  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.
