EQUATIONS FOR A POROUS CYLINDER

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 rand $\theta$ are the radial and azimuthal coordinates, it is not difficult to show that

$$\begin{eqnarray} \delta\tau_{rr} &=& He - 2\mu(e_\theta+e_z) - C\zeta, \ \delta\... ...r}\over{\partial z}} + {{\partial u_z}\over{\partial r}}\right), \end{eqnarray}$$ (201)

and (12) for $\delta p_f$ remains unchanged. The stress increments $\delta\tau_{zz}$, $\delta\tau_{\theta\theta}$,and $\delta\tau_{\theta z}$ are not of direct interest in the present application. The dilatations are given by

$$\begin{eqnarray} e = e_r + e_\theta + e_z, \end{eqnarray}$$ (202)

where

$$\begin{eqnarray} e_r = {{\partial u_r}\over{\partial r}}, \quad e_\theta = {{u_r... ...partial \theta}}, \quad e_z = {{\partial u_z}\over\partial z{}}. \end{eqnarray}$$ (203)

We redefine potential $\mbox{\boldmath$\beta$}$ in terms of two scalar potentials according to

$$\begin{eqnarray} \mbox{\boldmath$\beta$}= \hat{\bf z}\beta_1 + \nabla\times (\hat{\bf z}\beta_2), \end{eqnarray}$$ (204)

where both $\beta_i$ satisfy

$$\begin{eqnarray} (\nabla^2 + k_s^2)\beta_i = 0 \qquad\hbox{for}\qquad i = 1,2. \end{eqnarray}$$ (205)

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

$$\begin{eqnarray} A_{\pm} &=& \alpha_\pm J_0(j_\pm)\exp i(k_z z-\omega t), \ \bet... ..., \ \beta_2 &=& (\alpha_s/ik_z)J_0(j_s)\exp i(k_z z - \omega t), \end{eqnarray}$$ (206)

where

$$\begin{eqnarray} j_\pm = k_{\pm r} r, \qquad j_s = k_{sr}r \end{eqnarray}$$ (207)

and

$$\begin{eqnarray} k_{\pm r}^2 = k_\pm^2 - k_z^2, \qquad k_{sr}^2 = k_s^2 - k_z^2. \end{eqnarray}$$ (208)

J₀ is the zero-order Bessel function of the first kind. The coefficients $\alpha_\pm$, $\alpha_s$, $\gamma_s$, 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 $k_\pm$ and k_s take different values from the those in the central region, indicated by $k_\pm^*$ 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 J₀ and Y₀ (the Bessel function of the second kind, sometimes known as the Neumann function). In the outer shell, we have four coefficients apiece for J₀ and Y₀, all of which must also be determined by the boundary conditions.

Noting that

$$\begin{eqnarray} \Upsilon &=& (A_+-A_-)/(\Gamma_+-\Gamma_-),\nonumber\ \psi &=& (A_+\Gamma_--A_-\Gamma_+)/(\Gamma_--\Gamma_+) \end{eqnarray}$$

from the definitions of $A_\pm$,and substituting (40)-(42) and (45) into (23), and the result into (12) and (32)-(34), we finally obtain

$$\begin{eqnarray} \delta\tau_{r\theta} &\!\!=\!\!& m_{11}\gamma_s \equiv -\mu k_{... ...a\tau_{rz} &=& a_{31}\alpha_+ + a_{32}\alpha_- + a_{33}\alpha_s, \end{eqnarray}$$ (209)

where

$$\begin{eqnarray} (\Gamma_+ - \Gamma_-)a_{11} = \left[(C\Gamma_- -H)k_+^2 + 2\mu k_z^2\right]J_0(j_+) \nonumber\ + 2\mu k_{+r}J_1(j_+)/r,\qquad \end{eqnarray}$$

$$\begin{eqnarray} (\Gamma_+ - \Gamma_-)a_{12} = - 2\mu k_{-r}^2J_1(j_-)/j_-\qquad\nonumber\ +\left[(H-C\Gamma_+)k_-^2 - 2\mu k_z^2\right]J_0(j_-), \end{eqnarray}$$

$$\begin{eqnarray} a_{13} = 2\mu k_{sr}^2J_2(j_s), \end{eqnarray}$$ (210)

$$\begin{eqnarray} (\Gamma_+ - \Gamma_-)a_{21} = (M\Gamma_- - C)k_+^2J_0(j_+), \end{eqnarray}$$ (211)

$$\begin{eqnarray} (\Gamma_+ - \Gamma_-)a_{22} = (C-M\Gamma_+)k_-^2J_0(j_-), \end{eqnarray}$$ (212)

$$\begin{eqnarray} (\Gamma_+ - \Gamma_-)a_{31} = -2i\mu k_zk_{+r}J_1(j_+), \end{eqnarray}$$ (213)

$$\begin{eqnarray} (\Gamma_+ - \Gamma_-)a_{32} = 2i\mu k_zk_{-r}J_1(j_-), \end{eqnarray}$$ (214)

$$\begin{eqnarray} ik_z a_{33} = -\mu(k_s^2-2k_z^2)k_{sr}J_1(j_s), \end{eqnarray}$$ (215)

and a₂₃=0. There is an implicit factor of $\exp i(k_z z - \omega t)$on the right-hand side of (46)-(49).

Berryman (1983) has shown that a₁₁, a₁₃, a₃₁, and a₃₃ reduce in the limit $\phi \to 0$ to the corresponding results for isotropic elastic cylinders by Pochhammer (1876), Chree (1886, 1889), Love (1941), and Bancroft (1941), as they should.