Resonant waves in elastic structured media: Dynamic homogenisation versus Green's functions

We address an important issue of dynamic homogenisation in vector elasticity for a doubly periodic mass-spring elastic lattice. The notion of logarithmically growing resonant waves is used in the analysis of star-shaped wave forms induced by an oscillating point force. We note that the dispersion surfaces for Floquet-Bloch waves in the elastic lattice may contain critical points of the saddle type. Based on the local quadratic approximations of a dispersion surface, where the radian frequency is considered as a function of wave vector components, we deduce properties of a transient asymptotic solution associated with the contribution of the point source to the wave form. The notion of local Green's functions is used to describe localised wave forms corresponding to the resonant frequency. The special feature of the problem is that, at the same resonant frequency, the Taylor quadratic approximations for different groups of the critical points on the dispersion surfaces (and hence different Floquet-Bloch vectors) are different. Thus, it is shown that for the vector case of micro-structured elastic medium there is no uniformly defined dynamic homogenisation procedure for a given resonant frequency. Instead, the continuous approximation of the wave field can be obtained through the asymptotic analysis of the lattice Green's functions, presented in this paper.