An Analysis of Nonlinear Thickness-shear Vibrations of Quartz Crystal   Plates by Two-Dimensional Finite Element Method
  A nonlinear analysis of high-frequency thickness-shear vibrations of AT-cut quartz crystal plates is presented with the two-dimensional finite element method. We expanded both kinematic and constitutive nonlinear Mindlin plate equations and then truncated them to the first-order equations as an approximation, which is used later for the formulation of nonlinear finite element analysis with all zeroth- and first-order displacements and electric potentials. The matrix equation of motion is established with the first-order harmonic approximation and the generalized nonlinear eigensystem is solved by a direct iterative procedure. A backbone curve and corresponding mode shapes are obtained and analyzed. The nonlinear finite element program is developed based on earlier linear edition and can be utilized to predict nonlinear characteristics of miniaturized quartz crystal resonators in the design process. 
Ji Wang, Yangyang Chen, Rongxing Wu, Lihong Wang, Huimin Jing, Jianke   Du, Yuantai Hu, Guoqing Li
cond-mat.mtrl-sci