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To work most easily in cylindrical geometry, we rewrite the
stress-strain relations (06)-(12) in cylindrical
coordinates. If *z* is the coordinate along the cylinder axis while
*r*and are the radial and azimuthal coordinates, it is not
difficult to show that

| |
(17) |

and (12) for remains unchanged.
The stress increments , ,and are not of direct interest in the present
application. The dilatations are given by
| |
(18) |

where
| |
(19) |

We redefine potential in terms of two scalar potentials
according to

| |
(20) |

where both satisfy
| |
(21) |

For the problem of interest here, we will have two distinct regions:
The first region is a cylinder centered at the origin, within which
solutions of (25) and (39) must be finite at the origin.
Results take the form

| |
(22) |

where
| |
(23) |

and
| |
(24) |

*J*_{0} is the zero-order Bessel function of the first kind.
The coefficients , , , are constants to
be determined from the boundary conditions.
The second region is a cylindrical shell around the first region.
In this region, the factors and *k*_{s} take different values
from the those in the central region, indicated by and
*k*_{s}^{*} (where * means air-filled, and does *not* ever mean complex
conjugate in this paper). Furthermore, two linearly
independent solutions of the equations are allowed, *i.e.*, both *J*_{0} and
*Y*_{0} (the Bessel function of the second kind, sometimes known
as the Neumann function). In the outer shell, we have four coefficients
apiece for *J*_{0} and *Y*_{0}, all of which must also be determined by
the boundary conditions.

Noting that

from the definitions of ,and substituting (40)-(42) and (45)
into (23), and the result into (12) and
(32)-(34), we finally obtain
| |
(25) |

where
| |
(26) |

| |
(27) |

| |
(28) |

| |
(29) |

| |
(30) |

| |
(31) |

and *a*_{23}=0.
There is an implicit factor of on the right-hand side of (46)-(49).
Berryman (1983) has shown that *a*_{11}, *a*_{13}, *a*_{31}, and
*a*_{33} reduce in the limit to the corresponding
results for isotropic elastic cylinders by
Pochhammer (1876), Chree (1886, 1889), Love (1941), and
Bancroft (1941),
as they should.

** Next:** BOUNDARY CONDITIONS
** Up:** Berryman and Pride: Cylinder
** Previous:** EQUATIONS OF POROELASTICITY
Stanford Exploration Project

11/11/2002