B08
Curvature Effects in Molecular and Spin Systems

How do the well known phenomenological continuum theories of solid mechanics (linear and nonlinear elasticity theory, plasticity theory, fracture mechanics) emerge from discrete, atomistic models? This fundamental question lies at the heart of a great deal of current research in materials science and materials engineering, yet remains very poorly understood on a mathematical level. A key bottleneck is that we don't understand crystallization, that is to say the fact that under many conditions, atoms self-assemble into crystalline order and special geometric shapes. This is a main bottleneck because all the basic phenomena in solid mechanics (dislocations, grains, fracture, plastic and elastic deformation) are small or localized breakdowns of perfect crystalline order. The goal of the project is to address the phenomenon of crystallization, and its breakdowns, from the point of view of energy minimization. In particular, we aim to extend available results on crystalline order and shape from purely combinatorial energies to soft potentials which allow for elastic modes, and develop methods for the rigorous passage from these discrete models to continuum surface energy functionals and elastic energy functionals. Our mathematical approach will rely on combining methods from three areas: (i) atomistic mechanics and its recently developed generalized convexity notions, (ii) Gamma convergence techniques from the calculus of variations, and - crucially and as far as we know for the first time in our context - (iii) discrete differential geometry, which is a central theme in other projects of the SFB-Transregio.