Some comments on the dispersion relation for periodically layered pre-stressed elastic media

Abstract

In this paper the dispersion relation associated with harmonic waves propagating in a periodically layered structure is derived and analysed. Specifically, the structure is made up of repeating unit cells, with each layer composed of an incompressible, pre-stressed elastic material, each interface perfectly bonded and the upper and lower surfaces of the structure free of incremental traction. The complexity of the problem is reduced using an approach involving the Cayley–Hamilton theorem. A numerical method is also used which eliminates positive exponential functions, thereby considerably reducing the complexity of solving the dispersion relation numerically. Numerical solutions are presented in respect of both a two-ply and symmetric four-ply unit cell. An interesting feature of these solutions is the grouping together of harmonics as the number of unit cells increases. In the case of n unit cells, n−1 harmonics group together in the moderate wave number region, with an additional harmonic joining the group at a higher wave number.