Multi-Scale, Multi-Criteria Design of Meta-Materials with Offset Periodicity

Meta-materials are a class of artificial materials with a wide range of bulk properties, completely different from the base material they are made of. Some notable examples include negative Poisson\u27s ratio materials, materials designed for specific electromagnetic, acoustic, or thermal properties. The term meta-material in the context of this research refers to a continuous, heterogeneous structure with prescribed elastic properties. Such meta-materials are designed using Topology Optimization (TO). Tools like SIMP interpolation, mesh filtering and continuation methods are used to address the numerical issues with Topology Optimization. The most popular tool to design such materials is Asymptotic Homogenization. However, it has its limitations. Homogenization requires the meta-material to obey periodicity and scaling requirements. Dr. Chris Czech in his Ph.D. dissertation proposes a way to design meta-materials that may, due to manufacturing limitations, break the scaling requirements. Using Volume Averaging, he designs thin-layered meta-materials for use in the shear beam of a non-pneumatic wheel. By offsetting the said meta-material layers by a half-width of the Unit Cell, auxetic honeycomb-like geometry was obtained. This was the first time such a shape was observed as the result of Topology Optimization targeting the effective shear modulus. This research will further study the offset periodicity by considering offsets other than just zero or half-widths. The same shear beam of a non-pneumatic wheel is designed using such offsets. The optimization formulations in literature and the ones proposed by Dr. Czech have been extensively studied and used for single-criteria design problems. This research demonstrates the use of these formulations for the design of meta-materials with multiple prescribed elastic properties, such as prescribed behaviors in shear and in tension or compression. This research also identifies a possible physical limitation in the combinations of elastic properties that can be achieved for meta-materials when designed using Topology Optimizatio