The aim of this paper is to demonstrate how the principles of the Freedom, Actuation, and Constraint Topologies (FACT) synthesis approach may be applied to the design of compliant microstructural architectures that possess extreme or unusual thermal expansion properties (e.g., zero or large negative thermal expansion coefficients). FACT provides designers with a comprehensive library of geometric shapes, which may be used to visualize the regions wherein various microstructural elements can be placed for achieving desired bulk material properties. In this way, designers can rapidly consider and compare every microstructural concept that best satisfies the design requirements before selecting the final design. A screw-theory-based analytical tool is also provided in this paper to help designers calculate and optimize the thermal properties of microstructural concepts, which are generated using FACT. As a case study, this tool is used to calculate the negative thermal expansion coefficient of a microstructural architecture synthesized using FACT.

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