Airy stress function

Summary

An Airy stress function is a scalar function $\varphi$ of two variables whose second derivatives give the components of a plane stress field. Stresses defined this way automatically satisfy the equilibrium equations without body force. For an isotropic material in plane stress or plane strain, compatibility reduces to the biharmonic equation $\Delta^2 \varphi = 0$. So a two-dimensional elasticity problem becomes a problem for one scalar function.