the hydrostatic pressure is, the greater the viscos-
proportional to temperature. It has been shown
ity coefficient, the lower the strain rate of ice, and
that the hydrostatic pressure being applied to ice
the higher the ice strength will be. For p > σmax the
decreases its melting temperature and thus its
strength under triaxial compression.
higher the hydrostatic pressure is, the smaller the
In the present report a different approach has
viscosity coefficient, the higher the strain rate,
been undertaken. The authors considered the ice
and the lower the ice strength will be. This is
strength as a function of two variables: the hydro-
confirmed well by test data (Jones 1982, Jones and
static pressure and temperature, i.e.,
Chew 1983).
Equation 2 can be rewritten in terms of maxi-
τ * = τ * (p, T ) ;
γ i = Const.
mum (peak) shear strength τ * = τ i and presented
˙
(1d)
i
i
i
as a product of two independent functions: a yield
At a constant strain rate the strength of ice in a
function τio(p) and a nondimensional function
multiaxial stress state is described by the para-
Φ (γ i ) of the constant strain rate
˙
bolic yield criterion with three parameters: the
τ * (p, γ i ) = τio (p) Φ (γ i ) .
cohesion, and the friction angle, which are differ-
˙
˙
(2a)
i
ent nonlinear functions of temperature, and the
ice melting pressure. Then a strength criterion of
Function Φ (γ i ) has been selected in its simplest
˙
ice, which takes into account the combined effect
form:
of all three variables in eq 1, the strain rate, hy-
drostatic pressure and temperature, is obtained
γ i to 1/n γ i 1/n
˙
˙
Φ (γ i ) =
=
˙
˙
(3)
by combining eq 1b and 1d.
γ io
γ io
where γ i = applied octahedral constant shear
˙
STRAIN RATE EFFECT
strain rate
γ io = γ io / to = instantaneous (referenced)
˙
A constitutive equation for steady-state creep
octahedral shear strain rate
of homogeneous and isotropic ice in a multiaxial
γ io = C = instantaneous octahedral shear strain
~
stress state at constant strain rate and constant
n = dimensionless parameter; for poly-
temperature takes the following form (Fish 1991,
crystalline ice, n ≈ 4 (Jones 1982)
1992, 1993):
to = temperature-dependent time to fail-
ure, i.e., the time interval between
n
C τi
~
the initiation of the test conducted at
γi =
˙
to τio
.
(2)
γ i = γ io and the moment when the
˙
˙
ice strength reaches a maximum
(peak) value.
It should be emphasized that eq 2 is fundamentally
different from the Norton-Glen power flow law
The temperature dependency of to is given by
(eq 1c), although it contains a power function of
stress with exponent n.
E
h
~
In contrast to eq 1c parameters C and to in eq 2
to =
exp
(4)
RT
kT
have a definite physical meaning, and the denomi-
nator of the stress function is a temperature-de-
where E
=
activation energy
pendent yield criterion (eq 5 below). This yield
R
=
gas constant
criterion, which is a function of the first invariant
h
=
Planck's constant
of the stress tensor and temperature, relates the
k
=
Boltzmann's constant.
minimum shear strain rate and the shear stress in
the whole spectrum of hydrostatic pressures (mean
In eq 3 the "instantaneous" (referenced) octahe-
normal stresses).
dral strain rate γ io is defined as a strain rate at
˙
Equation 2 implies that the nonlinear viscosity
which the shear strength of ice reaches a maxi-
coefficient of ice
mum value, separating two different modes of
~
failure: the brittle mode dominated by the cleavage
ηo (p, T) = toτ n / C
io
mechanism of failure and the ductile mode domi-
is a function of the hydrostatic pressure and tem-
nated by the shear mechanism. At strain rates
perature. For p < σmax one can see that the higher
( γ i > γ io ) the ice strength either decreases (Gagnon
˙
˙
3