  Vertex Models, as used to describe cellular tissue, have an energy controlled
by deviations of each cell area and perimeter from target values. The
constrained nonlinear relation between area and perimeter leads to new
mechanical response. Here we provide a mean-field treatment of a highly
simplified model: a uniform network of regular polygons with no topological
rearrangements. Since all polygons deform in the same way, we only need to
analyze the ground states and the response to deformations of a single polygon
(cell). The model exhibits the known transition between a fluid/compatible
state, where the cell can accommodate both target area and perimeter, and a
rigid/incompatible state. %The rigid solid-like state has a single gapped
ground state. We calculate and measure the mechanical resistance to various
deformation protocols and discover that at the onset of rigidity, where a
single zero-energy ground-state exists, %We show that in the incompatible
state, where a single frustrated ground-state exists, linear elasticity fails
to describe the mechanical response to even infinitesimal deformations. In
particular we identify a breakdown of reciprocity expressed via different
moduli for compressive and tensile loads, implying non-analyticity of the
energy functional. We give a pictorial representation in configuration space
that reveals that the complex elastic response of the Vertex Model arises from
the presence of two distinct sets of reference states (associated with target
area and target perimeter).
