Example 1 - Beltrami solution

edit

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.

Solution

edit

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.