Cellular materials are ubiquitous in our world being found in natural and engineered systems as structural materials, sound and energy absorbers, heat insulators and more. Stochastic foams made of polymers, metals and even ceramics find wide use due to their novel properties when compared to monolithic materials. Properties of these so called hybrid materials, those that combine materials or materials and space, are derived from the localization of thermomechanical stresses and strains on the mesoscale as a function of cell topology. The effects of localization can only be generalized in stochastic materials arising from their inherent potential complexity, possessing variations in local chemistry, microstructural inhomogeneity and topological variations. Ordered cellular materials on the other hand, such as lattices and honeycombs, make for much easier study, often requiring analysis of only a single unit-cell.

Theoretical bounds predict that hybrid materials have the potential to push design envelopes offering lighter stiffer and stronger materials. Hybrid materials can achieve very low and even negative coefficients of thermal expansion (CTE) while retaining a relatively high stiffness --

properties completely unmatched by monolithic materials. In the first chapter of this thesis a two-dimensional lattice is detailed that possess near maximum stiffness, relative to the tightest theoretical bound, and low, zero and even appreciably negative thermal expansion. Its CTE and stiffness are given in closed form as a function of geometric parameters and the material properties. This result is confirmed with finite elements (FE) and experiment. In the second chapter the compressive stiffness of three-dimensional ordered foams, both closed and open cell, are predicted with FE and the results placed in property space in terms of stiffness and density. A novel structure is identified that effectively achieves theoretical bounds for Young's, shear and bulk modulus simultaneously, over a wide range of relative densities, greatly expanding the property space of available materials with a pragmatic manufacturable structure.

A variety of other novel and previously studied ordered foam topologies are also presented that are largely representative of the spectrum of performance of such materials, shedding insight into the behavior of all cellular materials.