Given:
Beltrami's solution for the equations of equilibrium states that if
-
where is a stress function, then
-
Airy's stress function is a special form of , given by (in 3 3 matrix notation)
-
Show:
Verify that the stresses when expressed in terms of Airy's stress function satisfy equilibrium.
In index notation, Beltrami's solution can be written as
-
For the Airy's stress function, the only non-zero terms of are which can have nine values. Therefore,
-
Since for , the above set of equations reduces to
-
Now, is non-zero only if , and is non-zero
only if . Therefore, the above equations further reduce to
-
Therefore, (using the values of , and the fact that the order of differentiation does not change the final result), we get
-
The equations of equilibrium (in the absence of body forces) are given by
-
or,
-
Plugging the stresses in terms of into the above equations gives,
-
Noting that the order of differentiation is irrelevant, we see that
equilibrium is satisfied by the Airy stress function.