Semianalytical geometry-property relationships for some generalized classes of pentamodelike additively manufactured mechanical metamaterials

We study a special class of mechanical metamaterials, namely lattices, based on beams (struts) with nonuniform cross sections, of which pentamode mechanical metamaterials are a special case. Five symmetric beam types including simple cylinder, concave double cone, convex double cone, concave hyperbolic, and convex hyperbolic are considered. Three types of loads including lateral force, axial force, and moment are applied to the free end of cantilever beams with the five aforementioned geometries and their responses are compared in terms of displacement, normal stress variation along the beam length, and shear stress variation along the beam length. Using the displacement diagrams obtained for different strut geometries and loading conditions, semi-analytical relationships are derived for the displacement at the end of cantilever beams. The semi-analytical relationships are then used to calculate the elastic modulus and yield strength of lattice structures based on two types of unit cells, namely diamond and cube. We also build numerical models and perform experiments to benchmark our semi-analytical results. Comparison of the analytical, numerical, and experimental results demonstrate the accuracy of the semi-analytical relationships presented for the mechanical properties of this class of mechanical metamaterials. Moreover, in both the cube- and diamond-based structures, increasing the α value (i.e., the ratio of the largest to the smallest cross-section radius in each strut) at a constant relative density decreases the elastic modulus, yield strength, the initial maximum stress, and the plateau stress.