# Airy stress function with body force
## 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,
    ${\begin{aligned}\sigma _{11,1}+\sigma _{21,2}+f_{1}&=0\qquad {\text{(6)}}\\\sigma _{12,1}+\sigma _{22,2}+f_{2}&=0\qquad {\text{(7)}}\end{aligned}}$
In terms of $\varphi$,
    ${\begin{aligned}\varphi _{,122}+V_{,1}-\varphi _{,212}+f_{1}&=0\qquad {\text{(8)}}\\-\varphi _{,112}+\varphi _{,211}+V_{,2}+f_{2}&=0\qquad {\text{(9)}}\end{aligned}}$
or,
    ${\begin{aligned}V_{,1}+f_{1}&=0\qquad {\text{(10)}}\\V_{,2}+f_{2}&=0\qquad {\text{(11)}}\end{aligned}}$
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
    ${\begin{aligned}\left({\cfrac {\partial ^{2}}{\partial r^{2}}}+{\cfrac {1}{r}}{\cfrac {\partial }{\partial r}}+{\cfrac {1}{r^{2}}}{\cfrac {\partial ^{2}}{\partial \theta ^{2}}}\right)&\left({\cfrac {\partial ^{2}\varphi }{\partial r^{2}}}+{\cfrac {1}{r}}{\cfrac {\partial \varphi }{\partial r}}+{\cfrac {1}{r^{2}}}{\cfrac {\partial ^{2}\varphi }{\partial \theta ^{2}}}\right)+\\&\left(2-{\cfrac {1}{\alpha }}\right)\left({\cfrac {\partial ^{2}V}{\partial r^{2}}}+{\cfrac {1}{r}}{\cfrac {\partial V}{\partial r}}+{\cfrac {1}{r^{2}}}{\cfrac {\partial ^{2}V}{\partial \theta ^{2}}}\right)=0\qquad {\text{(27)}}\end{aligned}}$

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