Mechanics of Lake Ice - Freezing of Lakes and the Evolution of their Ice Cover - page 141

Fig. 5.4 The basic rheology models (in one dimension) for the stress σ as a function of the strain ʵ
and the strain-rate e ; E is elastic modulus, ʷ is viscosity, and σ Y is yield strength

dimensional cases are illustrated in Fig. 5.4 . The linear elastic model assumes that stress is
proportional to strain, while in the linear viscous model stress is proportional to the strain-
rate; the proportionality coef s

modulus) and viscosity. Rubber and water 1 are good material examples of linear elastic
and viscous media. An ideal plastic medium collapses once the stress achieves yield
strength (children

cients are, respectively, the elastic modulus (or Young '

s modelling wax serves as an example of a plastic medium).
Mechanical analogues for these models include spring balance for linear elasticity,
dashpot 2 for linear viscosity, and static friction for plasticity.
The behaviour of a one-dimensional linear elastic beam is determined by Young

'

' s

modulus E:

r ¼ E e ð 5 : 4 Þ

s modulus appears as the slope angle in Fig. 5.5 . For an isotropic three-
(or two-) dimensional case, one additional parameter is required for shear. Usually this is
treated using the Poisson

Thus Young '

ʵ xx is
seen in the loading direction but contraction appears in the perpendicular directions, and
the Poisson

' s ratio ʼ . For a beam in uniaxial (x) tension, tensile strain

' is: ratio is:

e yy
e xx ¼

e zz
e xx

l ¼ ð 5 : 5 Þ

Once Young ' s modulus and Poisson ' s ratio are known, the bulk modulus K and the

shear modulus G are obtained as:

E
3 ð 1 2 lÞ ; G ¼

E
2 ð 1 þ lÞ

ð : Þ

K ¼ 5 6

Linear elastic rheology is written in three-dimensional form as

1 In laminar flow; in turbulent flow stresses are properties of the flow, not properties of the medium.
2 Such as a door stroke compressor, which dampens or softens the movements of a door.