Small amplitude waves in a pre-stressed compressible elastic layer with one fixed and one free face

We address the problem of wave propagation in a pre-stressed elastic layer with mixed boundary conditions, the layer having one fixed and one free face. Numerical analysis provides a good initial insight into the influence of these boundary conditions on dispersion characteristics. In the long wave regime, there is clearly no evidence of low-frequency motion and thus an absence of any long wave fundamental mode-like features. In the short wave regime, however, the dispersion relations does show evidence of low-frequency dispersion phenomena. The first harmonic’s short wave phase speed limit is shown to be distinct from that of all other harmonics; this coincides with the associated Rayleigh surface wave speed. The short wave analysis is completed with the derivation of approximate solutions for the higher harmonics. Asymptotic long wave approximations of the dispersion relation are then obtained for motion within the vicinity of the thickness stretch and thickness shear resonance frequencies. These approximations are required to obtain the relative asymptotic orders of the displacement components for frequencies within the vicinity of either the shear or stretch resonance frequencies. This enables an analogue of the asymptotic stress-strain state to be established through asymptotic integration

Dispersion phenomena in symmetric pre-stressed layered elastic structures

The dispersion relation associated with a symmetric three layer structure, composed of compressible, pre-stressed elastic layers, is derived. This mathematically elaborate transcendental equation gives phase speed as an implicit function of wave number. Numerical solutions are established to show a wide range of dispersion behaviour which is delicately dependent on the material parameters and pre-stress in each layer. Particularly interesting behaviour is observed within the short wave (high wave number) regime, with six possible cases of short wave liming behaviour shown possible. Within each of these, a short wave asymptotic analysis is carried out, resulting in a set of approximations which provide explicit relationships between phase speed and wave number. It is envisaged that these approximations may prove helpful to approximate numerical truncation errors associated with impact response, as well as providing excellent first approximations for particularly (numerically) challenging sets of material parameters