The Navier-Cauchy equation describes elastodynamics: the movements inside an elastic solid in response to external forces and/or internal stresses.
For a particle of the solid, whose position is given by the displacement field , Newton’s second law is as follows, where and are the particle’s mass and volume, respectively:
Where is the mass density, and is the effective force density, defined from the Cauchy stress tensor like so, with being an external body force, e.g. from gravity:
We can therefore write Newton’s second law as follows, while switching to index notation, where is the partial derivative with respect to the th coordinate:
The components of the Cauchy stress tensor are given by Hooke’s law, where and are the Lamé coefficients, which describe the material:
In turn, the components of the Cauchy strain tensor are defined as follows, where are once again the components of the displacement vector :
To derive the Navier-Cauchy equation, we start by inserting Hooke’s law into Newton’s law:
And then into this we insert the definition of the strain components , yielding:
Rearranging this a bit leads us to the Navier-Cauchy equation written in index notation:
Traditionally, it is written in vector notation instead, in which case it looks like this:
A special case is the Navier-Cauchy equilibrium equation, where the left-hand side is just zero. That version describes elastostatics: the deformation of a solid at rest.
- B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.