Three-dimensional hierarchical metamaterials incorporating multi-directional programmable thermal expansion

The dimensional stability under the temperature variation is of great significance for the temperature-sensitive structures, motivating the development of metamaterials with the programmable coefficient of thermal expansion (CTE). Here, all the configurations, totally including 51 types for the bi-material pyramid unit cells, were comprehensively devised to attain the widely programmable CTE in single direction. Furthermore, by using an originally developed matrix transformation method, multiple classes of the pyramid-based hierarchical metamaterials were systematically devised. These hierarchical metamaterials incorporated the unidirectional, transversal and isotropic CTEs in the multiple directions. The closed-form expressions of the CTE and relative density for the devised pyramid unit cells and hierarchical metamaterials were analytically established. The theoretical analysis was well verified by the performed numerical modeling, and confirmed that the large ranges of the programmable CTEs could be obtained by rationally adjusting the material and geometry parameters. Besides, the comprehensive comparison of the specific CTE identified that the programmable CTE and low relative density could be well balanced through the devised hierarchical metamaterials. The coupling effect of both material and geometrical parameters on the programmable CTE was figured out, providing a proper guideline to design the metamaterials with both light weight and desirable CTE.