  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 $A_0$ and a target perimeter $P_0$
for the component cells of the tissue. As the dimensionless target shape index
$s_0 = \frac{P_0}{\sqrt{A_0}}$ is varied, we find a transition from a soft
elastic regime for 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 $s_0^c$ 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.
