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.

References

References
1.
Mindlin
,
R. C.
, and
Medick
,
M. A.
,
1959
, “
Extensional Vibrations of Elastic Plates
,”
ASME J. Appl. Mech.,
26
(
2
), pp.
145
151
.
2.
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
.
3.
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
.
4.
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
.
5.
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
.
6.
Messina
,
A.
,
2001
, “
Two Generalized Higher Order Theories in Free Vibration Studies of Multilayered Plates
,”
J. Sound Vib.
,
242
(
1
), pp.
125
150
.
7.
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
.
8.
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
.
9.
Soedel
,
W.
,
2004
,
Vibrations of Shells and Plates
,
CRC Press
,
New York
.
10.
Mindlin
,
R. D.
,
2006
,
An Introduction to the Mathematical Theory of Vibrations of Elastic Plates
,
World Scientific
,
NJ
.
11.
Leissa
,
A. W.
, and
Qatu
,
M. S.
,
2011
,
Vibrations of Continuous Systems
,
McGraw-Hill
,
New York
.
12.
Elishakoff
,
I.
,
2019
,
Handbook on Timoshenko-Ehrenfest Beam and Uflyand-Mindlin Plate Theories
,
World Scientific
,
Singapore
.
13.
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
.
14.
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
.
15.
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
.
16.
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
.
17.
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
.
18.
Pian
,
T. H. H.
, and
Lee
,
S. W.
,
1976
, “
Notes on Finite Elements for Nearly Incompressible Materials
,”
AIAA J.
,
14
(
6
), pp.
824
826
.
19.
Hughes
,
T. J. R.
,
2012
,
The Finite Element Method: Linear Static and Dynamic Finite Element Analysis
,
Prentice-Hall, Inc., Englewood Cliffs
,
NJ
.
20.
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
.
21.
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
.
22.
Herrmann
,
L. R.
,
1965
, “
Elasticity Equations for Incompressible and Nearly Incompressible Materials by a Variational Theorem
,”
AIAA J.
,
3
(
10
), pp.
1896
1900
.
23.
Batra
,
R. C.
,
1980
, “
Finite Plane Strain Deformations of Rubberlike Materials
,”
Int. J. Numer. Methods Eng.,
15
(
1
), pp.
145
160
.
24.
Moler
,
C. B.
, and
Stewart
,
G. W.
,
1973
, “
An Algorithm for Generalized Matrix Eigenvalue Problems
,”
SIAM J. Numer. Anal.
,
10
(
2
), pp.
241
256
.
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