On the accuracy of plate theories for the prediction of unwanted modes near the fundamental thickness shear mode

The first order plate theories with correction factors are generally assumed to predict accurately the plate modes which have half wave lengths greater than the plate thickness, and at frequencies up to twenty percent higher than the fundamental thickness shear frequency. This assumption is assessed by comparing the straight crested wave solutions of the plate theories with those of the three-dimensional elastic equations of motion. The frequency spectra for bandwidths of resonant frequencies versus the aspect ratio of length to thickness are compared for three sets of plate equations: The first order Mindlin plate equations, the third order Mindlin plate equations, and the third order Lee and Nikodem plate equations. The finite element results for a quartz SC-cut strip with free edges show that Mindlin´s first order plate equations, and Lee and Nikodem´s third order plate equations do not yield an accurate frequency spectra of the modes in the vicinity of the fundamental thickness shear mode, although the thickness shear mode itself is predicted accurately. The third order Mindlin plate equations without correction factors, on the other hand, predict well the frequency spectrum in the vicinity. The first order Mindlin plate theory is found to yield accurate frequency spectra for normalized frequencies less than 0.1, which is lower than previously assumed. At normalized frequencies greater than 0.1, deviations are seen in the frequency spectra, starting with the modal branches which are more steeply inclined