---
tag: GQ3K59MRJP
kind: concept
name: Lamé equations of elasticity
msc: [74B05, 35Q74]
specializes: [GC5BDZ6QBC]
sources:
  - url: https://en.wikipedia.org/wiki/Linear_elasticity
---

## Summary

The Lamé equations, also called the Navier–Cauchy equations, govern small deformations of a homogeneous isotropic elastic solid.
Their unknown is the displacement field $u$, which tells how far each point of the body moves.
The material enters through two constants, the Lamé parameters $\lambda$ and $\mu$.
In the static case they form an elliptic system, and the time-dependent version predicts two kinds of elastic waves, pressure waves and shear waves.

## Explanations

### L1 human

Press on a rubber block and every point inside it moves a little.
The Lamé equations say how these small displacements balance the internal stresses and the outside forces.
The parameter $\mu$ measures resistance to shearing and $\lambda$ adds resistance to change of volume.
A displacement that just moves or turns the whole body without bending it costs no stress and solves the equations with no force.
Earthquake waves are solutions of the time-dependent equations: the faster pressure waves arrive first and the shear waves after them.