Sea Ice

Part II. Estimating the Full-Scale Tensile, Flexural,

and Compressive Strength of First-Year Ice

Austin Kovacs September 1996

CRREL REPORT 96-11

r = 0.941 2

S D = 0.25

σ B = 1970ε φ 0.32 – 0.95

C

·

90

80

70

60

50

40

10– 8

101

0 10

– 1 10

– 6 10

– 4 10

φ B , Bulk Porosity (‰)

σ , Horizontal

Compressive Strength (MPa) C

ε, Strain Rate (s ) – 1 ·

a
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Cover: Exxon ice floe σc 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. Methods for estimating full-thickness ice sheet strength, based

on large-scale field tests, are presented. The relationships 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 σc of first-year sea ice as a function of the applied strain rate and bulk porosity in the

form 

σ εφ c 2

l/

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

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

ε ̇

and 

φB are 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.
CRREL Report 96-11

Sea Ice

Part 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 Corps


of Engineers

Cold Regions Research &


Engineering Laboratory
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 creep–failure 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.
CONTENTS

Page

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

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

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

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

Ice–structure 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 and

temperature ........................................................................................... 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 unconfined compressive strength vs. Exxon’s field-measured values .... 8

13. Ice floe bulk porosity vs. average temperature and bulk salinity ...... 9

14. Modified Timco–Frederking ice floe σc

 strengths vs. their

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

15. Modified Timco–Frederking ice floe σc strengths vs. Exxon’s

field-measured σc values ...................................................................... 10

iii
INTRODUCTION

It is well known that the strength of sea ice is to

a large extent dependent on brine volume, and

that brine volume is a function of ice salinity and

temperature. Brine volume may be estimated using the empirical expressions of Frankenstein and

Garner (1967) or the theoretical density–volume

fraction equations of Cox and Weeks (1983). The

former is by far the simplest method, because the

only inputs required in the brine volume equations are ice salinity and temperature. Since the

latter approach is based on phase relationships, it

allows for the calculation of the gas volume and

thus the total ice porosity and the brine-free ice

density. With this method, the bulk density of the

sea ice is also needed and therefore must be determined. In principle this is a simple task,

but it can be difficult to do in the field.

Today, ice core weight is measured by electronic scales rather than triple beam balances, which are subject to uneven support

surface and wind effect errors when used

at unsheltered sites. Electronic scale weight

measurements can be highly representative

of the in-situ ice if brine drainage has not

occurred during coring or after the ice core

was removed from the bore hole. Determining ice core volume is a bit more troublesome. Ice core length should be measured

along the center axis using a special caliper

or gauging device (Kovacs 1993). A meter

rule should not be used for this purpose (it

often is), because the ends of the ice sample

may not be parallel to each other. The more

difficult measurement is ice core diameter.

Again, a ruler should not be used. A pie

tape (Kovacs 1993) or a caliper with long,

wide tongs is generally satisfactory, provided the

core is ridge-free and of uniform diameter. Where

this is not the case, the diameter measured will

not be accurate and the volume determination

will be off. This is illustrated in Figure 1, which

gives a comparison between brine volume determinations made on ~10-cm-long sections from

several first-year Arctic sea-ice cores using the

equations of Cox and Weeks and Frankenstein

and Garner. The regression curve through the data

in Figure 1 did not include the outlier (+). The

outlier indicates where a measurement error occurred, perhaps because of a volume measurement error.

It is recommended that when the Cox and

Weeks (1983) equations are used, and they must

be for determining gas volume, total porosity, and

Sea Ice

Part 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 the

Frankenstein and Garner methods for calculating sea-ice brine

volume. The data point represented by + is considered an

outlier.

250

200

150

100

50

0 50 100 150 200

+

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

ν , Ice Brine Volume (Cox-Weeks) ( ‰) B (C-W)

r = 1.000 2

S D = 1.15

νB (C-W) = – 1.270 + 1.026 νB (F-G)
brine-free ice density, the Frankenstein and Garner (1967) equation also be used to estimate the

ice brine volume. When the two brine-volume

estimates are off by more than 2‰, a measurement or data logging error has occurred. If the

problem cannot be resolved, the ice property values calculated with the Cox and Weeks equations

should not be used. It should be noted that the

Frankenstein and Garner equations for determining brine volume are not valid below –23°C. Only

the Cox and Weeks equations should be used

below this temperature.

The physical and electromagnetic properties of

21 winter Beaufort Sea ice cores, obtained by the

author on various field programs, were determined for each 10-cm increment of ice core. Physical property calculations were made using the

mathematical expressions of Cox and Weeks

(1983). The electromagnetic properties of the ice

were determined using the procedures of Kovacs

et al. (1987) as modified by Kovacs and Morey

(1987). Examples of the calculations are given in

Tables 1 and 2. Also assembled were the authors’

data from an additional 23 Beaufort Sea ice cores,

where only the salinity and temperature of the

winter ice were determined for each 10-cm increment of core. For these data, only the brine volume of the ice could be calculated using the equations of Frankenstein and Garner (1967).

From the above data, a number of winter ice

floe property trends can be seen. Ice floe bulk

density vs. thickness is shown in Figure 2. The

bulk density is seen to decrease with ice floe thickness. For the first-year sea ice, the decrease is

associated 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

ρ , Bulk Density (Mg/m ) 3

B

0.10

0.08

0.06

0.04

0.02

0

σ , Bulk DC Conductivity (S/m) B

First-Year Ice

Multiyear Ice

+

x ρB σB

0.88

+

+

+

+

+

+

+

+ +

+

+++

+

x

+

x

x

x

r = 0.919 2

S D = 0.002

ρ = 0.9363 – 0.0018TF

0.5

B

– T /154 F σ = 0.147e B

r = 0.915 2

S D = 0.008

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

processes, which reduce the volume fraction of the heavier brine

entrained within the ice (Kovacs

1996). The lower multiyear ice

densities are the result of this ice

containing proportionally less

brine and more gas, especially in

the freeboard portion, which is for

all intents and purposes low-density fresh ice. While density is the

property that a material’s strength

is often related to, as with the

strength of snow (Kovacs et al.

1969), it is not used for this purpose in sea-ice mechanics. Traditionally, sea-ice failure strength,

elastic modulus, and such have

been referenced to the brine volume of the ice. This brine volume

referencing is still the norm for sea-ice tensile or

flexural strength test presentations. The reasoning is that the strength of a material is a function

of its solidity. Therefore, the more brine volume

there is, the less solid the ice, and the weaker it

will be. This method, of course, does not take into

account the added porosity associated with the

small gas inclusions. Since brine volume, salinity,

and temperature are interrelated, brine volume

needs to be correlated with these two parameters,

as shown in Figure 3. This figure clearly illustrates that, at any given bulk salinity SB, the bulk

brine volume νB of an ice sheet is dependent on

the average ice sheet temperature TA. While this

presentation is enlightening, it does not serve the

user very well. More useful is the presentation in

Figure 4, where the relationship provided allows

for the determination of the bulk brine volume

0 246

s B , Bulk Salinity (‰)

8

100

80

60

40

B

20 ν , Bulk Brine Volume (‰)

– 3°C

– 11°C

Figure 3. Sea-ice bulk brine volume vs. bulk salinity

with average ice floe temperature as a parameter.

2
3

Table 2. Physical and electromagnetic properties of 1.93-m-thick sea ice at 100 MHz.*

Core Core Core Melt Ice Bulk Ice Ice Brine vol Air Total Brine Brine Loss Ice conductivity Ice Loss Phase

Depth diam. length weight cond. temp den. den. sal. C-W F-G vol. porosity cond. dielectric const. tang. DC 100 MHz dielectric const. Attn tang. vel.

(m) (cm) (cm) (gr) mS/m (–C°) Mg/m3 Mg/m3 ‰‰ ‰ ‰ ‰ σb  ̇ε b  ̇ ̇ε b φb σDC σi εm  ̇ε m  ̇ ̇ε m (dB/m) φi m/ns

0.05 7.85 9.80 422.6 10.10 –4.7 0.891 0.830 5.67 57.5 60.6 37.90 95.4 5.24 66.67 941.67 14.125 0.047 0.047 7.19 4.65 0.057 28.92 1.832 0.1120.15 7.85 9.79 423.9 11.97 –5.3 0.895 0.828 6.81 62.2 65.3 35.24 97.4 5.60 64.89 1007.48 15.526 0.057 0.058 8.13 4.82 0.059 33.07 2.150 0.1050.25 7.85 9.82 428.2 10.92 –5.5 0.901 0.842 6.16 54.9 57.2 27.29 82.2 5.71 64.32 1027.33 15.971 0.048 0.048 7.22 4.65 0.051 29.13 1.851 0.1120.35 7.85 9.84 428.7 10.22 –5.5 0.900 0.845 5.74 51.1 53.2 27.49 78.5 5.71 64.32 1027.33 15.971 0.042 0.042 6.72 4.54 0.048 26.81 1.683 0.1160.45 7.85 9.75 425.5 10.97 –5.4 0.902 0.841 6.19 56.1 58.4 26.60 82.7 5.66 64.60 1017.53 15.570 0.049 0.049 7.35 4.69 0.053 29.65 1.886 0.1110.55 7.85 9.72 424.5 10.26 –5.2 0.902 0.844 5.76 54.0 56.2 25.37 79.4 5.54 65.18 997.17 15.300 0.045 0.045 7.01 4.63 0.052 27.99 1.758 0.1130.65 7.85 9.77 426.3 9.06 –4.9 0.902 0.848 5.04 49.8 51.9 25.31 75.1 5.36 66.06 964.68 14.603 0.038 0.038 6.38 4.51 0.050 24.84 1.529 0.1190.75 7.85 9.84 430.1 8.30 –4.7 0.903 0.853 4.59 47.2 49.1 23.02 70.2 5.24 66.67 941.67 14.125 0.034 0.034 6.03 4.44 0.048 22.83 1.386 0.1220.85 7.85 9.85 429.7 7.92 –4.4 0.901 0.851 4.36 47.6 49.6 24.76 72.4 5.03 67.61 905.06 13.386 0.033 0.033 5.97 4.45 0.051 22.38 1.351 0.1231.05 7.85 9.76 430.4 7.61 –4.0 0.911 0.858 4.18 50.4 51.9 14.19 64.6 4.74 68.93 852.16 12.363 0.034 0.035 6.15 4.57 0.058 22.83 1.362 0.1211.15 7.85 9.80 432.3 7.99 –3.7 0.911 0.851 4.40 57.2 58.7 14.63 71.9 4.50 69.97 809.27 11.567 0.040 0.040 6.74 4.77 0.069 25.53 1.530 0.1161.25 7.85 9.71 428.3 8.21 –3.5 0.911 0.846 4.53 62.2 63.7 15.21 77.4 4.33 70.68 779.08 11.022 0.044 0.045 7.17 4.91 0.078 27.33 1.640 0.1121.35 7.85 9.81 432.7 8.46 –3.2 0.911 0.838 4.68 70.1 71.5 16.03 86.1 4.06 71.79 731.32 10.187 0.051 0.051 7.84 5.13 0.093 29.89 1.794 0.1071.45 7.85 9.66 425.3 8.25 –3.0 0.910 0.834 4.56 72.5 74.0 17.97 90.5 3.88 72.56 687.76 9.616 0.051 0.052 7.93 5.19 0.101 30.05 1.792 0.1071.55 7.85 9.78 431.0 7.69 –2.8 0.911 0.836 4.23 72.1 73.2 16.71 88.9 3.68 73.35 662.79 9.036 0.048 0.049 7.69 5.18 0.106 28.75 1.691 0.1081.65 7.85 9.87 435.2 7.20 –2.5 0.911 0.833 3.94 75.4 76.0 16.23 91.7 3.38 74.58 607.61 8.147 0.047 0.048 7.72 5.28 0.120 28.37 1.642 0.1081.75 7.85 9.84 432.5 6.87 –2.3 0.908 0.828 3.75 77.9 78.3 19.44 97.4 3.16 75.43 568.93 7.542 0.047 0.048 7.72 5.33 0.131 28.07 1.608 0.1081.85 7.85 9.43 415.9 8.72 –2.0 0.911 0.791 4.84 116.9 116.1 20.41 137.3 2.82 76.76 507.98 6.618 0.082 0.083 11.37 6.45 0.221 40.30 2.315 0.089

Avg 7.85 9.77 428.0 8.93 –4.0 0.905 0.839 4.97 63.1 64.7 22.43 85.5 4.65 69.13 835.94 12.270 0.047 00.047 7.35 4.90 0.080 28.15 1.711 0.112

C-W = Cox and Weeks (1983) equations F-G = Frankenstein and Garner (1967) equations σb, σDC and σi = S/m

* In the electromagnetic equations (Kovacs and Morey 1987), parameters n and m were 0.1 and 1.65, respectively.

Table 1. Physical and electromagnetic properties of 1.23-m-thick sea ice at 100 MHz.*

Core Core Core Melt Ice Bulk Ice Ice Brine vol Air Total Brine Brine Loss Ice conductivity Ice Loss Phase

Depth diam. length weight cond. temp den. den. sal. C-W F-G vol porosity cond. dielectric const. tang. DC 100 MHz dielectric const. Attn tang. vel.

(m) (cm) (cm) (gr) mS/m (–C°) Mg/m3 Mg/m3 ‰‰ ‰ ‰ ‰ σb  ̇ε b  ̇ ̇ε b φb σDC σi εm  ̇ε m  ̇ ̇ε m (dB/m) φi m/ns

0.05 7.85 9.80 431.3 18.05 –7.6 0.909 0.830 10.63 72.3 74.1 23.26 95.6 6.57 59.14 1181.03 19.970 0.086 0.086 10.78 5.15 0.059 43.14 3.025 0.0910.15 7.85 9.79 432.9 16.23 –7.3 0.914 0.840 9.47 66.9 68.4 17.24 84.2 6.48 59.81 1164.62 19.473 0.075 0.075 9.74 5.04 0.057 39.39 2.679 0.0960.25 7.85 9.83 437.0 15.31 –6.8 0.919 0.846 8.89 67.1 68.3 11.46 78.5 6.30 60.97 1133.42 18.590 0.073 0.073 9.61 5.07 0.059 38.71 2.601 0.0970.35 7.85 9.73 432.0 13.43 –6.4 0.917 0.851 7.71 61.2 62.5 11.30 72.5 6.14 61.95 1104.79 17.834 0.061 0.061 8.50 4.90 0.055 34.50 2.257 0.1030.45 7.85 9.78 431.7 12.93 –5.9 0.912 0.844 7.40 62.7 64.5 16.92 79.6 5.92 63.24 1064.09 16.827 0.061 0.062 8.53 4.92 0.058 34.53 2.254 0.1030.65 7.85 9.81 437.2 11.92 –4.9 0.921 0.848 6.77 68.4 69.8 7.33 75.7 5.36 66.06 964.68 14.603 0.064 0.065 8.92 5.13 0.071 35.40 2.267 0.1000.75 7.85 9.83 436.3 11.63 –4.2 0.917 0.836 6.60 76.5 78.3 11.94 88.4 4.89 68.26 879.21 12.880 0.070 0.071 9.59 5.34 0.086 37.45 2.387 0.0970.85 7.85 9.79 435.5 8.80 –3.4 0.919 0.846 4.88 69.5 70.5 7.71 77.2 4.24 71.05 763.50 10.746 0.052 0.053 7.97 5.15 0.090 30.50 1.838 0.1060.95 7.85 9.78 434.8 9.21 –2.9 0.919 0.830 5.13 85.3 86.0 9.86 95.1 3.78 72.95 680.46 9.327 0.065 0.066 9.36 5.60 0.124 35.22 2.114 0.0981.05 7.85 9.80 436.0 10.60 –2.4 0.919 0.795 5.97 120.3 119.7 12.93 133.2 3.27 75.00 588.46 7.846 0.099 0.100 12.93 6.61 0.202 45.71 2.734 0.0831.15 7.85 9.76 438.2 12.30 –1.9 0.928 0.740 7.01 182.2 177.9 10.20 192.4 2.70 77.22 486.87 6.305 0.163 0.165 19.69 8.49 0.376 60.86 3.499 0.068

Avg 7.85 9.79 434.8 12.76 –4.9 0.918 0.828 7.31 84.8 85.4 12.74 97.5 5.06 66.88 910.10 14.037 0.079 0.080 10.51 5.58 0.112 39.58 2.514 0.095

C-W = Cox and Weeks (1983) equations F-G = Frankenstein and Garner (1967) equations σb, σDC and σi = S/m

* In the electromagnetic equations (Kovacs and Morey 1987), parameters n and m were 0.1 and 1.65, respectively.
100 

80 

60 

40 

20 

0 

νB = 41.64 S0.88 T–0.67

B A 

r

2

 = 0.995

S D = 1.73

νB, Bulk Brine Volume (‰)

TA, Avg. Floe Temp. (|C°|)

SB, Bulk Salinity (‰)

8

6

4

2

4 

0

6 

8 

10 

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

temperature and bulk salinity.

vs. ice-sheet average temperature and bulk salinity. It should be realized that Figure 3 is an end

view of the 3-D data presentation in Figure 4. In

Figure 3, the points with no vertical tails are within

one standard deviation (SD) of the brine volume

surface shown in Figure 4. Points with tails lie

between one and two standard deviations of the

brine volume surface, which is located at the tip

of the vertical line.

TENSILE AND

FLEXURAL STRENGTH

The equation for predicting the bulk brine volume νB in ‰ as a function of SB and the absolute

value of TA (Fig. 4),

 

νB B

0.88

A

–0.67 = 41 64 . S T , (1)

along with the expressions of Kovacs (1996)

for estimating the bulk salinity (‰) of firstyear sea ice vs. ice floe thickness TF (cm)

(Fig. 5),

 

S T B F = + 4 606 91 603 . . /, (2)

offer the opportunity to make other ice property assessments, which in turn can be used to

estimate ice sheet strength.

For example, the horizontal uniaxial tensile

strength σt of an ice sheet could be estimated

using the empirical equation of Dykins (1970):

 

σ ν t B = − 0 816 0 069 0 5 .. . . (3)

Similarly, ice floe flexural strength σf may

be estimated using the empirical equation

(Timco and O’Brien 1994)

 

σ ν f

–5.88 = 1 76 e B . . (4)

These expressions could be used as follows. Ice thickness is determined by drillhole measurement. Then from eq 2 the bulk

salinity of the ice is estimated. Ice temperature is determined next at 20–25 cm below

the ice surface. A linear temperature gradient to the ice/water interface (~1.7°C) is

assumed, and the average ice sheet temperature is calculated. The bulk salinity and

average temperature values are then used

in eq 1 to determine the bulk brine volume.

This value can now be used in eq 3 or 4 to

determine σt or σf, respectively. It should

be noted that Gavrilo et al. (1995) compared various published expressions for determining the flexural strength of sea ice and found

that eq 4 gave σf values 2 to 4 times higher than

other research results. Further evaluation of the

various σf equations may be in order.

The above procedure still requires a drill-hole

thickness and a near-surface ice temperature measurement. These measurements may not be necessary. Using the formulations for determining

the electromagnetic properties of sea ice (Kovacs

and Morey 1987), the low-frequency (DC to ~100

MHz) bulk conductivity* σB of a model sea-ice

Figure 5. Combined Arctic and Antarctic ice floe bulk

salinity vs. thickness (from Kovacs 1996).

20

10

0 50 100 150 200

T , Floe Thickness (cm) F

S , Bulk Salinity (‰) B 30

r = 0.730 2

S = 4.606 + 91.603/T B F

S D = 1.47

* Sea-ice conductivity is approximately constant between DC

and 100 MHz (see Tables 1 and 2).

4
sheet was determined along with its brine

volume νB vs. ice temperature and salinity.

An ice sheet bulk density of 0.911 Mg/m3

was used in the determinations. The calculated results are graphically shown in Figure

6. There is a decrease in brine volume with

decreasing ice temperature. In addition to

ice conductivity, Figure 6 indicates that either ice salinity or temperature is required to

estimate νΒ.

The complex interrelationship between the

model sea-ice temperature, salinity, brine volume, and conductivity is well revealed in

Figure 6. The results suggest that determining σB of natural sea ice remotely will not, of

itself, allow νB to be estimated. Not revealed

is the effect of density, which was fixed at

0.911 Mg/m3 for this example. The decrease

in the bulk density and conductivity with

increasing ice floe thickness as revealed in Figure 2 affects all other ice

properties. It turns out that this effect is favorable to the remote sounding of sea ice. For example, Kovacs

and Morey (1991) and Kovacs et al.

(1996) have shown that a portable

electromagnetic induction sounding

system can be used to measure remotely an apparent bulk conductivity

from which an estimate of winter

Beaufort Sea ice thickness can be

made (Fig. 7). When the νB vs. σB at

DC values listed in Tables 1 and 2

and from 20 other ice floes were compared, it was found that the calculated bulk conductivity of Beaufort

Sea ice was directly related to νB, as

shown in Figure 8. Therefore, in principle, it should be possible to estimate both undeformed winter Arctic

ice floe thickness and bulk brine volume remotely using a conductivity

measurement system. Before the latter is possible, the relation between

the field- measured apparent bulk

conductivity, which is highly affected by the underlying seawater, and

the bulk brine volume of sea ice floes

of various thicknesses will need to

be determined. Conductivity measurement at a frequency well above

50 kHz is likely to be necessary to

separate the sea-ice conductivity conSalinity = 7‰

4‰

70

60

50

40

30

20

10

0.10

0.08

0.06

0.04

0.02

0

– 20 – 16 – 12 – 8 – 4

T, Temperature (°C)

ν , Brine Volume (‰) B

σ , Bulk DC Conductivity (S/m) B

70

60

50

40

30

20

0

0.08 0.06 0.04 0.02

– 20 – 16 – 12 – 8 – 4

T, Temperature (°C)

4‰

Salinity = 7‰

S D = 0.98

r = 0.996 2

ν = 9.02 +217.09/T + 528.48σ 2 B B

ν , Brine Volume (‰) B

σ , Bulk DC Conductivity (S/m)

B

Figure 6. 3-D presentation of sea-ice brine volume vs. DC conductivity and temperature with salinity as a parameter for an ice density of

0.911 Mg/m3 (a) and a boresight view from the above temperature

window (b).

Figure 7. EM-31 instrument conductivity reading at ~10 kHz

vs. ice floe snow plus ice thickness (from Kovacs et al. 1996).

0.8

0.6

0.4

0.2

0 24 68

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

Instrument Conductivity Reading (S/m)

5
tribution from the seawater conductivity contribution. A dual high and low frequency

system or a wideband measurement system

could be used.

In the interim, an off-the-shelf electromagnetic induction system operating at 10 kHz

can be used to estimate ice thickness (Fig. 8),

and the bulk salinity of the ice is estimated

using eq 2. After determining the average ice

sheet temperature, as previously described,

eq 1 can be used to calculate the bulk brine

volume. Ice strength may then be calculated

from eq 3 or 4.

The relation between sea-ice tensile or flexural strength, brine volume, and the calculated bulk DC conductivity of natural sea ice

is shown in Figures 9 and 10. These figures

are the result of using the equation in Figure

6 and eq 3 and 4, respectively. In the future,

using similar equations, an algorithm

could be added to a portable electromagnetic induction sounding system

that would provide both ice thickness

and strength from the apparent sea-ice

conductivity measurements.

COMPRESSIVE STRENGTH

Estimating the compressive strength

of sea ice is different. While the uniaxial

tensile or flexural strength of sea ice

does not reveal a significant strain rate

dependence, at the strain rates of interest for most ice engineering problems,

the compressive strength does. In addition, in reporting sea-ice compression

test results, the recent, and correct,

method is to relate strength to total porosity: brine volume plus gas volume.

Recently, Timco and Frederking (1990)

developed empirical equations for predicting the large-scale uniaxial unconfined compressive strength of Arctic sea

ice. Their equations are based on smallscale strength tests and the various controlling factors, such as ice structure,

brine volume, porosity, loading direction (horizontal–vertical), and strain

rate. Their strength equation for horizontally loaded columnar sea ice is

 

σε φ c B = − 37 1 270 [ ] ( ) 0 22 0 5  ̇ / . . , (5)

80

60

40

20

0 0.02 0.04 0.06 0.08

ν , Bulk Brine Volume (‰)

B

σ , Bulk DC Conductivity (S/m) B

0.10

100

S D = 3.169

r = 0.984 2

ν = 584 B σ 

0.74

B

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

calculated bulk DC conductivity of the ice.

Figure 10. Ice sheet flexural strength vs. bulk DC conductivity and brine volume.

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

conductivity and brine volume.

σB, Bulk DC Conductivity (S/m) 0

 25

 50

 75

 100

νB, Bulk Brine Volume (‰)

0.10 

0.08 

0.06 

0.04 

0.02 

σt, Horizontal Tensile


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

σB, Bulk DC Conductivity (S/m) νB, Bulk Brine Volume (‰)

σf, Flexural Strength (MPa)

1.0 

0.8 

0.6 

0.4 

0.2 

0 

 1.0 

 0.8 

 0.6 

 0.4 

 0.2 

 0 

6
where σc = horizontal uniaxial unconfined compressive strength (MPa)

 

ε ̇ = strain rate, s–1

φB = bulk ice porosity (‰).

Timco and Frederking compared σc values determined from eq 5 with in-situ, full ice sheet

thickness, horizontal, uniaxial unconfined compression tests performed by Exxon on Beaufort

Sea ice. The Exxon large-scale test data comprised

ice thickness, average ice temperature, loading

strain rate, and the measured strength (Table 3).

Since the bulk porosity of the ice sheets tested

was not provided, the authors had to make these

estimates. To do this, they first included additional data in Figure 11 and, following the same

approach as Cox and Weeks (1974), ran two linear

regression curves through the data. Only slightly

different 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 Frederking

determined SB for each ice sheet tested (Table 3).

Then the bulk density of the sea ice was selected

to be 0.907 Mg/m3. With these determinations

and the average ice floe temperature given in

Table 3, the expressions provided by Cox and

Weeks (1983) were used to calculate the bulk porosity of each ice sheet. Note that these values

were not provided by Timco and Frederking in

their paper. The missing bulk porosity values were

recalculated along with the bulk brine volume for

each ice sheet. The values are listed in the left

column of the brine volume and porosity data

columns in Table 3. The σc strengths measured by

Exxon and calculated by Timco and Frederking

are compared in Table 3 and shown graphically in

Figure 12. The regression curve in Figure 13 shows

that the calculated values are generally higher

than the measured ones. This offset may be due to

small- to large-scale scaling ambiguities.

To bring eq 5 into better agreement with the

Exxon test results, the bulk salinity and brine volume of each ice sheet were determined using eq 2

and 1, respectively. Next, the SB, TA, and φB data

from Tables 1 and 2 and others referenced were

plotted as shown in Figure 13a. A boresight view

of the data as seen looking into the salinity “window” 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 Calculated

cm –°C‰ ‰ ‰ s–1 MPa MPa

156 10.6 5.501 5.202 38.19 36.533 52.87 51.184 7.90e–7 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 1.10e–6 0.49 1.03 0.46 0.57 0.60

120 12.9 6.08 5.37 34.18 32.94 49.20 47.97 1.27e–6 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 9.20e–6 1.00 1.62 1.12 1.10 1.17

152 10.5 5.57 5.21 38.50 36.82 53.16 51.94 1.42e–5 1.30 1.81 1.30 1.26 1.35

166 4.2 5.34 5.16 70.67 67.46 82.66 79.20 9.00e–5 1.82 2.15 1.86 1.79 1.60

147 5.9 5.65 5.23 56.83 54.36 69.96 67.28 9.10e–5 1.86 2.36 2.04 1.95 1.88

154 4.3 5.54 5.20 69.91 66.86 81.96 78.65 1.33e–4 2.10 2.33 2.09 2.03 1.83

146 11.8 5.66 5.24 35.72 34.23 50.61 49.12 1.56e–4 2.86 3.11 2.70 2.71 3.00

130 5.1 5.92 5.32 63.26 60.85 75.86 73.18 2.16e–4 1.88 2.66 2.50 2.45 2.28

161 7.3 5.42 5.18 48.89 46.74 62.67 60.37 3.40e–4 3.53 3.41 3.08 3.08 3.17

129 3.4 5.94 5.32 83.08 79.84 93.96 90.54 7.50e–4 2.62 2.86 3.13 3.24 2.77

ˇ ˇˇ ˆ ˇ ˇˇ ˆ

ˇ ˇˇ ˆ

ˇ ˇˇ ˇ ˆ

a = EXXON Test Data 5 = Timco & Frederking, eq 5

1 = Timco & Frederking eq 6 or 7 6 = Timco & Frederking, revised, eq 8

2 = Eq 2 7 = Figure 18’s equation

3 = Eq 1 8 = Eq 10

4 = Figure 13’s equation

7
4

3

2

1

0 1234

σC ( Ex) , Exxon’s Horizontal Unconfined Compressive Strength (MPa)

r = 0.942 2

σ = 0.621 + 0.855 C (T-F) σC (Ex)

σ , Timco-Frederking Horizontal

Unconfined Compressive Strength (MPa)

C (T-F)

and Frederking. However, the modified method

for determining σc gives values that are still offset

from the Exxon field test results (Fig. 15). To alleviate this offset, eq 5 was revised to read

 

σε φ c B

0.93 = − ( ) 



 



 37 1 270 0 65 − 0 22 0 5  ̇ / .. . . (8)

This revision brings the σc values (Table 3) into

better agreement with the field determinations,

as shown in Figure 16.

If the DC conductivity of the ice could be determined remotely along with ice sheet thickness,

then SB could be determined from eq 2 and νB

estimated from an expression similar to that given

in Figure 6 and φB from Figure 17. The data in

reveals the interrelationship of average ice sheet

temperature and bulk salinity with the bulk porosity.

Using the SB values determined from eq 2, the

average ice floe temperature listed in Table 3, and

the equation provided in Figure 13, φB was estimated for each of the Exxon ice sheets. These νB,

SB, and φB values are listed in Table 3 as the right

column under each respective heading. Using the

new φB values and the related strain rates in Table

3, σc was recalculated using eq 5. These so-called

modified σc values were then compared (Fig. 14)

with the σc estimates made by Timco and

Frederking (Table 3). As can be seen in Figure 14,

using the above φB estimates in eq 5 gives σc

values that are comparable with those of Timco

Figure 12. Timco and Frederking estimated full ice sheet horizontal unconfined compressive strength vs. Exxon’s

field-measured values.

kov 5/4

16

12

8

4

0 100 200 300 400

S , Bulk Salinity (‰) B

T , Floe Thickness (cm) F

First-Year Ice

Multiyear Ice S = 14.237 – 0.194T B F

S = 7.879 – 0.016T B F

Figure 11. Sea-ice bulk salinity vs.

floe thickness (after Cox and Weeks

1974).

8
alyzed using a two-parameter power function.

The relationship is a further development of the

Norton (1929) secondary creep rate equation containing a power function of stress:

 

ε σ  ̇ = B n,

where B is an empirical parameter that takes into

account ice structure, temperature, activation energy, and other factors, σ is the applied stress, and

the empirical exponent n is generally found to be

about 3 for ice. Rearranging for σ, the above equation becomes

 

σ ε = B n 1

1 ̇ / ,

where B1 = B–1/n. The former expression was used

by Glen (1955, 1958) to evaluate the creep behavior of solid, not porous, freshwater ice. Since the

ice porosity has a profound effect on σ, the above

equation must be modified to include a porosity

parameter φ as follows:

 

σ εφ = B n m 2

1 ̇ / , (9)

where B2 = B1 φ–m and m is an empirical exponent.

Equation 9 was evaluated by substituting the

σc values obtained from eq 8 for σ in eq 9 and

using the related controlling parameters φB and ε ̇

given in Table 3. The result is shown in Figure 18.

Equation 9 is shown to statistically fit the data

extremely well with an r2 value of 0.993.

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

better determined. Even though the failure surface fits the data with a high r2 value of 0.993, the

data suggest that this surface is not a straight line

as viewed from the strain rate window in Figure

σ , Timco-Frederking

Horizontal Unconfined Compressive Strength (MPa)

C (T-F)

3

2

1

0 123

σ , Modified Horizontal

Unconfined Compressive Strength (MPa)

C (M)

4

4

r = 0.989 2

σ = – 0.016 + 1.007 C (M) σC (T-F)

S D = 0.09

Figure 13. Ice floe bulk porosity vs. average temperature

and bulk salinity (a) and boresight view from the above

bulk salinity window (b).

Figure 17 are from Tables 1 and 2 and the sources

cited above. This figure indicates that nondeformed Beaufort Sea ice floes have a gas volume

between about 1 and 2%.

The σc values calculated using eq 8 in relation

to the controlling parameters 

ε ̇ and φB can be anFigure 14. Modified Timco–Frederking ice

floe σc strengths vs. their original values.

120

100

80

60

40

20

–2 °C

–11 °C

0 24 6 8

SB, Bulk Salinity (‰)

b

φ B, Bulk Porosity (‰)

a

120

80

60

40

20

0

2

4

6

8

2

4

6

10

8

B A

r2 = 0.995

S D = 1.57

TA , Avg. Floe Temp. (|C° |)

S B, Bulk Salinity (‰)

φ B = 19.37 + 36.18 S0.91 T–0.69

φ B, Bulk Porosity (‰)100

9
4

3

2

1

0 1 2 34

σ C (Ex) , Exxon’s Horizontal Unconfined Compressive Strength (MPa)

σ , Modified Horizontal

Unconfined Compressive Strength (MPa)

C (M)

2 r = 0.921

σ C(M) = 0.615 + 0.862 σ C (Ex)

S D = 0.25

120

80

40

0 20 40 60 80 100

ν , Bulk Brine Volume (‰) B

φ , Bulk Porosity (‰) B

φ = 17.852 + 0.917ν B B

r = 0.983 2

S D = 2.44

4

3

2

1

0 1234

σ , Revised Horizontal

Unconfined Compressive Strength (MPa)

C(R)

σ C (Ex) , Exxon’s Horizontal Unconfined Compressive Strength (MPa)

r = 0.898 2

σ C(R) = 0.008 + 1.007σC (Ex)

S D = 0.25

Figure 15. Modified Timco–Frederking ice

floe σc strengths vs. Exxon’s field-measured

σc values.

Figure 16. Ice floe σc strengths calculated

using eq 8 vs. Exxon’s field-measured σc

values.

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

brine volume.

10
18b. The three data points at a strain rate of

about 10–6 are shown to be offset from their

predicted failure surface position at the tip of

their vertical tails, suggesting that the failure

surface curves downward. This is not surprising if B contains a nonlinear viscosity component. In this case, while stress is still a function of strain rate, it is not directly proportional

(Mellor 1986), at least at strain rates between

about 10–5 and 10–3 as indicated by the stress

vs. strain rate curves for fresh and saline ice

given in Sanderson (1988). The Exxon data

fall in the ductile-to-brittle strain rate transition region where the creep power-law may

not be valid, that is, ε σ  ̇ ≠ B n because n is no

longer a constant.

A statistically more representative failure

surface for the data is shown in Figure 19a.

This curved surface fits the data with a correlation coefficient of r2 = 1.000. A boresight

view of the data from the strain rate window

in Figure 19a is shown in Figure 19b. Based on

the data presented by Sanderson (1988), below a strain rate of about 10–6 the failure surface

should be linear, a ≈ 3. However, the existing data

are insufficient for developing this trend. In the

interim, for strain rates between 10–4 and 10–3, the

equation given in Figure 18 is preferred because it

does not contain a second constant. The equation

for the failure surface when νB is substituted for

φB in Figure 18 is σ εν c B = 150 75 0 31 0 36 .  ̇ . – . . In this

expression, νB implicitly accounts for the bulk gas

volume or the porosity. In using either σc expression, the only parameters needed are the loading strain rate and the bulk ice porosity or brine

volume. The latter may one day be estimated from

remote conductivity measurements along with ice

thickness using a helicopter-borne electromagnetic

induction sounding system (Kovacs and Holladay

1990).

A comparison of the σc data derived from using eq 8 with the horizontal uniaxial unconfined

compressive strength data of others should be of

interest. The data compiled by Sanderson (1988;

see his Fig. 4.8) are shown in Figure 20 for the

strain rate range of our data. In this standard

100

φ = 20‰ B

101

100

10– 1

10– 8 10– 7 10– 6 10– 5 10– 4 10– 3

b

ε, Strain Rate (s ) · – 1

σ , Horizontal

Compressive Strength (MPa)

C

80

60

40

10– 8

101

0 10

– 1 10

– 6 10

– 4 10

σ , Horizontal

Compressive Strength (MPa) C

ε, Strain Rate (s ) – 1

20

·

r = 1.000 2

S D = 0.02

σ = – 0.73 + 87.44ε φB

0.21 – 0.35

C

·

a

φ , Bulk Porosity (‰) B

.

ε, Strain Rate (s –1

)

10

1

0.1

80

60

40

20 10–8

10–6

10–4

σC, Horizontal

Compressive Strength (MPa)

φ B, Bulk Porosity (‰)

a

10

1

0.1

σC, Horizontal

Compressive Strength (MPa)

ε , Strain Rate (s –1) . 10–8 10–6 10–4 10–7 10–5 10–3

φ B = 20

100

b

r

2

 = 0.993

S D = 0.09

σ C = 295.22 ε 0.91 φ B

–0.5 .

Figure 19. Equation-8-determined ice floe σc strengths vs.

bulk porosity and strain rate (a) and the boresight view of

the above from the strain rate window (b).

Figure 18. Equation-8-determined ice floe σc strengths

vs. bulk porosity and strain rate (a) and the boresight

view from the above strain rate window (b).

11
presentation, no corrections have been made to

the data to account for differences in test technique, ice porosity, grain size, or crystal orientation. This is not unusual since some of these parameters were not measured or provided in the

original source material, as is the case for the

Exxon data used in this report. For this reason

there is considerable scatter in the data. The upper regression curve shown in Figure 20 fits the

data from Sanderson; the lower curve represents

the data generated from eq 8. The small amount

of data from this study coupled with the unknowns mentioned above does not allow a definitive statement to be made on the difference between the two lines. Clearly, the data from this

study fit well within the scatter of the data from

Sanderson. Both sets of data are statistically better represented by the nonlinear curves shown in

Figure 21. Here again, the failure surface appears

to be a non-power-law function of the strain rate.

The Timco and Frederking (1990) study, which

resulted in the formulation of eq 6, is based on

small-scale test results. How applicable this equation is to full-scale ice force assessments can be

questioned on the basis of small- to large-scale

scaling uncertainties. To avoid this problem, the

full-scale Exxon σc data were directly analyzed

using the values listed in the right-hand bulk porosity column in Table 3. Applying eq 9 to analyze

the σc and ε ̇ data of Exxon vs. the above-related

φB values (Fig. 22a) gives the following expression for the failure surface:

Figure 21. Ice floe σc values calculated using eq 8 from

this study compared with the data provided by Sanderson

(1988). The data were fitted with non-log-log power

equations.

10 1

10 – 7

σ , Horizontal Compressive Strength (MPa) C

·

ε, Strain Rate (s ) – 1

– 6 10 – 5 10 – 4 10 – 3 10

0 10

– 1 10

2 r = 0.280

S D = 1.43

σ = 8.76 – 2.98ε – 0.07

C

·

S D = 0.18

r = 0.972 2

· σ = – 1.43 + 12.93ε 0.14

C

Figure 20. Ice floe σc values calculated using eq 8 from

the study compared with the data provided by

Sanderson (1988). The data were fitted with a singleterm power equation.

101

10– 7

σ , Horizontal Compressive Strength (MPa) C

·

ε, Strain Rate (s ) – 1

– 6 10 – 5 10 – 4 10 – 3 10

0 10

– 1 10

r = 0.958 2

S D = 0.22

σ = 23.02 ε

· 0.263 C

r = 0.261 2

S D = 1.44

· C 0.177 σ = 14.07ε

Sanderson (1988)

This Study

 

σ εφ c B = − 1970 0 32 0 95  ̇ . . . (10)

In the boresight view of the data from the strain

rate window in Figure 22b, the “outlier” with

the long offset tail at a strain rate of 7.9 × 10–7s–1

is very apparent. However, removing this data

point and reanalyzing the remaining data gives

a new equation whose calculated σc values do

not vary by more than 0.04 MPa from those calculated by eq 10.

The form of the expression for the curved failure surface shown in Figure 17 does not fit the

new data set as well as the power type equation

in Figure 22. This may be seen by comparing the

statistics listed in Figure 22 with those in Figure

23. The boresight view of the failure plane from

the strain rate window (Fig. 23b) also indicates no

fit improvement for the outlier over the failure

surface shown in Figure 22b.

Substituting the new νB data for the φB data

and reanalyzing as before gives the results shown

in Figure 24. The expression for the failure surface

is now

 

σ εν c B = − 523 24 0 32 0 67 .  ̇ . . . (11)

As with the bulk porosity data, applying a curved

failure surface through the σc vs. bulk brine volume and 

ε ̇ data does not result in a statistically

better curve fit.

It is interesting that the σc vs. ε ̇ trend, as obtained with eq 8 and from Sanderson’s data, ap12
r = 0.941 2

S D = 0.25

σ B = 1970ε φ 0.32 – 0.95

C

·

90

80

70

60

50

40

10– 8

101

0 10

– 1 10 – 6 10– 4 10

φ B , Bulk Porosity (‰)

σ , Horizontal

Compressive Strength (MPa) C

ε, Strain Rate (s ) – 1 ·

a

101

b

100

φ = 40‰ B

σ , Horizontal

Compressive Strength (MPa) C

0 10

– 1 10 – 8 10 – 7 10 – 6 10 – 5 10 – 4 10 – 3 10

ε, Strain Rate (s ) · – 1

Figure 22. Exxon ice floe σc values vs. bulk porosity and

strain rate (a) and the boresight view of the above from the

strain rate window (b). Data were fitted with a two-term

power function, i.e., eq 9.

10– 8

10– 6 10– 4

80

60

40

φ , Bulk Porosity (‰)

B

101

100

10– 1

σ , Horizontal

Compressive Strength (MPa)

C

ε, Strain Rate (s ) – 1 ·

r = 0.929 2

S D = 0.29

σ = – 0.58 + 271.20ε φB

· 0.22 – 0.63

C

a

100

φ = 40‰ B

101

100

10– 1

10– 8 10– 7 10– 6 10– 5 10– 4

ε, Strain Rate (s ) · – 1

10– 3

σ , Horizontal

Compressive Strength (MPa)

C

b

Figure 23. Ice floe σc values vs. bulk porosity and strain

rate (a) and the boresight view of the above from the strain

rate window (b). Data were fitted with a non-log-log

power equation.

60

40

σ = 523.24ε ν · C B

0.32 – 0.67

r = 0.938 2

S D = 0.26

a

σ , Horizontal 

Compressive Strength (MPa) C

10 – 6 10– 4

ν , 

B Bulk Brine Volume (‰)

101

100

10– 1

20 10– 8

ε, Strain Rate (s ) · – 1

100

ν = 20‰ B

b

10– 8 10 – 7 10– 6 10– 5 10– 4 10– 3

ε, Strain Rate (s ) · – 1

σ , Horizontal

Compressive Strength (MPa)

C

101

100

10– 1

Figure 24. Ice floe σc values vs. bulk brine volume and

strain rate (a) and the boresight view of the above from the

strain rate window (b).

13
pears to favor a curved failure surface (Fig. 17

and 19), while the trend resulting from the analysis of the Exxon σc vs. 

ε ̇ data does not show a

strong preference. Additional full ice sheet tests

should help to further define the failure surface,

especially outside the area of the current data.

Within the range of the current data, 10–6s–1 < ε ̇ <

10–3s–1 and ~25‰ < νB < 80‰, eq 11 well represents the σc strength of sea ice as determined

from full-scale in-situ tests. This is equally true

for eq 10 and its counterpart in Figure 23a, in

which φB is a parameter. Nevertheless, eq 10 is

preferred because of its simpler form; it does not

contain a secondary constant. In addition, outside the range of this field data, the latter equation predicts an unrealistic zero ice strength between strain rates of about 10–7 and 10–8s–1.

The relationship of bulk porosity and strain

rate to sea ice strength is clearly revealed in Figure 25. The effect of φB on σc is shown to be significantly greater at the higher strain rates. As would

be expected, the same trend is maintained when

νB is substituted for φB (Fig. 26).

If one assumes that n and m are 3 and –1 respectively, then eq 10 can be simplified to

 

σ εφ c B = × − 2 7 103 13 1 .  ̇ . / (12)

σc values calculated from eqs 10 and 12 were

compared and found to cross-correlate with an r2

= 1.000 and a standard deviation of 0.012 over the

range of the Exxon full-scale test data.

By rearranging eq 9, one obtains an equation

for the secondary creep of sea ice:

 

ε σφ  ̇ = B n w 3 , (13)

where B B n 3 2 = − and w = –mn.

To take into account the creep process of ice

with time, eq 13 can be modified to describe primary creep as follows:

 

ε σφ  ̇ λ = B t n w 4 B , (14)

where parameter B4 will be (and the magnitude of

exponents n and m may be) different from those

in eq 13, and λ is an exponent.

For a more complete description of the creep

process of ice under a complex stress–strain state,

the reader is referred to the recent work of Fish

(1993).

ICE–STRUCTURE

INTERACTION FORCE

As is well recognized by engineers engaged in

estimating sea-ice forces on offshore structures,

many variables enter into the calculations. In the

following, an estimate is made of the horizontal

force that could develop between a cold first-year

sea-ice sheet and an offshore structure in the southern Beaufort Sea. In this scenario only the structure width (90 m) and the degree-days of seawater freezing since a stable ice cover formed (5000)

are known. From the work of Anderson (1961),

Wen et al. (1991), and R. Lewellen (pers. comm.),

the relationship between Arctic ice floe thickness

and the cumulative freezing degree-days since a

stable ice cover formed was determined (Fig. 27).

This curve in Figure 27 is based on sea-ice field

measurements with a natural snow cover. From

the equation for the curve in Figure 27, the expected ice thickness for 5000 freezing degree-days

is 176 cm, and from eq 2 the bulk salinity of the ice

5

4

3

2

1

0

100 80 60 40

φ , Bulk Porosity (‰) B

σ , Horizontal

Compressive Strength (MPa) C

10– 7

– 3 ε = 10 ·

Figure 26. Boresight view from the bulk brine volume

window in Figure 24a.

5

4

3

2

1

0

80 60 40 20

ν , Bulk Brine Volume (‰) B

σ , Horizontal

Compressive Strength (MPa) C

10– 7

– 3 ε = 10 ·

Figure 25. Boresight view from the bulk porosity window in Figure 22a.

14
sheet is estimated to be 5.15‰. We assume the ice

sheet has an average temperature of

–10°C. This is a reasonable value for cold, latewinter first-year sea ice along the Alaska Beaufort

Sea coast. The bulk porosity is found to be 52.20‰

from the equation, φB B a = + − 19 37 36 18 0 91 0 69 . . . . S T ,

given in Figure 13a.

The peak failure stress in sea ice occurs at a

strain rate of about 10–3s–1 for small-scale ice

samples with an aspect ratio less than 2. As the

sample size or the volume of ice under load increases, the peak stress occurs at a lower strain

rate (Sanderson 1988, Gavrilo 1995). At full-scale

ice structure interaction when the aspect ratio is

greater than 50, the peak stress appears to occur

at an effective strain rate 

ε ̇ e around 10–5 s–1. Since

the structure width d is very much greater than

TF, plane stress conditions apply and the apparent ice sheet velocity V at this strain rate is

(Sanderson 1988)

V = 

ε ̇ . e 0 44d (15)

or 1.4 m/h. Using 52.2‰ for φB and 10–5 s–1 for 

ε ̇

in eq 5 gives 1.6 MPa for the ice sheet peak horizontal uniaxial unconfined compressive strength.

For plane stress conditions, the indentation force

F between a vertical structure and the level ice

sheet may be estimated using the reference stress

expression (Sanderson 1988)

F = 1.15 dTF σc. (16)

For d = 90 m, TF = 1.76 m and σc = 1.6 MPa as

given above, the global ice force against the structure is estimated to be 291 MN or ≈ 3.24

MN/m. Since most estimates for the integrated limiting pack ice force lie between 104 and 105 N/m (Parmerter and

Coon 1973, Rothrock 1975, Hibler 1980,

Nevel 1983 and Croasdale 1984), the

pack ice driving force would seem to be

the design limiting-force load. However,

since the pack ice driving force can be

distributed across a floe that is much

wider than the structure, the concentrated force at the structure can easily

exceed the stress required to fail the ice.

Therefore, for this model and no other

factors being considered, the design

force would be the calculated value.

The constants given in eq 15 and 16

are best estimates and may need to be

revised (Sanderson 1988). The estimated

force of 291 MN is considered an upper limit for

the conditions given. Indeed, for a similar ice sheet

thickness and structure width, the ice force developed against the structure due to nonsimultaneous

ice crushing failure appears to be less than half

the above calculated indentation force (Wright

and Timco 1994).

SUMMARY

A constitutive relationship, in the form

 

σ εφ c B = B ln m

2  ̇ / , was developed for predicting the

unconfined compressive strength of first-year seaice floes as a function of only two parameters, the

applied strain rate and the ice floe bulk porosity.

This equation was evaluated using full sea-ice

sheet thickness, horizontal unconfined compression test data, so no uncertain small- to largescale scaling laws were involved. The magnitudes

of the parameters in the equation were found to

be σc = 2.7 × 10–3 

 ̇ / ε1 3 φ–1. Using this relationship,

an example is given for estimating the design

sea-ice crushing force against a 90-m-wide offshore structure.

Relationships are also provided for determining ice sheet tensile and flexural strength. The

relation between sea-ice bulk conductivity and

strength was explored. First-year sea-ice thickness can currently be measured using surface or

airborne systems. It is predicted that one day ice

sheet conductivity will also be remotely measured

and that this will lead to the capability of remotely estimating sea-ice strength.

Two widely used methods for calculating the

Figure 27. Sea-ice thickness vs. seawater freezing degree-days after

a stable ice cover has formed.

200

160

120

80

40

0 1000 2000 3000 4000 5000

θ, Cumulative -°C Freezing Degree-days After Ice Formation 

T , Floe Thickness (cm) F

TF = 2 θ 0.505

θ = ∫ (T – T ) dt

t = Time (days)

T = Seawater Freezing Temp.

T = Mean Daily Air Temp. < T 

t

o w a

w

a w

15
brine volume of sea ice were discussed. Both methods were found to provide similar results, provided due diligence is exercised when making

the required supporting measurements.

LITERATURE CITED

Anderson, D.L. (1961) Growth rate of sea ice.

Journal of Glaciology, 3(30): 1170–1172.

Cox, G.F.N. and W.F. Weeks (1974) Salinity variations in sea ice. Journal of Glaciology, 13(67): 109–

120.

Cox, G.F.N. and W.F. Weeks (1983) Equations for

determining the gas and brine volumes in sea-ice

samples. Journal of Glaciology, 29(102): 306–316.

Croasdale, K.R. (1984) The limiting driving force

approach to ice loads. In Proceedings of the 16th

Offshore Technology Conference, 7–9 May, Houston,

Texas, vol. 2, p. 57–64.

Dykins, J.E. (1970) Ice engineering: Tensile properties of saline ice grown in a confined system.

NCEL Technical Report R-720. Port Hueneme,

California: U.S. Naval Civil Engineering Laboratory.

Fish, A.M. (1993) Combined creep and yield model

of ice under multiaxial stress. Journal of Offshore

and Polar Engineering, 3(2): 130–138.

Frankenstein, G. and R. Garner (1967) Equations

for determining the brine volume of sea ice from

–0.5° to –22.9°C. Journal of Glaciology, 6(48): 943–

944.

Gavrilo, V.P., S.M. Kovalev, G.A. Lebedev and

O.A. Nedoshivin (1995) A method for estimating

sea ice flexural strength based on hydrometeorological data. In Proceedings of the 5th International

Offshore and Polar Engineering Conference, The

Hague, The Netherlands (J.S. Chang, M. Sayed and

A.M. Gresnigt, Eds.), vol. 2, p. 521–528. Golden,

Colorado: International Society of Offshore and

Polar Engineers (ISOPE).

Glen, J.W. (1955) The creep of polycrystalline

ice. Proceedings of the Royal Society, (London),

228A(1175): 519–538.

Glen, J.W. (1958) The flow law of ice. In Proceedings of the Chamonix Symposium, 16–24 June, September. IASH Publication 47, p. 171–183. Gentbrugge, Belgium.

Hibler, W.D. III (1980) Modeling a variable thickness sea ice cover. Monthly Weather Review, 108(12):

1943–1973.

Kovacs, A. (1993) Axial double-ball test versus

the uniaxial unconfined compression test for measuring the compressive strength of freshwater and

sea ice. USA Cold Regions Research and Engineering Laboratory, CRREL Report 93-25.

Kovacs, A. (1996) Sea ice: Part I: Bulk salinity vs.

ice floe thickness. USA Cold Regions Research

and Engineering Laboratory, CRREL Report 96-7.

Kovacs, A., D. Diemand and J.J. Bayer, Jr. (1996)

Electromagnetic induction sounding of sea ice

thickness. USA Cold Regions Research and Engineering Laboratory, CRREL Report 96-6.

Kovacs, A. and J.S. Holladay (1990) Sea ice thickness measurement using a small airborne electromagnetic sounding system. Geophysics, 55(10):

1327–1337

Kovacs, A. and R.M. Morey (1987) Electromagnetic measurements of a second-year sea ice floe.

In Proceedings of the 9th Conference on Port and

Ocean Engineering Under Arctic Conditions (POAC–

87), 17–21 August, University of Alaska, Fairbanks

(W.M Sackinger and M.O. Jeffries, Eds.), vol. 1.

Fairbanks: Geophysical Institute, p. 121–136.

Kovacs, A. and R.M. Morey (1991) Sounding of

sea ice thickness using a portable electromagnetic

induction instrument. Geophysics, 54(12): 1992–

1998.

Kovacs, A., R.M. Morey and G.F.N. Cox (1987)

Modeling the electromagnetic property trends in

sea ice, Part I. Cold Regions Science and Technology,

14(3): 207–235.

Kovacs, A., W.F. Weeks and F. Michitti (1969)

Variation of some mechanical properties of polar

snow, Camp Century, Greenland. USA Cold Regions Research and Engineering Laboratory, Research Report 276.

Mellor, M. (1986) Mechanical behavior of sea ice.

In Geophysics of Sea Ice (N. Untersteiner, Ed.). New

York: Plenum Press, p. 275–281.

Nevel, D.E. (1983) Pressure ridge forces. In Proceedings of the 7th International Conference on Port

and Ocean Engineering Under Arctic Conditions,

Espoo, Finland. Technical Research Center of Finland, vol. 1, p. 212–220.

Norton, H.F. (1929) Creep of Steel at High Temperatures. New York: McGraw-Hill.

Parmerter, R.R. and M.D. Coon (1973) Model of

pressure ridge formation in sea ice. Journal of Geophysical Research, 77(13): 6565–6575.

Rothrock, D.A. (1975) The energetics of the plastic deformation of pack ice by ridging. Journal of

Geophysical Research, 80(33): 4514–4519.

Sanderson, T.J.O. (1988) Ice Mechanics: Risks to

Offshore Structures. London: Graham and Trotman.

16
Timco, G.W. and R.M.W. Frederking (1990) Compressive strength of sea ice sheets. Cold Regions

Science and Technology, 17(3): 227–240.

Timco, G.W. and S. O’Brien (1994) Flexural

strength equation for sea ice. Cold Regions Science

and Technology, 22(3): 285–298.

Wen, T., G.R. Garrison, R.E. Francois, R.P. Stein

and W.J. Felton (1991) Sound speed, reflectivity,

and absorption measurements in Arctic ice in 1988.

Applied Physics Laboratory, University of Washington, Technical Report APL-UW-TR-9005.

Wright, B.D. and G.W. Timco (1994) A review of

ice forces and failure modes on the Molikpaq. In

Proceedings of the 12th IAHR Symposium on Ice, 23–

26 August, Trondheim, Norway. International Association of Hydraulic Research, vol. 2, p. 816–825.

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September 1996

Sea Ice: Part II. Estimating the Full-Scale Tensile, Flexural, and

Compressive Strength of First-Year Ice

Austin Kovacs

U.S. Army Cold Regions Research and Engineering Laboratory

72 Lyme Road CRREL Report 96-11

Hanover, New Hampshire 03755-1290

U.S. Army Cold Regions Research and Engineering Laboratory

72 Lyme Road

Hanover, New Hampshire 03755-1290

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Available from NTIS, Springfield, Virginia 22161

26

Porosity Salinity Sea ice Strain rate

UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL

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. Methods for estimating full-thickness ice sheet strength, based on

large-scale field tests, are presented. The relationships 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 σc of first-year sea ice as a function of the applied strain rate and bulk porosity in the form

 

σ εφ c B = B l n m 2  ̇ / , where parameters B2, n, and m are about 2.7 × 103, 3, and –1, respectively, and 

ε ̇ and φB are

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.