Strain Tensor

There are mainly two types of strain tensors; Lagrange Strain Tensor and Infinitesimal Strain Tensor which denote $\nabla u$, the gradient of displacement $u$ Lagrange Strain Tensor is defined by \[\begin{align} & {{E}^{*}}=\frac{1}{2}[\nabla u+{{(\nabla u)}^{T}}+{{(\nabla u)}^{T}}(\nabla u)] \\ & \\ \end{align}\] and Infinitesimal Strain Tensor is defined by \[\begin{align} & {{E}}=\frac{1}{2}[\nabla u+{{(\nabla u)}^{T}}] \\ & \\ \end{align}\] which is the strain tensor assuming for small deformation where its component \[\begin{align} & {{E}_{ij}}=\frac{1}{2}(\frac{\partial {{u}_{i}}}{\partial {{X}_{j}}}+\frac{\partial {{u}_{j}}}{\partial {{X}_{i}}}) \\ & \\ \end{align}\] Therefore, the Lagrange Strain Tensor for a) rectangular coordinates: \[\begin{align} & [E]=\left[ \begin{matrix} \frac{\partial {{u}_{1}}}{\partial {{X}_{1}}} & \frac{1}{2}(\frac{\partial {{u}_{1}}}{\partial {{X}_{2}}}+\frac{\partial {{u}_{2}}}{\partial {{X}_{1}}}) & \frac{1}{2}(\frac{\partial {{u}_{1}}}{\partial {{X}_{3}}}+\frac{\partial {{u}_{3}}}{\partial {{X}_{1}}}) \\ \frac{1}{2}(\frac{\partial {{u}_{2}}}{\partial {{X}_{1}}}+\frac{\partial {{u}_{1}}}{\partial {{X}_{2}}}) & \frac{\partial {{u}_{2}}}{\partial {{X}_{2}}} & \frac{1}{2}(\frac{\partial {{u}_{2}}}{\partial {{X}_{3}}}+\frac{\partial {{u}_{3}}}{\partial {{X}_{2}}}) \\ \frac{1}{2}(\frac{\partial {{u}_{3}}}{\partial {{X}_{1}}}+\frac{\partial {{u}_{1}}}{\partial {{X}_{3}}}) & \frac{1}{2}(\frac{\partial {{u}_{3}}}{\partial {{X}_{2}}}+\frac{\partial {{u}_{2}}}{\partial {{X}_{3}}}) & \frac{\partial {{u}_{3}}}{\partial {{X}_{3}}} \\ \end{matrix} \right] \\ & \\ \end{align}\] b) cylindrical coordinates: \[\begin{align} & [E]=\left[ \begin{matrix} \frac{\partial {{u}_{r}}}{\partial r} & \frac{1}{2}(\frac{1}{r}\frac{\partial {{u}_{r}}}{\partial \theta }-\frac{{{u}_{\theta }}}{r}+\frac{\partial {{u}_{\theta }}}{\partial r}) & \frac{1}{2}(\frac{\partial {{u}_{r}}}{\partial z}+\frac{\partial {{u}_{z}}}{\partial r}) \\ \frac{1}{2}(\frac{1}{r}\frac{\partial {{u}_{r}}}{\partial \theta }-\frac{{{u}_{\theta }}}{r}+\frac{\partial {{u}_{\theta }}}{\partial r}) & \frac{1}{r}\frac{\partial {{u}_{\theta }}}{\partial \theta }+\frac{{{u}_{r}}}{r} & \frac{1}{2}(\frac{\partial {{u}_{\theta }}}{\partial z}+\frac{1}{r}\frac{\partial {{u}_{z}}}{\partial \theta }) \\ \frac{1}{2}(\frac{\partial {{u}_{r}}}{\partial z}+\frac{\partial {{u}_{z}}}{\partial r}) & \frac{1}{2}(\frac{\partial {{u}_{\theta }}}{\partial z}+\frac{1}{r}\frac{\partial {{u}_{z}}}{\partial \theta }) & \frac{\partial {{u}_{z}}}{\partial z} \\ \end{matrix} \right] \\ & \\ \end{align}\] c) Spherical Coordinates: \[\begin{align} & [E]=\left[ \begin{matrix} \frac{\partial {{u}_{r}}}{\partial r} & \frac{1}{2}(\frac{1}{r}\frac{\partial {{u}_{r}}}{\partial \theta }-\frac{{{u}_{\theta }}}{r}+\frac{\partial {{u}_{\theta }}}{\partial r}) & \frac{1}{2}(\frac{1}{r\sin \theta }\frac{\partial {{u}_{r}}}{\partial \phi }-\frac{{{u}_{\phi }}}{r}+\frac{\partial {{u}_{\phi }}}{\partial r}) \\ {{E}_{21}}={{E}_{12}} & \frac{1}{r}\frac{\partial {{u}_{\theta }}}{\partial \theta }+\frac{{{u}_{r}}}{r} & \frac{1}{2}(\frac{1}{r\sin \theta }\frac{\partial {{u}_{\theta }}}{\partial \phi }-\frac{{{u}_{\phi }}\cot \theta }{r}+\frac{1}{r}\frac{\partial {{u}_{\phi }}}{\partial \theta }) \\ {{E}_{31}}={{E}_{13}} & {{E}_{32}}={{E}_{23}} & \frac{1}{r\sin \theta }\frac{\partial {{u}_{\phi }}}{\partial \phi }-\frac{{{u}_{r}}}{r}+\frac{{{u}_{\theta }}\cot \theta }{r} \\ \end{matrix} \right] \\ & \\ & \\ \end{align}\] resource: ISBN: 978-0-7506-8560-3