EN224: Linear Elasticity
Division of Engineering
6.4 Solving Plane Problems Using Airy Stress Functions
We proceed to develop some techniques for solving plane linear elastostatic boundary value problems.
Our initial approach is analogous to the procedure we developed to solve torsion problems. We will generate the stresses from a scalar potential, chosen so as to satisfy the equilibrium equations automatically. The governing equation for our potential then follows from the stress equations of compatibility.
This procedure actually only works if the body forces can also be derived from a scalar potential. Thus, assume
where is a scalar potential on
Let be a scalar potential on , and let
Furthermore, let satisfy
Proof: Add the first two equations defining stresses in terms of the Airy function, differentiate and substitute.
Then one may always find a scalar potential such that
Proof: Write the equations of equilibrium as
Thus, the Airy stress function is complete: all 2D elastostatic states may be derived from an Airy potential.
Airy Functions in Cylindrical-Polar Coordinates
Boundary value problems involving cylindrical regions are best solved using Cylindrical-polar coordinates. It is worth recording the governing equations for this coordinate system.
The state of stress is related to the Airy function by