The shear coefficient for quartz crystal of rectangular cross section in Timoshenko´s beam theory

This paper describes the shear coefficient K of transverse vibration for quartz crystal that is an anisotropic material. In this paper, we derive the shear coefficient K for quartz crystal of rectangular cross section in Timoshenko´s beam theory of three dimensional elasticity. The theoretical result gives K as a function of the transformed elastic stiffness constants c´₂₂, cˆ₄₄, c´₆₆. For an isotropic material it agrees completely with that derived by G.R. Cowper (1966). In addition, a value of K and its temperature coefficient versus cut angle is calculated, so that K has a value of 0.796 to 0.867 and its temperature coefficient of the first order varies from -15.3×10^-6/°C to +13.2×10^-6/°C, when a Z-plate of quartz crystal is rotated with a cut angle θ of 0 to 180 degrees about the x-axis. The values of K are then compared with those obtained by other authors