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: where
is the stress tensor, and the Beltrami-Michell compatibility equations: 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.
It can be shown [1] that a complete solution to the equilibrium equations may be written as Using index notation: where
is the Levi-Civita pseudotensor, with all values equal to zero except those in which the indices are not repeated.
For the Beltrami stress tensor to satisfy the Beltrami-Michell compatibility equations in addition to the equilibrium equations, it is further required that
Substituting the expressions for the stress into the Beltrami–Michell equations yields the expression of the elastostatic problem in terms of the stress functions:[3] These must also yield a stress tensor which obeys the specified boundary conditions.
are values of body forces in relevant direction.
In polar coordinates the expressions are: The Morera stress functions are defined by assuming that the Beltrami stress tensor
tensor is restricted to be of the form [2] 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.