广义材料性质沿厚度渐变的无限大板的厚度剪切振动频率
Thickness-shear Vibration Frequencies of an Infinite Plate with a Generalized Material Property Grading along the Thickness
摘要：
对于厚度剪切型石英晶体谐振器，振动频率和振型是谐振器在电路应用中的关键特征，它们反映了石英板的基本材料和结构特性及其在各种因素（例如可引起晶体坯的表面和内部损伤以及退化的侵蚀性气体和液体）下随时间的变化。累积的效应最终将改变弹性常数和刚度方面的表面条件，更重要的是，这些性质沿厚度的梯度。这是一种典型的功能梯度材料（FGM）结构，近年来已被广泛研究的结构应用下的多重载荷，如热和电磁场。对于声波谐振器，这样的研究是同样重要的，并且波在FGM结构中的传播可以用于基于声波的传感器（例如石英晶体微天平（QCM））的性能、可靠性和寿命的评估和评定。本文研究了AT切石英晶体的特性沿厚度方向变化的功能梯度材料板的厚度剪切振动，其振动模式由厚度坐标的正弦函数和余弦函数的三角函数表示。利用板变形的傅里叶展开法得到了解。我们还得到了频率的变化的基本和泛音模式，这是强耦合的谐振器结构的性能变化或设计的评估，以利用FGM在新的应用。
英文摘要：
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.
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