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

Now also divide the solution vector and forcing function into
and
where
= [i], C_ = [y], and F1 = r"] contains the first ti 3 compon-nts of
- \f2
Substituting these new variables and expanding the matrix products trans-
E.
forms the governing equation (38) into
This expression showsdirectly how the velocity and stress interact along
each of the four characteristic curves and the influence of advection and
the driving force.
vectors R
We consider equations (59) and (60) to determine explicitly the eigen-
-01
(alternately Ga and 8,).
Several general properties of the
eigenvectors R associated with the stress characteristic (a = 1,2 and
-a
det ca = 0) may be identified. Since c is singular we use (60) to find
-a
R that are nonzero. Then, if all eigenvalues are distinct, C is not singu-
-a -a
lar, and so the nonhomogeneous albegraic equations (59) may be solved for
S . A comparable consideration of the UeZocity characteristics (a = 3,4 and
-a
det c, = 0) requires that (60) give G, = 0 because c is nonsingdar. Using
(59) and the singularity of C provides nonzero S from the homogeneous
-a -a
equations.
For materials for which a normal flow rule is assumed, the stress and
velocity characteristics coincide and both C and c are singular simultane-
-a -a
ously. From (60) we find a nonzero vector R But then in (59) we find a
-a'
nonhomogeneous system of equations with a singular coefficient matrix C
-a'
While it is possible that values of R and S exist for this case, special
-a -a
care must be exercised to determine the eigenvectors.
*
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