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We address an important issue of a dynamic homogenisation in vectorelasticity for a doubly periodic mass-spring elastic lattice. The notion oflogarithmically growing resonant waves is used in a complete analysis ofstar-shaped wave forms induced by an oscillating point force. We note that thedispersion surfaces for Floquet-Bloch waves in an elastic lattice main containcritical points of the saddle type. Based on the local quadratic approximationsof the frequency, as a function of wave vector components, we deduce propertiesof a transient asymptotic solution as the contribution of the point source tothe wave form. In this way, we describe local Green's functions as localizedwave forms corresponding to the resonant frequency. The peculiarity of theproblem lies in the fact that, at the same resonant frequency, the Taylorquadratic approximations for different groups of the resonant points aredifferent, and hence we deal with different local Green's functions. Thus,there is no uniformly defined homogenisation procedure for a given resonantfrequency. Instead, the continuous approximation of the wave field can beobtained through the asymptotic analysis of the lattice Green's functions.

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