The edge waves on a Kirchhoff plate bilaterally supported by a two-parameter elastic foundation

In this paper, the bending waves propagating along the edge of a semi-infinite Kirchhoff plate resting on a two-parameter Pasternak elastic foundation are studied. Two geometries of the foundation are considered: either it is infinite or it is semi-infinite with the edges of the plate and of the foundation coinciding. Dispersion relations along with phase and group velocity expressions are obtained. It is shown that the semi-infinite foundation setup exhibits a cut-off frequency which is the same as for a Winkler foundation. The phase velocity possesses a minimum which corresponds to the critical velocity of a moving load. The infinite foundation exhibits a cut-off frequency which depends on its relative stiffness and occurs at a nonzero wavenumber, which is in fact hardly observed in elastodynamics. As a result, the associated phase velocity minimum is admissible only up to a limiting value of the stiffness. In the case of a foundation with small stiffness, asymptotic expansions are derived and beam-like one-dimensional equivalent models are deduced accordingly. It is demonstrated that for the infinite foundation the related nonclassical beam-like model comprises a pseudo-differential operator

Approximate analysis of surface wave-structure interaction

Surface wave-structure interaction is studied starting from a specialised approximate formulation involving a hyperbolic equation for the Rayleigh wave along with pseudostatic elliptic equations over the interior of an elastic half-space. The validity of the proposed approach for modelling a point contact is analysed. Explicit dispersion relations are derived for smooth contact stresses arising from averaging the effect of a regular array of spring-mass oscillators and also of elastic rods attached to the surface. Comparison with the exact solution of the associated plane time-harmonic problem in elasticity for the array of rods demonstrates a high efficiency of the developed methodology

Perturbed rigid body motions of an elastic rectangle

Plane and anti-plane dynamic problems for an elastic rectangle loaded along its sides are considered. Low-frequency perturbations to rigid body translations are calculated. The derivation involves the solutions of non-homogeneous boundary value problems for harmonic and bi-harmonic equations. The explicit solution for the harmonic problem for transverse anti-plane translation is expressed through Fourier series. The bi-harmonic problem corresponding to the longitudinal in-plane translation is studied in greater detail for an elongated rectangle, which also may be treated using the elementary theory for plate extension. The derived perturbations incorporate the variations of displacements and stresses over the interior of the rectangle, including the case of self-equilibrated loading. The latter is obviously outside the range of validity of the classical rigid body framework