Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions
Please always quote using this URN:urn:nbn:de:0296-matheon-9154
- We show that many couplings between parabolic systems for processes in solids can be formulated as a gradient system with respect to the total free energy or the total entropy. This includes Allen-Cahn, Cahn-Hilliard, and reaction-diffusion systems and the heat equation. For this, we write the coupled system as an Onsager system $(X,\Phi,K)$ defining the evolution $\dot{U} = -K(U)D\Phi(U)$. Here $\Phi$ is the driving functional, while the Onsager operator $K(U)$ is symmetric and positive semidefinite. If the inverse $G = K^{-1}$ exists, the triple $(X,\Phi,G)$ defines a gradient system. Onsager systems are well suited to model bulk-interface interactions by using the dual dissipation potential $\Psi^*(U,\Xi) = 1/2 <\Xi,K(U)\Xi>$. Then, the two functionals $\Phi$ and $\Psi^*$ can be written as a sum of a volume integral and a surface integral, respectively. The latter may contain interactions of the driving forces in the interface as well as the traces of the driving forces from the bulk. Thus, capture and escape mechanisms like thermionic emission appear naturally in Onsager systems, namely simply through integration by parts.
Author: | Alexander Mielke |
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URN: | urn:nbn:de:0296-matheon-9154 |
Referee: | Frank Schmidt |
Document Type: | Preprint, Research Center Matheon |
Language: | English |
Date of first Publication: | 2011/11/24 |
Release Date: | 2011/11/24 |
Tag: | |
Institute: | Humboldt-Universität zu Berlin |
Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS) | |
MSC-Classification: | 35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Kxx Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35] / 35K57 Reaction-diffusion equations |
74-XX MECHANICS OF DEFORMABLE SOLIDS / 74Fxx Coupling of solid mechanics with other effects / 74F25 Chemical and reactive effects | |
80-XX CLASSICAL THERMODYNAMICS, HEAT TRANSFER (For thermodynamics of solids, see 74A15) / 80Axx Thermodynamics and heat transfer / 80A17 Thermodynamics of continua [See also 74A15] | |
82-XX STATISTICAL MECHANICS, STRUCTURE OF MATTER / 82Bxx Equilibrium statistical mechanics / 82B35 Irreversible thermodynamics, including Onsager-Machlup theory [See also 92E20] | |
Preprint Number: | 825 |