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A
The equation of an ellipse centered at the origin
whose semimajor and semiminor axes are of
lengths a and b and whose angle of rotation with respect to the
xaxis in the (x,y)plane is is given by
 
(35) 
For comparison, when , , and
a stress of magnitude is applied
to a poroelastic system, the energy stored in the anisotropic media of
interest here [using (16) and (18)] is given by
 

 (36) 
In the second equation , and r_{0} in an
arbitrary number (say unity) having the dimensions of stress
(i.e., dimensions of Pa).
It is not hard to see that, when ,the two equations (35) and (36) have the same
functional form and, therefore, that contours of constant energy
in the complex (z = x +iy) plane are ellipses. Furthermore, we
can solve for the
parameters of the ellipse by setting U = 1 (in arbitrary units for now)
in (36) and then factoring r^{2} out of both equations. We find that
 

 (37) 
 
These three equations can be inverted for the parameters of the
ellipse, giving:
 

 (38) 
 
Although contours of constant energy are of some interest, it is
probably more useful to our intuition for the poroelastic application to
think instead about contours associated with applied stresses and
strains of unit magnitude, i.e., for r = 1 (in appropriate
units) and varying from to [again see definition
(18)]. We then have the
important function . [Note that, when varies
instead between and , we just get a copy of the behavior
for between and .The only difference is that the stress and strain vectors have an
overall minus sign relative to those on the other halfcircle. For a
linear system, such an overall phase factor of unit magnitude is
irrelevant to the mechanics of the problem.] Then, if we set
and plot in the complex plane, we will have a plot of the ellipse of
interest with R determined analytically by
 
(39) 
We call R the magnitude of the normalized stress (i.e.,
normalized with respect to r_{0}).
The analysis just outlined can then be repeated for the stiffness
matrix and applied strain vectors. The mathematics is completely
analogous to the case already discussed, so we will not repeat it
here. Since strain is already a dimensionless quantity, the factor
that plays the same role as r_{0} above can in this case be chosen
to be unity if desired, as the main purpose of the factor r_{0} above was
to keep track of the dimensions of the stress components.
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Stanford Exploration Project
5/23/2004