Dynamic wave propagation in micro-torus structures: Implementing a 3D physically realistic theory

Abstract

Abstract This paper makes strides to analyze the wave propagation in micro-torus structures with circular cross-sections using the modified couple stress theory (MCST). This physically realistic theory encompasses the material size effect The fundamental equilibrium equations of MCST in the mathematical framework of toroidal coordinates are first reformulated. Navier’s equations resulting from the implementation of MCST for micro-torus structures are treated by employing the Helmholtz decomposition technique. Several micro-torus problems using MCST in the 3D elasticity framework are solved numerically by a finite element method (FEM) to find eigenvalues and eigenfunctions from the elastic wave propagation standpoints. Wave propagation analysis within micro-torus structures and the effects of material length scale parameter on the vibrational response of various geometries, such as a quarter and a half torus along with a cylinder, are investigated. Opposed to the classical continuum theory that fails to describe the dispersion of waves at higher frequencies within micro-tours structures, the MCST can demonstrate the experimentally reported dispersive wave behavior. The findings can be potentially implemented to analyze and design a variety of sensors and actuators as well as medical, telecommunication, and electronic devices. This paper also develops an accurate estimation for the biharmonic and Laplace’s toroidal coordinates to study charge distribution in isolated conducting graphene micro-torus structures. Finally, the resulting electrostatic potential and the charge density in a conducting torus surface for electrostatics applications are determined in terms of Legendre functions. Regarding the different cases of arbitrary charge distribution, findings are helpful to control the electronic properties of graphene micro torus and rational design of nano-structures used in nanoscale transistor channels and superconductors. Treating the Laplace equation on a nano torus assists to more precisely realize the high-frequency localization of Laplacian eigenfunctions, i.e., how an eigenfunction is distributed in a small region of the torus domain and decays rapidly outside the region.