Geometric Frustration and Solid-Solid Transitions in Model 2D Tissue

We study the mechanical behavior of two-dimensional cellular tissues by formulating the continuum limit of discrete vertex models based on an energy that penalizes departures from a target area A₀ and a target perimeter P₀ for the component cells of the tissue. As the dimensionless target shape index s₀=(P₀/√{A₀ }) is varied, we find a transition from a soft elastic regime for a compatible target perimeter and area to a stiffer nonlinear elastic regime frustrated by geometric incompatibility. We show that the ground state in the soft regime has a family of degenerate solutions associated with zero modes for the target area and perimeter. The onset of geometric incompatibility at a critical s₀^c lifts this degeneracy. The resultant energy gap leads to a nonlinear elastic response distinct from that obtained in classical elasticity models. We draw an analogy between cellular tissues and anelastic deformations in solids.