The emergence of torsion in the continuum limit of distributed edge-dislocations

2014/10/31 by Raz Kupferman, Cy Maor · 20 citations
Computer Science · Engineering · Materials Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Composite Material Mechanics #Curvature #Geometry #Gravitational singularity #Homogenization (climate) #Limit (mathematics) #Limit of a sequence #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Nonlocal and gradient elasticity in micro/nano structures #Pure mathematics #Torsion (gastropod) #Zero (linguistics) #math.DG

paper · pdf · doi:10.3934/jgm.2015.7.361

published in The Journal of Geometric Mechanics 7(3), 361-387 (American Institute of Mathematical Sciences) · See an erratum regarding Definition 4.1 and Lemma 4.8 in arxiv:1701.08903 (v3 is the same as v2 but for this comment)

openalex publication_date 2015/01/01 · arxiv created 2018/02/22 · arxiv updated 2019/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We present a rigorous homogenization theorem for distributed edge-dislocations.We construct a sequence of locally-flat 2D Riemannian manifolds with dislocation-type singularities. We show that this sequence converges, as the dislocations become denser, to a flat non-singular Weitzenböck manifold, i.e. a flat manifold endowed with a metrically-consistent connection with zero curvature and non-zero torsion.In the process, we introduce a new notion of convergence of Weitzenböck manifolds, which is relevant to this class of homogenization problems.

Citations

• The emergence of torsion in the continuum limit of distributed edge-dislocations - erratum
• Some geometrical relations in dislocated crystals
• On the geometric structures of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations
• A theorem on geometric rigidity and the derivation of nonlinear plate theory from three‐dimensional elasticity
• Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry

Cited by

• Riemannian surfaces with torsion as homogenization limits of locally Euclidean surfaces with dislocation-type singularities
• Variational convergence of discrete geometrically-incompatible elastic models