Geoscience Reference
In-Depth Information
Fig. 5.12
Drift ice yield
curves: Coulomb or triangular
(Coon et al. 1974), teardrop
(Coon et al. 1974), and elliptic
(Hibler 1979)
Example 5.7.
The elliptic yield curve has the advantage that the failure yield can be
expressed in closed form. The yield curve is de
ned as (see Fig.
5.12
):
!
2
!
2
r
1
þ
2
P
1
r
2
þ
2
P
1
F
r
1
; r
2
ð
Þ
¼
þ
1
2
P
2e
P
where P is the compressive strength of ice and e is the ratio of compressive strength to
shear strength. Using the associated
flow rule (Eq.
5.37
) we can get the solution
P
2
P
e
I
I þ
e
2
e e
I
I
r
¼
p
e
I
þ
e
2
e
II
ð
Þ
I
2
where
ʵ
II
are the strain invariants equal to the sum and difference of the principal
strains. Clearly the stress is independent of the absolute level of the strain. For spherical
compressive strain,
ʵ
I
and
ʵ
I
< 0 and
ʵ
II
= 0, we have,
σ
=
-
P
I
. In the case of pure shear,
ʵ
I
=0,
we have
r
xx
¼
r
yy
¼
1
=
2
P and
r
xy
¼
r
yx
¼
e=e
II
ð
Þ
1
=
2
e
P
. Since it
is considered that
shear strength is signi
cant but
less than the compressive strength, we must have
1<e <<
; normally e = 2 is chosen.
Inside the yield curve, the ice may be assumed to behave in a rigid, elastic or viscous
manner. The linear elastic model gives the stress proportional to the strain:
∞
r
¼
rðe;
K
;
G
Þ
¼K
ð
tr
eÞ
I
þ
2G
e
0
;
F
ðr
1
; r
2
Þ
\
0
ð
5
:
38
Þ
e
0
¼
e e
I
I
where
is the deviatoric strain. The dimension of two-dimensional stress and
the moduli K and G is force/length. The magnitude of the drift ice elastic moduli is
Search WWH ::
Custom Search