Mechanics of Lake Ice - Freezing of Lakes and the Evolution of their Ice Cover - page 137

5.1 Rheology

The science of rheology examines the relationships between the internal stress,
mechanical deformation, and material properties of media. Material properties are such as
density, temperature and porosity, while deformation is speci

ed by strain and its time

history.
The present chapter begins with a general introduction into rheology. There are two
principal reasons for going to the basics. First, natural ice has quite complicated rheology,
and, secondly, for students and scientists in lake physics or limnology, rheological
questions are not so familiar, because lake water is a linear Newtonian

fluid, and its

mechanics obeys the well-established Navier Stokes equation. Natural lake ice is a

polycrystalline medium, and its mechanical behaviour depends on the time and space
scales under consideration.
In mechanics, lake ice sheet is normally considered as a continuum. The basic rheology
models

-

are all applicable in lake ice processes. For a
continuous sheet of polycrystalline ice, the size, shape and orientation of ice crystals and
the impurities between crystals in

— elastic, viscous, and plastic media —

uence on the mechanical properties. In the case of

broken ice , i.e. the ice cover consists of ice blocks or floes (Fig. 5.1 ), the granular medium

approach is taken where individual blocks or . When

the number of grains is large, a continuum approximation can be taken for length scales
much longer than the grain size. Important applications of granular models are mechanics
of ridge formation from ice blocks and drifting of ice

floes represent the individual ' grains '

floes on a lake surface.
Mechanics is treated here in the three-dimensional space. The Cartesian co-ordinates
are (x, y, z), with here x directed east, y directed north, and z directed up. The corre-
sponding unit vectors are denoted by i , j and k . The gradient operator is in the Cartesian
system . Furthermore, we deal with three different mathe-

matical quantities: scalars are numbers, vectors have magnitude and direction, and (2nd
order) tensors have vector components on all the three coordinate planes. In the Cartesian
system, tensors can be represented as matrixes.

r ¼i o = o x þ j o = o y þ k o = o z

Fig. 5.1 Solid ice sheet and broken ice on lakes