HOMOGENIZATION IN DYNAMICS OF HETEROGENEOUS STRUCTURES

Homogenisation of a one-dimensional heterogeneous structures is split into two procedures: homogenisation of the spectral properties and hom ogenisation of the heterogeneous mechanical parameters. The governing equation for the dynamics of complex structures is derived and solved by means of the Dyson integral equation which delivers the mean wave f ield in the case of a Gaussian, statistically homogeneous field of het erogeneity. The mean field is analysed in detail for the three particu lar correlation functions; (i) an exponential one, (ii) a differentiab le one with a hidden periodicity and (iii) a nondifferentiable one wit h a hidden periodicity. The general case of a structure with an arbitr ary heterogeneity of small scale is considered and simple closed form expressions for the mean field and attenuation are derived. It is show n that the average rigidity, average mass density and average wave spe ed are determined by the overall parameters of the structure, while th e attenuation depends strongly upon the size and range of the secondar y systems.