  Rapid advances in additive manufacturing over the past decade have kindled
widespread interest in the rational design of metamaterials with unique
properties. However, many applications require multi-physics metamaterials,
where multiple properties are simultaneously optimized. This is challenging,
since different properties, such as mechanical and mass transport properties,
typically impose competing requirements on the nano-/micro-/meso-architecture
of metamaterials. Here, we propose a parametric metamaterial design strategy
that enables independent tuning of the effective permeability and elastic
properties. We apply hyperbolic tiling theory to devise simple templates based
on which triply periodic minimal surfaces (TPMS) are partitioned into hard and
soft regions. Through computational analyses, we demonstrate how the decoration
of hard, soft, and void phases within the TPMS substantially enhances their
permeability-elasticity property space and offers high tunability in the
elastic properties and anisotropy, at constant permeability. We also show that
this permeability-elasticity balance is well captured using simple scaling
laws. We then proceed to demonstrate the proposed concept through
multi-material additive manufacturing of representative specimens. Our
approach, which is generalizable to other designs, offers a route towards
multi-physics metamaterials that need to simultaneously carry a load and enable
mass transport, such as load-bearing heat exchangers or architected
tissue-substituting meta-biomaterials.
