Generalisation of elastic models for a layer with elastically restrained boundaries

A series of long wave asymptotic models for an isotropic layer are developed for the case of elastically restrained boundary conditions (ERBC). The dispersion relation and cut-off frequency equation are established for the case of symmetric boundary conditions restrained in the normal and tangential directions. A long-wave low frequency asymptotic model is developed to describe motion associated with the fundamental modes for small values of the restraint parameters. Four high frequency approximate models are developed which describe all possible asymptotic regimes connected with vibration within the vicinity of thickness resonances. One of the features of these models is that the cut-off frequencies are given implicitly for all families of the frequency spectrum. A special uniform asymptotic model is developed for the case of coalescing long wave limits which interact with each other. In this case it is not possible to use any of the typical techniques associated with the classic asymptotic models.