  Disordered hyperuniform heterogeneous materials are new, exotic amorphous
states of matter that behave like crystals in the manner in which they suppress
volume-fraction fluctuations at large length scales, and yet are statistically
isotropic with no Bragg peaks. It has recently been shown that disordered
hyperuniform dielectric two-dimensional cellular network solids possess
complete photonic band gaps comparable in size to photonic crystals, while at
the same time maintaining statistical isotropy, enabling waveguide geometries
not possible with photonic crystals. Motivated by these developments, we
explore other functionalities of various two-dimensional ordered and disordered
hyperuniform cellular networks, including their effective thermal or electrical
conductivities and elastic moduli. We establish the multifunctionality of a
class of such low-density networks by demonstrating that they maximize or
virtually maximize the effective conductivities and elastic moduli. This is
accomplished using the machinery of homogenization theory, including optimal
bounds and cross-property bounds, and statistical mechanics. We have identified
ordered and disordered hyperuniform low-weight cellular networks that are
multifunctional with respect to transport (e.g., heat dissipation and fluid
transport), mechanical and electromagnetic properties, which can be readily
fabricated using 3D printing and lithographic technologies.
