Editing Linear elasticity (section)

=== Spherical coordinate form ===
In spherical coordinates (<math>r,\theta,\phi</math>) the equations of motion are<ref name="Slau" />
<math display="block">\begin{align}
    & \frac{\partial \sigma_{rr}}{\partial r} + \cfrac{1}{r}\frac{\partial \sigma_{r\theta}}{\partial \theta} + \cfrac{1}{r\sin\theta}\frac{\partial \sigma_{r\phi}}{\partial \phi} + \cfrac{1}{r} (2\sigma_{rr}-\sigma_{\theta\theta}-\sigma_{\phi\phi}+\sigma_{r\theta}\cot\theta) + F_r = \rho~\frac{\partial^2 u_r}{\partial t^2} \\
    & \frac{\partial \sigma_{r\theta}}{\partial r} + \cfrac{1}{r}\frac{\partial \sigma_{\theta\theta}}{\partial \theta} + \cfrac{1}{r\sin\theta}\frac{\partial \sigma_{\theta \phi}}{\partial \phi} + \cfrac{1}{r}[(\sigma_{\theta\theta}-\sigma_{\phi\phi})\cot\theta + 3\sigma_{r\theta}] + F_\theta = \rho~\frac{\partial^2 u_\theta}{\partial t^2} \\
    & \frac{\partial \sigma_{r\phi}}{\partial r} + \cfrac{1}{r}\frac{\partial \sigma_{\theta \phi}}{\partial \theta} + \cfrac{1}{r\sin\theta}\frac{\partial \sigma_{\phi\phi}}{\partial \phi} + \cfrac{1}{r}(2\sigma_{\theta\phi}\cot\theta+3\sigma_{r\phi}) + F_\phi = \rho~\frac{\partial^2 u_\phi}{\partial t^2}
\end{align}</math>
[[File:3D Spherical.svg|thumb|240px|right|Spherical coordinates (''r'', '' θ'', ''φ'') as commonly used in ''physics'': radial distance ''r'', polar angle ''θ'' ([[theta]]), and azimuthal angle ''φ'' ([[phi]]). The symbol ''ρ'' ([[rho]]) is often used instead of ''r''.]]
The strain tensor in spherical coordinates is
<math display="block">\begin{align}
  \varepsilon_{rr} & = \frac{\partial u_r}{\partial r}\\
  \varepsilon_{\theta\theta}& = \frac{1}{r} \left(\frac{\partial u_\theta}{\partial \theta} + u_r\right)\\
 \varepsilon_{\phi\phi} & = \frac{1}{r\sin\theta} \left(\frac{\partial u_\phi}{\partial \phi} + u_r\sin\theta + u_\theta\cos\theta\right)\\
  \varepsilon_{r\theta} & = \frac{1}{2} \left(\frac{1}{r} \frac{\partial u_r}{\partial \theta} + \frac{\partial u_\theta}{\partial r} - \frac{u_\theta}{r}\right) \\
  \varepsilon_{\theta \phi}  & = \frac{1}{2r} \left[\frac{1}{\sin\theta}\frac{\partial u_\theta}{\partial \phi} +\left(\frac{\partial u_\phi}{\partial \theta} - u_\phi \cot\theta\right)\right]\\
  \varepsilon_{r \phi} & = \frac{1}{2} \left(\frac{1}{r \sin \theta} \frac{\partial u_r}{\partial \phi} + \frac{\partial u_\phi}{\partial r} - \frac{u_\phi}{r}\right).
\end{align}</math>