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

Then

Proof: Substitute…

Furthermore, let satisfy

Then

Proof: Add the first two equations defining stresses in terms of the Airy function, differentiate and substitute.

Conversely, let

Then one may always find a scalar potential such that

Proof: Write the equations of equilibrium as

Furthermore, if

Then

Proof: substitute.

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