# Rigidity and auxeticity transitions in networks with strong bond-bending interactions

## TL;DR

This paper investigates the elasticity and phase transitions of 2D networks with strong bond-bending interactions, revealing a stiffening and auxeticity transition near the isostatic point, with implications for designing tunable materials.

## Contribution

It introduces a theoretical framework for analyzing networks with dominant bond-bending forces and identifies a new auxeticity transition in such systems.

## Key findings

- Existence of a generic isostatic point at z_c=4 with bond-bending interactions
- Emergence of a strain-induced stiffening transition near the isostatic point
- Identification of an auxeticity transition with Poisson's ratio approaching -1

## Abstract

A widely-studied model for gels or biopolymeric fibrous materials are networks with central force interactions, such as Hookean springs. Less commonly studied are materials whose mechanics are dominated by non-central force interactions such as bond-bending potentials. Inspired by recent experimental advancements in designing colloidal gels with tunable interactions, we study the micro- and macroscopic elasticity of two-dimensional planar graphs with strong bond bending potentials, in addition to weak central forces. We introduce a theoretical framework that allows us to directly investigate the limit in which the ratio of characteristic central-force to bending stiffnesses vanishes. In this limit we show that a generic isostatic point exists at $z_c=4$, coinciding with the isostatic point of frames with central force interactions in two dimensions. We further demonstrate the emergence of a stiffening transition when the coordination is increased towards the isostatic point, which shares similarities with the strain-induced stiffening transition observed in biopolymeric fibrous materials, and coincides with an auxeticity transition above which the material's Poisson's ratio approaches -1 when bond-bending interactions dominate.