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

lines through each point.
ways to traverse each line.
period TT. This fact is reflected in the expressions. We see that real char-
acteristic roots arise from (56) whenever IF’] - < 1. Thus, two distinct
directions exist except at the upper limit.
directions exist, the system of equations is hyperbolic.
are identical, the system of equations is parabolic.
When two distinct characteristic
Where the two roots
Finally. whenever
Ih’l > 1, no real roots may exist and the system of equations is elliptic.
As stated in the introduction, these different characterizations of the
equations are important primarily because discontinuities in the solutions
can exist only across real characteristic lines.
We cannot distinguish between the two possible
Therefore, any result must be repetitive over
To help visualize the patterns of characteristic curves that may appear,
the results of equations (50) and (55) are displayed in Table 1 and graphi-
cally in Figures 4 and 5.
In each of the six examples a homogeneous stress
state is assumed and the x-axis (horizontal) is the direction of maximum
principal stress (y = 0).
Since the plastic response is independent of the
Fig. 4. Stress invariants and stretching vectors associated with six
sample conditions: (a) isotropic contracting, (b) uniaxial contracting,
(c) shearing and closing, (d) pure shearing, (e) shearing
and opening, (f) uniaxial extending.
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