Editing Linear elasticity (section)

===Cylindrical coordinate form===
In cylindrical coordinates (<math>r,\theta,z</math>) the equations of motion are<ref name="Slau" />
<math display="block">\begin{align}
  & \frac{\partial \sigma_{rr}}{\partial r} + \frac{1}{r}\frac{\partial \sigma_{r\theta}}{\partial \theta} + \frac{\partial \sigma_{rz}}{\partial z} + \cfrac{1}{r}(\sigma_{rr}-\sigma_{\theta\theta}) + F_r = \rho~\frac{\partial^2 u_r}{\partial t^2} \\
  & \frac{\partial \sigma_{r\theta}}{\partial r} + \frac{1}{r} \frac{\partial \sigma_{\theta\theta}}{\partial \theta} + \frac{\partial \sigma_{\theta z}}{\partial z} + \frac{2}{r}\sigma_{r\theta} + F_\theta = \rho~\frac{\partial^2 u_\theta}{\partial t^2} \\
  & \frac{\partial \sigma_{rz}}{\partial r} + \frac{1}{r}\frac{\partial \sigma_{\theta z}}{\partial \theta} + \frac{\partial \sigma_{zz}}{\partial z} + \frac{1}{r} \sigma_{rz} + F_z = \rho~\frac{\partial^2 u_z}{\partial t^2}
\end{align}</math>
The strain-displacement relations are
<math display="block">\begin{align}
  \varepsilon_{rr} & = \frac{\partial u_r}{\partial r} ~;~~
  \varepsilon_{\theta\theta}  = \frac{1}{r} \left(\cfrac{\partial u_\theta}{\partial \theta} + u_r\right) ~;~~
  \varepsilon_{zz}  = \frac{\partial u_z}{\partial z} \\
  \varepsilon_{r\theta} & = \frac{1}{2} \left(\cfrac{1}{r}\cfrac{\partial u_r}{\partial \theta} + \cfrac{\partial u_\theta}{\partial r}- \cfrac{u_\theta}{r}\right) ~;~~
  \varepsilon_{\theta z}  = \cfrac{1}{2} \left(\cfrac{\partial u_\theta}{\partial z} + \cfrac{1}{r}\cfrac{\partial u_z}{\partial \theta}\right) ~;~~
  \varepsilon_{zr} = \cfrac{1}{2} \left(\cfrac{\partial u_r}{\partial z} + \cfrac{\partial u_z}{\partial r}\right) 
\end{align}</math>
and the constitutive relations are the same as in Cartesian coordinates, except that the indices 1,2,3 now stand for <math>r</math>,<math>\theta</math>,<math>z</math>, respectively.