In this article, five fundamental unit cells namely BCC, hexagonal packing, rhombicuboctahedron, diamond, and truncated octahedron, for which there was a lack of accurate analytical yield stress relationship in the literature, were studied analytically and numerically using the Timoshenko and Euler–Bernoulli beam theories. Two different approaches based on force or displacement methods were considered. The results of the analytical models were validated against finite element models results as well as experimental data from literature. The results demonstrated that the newly developed analytical relationships based on Timoshenko beam theory give more accurate results as compared to Euler–Bernoulli beam theory, especially in higher values of relative densities.

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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.@en