The time-harmonic antiplane elastic response of a constrained layer

The purpose of this paper is to examine an idealized model of an elastic structure that is simple enough to be amenable to analytical investigation yet offers a methodology that can be applied in more complex situations. A thick elastic plate, forced or constrained at a finite number of (parallel) discrete lines on one of its faces, is studied. Each constraint demands that the displacement vanishes over a small but finite-sized region of the surface, and asymptotic analysis is employed to determine the vibration field (consisting in this model of out-of-plane shear waves) both globally, away from the constraints, and in their vicinity. Analysis indicates that it is essential to model the constraint boundary condition accurately. However, perhaps surprisingly, this result is found to be asymptotically equivalent to taking a rib of vanishing thickness, but with the zero displacement condition applied at a single point, suitably shifted along the surface. It is noted that for point rather than distributed constraints, a discrete Fourier transform solution returns a finite, but erroneous, value at the constraint location. This value is not associated with any physical field, but instead depends only on the truncation employed. Finally, the effect of a number of constraints, and their location, on the propagation of shear waves, is investigated.