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Research Papers

Vibrations of an Incompressible Linearly Elastic Plate Using Discontinuous Finite Element Basis Functions for Pressure

[+] Author and Article Information
Lisha Yuan

Department of Biomedical Engineering and Mechanics,
M/C 0219, Virginia Polytechnic Institute and State University,
Blacksburg, VA 24061
e-mail: lishay@vt.edu

Romesh C. Batra

Honorary Member and Fellow,
Department of Biomedical Engineering and Mechanics,
M/C 0219, Virginia Polytechnic Institute and State University,
Blacksburg, VA 24061
e-mail: rbatra@vt.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the Journal of Vibration and Acoustics. Manuscript received February 12, 2019; final manuscript received May 10, 2019; published online June 19, 2019. Assoc. Editor: Julian Rimoli.

J. Vib. Acoust 141(5), 051016 (Jun 19, 2019) (13 pages) Paper No: VIB-19-1070; doi: 10.1115/1.4043816 History: Received February 12, 2019; Accepted May 13, 2019

We numerically analyze, with the finite element method, free vibrations of incompressible rectangular plates under different boundary conditions with a third-order shear and normal deformable theory (TSNDT) derived by Batra. The displacements are taken as unknowns at the nodes of a 9-node quadrilateral element and the hydrostatic pressure at four interior nodes. The plate theory satisfies the incompressibility condition, and the basis functions satisfy the Babuska-Brezzi condition. Because of the singular mass matrix, Moler's QZ algorithm (also known as the generalized Schur decomposition) is used to solve the resulting eigenvalue problem. Computed results for simply supported, clamped, and clamped-free rectangular isotropic plates agree well with the corresponding analytical frequencies of simply supported plates and with those found using the commercial software, abaqus, for other edge conditions. In-plane modes of vibrations are clearly discerned from mode shapes of square plates of aspect ratio 1/8 for all three boundary conditions. The magnitude of the transverse normal strain at a point is found to equal the sum of the two axial strains implying that higher-order plate theories that assume null transverse normal strain will very likely not provide good solutions for plates made of rubberlike materials that are generally taken to be incompressible. We have also compared the presently computed through-the-thickness distributions of stresses and the hydrostatic pressure with those found using abaqus.

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References

Mindlin, R. C., and Medick, M. A., 1959, “Extensional Vibrations of Elastic Plates,” ASME J. Appl. Mech., 26(2), pp. 145–151.
Srinivas, S., Rao, C. V. J., and Rao, A. K., 1970, “An Exact Analysis for Vibration of Simply Supported Homogeneous and Laminate Thick Rectangular Plates,” J. Sound Vib., 12(2), pp. 187–199. [CrossRef]
Srinivas, S., and Rao, A. K., 1970, “Bending, Vibration and Buckling of Simply Supported Thick Orthotropic Rectangular Plates and Laminates,” Int. J. Solids Struct., 6(11), pp. 1463–1481. [CrossRef]
Hanna, N. F., and Leissa, A. W., 1994, “A Higher Order Shear Deformation Theory for the Vibration of Thick Plates,” J. Sound Vib., 170(4), pp. 545–555. [CrossRef]
Carrera, E., 1999, “A Study of Transverse Normal Stress Effect on Vibration of Multilayered Plates and Shells,” J. Sound Vib., 225(5), pp. 803–829. [CrossRef]
Messina, A., 2001, “Two Generalized Higher Order Theories in Free Vibration Studies of Multilayered Plates,” J. Sound Vib., 242(1), pp. 125–150. [CrossRef]
Batra, R. C., and Aimmanee, S., 2005, “Vibration of Thick Isotropic Plates With Higher Order Shear and Normal Deformable Plate Theories,” Comput. Struct., 83(12–13), pp. 934–955. [CrossRef]
Batra, R. C., and Aimmanee, S., 2003, “Missing Frequencies in Previous Exact Solutions of Free Vibrations of Simply Supported Rectangular Plates,” J. Sound Vib., 265(4), pp. 887–896. [CrossRef]
Soedel, W., 2004, Vibrations of Shells and Plates, CRC Press, New York.
Mindlin, R. D., 2006, An Introduction to the Mathematical Theory of Vibrations of Elastic Plates, World Scientific, NJ.
Leissa, A. W., and Qatu, M. S., 2011, Vibrations of Continuous Systems, McGraw-Hill, New York.
Elishakoff, I., 2019, Handbook on Timoshenko-Ehrenfest Beam and Uflyand-Mindlin Plate Theories, World Scientific, Singapore.
Chattopadhyay, A. P., and Batra, R. C., 2019, “Free and Forced Vibrations of Monolithic and Composite Rectangular Plates With Interior Constrained Points,” ASME J. Vib. Acoust., 141(1), p. 011018. [CrossRef]
Du, J., Li, W. L., Jin, G., Yang, T., and Liu, Z., 2007, “An Analytical Method for the In-Plane Vibration Analysis of Rectangular Plates With Elastically Restrained Edges,” J. Sound Vib., 306(3–5), pp. 908–927. [CrossRef]
Aimmanee, S., and Batra, R. C., 2007, “Analytical Solution for Vibration of an Incompressible Isotropic Linear Elastic Rectangular Plate, and Frequencies Missed in Previous Solutions,” J. Sound Vib., 302(3), pp. 613–620. [CrossRef]
Batra, R. C., and Aimmanee, S., 2007, “Vibration of an Incompressible Isotropic Linear Elastic Rectangular Plate With a Higher-Order Shear and Normal Deformable Theory,” J. Sound Vib., 307(3–5), pp. 961–971. [CrossRef]
Batra, R. C., Vidoli, S., and Vestroni, F., 2002, “Plane Wave Solutions and Modal Analysis in Higher Order Shear and Normal Deformable Plate Theories,” J. Sound Vib., 257(1), pp. 63–88. [CrossRef]
Pian, T. H. H., and Lee, S. W., 1976, “Notes on Finite Elements for Nearly Incompressible Materials,” AIAA J., 14(6), pp. 824–826. [CrossRef]
Hughes, T. J. R., 2012, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Inc., Englewood Cliffs, NJ.
Batra, R. C., 2007, “Higher-Order Shear and Normal Deformable Theory for Functionally Graded Incompressible Linear Elastic Plates,” Thin Wall. Struct., 45(12), pp. 974–982. [CrossRef]
Mohammadi, M., Mohseni, E., and Moeinfar, M., 2019, “Bending, Buckling and Free Vibration Analysis of Incompressible Functionally Graded Plates Using Higher Order Shear and Normal Deformable Plate Theory,” Appl. Math. Model., 69, pp. 47–62. [CrossRef]
Herrmann, L. R., 1965, “Elasticity Equations for Incompressible and Nearly Incompressible Materials by a Variational Theorem,” AIAA J., 3(10), pp. 1896–1900. [CrossRef]
Batra, R. C., 1980, “Finite Plane Strain Deformations of Rubberlike Materials,” Int. J. Numer. Methods Eng., 15(1), pp. 145–160. [CrossRef]
Moler, C. B., and Stewart, G. W., 1973, “An Algorithm for Generalized Matrix Eigenvalue Problems,” SIAM J. Numer. Anal., 10(2), pp. 241–256. [CrossRef]

Figures

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Fig. 1

Plate geometry and coordinate axes

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Fig. 2

Location of pressure and displacement nodes in an element

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Fig. 3

First ten mode shapes of a clamped incompressible square plate with h/Lx = 1/8

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Fig. 4

For a clamped incompressible square plate with h/Lx = 1/8 vibrating in the first mode: (a) through-the-thickness distribution of the transverse displacement, w, at the point (33Lx/64, 33Ly/64), and (b) variation on the midplane of w along the horizontal line, x/Lx = 0.5. The two curves in (b) overlap each other.

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Fig. 5

For a clamped incompressible square plate with h/Lx = 1/8 vibrating in the first mode, through-the-thickness distributions at (33Lx/64, 33Ly/64) of (a) normal and volumetric strains, and (b) the hydrostatic pressure/shear modulus

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Fig. 6

For a clamped incompressible square plate with h/Lx = 1/8 vibrating in the first mode, through-the-thickness distributions at (33Lx/64, 33Ly/64) of (a) σ11/μ, (b) σ22/μ, (c) σ13/μ, (d) σ23/μ, and (e) σ12/μ

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Fig. 7

First ten mode shapes of a clamped-free incompressible square plate with h/Lx = 1/8

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Fig. 8

For a clamped-free incompressible square plate with h/Lx = 1/8 vibrating in the first mode: (a) through-the-thickness distributions of the transverse displacement, w, at the point (33Lx/64, 33Ly/64), and (b) the variation of w on the midplane along the horizontal line y/Ly = 0.5

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Fig. 9

For a clamped-free incompressible square plate with h/Lx = 1/8 vibrating in the first mode, through-the-thickness distributions at (33Lx/64, 33Ly/64) of (a) normal and volumetric strains and (b) the hydrostatic pressure/shear modulus

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Fig. 10

For a clamped-free incompressible square plate with h/Lx = 1/8 vibrating in the first mode, through-the-thickness distributions at (33Lx/64, 33Ly/64) of (a) σ11/μ, (b) σ22/μ, (c) σ13/μ, (d) σ23/μ, and (e) σ12/μ

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Fig. 11

For simply supported, clamped, and clamped-free square plates, variation with the aspect ratio of the nondimensional fundamental frequency Ω

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