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

arbitrarily assign the values of 1 and 2 to index a for these two directions.
Similarly, the elements of arise in the flow rule and involve velocity components.
Therefore, we call the directions obtained from equation (45) the
velocity characteristics and assign to index a the values 3 and 4 for these
roots.
Equation (44) is expanded by replacing X by K
a a
gent by sine and cosine. It becomes
and expressing the tan-
I[-(l+b' cos 2y) sin K - b' sin 2y cos K ][2b cos 2y sin K~ - 2b sin 2y cos K ~ ]
a
a
-[b ' sin 2y sin K - (1-b ' cos 2y) cos K ][-2b sin 2y sin K - 2b cos 2y cos K ] = o
a a a a
which reduces to
(46)
(b' +cos 2y) sin2 K ~ -
2 sin 2y sin K
a cos K
a + (b' - cos 2y) cos2 K a = o (47)
Rut substituting the double angle formulas for sin2 K a' cos2 K a and sir, K a cos K a
provides
(b'+cos 2y) (1-cos 2 ~ - ~ 2sin ) 2ysin 2~ + (b'- cos2y)(1+cos 2~ a ) = 0 (48)
a
which simplifies to
b' - cos 2y cos 2~ - sin 2y sin 2~ = 0 (49)
a
a
Finally, using the trigonometric expansion for the cosine of the difference
of two angles gives
1) , then
If the slope of the yield curve b' is replaced by the angle f3
(see Fig.
tan P = cos 2(y - K ~ ) a = l,2 (51)
relates all the angles that define the stress characteristic directions.
Similarly, equation (45) may be expanded to the form
12 8