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

a
the stress derivative dCldS remain as in equation (75). However, the veloc-
-
ity derivatives dUldSa also appear in the stress characteristic equation so
that uncoupling does not occur. The coefficient of the velocity derivative
dlJ/dSa - is
T T T T
-m u Ra + sa 421 = -m u R + m(X u-v) R C-' A21
-a c1 -a -a -
Some simplification of this coefficient occurs if we combine terms and sub-
stitute Ca = Aa 421 - G21e
Then the equation governing solutions along
stress characteristics may be written as (a = 1,2)
g21 =
B? - COS 2y
B' 1- COS 2y
2 sin 2y
It is clearly possible to express the above matrix equation in component
form. However, we shall not write out the component equations, simply
because the coefficients are quite lengthy and we do not intend to use the
result explicitly. The above form of the equation is useful in our analysis,
though, because it provides a concise description showing which terms appear.
This general result is expected to be useful in future work when results for
a normal flow rule are studied.
Taking the Limit to the Normal Flow Rule
The normal flow rule may be represented In the previous development by
letting B' = br so that the potential function defining the direction of
flow 9 is identical to the yield surface 4. It is seen that the character-
istic directions (given by ha, a = 1,2,3,4) are not distinct in this case.
That is, the stress and velocity characteristics coincide (A3 = Al, A!, = X2).
In this case ca and ca are singular simultaneously.
The previous analysis
breaks down when attempting to determine the eigenvectors associated with
146