# Variational Convergence of Discrete Geometrically-Incompatible Elastic Models

## TL;DR

This paper develops a continuum elastic model as a limit of discrete models on curved manifolds, revealing that stress-free states only occur when the underlying geometry is flat, with implications for incompatible elasticity.

## Contribution

It introduces a variational limit framework for discrete incompatible elastic models on curved manifolds, establishing a fundamental rigidity property related to curvature.

## Key findings

- Stress-free configurations only exist if the manifold is flat.
- The continuum limit captures the geometric incompatibility.
- Results generalize to higher dimensions.

## 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 $(M,\mathfrak{g})$, endowed with a flat, symmetric connection $\nabla$. The metric $\mathfrak{g}$ determines local equilibrium distances between neighboring points; the connection $\nabla$ 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 $\mathfrak{g}$ is flat, i.e., has zero Riemann curvature. Our analysis focuses on two-dimensional systems, however, all our results readily generalize to higher dimensions.