Analytical Homogenization for Dynamic Analysis of Composite Membranes with Circular Inclusions in Hexagonal Lattice Structures

Free vibrations of a composite membrane with a hexagonal lattice circular inclusion are investigated. We aim at a study of the lower frequency spectrum, i.e. it is assumed that the minimum space period of the eigenform is essentially larger than the characteristic dimension of the cell periodicity of the analyzed structure. This implies a possibility of approximating the composite structure by the homogenized one with effective characteristics. The latter is yielded by the multi-scale homogenization approach.

Introduction of slow and fast variables yields both counterpart quasistatic local problem regarding the periodically repeated cell and global (homogenized) dynamic problem for homogeneous material with effective properties. The most complicated part of this approach concerns in finding a solution to the local problem. It has been analytically found in the presented paper for relatively large inclusion sizes using lubrication approach. The performed numerical validation of the obtained results shows high accuracy of the implemented approach.