Surface Waves in a Microstructural Couple Stress Half Space under the Extended Mindlin’s Restrained Boundary Conditions

Abstract

Boundary conditions play a crucial role from theoretical and experimental perspectives in comprehending the dynamics of wave propagation in elastic solids. Stress free boundary conditions, are achieved by setting the surface tractions to zero, while for rigid surface boundary conditions displacements, are equated to zero. Despite utility of these two types of boundary conditions, it is crucial to acknowledge that these conditions are the extreme idealizations, and real-world scenarios often fall somewhere between the extremes of stress free and rigid surface boundary conditions. In the current investigation, the elastically restrained boundary conditions (ERBC) are proposed to examine the propagation of plane waves at the surface of semi-infinite elastic media. The elastically restrained boundary conditions act as an intermediate link between traction-free and rigid surface conditions. In the study, a microstructural couple stress half-space is considered to comprehend the propagation of surface waves. The impacts of boundary conditions are depicted through three parameters called as normal stiffness \(\left( {{{k}_{n}}} \right)\), shear stiffness \(\left( {{{k}_{t}}} \right)\), and rotational stiffness \(\left( {{{k}_{r}}} \right)\). The characteristic length scale parameter (l) within the couple stress model represents the influence of microstructural effects. Dispersion relations have been derived analytically and the classical cases of stress-free (Rayleigh type wave), rigid surface boundary conditions are obtained as the special cases. Mathematical results have been illustrated graphically.