Airy stress function

Definition
The Airy stress function ($$\varphi$$):
 * Scalar potential function that can be used to find the stress.
 * Satisfies equilibrium in the absence of body forces.
 * Only for two-dimensional problems (plane stress/plane strain).

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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} $$
 * }

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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} $$
 * }

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 $$    \nabla^2 {\psi} = 0 ~, \psi_{,12} = \nabla^2 {\varphi} $$

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

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