# Airy stress function with body force potential

If a body force exists, the Airy stress function ($\varphi$) has to be combined with a body force potential ($V$). Thus,

$\sigma_{11} = \varphi_{,22} + V ~;~~ \sigma_{22} = \varphi_{,11} + V ~;~~ \sigma_{12} = -\varphi_{,12} \qquad \text{(3)}$

or,

$\sigma_{11} = \cfrac{\partial^2\varphi}{\partial x_2^2} + V ~;~~ \sigma_{22} = \cfrac{\partial^2\varphi}{\partial x_1^2} + V ~;~~ \sigma_{12} = -\cfrac{\partial^2\varphi}{\partial x_1 \partial x_2} \qquad \text{(4)}$

## Is equilibrium still satisfied ?

Recall the equilibrium equation in two dimensions:

$\boldsymbol{\nabla}\bullet{\boldsymbol{\sigma}} + \mathbf{f} = 0 ~;~~ \sigma_{\beta\alpha, \beta} + f_{\alpha} = 0 \qquad \text{(5)}$

or,

\begin{align} \sigma_{11,1} + \sigma_{21,2} + f_1 & = 0 \qquad \text{(6)}\\ \sigma_{12,1} + \sigma_{22,2} + f_2 & = 0 \qquad \text{(7)} \end{align}

In terms of $\varphi$,

\begin{align} \varphi_{,122} + V_{,1} - \varphi_{,212} + f_1 & = 0 \qquad \text{(8)}\\ -\varphi_{,112} + \varphi_{,211} + V_{,2} + f_2 & = 0 \qquad \text{(9)} \end{align}

or,

\begin{align} V_{,1} + f_1 & = 0 \qquad \text{(10)}\\ V_{,2} + f_2 & = 0 \qquad \text{(11)} \end{align}

Therefore,

$f_1 = - V_{,1} ~;~~ f_2 = - V_{,2} \qquad \text{(12)}$

or,

$f_1 = - \cfrac{\partial V}{\partial x_1} ~;~~ f_2 = - \cfrac{\partial V}{\partial x_2}\qquad \text{(13)}$

Hence,

$\mathbf{f} = -\boldsymbol{\nabla}{V}\qquad \text{(14)}$
• Equilibrium is satisfied only if the body force field can be expressed as the gradient of a scalar potential.
• A force field that can be expressed as the gradient of a scalar potential is called conservative.

## What condition is needed to satisfy compatibility ?

Recall that the compatibility condition in terms of the stresses can be written as

$\nabla^2{\sigma_{\gamma\gamma}} = -\cfrac{1}{\alpha} f_{\gamma,\gamma} \qquad \text{(15)}$

where,

$\alpha = \begin{cases} 1-\nu & \rm{for~ plane~ strain} \\ \cfrac{1}{1+\nu} & \rm{for~ plane~ stress} \end{cases} \qquad \text{(16)}$

or,

$\nabla^2{(\sigma_{11}+\sigma_{22})} + \cfrac{1}{\alpha}(f_{1,1}+f_{2,2}) = 0 \qquad \text{(17)}$

or,

$\sigma_{11,11}+\sigma_{11,22}+\sigma_{22,11}+\sigma_{22,22} + \cfrac{1}{\alpha}(f_{1,1}+f_{2,2}) = 0 \qquad \text{(18)}$

which is the same as,

$\cfrac{\partial^2\sigma_{11}}{\partial x_1^2}+ \cfrac{\partial^2\sigma_{11}}{\partial x_2^2}+ \cfrac{\partial^2\sigma_{22}}{\partial x_1^2}+ \cfrac{\partial^2\sigma_{22}}{\partial x_2^2}+ \cfrac{1}{\alpha}(\cfrac{\partial f_1}{\partial x_1}+\cfrac{\partial f_2}{\partial x_2}) = 0 \qquad \text{(19)}$

Plug in the stress potential and the body force potential in equation (18) to get

$\varphi_{,2211}+V_{,11}+\varphi_{,2222}+V_{,22}+ \varphi_{,1111}+V_{,11}+\varphi_{,1122}+V_{,22}+ \cfrac{1}{\alpha}(-V_{,11}-V_{,22}) = 0 \qquad \text{(20)}$

or,

$2\varphi_{,2211}+2V_{,11}+\varphi_{,2222}+2V_{,22}+ \varphi_{,1111}+ \cfrac{1}{\alpha}(-V_{,11}-V_{,22}) = 0 \qquad \text{(21)}$

Rearrange to get

$\varphi_{,1111} + 2\varphi_{,2211} + \varphi_{,2222} + \left(2 - \cfrac{1}{\alpha}\right) (V_{,11} + V_{,22}) = 0 \qquad \text{(22)}$

which is the same as,

$\cfrac{\partial^4\varphi}{\partial x_1^4} + 2\cfrac{\partial^4 \varphi}{\partial x_1^2 \partial x_2^2} + \cfrac{\partial^4\varphi}{\partial x_2^4} + \left(2 - \cfrac{1}{\alpha}\right) \left(\cfrac{\partial^2 V}{\partial x_1^2} + \cfrac{\partial^2 V}{\partial x_2^2}\right) = 0 \qquad \text{(23)}$

Therefore,

$\nabla^4{\varphi} + \left(2 - \cfrac{1}{\alpha}\right) \nabla^2{V} = 0 \qquad \text{(24)}$

Compatibility is satisfied only if equation (24) is satisfied.

# Equations for Airy stress function with body force potential

The relation between the Cauchy stress and the Airy stress function is (in direct tensor notation)

$\boldsymbol{\sigma} = \boldsymbol{\nabla}\times{\boldsymbol{\nabla}\times{\varphi}}$

The relation between the body force and the body force potential is

$\mathbf{f} = -\boldsymbol{\nabla}{V}$

We also have to satisfy the compatibility condition for the Airy stress function to be a true stress potential, i.e.,

$\nabla^4{\varphi} + \left(2 - \cfrac{1}{\alpha}\right) \nabla^2{V} = 0$

## In rectangular Cartesian coordinates

In rectangular Cartesian coordinates, the relation between the Cauchy stress components and the Airy stress function + body force potential can be written as

$\sigma_{11} = \varphi_{,22} + V ~;~~ \sigma_{22} = \varphi_{,11} + V ~;~~ \sigma_{12} = -\varphi_{,12}$

or,

$\sigma_{xx} = \cfrac{\partial^2\varphi}{\partial y^2} + V ~;~~ \sigma_{yy} = \cfrac{\partial^2\varphi}{\partial x^2} + V ~;~~ \sigma_{xy} = -\cfrac{\partial^2\varphi}{\partial x \partial y}$

The relation between the body force components and the body force potential are:

$f_1 = - V_{,1} ~;~~ f_2 = - V_{,2}$

or,

$f_x = - \cfrac{\partial V}{\partial x} ~;~~ f_y = - \cfrac{\partial V}{\partial y}$

The compatibility condition is written as

$\varphi_{,1111} + 2\varphi_{,2211} + \varphi_{,2222} + \left(2 - \cfrac{1}{\alpha}\right) (V_{,11} + V_{,22}) = 0$

or,

$\cfrac{\partial^4 \varphi}{\partial x^4} + 2\cfrac{\partial^4 \varphi}{\partial x^2 \partial y^2} + \cfrac{\partial^4 \varphi}{\partial y^4} + \left(2 - \cfrac{1}{\alpha}\right) \left(\cfrac{\partial^2 V}{\partial x^2} + \cfrac{\partial^2 V}{\partial y^2}\right) = 0$

## In cylindrical coordinates

In cylindrical coordinates, the relation between the Cauchy stresses and the Airy stress function + body force potential can be written as

$\sigma_{rr} = \cfrac{1}{r}\cfrac{\partial\varphi}{\partial r} + \cfrac{1}{r^2}\cfrac{\partial^2\varphi}{\partial \theta^2} + V ~;~~ \sigma_{\theta\theta} = \cfrac{\partial^2\varphi}{\partial r^2} + V ~;~~ \sigma_{r\theta} = -\cfrac{\partial}{\partial r} \left(\cfrac{1}{r}\cfrac{\partial\varphi}{\partial \theta}\right) \qquad \text{(25)}$

The components of the body force are related to the body force potential via

$f_r = - \cfrac{\partial V}{\partial r} ~;~~ f_{\theta} = - \cfrac{1}{r}\cfrac{\partial V}{\partial \theta} \qquad \text{(26)}$

The compatibility condition can be expressed as

\begin{align} \left(\cfrac{\partial^2}{\partial r^2} + \cfrac{1}{r}\cfrac{\partial}{\partial r} + \cfrac{1}{r^2}\cfrac{\partial^2}{\partial \theta^2}\right) & \left(\cfrac{\partial^2\varphi}{\partial r^2} + \cfrac{1}{r}\cfrac{\partial\varphi}{\partial r} + \cfrac{1}{r^2}\cfrac{\partial^2\varphi}{\partial \theta^2}\right) + \\ &\left(2 - \cfrac{1}{\alpha}\right) \left(\cfrac{\partial^2 V}{\partial r^2} + \cfrac{1}{r}\cfrac{\partial V}{\partial r} + \cfrac{1}{r^2}\cfrac{\partial^2 V}{\partial \theta^2}\right) = 0 \qquad \text{(27)} \end{align}

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