Thermoelastic relaxation in elastic structures with applications to thin plates

TL;DR

This paper presents a new method to calculate thermoelastic damping in vibrating elastic solids, especially thin plates, by linking thermal diffusion with local flexural deformation, and extends Zener's theory for practical applications in MEMS and NEMS devices.

Contribution

It introduces a direct calculation approach for thermoelastic damping in elastic structures, incorporating local curvature effects and transverse thermal diffusion, generalizing Zener's theory.

Findings

Thermoelastic damping depends on local principal curvatures of the plate.

Maximum damping occurs at points with equal principal curvatures; zero at saddle points.

Transverse thermal diffusion effects are negligible for sufficiently thick plates.

Abstract

A new result enables direct calculation of thermoelastic damping in vibrating elastic solids. The mechanism for energy loss is thermal diffusion caused by inhomogeneous deformation, flexure in thin plates. The general result is combined with the Kirchhoff assumption to obtain a new equation for the flexural vibration of thin plates incorporating thermoelastic loss as a damping term. The thermal relaxation loss is inhomogeneous and depends upon the local state of vibrating flexure, specifically, the principal curvatures at a given point on the plate. Thermal loss is zero at points where the principal curvatures are equal and opposite, that is, saddle shaped or pure anticlastic deformation. Conversely, loss is maximum at points where the curvatures are equal, that is, synclastic or spherical flexure. The influence of modal urvature on the thermoelastic damping is described through a modal pparticipation factor. The effect of transverse thermal diffusion on plane wave propagation is also examined. It is shown that transverse diffusion effects are always small provided the plate thickness is far greater than the thermal phonon mean free path, a requirement for the validity of the classical theory of heat transport. These results generalize Zener's theory of thermoelastic loss in beams and are useful in predicting mode widths in MEMS and NEMS oscillators.