To clarify the situation, we show some examples in Figures 1-8.
The details of the analysis that produces these figures are summarized
in the Appendix. The main point is that, for the compliance version
of the analysis, the contours of constant energy are ellipses when the
vector *f* in (18) is interpreted as a stress. Analogously, when the
vector is treated as a strain, the contours of constant energy are
ellipses for the dual (or stiffness) formulation. If we choose to
think of these figures as diagrams in the complex plane, then we note
that -- while circles and lines transform to circles and lines when
transforming back and forth between these two planes -- the shapes of
ellipses are not preserved (except, of course, in the special case
- which is precisely that of isotropy - when the ellipses degenerate
to circles). Eigenvectors are determined by the directions in which
the points of contact of these two curves lie (indicated by red
circles).

Figure 1

Figure 2

Figures 1 and 2 present an example based on a glassy material.
Typical values for the bulk and shear moduli of glass were used:
*K*_{m} = 46.3 GPa and *G*_{m} = 30.5 GPa, respectively. The value
of the Biot-Willis coefficient was arbitrarily chosen as ,so *K*_{dr} = 18.52 GPa. Taking Poisson's ratio as , we
have *G*_{dr} = 13.89 GPa. Skempton's coefficient was chosen for
simplicity to be in this and all the other examples as well.
(This choice is extreme because it implies that *K*_{u} = *K*_{m}. But,
since our interest here is in analysis of the undrained shear modulus,
the study of this limit is particularly useful to us.)
The most anisotropic choices of and were
used that would not produce absurd (negative) values of the diagonal
coefficients for either *s ^{*}*

For the remaining three sets of examples, the values used for the moduli of the samples are taken from results contained in Berryman (2004a), wherein it was shown how certain laboratory data could be fit using an elastic differential effective medium scheme. These results are summarized in the TABLE.

Figure 3

Figure 4

Figures 3 and 4 present results for Sierra White granite.
Laboratory data on this material were presented by Murphy (1982).
The values chosen for and were
and .The value of the energy used for normalization was
GPa. Computed values for the effective and undrained
shear moduli were *G*_{eff} = 39.8 GPa and *G*_{u} = 28.3 GPa.

Figure 5

Figure 6

Figures 5 and 6 present results for Schuler-Cotton Valley sandstone.
Laboratory data on this material were also presented by Murphy (1982).
The values chosen for and were
and .The value of the energy used for normalization was
GPa. Computed values for the effective and undrained
shear moduli were *G*_{eff} = 35.8 GPa and *G*_{u} = 17.7 GPa.

Figure 7

Figure 8

Figures 7 and 8 present results for Spirit River sandstone.
Laboratory data on this material were presented by Knight and
Nolen-Hoeksema (1990). The values chosen for and were
and .The value of the energy used for normalization was
GPa. Computed values for the effective and undrained
shear moduli were *G*_{eff} = 20.11 GPa and *G*_{u} = 12.41 GPa.

5/23/2004