Given:
Beltrami's solution for the equations of equilibrium states that if
![{\displaystyle {\boldsymbol {\sigma }}={\boldsymbol {\nabla }}\times {\boldsymbol {\nabla }}\times \mathbf {A} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0701178a3867b3423826177dddd7b1784a66aef5)
where
is a stress function, then
![{\displaystyle {\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}=0~;~~{\boldsymbol {\sigma }}={\boldsymbol {\sigma }}^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e336c86defd2e8ae77c32a9973f615c1b85a3ac3)
Airy's stress function is a special form of
, given by (in 3
3 matrix notation)
![{\displaystyle \left[A\right]={\begin{bmatrix}0&0&0\\0&0&0\\0&0&\varphi \end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca557b9ba53ed9779d30cf529535c740b3bbf543)
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
![{\displaystyle \sigma _{ij}=e_{imn}~e_{jpq}~A_{mp,~nq}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/933422cb1bb1a689d6e2d5fa5491f57f2636c190)
For the Airy's stress function, the only non-zero terms of
are
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5da352b964bcd6194d09cc12223e9872eeb3dbee)
Since
for
, 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/930fcbe31b8e1815433dd7fa3a73dcf43c8d45c6)
Now,
is non-zero only if
, and
is non-zero
only if
. 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23b3f605ef476b269309866518d03f8279d743ee)
Therefore, (using the values of
,
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7468fee996366074256315cbbf35b6d2787578b6)
The equations of equilibrium (in the absence of body forces) are given by
![{\displaystyle \sigma _{ji,j}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f26bbdda7b31ccd4f1649aa567f47aff41676a1f)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0f8660eddefd8c98ca6c2383595028b03c9d4c8)
Plugging the stresses in terms of
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb249fa7cc722ff427af8f487c487a1f4cff00db)
Noting that the order of differentiation is irrelevant, we see that
equilibrium is satisfied by the Airy stress function.