Abstract

We address the grand question of identifying the set of possible elasticity tensors (including anisotropic ones) of 3d-printed materials constructed from a given elastic material with known elastic constants. We identify many almost optimal geometries with elasticity tensors arbitrarily near the boundary of what one can achieve. We characterize many parts of the surface of the set of possible elasticity tensors. This is no easy task as completely anisotropic 3d-elasticity tensors live in an 18-dimensional space of invariants, much more than the two invariants (bulk and shear moduli) that characterize isotropic elasticity tensors. We completely characterize the set of possible (average stress, average strain) pairs that can exist in these porous materials. Unfortunately, the geometries we find are rather extreme but this should motivate the search for more realistic ones that come close to having the desired elasticity tensors. Also, not all parts of the surface are characterized for elastically isotropic composites. Further progress will require new ideas. This is joint work with Marc Briane, Mohamed Camar-Eddine, and Davit Harutyunyan.