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Linear elasticity

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 studies - with mathematical methods - 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