Technical Brief

A Refinement of Mindlin Plate Theory Using Simultaneous Rotary Inertia and Shear Correction Factors

[+] Author and Article Information
Andrew N. Norris

Mechanical and Aerospace Engineering,
Rutgers University,
98 Brett Road,
Piscataway, NJ 08854
e-mail: norris@rutgers.edu

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received February 15, 2017; final manuscript received December 17, 2017; published online February 9, 2018. Assoc. Editor: A. Srikantha Phani.

J. Vib. Acoust 140(3), 034503 (Feb 09, 2018) (4 pages) Paper No: VIB-17-1065; doi: 10.1115/1.4038956 History: Received February 15, 2017; Revised December 17, 2017

We revisit Mindlin's theory for flexural dynamics of plates using two correction factors, one for shear and one for rotary inertia. Mindlin himself derived and considered his equations with both correction factors, but never with the two simultaneously. Here, we derive optimal values of both factors by matching the Mindlin frequency–wavenumber branches with the exact Rayleigh–Lamb dispersion relations. The thickness shear resonance frequency is obtained if the factors are proportional but otherwise arbitrary. This degree-of-freedom allows matching of the main flexural mode dispersion with the exact Lamb wave at either low or high frequency by choosing the shear correction factor as a function of Poisson's ratio. At high frequency, the shear factor takes the value found by Mindlin, while at low frequency, it assumes a new explicit form, which is recommended for flexural wave modeling.

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Grahic Jump Location
Fig. 3

Exact and Mindlin wavenumbers for (κ, λ) = (κ0, 1), ν = 0.35 The abscissa and ordinate are the same as in Fig. 1

Grahic Jump Location
Fig. 2

The relative error in the F1 flexural wavenumber k1 compared with the exact Rayleigh–Lamb wavenumber k, as a function of nondimensional frequency kTh for five combinations of κ, λ. The curves represent k1 calculated from Eq. (10a) with shear correction factors only and with both correction factors κ, λ satisfying the constraint (17).

Grahic Jump Location
Fig. 1

The wavenumbers of the three exact branches, F1, F2, S, and the Mindlin wavenumbers k1, k2, k3 from Eq. (10) for (κ, λ) = (κ1, 1), with ν = 0.35 The abscissa shows the real and imaginary parts of the nondimensional wavenumber kh for the separate branches. The ordinate is nondimensional frequency defined by kT of Eq. (11).

Grahic Jump Location
Fig. 4

The shear correction factors κ0, κ1, κ2 from Eqs. (12), (15), and (21), respectively, and the high frequency F1 correction factor κR, as functions of Poisson's ratio ν

Grahic Jump Location
Fig. 5

The exact and Mindlin wavenumbers for (κ, λ) = (κ2, κ2/κ0), with ν = 0.35 The abscissa and ordinate are the same as in Fig.1




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