Mindlin plate theory: best shear coefficient and higher spectra validity

Abstract

Mindlin plate theory predicts three frequency spectra or, equivalently, three branches to a phase velocity dispersion diagram, the lowest of which—thew1mode—provides rotatory inertia and shear deformation corrections to classical thin plate theory. Employing consistent truncation procedures to both the Mindlin and the exact Rayleigh–Lamb frequency equations, valid for long wavelength and low phase velocity , one finds thatw1mode agreement is achieved when the shear coefficient takes the valuek=5/(6?v); the Mindlin prediction is then less than ?0·5% in error when the wavelength is equal to the plate thickness, and less than +1% in error as wavelength approaches zero. The previously dismissed MindlinHmode is seen to be in exact frequency (or phase velocity) agreement with the second slowestSHwave in the infinite plate, as long as the shear coefficient for this mode takes the valuek=?2/12. However thew2mode, as with the second frequency spectrum of Timoshenko beams, should be regarded as the inevitable, but meaningless, consequence of an otherwise remarkable approximate engineering theory