Elasticity/Airy example 1

Example 1 - Beltrami solution

Given:

Beltrami's solution for the equations of equilibrium states that if

$\mathit{\sigma }=\mathrm{\nabla }\times \mathrm{\nabla }\times 𝐀$

where $𝐀$ is a stress function, then

$\mathrm{\nabla }\bullet \mathit{\sigma }=0;\mathit{\sigma }={\mathit{\sigma }}^T$

Airy's stress function is a special form of $𝐀$, given by (in 3$\times$3 matrix notation)

$\left[A\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& \phi \end{array}\right]$

Show:

Verify that the stresses when expressed in terms of Airy's stress function satisfy equilibrium.

Solution

In index notation, Beltrami's solution can be written as

${\sigma }_{ij}=e_{imn}{e}_{jpq}{A}_{mp,nq}$

For the Airy's stress function, the only non-zero terms of $A_{mp,nq}$ are $A_{33,nq}={\phi }_{,nq}$ which can have nine values. Therefore,

$\begin{array}{cc}{\sigma }_{11}& =e_{13n}{e}_{13q}{\phi }_{,nq}\\ {\sigma }_{22}& =e_{23n}{e}_{23q}{\phi }_{,nq}\\ {\sigma }_{33}& =e_{33n}{e}_{33q}{\phi }_{,nq}\\ {\sigma }_{23}& =e_{23n}{e}_{33q}{\phi }_{,nq}\\ {\sigma }_{31}& =e_{33n}{e}_{13q}{\phi }_{,nq}\\ {\sigma }_{12}& =e_{13n}{e}_{23q}{\phi }_{,nq}\end{array}$

Since $e_{33k}=0$ for $k=1,2,3$, the above set of equations reduces to

$\begin{array}{cc}{\sigma }_{11}& =e_{13n}{e}_{13q}{\phi }_{,nq}\\ {\sigma }_{22}& =e_{23n}{e}_{23q}{\phi }_{,nq}\\ {\sigma }_{33}& =0\\ {\sigma }_{23}& =0\\ {\sigma }_{31}& =0\\ {\sigma }_{12}& =e_{13n}{e}_{23q}{\phi }_{,nq}\end{array}$

Now, $e_{13k}$ is non-zero only if $k=2$, and $e_{23k}$ is non-zero only if $k=1$. Therefore, the above equations further reduce to

$\begin{array}{cc}{\sigma }_{11}& =e_{132}{e}_{132}{\phi }_{,22}\\ {\sigma }_{22}& =e_{231}{e}_{231}{\phi }_{,11}\\ {\sigma }_{33}& =0\\ {\sigma }_{23}& =0\\ {\sigma }_{31}& =0\\ {\sigma }_{12}& =e_{132}{e}_{231}{\phi }_{,21}\end{array}$

Therefore, (using the values of $e_{132}$, $e_{231}$ and the fact that the order of differentiation does not change the final result), we get

$\begin{array}{cc}{\sigma }_{11}& ={\phi }_{,22}\\ {\sigma }_{22}& ={\phi }_{,11}\\ {\sigma }_{33}& =0\\ {\sigma }_{23}& =0\\ {\sigma }_{31}& =0\\ {\sigma }_{12}& =-{\phi }_{,12}\end{array}$

The equations of equilibrium (in the absence of body forces) are given by

${\sigma }_{ji,j}=0$

or,

$\begin{array}{cc}{\sigma }_{11,1}+{\sigma }_{21,2}+{\sigma }_{31,3}& =0\\ {\sigma }_{12,1}+{\sigma }_{22,2}+{\sigma }_{32,3}& =0\\ {\sigma }_{13,1}+{\sigma }_{23,2}+{\sigma }_{33,3}& =0\end{array}$

Plugging the stresses in terms of $\phi$ into the above equations gives,

$\begin{array}{cc}{\phi }_{,221}-{\phi }_{,122}+0& =0\\ -{\phi }_{,121}+{\phi }_{,112}+0& =0\\ 0+0+0& =0\end{array}$

Noting that the order of differentiation is irrelevant, we see that equilibrium is satisfied by the Airy stress function.