On the reflection of waves in half-spaces of microstructured materials governed by dipolar gradient elasticity

Abstract

The present work studies the propagation and reflection of plane waves in a body having the form of a half-space. It is assumed that the mechanical response of this body is governed by dipolar gradient elasticity. Our aim is to investigate the effect of boundaries on the elastic wave motion in a medium with microstructure and, thus, to determine possible deviations from the predictions of classical linear elastodynamics. The use of the theory of gradient elasticity is intended to model the response of materials with microstructure and incorporate size effects into stress analysis in a manner that the classical theory cannot afford. Here, a simple but yet rigorous version of the Toupin–Mindlin generalized continuum theory is employed that also includes micro-inertial effects. Our results show significant departure from those of classical elastodynamic theory. Indeed, it is observed that an incident dilatational or distortional wave at the traction-free plane boundary gives rise to four reflected waves, instead of the usual two waves predicted by the classical theory. It is shown that the amplitudes, the angles of reflection, and the phase shift of the reflected waves depend significantly upon the material microstructure. This dependence becomes more pronounced at shorter wavelengths.