Closed-form analytical relationships for pentamode metamaterials

Pentamode metamaterials are a class of extremal materials exhibiting fluid-like mechanical behavior. The mechanical properties of pentamode metamaterials arise from their unique micro-architecture, rather than their constituent material. In this research, we present closed-form analytical relationships for the elastic modulus and Poisson’s ratio of pentamode lattice structures with double-cone struts based on cubic diamond morphology. To validate our analytical solutions, we performed numerical simulations and experimental tests, which confirmed the accuracy of the derived relationships. Our findings indicate that increasing the smaller diameter ($d$) and the larger-to-smaller diameter ratio ($\alpha$) of the double-cones increases the elastic modulus of pentamode metamaterials. However, within the considered range of $d$ and $\alpha$, the Poisson’s ratio is nearly constant and lies within the range of approximately 0.5. These analytical relationships provide valuable insight into the mechanical behavior of pentamode metamaterials, which can aid in the design and optimization of new materials with unique properties.