On localised vibrations in incompressible pre-stressed transversely isotropic elastic solids

This paper is concerned with 2D localised vibration in incompressible pre-stressed fibre-reinforced elastic solids and the closely related problem of surface wave propagation in such materials. An appropriate constitutive model is derived and its stability discussed within the context of the strong ellipticity condition. Surface wave propagation in an associated half-space is considered, with the particular cases of propagation along a principal direction of primary deformation and that of almost inextensible fibres also investigated. The problems of free and forced edge vibration of a semi-infinite strip are analysed, revealing a link between the natural edge frequencies and the associated Rayleigh surface wave speed

Dispersion of elastic waves in laminated glass

Elastic sandwich-type structures with high-contrast material and geometrical properties have numerous applications in modern engineering, including, in particular, laminated glass, photovoltaic panels, precipitator plates in gas filters, etc. Multi-parametric modelling of such structures assumes taking into consideration various types of contrast in stiffness, density and thickness. The present contribution is concerned with analysis of low-frequency dispersion of elastic waves in case of an antisymmetric motion of a three-layered symmetric plate, modelling laminated glass. The conditions on material and geometrical parameters, leading to the lowest non-zero thickness shear resonance frequency tending to zero, are formulated. In this case the dispersion relation possesses two low-frequency modes instead of a single fundamental low-frequency mode, which is typical for a homogeneous plate. A two-mode uniform asymptotic approximation is constructed, along with local approximations for the fundamental mode and the first shear harmonic

Reduced model for the surface dynamics of a generally anisotropic elastic half-space

Near-surface resonance phenomena often arise in semi-infinite solids. For instance, when a moving load with a speed v close to the surface wave speed vR is applied to the surface of an elastic half-space, it will give rise to a large-amplitude disturbance inversely proportional to v − vR. The latter can be determined by a multiple-scale approach using an extra slow time variable. It has also been shown for isotropic elastic half-spaces that the reduced governing equation thus derived is capable of describing the surface wave contribution even for arbitrary dynamic loading. In this paper, we first derive the analogous evolution equation for a generally anisotropic elastic half-space, and then assess its applicability in the study of travelling waves in a half-space that is coated with a continuous array of spring-like vertical resonators or bonded to an elastic layer of different properties. Our results are validated by comparison with previously known results, and illustrative calculations are carried out for a fibre-reinforced half-space and a coated half-space that is subjected to a finite deformation

Explicit model for surface waves in a pre-stressed, compressible elastic half-space

The paper is concerned with the derivation of the hyperbolic-elliptic asymptotic model for surface wave in a pre-stressed, compressible, elastic half-space, within the framework of plane-strain assumption. The consideration extends the existing methodology of asymptotic theories for Rayleigh and Rayleigh-type waves induced by surface/edge loading, and oriented to extraction of the contribution of studied waves to the overall dynamic response. The methodology relies on the slow-time perturbation around the eigensolution, or, equivalently, accounting for the contribution of the poles of the studied wave. As a result, the vector problem of elasticity is reduced to a scalar one for the scaled Laplace equation in terms of the auxiliary function, with the boundary condition is formulated as a hyperbolic equation with the forcing terms. Moreover, hyperbolic equations for surface displacements are also presented. Scalar hyperbolic equations for surface displacements could potentially be beneficial for further development of methods of non-destructive evaluation

On Rayleigh wave field induced by surface stresses under the effect of gravity

The paper is concerned with development of the asymptotic formulation for surface wave field induced by vertical surface stress under the effect of gravity in the short-wave region. The approach relies on the methodology of hyperbolic-elliptic models for the Rayleigh wave and results in a regularly perturbed hyperbolic equation on the surface acting as a boundary condition for the elliptic equation governing decay over the interior. A special value of the Poisson's ratio v = 0.25 is pointed out, at which the effect of gravity disappears at leading order

On surface wave fields arising in soil-structure interaction problems

Abstract The paper aims at generalization of the specialized formulation, originally developed for the surface wave fields induced by prescribed surface stresses. We extend this formulation to soil-structure interaction problems with unknown contact stresses and internal sources. The problem for an internal source embedded in an elastic half-plane is reduced to that for prescribed surface stresses by considering the point source solution for an unbounded medium. In this case the sub-problems corresponding to normal and tangential stresses assume a separate treatment. Then, we analyze interaction of an elastic half-plane with a one degree of freedom mass-spring system. The focus is on a near-resonant regime investigated by a perturbation technique

Explicit model for bending edge wave on an elastic orthotropic plate supported by the Winkler–Fuss foundation

The paper is concerned with a bending edge wave on a thin orthotropic elastic plate resting on a Winkler–Fuss foundation. The main focus of the contribution is on derivation of a specialised reduced model accounting for the contribution of the bending edge wave to the overall dynamic response, allowing simplified analysis for a number of dynamic problems. The developed formulation includes an elliptic equation associated with decay over the interior, and a beam-like equation on the edge governing wave propagation accounting for both bending moment and modified shear force excitation, thus highlighting a dual parabolic-elliptic nature of the bending edge wave. A model example illustrates the benefits of the approach