  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.
