CRREL REPORT 97-6

Ice Strength as a

Function of Hydrostatic

Pressure and Temperature

Anatoly M. Fish and Yuri K. Zaretsky October 1997

* *

τi = J2

τ max 1

τ max 2

τ max 3

φ 1 φ 2

φ 3

c1

c3

c2

– p p3 p2 p1 p

0 σmax 3 σmax 2 σmax 1

*

M

τ i = (c + bp) – (c + bp*) p

p* ––

 2

T1

T2

T3

b1

τi, Shear Strength

p = , Hydrostatic Pressure I1

3
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Cover: Strength envelopes of ice at various temperatures: c, φ, p*, and T are the ice cohesion, friction angle, ice

melting pressure, and absolute temperature, respectively.

Abstract: A temperature model has been developed

that describes the ice strength in a multiaxial stress

state over a wide spectrum of negative temperatures. The model takes into account the anomalous

behavior of ice under high hydrostatic pressure,

when its strength reaches a maximum, and then

gradually decreases with the pressure increase. It

has been shown that strength of ice under high hydrostatic pressure is described by a parabolic yield

criterion with only three fundamental parameters, ice

cohesion, internal friction angle, and ice melting pressure, which all have a definite physical meaning and

are functions of temperature. The model has been

verified using test data on the strength of iceberg ice

and laboratory-made polycrystalline freshwater ice

under triaxial compression at strain rates between 10–3

and 10–5 s–1 over the temperature range between –1°

and –40°C.
CRREL Report 97-6

Ice Strength as a

Function of Hydrostatic

Pressure and Temperature

Anatoly M. Fish and Yuri K. Zaretsky October 1997

Prepared for

OFFICE OF THE CHIEF OF ENGINEERS

Approved for public release; distribution is unlimited.

US Army Corps

of Engineers

Cold Regions Research &

Engineering Laboratory

®®

Cold Regions Research &

Engineering Laboratory
PREFACE

This report was prepared by Dr. Anatoly M. Fish, Research Civil Engineer, of the

Civil Engineering Research Division, Research and Engineering Directorate, U.S.

Army Cold Regions Research and Engineering Laboratory; and Dr. Yuri K. Zaretsky,

Director, Institute of Geomechanics and Hydrostructures, Moscow, Russia.

Funding of this research was provided by DA Project 4A762784AT42, Cold Regions

Engineering Technology, Work Package 201, Conventional Facilities in Cold Regions,

Work Unit CA-D13, Geotechnical Structures in Cold Regions.

The authors extend their appreciation to Dr. Devinder Sodhi, Dr. Patrick Black,

and Dr. Stephen Ketcham of CRREL for technical review of this report and their

valuable comments.

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.

ii
CONTENTS

Page

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

Nomenclature ....................................................................................................... iv

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

Strain rate effect ................................................................................................... 3

Effect of hydrostatic pressure ............................................................................ 4

Temperature effect............................................................................................... 5

Ice melting pressure ....................................................................................... 5

Ice cohesion ...................................................................................................... 5

Angle of internal friction ............................................................................... 6

Temperature criteria of ice strength.................................................................. 6

Test data ............................................................................................................... 7

Parameter evaluation .......................................................................................... 8

Ice cohesion .......................................................................................................... 9

Angle of internal friction .................................................................................... 10

Strength predictions ............................................................................................ 11

Summary and conclusions ................................................................................. 13

Literature cited ..................................................................................................... 13

Abstract ............................................................................................................... 15

ILLUSTRATIONS

Figure

1. Strength envelopes of ice at various temperatures ............................. 2

2. Temperature diagrams of the strength parameters of ice ................. 6

3 Strength test data of ice under triaxial compression .......................... 7

4. Temperature dependencies of the strength parameters of ice .......... 8

5. Determination of parameters co and α .................................................. 9

6. Determination of parameters bo and β .................................................... 9

7. Angle of internal friction of ice as a function of temperature ........... 10

8. Strength of ice as a function of temperature and confining

pressure ................................................................................................ 11

9. Predicted and test magnitudes of ice strength under triaxial

compression at various temperatures .............................................. 12

10. Strength of ice in the range of high hydrostatic pressures at

 

ε≈ ×  ̇ − − 5 10 3 1 s ....................................................................................... 12

TABLES

Table

1. Strength parameters of ice at various temperatures........................... 8

2. Strength predictions of Labrador iceberg ice ...................................... 11

iii
NOMENCLATURE

A ice melting coefficient

B flow equation parameter

b = tan φ, internal friction parameter

bo internal friction parameter at freezing

b1 component of the internal friction parameter at T < Tm

 

C ̃ material constant

c ice cohesion

co ice cohesion at freezing

c1 component of ice cohesion at T < Tm

d grain diameter

E activation energy

h Planck’s constant

h2 negative hydrostatic pressure at which the ice shear strength equals zero

I1 first invariant of the stress tensor

J2 second invariant of the stress deviator

k Boltzmann’s constant

Lm specific heat of melting of the unit mass

n strain hardening parameter

p hydrostatic pressure

p* ice melting pressure

R gas constant

Sl

, Ss unit mass entropy of the liquid and solid state, respectively

T absolute temperature

Tm ice melting temperature

to failure time at strain rate, 

γ γ  ̇  ̇ i io =

Vl, Vs unit mass volume of the material in the liquid and solid state, respectively

α empirical parameter of the cohesion

β, β1 parameters of the internal friction angle

 

γ ̇

i octahedral strain rate

 

γ ̇

io octahedral (referenced) strain rate

γi octahedral strain

γio octahedral (referenced) strain

ε axial strain

εo axial (referenced) strain

 ε ̇ = 

ε ̇ 1 axial strain rate

 

ε ̇ o axial strain rate (referenced)

η viscosity coefficient

θ temperature, Celsius

σmax hydrostatic pressure at which the shear strength of ice reaches a maximum

value

σ1,2,3 principal stress

τi octahedral shear stress (strength)

τmax maximum shear stress (strength)

φ internal friction angle

φ1 component of internal friction angle at T < Tm

φο internal friction angle at freezing

ω parameter of the internal friction angle at T < Tm

iv
INTRODUCTION

Studies of ice strength in the second half of this

century have attracted efforts of a number of researchers. Their attention has been focused on

investigation of ice strength as a function of the

strain rate, temperature, grain size, and other factors mainly in simple stress-strain state, uniaxial

compression. During the last two decades, however, the attention of researchers was shifted to

investigation of the mechanical behavior of ice,

particularly the strength of ice, in a complex stressstrain state such as triaxial compression.

In a general case, strength of ice in a multiaxial

stress state, when other conditions (the type of

ice, its structure, the grain size, etc.) are equal, can

be presented as

 

τ τγ i ii * * = ( )  ̇ , , p T (1)

where 

τ i

* = 

τ i = J2 = octahedral (peak) shear

stress

J2 = second invariant of the stress deviator

γ ̇i = octahedral constant shear strain rate

p = I1/3 = hydrostatic pressure

I1 = first invariant of the stress tensor

T = ice temperature, K.

The first studies of the effect of low confining

pressure on creep strength of ice under triaxial

(σ2 = σ3) compression were carried out by Sayles

(1974). The studies showed that the ice strength is

a nonlinear function of the confining pressure.

More detailed investigations of the effect of the

strain rate and high confining pressures on the ice

strength under triaxial (σ2 = σ3) compression were

performed by Jones (1978, 1982). The studies, carried out over a wide range of constant strain rates

and confining pressures, revealed the anomalous

behavior of ice when its shear strength (at a certain magnitude of confining pressure) reaches a

maximum attributable to melting of ice, and then

gradually decreases with the pressure increase.

Jones showed that the ice strength is a nonlinear

(power) function of the axial (constant) strain rate

(see eq 1c below). Further experimental studies

with various types of ice revealed effects of temperature, salinity, structure, and other factors on

the ice strength (Hausler 1983, Nadreau and

Michel 1986, Richter-Menge et al. 1986, Timco and

Frederking 1986, Nadreau et al. 1991, Rist and

Murrell 1994, Gagnon and Gammon 1995, Weiss

and Schulson 1995 and others).

At the same time mathematical models were

also developed. Thus, Reinicke and Ralston (1977)

and Hausler (1986) applied a parabolic equation

of Smith (1974), developed originally for rocks, to

describe the strength dependency of ice upon the

hydrostatic pressure (in our notation):

 

τ τ i i * * = ( ) p . (1a)

The absolute magnitudes and the physical meaning of the parameters in the Smith equation have

not been defined.

Nadreau proposed a model (Nadreau and

Michel 1986) that described the nonlinear dependency of ice strength on confining pressure (σ2 =

σ3) by means of a third-order polynominal function with four phenomenological parameters, to

be determined from a series of triaxial tests of ice

at different strain rates, and the principal parameter—the ice melting pressure p*. This pressure

was suggested to be determined either from the

ice state diagram or to be calculated by an empiri-

* Denotes failure stress cal equation.

Ice Strength as a Function of

Hydrostatic Pressure and Temperature

ANATOLY M. FISH AND YURI K. ZARETSKY
In the model of Fish (1991, 1992, 1993) the relationship

 

ττ γ ii i Const * * = ( ) p T ,  ̇ ; = (1b)

was described by a product of two functions: 1) a

parabolic function—extended by Fish (1991) the

Drucker-Prager yield criterion, and 2) a normalized (dimensionless) power function of the strain

rate. The yield criterion describes the strength

dependency of ice on the hydrostatic pressure by

means of three parameters: the cohesion c, the

friction angle φ and the magnitude of the hydrostatic pressure σmax, at which the shear strength

of ice reaches a maximum τmax, related to the ice

melting pressure p* (Fig. 1). The model described

Jones’ (1982) test data well. It was shown that the

ice cohesion and the friction angle are functions

of the strain rate.

The temperature effect on ice strength under

triaxial compression was studied by Rist and

Murrell (1994) and Gagnon and Gammon (1995)

using the Arrhenius-type (Norton-Glen) equation,

 

ε τ  ̇ = − B E RT exp / ( ) i

n (1c)

where 

ε ̇ = axial (constant) strain rate

B = empirical parameter

E = activation energy

R = universal gas constant

T = absolute temperature, K

n ≈ 4 = strain hardening parameter (Jones

1982),

in which the temperature variations of the ice

strength are characterized by only one parameter—the activation energy. The apparent activation energy magnitudes were found as E = 69 kJ

mol–1 in the temperature range between –20° and

–40°C (Rist and Murrell 1994), and E = 101 kJ

mol–1 in the temperature range between –1° and

–16°C (Gagnon and Gammon 1995). The higher

activation energy magnitudes were explained by

grain-boundary softening associated with the presence of liquid on grain boundaries.

Equation 1c implies that the nonlinear (with

regard to stress) viscosity coefficient of ice,

η(T) = 1/B exp(E/RT) ,

is independent of the hydrostatic pressure. In other

words, ice is assumed to be incompressible.

It has been shown earlier (Fish 1991) that the

ice strength in a multiaxial stress state is characterized by two or more temperature-dependent

parameters that can affect the magnitude of the

apparent activation energy: the ice cohesion, the

friction angle, and the hydrostatic pressure. So

the predicted values of the ice strength, calculated by eq 1c, may deviate considerably from

those obtained in the tests.

A mathematical model (Zaretsky and Fish

1996a, 1996b) takes into account the effect of all

three variables in eq 1. The ice strength is described by three parameters: the ice cohesion, friction angle and ice melting pressure. The ice cohesion and the friction parameter are assumed to be

* *

τi = J2

τ max 1

τ max 2

τ max 3

φ1 φ2

φ3

c1

c3

c2

– p p3 p2 p1 p

0 σmax 3 σmax 2 σmax 1

*

M

p = –

I1

– , Hydrostatic Pressure 3

τ i = (c + bp) – (c + bp*) p

p* ––

2

τ max = c + σmax b

—2

σmax = + p* c

—b

p*2

–––2

–1

p* = p*(T); c = c(T)

b = b(T)

b(T) = tan φ(T)

T1

T2

T3

τ i, Shear Stress

b1

Figure 1. Strength envelopes of ice at various temperatures: c, φ, p*, and T are the ice cohesion,

friction angle, ice melting pressure, and absolute temperature, respectively.

2
proportional to temperature. It has been shown

that the hydrostatic pressure being applied to ice

decreases its melting temperature and thus its

strength under triaxial compression.

In the present report a different approach has

been undertaken. The authors considered the ice

strength as a function of two variables: the hydrostatic pressure and temperature, i.e.,

 

ττ γ ii i Const * * = ( ) p T, ;  ̇ = . (1d)

At a constant strain rate the strength of ice in a

multiaxial stress state is described by the parabolic yield criterion with three parameters: the

cohesion, and the friction angle, which are different nonlinear functions of temperature, and the

ice melting pressure. Then a strength criterion of

ice, which takes into account the combined effect

of all three variables in eq 1, the strain rate, hydrostatic pressure and temperature, is obtained

by combining eq 1b and 1d.

STRAIN RATE EFFECT

A constitutive equation for steady-state creep

of homogeneous and isotropic ice in a multiaxial

stress state at constant strain rate and constant

temperature takes the following form (Fish 1991,

1992, 1993):

 

 ̇  ̃ γ

τ

τ

i o

i

io

= 



 



 C

t

n

. (2)

It should be emphasized that eq 2 is fundamentally

different from the Norton-Glen power flow law

(eq 1c), although it contains a power function of

stress with exponent n.

In contrast to eq 1c parameters 

C ̃ and to in eq 2

have a definite physical meaning, and the denominator of the stress function is a temperature-dependent yield criterion (eq 5 below). This yield

criterion, which is a function of the first invariant

of the stress tensor and temperature, relates the

minimum shear strain rate and the shear stress in

the whole spectrum of hydrostatic pressures (mean

normal stresses).

Equation 2 implies that the nonlinear viscosity

coefficient of ice

 

η τ o o io (, ) /  ̃ pT t C n =

is a function of the hydrostatic pressure and temperature. For p < σmax one can see that the higher

the hydrostatic pressure is, the greater the viscosity coefficient, the lower the strain rate of ice, and

the higher the ice strength will be. For p > σmax the

higher the hydrostatic pressure is, the smaller the

viscosity coefficient, the higher the strain rate,

and the lower the ice strength will be. This is

confirmed well by test data (Jones 1982, Jones and

Chew 1983).

Equation 2 can be rewritten in terms of maximum (peak) shear strength 

τ τ i i * = and presented

as a product of two independent functions: a yield

function τio(p) and a nondimensional function

 

Φ(γ ̇ )i of the constant strain rate

 

τγτ γ i i io i * ( ) p p ,  ̇ = Φ () ( ̇ ) . (2a)

Function 

Φ(γ ̇ )i has been selected in its simplest

form:

 

Φ( ̇ )  ̇  ̇

 ̇

/ /

γ γ

γ

γ

γ

i

i o

io

i

io

= 



 



 = 



 



 t 1 1 n n

(3)

where 

γ ̇ i = applied octahedral constant shear

strain rate

 

γ ̇ io = γ io o / t = instantaneous (referenced)

octahedral shear strain rate

 

γ io = C ̃ = instantaneous octahedral shear strain

n = dimensionless parameter; for polycrystalline ice, n ≈ 4 (Jones 1982)

t

o = temperature-dependent time to failure, i.e., the time interval between

the initiation of the test conducted at

 

γ γ  ̇  ̇ i = io and the moment when the

ice strength reaches a maximum

(peak) value.

The temperature dependency of t

o is given by

 

t

h

kT

E

RT o = 



 

 exp  (4)

where E = activation energy

R = gas constant

h = Planck’s constant

k = Boltzmann’s constant.

In eq 3 the “instantaneous” (referenced) octahedral strain rate 

γ ̇ io is defined as a strain rate at

which the shear strength of ice reaches a maximum value, separating two different modes of

failure: the brittle mode dominated by the cleavage

mechanism of failure and the ductile mode dominated by the shear mechanism. At strain rates

(

 

γ γ  ̇  ̇ i io > ) the ice strength either decreases (Gagnon

3
and Gammon 1995) or remains unchanged (Jones

1982). Apparently, the magnitude of the instantaneous strain rate depends on the type of ice, its

structure and other factors and varies between

10–1 and 10–3 s–1. Since the magnitude of ice

strength is greatly affected by the strain rate, selection of an adequate value of this referenced

strain rate is extremely important for an accurate

prediction of the ice strength at lower strain rates.

In experimental studies of ice under triaxial

(biaxial) compression, the radial strains are assumed to be small and are usually ignored. In this

case eq 3 can be replaced by

 

Φ Φ ( ̇ ) ( ̇)  ̇  ̇

 ̇

/ /

γ ε

ε

ε

ε

ε

i o

o o

= = 



 



 = 



 



 t 1 1 n n

(3a)

where 

ε ̇ = axial strain rate

 

ε ε  ̇ , o o = instantaneous (referenced) axial strain

rate and strain respectively.

 

to =ε ε o o / ̇ = given by eq 4.

Thus, the strain rate function varies in the limits

 

1 0 ≥ =≥ Φ Φ (γ ε  ̇ ) ( ̇) . i

The effect of the strain rate on the strength of ice

can be excluded from consideration by selecting

the strength test data of ice corresponding to

 

Φ(ε =  ̇) 1.

EFFECT OF HYDROSTATIC PRESSURE

When applied strain rate is equal to the instantaneous strain rate 

γ γ εε  ̇  ̇  ̇  ̇ i io o = = or , in eq 2a and

3a, 

Φ Φ (γ ε  ̇ ) ( ̇ ) io o = = 1, and the strength dependency of ice upon the hydrostatic pressure is described by the parabolic yield criterion (Fish 1991)

depicted in Figure 1,

 

τ

σ

io max

( ) p c bp b=+ − p 2

2 (5)

or

 

τ io( ) ( ) ( *) * p c bp c bp p

p

=+ −+ 











2

(5a)

where c = ice cohesion on the octahedral plane

b = tan φ, where φ is the angle of internal

friction of ice on the octahedral plane

p = 

( ) σσσ 123 + + /3 = hydrostatic pressure (mean normal stress)

 

σσσ 123 , , = principal stresses,

τi = 

1

1 2

2

2 3

2

6 [( ) σσ σσ − + − ( )

1 3

2 1 2

+ − ( ) σ σ ] /

= octahedral shear stress (resultant)

σmax = magnitude of the hydrostatic pressure at which the shear strength

reaches a maximum

 

τ σ max max = +c

b

2 (6)

 

σmax =

+ 



 





p

c

b p

*

*

2

2 (7)

p* = ice melting pressure at which the

shear strength of ice equals to zero.

Subscript (o) at τio in eq 5 indicates that parameters c and b are referred to the instantaneous

condition when the applied strain rate is equal to

the instantaneous strain rate.

Note that the yield curve may also intersect the

hydrostatic axis in the domain of the negative

hydrostatic pressures (–p) at point h2 (not shown

in Fig. 1), the abscissa of which is equal to

 

h p

b

c p

2

1

= −

+

*

*

. (8)

Equation 5 can be considered an extended Von

Mises–Drucker-Prager yield criterion. At low

stress level p << σmax the third term in the right

side of eq 5 approaches zero and eq 5 transforms

into the Drucker-Prager (1952) (extended Von

Mises) yield criterion:

τio = c + bp . (9)

For frictionless materials (b = 0) eq 5 reduces to

the Von Mises yield criterion

τio = c . (10)

Thus, in a multiaxial stress state, the strength

of ice as well as its strength characteristics are

functions of only three parameters: p*, c and b

which all have a definite physical meaning and are

easily determined from test data. Studies show

that all these parameters are functions of temperature, i.e.,

 

p p T c cT b bT * * ( ), ( ), ( ) = == .

Consequently, the shear strength of ice as well as

the strength characteristics of ice are also functions of temperature:

4
 

ττ τ τ io io max max = = ( ), ( ), T T

σ σ max max = = ( ), ( ). T h hT 2 2

Thus, if a series of strength tests of ice is carried

out at a constant strain rate instead of one yield

curve, one obtains a family of curves for various

temperatures (Fig. 1). It should be emphasized

that parameters c, b and τmax are also strain-rate

dependent, while parameters σmax, p* and h2 are

independent of the strain rate (Fish 1992, 1993).

TEMPERATURE EFFECT

Ice melting pressure

It is well known that the ice melting temperature, as well as the melting temperature of other

crystalline materials, is a function of the hydrostatic pressure. This pressure can be determined

(Zaretsky and Fish 1996a) from the Clapeyron

equation, according to which a small change in

the equilibrious melting temperature of a solid

∆Tm attributable to a small change in the hydrostatic pressure ∆p can be calculated from the relationship

 

dT V V

S S dp T V VL m dp s

s m sm

= −

− = l −

l

l (11)

whereVl and Vs = unit mass volume of the material in the liquid and in the

solid state, respectively

Sl and Ss = unit mass entropy of the liquid and of the solid state, respectively

Lm = specific heat of melting of the

unit mass.

Since at melting the volume 

V V dT l < < s m , 0, i.e.,

the equilibrious melting temperature of ice Tm

decreases as well,

 

dT Adp m = − . (12)

Thus, for ice when p = 0, Tm = 273.1 K, Vl = 10–3

m3 kg–1, Vs = 1.09 × 10–3 m3 kg–1, Lm = 3.336 × 105

J kg–1, and parameter A for ice at temperature

θ = 0°C is equal to

A = 0.074 K/MPa.

The magnitude of parameter A coincides with

that calculated by Barnes et al. (1971). It is not

difficult at this point to calculate the critical hydrostatic pressure p* at which the shear resistance

of ice equals zero, i.e.,

 

T T Ap m m += − θ * (13)

and

 

p A * = −θ (14)

where θ = ice temperature (°C) and Tm = 273.1 K

is the ice melting temperature at the atmospheric

pressure. Note that the ice melting pressure is

unrelated to the grain size or the structure of ice.

The magnitudes of ice melting pressures p* for

various temperatures are presented in Table 1.

Ice cohesion

The ice cohesion defines the ice strength when

the hydrostatic pressure p = 0. Ice strength is a

stochastic event, a culmination point of failure of

intermolecular bonds and growth of cracks. A certain number of these bonds in the unit volume of

ice are formed during freezing of water at temperature 0°C. Further temperature decrease brings

about formation of new bonds, attributable to

freezing of the liquid phase on grain boundaries,

and a sharp increase of the ice strength (cohesion). Thus, one may conclude that the ice cohesion is a function of temperature and consists of a

sum of two components:

 

cT c c T ( ) = + o 1 ( ) (15)

where co is the component of the ice cohesion

brought about at the time of ice formation at 0°C

and c1(T) is the temperature-dependent component of the cohesion brought about by freezing of

liquid phase at temperature below 0°C.

Since the physical nature of co and c1 is the

same, and taking into account that at temperature

T = Tm, c1 = 0, the temperature dependency of

component c1 can be presented in the form

 

cT c TT 1 () / = − ( ) 1 o m α (16)

where α is a parameter. Combining eq 15 and 16

 

cT c T T ( ) ( / ). = +− o m [ ] 1 1 α (17)

Equation 17 establishes a linear dependency of

the ice cohesion upon temperature. Test data show

that such a relationship is valid in the domain of

relatively low temperatures below –20°C. In the

range of moderate temperature, this relationship becomes nonlinear and somewhat better

5
described by

 

cT ce T T ( ) ( /) = − o α m μ 1 (18)

or

 

c ce T ( ) | |/ θ α θ = o m (18a)

in which μ = 1 and θ= (Tm – T ) is ice temperature (°C).

Equations 18 and 18a can be considered approximate forms of eq 17. Note that at freezing

 

TT c c == ° = m o 273 1 0 0 . ( ); ( ) K C

when

 

T c ce =− ° = 0 273 1 ( . ); * . C o α

The temperature diagram of parameter c is shown

in Figure 2a.

Angle of internal friction

The parameter of internal friction of ice b(T)

can also be presented as a sum of two components:

 

bT b b T () () = + o 1 (19)

where bo = tan φo and φo is the angle of internal

friction of ice brought about at the time of ice

formation at 0°C; b1(T) = tan φ1 (T), and φ1 (T) is

the angle of internal friction attributable to the

increased viscosity of the liquid phase (lubricant)

at grain boundaries at temperatures below 0°C.

One may assume that the component b1(T) is

proportional to bo and to the temperature melting

function, i.e.,

 

bT b TT 1() / = − ( ) 1 o m β (20)

where β is a parameter. Combining eq 19 and 20

 

bT b T T () / . = +− o m [ ] 1 1 β( ) (21)

To simplify the parameter evaluation procedure,

eq 21 can approximately be presented in the form

 

bT be T T ( ) / = ( ) − o β m

ν 1 (22)

or

 

b be ( ) θ β θ ν= o 1 (22a)

where 

β β ν 1 = /Tm . It was found that ν = 1/2 is in

agreement with test data.

At freezing when

 

TT b b == ° = m o 273 1 0 0 . ( ); ( ) K C

at the absolute zero

 

T b be =− ° = 0 273 1 ( . ); C o * β

The temperature diagram of parameter b(T) is

given in Figure 2b. Note that at very low temperatures certain adjustments of eq 15 through 22

will probably be required.

TEMPERATURE CRITERIA

OF ICE STRENGTH

Since temperature dependencies of the ice

strength parameters have been established, eq 1d

taking into account eq 5 can be presented in an

explicit form

 

τ

σ

io max

(, ) () () ( )

( ) pT cT bT p b TT

=+ − p 2

2 (23)

in which c(T), b(T) and σmax(T) are given by eq 7

and 14 through eq 22a. Equation 23 is a temperature criterion of ice strength in a multiaxial stress

state. To take into account the strain rate effect, eq

b. b*

b0

0 Tm

T, Temperature (K)

b, Friction Parameter

c*

c0

0 Tm

c(T) = c0 eα(1 – T/T m )

μ

a.

c, Cohesion

T, Temperature (K)

b(T) = b0 eβ(1 – T/T m )

ν

a. Ice cohesion, c(T).

b. Friction parameter b(T).

Figure 2. Temperature diagrams of the

strength parameters of ice.

6
23 should be combined with eq 2a through 4.

Then eq 1 can be written as

 

τε τ ε

σ

ε

ε

i

* io

max o

( ,  ̇ , ) (, ) ( ̇)

() () ( )

( )

 ̇

 ̇

/

p T pT

cT bT p b T

T p

n

=

=+ − 



 









 





Φ

2

2

1 (24)

In the range of the hydrostatic pressures p <<

σmax, eq 24 takes the form

 

τ ε

ε

ε

i

*

o

( ,  ̇ ,) () ()  ̇

 ̇

/

p T cT bTp

n

= + [ ] 



 





1

(25)

when p = 0 (pure shear)

 

τε ε

ε

ε i

*

o

( ̇ ,) ( ̇ ,) ()  ̇

 ̇

/

T c T cT

n

= = 



 





1

(26)

Accordingly

 

b T bT

n

( ̇ ,) ()  ̇

 ̇

/

ε

ε

ε = 



 





o

1

(25a)

where parameters c(T) and b(T) are corresponding to the strain rate 

ε ε  ̇  ̇ = . o

If the radial strain rates are taken into account

function 

Φ(ε ̇) in eq 24, 25 and 26 should be replaced by function 

Φ(γ ̇).

TEST DATA

The validity of the temperature dependencies

of the strength parameters of ice presented above

was verified using test data of Gagnon and

Gammon (1995). The triaxial (σ2 = σ3) compression tests were carried out using cylindrical specimens of Labrador iceberg ice, 9.58 cm in diameter

and 26 cm long. Mean grain diameter d ≈ 8.1 mm.

The test temperature varied between –1°C and

 –16°C. The tests were conducted at constant axial

strain rates between 10–2 and 10–5 s–1. Test data

corresponding to the strain rate 

ε ̇ o ≈ × s − − 5 10 3 1

were selected in the following analysis to evaluate the strength parameters of ice. This strain rate

was defined by the authors of this report as the

instantaneous strain rate because the test strength

magnitudes at this rate reached their maximum

values. The tests were carried out at four different

confining pressures: 1.38, 3.45, 6.89 and 13.79 MPa.

The test data replotted by the authors of this report in terms of the shear strength 

τ σσ i = − ( ) 1 3 / 3

vs. the hydrostatic pressure 

p = ++ ( ) σσσ 123 /3

are presented in Figure 3. Note that each point in

Figure 3 represents the mean of five tests.

In Figure 3 for comparison tests, data are presented on triaxial compression of freshwater polycrystalline ice obtained by Jones (1982). The ran12

6

0 15 30 45

p, Hydrostatic Pressure (MPa)

–16°C

–11.8

–11

–6

–1

Labrador Ice

(data from

Gagnon and

Gammon, 1995)

Polycrystalline Ice

(data from Jones,

1982)

σ3 Trajectory

σ1 = σ3

ε

.

1 ≈ 5 × 10–3 s–1

√

–

3

18

σ3 = 1.38 MPa σ3 = 6.89 MPa

τi = (

σ1 – 

σ3)/ 3,Shear Strength (MPa) Figure 3. Strength test data of ice under triaxial compression. Data from Gagnon

and Gammon (1995) and Jones (1982).

τi = (

σ1 – 

σ3)/√3, Shear Strength (MPa)

= 5 × 10–3 s–1

σ2 = σ3

7
domly oriented laboratory-made ice

samples were 20 mm in diameter and 60

mm long with mean grain diameter less

than 1 mm. The test data selected for

comparison refer to the axial constant

strain rate 

ε= ×  ̇ . − − 5 4 10 3 1 s . The test temperature was –11.8 ± 0.9°C. The confining pressure (σ2 = σ3) varied between ~5

and ~ 85 MPa.

In Figure 10 (below), results of triaxial

constant strain rate compression tests

are presented of freshwater, randomly

oriented, polycrystalline ice carried out

by Rist and Murrell (1994). In these tests

laboratory-made samples of ice were 40 mm in

diameter and 100 mm long. Mean grain diameter

was ~ 1.7 mm. The tests were conducted at constant strain rates from 10–2 to 10–5 s–1; the test

temperatures varied between –20° and –45°C. The

data selected for this analysis are referred to as

the axial strain rate, 

ε ̇ ~ 10− − 3 1 s .

PARAMETER EVALUATION

Temperature variations of ice strength in a multiaxial stress state were shown to be defined by

three temperature-dependent parameters: the ice

melting pressure p(T), ice cohesion c(T) and friction parameter b(T). The magnitudes of

the ice melting pressure p(T) were calculated by eq 14 for the test temperatures and are included in Table 1. The

magnitudes of p* for θ = –11.8°C was

found to be in agreement with p* = 123.7

MPa obtained earlier (Fish 1991) from

the analysis of Jones’ data and with that

calculated by an empirical equation of

Hallam and Nadreau (1988).

The temperature dependencies of parameters c(θ) and b(θ) can be determined

based upon the considerations that in

the range of low hydrostatic pressures p

<< σmax (test pressures in Fig. 3). As a

first approximation, eq 5 can be replaced

by eq 9. Then, the slopes of the straight

lines and their intersects with the ordinate axis obtained by a regression analysis of the test data define the magnitudes of friction parameter b(θ) and

cohesion c(θ) for corresponding temperatures. The results of the analysis are

presented in Figure 4.

One can see in Figure 4 that temperature dependencies of parameters c and b

are indeed nonlinear and can be described by eq

18a and 22a. The evaluation procedures of parameters that enter these equations are shown in

Figures 5 and 6. It was found that for the Labrador iceberg ice

co = 5.37 MPa ; bo = 0.04

α = 9.61 ; β = 4.69.

Using these parameters, we calculated the magnitudes of the ice cohesion c(θ), the friction parameter b(θ) and the friction angle φ(θ) by eq 18a and

22a (Table 1). The magnitudes of σmax(θ) and

Table 1. Strength parameters of ice at various temperatures*.

θ°C c (MPa) b ~ φo σmax (MPa) τmax (MPa) p* (MPa)

–40 12.93† 0.241 14 245.87 43.52 540.54

–16 9.43 0.125 7 80.15 14.44 216.22

–11.8 8.14 0.106 6 53.81 10.99 159.46

–11 7.91 0.103 6 49.0 10.43 148.65

–6 6.63 0.08 4°30′ 20.05 7.43 81.08

–1 5.56 0.053 3 (4.79)** (5.69)** (36.85)**

0 5.37†† 0.04†† 2††

* Data in this table for strain rate 

 ̇

ε≈ × . − − 5 10 3 1 s † Calculated by eq 17.

** From the best fit of eq 5.

†† At freezing.

θ, Temperature (°C)

16

12

8

4

0 – 10 – 20 – 30

b (θ)

c (θ)

b0

Labrador Ice

Polycrystalline Ice

Predicted

c0

c, Cohesion (MPa)

16 × 10–2

12

8

4

b, Friction Parameter

0

ε

.

 ~ 5 × 10–3 s–1

Figure 4. Temperature dependencies of the strength parameters of

ice. Data from Gagnon and Gammon (1995) and Jones (1982).

8
τmax(θ) in Table 1 were calculated by eq 6 and 7

using corresponding values of c(θ), b(θ) and p*(θ).

Predicted temperature dependencies of parameters c and b are presented in Figure 4. From Table

1 and Figure 4, it follows that major changes of

parameters b and c, and ice strength take place at

temperature below –10°C. Between –10° and

–12°C the ice strength changes insignificantly.

It is curious to compare, at least approximately,

parameter b(T) and the kinetic friction coefficient

μc of ice, although the physical nature of these

two parameters is quite different. Thus, Jones et

al. (1991) reported that for θ = –10°C and for the

sliding velocity of 10–3 m s–1, μc ≈ 0.13. In Table 1

we found that for –10°C, b = 0.098 that is in agreement with the test data.

Comparing parameter bo with the kinetic dry

friction coefficient μd of ice on ice is particularly

interesting. Since the latter is temperature-dependent, the comparison should be made for a low

ice temperature when the effect of the liquid phase

(lubricant) is minimal. Thus, from Casassa et al.

(1991) we find that for θ = –35°C, μd = 0.019,

which is in correlation with bo = 0.04 obtained

above. Obviously a more accurate comparison

would require certain adjustments of both parameters to account for the differences in the test

velocities.

Note that a seeming contradiction may arise

when one compares the ice strength values in

Table 1 with eq 5a. From Table 1 it follows that ice

possesses a certain strength at 0°C. On the other

hand, from a comparison of eq 5a and 14 one

concludes that the shear strength of ice at this

temperature should be zero. Such a contradiction

arises from a peculiar property of ice: ice is formed

and melts at the same temperature, 0°C. This contradiction is easily eliminated when one remembers that melting of ice (in accordance with eq 5a)

will take place over a certain time.

ICE COHESION

Ice cohesion is the principal strength parameter of ice; therefore, it is important to examine

the relationship of ice cohesion with the other

physical characteristics of ice, such as the grain

size, ice structure, etc. Although we have limited

information on the effects of these factors on the

behavior of ice under triaxial compression, some

preliminary conclusions can still be made. Note,

in Figures 4, 5 and 6, test points corresponding to

Jones’ (1982) tests on polycrystalline ice with the

grain size of less than 1 mm follow the same

trends as the test data of Gagnon and Gammon

on the Labrador ice with an average grain size of

8.1 mm, i.e., eight times larger than the ice in

Jones’ tests. Such correlation suggests that the ice

cohesion magnitudes may not be sensitive to variations of the grain size.

To make sure that this conclusion extends

beyond the temperature range between –1° and

–16°C, and that it is valid in the strain rate range

different from ~ 5 × 10–3 s–1 used above in the

evaluation of the cohesion, a comparative analyln c

10

8

6

4

2

0 4 8 12 16 20

(Tm – T ) = |θ|, Temperature (°C)

c = c0e

α (Tm – T)/Tm

ln c0

α/Tm

Figure 5. Determination of parameters co and α.

Figure 6. Determination of parameters bo and β.

0

– 2

– 4

– 6

– 8

– 10

4 8 12 16 20

ln b0

(Tm – T )

1/2 = |θ |1/2

b = b0e

β1 (Tm – T)1/

2

β 1 = 1/

2 Tm

–––

β

ln b

9
sis of ice strength was carried out using test data

of Rist and Murrell (1994). This polycrystalline ice

had a grain size of 1.7 mm. The comparison was

performed for temperature –20°C and the strain

rate range between ~10–3 s–1 and ~ 10–5 s–1. The

results of the analysis confirmed the above conclusion. At the same time our studies indicate

that the ice cohesion magnitude is strongly dependent on the ice structure.

ANGLE OF INTERNAL FRICTION

From the above studies an important conclusion can also be made regarding the physical nature of the angle of internal friction of ice. While

the temperature dependency of the friction parameter b(θ) = tan φ(θ) is nonlinear (Fig. 4), the

test data from Table 1 plotted in Figure 7 suggest

that the friction angle is a simplest linear function

of temperature:

φ(θ) = φo + ω|θ| (27)

where φo = 2°50′, ω = 16.4′ degree–1 andθ= (Tm

– T ) is ice temperature (°C). It should be remembered that the magnitudes of the parameters in eq

27 refer to a strain rate 

ε ̇ ~ 10− − 3 1 s .

Note that the magnitudes of the internal friction angles φ in Table 1 and Figure 7 were calculated based mainly upon test data on the Labrador iceberg ice (d = 8.1 mm) at the temperature

range between –1° and –16°C. These data were

extrapolated to calculate the friction angle φ and

the ice strength for temperature –40°C (Fig. 7) and

compared (see Fig. 10 below) with test data of

Rist and Murrell (1994) obtained for this temperature and for the grain size of 1.7 mm.

One can see that despite the considerable difference (almost five times) in the grain sizes of

these two types of ice and the difference in the

test temperatures, the predicted angle of internal

friction correlates well with the test data. We find

the friction angle magnitudes are unaffected by

variations of the grain size of ice.

Thus, eq 27 can be used for calculations of

parameter b(T) in the above equations and for

prediction of the ice strength over a wide spectrum of temperatures between 0° and –40°C. It

should be remembered that the ice friction angle

is a function of the strain rate as well. It decreases

rapidly with the strain rate decrease. Fish (1991,

1993) showed that at low strain rates ~10-7 s–1 and

below, ice at –10°C can be considered as an ideally cohesive (φ = 0) material, the strength of which

is defined by eq 10 and 26. At high temperatures

the internal friction angle is small and so is its

effect on the ice strength. However, at low temperatures or at high strain rates and high confining pressures, the effect of the internal friction

angle on the strength of ice can be considerable.

Since parameters c, b and p* were found to be

unaffected by variations of the grain size. Table 1

data can be used to predict strength of various

Figure 7. Angle of internal friction of ice as a function of temperature. Data from

Gagnon and Gammon (1995), Jones (1982), and Rist and Murrell (1994).

0 –10 –20 –30 –40

θ, Temperature (°C)

Data from

Gagnon and Gammon (1995)

Jones (1982)

Rist and Murrell (1994) (predicted)

ω

Extrapolated

ε ≈ 10–3 s–1

24°

16°

8°

φ, Angle of Internal Friction 

φο

.

10
 

σ σ

ε

ε

1 3

1

−= + 3( )



 



 c bp

n  ̇

 ̇ .

/

o

(25a)

In Table 2 the results of such calculations are presented for n = 4, 

ε ̇ o = × − − 5 10 3 1 s , two different

confining pressures (assuming that p = σ3), and

two different strain rates together with the test

results from Table 2 of Gagnon and Gammon

(1995).

From Table 2 it follows that predicted values of

ice strength practically coincide with the test data.

Unfortunately only two tests of this ice type were

available for comparison.

In Figure 9 predicted dependencies of the ice

strength are presented calculated by eq 5 using

test values of hydrostatic pressures and the

strength parameters from Table 1. One can see

that in the moderate temperature range between

–6° and –16°C the calculated values of the ice

strength are in good agreement with test data

of Gagnon and Gammon (1995). An exception

is the test data for –1°C for which the magnitude of the ice melting pressure p* = 13.51

MPa calculated by eq 14 turned out to be

considerably smaller than p* = 36.85 MPa obtained from the best fit of the experimental

points in eq 5. Apparently, as the ice temperature approaches the ice melting temperature,

the relationship between the ice melting pressure and temperature becomes nonlinear. Further studies of the temperature-dependency

of the ice melting pressure are obviously necessary.

Equation 5 and the data in Table 1 were

used to predict the ice strength over the range

of high hydrostatic pressures. The results of

such an extrapolation are presented in Figure

10 and compared with test data of Jones (1982)

for temperature –11.8°C, and of Rist and

Murrell (1994) for temperature –40°C. In the

latter case parameter c was calculated by eq

17 to fit the test data, suggesting that further

studies of the temperature dependency of ice

cohesion in the temperature domain below

–20°C are required. At the same time the gentypes of iceberg ice and freshwater randomly oriented polycrystalline ice in a multiaxial stress state.

STRENGTH PREDICTIONS

Let us verify the accuracy of strength prediction by comparing the computed and the test results of ice strength as functions of temperature,

strain rate and hydrostatic pressure.

For low hydrostatic pressures p << σmax and

the strain rate 

ε ε  ̇  ̇ = o , ice strength can be calculated by eq 9 using data of Table 1 and the hydrostatic pressure magnitudes p from Figure 3, corresponding to confining pressures σ3 = 1.38 MPa

and 6.89 MPa. The results of such calculations for

the temperature range between –1° and –16°C are

presented in Figure 8 together with the test data

from Gagnon and Gammon (1995). One can conclude that the predicted ice strength values are in

agreement with the test data.

The data in Table 1 can also be used to evaluate

the ice strength in those cases when the strain rate

 

ε ε  ̇  ̇ < o. For p << σmax the ice strength is calculated

by eq 25. To simplify the comparison with the

published data, the latter can be rewritten in terms

of deviator stress:

14

12

10

8

6

4

2

0 – 4 – 8 – 12 – 16 – 20

θ, Temperature (°C)

σ3 = 6.89 MPa

σ3 = 1.38 MPa

σ2 = σ3

ε

.

≈ 5 × 10– 3 s– 1

Predicted

– 24

Labrador Iceberg Ice

———————————

Data from Gagnon and Gammon (1995)

τi = (

σ1 – 

σ3)/ 3,Shear Strength MP ( a)

16

Figure 8. Strength of ice as a function of temperature and

confining pressure. Data from Gagnon and Gammon (1995).

Table 2. Strength predictions of Labrador iceberg ice.

Confining Strain

 θ pressure rate 

ε ̇ σ1 − σ3 (MPa)

(°C) (MPa) (s–1) Measured Predicted Ice type

–11 1.38 4.82 10–5 3.50 4.36 Labrador ice

–11 6.89 5.96 10–5 4.75 4.93 Labrador ice

τi = (

σ1 – 

σ3)/√3, Shear Strength (MPa)

11
– 16°C

– 11.8

– 11

– 6

– 1

12

6

0 15 30 45

Labrador Ice

(data from

Gagnon and

Gammon, 1995)

Polycrystalline Ice

(data from Jones,

1982)

Predicted

σ1 = σ3

ε

.

≈ 5 × 10–3 s–1

p, Hydrostatic Pressure (MPa)

τi = (

σ1 – 

σ3)/ 3,Shear Strength MP ( a)

18

Figure 9. Predicted and test magnitudes of ice strength under triaxial compression at various temperatures. Data from Gagnon and Gammon (1995), and Jones

(1982).

24

18

12

6

0 100 200 300

p, Hydrostatic Pressure (MPa)

– 11

– 6

– 1

– 16°C

p* = 81.08 p* = 148.65 p* = 216.22 MPa

θ (°C) Data from:

Polycrystalline Ice

Labrador Ice

Rist and Murrell

(1994)

Jones (1982)

Predicted

Gagnon and

Gammon (1995)

– 40

–11.8

– 16

– 11

– 6

– 1

– 40°C

τi = (

σ1 – 

σ3)/ 3,Shear Strength MP ( a) Figure 10. Strength of ice in the range of high hydrostatic pressures at

 

ε≈ ×  ̇ − − 5 10 3 1 s . Data from Jones (1982), Gagnon and Gammon (1995), and

Rist and Murrell (1994).

τi = (

σ1 – 

σ3)/√3, Shear Strength (MPa)

τi = (

σ1 – 

σ3)/√3, Shear Strength (MPa)

= 5 × 10–3 s–1

σ2 = σ3

1995)

12
eral trend of predicted values of ice strength as

functions of the hydrostatic pressure is well confirmed by Jones’ test data.

SUMMARY AND CONCLUSIONS

1. A temperature model has been developed

that describes the ice strength in a multiaxial stress

state over a wide spectrum of negative (subfreezing) temperatures.

2. The strength dependency of ice on the hydrostatic pressure and temperature is well described by eq 5a, which is a parabolic yield criterion:

 

τ io( , ) ( ) ( *) * p T c bp c bp p

p

=+ −+ 











2

(5a)

with only three fundamental physically wellfounded parameters: the ice cohesion c(T), angle

of internal friction φ(T) or b(T), and ice melting

pressure p*(T). They all have a definite physical

meaning and are functions of temperature.

3. The temperature model was developed

based upon data on over 100 triaxial compression

tests of iceberg ice and laboratory-made freshwater randomly oriented polycrystalline ice with

a grain diameter between ~ 1 mm and ~ 8 mm in the

temperature range between –1° and –40°C at the

range of strain rates between 10–3 s–1 and 10–5 s–1.

4. Further studies are required of the temperature dependencies of the ice strength parameters

for various types of ice, particularly the ice melting pressure over a wider spectrum of temperatures, strain rates, and confining pressures.

LITERATURE CITED

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Rist, M.A., and S.A.F. Murrell (1994) Ice triaxial

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VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503.

October 1997

Ice Strength as a Function of Hydrostatic Pressure and Temperature PE: 6.27.84A

PR: 4A762784AT42

WP: 201

WU: CA-D13

Anatoly M. Fish and Yuri K. Zaretsky

U.S. Army Cold Regions Research and Engineering Laboratory

72 Lyme Road CRREL Report 97-6

Hanover, New Hampshire 03755-1290

Office of the Chief of Engineers

Washington, D.C. 20314-1000

For conversion of SI units to non-SI units of measurement, consult ASTM Standard E380-93, Standard Practice for Use of the

International System of Units, published by the American Society for Testing and Materials, 1916 Race St., Philadelphia, Pa.

19103.

Approved for public release; distribution is unlimited.

Available from NTIS, Springfield, Virginia 22161

23

Freshwater ice Temperature model

Ice melting pressure Triaxial compression tests

UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL

A temperature model has been developed that describes the ice strength in a multiaxial stress state over a wide

spectrum of negative temperatures. The model takes into account the anomalous behavior of ice under high

hydrostatic pressure, when its strength reaches a maximum, and then gradually decreases with the pressure

increase. It has been shown that strength of ice under high hydrostatic pressure is described by a parabolic yield

criterion with only three fundamental parameters, ice cohesion, internal friction angle, and ice melting pressure,

which all have a definite physical meaning and are functions of temperature. The model has been verified using

test data on the strength of iceberg ice and laboratory-made polycrystalline freshwater ice under triaxial

compression at strain rates between 10–3 and 10–5 s–1 over the temperature range between –1° and –40°C.