A minimal-length approach unifies rigidity in under-constrained materials

TL;DR

This paper introduces a geometric criterion based on minimal length to unify understanding of rigidity in under-constrained materials, predicting elastic properties and transitions with potential for experimental measurement.

Contribution

It presents a universal geometric approach to determine rigidity and elastic properties in under-constrained materials, linking local geometry to macroscopic behavior.

Findings

Predicts the onset of rigidity using a minimal length criterion.

Quantifies the magnitudes of elastic moduli discontinuities at the transition.

Identifies a universal factor of three relating shear modulus, shear stress, and critical strain.

Abstract

We present a novel approach to understand geometric-incompatibility-induced rigidity in under-constrained materials, including sub-isostatic 2D spring networks and 2D and 3D vertex models for dense biological tissues. We show that in all these models a geometric criterion, represented by a minimal length $\bar\ell_\mathrm{min}$, determines the onset of prestresses and rigidity. This allows us to predict not only the correct scalings for the elastic material properties, but also the precise {\em magnitudes} for bulk modulus and shear modulus discontinuities at the rigidity transition as well as the magnitude of the Poynting effect. We also predict from first principles that the ratio of the excess shear modulus to the shear stress should be inversely proportional to the critical strain with a prefactor of three, and propose that this factor of three is a general hallmark of geometrically induced rigidity in under-constrained materials and could be used to distinguish this effect from nonlinear mechanics of single components in experiments. Lastly, our results may lay important foundations for ways to estimate $\bar\ell_\mathrm{min}$ from measurements of local geometric structure, and thus help develop methods to characterize large-scale mechanical properties from imaging data.