Timoshenko's Shear Coefficient for Transverse Vibrations of Trigonal Crystals of Rectangular Cross Section

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  • Timoshenko′s Shear Coefficient for Transverse Vibrations of Trigonal Crystals of Rectangular Cross Section
  • Timoshenko s Shear Coefficient for Tran

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In this paper, we derive the shear coefficient K for trigonal crystals 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 c22, c44, c66. For an isotropic material it agrees completely with that derived by G. R. Cowper. In addition, values of K versus cut angle for quartz crystal, lithium tantalate, and lithium niobate which belong to the trigonal system are calculated, so that K has a value of 0.796-0.867 for quartz crystal, 0.834- 0.857 for lithium tantalate, and 0.831-0.863 for lithium niobate, when Z-plates of the crystals are rotated with a cut angle θof 0-180°about the x-axis. The values of K are then compared with those for isotropic materials obtained by other authors.

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