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In contrast to engineering plastics, for which elastoplastic models are commonly used,
Hibler's model is viscous-plastic and treats the ice as a kind of isotropic fluid. The
constitutive law is cast in terms of its viscosity which is made highly dependent on strain
rate so as to incorporate plastic yield. Here a concise exposition is given since it is
required for reference.
Again in spherical co-ordinates, the tensor of strain rates is given by
1 du i tan 7
£ il “ *
R cos q> dX R
1
£¡2 =—(
1
ôv. 1 du, tan co
+ —+ —^u,)
£22
2 R cos#? dX R dcp
1 3v ( .
R dcp
R
The constitutive equations read
<J n = %(£u + £22) + TJ (£\\~ £22) — P/2
cr 12 = 2rj£\2
<J22 = ¿¡(£22 + £u) + ^{.£22 — in) - P/2
where bulk viscosity Ç and shear viscosity rj depend on strain rate. (As a typical feature
of Hibler’s model the hydrostatic pressure P/2 is related to the compressive yield
strength P and shear strength P/(2e).) The non-linear constitutive law is stated in terms
of the following invariants of the strain rate tensor.
£ I ~ ^11 + £ 22
£ I1 ~ i( £ U~ £ 22) 2 + (2 £ n) 2 ) U2
A = (£ 1 2 + (£ II /e) 2 ) m
While Hibler’s model is perfectly plastic, we employ viscoplasticity for régularisation. Let
us introduce a small non-negative parameter m and put for bulk viscosity and shear
viscosity
_ l + crmax(A/A 0 -1,0) P
max(A,A 0 ) 2
(As is easily seen, Hibler’s constitutive law is recovered as the special case m = 0.)
The entire ice dynamics problem, or system of equations, is highly non-linear. There are
three types of non-linearity originating from dynamics, constitutive response and
transport, by far the stiffest of which is attributable to plasticity.