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Metamaterials are engineered materials with unusual properties deriving mainly from their micro-architectures rather than composition. They have attracted increasing attention owing to their tailorable multi-functional properties that cannot be exhibited by naturally-occurring or traditionally-manufactured materials. Materials with a negative Poisson’s ratio (PR) and materials with a non-positive coefficient of thermal expansion (CTE) are examples of mechanical metamaterials. In this dissertation, metamaterials with negative Poisson’s ratios and negative coefficients of thermal expansion are designed using both heuristic approaches and topology optimization methods.

Firstly, based on a star-shaped re-entrant structure, three two-dimensional (2-D) periodic cellular materials are developed, which can be tailored to exhibit a negative Poisson’s ratio by adjusting geometrical parameters and strut connections between unit cells. Castigliano’s second theorem is employed to obtain analytical formulas for the effective Poisson’s ratio and Young’s modulus. In addition, by using two metallic materials with different coefficients of thermal expansion (CTE), four planar bi-material lattice metamaterials are designed based on a re-entrant structure unit, which can achieve both negative Poisson’s ratios and non-positive CTEs at the same time. Furthermore, three dimensional (3-D) metamaterials are designed via spatial tessellations of 2-D unit vii cells. Moreover, interpenetrating phase composites with negative Poisson’s ratios are obtained by embedding a 3-D re-entrant structure into a matrix phase. By choosing different tessellation patterns in the 3-D space, three different bi-material metamaterials are proposed, which can attain negative Poisson’s ratio and non-positive CTE simultaneously. For all these designs, detailed parametric studies are conducted.

Secondly, topology optimization is used to obtain optimal material distributions in the problem domain in order to create metamaterials with unusual material properties. A parametric level set method combined with a meshfree method is employed to design cellular metamaterials with a maximum bulk modulus, a maximum shear modulus or a minimum Poisson’s ratio in which both the case of one solid material with a void phase and the case of two solid materials with a void phase are considered. In addition, the parametric level set method is utilized to generate 2-D and 3-D metamaterials with unusual thermomechanical properties. The optimization problems for achieving minimum anisotropic or isotropic CTEs with a prescribed Poisson’s ratio constraint are formulated, and a few novel microstructures with distinct material interfaces between two phases are produced. For the final optimized metamaterials with negative Poisson’s ratios, the re-entrant and chiral structures are observed even though the optimization starts from initial distributions without such features.

Degree Date

Fall 12-16-2017

Document Type



Mechanical Engineering


Xin-Lin Gao

Number of Pages




Creative Commons License

Creative Commons Attribution-Noncommercial 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial 4.0 License

Available for download on Monday, December 16, 2019