Variational convergence of discrete geometrically-incompatible elastic models

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, g), endowed with a flat, symmetric connection ∇. The metric g 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 g is flat, i.e., has zero Riemann curvature. Our analysis focuses on two-dimensional systems, however, all our results readily generalize to higher dimensions.

Discrete-to-continuum limits of planar disclinations

“…A simple example of Ψ is Ψ(a) = |a − 1|. These conditions are more general than those in Section 3.3 of [33].…”

“…For large deformations, the recent work in [33] presents and anlyzes an atomistic model of a stretchable hexagonal lattice defined over a smooth manifold, in which the main result is the continuum limit in the form of Γ-convergence. Since the approach in [33] is designed to describe the energetics of highly distorted membranes, we follow a similar approach. However, the choice in [33] that the number of bonds per atom is always 6 makes the result not applicable to disclinations.…”

“…Since the approach in [33] is designed to describe the energetics of highly distorted membranes, we follow a similar approach. However, the choice in [33] that the number of bonds per atom is always 6 makes the result not applicable to disclinations.…”

“…As in [33], we assign an energy E ε to a deformed lattice, where ε is the lattice spacing. Two examples of such deformed lattices are illustrated by the dotted nodes in Figure 1.…”

Anomalous elasticity of cellular tissue vertex model

“…To introduce our pictorial representation of deforma- tions in configuration space we first consider a common microscopic model for elastic solids, which is a lattice of masses and springs. In 2𝐷 a triangular lattice of identical masses and springs leads, in the coarse-grained limit, to homogeneous and isotropic linear elasticity [32,38]. As discussed before, the response to uniform loads is equivalent to the response of a single triangle.…”