Limiting Theories in Material Science: Mathematical derivation and Analysis

• Applicant: Professor Dr. Peter Bella
• Subject Area: Mathematics
• Term: from 2016 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 313878761

Recent technological advances have allowed for engineering of materials at ever decreasing scales toward broad applications, ultrathin films being one of the examples. The presence of different length scales often makes the numerical study of models in material science prohibitively expensive. Instead of treating them numerically one first studies them analytically to obtain some understanding of their solutions and then uses the acquired insight to pave the way for development of more effective numerical methods. Our goal is to rigorously analyze a few such problems. In the first part of the project we study the wrinkling patterns in compressed thin elastic sheets. In some situations the wrinkling could be non-uniform and the actual pattern shows branching. To understand wrinkling microstructure we consider a variational viewpoint, and identify and analytically study the next-order expansion of the energy (the subdominant energy) in the limit of vanishing sheet thickness. Within this framework we will study several physical situations, a model describing graphene nanoribbons being one of them. In the second part we study elliptic systems with random and rapidly oscillating coefficients, with the application to the model describing heterogeneous linearly-elastic materials in mind. Though the microscopic behavior could be quite complicated, due to stochastic cancellations the macroscopic behavior should be much simpler and deterministic, a process called homogenization. We will use PDE methods to study quantitative aspects of the stochastic homogenization for elliptic systems. The last area of research concerns behavior of compressible viscous fluids in domains with rough boundaries. Rather than study the problem in a rough domain, one poses the problem in a smooth domain where the roughness of the original boundary is reduced to an effective boundary law. Using the concept of relative energy inequality for dissipative solutions to the Navier-Stokes system our aim is to rigorously derive these effective boundary conditions and analyze the error one makes by taking this approach.

• DFG Programme: Independent Junior Research Groups