2013/10/26 by A. B. Movchan, Movchan, Alexander, Leonid I. Slepyan +1
Computer Science · Engineering · #Acoustic Wave Phenomena Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1310.7089
openalex publication_date 2013/10/26 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
We address an important issue of a dynamic homogenisation in vector\nelasticity for a doubly periodic mass-spring elastic lattice. The notion of\nlogarithmically growing resonant waves is used in a complete analysis of\nstar-shaped wave forms induced by an oscillating point force. We note that the\ndispersion surfaces for Floquet-Bloch waves in an elastic lattice main contain\ncritical points of the saddle type. Based on the local quadratic approximations\nof the frequency, as a function of wave vector components, we deduce properties\nof a transient asymptotic solution as the contribution of the point source to\nthe wave form. In this way, we describe local Green's functions as localized\nwave forms corresponding to the resonant frequency. The peculiarity of the\nproblem lies in the fact that, at the same resonant frequency, the Taylor\nquadratic approximations for different groups of the resonant points are\ndifferent, and hence we deal with different local Green's functions. Thus,\nthere is no uniformly defined homogenisation procedure for a given resonant\nfrequency. Instead, the continuous approximation of the wave field can be\nobtained through the asymptotic analysis of the lattice Green's functions.\n