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In tensor notation, the relationship between components of stress and strain *u*_{k,l} is given by

| |
(1) |

where *C*_{ijkl} is the stiffness tensor,
and repeated indices on the right hand side of (3)
are summed. In (1), *u*_{k} is the *k*th Cartesian
component of the displacement vector
, and .Whereas for an isotropic elastic medium
the stiffness tensor has the form
| |
(2) |

depending on only two parameters (the Lamé constants, and ),
this tensor can have up to 21 independent constants for general
anisotropic elastic media.
The stiffness tensor has pairwise symmetry in its indices such that
*C*_{ijkl} = *C*_{jikl} and *C*_{ijkl} = *C*_{ijlk}, which will be used later
to simplify the resulting equations.
The general equation of motion for elastic wave propagation
through an anisotropic medium is given by

| |
(3) |

where is the second time derivative of the *i*th Cartesian
omponent of the displacement vector
and is the density (assumed constant).
Equation (3) is a statement that the product
of mass times acceleration of a particle is
determined by the internal stress force . For the present
purposes, we are more interested in the quasistatic limit of this equation,
in which case the left-hand side of (3) vanishes and
the equation to be satisfied is just the force equilibrium equation
| |
(4) |

A commonly used simplification of the notation for elastic analysis
is given by introducing the strain tensor, where

| |
(5) |

Then, using one version of the Voigt convention,
in which the pairwise symmetries of the stiffness tensor indices
are used to reduce the number of indices from 4 to 2 using the rules
, , , ,, and ,we have
| |
(6) |

Although the Voigt convention introduces no
restrictions on the stiffness tensor, we have chosen to limit discussion
to the form in (6), which is not completely general.
Of the 36 coefficients (of which 21 are generally independent), we choose
to treat only those cases for which the 12 coefficients shown (of which nine
are generally independent) are nonzero. This form includes all orthorhombic,
cubic, hexagonal, and isotropic systems, while excluding triclinic,
monoclinic, trigonal, and some tetragonal systems, since each of the latter
contains additional off-diagonal constants that may be nonzero.
Nevertheless, we will restrict our discussion to
(6) or to the still simpler
case of transversely isotropic (TI) materials.
For TI materials whose symmetry axis is in the *x*_{3} direction,
another common choice of notation is
, ,,, , and .There is also one further constraint on the constants that *a* = *b* + 2*m*,
following from rotational symmetry in the *x*_{1}x_{2}-plane.
In such materials, (6) may be replaced by

| |
(7) |

in which the matrix has the same symmetry as hexagonal systems and
of which isotropic symmetry is a special case
(having , , and ).

** Next:** BACKUS AVERAGING OF FINE
** Up:** Berryman: Poroelastic shear modulus
** Previous:** INTRODUCTION
Stanford Exploration Project

7/8/2003