Elasticity/Airy example 1

Example 1 - Beltrami solution

Given:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

or,

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

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

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

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

Elasticity