# Compatibility Conditions for Discrete Planar Structure

## TL;DR

This paper investigates the mathematical conditions necessary for planar network structures to remain compatible during deformation, linking discrete structural compatibility to continuum mechanics and topological invariants.

## Contribution

It introduces polynomial compatibility equations for triangulated structures and explores their continuum limits, providing new insights into structural resilience and rigidity.

## Key findings

- Compatibility equations expressed as polynomial relations in edge lengths.
- Continuum limits of compatibility conditions match those in continuum mechanics.
- The number of compatibility conditions measures structural resilience.

## Abstract

Compatibility conditions are investigated for planar network structures consisting of nodes and connecting bars; these conditions restrict the elongations of bars and are analogous to the compatibility conditions of deformation in continuum mechanics. The requirement that the deformations remain planar imposes compatibility. Compatibility for structures with prescribed lengths and its linearization is considered. For triangulated structures, compatibility is expressed as a polynomial equation in the lengths of edges of the star domain surrounding each interior node. The continuum limits of the conditions coincide with those in the continuum problems. The compatibility equations may be summed along a closed curve to give conditions analogous to Gauss-Bonnet integral formula. There are rigid trusses without compatibility conditions in contrast to continuous materials. The compatibility equations around a hole involve the edges in the neighborhood surrounding the hole. The number of compatibility conditions is the number of bars that may be removed from a structure while keeping it rigid; this number measures the structural resilience. An asymptotic density of compatibility conditions is analyzed.