The mechanics of a foam typically depends on the bubble geometry, topology,
and the material at hand, be it metallic or polymeric, for example. While the
foam energy functional for each bubble is typically minimization of surface
area for a given volume, biology provides us with a wealth of additional energy
functionals, should one consider biological cells as a foam-like material.
Here, we focus on a mean field approach to obtain the elastic moduli, within
linear response, for an ordered, three-dimensional vertex model using the
space-filling shape of a truncated octahedron and whose energy functional is
characterized by a restoring surface area spring and a restoring volume spring.
The tuning of the three-dimensional shape index exhibits a rigidity transition
via a compatible-incompatible transition. Specifically, for smaller shape
indices, both the target surface area and volume cannot be achieved, while
beyond some critical value of the three-dimensional shape index, they can be,
resulting in a zero-energy state. As the elastic moduli depend on curvatures of
the energy when the system, we obtain these as well. In addition to
analytically determining the location of the transition in mean field, we find
that the rigidity transition and the elastic moduli depend on the
parameterization of the cell shape with this effect being more pronounced in
three dimensions given the array of shapes that a polyhedron can take on (as
compared to a polygon). We also uncover nontrivial dependence on the
deformation protocol in which some deformations result in affine motion of the
vertices, while others result in nonaffine motion. Such dependencies on the
shape parameterization and deformation protocol give rise to a nontrivial shape
landscape and, therefore, nontrivial mechanical response even in the absence of
topology changes.Comment: 18 pages, 24 figure