Effective Shear Modulus for Flexural and Extensional Waves in an Unloaded Thick Plate

摘要

Thin-plate theory suffers from failure to properly predict the correct phase speed of bending waves and extensional waves in the limit of high frequency or large thickness. Mindlin has adapted the basic ideas that Timoshenko developed for bars to the case of bending waves in plates to partly correct this situation. Kane and Mindlin have given a similar correction for extensional waves in plates. In this report a new derivation of such correction factors is presented that is based on a comparison of the approximate theory with the results of exact elasticity theory (Lamb waves). Explicit equations are given to compute the effective shear modulus in antisymmetric waves. It is shown analytically that the phase speed calculated with these correction factors for the shear modulus asymptotically approaches the Rayleigh wave speed for high frequency. The same principle of comparison of approximate theory with exact elasticity theory is applied to the other terms in the equations of motion besides those depending on the shear modulus. This has not been entirely successful, but it promises, among other things, to improve the identification of the second root in the quadratic dispersion relation with the first order antisymmetric and symmetric Lamb waves. (Author)