Dispersion of elastic waves in a layer interacting with a Winkler foundation

Abstract

Dispersion of plane harmonic waves in an elastic layer interacting with one or twosided Winkler foundation is analysed. The long-wave low frequency polynomial approximations of the full transcendental dispersion relations are derived for a relatively soft foundation. The validity of the conventional engineering formulation of a Kirchhoff plate resting on an elastic foundation is investigated. It is shown that this formulation has to be refined near the cutoff frequency of bending waves. The associated near cutoff expansion is obtained for both cases. A simple explicit formula demonstrating veering of bending and extensional waves is presented for a one-sided foundation.

Elastodynamics of a coated half-space under a sliding contact

“…Comparison of mid-plane displacements performed using finite element software shows that for a sliding contact the tangential displacement significantly dominates over the vertical one, whereas for a fixed lower face both displacements seem to be of the same order. This is in line with initial physical insight, as well as observations of an extra longitudinal mode in Achenbach and Keshava [15], and also more recent findings reported for an elastic contact in Erbaş et al [27,28].…”

A new Rayleigh-like wave in guided propagation of antiplane waves in couple stress materials

“…For example, in [18], in the context of sagittal propagation, it is claimed that 'as the frequency increases, all modes converge to the Rayleigh wave propagation speed'. Upon considering equation (3.43) and the limit behaviour (3.8), the asymptotic model [37] for symmetric antiplane waves in the long-wave low-frequency (LWLF) range is, to leading order in Ω, . Symmetric antiplane RL waves (solid, black) and even SH partial waves (dashed, red) frequency spectrum (η = 0.1, 0 = 0.1, H = 10).…”

“…For example, in [18], in the context of sagittal propagation, it is claimed that 'as the frequency increases, all modes converge to the Rayleigh wave propagation speed'. Upon considering equation (3.43) and the limit behaviour (3.8), the asymptotic model [37] for symmetric antiplane waves in the long-wave low-frequency (LWLF) range is, to leading order in Ω, In the SWHF limit, all branches but the first asymptote to the bulk SH wavenumber κ = δ; instead, the first branch approaches the Rayleigh wavenumber κ R > δ from above (i.e. from lower speed).…”

Asymptotically consistent size-dependent plate models based on the couple-stress theory with micro-inertia

“…12. Thus, we have κ = O( ) and we disregard the closest neighbourhood of the cut-on frequency, that is to say that Ω − Ω o0 ∼ , see Erbaş et al (2018) for more details in the context of a Kirchhoff plates resting on a Winkler foundation.…”