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).
Highlights
Homogenization of materials with discrete microstructures is becoming increasingly important due to widespread applications of cellular structures and lattice-based metamaterials (e.g., [1,2,3,4,5,6,7,8,9,10,11]).Various homogenization methods have been developed using classical elasticity
2-D and 3-D cellular materials with and without periodicity constraints is developed, which is built on classical elasticity and has no restriction on shape, symmetry or number of struts in a unit cell
Equation (54) does not contain any unknown kinematic variable. It follows from Equations (32) and (54) that the effective stiffness matrix C for the 2-D cellular material can be obtained from the coefficient matrix of the following constitutive equations:
Summary
Homogenization of materials with discrete microstructures is becoming increasingly important due to widespread applications of cellular structures and lattice-based metamaterials (e.g., [1,2,3,4,5,6,7,8,9,10,11]). Norris [18] proposed a homogenization method for periodic lattice structures and derived analytical formulas for effective properties of elastic networks, including pentamode metamaterials. 2-D and 3-D cellular materials with and without periodicity constraints is developed, which is built on classical elasticity and has no restriction on shape, symmetry or number of struts in a unit cell In this new approach, the nodal equilibrium equations are enforced at all boundary and interior nodes, unlike in existing classical elasticity-based methods where the nodal equilibrium is imposed only at the boundary nodes and, as a result, the equilibrium equations are often not satisfied at the interior nodes by the approximate solutions obtained (e.g., [35]).
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