# Airy stress function

## Definition

The Airy stress function ($\varphi$):

• Scalar potential function that can be used to find the stress tensor.
• Satisfies equilibrium in the absence of body forces.
• Only for two-dimensional problems (plane stress/plane strain).

### Airy stress function in rectangular Cartesian coordinates

If the coordinate basis is rectangular Cartesian $(\mathbf{e}_1,~\mathbf{e}_2)$ with coordinates denoted by $(x_1, ~x_2)$ then the Airy stress function $(\varphi)$ is related to the components of the Cauchy stress tensor $(\boldsymbol{\sigma})$ by

\begin{align} \sigma_{11} &= \varphi_{,22} = \cfrac{\partial^2 \varphi}{\partial x_2^2} \\ \sigma_{22} &= \varphi_{,11} = \cfrac{\partial^2 \varphi}{\partial x_1^2} \\ \sigma_{12} &= -\varphi_{,12}= -\cfrac{\partial^2 \varphi}{\partial x_1 \partial x_2} \end{align}

Alternatively, if we write the basis as $(\mathbf{e}_x, \mathbf{e}_y)$ and the coordinates as $(x, y)\,$, then the Cauchy stress components are related to the Airy stress function by

\begin{align} \sigma_{xx} &= \cfrac{\partial^2 \varphi}{\partial y^2} \\ \sigma_{yy} &= \cfrac{\partial^2 \varphi}{\partial x^2} \\ \sigma_{xy} &= -\cfrac{\partial^2 \varphi}{\partial x \partial y} \end{align}

### Airy stress function in polar coordinates

In polar basis $(\mathbf{e}_r, \mathbf{e}_\theta)$ with co-ordinates $(r, \theta)\,$, the Airy stress function is related to the components of the Cauchy stress via

\begin{align} \sigma_{rr} & = \cfrac{1}{r}\cfrac{\partial\varphi}{\partial r} + \cfrac{1}{r^2}\cfrac{\partial^2\varphi}{\partial \theta^2} \\ \sigma_{\theta\theta} & = \cfrac{\partial^2\varphi}{\partial r^2} \\ \sigma_{r\theta} & = -\cfrac{\partial}{\partial r} \left(\cfrac{1}{r}\cfrac{\partial\varphi}{\partial \theta}\right) \end{align}
 Something to think about ... Do you think the Airy stress function can be extended to three dimensions?

## Stress equation of compatibility in 2-D

In the absence of body forces,

$\nabla^2{(\sigma_{11} + \sigma_{22})} = 0$

or,

$\sigma_{11,11} + \sigma_{11,22} + \sigma_{22,11} + \sigma_{22,22} = 0 \,$
• Note that the stress field is independent of material properties in the absence of body forces (or homogeneous body forces).
• Therefore, the plane strain and plane stress solutions are the same if the boundary conditions are expressed as traction BCS.

In terms of the Airy stress function

$\varphi_{,1122} + \varphi_{,2222} + \varphi_{,1111} + \varphi_{,1122} = 0 \,$

or,

$\cfrac{\partial^4 \varphi}{\partial x_1^4} + 2\cfrac{\partial^4 \varphi}{\partial x_1^2 \partial x_2^2} + \cfrac{\partial^4 \varphi}{\partial x_2^4} = 0$

or,

$\nabla^4\varphi = 0 \,$
• The stress function $(\varphi)$ is biharmonic.
• Any polynomial in $x_1$ and $x_2$ of degree less than four is biharmonic.
• Stress fields that are derived from an Airy stress function which satisfies the biharmonic equation will satisfy equilibrium and correspond to compatible strain fields.

## Some biharmonic Airy stress functions

In cylindrical co-ordinates, some biharmonic functions that may be used as Airy stress functions are

\begin{align} \varphi &= C\theta \\ \varphi &= Cr^2\theta \\ \varphi &= Cr\theta\cos\theta \\ \varphi &= Cr\theta\sin\theta \\ \varphi &= f_n(r)\cos(n\theta) \\ \varphi &= f_n(r)\sin(n\theta) \\ \end{align}

where

\begin{align} f_0(r) & = a_0 r^2 + b_0 r^2 \ln r + c_0 + d_0 \ln r \\ f_1(r) & = a_1 r^3 + b_1 r + c_1 r \ln r + d_1 r^{-1} \\ f_n(r) & = a_n r^{n+2} + b_n r^n + c_n r^{-n+2} + d_n r^{-n} ~,~~n > 1 \end{align}

## Displacements in terms of scalar potentials

If the body force is negligible, then the displacements components in 2-D can be expressed as

$2\mu u_1 = -\varphi_{,1} + \alpha \psi_{,2} ~,~~ 2\mu u_2 = -\varphi_{,2} + \alpha \psi_{,1}$

where,

$\alpha = \begin{cases} 1-\nu & \text{for plane strain} \\ \cfrac{1}{1+\nu} & \text{for plane stress} \end{cases}$

and $\psi(x_1,x_2)$ is a scalar displacement potential function that satisfies the conditions

$\nabla^4 {\psi} = 0 ~,~~ \psi_{,12} = \nabla^4 {\varphi}$

To prove the above, you have to use the plane strain/stress constitutive relations

$\begin{matrix} \sigma_{\alpha\beta} & = 2\mu\left[\varepsilon_{\alpha\beta} + \left(\cfrac{1-\alpha}{2\alpha-1}\right) \varepsilon_{\gamma\gamma}\delta_{\alpha\beta}\right] \\ \varepsilon_{\alpha\beta} & = \cfrac{1}{2\mu}\left[\sigma_{\alpha\beta} + \left(1-\alpha\right) \sigma_{\gamma\gamma}\delta_{\alpha\beta}\right] \end{matrix}$

Note also that the plane stress/strain compatibility equations can be written as

$\nabla^2{\sigma_{\gamma\gamma}} = -\cfrac{1}{\alpha} f_{\gamma,\gamma}$

## Related Content

Introduction to Elasticity