Torsional and SH surface waves in an isotropic and homogenous elastic half-space characterized by the Toupin-Mindlin gradient theory

TL;DR

This paper demonstrates that the complete Toupin-Mindlin gradient elasticity theory predicts the existence of torsional and SH surface waves in a homogeneous, isotropic half-space, contrasting with classical theory and highlighting the role of microstructural parameters.

Contribution

It proves that the full Toupin-Mindlin gradient elasticity theory can predict surface waves in isotropic, homogeneous materials without anisotropy or layered structures.

Findings

Surface waves are dispersive and propagate at any frequency.

The character of dispersion depends on microstructural parameters.

Surface waves do not have a cut-off frequency in this model.

Abstract

The existence of torsional and SH surface waves in a half-space of a homogeneous and isotropic material is shown to be possible in the context of the complete Toupin-Mindlin theory of gradient elasticity. This finding is in marked contrast with the well-known result of the classical theory, where such waves do not exist in a homogeneous (isotropic or anisotropic) half-space. In the context of the classical theory, this weakness is usually circumvented by modeling the half-space as a layered structure or as having non-homogeneous properties. On the other hand, employing a simplified version of gradient elasticity (including only one microstructural parameter and an additional surface-energy term), Vardoulakis and Georgiadis (1997), and Georgiadis et al. (2000), showed that such surface waves may exist in a homogeneous half-space only if a certain type of gradient anisotropy is included in the formulation. On the contrary, in the present work, we prove that the complete Toupin-Mindlin theory of isotropic gradient elasticity (with five microstructural parameters) is capable of predicting torsional and SH surface waves in a purely isotropic and homogeneous material. In fact, it is shown that torsional and SH surface waves are dispersive and can propagate at any frequency (i.e. no cut-off frequencies appear). The character of the dispersion (either normal or anomalous) depends strongly upon the microstructural characteristics.