Editing Infinitesimal strain theory (section)

== Strain tensor in non-Cartesian coordinates ==

=== Strain tensor in cylindrical coordinates ===
In [[cylindrical polar coordinates]] (<math>r, \theta, z</math>), the displacement vector can be written as
<math display="block"> \mathbf{u} = u_r~\mathbf{e}_r + u_\theta~\mathbf{e}_\theta + u_z~\mathbf{e}_z </math>
The components of the strain tensor in a cylindrical coordinate system are given by:<ref name=Slaughter>{{cite book |last1=Slaughter |first1=William S. |title=The Linearized Theory of Elasticity |date=2002 |publisher=Springer Science+Business Media |location=New York |isbn=9781461266082 |doi=10.1007/978-1-4612-0093-2}}</ref>
<math display="block">\begin{align}
    \varepsilon_{rr} & = \cfrac{\partial u_r}{\partial r} \\
    \varepsilon_{\theta\theta} & = \cfrac{1}{r}\left(\cfrac{\partial u_\theta}{\partial \theta} + u_r\right) \\
    \varepsilon_{zz} & = \cfrac{\partial u_z}{\partial z} \\
    \varepsilon_{r\theta} & = \cfrac{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>

=== Strain tensor in spherical coordinates ===
[[File:3D Spherical.svg|class=skin-invert-image|thumb|240px|right|Spherical coordinates (''r'', ''&theta;'', ''&phi;'') as commonly used in ''physics'': radial distance ''r'', polar angle ''&theta;'' ([[theta]]), and azimuthal angle ''&phi;'' ([[phi]]). The symbol ''&rho;'' ([[rho]]) is often used instead of ''r''.]]
In [[spherical coordinates]] (<math>r, \theta, \phi</math>), the displacement vector can be written as
<math display="block">
   \mathbf{u} = u_r~\mathbf{e}_r + u_\theta~\mathbf{e}_\theta + u_\phi~\mathbf{e}_\phi
 </math>
The components of the strain tensor in a spherical coordinate system are given by <ref name=Slaughter/>
<math display="block">\begin{align}
    \varepsilon_{rr} & = \cfrac{\partial u_r}{\partial r} \\
    \varepsilon_{\theta\theta} & = \cfrac{1}{r}\left(\cfrac{\partial u_\theta}{\partial \theta} + u_r\right) \\
    \varepsilon_{\phi\phi} & = \cfrac{1}{r\sin\theta}\left(\cfrac{\partial u_\phi}{\partial \phi} + u_r\sin\theta + u_\theta\cos\theta\right)\\
    \varepsilon_{r\theta} & = \cfrac{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 \phi} & = \cfrac{1}{2r}\left(\cfrac{1}{\sin\theta}\cfrac{\partial u_\theta}{\partial \phi} + \cfrac{\partial u_\phi}{\partial \theta} - u_\phi\cot\theta\right) \\
    \varepsilon_{\phi r} & = \cfrac{1}{2}\left(\cfrac{1}{r\sin\theta}\cfrac{\partial u_r}{\partial \phi} + \cfrac{\partial u_\phi}{\partial r} - \cfrac{u_\phi}{r}\right) 
\end{align} </math>