# Introduction to Elasticity/Airy example 1

## Example 1 - Beltrami solution

Given:

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

${\displaystyle {\boldsymbol {\sigma }}={\boldsymbol {\nabla }}\times {\boldsymbol {\nabla }}\times \mathbf {A} }$

where ${\displaystyle \mathbf {A} }$ is a stress function, then

${\displaystyle {\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}=0~;~~{\boldsymbol {\sigma }}={\boldsymbol {\sigma }}^{T}}$

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

${\displaystyle \left[A\right]={\begin{bmatrix}0&0&0\\0&0&0\\0&0&\varphi \end{bmatrix}}}$

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

${\displaystyle \sigma _{ij}=e_{imn}~e_{jpq}~A_{mp,~nq}}$

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

{\displaystyle {\begin{aligned}\sigma _{11}&=e_{13n}~e_{13q}~\varphi _{,~nq}\\\sigma _{22}&=e_{23n}~e_{23q}~\varphi _{,~nq}\\\sigma _{33}&=e_{33n}~e_{33q}~\varphi _{,~nq}\\\sigma _{23}&=e_{23n}~e_{33q}~\varphi _{,~nq}\\\sigma _{31}&=e_{33n}~e_{13q}~\varphi _{,~nq}\\\sigma _{12}&=e_{13n}~e_{23q}~\varphi _{,~nq}\end{aligned}}}

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

{\displaystyle {\begin{aligned}\sigma _{11}&=e_{13n}~e_{13q}~\varphi _{,~nq}\\\sigma _{22}&=e_{23n}~e_{23q}~\varphi _{,~nq}\\\sigma _{33}&=0\\\sigma _{23}&=0\\\sigma _{31}&=0\\\sigma _{12}&=e_{13n}~e_{23q}~\varphi _{,~nq}\end{aligned}}}

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

{\displaystyle {\begin{aligned}\sigma _{11}&=e_{132}~e_{132}~\varphi _{,~22}\\\sigma _{22}&=e_{231}~e_{231}~\varphi _{,~11}\\\sigma _{33}&=0\\\sigma _{23}&=0\\\sigma _{31}&=0\\\sigma _{12}&=e_{132}~e_{231}~\varphi _{,~21}\end{aligned}}}

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

{\displaystyle {\begin{aligned}\sigma _{11}&=\varphi _{,~22}\\\sigma _{22}&=\varphi _{,~11}\\\sigma _{33}&=0\\\sigma _{23}&=0\\\sigma _{31}&=0\\\sigma _{12}&=-\varphi _{,~12}\end{aligned}}}

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

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

or,

{\displaystyle {\begin{aligned}\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{aligned}}}

Plugging the stresses in terms of ${\displaystyle \varphi }$ into the above equations gives,

{\displaystyle {\begin{aligned}\varphi _{,221}-\varphi _{,122}+0&=0\\-\varphi _{,121}+\varphi _{,112}+0&=0\\0+0+0&=0\end{aligned}}}

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