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In an elastic region, if the deformation is less than 10^{-3},
nonlinearity is assumed to be due to geometry and not due to
a nonlinear stress-strain relation.
However the boundary between geometric and material nonlinearity
cannot be drawn very clearly.
Following Biot 1965,
it is possible to compute the material deformation to second order as:
| |
(1) |

| |
(2) |

Neglecting the rotational tensor components in (1)
results in the usual definition of the elastic symmetric strain tensor *e*.
(All doubly appearing indices are summed over and the ranges are always
*i*=1,2,3.)
Following Fung 1965
we have a similar formulation for the Eulerian strain tensor:

| |
(3) |

Both equation (1) and (3)
describe the deformation of a medium to a
higher order than the usual elastic strain tensor *e*. The deformation
is a pure geometrical property and not a material property.
Consequently the linear Hooke's law:
| |
(4) |

remains unmodified. The relation between stress and strain components
is still linear. It is the computation of that changes in
each case. When considering the displacement gradients, we can see that
higher order gradient components are related to stress
components nonlinearly. However the parameters which link them,
the stiffness coefficients *c*_{ijkl}, are still linear elastic parameters.
All previous equations lead to an elastic wave equation of the form
| |
(5) |

or in the geometrically nonlinear case to a slight modification of the
previous expression:
| |
(6) |

where is the geometrically nonlinear derivative operator.

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** Up:** ABANDONING LINEARITY
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Stanford Exploration Project

11/16/1997