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:
Contents
- Beltrami stress functions
- Maxwell stress functions
- Airy stress function
- Morera stress functions
- Prandtl stress function
- References
where
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
Maxwell stress functions
The Maxwell stress functions are defined by assuming that the Beltrami stress tensor
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
Where
In polar coordinates the expressions are:
Morera stress functions
The Morera stress functions are defined by assuming that the Beltrami stress tensor
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.