AIDJEX Bulletin #40 - Polar Science Center - University of Washington

equations is classified as hyperbolic, parabolic, or elliptic, depending on
the number of real characteristic directions that exist. In plasticity
problems this will be determined by the state of stress and the type and
shape of yield surface that is used. In statically indeterminate problems
dealing with materials described by a normal flow rule theremaybearepeated
root in the characteristic determinant. To the authors' knowledge, this
problem has not been dealt with in the plasticity literature. It is recognized
by Martin (1975), who shows that the problem may be circumvented by
neglecting the advection term in momentum balance. This has the effect of
uncoupling the stress equations from the velocity equations so that two
independent systems may be solved.
In the field of plasticity of metals, Hill (1950) solves problems with
no body forces or advection. He discu6ses the possibility of parabolic and
elliptic systems and obtains characteristic relations in the hyperbolic case.
The well-known theorem due to Hencky for construction of slipline networks
results from this assumption. Martin (1975) solves the same typeofproblems,
but uses the null-vector technique, which is more easily generalized to
include body forces and advection.
The important distinction between plane strain and plane stress in
plasticity is discussed by Hodge (1950), and Prager and Hodge (1951), as
well as Hill (1950). It must be remembered that plane stress and strain are
special cases of three-dimensional models, and our sea ice model is not
derived from a three-dimensional model; it is a two-dimensional model.
In plane strain the von Mises and Tresca yield conditions coincide,
while in plane stress they do not. In plane strain the characteristics of
the stress equations exist at every point; they form an orthogonal network
and are identical with the lines of maximum shearing stress and maximum
shearing strain. In plane stress, if the Tresca yield condition is used
together with the stress-strain law of von Mises, the stress equations may
be hyperbolic in part of the plastic domain and parabolic in the rest.
Characteristics of the velocity equations differ from those of the stress
equations in this case. If the von Mises condition is used in plane stress,
the stress and velocity characteristics coincide, but the system may be
hyperbolic, parabolic, or elliptic.
113