Variational models in Materials Science: from crystalline structures to continuum theories

• Applicant: Dr. Leonard Kreutz, Ph.D.
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 509436910

The overall aim of the project is to study discrete and continuous models arising from Materials Science in a variational framework. In particular, the project purpose is to derive effective theories for thin materials starting from discrete models, investigate the competition between bulk and surface energies, and advance the approach of molecular mechanics in a rigorous way.The project tackles three major themes:• Discrete thin structuresDiscrete-to-continuum theories for nanotubes;Effective behavior of discrete thin sheets.• Stress-driven rearrangement instabilities (SDRI) Phase-field approximation & linearization of SDRI models;Dimension reduction in the framework of SDRI.• PolycrystalsContinuum models for grain boundary energies;Discrete-to-continuum analysis for polycrystalline graphene.The problems will be investigated using advanced tools from the calculus of variations including Γ-convergence and quantitative geometric rigidity estimates, techniques from free discontinuity problems and geometric measure theory, as well as discrete mathematics. Besides its applications to Materials Science, the proposed project will further develop the mathematical theory by deepening the understanding of discrete and continuum models, their relation, as well as the derivation of effective theories with particular emphasis on dimension reduction problems.

• DFG Programme: Independent Junior Research Groups