Variational convergence of discrete geometrically-incompatible elastic models

2017/04/30 by Raz Kupferman, Cy Maor · 4 citations
Materials Science · Mathematics · #Connection (principal bundle) #Elasticity (physics) #Geometric Analysis and Curvature Flows #Lattice (music) #Limit (mathematics) #Manifold (fluid mechanics) #Nonlinear Partial Differential Equations #Nonlocal and gradient elasticity in micro/nano structures #Riemannian manifold #Rigidity (electromagnetism) #Zero (linguistics) #math.AP #math.DG #msc:53Z05 #msc:74B20

paper · pdf · doi:10.1007/s00526-018-1306-1

published in Calculus of Variations and Partial Differential Equations 57(2) (Springer Science+Business Media) · v3: a more concise version (similar to the published version); proof of Proposition 4.4 corrected, Lemma A.4 added

openalex created_date 2017/06/23 · openalex publication_date 2018/02/14 · arxiv created 2019/01/20 · arxiv updated 2019/01/23 · openalex updated_date 2026/08/05

Abstract

We derive a continuum model for incompatible elasticity as a variational limit of a family of discrete nearest-neighbor elastic models. The discrete models are based on discretizations of a smooth Riemannian manifold (M,\mathfrakg), endowed with a flat, symmetric connection ∇. The metric \mathfrakg determines local equilibrium distances between neighboring points; the connection ∇ induces a lattice structure shared by all the discrete models. The limit model satisfies a fundamental rigidity property: there are no stress-free configurations, unless \mathfrakg is flat, i.e., has zero Riemann curvature. Our analysis focuses on two-dimensional systems, however, all our results readily generalize to higher dimensions.

Citations

• The emergence of torsion in the continuum limit of distributed edge-dislocations
• Some geometrical relations in dislocated crystals
• Riemann–Cartan Geometry of Nonlinear Dislocation Mechanics
• 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
• Riemannian surfaces with torsion as homogenization limits of locally Euclidean surfaces with dislocation-type singularities