In this post, we provide a concise summary of the field equations of solid mechanics as they pertain to cylindrical and spherical coordinates. These representations are particularly beneficial in deriving the differential equations of axisymmetric objects and obtaining analytical solutions for displacements and stresses.
\[\rho \dot{\boldsymbol{v}} = \boldsymbol{\nabla} \cdot \boldsymbol{\sigma} + \rho \boldsymbol{b}\]
\[\begin{split} \rho \dot{v}_r &= \frac{\partial \sigma_{rr}}{\partial r} + \frac{1}{r}\frac{\partial \sigma_{r\theta}}{\partial \theta} + \frac{\partial \sigma_{rz}}{\partial z} + \frac{1}{r}\left(\sigma_{rr}-\sigma_{\theta\theta}\right) + \rho b_r \\ \rho \dot{v}_{\theta} &= \frac{\partial \sigma_{\theta r}}{\partial r} + \frac{1}{r}\frac{\partial \sigma_{\theta\theta}}{\partial \theta} + \frac{\partial \sigma_{\theta z}}{\partial z} + \frac{2}{r}\sigma_{\theta r} + \rho b_{\theta}\\ \rho \dot{v}_{z} &= \frac{\partial \sigma_{zr}}{\partial r} + \frac{1}{r}\frac{\partial \sigma_{z\theta}}{\partial \theta} + \frac{\partial \sigma_{zz}}{\partial z} + \frac{1}{r}\sigma_{zr} + \rho b_z\\ \end{split}\tag{1}\]
\[\begin{split} \dot{v}_r &= \frac{\partial v_r}{\partial t} + v_r\frac{\partial v_r}{\partial r} + \frac{v_{\theta}}{r}\frac{\partial v_r}{\partial \theta} + v_z\frac{\partial v_r}{\partial z} - \frac{1}{r}v_\theta^2\\ \dot{v}_{\theta} &= \frac{\partial v_{\theta}}{\partial t} + v_r\frac{\partial v_{\theta}}{\partial r} + \frac{v_{\theta}}{r}\frac{\partial v_{\theta}}{\partial \theta} + v_z\frac{\partial v_{\theta}}{\partial z} - \frac{1}{r}v_rv_{\theta}\\ \dot{v}_z &= \frac{\partial v_z}{\partial t} + v_r \frac{\partial v_z}{\partial r} + \frac{v_{\theta}}{r}\frac{\partial v_z}{\partial \theta} + v_z \frac{\partial v_z}{\partial z}\\ \end{split}\tag{2}\]
\[\boldsymbol{L} = \begin{bmatrix} \frac{\partial v_r}{\partial r} & \frac{1}{r}\frac{v_r}{\partial \theta}-\frac{v_{\theta}}{r} & \frac{\partial v_r}{\partial z}\\ \frac{\partial v_{\theta}}{\partial r} & \frac{1}{r}\frac{\partial v_{\theta}}{\partial \theta}+\frac{v_r}{r} & \frac{\partial v_{\theta}}{\partial z}\\ \frac{\partial v_z}{\partial r} & \frac{1}{r}\frac{\partial v_z}{\partial \theta} & \frac{\partial v_z}{\partial z} \end{bmatrix}\tag{3}\]
\[\rho \dot{\boldsymbol{v}} = \boldsymbol{\nabla} \cdot \boldsymbol{\sigma} + \rho \boldsymbol{b}\]
\[\begin{split} \rho\dot{v}_r &= \frac{\partial \sigma_{rr}}{\partial r} + \frac{1}{r}\frac{\partial \sigma_{r\theta}}{\partial \theta} + \frac{1}{r\sin\theta}\frac{\partial \sigma_{r\phi}}{\partial \phi} + \frac{1}{r}\left(2\sigma_{r\theta} - \sigma_{\theta\theta}-\sigma_{\phi\phi}+\sigma_{r\theta}\cot\phi\right) + \rho b_r\\ \rho\dot{v}_{\theta} &= \frac{\partial \sigma_{\theta r}}{\partial r} + \frac{1}{r}\frac{\partial \sigma_{\theta\theta}}{\partial \theta}+\frac{1}{r\sin\theta} \frac{\partial \sigma_{\theta\phi}}{\partial \phi} + \frac{1}{r}\left(3\sigma_{r\theta} +(\sigma_{\theta\theta}-\sigma_{\phi\phi})\cot\theta\right) + \rho b_{\theta}\\ \rho\dot{v}_{\phi} &= \frac{\partial \sigma_{\phi r}}{\partial r} + \frac{1}{r}\frac{\partial \sigma_{\phi\theta}}{\partial \theta} + \frac{1}{r\sin\theta}\frac{\partial \sigma_{\phi\phi}}{\partial \phi} + \frac{1}{r}\left(3\sigma_{r\phi} + 2\sigma_{\theta\phi}\cot\theta\right) + \rho b_{\phi}\\ \end{split}\tag{4}\]
\[\begin{split} \dot{v}_r &= \frac{\partial v_r}{\partial t} + v_r \frac{\partial v_r}{\partial r} + \frac{\partial v_r}{\partial \theta} + \frac{v_{\phi}}{r\sin\theta}\frac{\partial v_r}{\partial \phi} - \frac{v_{\phi}^2 - v_{\theta}^2}{r}\\ \dot{v}_{\theta} &= \frac{\partial v_{\theta}}{\partial t} + v_r \frac{\partial v_{\theta}}{\partial r} + \frac{v_{\theta}}{r}\frac{\partial v_{\theta}}{\partial \theta} + \frac{v_{\phi}}{r\sin\theta}\frac{\partial v_{\theta}}{\partial \phi} + \frac{v_rv_{\theta}}{r} - \frac{v_{\phi}^2}{r}\cot\theta\\ \dot{v}_{\phi} &= \frac{\partial v_{\phi}}{\partial t} + v_r\frac{\partial v_{\phi}}{\partial r} + \frac{v_{\theta}}{r}\frac{\partial v_{\phi}}{\partial \phi} + \frac{v_{\phi}}{r\sin\theta}\frac{\partial v_{\phi}}{\partial \phi} + \frac{v_rv_{\phi}}{r} + \frac{v_{\theta}v_{\phi}}{r}\cot \theta\\ \end{split}\tag{5}\]
\[\boldsymbol{L} = \begin{bmatrix} \frac{\partial v_r}{\partial r} & \frac{1}{r}\frac{\partial v_r}{\partial \theta}-\frac{v_{\theta}}{r} & \frac{1}{r\sin\theta}\frac{\partial v_r}{\partial \phi} - \frac{v_{\phi}}{r}\\ \frac{\partial v_{\theta}}{\partial r} & \frac{1}{r}\frac{\partial v_{\theta}}{\partial \theta} + \frac{v_r}{r} & \frac{1}{r\sin\theta}\frac{\partial v_{\theta}}{\partial \phi} - \frac{v_{\phi}}{r}\cot \phi\\ \frac{\partial v_{\phi}}{\partial r} & \frac{1}{r}\frac{\partial v_{\phi}}{\partial \theta} & \frac{1}{r\sin\theta}\frac{\partial v_{\phi}}{\partial \phi} + \frac{v_{\theta}}{r}\cot\theta + \frac{v_r}{r} \end{bmatrix}\tag{6}\]