Geometric frustration and solid-solid transitions in model 2D tissue

TL;DR

This paper investigates how geometric frustration influences the mechanical properties and phase transitions in 2D cellular tissues modeled by vertex models, revealing a transition from soft to nonlinear elastic regimes.

Contribution

It introduces a continuum vertex model framework to analyze tissue mechanics, highlighting the role of geometric incompatibility and degeneracy lifting in elastic behavior.

Findings

Transition from soft to nonlinear elastic regime at critical shape index

Degenerate ground states in compatible regimes with zero modes

Energy gap emergence causes nonlinear elasticity distinct from classical models

Abstract

We study the mechanical behavior of two-dimensional cellular tissues by formulating the continuum limit of discrete vertex models based on an energy that penalizes departures from a target area $A_0$ and a target perimeter $P_0$ for the component cells of the tissue. As the dimensionless target shape index $s_0 = \frac{P_0}{\sqrt{A_0}}$ is varied, we find a transition from a soft elastic regime for compatible target perimeter and area to a stiffer nonlinear elastic regime frustrated by geometric incompatibility. We show that the ground state in the soft regime has a family of degenerate solutions associated with zero modes for the target area and perimeter. The onset of geometric incompatibility at a critical $s_0^c$ lifts this degeneracy. The resultant energy gap leads to a nonlinear elastic response distinct from that obtained in classical elasticity models. We draw an analogy between cellular tissues and anelastic deformations in solids.