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 $P_0^2/A_0$ below a critical threshold, which corresponds to an isoperimetric constraint, the system is residually-stressed—there are no zero-energy states. For $P_0^2/A_0$ 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.

我们考虑二维细胞顶点模型，对上皮组织的力学建模。平面构造的能量惩罚了每个单元中与参考周长P ₀和参考面积A _0的偏差。我们研究该模型的变化极限，因为像元大小趋于零，从而获得了连续变化模型。对于$P_0^2/{\mathrm{一种}}_0$低于临界阈值（对应于等压约束）时，系统处于残余应力状态-没有零能量状态。对于$P_0^2/{\mathrm{一种}}_0$ 高于该阈值，零能态高度简并，尤其允许形成微结构，而这些结构并没有被正式的长波扩展所捕获。