Mechanics of Lake Ice - Freezing of Lakes and the Evolution of their Ice Cover - page 142

E
1 þ l e þ

l
1 2 l tr ð I

¼ K tr ð I þ 2G e 1
3

r ¼ tr ð I ð 5 : 7 Þ

In isotropic media, the principal axes of the stress and strain tensors are parallel. In two-
dimensional case, in the last term 1 = 2

is taken instead of 1 = 2

.
In a viscous medium, the stress depends on the rate of strain, and the linear viscous
rheology is as Eq. ( 5.7 ) with

e

ʵ replaced by the strain-rate . The basic case is the simple

shear ), where the viscous stress is proportional to the rate of shear strain, and

the proportionality coef

ow u = u ( y

cient is the viscosity ʷ :

r xy ¼ 2 ge xy ¼ g du
dy

ð 5 : 8 Þ

For compressible fluids, a second viscosity or bulk viscosity ( ʶ ) is added to account for

spherical stresses:

e 1
3

r ¼ f tr ð I þ 2 g ð I tr ð 5 : 9 Þ

In ice applications, non-linear viscous rheologies are used, obtained from Eq. ( 5.9 )by
allowing the viscosities to depend on the invariants of the strain-rate tensor.
An ideal plastic medium fails at yield stress. In one dimension,

e ¼ 0 ; r \ r Y ; e 6 ¼ 0 ; r ¼ r Y ð 5 : 10 Þ

σ Y is the yield stress. In two (three) dimensions, the yield point is replaced by a
yield curve (yield surface), which describes the yield stress as a function of strain
invariants. At yield, the actual strain evolution depends on inertia and external forces
acting on the system. In stable plastic medium strain hardening takes place. The yield
strength increases under strain, and more stress is needed for further deformation. To
continue the modelling wax example, more and more force would be needed to make a
wax ball smaller, and therefore for a constant load a stationary steady state results. In
unstable plastic medium strain softening takes place, and even if forcing is lowered the
strain may continue.
Basic rheological models can be expanded for more complex ones (e.g., Mellor 1986).
First, linearities can be changed to general non-linear laws in the elastic and viscous
models. Second, models can be combined. E.g., linear elastic and linear viscous models in
series give the Maxwell medium, while combining them parallel gives the Kelvin

where

Voigt
medium. For a constant load, Maxwell medium has an immediate elastic response fol-
lowed by linear viscous

-

flow, while a Kelvin Voigt medium flows in a viscous manner

-

toward an asymptotic determined by the elastic part.