Riemannian surfaces with torsion as homogenization limits of locally Euclidean surfaces with dislocation-type singularities

2014/10/31 by Raz Kupferman, Cy Maor · 9 citations
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Affine transformation #Composite Material Mechanics #Condensed matter physics #Dislocation #Euclidean geometry #Geometric Analysis and Curvature Flows #Geometry #Gravitational singularity #Homogenization (climate) #Mathematical analysis #Mathematics #Physics #Pure mathematics #Riemannian manifold #Singularity #Torsion (gastropod) #math.DG

paper · pdf · doi:10.1017/s0308210515000773

published in Proceedings of the Royal Society of Edinburgh Section A Mathematics 146(4), 741-768 (Cambridge University Press)

arxiv created 2015/06/28 · openalex publication_date 2016/07/01 · openalex created_date 2016/07/22 · arxiv updated 2019/01/23 · openalex updated_date 2026/08/05

Abstract

We reconcile two classical models of edge dislocations in solids. The first, from the early 1900s, models isolated edge dislocations as line singularities in locally Euclidean manifolds. The second, from the 1950s, models continuously distributed edge dislocations as smooth manifolds endowed with non-symmetric affine connections (equivalently, endowed with torsion fields). In both models, the solid is modelled as a Weitzenböck manifold. We prove, using a weak notion of convergence, that the second model can be obtained rigorously as a homogenization limit of the first model as the density of singular edge dislocation tends to infinity.

Citations

• The emergence of torsion in the continuum limit of distributed edge-dislocations
• Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry
• A Panoramic View of Riemannian Geometry

Cited by

• Variational convergence of discrete geometrically-incompatible elastic models