Thickness-Shear Frequencies of an Infinite Quartz Plate with Graded Material Properties Across the Thickness

For quartz crystal resonators of thickness-shear type, the vibration frequency and mode shapes reflect the basic material and structural properties of a quartz crystal plate and the variation with time under attack by erosive gases and liquids that can cause surface and internal degradation in a graded form. Such gradual effects, in turn, will change surface conditions through elastic constants and stiffness and more importantly the gradient of such properties across the thickness with certain depth. Now we study the thickness-shear vibrations of functionally graded material plates with properties of the popular AT-cut quartz crystal varying across the thickness in a general pattern represented by both sine and cosine components of the thickness. The vibration solutions are obtained through the Fourier expansion of the plate deformation.