Modal analysis with Reissner-Mindlin plate theory
Our goal is to have a modal analysis (eigenvalue problem) with the Reissner Mindlin plate formulation.
Initially, please check the example in Section 8.1 of the book Numerical tours of continuum mechanics using FEniCS - 2021, which solves the Reissner Mindlin plate under a static load.
I recommend you to follow their steps in the book and implement them yourself. If something doesn't work, you can get working codes from here https://comet-fenics.readthedocs.io/en/latest/index.html
After understanding a bit the code and the example, please try to implement a modal analysis of the Reissner Mindlin plate formulation following this paper Verma_-2020-_Free_Vibration_Analysis_of_Reissner_Mindlin_Plates_Using_FEniCS.pdf. In it, Verma uses a similar formulation to the one implemented in Section 8.1, with the additional terms for the kinetic energy.
You can also check section 2.6 of the book to see how an eigenvalue problem is solved in Fenics. Alternatively, you can also the Jupyter notebook available in this repository with my implementation of the eigenvalue problem using the Kirchhoff equation.
To get familiar with the Reissner-Mindlin plate theory, you can take a look at the Wikipedia page https://en.wikipedia.org/wiki/Mindlin%E2%80%93Reissner_plate_theory, and this book chapter goes a bit more in-depthOnate_-THICK_-_THIN_PLATES._REISSNER-MINDLIN_THEORY.pdf.