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Accuracy of crystal plate theories for high-frequency vibrations in the range of 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 wavelengths greater than the plate thickness, and at frequencies up to 20% 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 of plate 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 yield less accurate frequency spectra of the modes in the vicinity of the fundamental thickness shear mode than the third-order Mindlin plate equations without correction factors. The degree of inaccuracy increases with the ratio of plate length to thickness, and the slope of the modal branches in the frequency spectra. For a plate length to thickness ratio of 31 to 33, the first-order plate theory is found to yield accurate frequency spectra for normalized frequencies less than 0.3, which is lower than previously assumed. At normalized frequencies greater than 0.3, deviations are seen in the frequency spectra, starting with the modal branches which are more steeply inclined.