A minimal-length approach unifies rigidity in underconstrained materials

Matthias Merkel*, Karsten Baumgarten, Brian P. Tighe, M. Lisa Manning

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

62 Citations (Scopus)
66 Downloads (Pure)


We present an approach to understand geometric-incompatibility–induced rigidity in underconstrained materials, including subisostatic 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 ℓ¯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 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 3. We propose that this factor of 3 is a general hallmark of geometrically induced rigidity in underconstrained materials and could be used to distinguish this effect from nonlinear mechanics of single components in experiments. Finally, our results may lay important foundations for ways to estimate ℓ¯min from measurements of local geometric structure and thus help develop methods to characterize large-scale mechanical properties from imaging data.
Original languageEnglish
Pages (from-to)6560-6568
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number14
Publication statusPublished - 2019

Bibliographical note

Accepted Author Manuscript


  • Biopolymer networks
  • Constraint counting
  • Rigidity
  • Strain stiffening
  • Vertex model


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