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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
to elastic strains *e*_{ij} is given by

| |
(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 *x*_{3}
direction as is the normal choice in the acoustics and geophysics
literature, then , , , *e*_{11},
*e*_{22}, and *e*_{12} are continuous and in fact
constant throughout such a laminated material. If the constancy of
*e*_{11}, *e*_{22}, and *e*_{12} were not so, the layers would necessarily
experience relative slip; while if the constancy of ,, and 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

| |
(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 ,
assuming as mentioned previously that
the variation is along the *z* or *x*_{3} direction, we find,
using the notation of (7),

which can then be solved to yield the expressions
| |
(10) |

| |
(11) |

| |
(12) |

| |
(13) |

| |
(14) |

and
| |
(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 is a constant, regardless of the behavior of
.Another fact that can easily be checked is that *a* = *b* + 2*m*, 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:** BULK MODULUS K AND K_{eff}
** Up:** Berryman: Poroelastic shear modulus
** Previous:** NOTATION FOR ELASTIC ANALYSYS
Stanford Exploration Project

7/8/2003