We consider a two-dimensional cellular vertex model, modeling the mechanics of epithelial tissues. The energy of a planar configuration penalizes deviations in each cell from a reference perimeter P0 and a reference area A0. We study the variational limit of this model as the cell size tends to zero, obtaining a continuum variational model. For P02/A0 below a critical threshold, which corresponds to an isoperimetric constraint, the system is residually-stressed—there are no zero-energy states. For P02/A0 above this threshold, the zero-energy states are highly degenerate, allowing in particular for the formation of microstructures, which are not captured by formal long-wavelength expansions.
Bibliographical noteFunding Information:
We thank B. Dacorogna, R. Kohn, C. Maor and F. Rindler for helpful advice. We are grateful to S. Armon for illuminating discussions on the mechanics of Trichoplax Adherence. RK was partially funded by the Israel Science Foundation (Grant No. 1035/17 ). MM was partially funded by the Israel Science Foundation (Grant No. 1441/19 ).
- Cellular models
- Incompatible elasticity