On the use of finite element method in resonant ultrasound spectroscopy

Abstract

The Rayleigh–Ritz method is conventionally used in Resonant Ultrasound Spectroscopy (RUS) for the determination of the elastic constants of materials. It limits the application of RUS only for samples in simple shapes. In order to use RUS to characterize the complicated MEMS structures, the Finite Element Method (FEM) in RUS has been studied in this paper. Starting from Hamiltonian’s principle, FEM is compared with the Rayleigh–Ritz method for RUS application. As numerical tests, the accuracy of FEM is checked by using an isotropic quartz cube and an anisotropic corundum rectangular parallelepiped. The results show that for isotropic samples FEM converges very fast and resolution of calculated natural frequencies can be as high as 0.4%. Then FEM is used to calculate natural frequencies and make mode analysis of a micromachined silicon nitride membrane achored on silicon substrate. Experimentally, the natural frequencies for those coupled membrane modes identified by FEM have been measured by RUS and used to determine Young’s modulus of silicon nitride, which is 197 GPa and verified by comparing with measured results from the Load-Deflection method. This illustrates that RUS combined with FEM can be used as a new tool for Non-Destructive Testing of thin films in MEMS structures.