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BACKUS AVERAGING OF FINE ELASTIC LAYERS

Backus (1962) presents an elegant method of producing the effective constants for a finely layered medium composed of either isotropic or anisotropic elastic layers. For simplicity, we will assume that the layers are isotropic, in which case the equation relating elastic stresses $\sigma_{ij}$ to elastic strains eij is given by
   \begin{eqnarray}
\left(
\begin{array}
{cccccc} \sigma_{11} \sigma_{22} \sigma_{3...
 ...{11} e_{22} e_{33} \
 e_{23} e_{31} e_{12} \\ end{array}\right).
 \end{eqnarray} (8)
The key idea presented by Backus is that these equations can be rearranged into a form where rapidly varying coefficients multiply slowly varying stresses or strains. For simple layering, we know physically (and can easily prove mathematically) that the normal stress and the tangential strains must be continuous at the boundaries between layers. If the layering direction is the z or x3 direction as is the normal choice in the acoustics and geophysics literature, then $\sigma_{33}$, $\sigma_{23}$, $\sigma_{31}$, e11, e22, and e12 are continuous and in fact constant throughout such a laminated material. If the constancy of e11, e22, and e12 were not so, the layers would necessarily experience relative slip; while if the constancy of $\sigma_{33}$,$\sigma_{23}$, and $\sigma_{31}$ were not so, then there would be force gradients across boundaries necessarily resulting in nonstatic material response to the lack of force equilibrium.

By making use of this elegant idea, we arrive at the following equation
   \begin{eqnarray}
\left(
\begin{array}
{cccccc} \sigma_{11} \sigma_{22} -e_{33} \...
 ...gma_{33} \
 \sigma_{23} \sigma_{31} e_{12} \\ end{array}\right),
 \end{eqnarray} (9)
which can be averaged essentially by inspection. Equation (9) can be viewed as a Legendre transform of the original equation, to a different set of dependent/independent variables in which both vectors have components with mixed physical significance, some being stresses and some being strains. Otherwise these equations are completely equivalent to the original ones in (8).

Performing the layer average using the symbol $\left<\cdot\right\gt$, assuming as mentioned previously that the variation is along the z or x3 direction, we find, using the notation of (7),
   \begin{eqnarray}
\left(
\begin{array}
{cccccc} \left<\sigma_{11}\right\gt \left<...
 ...gma_{33} \
 \sigma_{23} \sigma_{31} e_{12} \\ end{array}\right),
 \end{eqnarray}
which can then be solved to yield the expressions
   \begin{eqnarray}
a = \left<{{\lambda}\over{\lambda+2\mu}}\right\gt^2
 \left<{{1}...
 ...{-1}
 + 4\left<{{\mu(\lambda+\mu)}\over{\lambda+2\mu}}\right\gt,
 \end{eqnarray} (10)
   \begin{eqnarray}
b = \left<{{\lambda}\over{\lambda+2\mu}}\right\gt^2
 \left<{{1}...
 ...\gt^{-1}
 + 2 \left<{{\lambda\mu}\over{\lambda+ 2\mu}}\right\gt,
 \end{eqnarray} (11)
   \begin{eqnarray}
c = \left<{{1}\over{\lambda+2\mu}}\right\gt^{-1}
 \end{eqnarray} (12)
   \begin{eqnarray}
f = \left<{{\lambda}\over{\lambda+2\mu}}\right\gt
 \left<{{1}\over{\lambda+2\mu}}\right\gt^{-1},
 \end{eqnarray} (13)
   \begin{eqnarray}
l = \left<{{1}\over{\mu}}\right\gt^{-1}
 \end{eqnarray} (14)
and
   \begin{eqnarray}
m = \left<\mu\right\gt.
 \end{eqnarray} (15)
Equations (11)-(16) are the well-known results of Backus (1962) for layering of isotropic elastic materials. One very important fact that is known about these equations is that they reduce to isotropic results, having a=c, b=f, and l=m, if the shear modulus $\mu$ is a constant, regardless of the behavior of $\lambda$.Another fact that can easily be checked is that a = b + 2m, which is a general condition (mentioned earlier) that must be satisfied for all transversely isotropic materials and shows that there are only five independent constants.


next up previous print clean
Next: BULK MODULUS K AND Keff Up: Berryman: Poroelastic shear modulus Previous: NOTATION FOR ELASTIC ANALYSYS
Stanford Exploration Project
7/8/2003