The Linear elasticity reference article from the English Wikipedia on 24-Apr-2004
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Linear elasticity

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Table of contents
1 Linear elasticity
2 Basic equations
3 Wave equation
4 Plane waves
5 Isotropic media
6 References

Linear elasticity

The linear theory of elasticity models the macroscopic mechanical properties of solids assuming "small" deformations.

Basic equations

Linear elastodynamics is based on three tensor equations: 

where:

Wave equation

From the basic equations one gets the wave equation



where

 

is the acoustic differential operator, and  is Kronecker delta.

Plane waves

A plane wave has the form



with  of unit length. 
It is a solution of the wave equation with zero forcing, if and only if 

and  constitute an eigenvalue/eigenvector pair of the


acoustic algebraic operator



This propagation condition may be written as



where

denotes propagation direction
and  is phase velocity.

Isotropic media

In isotropic media, the elasticity tensor has the form



where 

is incompressibility, and



is rigidity.


Hence the acoustic algebraic operator becomes 



where 

denotes the tensor product, 



is the identity matrix, and




are the eigenvalues of 

with eigenvectors  parallel and orthogonal to the propagation direction 
, respectively.
In the seismological literature, the corresponding plane waves are called P-waves and S-waves (see Seismic wave).


References