Mechanics of Lake Ice - Freezing of Lakes and the Evolution of their Ice Cover - page 149

P ¼ h

s a

ð 5 : 20 Þ

L

ˁ a C a U 2 Wind stress depends quadratically on the wind speed U a , ˄ a = (see Eq. 4.9a ) ,

1.5 × 10 − 3 is the drag coef cient. At the wind speed of 15 m s − 1 , wind stress is where C a ≈

10 − 5 ; then, if L = 10 km, the lake ice is

˄ a * ½ Pa. With P* = 50 kPa, we have ˄ a /P*

*

stable as soon as h > 10 cm.
In general, the dependence of ice strength on ice thickness and length scale is for-
mulated as

n

h
h

Ph ; ðÞ ¼P ð l Þ ð 5 : 21 Þ

where h* is the scaling thickness and n, ½ 2, is the power of the thickness

dependence (Coon 1974; Hibler 1979). The power depends on the mode of breakage. The
length scale dependence scales as l − 1/2 (Sanderson 1988; Leppäranta 2011). In coastal
landfast ice basins, where the size corresponds to medium-size lakes, the power n = 2 has
given a good

≤ n ≤

fit (Leppäranta 2013). The compressive strength in engineering experiments

is 1 - 10 MPa (l * 1 - 10 m), and thus at the length scale of 10 - 100 km the strength

magnitude is 50 kPa.
The plastic strength of drift ice depends on the mode of deformation. In compressive
failure the ice breaks by crushing and n = 1, i.e.

P ¼ r c h ð 5 : 22a Þ

where σ c is the three-dimensional compressive strength of ice. Using the theory of plate on

elastic foundation, bending failure (Coon 1974) is obtained as

s

q w gEh
31 l 2

r

h
m

p 2
192

r B ¼ ¼ 10 : 8 kPa ð 5 : 22b Þ

ð Þ

Thus the crushing strength is by one order of magnitude larger than the bending
strength. In the case of buckling failure (see Ashton 1986), the limiting buckling force per
unit width is F b =

2 , where

ˁ w g λ ʻ is the characteristic length of ice beam on water

foundation. The stability condition is

s

31 l ð Þ
q w g Eh

s r c ¼ s a

q w g k 2 h 1 ¼

s a \ h
L

ð 5 : 22c Þ

p

4

Eh 3
q w g 31 l 2

k ¼ ð 5 : 22d Þ

ð Þ