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

[2 sin 2y sin K~ - - cos 2y) cos K I[ (;: ' + cos 2y) sin K - 2 sin 2y cos K ]
c1 a a
This expression reduces to
- [ - (P' +cos 2 ~ cos ) K,][(B'
- cos 2y) sin K 0. 3 = o
(52)
(B' +cos 2y) sin' K - 2 sin 2y sin K cos K + (B' - cos 2y) cos2 K = o (53)
a a a a
which is identical to equation (47) with b' replaced by B'.
find that
Therefore, we
The velocity characteristics are thus defined by a relationship similar to
that for the stress characteristics. If we substitute for B' in terms of the
angle 8 which relates shearing to dilating, then from equation (13)
tan (8 - n/2) = cos 2(y - K ) a= 3,4 (55)
a
These transcendental equations will be useful for interpreting the exis-
tence of characteristic directions. However, to transform the governing
equations into their appropriate forms along the characteristic coordinates
requires that we have an explicit expression for X . These expressions may
a
be found by considering equations (47) and (53) directly, substituting h
a
for tan K . The roots of the quadratic equations then are
a
a
= I-
sin 2y+ J 1 - (b')'
b'-+--cos 2Y
a = 1,2
and
sin 2y d 1 - (~')2
a I?' + cos 2Y
A =
a = 3,4 (57)
It is seen that when a non-normal flow rule is assumed, four distinct characteristic
roots and directions occur. However, when a normal flow rule is
assumed, two pairs of repeated roots arise and the stress characteristics
coincide with the velocity characteristics. In this latter case, the governing
equations along characteristic directions cannot be obtained from
equation (38) when advection is included because the formal result breaks down.
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