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

Abstract

Abstract. We address the problem of wave propagation in a pre-stressed elastic layer with mixed boundary conditions, the layer having one fixed the 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-strainstain to be established through asymptotic integration.

Rayleigh-type waves on a coated elastic half-space with a clamped surface

“…In contrast to traditional derivations usually dealing with low-frequency vibrations (the frequency range of interest is well below the first thickness resonance) of a coating with a traction-free upper face (e.g. [3][4][5][6]), we develop an asymptotic procedure also covering the highfrequency vibrations, similar to what has been done for a thin interfacial layer in [7], and earlier for thin-walled structures in [8][9][10], see also [11][12][13][14][15]. As a result, we are able to tackle higher-order Rayleigh-type waves with a sinusoidal variation across the thickness of the coating, which is a feature of a soft coating only.…”

Asymptotic analysis of 3D dynamic equations in linear elasticity for a thin layer resting on a Winkler foundation

“…The asymptotic near cut-off dispersion relation takes the form of a fraction, see (4.3) and (5.7), instead of a polynomial typical of a plate with free or fixed faces, e.g., see Kaplunov et al (1998) and also Rogerson et al (2007), Lashhab et al (2015). In this case, the pole K = √ 1 − ν/2 leads to a nonuniform behaviour.…”