Vibrations and damping of the eigenmodes of viscoelastic nanospheres with thermal conductivity

TL;DR

This paper derives exact analytical formulas for the eigenfrequencies and damping of viscoelastic nanospheres, considering thermal effects and damping contributions, revealing complex behaviors including non-monotonic damping and frequency shifts.

Contribution

It provides the first closed-form analytical expressions for eigenfrequencies and damping rates of viscoelastic nanospheres with thermal conductivity, including non-monotonic damping behaviors.

Findings

Damping rates increase monotonically with frequency for breathing and torsional modes.

Spheroidal modes exhibit non-monotonic damping behavior.

Frequency shifts due to damping are generally reduced, with some anomalous shifts observed.

Abstract

In this paper, we derive a series of exact analytical closed expressions to calculate the complex eigenfrequencies and the displacement for the corresponding eigenmodes of a viscoelastic (nano-)sphere in the presence of linear damping. Where possible, we provide closed expressions for damping rates, including the contributions from viscosity, as well as thermal conductivity and solutions of the heat equation. We assume an isolated system, such that no energy/heat transfer to the environment is allowed. We find monotonic behavior of the damping as a function of frequency for breathing and torsional modes, however, for spheroidal modes we find non-monotonicity. Furthermore, we analytically analyze the thermodynamic limit for all mode types. We also investigate the frequency shift and find expected behavior, i.e. a reduced eigenfrequency with damping than without damping for breathing and torsional modes. For spheroidal modes, however, we find non-monotonic shifts, corresponding to the damping. For some eigenfrequencies we find anomalous frequency shifts.