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

In this paper the dispersion relation associated with harmonic waves propag ating in a periodically layered structure is derived and analysed. Specific ally, the structure is made up of repeating unit cells, with each layer com posed of an incompressible, pre-stressed elastic material, each interface p erfectly bonded and the upper and lower surfaces of the structure free of i ncremental traction. The complexity of the problem is reduced using an appr oach involving the Cayley-Hamilton theorem. A numerical method is also used which eliminates positive exponential functions, thereby considerably redu cing the complexity of solving the dispersion relation numerically. Numeric al 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 to gether 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 reg ion, with an additional harmonic joining the group at a higher wave number. (C) 2001 Elsevier Science Ltd. All rights reserved.