Meso-scale method of asymptotic analysis of elastic vibrations in periodic and non-periodic multi-structures

Abstract

The method of meso-scale asymptotic approximations has proved to be very effective for the analysis of models of solids containing large clusters of defects, such as small inclusions or voids. Here, we present a new avenue where the method is extended to elastic multi-structures. Geometrically, a multi-structure makes a step up in the context of overall dimensions, compared to the dimensions of its individual constituents. The main mathematical challenge comes from the analysis of the junction regions assigned to the multi-structure itself. Attention is given to problems of vibration and on the coupling of vibration modes corresponding to displacements of different orientations. The method is demonstrated through the dynamic analysis of infinite or finite multi-scale asymmetric flexural systems consisting of a heavy beam connected to a non-periodic array of massless flexural resonators within some interval. In modelling the interaction between the beam and the resonators, we derive a vectorial system of partial differential equations through which the axial and flexural motions of the heavy beam are coupled. The solution of these equations is written explicitly in terms of Green’s functions having intensities determined from a linear algebraic system. The influence of the resonators on the heavy beam is investigated within the framework of scattering and eigenvalue problems. For large collections of resonators, dynamic homogenization approximations for the medium within the location of the resonant array are derived, leading to (i) the classical Rayleigh beam for symmetric systems and (ii) a generalized Rayleigh beam for asymmetric structures that support flexural–longitudinal wave coupling. Independent numerical simulations are also presented that demonstrate the accuracy of the analytical results.