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{align}

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

\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{align}$$ Now, $$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{align}

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

\sigma_{11} & = \varphi_{,~22} \\ \sigma_{22} & = \varphi_{,~11} \\ \sigma_{33} & = 0 \\ \sigma_{23} & = 0 \\ \sigma_{31} & = 0 \\ \sigma_{12} & = -\varphi_{,~12} \end{align}$$ The equations of equilibrium (in the absence of body forces) are given by

\sigma_{ji,j} = 0 $$ or,
 * $$\begin{align}

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

\varphi_{,221} - \varphi_{,122} + 0 & = 0 \\ -\varphi_{,121} + \varphi_{,112} + 0 & = 0 \\ 0 + 0 + 0 & = 0 \end{align}$$ Noting that the order of differentiation is irrelevant, we see that equilibrium is satisfied by the Airy stress function.