C05 - Effective models for materials and interfaces with multiple scales
Head(s): Prof. Dr. Alexander Mielke (WIAS)
Project member(s): Dr. Martin Heida, Artur Stephan
Participating institution(s): WIAS
This project provides analytical techniques for discrete or continuous material models that depend on one or several small parameters. Special emphasis is given to systems that have a variational structure such as static minimisation problems or gradient-flow equations systems. Methods of static or evolutionary Gamma convergence are employed and further investigated, in particular EDP-convergence (i.e., in the sense of the energy dissipation principle). The small parameter may determine material properties via small layers or periodic, fractal, or stochastic material properties with a small correlation length. Applications involve diffusion in strongly heterogeneous media, elastic bulk materials with embedded interfaces along which Coulomb friction and other processes may occur.
More precisely we consider the topics:
- Gradient systems and evolutionary Gamma convergence, which provide tools for deriving effective models for systems with many scales. The connection between large-deviation principles for microscopic stochastic models and the gradient structures for the macroscopic deterministic models.
- Homogenisation of discrete elliptic operators (linear or non-linear) on regular or random graphs with random coefficients. Fractal homogenisation of elliptic problems with transmission on a fractal set of interfaces.
- Mathematical and thermodynamical modeling of evolutionary processes in bulk materials and in materials with interfaces. Periodic, fractal and stochastic homogenisation of evolutionary systems with variational structure.
- Rate-independent and rate-and-state friction between elastic bodies and its justification via dimension reduction.
- Connections between discrete chemical master equations and continuum descriptions like the reaction-rate equation. Hybrid models for reaction kinetics and reaction-diffusion systems.