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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:
• dynamic equation

• constitutive equation (anisotropic Hooke's law)

• kinematic equation

where:
• is stress
• is body force
• is density
• is displacement
• is elasticity tensor
• is strain

## 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

## 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

• Gurtin M. E., Introduction to Continuum Mechanics, Academic Press 1981
• L. D. Landau & E. M. Lifschitz, Theory of Elasticity, Butterworth 1986