Thickness-shear vibration frequencies of an infinite plate with a generalized material property grading along the thickness

For quartz crystal resonators of thickness-shear type, the vibration frequency and mode shapes, which are key features of resonators in circuit applications, reflect the basic material and structural properties of the quartz plate and its variation with time under various factors such as erosive gases and liquids that can cause surface and internal damages and degradation of crystal blanks. The accumulated effects eventually will change the surface conditions in terms of elastic constants and stiffness and more importantly, the gradient of such properties along the thickness. This is a typical functionally graded materials (FGM) structure and has been studied extensively for structural applications under multiple loadings such as thermal and electromagnetic fields in recent years. For acoustic wave resonators, such studies are equally important and the wave propagation in FGM structures can be used in the evaluation and assessment of performance, reliability, and life of sensors based on acoustic waves such as the quartz crystal microbalances (QCM). Now we studied the thickness-shear vibrations of FGM plates with properties of AT-cut quartz crystal varying along the thickness in a general pattern represented by a trigonometric function with both sine and cosine functions of the thickness coordinate. The solutions are obtained by using Fourier expansion of the plate deformation. We also obtained the frequency changes of the fundamental and overtone modes which are strongly coupled for the evaluation of resonator structures with property variation or design to take advantages of FGM in novel applications.