Thickness-shear Vibrations Of Functionally Graded Materials(FGMs) Plates

Functionally graded materials(FGMs) as new type of materials are widely used in various fields of engineering. Its main principle is that we can change material properties by changing the volume or fraction of material components of structures, and change some aspects of the characteristics of structures like strength, heat resistance, and resonance. With the consideration of actual demand of structures, we can produce materials to satisfy specific needs of structures through analysis and design.In this paper, we first reviewed the research history of FGMs with focus on the analysis of resonant properties and problems of elastic wave propagation of FGMs plate. Moreover, we briefly introduced several typical elastic waves existed in elastic solid. The research of the characteristics of elastic waves was always active with important applications in engineering fields. Then we focused on quartz crystal resonators which are frequently subjected to surface damage and internal degradation in blanks in production and processing, or long-term service in the severe environment which can be the equivalent of FGMs plate, assuming that the gradient function for the material parameters along the thickness direction of the plate as continuous function. In analysis and calculation of vibration characteristics of the plate with thickness-shear mode, some factors of the partial differential equation were given by Fourier expansion. Finally, we utilized numerical examples to demonstrate the analysis and explanation of thickness-shear vibrations frequency characteristics of FGMs plates. First, we considered the gradient in the thickness direction of symmetric form. Material properties then are expressed as cosine function. Then, we expanded gradient form to include symmetric and asymmetric forms for more general variations. As a result, material properties can now be represented as the combination of cosine and sine functions, and the process of solution is also based on Fourier expansion. We analyzed and calculated the two cases of FGMs plates in thickness-shear vibrations and the frequencies, or frequency changes from a uniform plate in terms of the fundamental frequency can be used for the evaluation of the grading.