XXI ICTAM, 15–21 August 2004, Warsaw, Poland THERMOELASTIC RELAXATION IN THIN PLATES WITH APPLICATIONS TO MEMS AND NEMS OSCILLATORS

Equations governing thermoelastic loss in vibrating thin plates are derived, generalizing Zener’s theory of thermoelastic loss in beams. The mechanism for the loss of mechanical energy is thermal diffusion caused by inhomogeneous deformation, flexure in thin plates. It is shown that a critical plate thickness, h∗, exists which separates two distinct types of response, which may be called thermally thick and thin. For plates of thickness h > h∗ the diffusion, and hence the thermal component of the solution, is restricted to a 1-dimensional heat flux across the plate thickness. If the thickness is less than h∗ then in-plane thermal diffusion cannot be ignored, and may in fact dominate. Several general results are obtained for plates with h > h∗. Thus, using the Kirchhoff assumption for the elastic deformation, it is shown that the local thermal relaxation loss depends upon the local state of vibrating flexure, specifically, the principal curvatures at a given point on the plate. The thermal loss is zero at points where the principal curvatures are equal and opposite, i.e. saddle shaped deformation. Conversely, thermal loss is maximum at points where the curvatures are equal, or spherical flexure. An effective plate equation is derived that incorporates the thermoelastic loss as a damping term. The general form of the effective damping that is obtained for h > h∗ can be generalized to arbitrary thickness, in particular the case of thermally thin plates can be considered. These results are useful in predicting mode widths in MEMS and NEMS oscillators.