Periodic Reporting for period 1 - VarCrysDef (Variational coarse-graining of lattice energies for crystal defects: Applications to partial dislocations and stacking faults)
Reporting period: 2022-10-01 to 2024-09-30
Understanding the emergence, interaction, and motion of crystal defects is crucial for the prediction of possible failure or, on the contrary, strengthening of crystalline materials. Such defects can roughly be described as a local irregularity in the typical crystalline structure. A prominent example of crystal defects are partial dislocations and stacking faults, which typically appear in closed-packed crystalline structures, for example in Hexagonal-closed-packed (HCP) or Face-centered-cubic (FCC) crystals.
The main goal of the present action was to predict the emergence of those defects in a mathematically rigorous passage from microscopic/ atomistic models to macroscopic/ continuum models, thus providing a possible justification of phenomenological models used in continuum mechanics. Here our viewpoint was energetic and we employed tools from the Calculus of Variations and Geometric Measure Theory, in particular Gamma-convergence. The latter technique allows to derive effective limiting energies for atomistic energies when the interatomic distance vanishes. Such a procedure is often referred to as a variational coarse-graining.
The main goal of the present action was to predict the emergence of those defects in a mathematically rigorous passage from microscopic/ atomistic models to macroscopic/ continuum models, thus providing a possible justification of phenomenological models used in continuum mechanics. Here our viewpoint was energetic and we employed tools from the Calculus of Variations and Geometric Measure Theory, in particular Gamma-convergence. The latter technique allows to derive effective limiting energies for atomistic energies when the interatomic distance vanishes. Such a procedure is often referred to as a variational coarse-graining.
In the framework described above we were up to now able to carry out a variational coarse-graining procedure for a simplified two-dimensional discrete model for partial edge dislocations and stacking faults. Although our model does not account yet for all geometric properties of the HCP lattice, it still contains enough relevant information to capture the emergence of point- and line defects corresponding to partial dislocations and stacking faults, respectively, in the continuum limit. In particular, we were able to show that the limiting energy consists of three contributions: A core contribution concentrated around each limiting point defect, a surface contribution supported on the limiting line defects and an elastic contribution far from the limiting defects.
In the variational coarse graining result above we had to deal with the presence and interaction of defects of different dimensions (points and lines in our two dimensional model) in a passage from discrete to continuum. While an interaction of point and line defects has been addressed in the context of XY models and continuum Ginzburg-Landau vortices, our result is, to the best of our knowledge, the first one that addresses this in the context of partial edge dislocations and stacking faults. It will thus be also a starting point for further analysis on more complex discrete energies.