Hamiltonian and thermodynamic theory of solids and fluids

• Thesis title in Czech: Hamiltonovská a termodynamická teorie pevných látek a tekutin
• Thesis title in English: Hamiltonian and thermodynamic theory of solids and fluids
• Key words: Nerovnovážná termodynamika, GENERIC, eulerovský popis
• English key words: Non-equilibrium thermodynamics, GENERIC, Eulerian description
• Academic year of topic announcement: 2017/2018
• Thesis type: diploma thesis
• Thesis language: angličtina
• Department: Mathematical Institute of Charles University (32-MUUK)
• Supervisor: doc. RNDr. Michal Pavelka, Ph.D.
• Author: hidden - assigned and confirmed by the Study Dept.
• Date of registration: 22.01.2018
• Date of assignment: 22.01.2018
• Confirmed by Study dept. on: 01.02.2018
• Date and time of defence: 13.09.2019 08:00
• Date of electronic submission: 19.07.2019
• Date of submission of printed version: 19.07.2019
• Date of proceeded defence: 13.09.2019
• Opponents: doc. Ing. Václav Klika, Ph.D.
• Advisors: RNDr. Jaroslav Hron, Ph.D.

Guidelines

1. Eulerian formulation of the standard reversible part of continuum mechanics, in particular with a neo-Hookean model or linearized elasticity.
2. A short introduction to the GENERIC framework.
3. Hamiltonian formulation of kinematics of deformation and comparison to the standard Eulerian equations.
4. Analytical or numerical solutions of a simple elastic problem without free boundaries formulated in the Eulerian frame.
5. Addition of dissipation into the evolution equation for the distortion matrix or left Cauchy-Green tensor and numerical solution to the simple problem with dissipation - simulation of plasticity.
6. If possible, characteristics and eigenvalues could be calculated to verify hyperbolicity of the problem.
7. If possible, free boundaries and other solution techniques (Lagrangian, ALE, Eulerian and finite elements vs. finite volumes) could be discussed.

References

[1] Grmela, M., A framework for elasto-plastic hydrodynamics, Physics Letters A, 2003, 312, 136
[2] Godunov, S. K. and Peshkov, I. M, Thermodynamically Consistent Nonlinear Model of Elastoplastic Maxwell Medium, Computational Mathematics and Mathematical Physics, 2010, Vol. 50, No. 8, pp. 1409-1426
[3] Ilya Peshkov, Michal Pavelka, Evgeniy Romenski, Miroslav Grmela, Continuum Mechanics and Thermodynamics in the Hamilton and the Godunov-type Formulations, Accepted to Continuum Mechanics and Thermodynamics (2018).

Preliminary scope of work in English

Kinematics of classical continuum mechanics is based on the notion of a reference (Lagrangian) configuration, deformation of which is studied by means of Newton laws. The reference configuration, however, loses its physical meaning for example in the case of severe plastic deformations, first-order phase transitions or chemical reactions because the mapping from the reference configuration to the current configuration no longer exists or is forgotten by the material. A way to circumvent this obstacle is to focus on fully Eulerian theories of solids and fluids by means of an underlying Hamiltonian structure. One then obtains a fully Eulerian (everything is in the current configuration) thermodynamically consistent theory of solids and fluids, which can be transformed to each other.