Size-Dependent Thin Plate Dynamics: Investigating Anti-Plane and In-Plane Wave Propagation with Generalized Boundary Restraints

Abstract

Despite extensive research on plates with either traction-free boundaries or rigidly fixed faces, many real-world situations involve boundary conditions that fall between these two extremes. To bridge the gap between these fundamental cases—traction-free and fixed-boundary conditions—it is reasonable to assume that the fields at the boundaries follow a Hooke-type law, representing elastic restraints at the surfaces. These generalized boundary conditions, known as Elastically Restrained Boundary Conditions (ERBC), are applied in the normal, shear, and rotational directions to study anti-plane (SH) and in-plane (P-SV) wave phenomena in a microstructural elastic plate, modeled using the consistent couple stress theory. The ERBC incorporate stiffness coefficients to relate normal, tangential, and rotational stresses to the corresponding displacements within the plate. Analytical derivation of the dispersion relations is carried out to examine the wave propagation characteristics under varying boundary conditions. Special cases, such as stress-free boundary conditions (similar to Rayleigh-Lamb type waves), mixed boundary conditions, and rigid boundary conditions, emerge as limiting cases. The study explores the influence of the characteristic length scale parameter (l) introduced by the consistent couple stress model, along with stiffness coefficients like normal stiffness, shear stiffness, and rotational stiffness, on wave propagation. It also investigates the transition between rigidly fixed and stress-free boundary conditions, offering insights into how different boundary conditions affect wave behavior in a thin microstructural elastic plate.