 # 3D Elastic Beam: Kinematic & Dynamics

Hi - I’m quite new to Fenics. I read docs and tutorials a bit but I decided to ask my question here, hopefully to get more info. I would like to solve a 3d model for a beam. I found two examples which are partially similar to my problem: This* and This**.

MY questions: (1) Is it possible to solve the following coupled PDEs with Fenics? If yes, any suggestion as a starting point is appreciated a lot.
(2) Do I need to do a time discretization such as This**?

\boldsymbol{J}\boldsymbol{\omega}_t+\boldsymbol{\omega} \times \boldsymbol{J}\boldsymbol{\omega} +\boldsymbol{n}\times \boldsymbol{\Lambda}^{-1}\boldsymbol{\phi}_x-\boldsymbol{\Lambda}^{-1}\boldsymbol{\Lambda}_x\times \boldsymbol{m}-\boldsymbol{m}_x = 0
M\boldsymbol{\phi}_{tt}-\boldsymbol{\Lambda}(\boldsymbol{\Lambda}^{-1}\boldsymbol{\Lambda}_x\times \boldsymbol{n}) - \boldsymbol{\Lambda} \boldsymbol{n}_x+ \boldsymbol{f} =0

where M , \boldsymbol{J} \in \mathbb{R}^{3> \times 3} and \boldsymbol{f}\in \mathbb{R}^3 are known.

\boldsymbol{\phi} \in \mathbb{R}^3 and \boldsymbol{\Lambda} \in \mathbb{R}^{3\times 3} are unknown (Variables).

\boldsymbol{n} =\boldsymbol{J} \Lambda^T\boldsymbol{\phi}_x \in \mathbb{R}^3
\boldsymbol{m} = (\boldsymbol{J} \Lambda^T \Lambda_x)^∨ \in \mathbb{R}^3.
\boldsymbol{\omega} = (\Lambda^{-1} \Lambda_t)^∨ \in \mathbb{R}^3

is inverse of Hat operator to map back a vector from that matrix.

(\cdot)_x , (\cdot)_t , and (\cdot)_{tt} denote partial derivatives with respect to position, time, and the second partial derivative with respect to time, respectively.