0
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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Uflyand, Y. S. , 1948, “ The Propagation of Waves in the Transverse Vibrations of Bars and Plates,” Prikl. Mat. Mekh., 12, pp. 287–300.
Mindlin, R. D. , 1951, “ Influence of Rotary Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates,” ASME J. Appl. Mech., 18(1), pp. 31–38.
Mindlin, R. D. , 1961, “ High Frequency Vibrations of Crystal Plates,” Q. Appl. Math., 19, pp. 51–61. [CrossRef]
Mindlin, R. D. , and Yang, J. , 2006, An Introduction to the Mathematical Theory of Vibrations of Elastic Plates, World Scientific, Singapore. [CrossRef]
Norris, A. N. , Krylov, V. V. , and Abrahams, I. D. , 2000, “ Flexural Edge Waves and Comments on ‘A New Bending Wave Solution for the Classical Plate Equation’ [J. Acoust. Soc. Am. 104, 2220–2222 (1998)],” J. Acoust. Soc. Am., 107(3), pp. 1781–1784. [CrossRef] [PubMed]
Vemula, C. , and Norris, A. N. , 1997, “ Flexural Wave Propagation and Scattering on Thin Plates Using Mindlin Theory,” Wave Motion, 26(1), pp. 1–12. [CrossRef]
Hutchinson, J. R. , 1984, “ Vibrations of Thick Free Circular Plates, Exact Versus Approximate Solutions,” ASME J. Appl. Mech., 51(3), pp. 581–585. [CrossRef]
Stephen, N. , 1997, “ Mindlin Plate Theory: Best Shear Coefficient and Higher Spectra Validity,” J. Sound Vib., 202(4), pp. 539–553. [CrossRef]
Hull, A. J. , 2006, “ Mindlin Shear Coefficient Determination Using Model Comparison,” J. Sound Vib., 294(1–2), pp. 125–130. [CrossRef]
Lakawicz, J. M. , and Bottega, W. J. , 2017, “ Branch Dependent Shear Coefficients and Their Influence on the Free Vibration of Mindlin Plates,” J. Sound Vib., 389, pp. 202–223. [CrossRef]
Benscoter, S. U. , 1955, ‘ Review of ‘Timoshenko's Shear Coefficient for Flexural Vibrations of Beams,’ by R.D. Mindlin and H. Deresiewicz,” ASME Appl. Mech. Rev., 8, pp. 510–511.
Mindlin, R. D. , and Deresiewicz, H. , 1955, “ Timoshenko's Shear Coefficient for Flexural Vibrations of Beams,” Second U.S. National Congress on Applied Mechanics, Ann Arbor, MI, June 14–18, pp. 175–178.
Bresse, M. , 1859, Cours De Mécanique Appliquée, Mallet-Bachelier, Paris, France.
Graff, K. F. , 1991, Wave Motion in Elastic Solids, Dover, Mineola, NY.
Shames, I. H. , and Dym, C. L. , 1985, Energy and Finite Element Methods in Structural Mechanics, Taylor and Francis, Boca Raton, FL.
Norris, A. N. , 2003, “ Flexural Waves on Narrow Plates,” J. Acoust. Soc. Am., 113(5), pp. 2647–2658. [CrossRef] [PubMed]
Rayleigh, L. , 1885, “ On Waves Propagated Along the Plane Surface of an Elastic Solid,” Proc. London Math. Soc., s1–17(1), pp. 4–11. [CrossRef]

Figures

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. 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. 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. 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

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In