[(σ - σ )
and Gammon 1995) or remains unchanged (Jones
1
+ (σ 2 - σ 3 )
2
2
τi =
1982). Apparently, the magnitude of the instanta-
1
2
6
neous strain rate depends on the type of ice, its
]
2 1/2
structure and other factors and varies between
+ (σ1 - σ 3 )
strength is greatly affected by the strain rate, se-
= octahedral shear stress (resultant)
lection of an adequate value of this referenced
σmax
= magnitude of the hydrostatic pres-
strain rate is extremely important for an accurate
sure at which the shear strength
prediction of the ice strength at lower strain rates.
reaches a maximum
In experimental studies of ice under triaxial
b
τmax = c +
σmax
(6)
(biaxial) compression, the radial strains are as-
2
sumed to be small and are usually ignored. In this
p *2
case eq 3 can be replaced by
σmax =
c
(7)
2 + p *
b
ε to 1/n ε 1/n
˙
˙
Φ (γ i ) = Φ (ε) =
=
˙
˙
p* = ice melting pressure at which the
(3a)
εo
εo
˙
shear strength of ice equals to zero.
Subscript (o) at τio in eq 5 indicates that param-
where ε = axial strain rate
˙
εo , εo = instantaneous (referenced) axial strain
˙
eters c and b are referred to the instantaneous
rate and strain respectively.
condition when the applied strain rate is equal to
to = εo /εo = given by eq 4.
˙
the instantaneous strain rate.
Note that the yield curve may also intersect the
Thus, the strain rate function varies in the limits
hydrostatic axis in the domain of the negative
hydrostatic pressures (p) at point h2 (not shown
1 ≥ Φ (γ i ) = Φ (ε) ≥ 0.
˙
˙
in Fig. 1), the abscissa of which is equal to
-p *
The effect of the strain rate on the strength of ice
h2 =
.
can be excluded from consideration by selecting
b
(8)
1+ c p *
the strength test data of ice corresponding to
Φ (ε) = 1 .
˙
Equation 5 can be considered an extended Von
MisesDrucker-Prager yield criterion. At low
stress level p << σmax the third term in the right
EFFECT OF HYDROSTATIC PRESSURE
side of eq 5 approaches zero and eq 5 transforms
into the Drucker-Prager (1952) (extended Von
When applied strain rate is equal to the instan-
Mises) yield criterion:
taneous strain rate γ i = γ io or ε = εo , in eq 2a and
˙˙
˙
˙
τio = c + bp .
3a, Φ (γ io ) = Φ (εo ) = 1, and the strength depen-
(9)
˙
˙
dency of ice upon the hydrostatic pressure is de-
For frictionless materials (b = 0) eq 5 reduces to
scribed by the parabolic yield criterion (Fish 1991)
the Von Mises yield criterion
depicted in Figure 1,
τio = c .
(10)
b
τ io(p) = c + bp -
p2
(5)
2σmax
Thus, in a multiaxial stress state, the strength
of ice as well as its strength characteristics are
or
functions of only three parameters: p*, c and b
p 2
which all have a definite physical meaning and are
τ io(p) = (c + bp) - (c + bp*)
(5a)
p *
easily determined from test data. Studies show
that all these parameters are functions of tem-
where c = ice cohesion on the octahedral plane
perature, i.e.,
b = tan φ, where φ is the angle of internal
p* = p * (T),
c = c(T), b = b(T) .
friction of ice on the octahedral plane
p = (σ1 + σ 2 + σ 3 )/ 3 = hydrostatic pres-
Consequently, the shear strength of ice as well as
sure (mean normal stress)
the strength characteristics of ice are also func-
σ1 , σ 2 , σ 3 = principal stresses,
tions of temperature:
4
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