Abstract
Motivated by the unexpected appearance of shear horizontal Rayleigh surface waves, we investigate the mechanics of antiplane wave reflection and propagation in couple stress (CS) elastic materials. Surface waves arise by mode conversion at a free surface, whereby bulk travelling waves trigger inhomogeneous modes. Indeed, Rayleigh waves are perturbations of the travelling mode and stem from its reflection at grazing incidence. As is well known, they correspond to the real zeros of the Rayleigh function. Interestingly, we show that the same generating mechanism sustains a new inhomogeneous wave, corresponding to a purely imaginary zero of the Rayleigh function. This wave emerges from 'reflection' of a bulk standing mode: This produces a new type of Rayleigh-like wave that travels away from, as opposed to along, the free surface, with a speed lower than that of bulk shear waves. Besides, a third complex zero of the Rayleigh function may exist, which represents waves attenuating/exploding both along and away from the surface. Since none of these zeros correspond to leaky waves, a new classification of the Rayleigh zeros is proposed. Furthermore, we extend to CS elasticity Mindlin's boundary conditions, by which partial waves are identified, whose interference lends Rayleigh-Lamb guided waves. Finally, asymptotic analysis in the thin-plate limit provides equivalent one-dimensional models.
Highlights
The discovery of surface waves by Lord Rayleigh [1] revealed that bulk waves may interact with a free surface to produce a substantially different type of wave, that still propagates along the surface and yet it decays exponentially in the interior
For an elastic theory to support Rayleigh waves, there needs to exist a form of mode conversion from travelling to inhomogeneous waves upon reflection at a free surface
This mechanism is required to stand right at grazing incidence. It may happen beyond a certain critical angle of incidence, like in sagittal plane propagation of shear vertical (SV) waves within classical elasticity (CE), or, as in antiplane motion for couple stress (CS) materials, the inhomogeneous wave may appear for all incident angles
Summary
The discovery of surface waves by Lord Rayleigh [1] revealed that bulk waves may interact with a free surface to produce a substantially different type of wave, that still propagates along the surface and yet it decays exponentially in the interior. A combination of the above is considered in [8], dealing with piezoacoustic (Bleustein–Gulyaev) SH surface waves in a functionally graded material (FGM) This notwithstanding, no study appears in the literature investigating the mechanics of surface reflection in the presence of SH surface waves, in an attempt to single out the characteristic feature that triggers their appearance. A number of contributions have appeared in the literature investigating wave propagation in CS materials In their pioneering work [15], Graff and Pao consider wave reflection and propagation in the sagittal plane (i.e. plane-strain) of an isotropic CS half-space, in the absence of rotational inertia. We investigate a novel type of ‘reflection’ that involves standing waves and leads to a new Rayleigh-like wave, propagating in the interior of the material and exponentially exploding/decaying along the surface (§3c) This wave cannot exist on an infinite surface.
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