The Shoenberg-Muir equivalence

For the case of a 2-D elastic medium, the relations for finding the equivalent 1D Young modulus (equation (2)) needs to be extended for all the involved 2-D elastic moduli. This extension is provided by the group theory of Shoenberg and Muir (1989), which defines the equations for finding the equivalent elastic parameters.

The general elastic wave equation is given by  

$$\rho {\partial^2 u_i \over \partial t^2} - f_i = {\partial \tau_{ij} \over \partial x_j},$$ (4)

where indices i and j refers to the spatial components (x,y,z), f_i is the i component of the internal forces, and $\tau_{ij}$ is the ij component of the stress field (Auld, 1990). In the 2-D transverse isotropic case without internal sources, the x component of equation (4) becomes

$$\rho {\partial^2 u_x \over \partial t^2} = {\partial \tau_{xx} \over \partial x} + {\partial \tau_{xz} \over \partial z},$$

where

$$\tau_{xx} = c_{11} {\partial u_x \over \partial x} + c_{13}... ...\over \partial z} + {\partial u_z \over \partial x} \right).$$

Combining these equations we find for the x component

$$\begin{eqnarray} \rho {\partial^2 u_x \over \partial t^2} = {\partial \hat{c}_{1... ...er \partial z^2} + {\partial^2 u_z \over \partial x \partial z}),\end{eqnarray}$$ (5)

and for the z component

$$\begin{eqnarray} \rho {\partial^2 u_z \over \partial t^2} = {\partial \hat{c}_{3... ...er \partial x^2} + {\partial^2 u_x \over \partial z \partial x}).\end{eqnarray}$$ (6)

The equivalent stiffness constants $\hat{c}$ are obtained with the Shoenberg-Muir algebra. For displacements in the x direction,

$$\begin{eqnarray} \hat{c}^{ij}_{11} & = & {2 c^i_{11} c^j_{11} \over c^i_{11} + c... ...{13} \over c^i_{11}} + {c^j_{13} \over c^j_{11}} \right) \nonumber\end{eqnarray}$$ (7)

and for displacements in the z direction,

$$\begin{eqnarray} \hat{c}^{ij}_{33} & = & {2 c^i_{33} c^j_{33} \over c^i_{33} + c... ...13} \over c^i_{33}} + {c^j_{13} \over c^j_{33}} \right). \nonumber\end{eqnarray}$$ (8)