  There are two main classes of physics-based models for two-dimensional
cellular materials: packings of repulsive disks and the vertex model. These
models have several disadvantages. For example, disk interactions are typically
a function of particle overlap, yet the model assumes that the disks remain
circular during overlap. The shapes of the cells can vary in the vertex model,
however, the packing fraction is fixed at $\phi=1$. Here, we describe the
deformable particle model (DPM), where each particle is a polygon composed of a
large number of vertices. The total energy includes three terms: two quadratic
terms to penalize deviations from the preferred particle area $a_0$ and
perimeter $p_0$ and a repulsive interaction between DPM polygons that penalizes
overlaps. We performed simulations to study the onset of jamming in packings of
DPM polygons as a function of asphericity, ${\cal A} = p_0^2/4\pi a_0$. We show
that the packing fraction at jamming onset $\phi_J({\cal A})$ grows with
increasing ${\cal A}$, reaching confluence at ${\cal A} \approx 1.16$. ${\cal
A}^*$ corresponds to the value at which DPM polygons completely fill the cells
obtained from a surface-Voronoi tessellation. Further, we show that DPM
polygons develop invaginations for ${\cal A} > {\cal A}^*$ with excess
perimeter that grows linearly with ${\cal A}-{\cal A}^*$. We confirm that
packings of DPM polygons are solid-like over the full range of ${\cal A}$ by
showing that the shear modulus is nonzero.
