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:
where formula_2 is the stress tensor, and the Beltrami-Michell compatibility equations:
A general solution of these equations may be expressed in terms 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.
Beltrami stress functions.
It can be shown that a complete solution to the equilibrium equations may be written as
Using index notation:
where formula_6 is an arbitrary second-rank tensor field that is at least twice differentiable, and is known as the "Beltrami stress tensor". Its components are known as Beltrami stress functions. formula_7 is the Levi-Civita pseudotensor, with all values equal to zero except those in which the indices are not repeated. For a set of non-repeating indices the component value will be +1 for even permutations of the indices, and -1 for odd permutations. And formula_8 is the Nabla operator. For the Beltrami stress tensor to satisfy the Beltrami-Michell compatibility equations in addition to the equilibrium equations, it is further required that formula_6 is at least four times continuously differentiable.
Maxwell stress functions.
The Maxwell stress functions are defined by assuming that the Beltrami stress tensor formula_6 is restricted to be of the form.
The stress tensor which automatically obeys the equilibrium equation may now be written as:
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:
These must also yield a stress tensor which obeys the specified boundary conditions.
Airy stress function.
The Airy stress function is a special case of the Maxwell stress functions, in which it is assumed that A=B=0 and C is a function of x and y only. This stress function can therefore be used only for two-dimensional problems. In the elasticity literature, the stress function formula_13 is usually represented by formula_14 and the stresses are expressed as
Where formula_16 and formula_17 are values of body forces in relevant direction.
In polar coordinates the expressions are:
Morera stress functions.
The Morera stress functions are defined by assuming that the Beltrami stress tensor formula_6 tensor is restricted to be of the form 
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. Substituting the expressions for the stress into the Beltrami-Michell equations yields the expression of the elastostatic problem in terms of the stress functions:
Prandtl stress function.
The Prandtl stress function is a special case of the Morera stress functions, in which it is assumed that A=B=0 and C is a function of x and y only.