Continuum mechanics of a cellular tissue model

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.