Crystallization, Defects, and Lattice Elasticity

Understanding Crystallization

Many basic phenomena in solid mechanics like dislocations or plastic and elastic deformation are in fact discrete operations: small breakdowns of perfect crystalline order. The goal of this project is thereofore to address the phenomenon of crystallization, and its breakdowns, from the point of view of energy minimization.

Scientific Details+

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.


Crystalline Order, Surface Energy Densities and Wulff Shapes: Emergence from Atomistic Models

Author: Au Yeung, Yuen
Advisor: Gero Friesecke
Date: 2013
Download: external

Domain formation in magnetic polymer composites: an approach via stochastic homogenization

Authors: Alicandro, R. and Cicalese, M. and Ruf, M.
Journal: Arch. Rat. Mech. Anal., 218(2):945--984
Date: 2015
Download: arXiv

Frustrated ferromagnetic spin chains: a variational approach to chirality transitions

Authors: Cicalese, M. and Solombrino, F.
Journal: Journal of Nonlinear Science, 25(291-313)
Date: 2015

Twisted x-rays: incoming waveforms yielding discrete diffraction patterns for helical structures

Authors: G. Friesecke, R. D. James, and Jüstel, D.
Date: 2015
Download: arXiv

Metastability and dynamics of discrete topological singularities in two dimensions: a Gamma-convergence approach

Authors: R. Alicandro, L. De Luca, A. Garroni and Ponsiglione, M.
Journal: Archive for Rational Mechanics and Analysis, 214(1):269--330
Date: 2014


Prof. Dr. Gero Friesecke   +

Projects: B08
University: TU München
E-Mail: gf[at]ma.tum.de
Website: http://www-m7.ma.tum.de/bin/view/Analysis/GeroFriesecke

PhD Lucia De Luca   +

Projects: B08
University: TU München
E-Mail: deluca[at]ma.tum.de