Exact forms of effective elastic properties of frame-like periodic cellular solids

This study presents the exact forms of the effective elastic constants of arbitrary frame-like periodic cellular solids that can be modeled accurately using Euler beams. The forms are derived analytically by using the homogenization method based on equivalent strain energy. In the derivation, the cross sections of all struts in a frame-like periodic cellular solid with an arbitrary topology are set to be the same and are also set to have the same moment of inertia in all directions. The exact forms of the effective elastic constants are obtained in terms of some dimensionless factors, the characteristic length and volume of the unit cell, the area and moment of inertia of the struts, and Young's modulus of the base material. The dimensionless factors are dependent on the topology of the solid. In general, these factors can be functions of the area and moment of inertia of the struts. However, for numerous practical topologies, the factors are constant. In these cases, the obtained forms can be used as exact parametric forms. The constant factors for frame-like periodic cellular solids with a particular topology can be determined by exact curve fitting using numerical finite element results with different areas and moments of inertia of the struts. If exact fitting cannot be achieved beyond the fitting data, it follows that, for the topology being considered, some of the factors are not constant and the results of the curve fitting should not be used. To check the validity of the proposed exact forms of the effective elastic constants, they are used with several topologies of 2D and 3D frame-like periodic cellular solids. The obtained effective elastic constants are compared with exact solutions from elaborate symbolic finite element computations and/or the literature. When all dimensionless factors in the proposed exact forms are constant, the effective elastic constants obtained in this study are found to be exactly the same as the exact solutions.