Issue 29

A. Bacigalupo et alii, Frattura ed Integrità Strutturale, 29 (2014) 1-8; DOI: 10.3221/IGF-ESIS.29.01 7 micropolar media. Since the matrix of the coefficient in the eigenproblem (9) is hermitian, the eigenvalues are real, but not necessarily positive. Therefore, for some values of the model parameters is expected a reduced number of dispersive functions. In case of a tetrachiral lattice with internal resonant masses the free wave equation of motion takes the form:           1,11 1,22 2,11 2,22 ,1 ,2 1 1 1 1 1,11 1,22 2,11 2,22 ,1 ,2 2 2 1 2 ,11 ,22 1,1 2,2 1,2 2,1 1 1 1 2 1 2 2 2 2 2 2 2 ˆ ˆ ˆ d d d d u ku Bu Bu B k k v u u Bu Bu ku u k B k v u u S S Bu Bu ku ku k k I k u v v k u v v                                                         2 k I                   (10) In this case the secular equation system is written as follows: 2 2 2 1 2 2 2 1 2 2 1 2 2 2 2 2 2 ˆ 2 ˆ 0 0 ˆ ˆ 0 0 2 ˆ 0 0 ˆ ˆ ˆ 0 0 0 0 ˆ ˆ 0 0 0 0 ˆ ˆ 0 0 0 0 d d d d d d d d q k Bq iBq k kq k Bq ikq k Sq k iBq ikq k k I k k k k k k I                                                                                                          1 2 1 2 ˆ ˆ ˆ ˆ ˆ ˆ u u v v                                 0 (11) Again the solution of the eigenproblem (11) with hermitian matrix provides dispersion functions   q  , defined in the domain , q a a          . The analytical formulation here derived, which is based on a micropolar continuum model enriched with additional degrees of freedom of the resonating masses located inside the rings, will be developed in future research to analyze the wave propagation and to appreciate the influence of the geometrical and mechanical parameters of the microstructure and of the resonant devices on the frequency spectra and on the conditions of wave attenuation. The reliability of the results has to be evaluated from the comparison with the rigorous solutions obtained from a Floquet-Bloch analysis of the generalized beam lattice proposed in this paper. Finally, it should be noted that the validity of this model relies on the rigidity assumption of the rings [15], a condition that may be easily implemented in the physical model. A CKNOWLEDGEMENTS he Authors acknowledge financial support of the (MURST) Italian Department for University and Scientific and Technological Research in the framework of the research MIUR Prin09 project XWLFKW, Multi-scale modelling of materials and structures, coordinated by prof. A. Corigliano. Andrea Bacigalupo gratefully thanks financial T