Issue 29

A. Bacigalupo et alii, Frattura ed Integrità Strutturale, 29 (2014) 1-8; DOI: 10.3221/IGF-ESIS.29.01 6 moduli of the tetragonal system. The elasticity tensor depends on the chirality parameter, but unlike the hexachiral honeycomb, some elasticities are odd functions of  [15]. P LANE WAVES PROPAGATION he free wave motion in the hexachiral material is written in terms of the components of the generalized displacement field     1 2 1 2 T u u v v    U x in the following form             1,11 1,12 1,22 2,11 2,12 2,22 ,1 ,2 1 1 1 1 1,11 1,12 1,22 2,11 2,12 2,22 ,1 ,2 2 2 1 2 ,11 ,22 1,1 2,2 1,2 2 (2 ) 2 2 2 2 (2 ) 2 2 2 2 2 2 d d u Au k u Au k u Au A k k v u u Au k u Au k u Au u k A k v u u S S Au Au ku ku                                                                ,1 1 1 1 2 1 2 2 2 2 2 4 ˆ , ˆ , ˆ . d d k k I k u v v k u v v k I                                      (8) If a harmonic plane wave propagating along axis 1 x in an infinite planar micropolar medium is admitted, the generalized displacement field at a point is assumed in the following form   1 ˆ exp T i qx t        U U , where q and  denote the wave number and the circular frequency, respectively, and   1 2 1 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ T u u v v    U is the vector of the amplitudes   1 i   . Substituting the assumed generalized displacement field in the equation of motion (8) one obtains the secular equation system for the equivalent continuum model     2 2 2 1 2 2 2 1 2 2 1 2 2 2 2 2 2 2 ˆ 2 0 0 ˆ ˆ 2 0 0 ˆ 4 ˆ 2 2 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 Aq iAq k k k q Aq ikq k k Sq k iAq 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 (9) The solution of the eigenproblem (9), characterized by a hermitian matrix, provides dispersion functions   q  defined in the domain , q a a           , a being the characteristic size of the cell as shown in Figure 2. Since the model has six degrees of freedom, it follows that in the general case six plane waves may propagate, each characterized by a dispersion function. It follows that this model exhibits plane waves more complex than those obtained and discussed by Parfitt and Eringen [17] for isotropic achiral two dimensional micropolar solids and more recently by Khurana and Tomar [18] for 3D chiral T