Stress functions

In linear elasticity, the equations describing the deformation of an elastic body subject only to surface forces (or body forces that could be expressed as potentials) on the boundary are (using index notation) the equilibrium equation:

${\displaystyle \sigma _{ij,i}=0\,}$

where ${\displaystyle \sigma }$ is the stress tensor, and the Beltrami-Michell compatibility equations:

${\displaystyle \sigma _{ij,kk}+{\frac {1}{1+\nu }}\sigma _{kk,ij}=0}$

A general solution of these equations may be expressed in terms of the Beltrami stress tensor. Stress functions are derived as special cases of this Beltrami stress tensor which, although less general, sometimes will yield a more tractable method of solution for the elastic equations.