# Stress Functions - Maxwell Stress Functions

The Maxwell stress functions are defined by assuming that the Beltrami stress tensor tensor is restricted to be of the form.

$Phi_{ij}= begin{bmatrix} A&0&0\ 0&B&0\ 0&0&C end{bmatrix}$

The stress tensor which automatically obeys the equilibrium equation may now be written as:

 $sigma_x = frac{partial^2B}{partial z^2} + frac{partial^2C}{partial y^2}$ $sigma_{yz} =-frac{partial^2A}{partial y partial z}$ $sigma_y = frac{partial^2C}{partial x^2} + frac{partial^2A}{partial z^2}$ $sigma_{zx} = -frac{partial^2B}{partial z partial x}$ $sigma_z = frac{partial^2A}{partial y^2} + frac{partial^2B}{partial x^2}$ $sigma_{xy} =-frac{partial^2C}{partial x partial y}$

The solution to the elastostatic problem now consists of finding the three stress functions which give a stress tensor which obeys the Beltrami–Michell compatibility equations for stress. Substituting the expressions for the stress into the Beltrami-Michell equations yields the expression of the elastostatic problem in terms of the stress functions:

$nabla^4 A+nabla^4 B+nabla^4 C=3left( frac{partial^2 A}{partial x^2}+ frac{partial^2 B}{partial y^2}+ frac{partial^2 C}{partial z^2}right)/(2-nu),$

These must also yield a stress tensor which obeys the specified boundary conditions.