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REFERENCES

A The equation of an ellipse centered at the origin whose semi-major and semi-minor axes are of lengths a and b and whose angle of rotation with respect to the x-axis in the (x,y)-plane is $\psi$ is given by
   \begin{eqnarray} (x\cos\psi+y\sin\psi)^2/a^2 + (-x\sin\psi+y\cos\psi)^2/b^2 = 1. \end{eqnarray} (35)
For comparison, when $x = r\cos\theta$, $y = r\sin\theta$, and a stress of magnitude $r = \sqrt{x^2+y^2}$ is applied to a poroelastic system, the energy stored in the anisotropic media of interest here [using (16) and (18)] is given by
   \begin{eqnarray} r^2f^T(\theta)\Sigma^*f(\theta) \equiv U(r,\theta) = \qquad\qqu... ...heta\sin\theta + 2A_{33}\sin^2\theta\right] = R^2 U(r_0,\theta). \end{eqnarray}
(36)
In the second equation $R \equiv r/r_0$, and r0 in an arbitrary number (say unity) having the dimensions of stress (i.e., dimensions of Pa). It is not hard to see that, when $U(r,\theta) = const$,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 r2 out of both equations. We find that
   \begin{eqnarray} 3A_{11} = {{\cos^2\psi}\over{a^2}} + {{\sin^2\psi}\over{b^2}}, ... ... = {{\sin^2\psi}\over{a^2}} + {{\cos^2\psi}\over{b^2}}.\nonumber \end{eqnarray}
(37)
These three equations can be inverted for the parameters of the ellipse, giving:
   \begin{eqnarray} {{1}\over{a^2}} = {{3A_{11}\cos^2\psi - 6A_{33}\sin^2\psi}\over... ...qrt{2}A_{13}}\over{A_{11}-2A_{33}}}.\qquad\qquad\qquad \nonumber \end{eqnarray}
(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 $\theta$ varying from to $\pi$ [again see definition (18)]. We then have the important function $U(1,\theta)$. [Note that, when $\theta$ varies instead between $\pi$ and $2\pi$, we just get a copy of the behavior for $\theta$ between and $\pi$.The only difference is that the stress and strain vectors have an overall minus sign relative to those on the other half-circle. For a linear system, such an overall phase factor of unit magnitude is irrelevant to the mechanics of the problem.] Then, if we set $U(r,\theta) = const = R^2 U(r_0,\theta)$ and plot $z = R e^{i\theta}$in the complex plane, we will have a plot of the ellipse of interest with R determined analytically by
   \begin{eqnarray} R = \sqrt{U(r,\theta)/U(r_0,\theta)}= \sqrt{const/U(r_0,\theta)}. \end{eqnarray} (39)
We call R the magnitude of the normalized stress (i.e., normalized with respect to r0).

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 r0 above can in this case be chosen to be unity if desired, as the main purpose of the factor r0 above was to keep track of the dimensions of the stress components.

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Next: About this document ... Up: Berryman: Poroelastic fluid effects Previous: SUMMARY AND CONCLUSIONS
Stanford Exploration Project
5/23/2004