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.

中文翻译：

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沿厚度方向具有渐变材料特性的无限石英板的厚度剪切频率

对于厚度剪切型石英晶体谐振器，振动频率和振型反映了石英晶体板的基本材料和结构特性，以及随着时间推移而受到侵蚀性气体和液体侵蚀的变化，这些腐蚀和腐蚀可能导致表面和内部性能下降。形成。反过来，这种渐进效应将通过弹性常数和刚度，更重要的是，改变这些特性在整个厚度上具有一定深度的梯度。现在，我们研究功能梯度材料板的厚度-剪切振动，其中流行的AT切割石英晶体的特性在整个厚度范围内以厚度的正弦和余弦分量表示的一般模式变化。振动解是通过板变形的傅立叶展开获得的。