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

One way to circumvent the problems that arise when stress and velocity
characteristics coincide is to neglect advection. In this classical approach,
the stress and velocity are found by considering the two systems of equations
whose principal parts are uncoupled.
Systems Uncoupl ed by Neglecting Advection
If we consider the special case in which advection is neglected, the
results take a simpler form. The equations governing stress and the equations
governing velocity uncouple in their first derivatives so that if tractions
are presecribed everywhere along the boundary, and velocity-dependent body
forces are zero, a statically determinant problem is defined. In this case
the stress state may be found independently of the velocity field.
The equations (59) and (60) for finding JLa
reduce to
T
s c = g
-a -a
If we seek the stress characteristics given by a = 1,2 then
det 7 = 0 det -C, # 0 (69)
-a
and from the latterpropertywerequire thatS = fora=land2. On the other
a
hand, since ,C is singular we can find nonzero vectors e Thus, let
3 0-
and then find q and r from equation (68) expanded into
a a
components
(9, Fa)
Xa(l+b' cos2y) -b'si~2y -2X bsin2y- 2b cos 2y
a
X b ' sin 2y - (1 - b ' cos 2y) 2Xa b cos 2y -
2b sin 2y
= (0 0) (71)
If we arbitrarily choose the length of vectors aa by letting
r2 = 1 then it can be shown (after some algebra) that
= 1 and
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