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

In this paper, we derive the shear coefficient K for quartz crystal of rectangular cross section in Timoshenko's beam theory of three-dimensional (3-D) elasticity. The theoretical result gives K as a function of the transformed elastic stiffness constants C/sub 22//sup '/, c/spl circ//sub 44/, C/sub 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 0.796-0.867 and its temperature coefficient of the first order varies from -15.3/spl times/10/sup -6///spl deg/C to +13.2/spl times/10/sup -6///spl deg/C, when a Z-plate of quartz crystal is rotated with a cut angle /spl theta/ of 0-180/spl deg/ about the x axis. The values of K are then compared with those for isotropic materials obtained by other authors.