τio = τio (T), τmax = τmax (T),
drostatic pressure p* at which the shear resistance
of ice equals zero, i.e.,
σmax = σmax (T),
h2 = h2 (T).
Tm + θ = Tm - Ap*
(13)
Thus, if a series of strength tests of ice is carried
and
out at a constant strain rate instead of one yield
-θ
curve, one obtains a family of curves for various
p* =
(14)
temperatures (Fig. 1). It should be emphasized
A
that parameters c, b and τmax are also strain-rate
where θ = ice temperature (C) and Tm = 273.1 K
dependent, while parameters σmax, p* and h2 are
is the ice melting temperature at the atmospheric
independent of the strain rate (Fish 1992, 1993).
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
TEMPERATURE EFFECT
various temperatures are presented in Table 1.
Ice melting pressure
Ice cohesion
It is well known that the ice melting tempera-
The ice cohesion defines the ice strength when
ture, as well as the melting temperature of other
the hydrostatic pressure p = 0. Ice strength is a
crystalline materials, is a function of the hydro-
stochastic event, a culmination point of failure of
static pressure. This pressure can be determined
intermolecular bonds and growth of cracks. A cer-
(Zaretsky and Fish 1996a) from the Clapeyron
tain number of these bonds in the unit volume of
equation, according to which a small change in
ice are formed during freezing of water at tem-
the equilibrious melting temperature of a solid
perature 0C. Further temperature decrease brings
∆Tm attributable to a small change in the hydro-
static pressure ∆p can be calculated from the rela-
about formation of new bonds, attributable to
freezing of the liquid phase on grain boundaries,
tionship
and a sharp increase of the ice strength (cohe-
V - Vs
V - Vs
sion). Thus, one may conclude that the ice cohe-
dTm = l
dp = Tm l
dp
(11)
Sl - Ss
sion is a function of temperature and consists of a
Lm
sum of two components:
whereVl and Vs = unit mass volume of the ma-
c (T ) = co + c1 (T )
(15)
terial in the liquid and in the
solid state, respectively
where co is the component of the ice cohesion
brought about at the time of ice formation at 0C
uid and of the solid state, re-
and c1(T) is the temperature-dependent compo-
spectively
nent of the cohesion brought about by freezing of
Lm = specific heat of melting of the
liquid phase at temperature below 0C.
unit mass.
Since the physical nature of co and c1 is the
same, and taking into account that at temperature
Since at melting the volume Vl < Vs , dTm < 0 , i.e.,
T = Tm, c1 = 0, the temperature dependency of
the equilibrious melting temperature of ice Tm
component c1 can be presented in the form
decreases as well,
c1 (T) = coα (1 - T /Tm )
(16)
dTm = -Adp .
(12)
where α is a parameter. Combining eq 15 and 16
Thus, for ice when p = 0, Tm = 273.1 K, Vl = 103
m3 kg1, Vs = 1.09 103 m3 kg1, Lm = 3.336 105
c (T) = co[1 + α (1 - T /Tm )].
(17)
J kg1, and parameter A for ice at temperature
θ = 0C is equal to
Equation 17 establishes a linear dependency of
the ice cohesion upon temperature. Test data show
A = 0.074 K/MPa.
that such a relationship is valid in the domain of
relatively low temperatures below 20C. In the
The magnitude of parameter A coincides with
range of moderate temperature, this relation-
that calculated by Barnes et al. (1971). It is not
ship becomes nonlinear and somewhat better
difficult at this point to calculate the critical hy-
5
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