater conductivity con-tribution. A dual high and low frequencysystem or a wideband measurement systemcould be used.

In the interim, an off-the-shelf electromag-netic induction system operating at 10 kHzcan be used to estimate ice thickness (Fig. 8),and the bulk salinity of the ice is estimated

using eq 2. After determining the average icesheet temperature, as previously described,eq 1 can be used to calculate the bulk brinevolume. Ice strength may then be calculatedfrom eq 3 or 4.

The relation between sea-ice tensile or flex-ural strength, brine volume, and the calcu-lated bulk DC conductivity of natural sea iceis shown in Figures 9 and 10. These figuresare the result of using the equation in Figure6 and eq 3 and 4, respectively. In the future,

using similar equations, an algorithmcould be added to a portable electro-

magnetic induction sounding systemthat would provide both ice thicknessand strength from the apparent sea-iceconductivity measurements.

COMPRESSIVE STRENGTH

Estimating the compressive strengthof sea ice is different. While the uniaxialtensile or flexural strength of sea icedoes not reveal a significant strain ratedependence, at the strain rates of inter-est for most ice engineering problems,the compressive strength does. In addi-tion, in reporting sea-ice compressiontest results, the recent, and correct,method is to relate strength to total po-rosity: brine volume plus gas volume.Recently, Timco and Frederking (1990)developed empirical equations for pre-dicting the large-scale uniaxial uncon-fined compressive strength of Arctic seaice. Their equations are based on small-scale strength tests and the various con-

trolling factors, such as ice structure,brine volume, porosity, loading direc-tion (horizontalvertical), and strainrate. Their strength equation for hori-zontally loaded columnar sea ice is

c B= ( )[ ]37 1 2700 220 5

/..

, (5)

80

60

40

20

0 0.02 0.04 0.06 0.08

,Bu

lkBrineVolume()

B

, Bulk DC Conductivity (S/m)B

0.10

100

S D = 3.169r = 0.9842

= 584 B0.74

B

Figure 8. Measured sea-ice floe bulk brine volume vs. thecalculated bulk DC conductivity of the ice.

Figure 10. Ice sheet flexural strength vs. bulk DC conductiv-ity and brine volume.

Figure 9. Ice sheet horizontal tensile strength vs. bulk DCconductivity and brine volume.

B,BulkDCConductivity(S/m)

0 25

50

75

100

B,Bu

lkBrine

Volum

e()

0.100.0

80.060.0

40.02

t,

HorizontalTensile

Strength(MPa)

0.8

0.6

0.4

0.2

0

0.8

0.6

0.4

0.2

0

0.02

0.04

0.06

0.08

0.10 0

20 40

60 80

100

B,BulkDCConductivity(S/m)

B,Bu

lkBrine

Volum

e()

f,

FlexuralStrength(MPa) 1.0

0.8

0.6

0.4

0.2

0

1.0

0.8

0.6

0.4

0.2

0

6


12/23

where c = horizontal uniaxial unconfined com-pressive strength (MPa)

= strain rate, s1B = bulk ice porosity ().

Timco and Frederking compared c values de-termined from eq 5 with in-situ, full ice sheetthickness, horizontal, uniaxial unconfined com-pression tests performed by Exxon on BeaufortSea ice. The Exxon large-scale test data comprisedice thickness, average ice temperature, loadingstrain rate, and the measured strength (Table 3).Since the bulk porosity of the ice sheets testedwas not provided, the authors had to make theseestimates. To do this, they first included addi-tional data in Figure 11 and, following the sameapproach as Cox and Weeks (1974), ran two linearregression curves through the data. Only slightlydifferent from the Cox and Weeks equations (Fig.11), the new equations are:

SB = 13.4 0.174 TF (6)

for TF 0.34 cm; and

SB = 8.0 0.016 TF (7)

for TF > 0.34 cm.

From these equations, Timco and Frederkingdetermined SB for each ice sheet tested (Table 3).Then the bulk density of the sea ice was selectedto be 0.907 Mg/m3. With these determinationsand the average ice floe temperature given inTable 3, the expressions provided by Cox andWeeks (1983) were used to calculate the bulk po-rosity of each ice sheet. Note that these valueswere not provided by Timco and Frederking intheir paper. The missing bulk porosity values wererecalculated along with the bulk brine volume foreach ice sheet. The values are listed in the leftcolumn of the brine volume and porosity datacolumns in Table 3. The c strengths measured byExxon and calculated by Timco and Frederkingare compared in Table 3 and shown graphically inFigure 12. The regression curve in Figure 13 showsthat the calculated values are generally higherthan the measured ones. This offset may be due tosmall- to large-scale scaling ambiguities.

To bring eq 5 into better agreement with theExxon test results, the bulk salinity and brine vol-ume of each ice sheet were determined using eq 2and 1, respectively. Next, the SB, TA, and B datafrom Tables 1 and 2 and others referenced wereplotted as shown in Figure 13a. A boresight viewof the data as seen looking into the salinity win-dow is shown in Figure 13b. This view clearly

Table 3. Field-measured and calculated test data.

Ice Avg Strain Horizontal compressive strength

thick.a temp.a Bulk salinity Bulk brine volume Bulk porosity ratea Measureda Calculatedcm C s1 MPa MPa

156 10.6 5.501 5.202 38.19 36.533 52.87 51.184 7.90e7 0.81 0.965 0.386 0.517 0.548

154 10.5 5.54 5.20 38.44 36.26 53.10 51.39 1.10e6 0.49 1.03 0.46 0.57 0.60120 12.9 6.08 5.37 34.18 32.94 49.20 47.97 1.27e6 0.66 1.07 0.52 0.62 0.67

129 10.7 5.94 5.32 38.50 32.04 53.16 51.64 9.20e6 1.00 1.62 1.12 1.10 1.17152 10.5 5.57 5.21 38.50 36.82 53.16 51.94 1.42e5 1.30 1.81 1.30 1.26 1.35166 4.2 5.34 5.16 70.67 67.46 82.66 79.20 9.00e5 1.82 2.15 1.86 1.79 1.60147 5.9 5.65 5.23 56.83 54.36 69.96 67.28 9.10e5 1.86 2.36 2.04 1.95 1.88154 4.3 5.54 5.20 69.91 66.86 81.96 78.65 1.33e4 2.10 2.33 2.09 2.03 1.83146 11.8 5.66 5.24 35.72 34.23 50.61 49.12 1.56e4 2.86 3.11 2.70 2.71 3.00130 5.1 5.92 5.32 63.26 60.85 75.86 73.18 2.16e4 1.88 2.66 2.50 2.45 2.28161 7.3 5.42 5.18 48.89 46.74 62.67 60.37 3.40e4 3.53 3.41 3.08 3.08 3.17129 3.4 5.94 5.32 83.08 79.84 93.96 90.54 7.50e4 2.62 2.86 3.13 3.24 2.77

a = EXXON Test Data 5 = Timco & Frederking, eq 51 = Timco & Frederking eq 6 or 7 6 = Timco & Frederking, revised, eq 82 = Eq 2 7 = Figure 18s equation3 = Eq 1 8 = Eq 104 = Figure 13s equation

7


13/23

4

3

2

1

0 1 2 3 4

, Exxons Horizontal Unconfined Compressive Strength (MPa)C ( Ex)

r = 0.9422

= 0.621 + 0.855 C (Ex)C (T-F)

,Timco-FrederkingHorizontal

UnconfinedCompressiveStrength(MPa)

C(T-F)

and Frederking. However, the modified methodfor determining c gives values that are still offsetfrom the Exxon field test results (Fig. 15). To alle-viate this offset, eq 5 was revised to read

c B0.93= ( )

37 1 270 0 650 220 5

/ . ...

(8)

This revision brings the c values (Table 3) intobetter agreement with the field determinations,as shown in Figure 16.

If the DC conductivity of the ice could be de-termined remotely along with ice sheet thickness,then SB could be determined from eq 2 and Bestimated from an expression similar to that givenin Figure 6 and B from Figure 17. The data in

reveals the interrelationship of average ice sheettemperature and bulk salinity with the bulk po-rosity.

Using the SB values determined from eq 2, theaverage ice floe temperature listed in Table 3, andthe equation provided in Figure 13, B was esti-mated for each of the Exxon ice sheets. These B,SB, and B values are listed in Table 3 as the right

column under each respective heading. Using thenew B values and the related strain rates in Table3, c was recalculated using eq 5. These so-calledmodified c values were then compared (Fig. 14)with the c estimates made by Timco andFrederking (Table 3). As can be seen in Figure 14,using the above B estimates in eq 5 gives cvalues that are comparable with those of Timco

Figure 12. Timco and Frederking esti-

mated full ice sheet horizontal uncon-fined compressive strength vs. Exxonsfield-measured values.

16

12

8

4

0 100 200 300 400

S,BulkSalinity()

B


First-Year IceMultiyear IceS = 14.237 0.194TFB

S = 7.879 0.016TFB

Figure 11. Sea-ice bulk salinity vs.floe thickness (after Cox and Weeks1974).

8


14/23

alyzed using a two-parameter power function.The relationship is a further development of theNorton (1929) secondary creep rate equation con-taining a power function of stress:

= B n ,where B is an empirical parameter that takes into

account ice structure, temperature, activation en-ergy, and other factors, is the applied stress, andthe empirical exponent n is generally found to beabout 3 for ice. Rearranging for , the above equa-tion becomes

= B n1 1 / ,where B1 = B

1/n. The former expression was usedby Glen (1955, 1958) to evaluate the creep behav-ior of solid, not porous, freshwater ice. Since theice porosity has a profound effect on , the aboveequation must be modified to include a porosity

parameter

as follows:

= B n m2 1 / , (9)where B2 = B1

m and m is an empirical exponent.Equation 9 was evaluated by substituting the

c values obtained from eq 8 for in eq 9 andusing the related controlling parameters B and given in Table 3. The result is shown in Figure 18.Equation 9 is shown to statistically fit the dataextremely well with an r2 value of 0.993.

As more data become available, the failure sur-face and therefore the variables in eq 9 may be

better determined. Even though the failure sur-face fits the data with a high r2 value of 0.993, thedata suggest that this surface is not a straight lineas viewed from the strain rate window in Figure

, Timco-FrederkingHorizontal Unconfined Compressive Strength (MPa)

C (T-F)

3

2

1

0 1 2 3

,ModifiedHorizontal


C(M)

4

4

r = 0.9892 = 0.016 + 1.007 C (M) C (T-F)

S D = 0.09

Figure 13. Ice floe bulk porosity vs. average temperatureand bulk salinity (a) and boresight view from the abovebulk salinity window (b).

Figure 17 are from Tables 1 and 2 and the sourcescited above. This figure indicates that nonde-formed Beaufort Sea ice floes have a gas volume

between about 1 and 2%.The c values calculated using eq 8 in relation

to the controlling parameters and B can be an-

Figure 14. Modified TimcoFrederking icefloe c strengths vs. their original values.

120

100

80

60

40

20

2 C

11 C

0 2 4 6 8

SB

, Bulk Salinity ()

b

B,BulkPorosity()

a

120

80

60

40

20

0

24

68

24

6810

B A

r2 = 0.995

S D = 1.57

TA ,Avg.FloeTemp.(|C| ) S B,BulkSa

linity(

)

B

= 19.37 + 36.18 S0.91T0.69

B,BulkPorosity() 100100

9


15/23

4

3

2

1

0 1 2 3 4 , Exxons Horizontal Unconfined Compressive Strength (MPa)C (Ex)

,M

odifiedHorizontal

UnconfinedCom

pressiveStrength(MPa)

C(M)

2r = 0.921

= 0.615 + 0.862 C (Ex)C(M)

S D = 0.25

120

80

40

0 20 40 60 80 100

, Bulk Brine Volume ()B

,Bulk

Porosity()

B = 17.852 + 0.917BB

r = 0.9832

S D = 2.44

4

3

2

1

0 1 2 3 4

,RevisedHorizontal


C(R)

, Exxons Horizontal Unconfined Compressive Strength (MPa)C (Ex)

r = 0.8982 = 0.008 + 1.007C (Ex)C(R)

S D = 0.25

Figure 15. Modified TimcoFrederking icefloe c strengths vs. Exxons field-measuredc values.

Figure 16. Ice floe c strengths calculatedusing eq 8 vs. Exxons field-measured c

values.

Figure 17. Sea-ice floe bulk porosity vs. bulkbrine volume.

10


16/23

18b. The three data points at a strain rate ofabout 106 are shown to be offset from theirpredicted failure surface position at the tip oftheir vertical tails, suggesting that the failuresurface curves downward. This is not surpris-ing ifB contains a nonlinear viscosity compo-nent. In this case, while stress is still a func-tion of strain rate, it is not directly proportional(Mellor 1986), at least at strain rates betweenabout 105 and 103 as indicated by the stressvs. strain rate curves for fresh and saline icegiven in Sanderson (1988). The Exxon datafall in the ductile-to-brittle strain rate transi-tion region where the creep power-law maynot be valid, that is, B n because n is no

longer a constant.A statistically more representative failuresurface for the data is shown in Figure 19a.This curved surface fits the data with a cor-relation coefficient ofr2 = 1.000. A boresightview of the data from the strain rate windowin Figure 19a is shown in Figure 19b. Based onthe data presented by Sanderson (1988), be-

low a strain rate of about 106 the failure surfaceshould be linear, a 3. However, the existing dataare insufficient for developing this trend. In theinterim, for strain rates between 104 and 103, theequation given in Figure 18 is preferred because itdoes not contain a second constant. The equationfor the failure surface when B is substituted forB in Figure 18 is c B= 150 75

0 31 0 36. . . . In thisexpression,B implicitly accounts for the bulk gasvolume or the porosity. In using either c ex-pression, the only parameters needed are the load-ing strain rate and the bulk ice porosity or brinevolume. The latter may one day be estimated fromremote conductivity measurements along with icethickness using a helicopter-borne electromagnetic

induction sounding system (Kovacs and Holladay1990).A comparison of the c data derived from us-

ing eq 8 with the horizontal uniaxial unconfinedcompressive strength data of others should be ofinterest. The data compiled by Sanderson (1988;see his Fig. 4.8) are shown in Figure 20 for thestrain rate range of our data. In this standard

100

= 20B

101

100

10 1

10 8 10 7 10 6 10 5 10 4 10 3

b

, Strain Rate (s ) 1

,Horizon

tal

Com

pressive

Streng

th(MPa

)

C

8060

4010

8

101

010

110

610

410

,Horizon

tal

Compressive

Streng

th(MPa

)

C

,StrainRate

(s )1

20

r = 1.0002

S D = 0.02

= 0.73 + 87.44 B 0.350.21

C

a

,BulkPorosity()

B. ,St

rainRate(

s1)

10

1

0.1

8060

4020 10

8

106104

C,Horizontal

Compres

siveStrength(MPa)

B,BulkPorosity()

a

10

1

0.1

C,Horizontal

Comp

ressiveStrength(MPa)

, Strain Rate (s1).

108 106 104107 105 103

B

= 20

100

b

r2 = 0.993

S D = 0.09

C

= 295.22 0.91B0.5

.

Figure 19. Equation-8-determined ice floe c strengths vs.bulk porosity and strain rate (a) and the boresight view ofthe above from the strain rate window (b).

Figure 18. Equation-8-determined ice floe c strengthsvs. bulk porosity and strain rate (a) and the boresightview from the above strain rate window (b).

11


17/23

presentation, no corrections have been made tothe data to account for differences in test tech-nique, ice porosity, grain size, or crystal orienta-tion. This is not unusual since some of these pa-rameters were not measured or provided in theoriginal source material, as is the case for theExxon data used in this report. For this reasonthere is considerable scatter in the data. The up-per regression curve shown in Figure 20 fits thedata from Sanderson; the lower curve representsthe data generated from eq 8. The small amountof data from this study coupled with the un-knowns mentioned above does not allow a defini-tive statement to be made on the difference be-tween the two lines. Clearly, the data from thisstudy fit well within the scatter of the data fromSanderson. Both sets of data are statistically bet-ter represented by the nonlinear curves shown inFigure 21. Here again, the failure surface appearsto be a non-power-law function of the strain rate.

The Timco and Frederking (1990) study, whichresulted in the formulation of eq 6, is based onsmall-scale test results. How applicable this equa-tion is to full-scale ice force assessments can be

questioned on the basis of small- to large-scalescaling uncertainties. To avoid this problem, thefull-scale Exxon c data were directly analyzedusing the values listed in the right-hand bulk po-rosity column in Table 3. Applying eq 9 to analyzethe c and data of Exxon vs. the above-relatedB values (Fig. 22a) gives the following expres-sion for the failure surface:

Figure 21. Ice floe c values calculated using eq 8 fromthis study compared with the data provided by Sanderson(1988). The data were fitted with non-log-log powerequations.

101

10 7

,HorizontalCom


C

1, Strain Rate (s )

610

510

410

310

010

110

2r = 0.280

S D = 1.43

= 8.76 2.98 0.07

C

S D = 0.18r = 0.9722

= 1.43 + 12.93 0.14C

Figure 20. Ice floe c values calculated using eq 8 fromthe study compared with the data provided bySanderson (1988). The data were fitted with a single-term power equation.

101

10 7

,

HorizontalCom


C

1, Strain Rate (s )

610

510

410

310

010

110

r = 0.9582

S D = 0.22

= 23.02 0.263

C

r = 0.2612

S D = 1.44

C

0.177 = 14.07

Sanderson (1988)This Study

c B= 1970 0 32 0 95 .. . (10)In the boresight view of the data from the strainrate window in Figure 22b, the outlier withthe long offset tail at a strain rate of 7.9 107s1

is very apparent. However, removing this datapoint and reanalyzing the remaining data givesa new equation whose calculated c values donot vary by more than 0.04 MPa from those cal-culated by eq 10.

The form of the expression for the curved fail-

ure surface shown in Figure 17 does not fit thenew data set as well as the power type equationin Figure 22. This may be seen by comparing thestatistics listed in Figure 22 with those in Figure23. The boresight view of the failure plane fromthe strain rate window (Fig. 23b) also indicates nofit improvement for the outlier over the failuresurface shown in Figure 22b.

Substituting the new B data for the B dataand reanalyzing as before gives the results shownin Figure 24. The expression for the failure surfaceis now

c B= 523 24 0 32 0 67. .. . (11)As with the bulk porosity data, applying a curvedfailure surface through the c vs. bulk brine vol-ume and data does not result in a statistically

better curve fit.It is interesting that the c vs. trend, as ob-

tained with eq 8 and from Sandersons data, ap-

12


18/23

r = 0.9412

S D = 0.25

B = 1970 0.950.32

C

90 80 70 60 50 40 10 8

101

010

110

610

410

,BulkPorosity()B

,

Horizontal

Compressiv

eStrength(MPa)

C

,StrainRate

(s )1

a

101

b

100100

= 40B

,

Horizontal

Com


C

010

110 810 710 610 510 410 310


Figure 22. Exxon ice floe c values vs. bulk porosity andstrain rate (a) and the boresight view of the above from thestrain rate window (b). Data were fitted with a two-term

power function, i.e., eq 9.

10 8

10 6

10 4

8060

40 ,BulkPorosity()B

101

100

10 1

,Horizontal


C

,StrainRat

e(s)1

r = 0.9292

S D = 0.29

= 0.58 + 271.20 B 0.63 0.22

C

a

100

= 40B

101

100

10 1

10 8 10 7 10 6 10 5 10 4


10 3

,Horizontal

Com


C

b

Figure 23. Ice floe c values vs. bulk porosity and strainrate (a) and the boresight view of the above from the strainrate window (b). Data were fitted with a non-log-log

power equation.

6040

= 523.24

C B0.32 0.67

r = 0.9382

S D = 0.26

a

,Horizontal


C

10 610 4

,BulkBrineVolume()

B

101

100

10 1

20 10 8

,StrainRa

te(s)

1

100

= 20B

b

10 8 10 7 10 6 10 5 10 4 10 3


,Horizontal

CompressiveStrength

(MPa)

C

101

100

10 1

Figure 24. Ice floe c values vs. bulk brine volume andstrain rate (a) and the boresight view of the above from thestrain rate window (b).

13


19/23

pears to favor a curved failure surface (Fig. 17and 19), while the trend resulting from the analy-sis of the Exxon c vs. data does not show astrong preference. Additional full ice sheet testsshould help to further define the failure surface,especially outside the area of the current data.Within the range of the current data, 106s1

sea ice estimating strength of first year ice crrel 96-11

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