A Generalized Strain Energy-Based Homogenization Method for 2-D and 3-D Cellular Materials with and without Periodicity Constraints

Abstract

A generalized strain energy-based homogenization method for 2-D and 3-D cellular materials with and without periodicity constraints is proposed using Hill’s Lemma and the matrix method for spatial frames. In this new approach, the equilibrium equations are enforced at all boundary and interior nodes and each interior node is allowed to translate and rotate freely, which differ from existing methods where the equilibrium conditions are imposed only at the boundary nodes. The newly formulated homogenization method can be applied to cellular materials with or without symmetry. To illustrate the new method, four examples are studied: two for a 2-D cellular material and two for a 3-D pentamode metamaterial, with and without periodic constraints in each group. For the 2-D cellular material, an asymmetric microstructure with or without periodicity constraints is analyzed, and closed-form expressions of the effective stiffness components are obtained in both cases. For the 3-D pentamode metamaterial, a primitive diamond-shaped unit cell with or without periodicity constraints is considered. In each of these 3-D cases, two different representative cells in two orientations are examined. The homogenization analysis reveals that the pentamode metamaterial exhibits the cubic symmetry based on one representative cell, with the effective Poisson’s ratio v¯ being nearly 0.5. Moreover, it is revealed that the pentamode metamaterial with the cubic symmetry can be tailored to be a rubber-like material (with v¯ ≅0.5) or an auxetic material (with v¯< 0).

Two-Dimensional Pentamode Metamaterials: Properties, Manufacturing, and Applications

“…The pentamode behavior-where the material behaves like an ideal fluid that supports compressional waves but not shear waves-is critically dependent on the ratio of the structural dimensions to the wavelength. For a given material, the size of the unit cells and, by extension, the struts and joints that compose these cells, must be small enough so that the incoming wave does not resolve the individual structures, ensuring that the wave interacts with the metamaterial as if it were a homogeneous medium [58].…”