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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