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A minimal-length approach unifies rigidity in underconstrained materials.

Matthias Merkel1,2, Karsten Baumgarten3, Brian P Tighe3

  • 1Department of Physics, Syracuse University, Syracuse, NY 13244; mmerkel@syr.edu.

Proceedings of the National Academy of Sciences of the United States of America
|March 22, 2019
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A minimal length criterion predicts prestress and rigidity in underconstrained materials like spring networks and biological tissues. This geometric insight aids in characterizing material properties from structural data.

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biopolymer networksconstraint countingrigiditystrain stiffeningvertex model

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Area of Science:

  • Materials Science
  • Biophysics
  • Solid Mechanics

Background:

  • Underconstrained materials exhibit complex mechanical behaviors.
  • Geometric incompatibility is a key factor in material rigidity.
  • Existing models often lack a unified criterion for predicting rigidity onset.

Purpose of the Study:

  • To introduce a geometric criterion for understanding rigidity in underconstrained materials.
  • To predict prestress, elastic properties, and mechanical discontinuities.
  • To establish a general hallmark for geometrically induced rigidity.

Main Methods:

  • Analysis of subisostatic 2D spring networks.
  • Modeling of 2D and 3D vertex models for biological tissues.
  • Derivation of a minimal length criterion for rigidity onset.

Main Results:

  • A minimal length criterion ([Formula: see text]) governs prestress and rigidity.
  • Accurate prediction of elastic property scalings and modulus discontinuities.
  • Identification of a universal prefactor of 3 for excess shear modulus to shear stress ratio.

Conclusions:

  • Geometric incompatibility is a fundamental driver of rigidity in diverse underconstrained systems.
  • The minimal length criterion offers a predictive framework for material properties.
  • This work provides a foundation for characterizing large-scale mechanical properties from local structural imaging.